"Interaction of space and matter elements during melting and cristallisation of metals and alloys" - читать интересную книгу автора (Igor V. Gavrilin) Igor V. Gavrilin Melting and Crystallization of Metals
and Alloys Table of
Contents Chapter 1.
Melting and Crystallization. State of Issue 1.1. Thermodynamics of melting and crystallization 1.2. Structural theories of melting and crystallization and the
structure of liquid metals 1.3. Statistic and monatomic theories of liquid state of metals 1.4. Models of microinhomogeneous structure of liquid metals 1.6. The elements of the thermodynamic theory of crystalline
centers nucleation 1.7. The elements of the heterophase fluctuations theory as applied
to the theory of crystallization 1.9. Of the growth of crystals theory 1.10. Of the correlation of the structural theory of
crystallization with the theory of hardening 1.11. Statistic theory of crystallization 2.4. The correlation of melting centers with the centers of
crystallization Chapter 3.
The Mechanism of Metals and Alloys Melting Process 3.2. Calculating the elementary act of melting 3.3. The structure of liquid metals at the point of their formation 3.4. Calculating cluster dimensions in liquid metals at the
melting temperature 3.5. Calculating the dimensions of spatial elements in liquid
metals at the melting temperature 3.6. Calculating the energy of clusters and intercluster splits
motion in liquid 3.7. Calculating the point of metal melting 3.8. Calculating the content of activated atoms in liquid 3.10. The period of cluster existence Chapter 4.
The Change of Liquid Metals Structure at Heating and Cooling 4.2. The modification of cluster dimensions with the change of
temperature 5.1. Of the connection between the structure and characteristics of
liquid metals 5.2. The mechanism of fluidity in liquid metals 5.3. Viscosity in liquid metals 5.4. Self-diffusion in liquid metals 5.5. Comparing the effects of mass transfer in various states of
aggregation of matter in metals. 5.6. Admixture diffusion in liquid metals and alloys 5.7. Admixture diffusion in liquid iron 5.8. Of the change of coordinating numbers at melting 5.9. Of the change of the electrical resistivity of metals at
melting Chapter 6.
Of the Mechanism of Crystallization of Metals and Alloys 6.2. The formation of crystallization centers 6.3. The overcooling problem at crystallization 6.4. Spontaneous and forced crystalline centers nucleation in
liquid metals 6.5. The frequency of crystalline centers nucleation 6.6. Time factor at crystallization 6.7. The problem of mass crystalline centers nucleation 6.8. The competition theory of crystallization 6.9. Of the change in the volume of metals at melting and
crystallization 6.10. The formation of shrinkage cavities and blisters in metals
and alloys Chapter 7.
Alloy Formation and the Structure of Liquid Metals 7.1. Of the mechanism of the formation of liquid alloys 7.2. The point of metal dissolution and contact phenomena 7.3. The formation of alloy structure in liquid state 7.4. Cluster mixing stage in the formation of liquid alloys 7.5. The stage of atomic diffusive mixing 7.6. The stage of convective mixing in the formation of liquid
metals 7.7. Of the function of gravity in liquid metals formation 7.8. The production of locally inhomogeneous alloys 7.9. The formation of alloy hardening interval 7.10. The structure of liquid metals with the unrestricted
solubility of elements 7.12. The structure of liquid eutectic alloys Chapter 8.
The Structure and Crystallization of Liquid Cast Iron 8.1. General data on iron-carbon alloys 8.2. The formation of liquid cast iron structure 8.3. The structure of liquid cast iron 8.4. The peculiarities of cast iron crystallization. The formation
of gray cast iron 8.5. The formation of white cast iron 9.1. General data on modifying 9.4. The choice of the amount of modifiers of the first type.
Gas-like modifying mechanism 9.9. Of the characteristics and choice of second-type modifiers for
alloys Chapter 10
Experimental Research of Liquid Alloy Structure 10.3. Of the Brownian motion experiments 10.4. Sedimentation research procedures of the structure of liquid
alloys 10.5. Sedimentation in Pb-Sn liquid alloy 10.6. Sedimentation of the elements of matter in Bi-Cd liquid alloy 10.7. Sedimentation in Zn-Al liquid alloy 10.8. Sedimentation in Al-Si liquid alloys /149/ 10.9. Sedimentation in liquid casting bronze /152/ 10.10. Sedimentation in liquid cast iron /144/ 10.11. The influence of the sample height upon the degree of
inhomogeneity obtained in liquid alloys 10.12. Conglomeration in liquid alloys as a result of sedimentation 10.15. Of the mechanisms of metallurgical heredity Annotation A new general theory of the crystallization of metals
and alloys was worked out. It is based on a principle - never yet discovered -
of the structure of real physical bodies, metals as well as any others. According to the new principle, metals and alloys, as
well as any other bodies, contain not only the elements of matter, i.e. atoms,
but also the interior elements of space (unknown up to the present moment),
that have the characteristics of vacuum. The elements of matter and space
inside physical bodies are closely interconnected, being at a continuous
interplay. The features of both the matter and space elements determine any
characteristics of physical bodies. Each configuration of the elements of
matter conforms to the corresponding structure of the elements of space that is
inseparably linked with the former. The interior elements of matter and space
are indivisible in the sense that they do not exist separately. The division
between the interior and exterior spatial elements is relative. There exists the hierarchy of the
levels of real bodies structure and the hierarchy of their states, e.g. the
states of aggregation of matter. Each of these levels can be distinguished by
having its own pairs of elements of matter and space, interconnected and
interacting. The main type of interaction between spatial and
material elements at macrolevel is the flicker of various elements of space
inside real bodies during the process of heat oscillations of the elements of
matter and space and the respective flicker of connections between the elements
of matter. The flickering type of interplay between the elements of matter and
space essentially determines all the properties of real physical bodies,
including durability, density, plasticity, etc. of solids, fluidity and
viscosity etc. of liquids. Any state, except for its dominant form of the
elements of matter and space, includes latently the attributes of other
configurations of the elements of matter and space, inherent in other possible
states of the physical body. Dominant and latent elements of matter and space are
constantly interactive, mutually expulsive. Such extrusion provides the inner
rearrangement of a system in conformity to the environmental conditions,
secures the preparedness of a system for the change of the dominant
configuration of spatial and material elements, i.e. the change of the states
of aggregation of matter. The principles mentioned above enable the author to
initiate a new quantitative theory of the melting and crystallization of
metals. It was demonstrated that the process of melting
results from the interaction of the elements of matter and space in solid state
and, in its turn, is the process of the formation of new moving clusters,
dominant only in liquid state of the elements of matter - and flickering F
splits, dominant in liquid state of the elements of space only. A definition
and equation of the elementary act of melting were put forward. All the
parameters of the elements of matter and space in liquid metals were calculated
- their dimensions, the energy of oscillations, parameters of interaction, etc.
A theory of the structure of liquid metals was established on this basis. Considering the presence of paired interactive
elements of matter (clusters) and space (intercluster splits) in the structure
of liquid metals, the author formulates a new theory of crystallization, where
the elementary act of crystallization represents the act of accretion of any of
the two adjoining clusters and the shutting of the isolated element of space -
the intercluster split. Calculations show that the total area of similar space
elements in a mole of any liquid metal or alloy mounts from 100 to 500 sq.m.
So, crystallization is not accompanied by the forming of new section surfaces,
as it is usually accepted, but, on the contrary, by the closing of a huge area
of the interior surfaces of space elements. This observation helps solve the problem of the
critical radii of new-phase nuclei within the theory of crystallization. In its
turn, the elimination of the artificial problem of the critical radius
annihilates the necessity of the heterophase fluctuations theory. As a result, the new theory of crystallization is
entirely different from the already existent one with respect to its conformity
to the facts. In particular, current theory admits that the spontaneous
formation of crystals is actually impossible, since it requires a cooling of
several hundred degrees[1],
which can hardly be put into practice. The new theory states that
crystallization occurs easily and without hindrance; spontaneous
crystallization is always the main type of crystallization, and the influence
of external factors and admixtures accelerates or hampers spontaneous
crystallization, yet by no means replaces it. Overcooling, as the new theory
specifies, fulfills but a thermal function of a factor necessary for absorbing
latent heat of the elementary act of crystallization within the volume of
clusters crystallized. The process demands a cooling of but a few degrees,
sometimes - even 1/10 of a degree, which is observed in reality. Another theory closely associated with the theory of
crystallization - the theory of accretion of spatial elements in the process of
crystallization – was elaborated. It is founded on the new quantitative theory
of shrinkage. The author points out that during the process of crystallization
shrinkage is wholly determined by the presence of spatial elements in liquid
metals. A competition theory of crystal growth proposes that
the process of crystal growing involves not only separate atoms nor clusters as
the building material, but also microcrystals of various dimensions.
Corresponding calculations are made. It is shown that the structure of casting
closely relates to the factor of hardening rate; formulas are derived
respectively. There was suggested a new theory of liquid alloys
modifying. It allows for the actual complicated structure of liquid alloys and
the real crystallization mechanism. The methods of calculating the processes of
modifying, as well as the choice of modifiers, are supplied. Introduction Melting and crystallization
concern regular technological processes in the sphere of metallurgy and foundry
practice. The cycles of melting - alloying - thermal-temporal processing -
refining - teeming - crystallization are daily repeated time and again in each
of the thousands of foundry and metallurgic shops all over the world. These are
adjusted processes that seem to be explored to the minute detail. Everything
measurable referring to the processes indicated was measured long ago,
everything left to study underwent exploration and was entered into monographs,
manuals and textbooks. So what is the aim of writing this book? Strange as it might seem, with all the knowledge of many practical
details, there is no answer to the simplest though cardinal questions about the
essence of the processes of melting and crystallization. The questions are the
following: 1.
What is the cause of melting and
crystallization? 2.
What is the way they act? 3.
How are melting and crystallization connected,
what is the relation between liquid and solid metals? 4.
How do the mechanisms of melting and
crystallization influence the structure and characteristics of castings and
ingots? The book is meant to supply
answers to these questions. Indeed, it has not been found out
yet why solid metals start to melt while heating at a certain temperature. Why with the cooling of almost the
same temperature liquid metals start crystallizing? What causes these processes? How do they develop, in other
words, what is their mechanism? What is the structure of liquid
metals? And how is all that connected with
the structure and properties of solid metals? The knowledge of answers to these
questions is extremely important in scientific-theoretical respect, since it is
time to fill in these absolute blanks of science. It is useful with respect to
practice, too, as soon as the knowledge of answers to such questions enables us
to control the processes of melting and crystallization of metals and alloys
with more comprehension and efficiency. The research on characteristics
and structure of liquid metallic melts, the study of their relation to the structure
and properties of solid metals and alloys, the exploration of liquid-solid and
v. v. transition processes - these closely interconnected fields traditionally
attract a large number of researchers. A distinctive feature of the scientific
development of the sphere in question can be formulated as the extreme
unevenness of the research of its parts. I.e. solid metals are explored much
better than liquid ones. The lack of one synthetic idea integrating both
liquid and solid states, both melting and crystallization may be considered as another bar to
melting-crystallization research. Really, the processes of melting and
crystallization must be interrelated, if we follow simple logic. Moreover, they
are reversible at any moment and a priori must be, in general, the mirror image
of each other. Correspondingly, the theory of crystallization ought to be the
theory of melting at the same time, and v. v. Yet there is no unified general
theory of melting-crystallization with the exception of the remotest thermodynamic
approach which by no means allows looking into the details of these processes. We have a whole bunch of
crystallization theories whereas the incomparably lesser amount of theories of
melting fails to counterbalance them, as if those were completely
non-correlative processes. In this connection it is apparent that this is an
inadmissible thesis. A widely acknowledged phenomenon of metallurgical heredity
manifests itself in various parameters, i.e. in the alloy structure succession
before and after melting and crystallization. Such a phenomenon indicates the
existence of a certain structural parallelism of liquid and solid states. So a
thorough, well-balanced theory of melting-crystallization ought to explain the
mechanism of this congeniality. Notwithstanding the obviousness of
these points to every scientist and practical experts engaged in
melting-crystallization research, the response to them are almost a century
overdue, if the reading is taken from the initiation of Tamman’s theory, and
even more, counting from Gibbs’ works on thermodynamics of the nucleation of
new-phase centers. During the last century, numerous were the researchers who
channeled their energy into the field of melting-crystallization; numerous
issues were joined except for the problems raised above. The book furnishes clues to all of
them. In the author’s opinion, the existence of such
questions and problems underlines the priority of present difficulties. These
problems are not to be solved by ordinary tools. There is a demand for
fundamentally new approaches to these objectives, new ideas, new concepts and
new solutions. Alongside with the other provisions, the book is dedicated to
the working out of new conceptions, ideas and solutions in the sphere of
melting and crystallization. Necessity arising, the newfound data can outspring
the limits of the book. The book weds theory with practice everywhere - with
the application of the newborn theory to metallurgical practice and casting
production, in particular. Yet metallurgical and casting practice seems to be
overloaded with the core difficulties listed above. For instance, it is of
practical importance to know how structural ties inside solid and liquid metals
become established, how the primary casting characteristics like fluidity and
shrinkage get formed. Rather a troublesome circumstance for the foundry
science, for example, is the total lack of any theory of shrinkage at
crystallization to be coupled with the problem of shrinkage cavities formation
and that of porosity. A theory like that deploys below, supported by
numerical calculations. It will be shown that the wholly applied, to all
appearance, phenomenon of shrinkage, as many other processes, is inseparably
linked with the mechanism of melting and crystallization, the structure of
liquid and solid metals, the structure of physical bodies at large. Under the general theory of melting and
crystallization the theories of diffusion, viscosity, the change of
coordinating numbers and electrical resistivity at melting, then, the points of
melting, cooling, the structure of castings and ingots, the theory and
calculations of modifying processes were created and mastered as far as
experimental data can prove. A new theory of liquid cast iron structure and
crystallization was worked out. There is a description to a new set of experimental procedures for
the research of liquid alloy structure by means of capillary sedimentation
experiment; the results of testing a series of alloys by such means are
presented. Data concerning the possibility of alloy refining with the help of
the new set of methods are supplied. It is demonstrated that potentially the
new procedures are no inferior to the method of zone refining of metals. Some additional possibilities and
perspectives of the new approach are outlined in the conclusion. Chapter 1. Melting and Crystallization. State of Issue1.1. Thermodynamics of melting and crystallization The most general and undoubtedly correct description
of melting and crystallization processes originates from thermodynamics. Thermodynamics, by force of its specificity as a
phenomenological science, can give most generalized descriptions to phenomena
through simple ‘exterior’ parameters such as pressure, volume and temperature
without delving into the structure of the substance given and the mechanism of
this or that process. Both the strong and weak points of thermodynamics lie
there. Thermodynamics is unable to furnish data on the structure and mechanism
of the processes under consideration, which shapes the major blank gap in the
sphere of melting-crystallization. What can thermodynamics supply, at
any rate? It gives the most general description to processes and points out
their basic limitations, which is of utter importance. All morphological or
structural or other types of theories have to comply with thermodynamic
limitations, i.e. thermodynamics creates the most general criteria of truth, if
we can say so. So, if any theory fails to meet thermodynamic requirements, it
is erroneous in principle. However, if a theory conforms to the aforementioned
criteria, it does not always follow that it is automatically true. Compliance
with thermodynamic criteria is the immutable but rather an insufficient
condition of truth of any structural theory. Other criteria of truth refer to
truthfulness-to-facts level. Still, it is accepted to start
from thermodynamics in order to establish the generalized truth criteria
touched upon before. The first application of
thermodynamics to melting dates from the middle of 19th century. ‘Krystallizieren
und Schmelzen’ (1903) by Gustav Tamman may be regarded as the first classical
work on melting and crystallization. The crystalline structure of solids had
not been studied yet by that period so Tamman’s theory was by necessity
phenomenological. Thermodynamics of phase fluctuations, melting and
crystallization in particular, has developed considerably by present, which is
vividly corroborated by A.Ubbbelode’s works /1,2/. Taking into account the specific features of the work
given, let us survey the most general thermodynamic theses only that relate to
the core part of the research undertaken in this book. Balance between phases /1,2/. It follows from the theorems of classical
thermodynamics that at the equilibrium between any two states of any material
systems the free energies of the matter mass units in both of these states must
be equal. Mathematical correlation between the parameters of the two phases, e.
g. solid and liquid, coexisting in equilibrium, is determined by the condition GS =GL (1) where GS is Gibbs’ free energy index for solid phase; GL is Gibbs’ free
energy index for liquid phase. The term of phase here and below as applied to
liquids and solid bodies will denote, for the sake of abbreviation, solid or
liquid states of aggregation of matter. All the numerous thermodynamic theorems that relate
to melting and crystallization are derived, in any case, with the use of the
fundamental equality (1). According to Gibbs’ phase principle, if we take a
homogeneous substance forming a single-component system, solid, liquid and
gaseous phases can coexist at the sole combination of temperature and pressure
- at the so-called triple point. The majority of metals, and not only metals,
have such a low pressure of steam at the triple point that the temperature of
the triple point coincides, practically, with the melting temperature at the
ambient pressure of 1 atm. For example, the melting temperature of water and
ice is TM= 0.0000C (by definition) whereas the triple
point temperature is TTR=0.0100C. At higher pressures than that which corresponds to
the triple point, gaseous phase in a single-component system practically
disappears, while the melting temperature depends on the aggregate pressure. The change of Gibbs’ free energy of solid and liquid
states of aggregation of matter with temperature (at constant pressure) is
shown in Picture 1. It is clear that, in conformity with the aforesaid, these
two characteristic curves cross at a single point that corresponds to the
melting temperature. Within the area of temperatures lower than the melting
temperature the free energy of solid phase is minimal and there exists
stability in solid state. On the contrary, within the area of temperatures
higher than the melting temperature the free energy of liquid phase is of less
importance than the free energy of solid phase. Within this area liquid phase
is stable. Generally, in conformity to the laws of thermodynamics, the state
the free energy of which under any particular conditions is the least possible
will be stable under the conditions given.
We should admit that in literature there is no
unified opinion on the fundamental point under consideration, yet
contradictions are apparent. In particular, there circulates an opinion that
any ‘single-phase’ models and theories of melting and crystallization are
erroneous by definition, because they contradict the principle of
thermodynamics noted above, which we agreed to label as the two-phase
principle, or the principle of structural dualism, also pointed out by
A.R.Ubbelode /1,2/. Other authors construct their theories and models ignoring
this principle. Still, the principle of dualism, or, to be precise,
the principle of the constant coexistence of two certain complexes of
quantities that determine the states of aggregation of matter acts at least as
a fact of negation of ‘single-phase’ models and theories of melting and
crystallization. Nobody knows for sure what the adequate
melting-crystallization theory should be like, yet it follows from the
principle stated above that it ought to be a ‘two-phase’, or ‘dualistic’,
theory. To be more exact, it should be a ‘two-factor’ theory representing the
processes of existence and change of the states of aggregation of matter as the
corollary of the interaction between two certain factors or two complexes of
factors. Until present, it has been impossible to find any
convincing proofs of the real existence of such dualism. The argument is to broach the question whether these
quantities exist in reality, or they are just conceptual. Should thermodynamics
be understood directly, or roundabout ways of comprehension be scouted? As we see, even the simplest approach to the problems
of melting and crystallization, even within the stabilized area of
thermodynamics, turns out to transcend its seeming simplicity and clarity. At the minimum, the above-mentioned theses give rise
to one unsolved problem: are there any two factors, existing constantly, the
competition of which leads to the change of the states of aggregation of matter
under certain conditions? If it is so, what are those factors, unknown to
science? Questions generating... There ensues a general idea out of the comparison of
the thermodynamic parameters of a large number of various substances in solid
and liquid states. The attempts at using the volume measurement criterion as
the sole one are still being made, yet they look self-defeating if we recall
the existence of the so-called anomalous metals which do not only increase
their volume while melting but reduce it. That is quite a stumbling block, a
real hindrance to the application of many thermodynamic conceptions, i.e. the
principle of corresponding states, to melting. However, there remains the major thermodynamic
parameter that changes equally for all the known cases of melting. It is
entropy. For all the known substances in liquid state entropy is higher than in
solid state. Let us use this experimental fact for our purposes. The change of
entropy at melting can be represented by Boltzman equation /1/: Δ Sf =R ln Wl /Ws,
(2) where Wl is the number of independent ways of substance
realization in liquid state, Ws - the same for solid state. It was demonstrated by (2) that
the transition from solid to liquid state is accompanied by the increase in a
certain kind of disorderedness in a substance. Thermodynamics cannot answer the
question about the nature of such disorderedness. Nevertheless, this inference
is sufficiently important to be allowed for further. 1.2. Structural theories of melting and crystallization and the structure of liquid metals Proceeding from the problem specified, let us analyze
some of the most prominent theories of melting and crystallization for the
purpose of satisfying the condition of ‘two-factorness’. Like any other science, the
science of melting, crystallization and liquid state of metals developed by
accumulating, analyzing and systematizing the experimental data. Such a way is
characteristic of the stable period in the development of any science. With the
exception of the periods of stability, at times science undergoes periods of
discoveries, i.e. unpredictable qualitative leaps in the understanding of this
or that issue or even just the finding of new methods never known before, and
research areas. Scores of years have passed without bringing great changes for
the science of melting, crystallization and the liquid state of metals. The last substantial experimental discovery in this
sphere was made in the early 20th century together with the
development of X-rays- and further on - neutron diffraction. The transposition of substance structure research
with the help of penetrating radiation on particle floods from solid to liquid
state was undertaken in the 30-ies. Valuable works were accomplished by Stewart
/3,4/, Kirkwood /5,6/, Bernal /7,8/, Gingrich /9/ and others. However, the
first researcher to carry out systematic exploration of liquid metals structure
with the use of the method of x-rays dispersion by the surface of liquids, as
well as other methods, was V.I.Danilov /10,11/. In the works listed, the
likeness between the atomic structures of neighboring order in solid and liquid
metals in the vicinity of the melting temperature was emphasized with all definiteness,
as well as the gradual diffusion of the crystal-like structure of neighboring
order in liquids within the limits of overheating process. The result of such a discovery was the retreat from
the formerly predominant notions of the similarity of liquid metals structure
to the chaotic gas structure, that originate in Van-der-Vaals’ works /12/. Thus, the turn marking the transition from the
conception of the gas-like structure of neighboring order in liquids to the
ideas of their crystal-like structure took place. As we see, the historic logic of science development
deprived (for reasons unknown) liquid state of the right to the independence of
structure, though thermodynamics affirms unambiguously that the liquid state of
aggregation of matter, as any other state, is sufficiently independent and must
have a certain independent, specific structure. Yet this unsophisticated
inference never evolved, and liquids were looked upon further viewed as
something that has a dependent, transitional structure. Such an approach caused
much damage and hindered the development of the science of liquid metals
structure, melting and crystallization for a long period. In the course of time the development of the science
of liquid metals structure was distinguished by the rivalry, or eclecticism, of
the two approaches described above: the quasi-gaseous and quasi-crystalline
ones. Either of them was accumulating, and not without success, facts in its
favor, while the structural independence of liquid state was never prospected. For instance, there was analyzed a number of
experimental facts which testify to the proximity of liquid and solid metals
structure near the melting temperature. This is rather a negligible quantity of
enthalpies and entropies of fusion as compared with the same quantities for
evaporation /13/; a negligible change in the volume of metals at melting; a
slight change in heat capacity, heat conduction and electrical resistivity; a
qualitative similarity in the position of the first maximums and minimums on
the curves of reflected X-rays intensity in liquids with the position of lines
on X-rays photographs of powders of respective solid metals, etc. This
information is supported by the data on Hall’s coefficient, magnetic
characteristics, coordinating numbers, etc. However, the supporters of the
gaseousness of liquid metals structure did have a considerable amount of data
in their favor. In the first place, it is almost a
statistic distribution of atoms in liquid metals at heightened temperatures; a
possibility of continued transition of liquids into gas; a huge difference in
the mass transfer coefficients in solid and liquid metals, of fluidity, etc. Out of the entire array of
controversial facts there can be drawn a reliable conclusion: by nature, liquid
metals possess the features of both the semblance and dissimilarity to solid as
well as gaseous state. Yet no inferences about the structural independence of
liquid state were made, there was not even formulated the conception of the
structural independence of liquid state. From our point of view, such
ambivalence of liquid state characteristics is really one of the forms of its
structural independence and a means of manifesting the dualism that we touched
upon earlier. So any acceptable theory of melting and liquid state should
interpret the entire mass data available on liquid state, melting and
crystallization, and also the dual nature of liquid state structure and
properties. Such a theory must be able to explain the structural independence
of liquid state, too and its bonds with the structure of the neighboring states
of aggregation of matter. So far, some researchers are still trying to see into
the nature of liquid state, proceeding from its similarity to solid state,
while others stress the liquid and gaseous states affinity. Both the
approaches, as it follows from the fore-going material, may be appropriate,
because the same liquid reaches, by structure and properties, the indices of
solid state near the melting temperature and of gases near the point of
vaporization, or critical point. Both the approaches are one-sided, for they cannot
cover the whole range of liquid metals characteristics within the whole
temperature range of their existence. To define the structural peculiarities of different
states of aggregation of matter, liquid state in particular, the author
introduces the concept of a structural unit (an element) of liquid state. It is quite a convenient notion, since it lets us
single out the smallest part of liquid state, on the one hand, and classify the
existing structural theories on the basis of structural units or structural
elements these theories operate with, on the other hand. To bring up the problem of the structural elements of
liquid state is to the point, because any science of a system’s structure is
based on the concept of the smallest structural element of the system
specified, the element carrying the basic attributes of the system. For
instance, an atom (a molecule) is the smallest structural unit of matter in
chemistry. From the viewpoints of differentiation between
structural units of matter in liquid state there exist two large groups of
theories of melting and liquid state that occupy, directly or indirectly,
unlike positions with regard to this issue. The most numerous group of theories
premise the definition of a structural unit of matter that is accepted in
chemistry and treats an atom (a molecule) as the structural unit of any state
of aggregation of matter. The ideal model of liquid from this viewpoint is a
monatomic liquid. In reality, the most closely related to the model given are
liquids with weak interaction between their particles, i.e. liquid inert gases
near the critical point /14,15/. The other group of theories and liquid state models
proceeds from the fact that liquids have a complex structure which may consist
of particle groupings of various kinds /1,3,4,16,17/. It leads us to assume
that not only separate atoms but also some of their groupings act as structural
units (elements) of liquid state. In the conceptual aspect, it can be regarded as a
step forward in comparison with simplified monatomic models and theories of
liquid state. Why? The point is that the monatomic approach equalizes all the
states of aggregation of matter in the sense that they turn out to be
indistinguishable by their structural elements, which is false a priori,
because the properties of the states of aggregation of matter become apparent
at the level of certain particle aggregations but not that of separate
particles. The role of separate atoms and molecules in the
structure of liquids is not denied but one more level, one more state to the
understanding of the structure of the states of aggregation of matter is
introduced into this group of models and theories of liquid state. Since any groupings consist of
similar atoms and molecules, there arises a question whether the distinction
between the definitions of a structural unit of liquid state is essential. We can suggest the positive answer, because the
description level should be adequate to the subject described. Thus, atoms
include protons, neutrons and electrons; however, if we limited ourselves to
these concepts ignoring the idea of the atom, it would be extremely hard or
almost impossible to describe, for instance, the structure of molecules or
crystals. The fact is that any agglomeration of particles is more than a mere
agglomeration for the reason that it has a certain structure, which, as a rule,
is not a mere sum of particles. What is more, the particles under consideration
are tied between them and organized in space in a certain way. That is why the question about the structural
elements of liquid state seems to be the question of paramount importance, so
using it as the basis for the theories and models of liquid state classification
is quite defensible. Returning to the thermodynamic requirement of the
‘two-factorness’ of a system, let us note that the structural elements of
liquid as well as any other state should meet this requirement, i.e. the
structural element in question should consist, at the minimum, of two parts of
differing nature. Such duality possesses a totally general character; still, it
is undisclosed as yet. The nature of structural dualism is to be viewed below. 1.3. Statistic and monatomic theories of liquid state of metals Modern statistic theories of liquid state refer to
the first group of the classification on offer, since they make use,
principally, of the pair interparticle interaction concepts /12-15/. The data
on such interaction come out to be extracted from the information on X-rays-
and neutron diffraction of liquid metals. The assumption function is the
dependence of the diffused X-radiation intensity on the radiation angle, which
allows determining the statistic structural factor. The knowledge of the
structural factor, in its turn, lets us find, with the application of
Furie-transformation procedure, the relative function of radial distribution
and the aggregate correlative function, as well as calculate the coefficients
of self-diffusion, viscosity and some other properties of liquid metals if we
imply their monatomic structure. With the help of the well-known methods of Born-Green
/18/ and Percus-Yewick /19/ we can calculate pair potentials of interatomic
interaction out of the registered quantities. It may be admitted that there is
no reliable information on the nature of interatomic potential, diverse kinds
of constructed functions of pair potential are used. It is clear that the
results obtained differ greatly between themselves, as well as from the
experimental data, which diminishes the practical value of the indicated
methods /20/. The connection of the statistic theory of liquid
metals with melting is not established directly, displaying the weakness of the
theory under analysis. Frequently we come through an assertion that the
statistic theory does not serve as the model one to the extent that it uses the
experimental data on neutron or X-rays diffusion /20/. However, out of these
data there can be inferred nothing but the information about the pair function
of distribution, i.e. the pair interaction between the particles. It is
supposed implicitly that liquids consist of dispersive particles (atoms) only.
Consequently, the monatomic model of liquids structure starts functioning as
the basic approach in the statistic theory though in the most general, implicit
aspect. Concretizing the specificities of the first group
theories, i.e. those which adhere to the thesis that an atom (a molecule) is
the structural unit of liquid state, leads to the formation of the so-called
corpuscular models /12,14,21/. Among the widespread models there can be mentioned
the so-called hard sphere model /22,23/, the working out of which amasses a
large number of publications. In accordance with this model, atoms in liquid
metals impute the properties of hard spheres. The method in question as applied
to certain X-rays photographs looks effective enough; still, no-one succeeded
in achieving the correspondence between the experimental and calculated data
for the wide variety of liquid metals /23-25/. Because of computer expansion,
calculation procedures such as the method of molecular dynamics enabling to
experiment with the modeling of liquids using the statistic theory results and
methods, became popular /26, 27/. These methods are not applied to the sphere
of practical metallurgy and foundry works, their importance is so far purely
theoretic. To the group of monatomic models we should also refer
the widely known yet rarely used model of J.Bernal /8,28/, where the structure
of liquid is rated as a result of the disordered allocation of separate atoms
with the possible realization in the neighboring order of the symmetry of the
fifth order (nontranslatable) that cannot be realized in solid state. The model
under consideration reveals rather the author’s rich imagination than the real
state of things, yet we cannot deny it the right to existence. While the real
structure of liquid metals remains unexplored, any models and theories have a
right to existence. It must be noted that neither the model of hard
spheres nor that of J.Bernal allows modeling explicitly the process of melting
and crystallization, though such attempts are occasionally being made. There
was advanced an opinion that if computer memory volume amplified a bit
increasing the number of modeling steps, the problem solution would ensue by
itself /26/. It looks fairly improbable. Let us emphasize as the common drawback of all these
theories and models that none of them conforms to the principle of dualism or
two-phaseness analyzed above. A special position belongs to the models of melting
the most widely known of which is Lindeman’s model. Lindeman put forward a
hypothesis that melting sets in when atomic oscillation frequency and amplitude
in crystalline lattice points increase so much that the atomic bonds start
breaking. Further research did not corroborate Lindeman’s suppositions, since
calculations made under his theory issue the points of melting that approach
the vaporization point /1,2/. Still, if we take into account the extreme
scarcity of logical theories of melting, Lindeman’s model continues being
mentioned in reviews for scores of years, which we are in order to do by
tradition /29/. Let us observe that Lindeman’s model fails to meet
the thermodynamic requirement of dualism of melting and liquid state, because,
according to this model, there is only one reason for melting – it is an
increase in amplitude and frequency of atomic oscillations at metal heating,
and the cause in question does not compete, or interact, with any other factor. 1.4. Models of microinhomogeneous structure of liquid metals Among the second group of theoretical conceptions
that present complex views upon the structural unit of liquid substance we
should place diverse variants of models of microinhomogeneous structure of
liquid metals. This trend is intensely developing
during the last years /30-36/. The first works representing such a type belong
to Stewart /3-4/. Stewart was the first to introduce the cybotaxis concept that
turned out to be highly viable. Stewart’s cybotaxes are, in their own way,
short-lived crystals forming liquid. A divergent but analogous idea of liquid metals is
observed in works /31,36,37/. Liquid is viewed here as a double-structure
system which consists of relatively long-lived atomic microgroups with the
ordered structure of the neighboring order similar to solid state, and the
disordered zone with the chaotic arrangement of atoms. The eclectic nature of such views cannot but manifest
itself: monatomic and cluster models are forced together. It might not even be
considered an error if Gibbs’ phase principle was not disregarded here. One and
the same substance within a wide range of the single state of aggregation of
matter temperatures appears to form two different phases, which conflicts with
reality. The quantum mechanics rule of quantum objects indistinguishability
comes to be violated, too. It is evident that atoms in microgroups and within
the disordered zone must differ in certain characteristics, whereas quantum
mechanics states that under the same conditions atoms of the same kind must be
similar by their properties (i.e. indistinguishable). It is possible that there
is a way out of this disparity which remains, however, undisclosed. Hypothetical atomic microgroups in liquid metals
structure are termed differently in various works: cybotaxes, microcrystals,
microgroups, clusters, blocs, etc. Still, the meditation upon these works
enables us to infer that similar objects divergent by details only are meant.
That is why we shall use the term of ‘cluster’ to label the objects of such a
sort further, concretizing or enlarging it as required. The conclusions about the existence of clusters in
liquid metals are made on the basis of precision analysis of thin structure
curves of X-rays dispersion /34,40/, where the ordered areas dimensions
(approx.1 nm) as well as the atomic granulation type, are estimated on the
basis of the first maximum splitting, its intensity and width. In a number of
works, on the contrary, quite acceptable curves of X-rays dispersion are
diagrammed by means of calculations, proceeding from the microcrystal or
paracrystal models /41,42/. There was given a high-grade description to the
process of metal crystallization on the basis of the quasipolycrystalline model
of liquid metals structure by Arkharov-Novokhatsky /43,44/. An original description of the mechanism of melting
and the structural transitions in liquid metals can be found in Ye.S.Filippov’s
works /44,46/. The presence of such transitions is permitted and substantiated
by the repeated observation (carried out by numerous authors) of anomalies in
temperature and concentration dependencies of various structure-sensitive
properties of liquid metals and alloys /46/. Still, the quantities of the anomalies observed are
relatively low at times and located within the error limits of standard
measuring. Nevertheless, the data mass on the anomalies in structure-sensitive
properties of liquid metals and alloys is so impressive at present that on the
whole the conclusion about the existence of such anomalies seems sound. Their
presence is especially indubitable in relation to some alloys, yet it is not
proved by a series of works on pure metals /47/. Upon the whole, the existence
of certain analogues of phase transitions of the second order in liquid metals
and alloys appears to be logical in the presence of microgroups that have the
features of neighboring order in the arrangement of atoms inside such a group,
in their structure. If there is order, a transition from one of its forms to
others is possible. Clusters lacking, we consider phase transitions impossible,
on the contrary. So the problem of the existence of similar transitions serves,
for tens of years already, as a battleground between the supporters of
different approaches to the description of liquid metals and alloys nature. This discussion going on for almost a century took
place because of the possibility of a polysemantic interpretation of the
majority of the results obtained in most of the experiments with liquid metals,
which, in its turn, reflects the objective complexity of the liquid state
nature and its ambiguity. Till latest, for example, there were no direct
experiments that could definitely corroborate or disprove the hypothesis of
cluster, microinhomogeneous structure of liquid metals. Its postulational
character of the hypothesis under analysis, as well as the ambiguity of
suppositions advanced concerning the structure of liquids, can be regarded as
its main shortcomings /25/. In this connection, the prevailing position in the
theory of pure single-component liquid metals occupies the statistic theory
that denies cluster existence and the microgroups similar to them in liquid
metals /48/. A series of computer models of liquid metals structure, adherent
to the hard sphere model, sides with the theory mentioned. There were created
several successful computer methods of calculating certain characteristics of
some liquid metals, which have as their basis the curves of X-rays dispersion
and the supposition about the monatomic structure of liquid metals, or diverse
forms of interatomic interaction potential. Yet we do not have a right to
attribute the achieved results to the whole range of liquid metals. The situation in the alloy theory field is developing
for the most part in favor of the cluster approach, especially in the area of
studying the systems Fe – C, Fe – Si, Cr – C etc. that are of practical
importance /49–51/, and liquid eutectics /50,52–55/. It is explained by the presence
in this area of a far greater mass of experimental facts testifying to the
advantage of the cluster theory of liquid metals structure. Above all, there should be placed the well-known
experiments on the melts centrifuging that lead to the enrichment of the sample
remote from the rotation axis by the heavier component in all the cases. It
allows determining the dimensions of the areas enriched by this component,
proceeding from the experimental conditions. Experiments of such a kind were carried out systematically
for the first time in Russia by A.A.Vertman and A.M.Samarin /16–17/ to be
repeated periodically later by a series of researchers with certain
modifications but always with the same result / 55,57,58/. The definite
significance of these experiments consists in asserting that during the
experiment and under its conditions no appreciable change of the alloy
concentration along the sample was to take place in the centrifugal force
direction with the monatomic structure of melts. However, the change was continually occurring, so it
was possible to calculate the floating zone dimensions by the simple Stokes’
formula on the basis of this change. Since such experiments were abolishing the monatomic
hypothesis, their interpretation underwent a whirlwind of criticism by the
monatomic hypothesis advocates from the ground of the theory of regular
solutions /20,56/. Unfortunately, the criticism came out to be valid in a
number of items, because Vertman–Samarin experimental procedures were
imperfect. To our disappointment, a comprehensive response to the criticism in
question never followed, so the centrifuging trend was stigmatized doubtful and
the research in this sphere was suspended for a long time. Still, scientific
development proved that the trend referred to was right upon the whole,
notwithstanding the incompleteness of its methods, and is being gradually
reestablished at present as a reliable way of liquid alloys study /57,60–61/. The inferences in favor of cluster existence are
generally made on the grounds of the congeniality between the whole range of
liquids and solids listed above. It merits our attention to notice that a large
mass of data on liquid metals structure that is accepted as naturally
fundamental may be viewed ambivalently. These are the X-ray diffraction analysis data, in the
first place. They are successfully used in calculations and conclusions,
testifying to the cluster, as well as monatomic, structure of liquid metals. On
the basis of this we may state, to our regret, that so weighty a tool of X-ray
diffraction analysis, cogent in other cases, was unable to supply any precise
information upon the nature of liquid metals. The conclusion sequent to the aforesaid formulates
like this: the nature of liquid metals, in all probability, is not so plain and
definite as it happens to be presented in the already existing models, that
there must be some factors, unfound till now, which determine the nature under
research. Simpler speaking, there is something in liquid metals beside clusters
or monatoms. What is it? 1.5. The interrelation of monatomic and cluster approaches and the connection of various theories to experimental data Two main groups of theoretical concepts of liquid
metals and alloys structure that were accentuated above almost fail to have
common ground at present. Several statistic theory supporters see it clearly
enough. I.Z.Fisher argued convincingly /48/ that if a monatom is to be considered as the
structural unit of liquid state, the microgroup idea looks absolutely redundant.
Let us observe, for justice, that the premises of the afore-mentioned work by I.Z.Fisher
contain this conclusion already. In any case, the significance of Fisher’s work cannot
be assessed by the fact that it proves the absence of microgroups (clusters) in
real liquid yet it demonstrates the antagonism and incompatibility of the
monatomic and cluster approaches to the description of liquid metals and alloys
structure. The antagonism under analysis is not always accepted or even
understood in the works on liquid metals theory. There are some attempts at its
negation, the bringing together of the two approaches /62/, or assuming that
structural units may differ in the same liquid /43,44/. These works mark
nothing but the quantitative difference of the two approaches; the trends are
forced to reconciliation or joining. Still, the hypothesis of monatomic structure of
liquid metals does not require the cluster concept by its inner logic, which
was shown by I.Z.Fisher. In its turn, the cluster hypothesis implies an
insolvency of the monatomic idea for explaining the phase characteristics,
though it may be applied to interpret cluster structure. The acknowledgment of the antagonism of these
approaches could be more advisable at present stage than the attempts at linking
fundamentally disparate ideas. The cardinal incompatibility of these approaches, as
it was pointed out above, is that similar atoms cannot be at two different
states under constant conditions. It is quite obvious that atoms in clusters
and atoms in their free state (monatoms) are at divergent states. Therefore,
clusters and monatoms in liquid metals cannot exist simultaneously as
structural elements. Either must be held to. The alloy of these states is
totally unacceptable, if we proceed from the quantum principle of
indistinguishability of similar atoms. Hence, it is clear that the question of the choice of
the liquid state structural unit is sufficiently important. It is also evident
that this problem does not have any satisfactory solution so far. That is why
the analysis of the areas where this or that approach leads to the optimal
results plays a vital part in the model of liquid state choice. The previous analysis shows that the monatomic
approach (statistic theory, hard sphere model, molecular dynamics method, etc.)
agrees with experiment most thoroughly in the critical point area, or generally
at high temperatures remote from the melting temperature, for liquids with weak
interaction between their particles like liquid argon, and pure metals. The maximum conformity with the experimental data for
the cluster approach is achieved in the low-temperature area near the melting
temperature or liquidus point, for alloys, systems with strong interaction
between the particles, eutectic alloys. Undoubtedly, only the preferred spheres of relative
achievement are listed here. In certain cases these areas intersect, yet on the
whole the state of things is as presented. If we investigate the situation
given, the models of cluster structure of liquid metals and alloys are
preferably suited for metallurgists and castors in their practical work, since
these models describe the properties of liquid metals and alloys in the optimal
way within the practically important temperature interval. Unfortunately, the applied advances of the models in
question and their use for the practically important theories of
crystallization, hardening, the theory and practice of modifying and doping of
alloys have an exclusively qualitative nature and can be applied to practice to
the least degree possible. In the sphere of the statistic theory, practical
applications are also scanty. Though the heterophase fluctuations theory allows
to qualitatively describe the new phase nucleation process, it is
correspondingly weak in handling this area having no practical applications,
notwithstanding the more than a semi-centennial history of its development. Finally, both the major approaches to the description
of liquid metals and alloys structure – monatomic and cluster ones – are
subject to the lack of correspondence with the thermodynamic requirement of
two-factorness. Both of these approaches premise that the concept of monatoms,
or clusters, suffices to describe liquid state and the states of aggregation of
matter in general. Thermodynamics claims that it does not sound
exhaustive, that there must be at least one factor essential to liquid state.
The detection of this attendant factor is of paramount importance to the theory
of liquid metals and alloys structure at present. 1.6. The elements of the thermodynamic theory of crystalline centers nucleation In the structural aspect,
crystallization process starts from the formation of elementary microcrystals
that are termed crystalline centers. Since W.Gibbs, it has been accepted to
assume in the existent thermodynamic theory that the process indicated requires
energy consumption for the formation of solid and liquid phase section surface
/63,64/. Such a hypothesis seems valid from the angle of
common sense, yet it leads to a rather drastic assertion of the existence of a
certain critical radius of crystalline nuclei, which had distressing
consequences for the theory of crystallization. Let us consider the problem how the concept of the
critical dimensions of crystalline centers arose in present theory. It is surmised that the aggregate change in the free
energy of a system while forming a solid phase zone in liquid, amounts to the
sum total of the changes in the volumetric and surface energies of the system’s
zone specified /64–68/: ΔG = -ΔGv + ΔGs, where ΔGv is specific volumetric free energy; ΔGs is specific surface energy. The signs in the equation given constitute the main
theoretic problem. These signs have a physical significance and are introduced
for the reasons of common sense. The minus in front of the first term on the
right denotes that energy at solid phase formation at T = T0 evolves in correspondence with
the experimental fact of latent heat crystallization emittance. The plus in
front of the second term on the right signifies that, on the contrary, there
must be energy consumption for the formation of phase section surface. The latest conclusion in the stated case is based not
upon the experimental facts but common sense considerations that there must be
work (energy) consumption for the formation of phase section surface. This
consideration is true in many cases, so an inference that it is right in all
the cases was formerly made. As a result, it follows from the theory that the
process of solid phase center formation at the crystallization of metals has
some features of inherent contradiction. In particular, solid phase center formation is
recognized thermodynamically expedient whereas the surface formation of the
same phase is reckoned thermodynamically inexpedient. We are going to expand on
the present contradiction in Chapter 6. So far let us mark the fact of its
existence, - it gives rise to the conclusion of the existence of solid phase
nucleus critical dimensions further. Let us make a conjecture that a solid phase nucleus
is spherical. Then, the latest expression for ΔG will shape into ΔG = - ΔGv 4/3 pr3 + s 4pr2 The expression is graphically shown by the curve in
Fig.2.
Factually, such a character of the curve in Fig.2
stands for a thermodynamic prohibition of the nucleation of microcrystals with
the radius less than the critical one. In liquids, such crystals must
dissociate.
Let us consider the existent solutions to the problem
of the critical radius of crystallization centers. Willard Gibbs demonstrated
for the first time that, to form a critical dimension nucleus, it is necessary
to expend work (energy) A = Δ Gs, which equals to one third of the spare surface energy of the
nucleus mentioned: Δ Gs = 1/3 S Si si, where Si is
the specific surface of the ith zone of an equilibrium crystal at
nucleation; si is the
specific surface energy for the zone given. If a cube is the equilibrium form of a crystal, then Δ G = 8s r2 c,
(3) where rc is the radius of the sphere inscribed
into a cube of the critical dimension. To calculate the quantity of rc of a spherical
nucleus the following formula is used /65,67,68/: rc = 2s T0/ L Δ T.
(4) Graphically, the last expression corresponds to the
minimum in Pict.2. The concluding formula shows that the critical dimension
of a solid phase nucleus deflates with an increase in the melt overcooling ΔT = (T0 – T). The nucleus formation work ?G
decreases at the same time as the overcooling increase is taking place. For a separate nucleus in the form of a cube with
the edge a = 2 rc there is the following quantity of work that is to be expended for
the formation of the section surface of the nucleus given in accordance with
the present idea: Δ G = 1/3 Ss = 32 s 3 T0 / L2 (Δ
T)2 This expression confirms that, in correspondence with
present theory, the formation of the solid phase nucleus of the critical
dimension demands the expenditure of work (energy) and is therefore
thermodynamically inexpedient. Thus, the existent thermodynamic theory of solid
phase nucleation in liquids cannot surmount the theoretical bar it constructed
concerning the idea of the critical radius of the crystalline nucleus as well
as the thermodynamic inexpedience of the process of crystal growth with the
crystalline dimensions less than the critical ones. To overpass the
contradiction under analysis, the thermodynamic theory required either the
shift of the conception of the energy expenditure necessity for nucleation in
liquid, or extraneous help. It happened that the scientific development in the signalized
sphere chose the latter, i.e. the support from non-thermodynamic theories. This
support was rendered by the theory of heterophase fluctuations nucleation. 1.7. The elements of the heterophase fluctuations theory as applied to the theory of crystallization The noted contradictions in the thermodynamic theory
of nucleation remained a problem for quite a time and were formally mastered
only with the help of the non-thermodynamic, probabilistic by nature, theory of
heterophase fluctuations. The aforesaid theory was being created for scores of
years by the efforts of scientists innumerable, so it is termed by the names of
different authors. Among the most frequently-mentioned pioneers of this theory
we can enumerate Frenkel, Volmer, Weber, Bekker, During,
Eiring and several other authors occasionally /69–72/. In physics, fluctuations are any contingent
deviations from the average state and distribution of particles in any large
systems, which are determined by the chaotic thermal motion of the system's
particles. The measure of fluctuations is the average square of the difference
in any local value of any physical quantity in this system L' and the average
value of the same quantity for the whole system L. (Δ L)2 = (L – L')2 As a rule, fluctuations are small and the probability
of any fluctuation given decreases exponentially with an increase in its
quantity. If a system consists of N
independent parts, the relative fluctuation of any additive function of the
state L of the system specified is inversely proportionate to the square root
of the number of its (the system's) parts. The theory of heterophase fluctuations employs the
fact there can be fluctuations of any type, heterophase including
/69,70,73–75/. The latter means that as a result of contingent chaotic thermal
motion of atoms there may occasionally appear some zones in the melt that have
the atomic distribution similar to that of a crystal. It is supposed that within the limits of such
fluctuations neighboring order is casually realized, the order which is
characteristic of a crystal, so the surface of the section with the surrounding
melt is established. Theoretically, this supposition is quite possible. In the
theory in question, such fluctuations are identified with microcrystals. It means that the theory under consideration implies
rather groundlessly that the instantaneous contingent organization of a certain
atomic configuration in space suffices for this zone to acquire the structure
and properties of some other phase. Such a statement cannot be assessed as
potent or convincing. Fluctuations are unstable and transient by nature.
The period of their existence correlates with the duration of heat oscillations
of particles forming up liquid. In case of atomic fluctuations, the noted
period equals to 10-12 of a second. Fluctuations set in and fade
right away. Only under this condition the average state of the system remains
constant. The theory of heterophase fluctuations admits that
under certain conditions heterophase fluctuations may turn from the unstable
state into the stable one acting as crystallization centers. These conditions
are adopted from thermodynamics. The first similar condition in the heterophase
fluctuations theory reads as overcooling, since a stable existence of zones
with solid crystal structure is thermodynamically possible only in the melt
overcooled below the melting temperature. The greater overcooling gets, the
less the critical radius of nucleation is, in accordance with (4).
Correspondingly, the less must be the dimension of a heterophase fluctuation
and the greater the probability of such a fluctuation to set in. The second condition is
that of the critical dimension of fluctuations under analysis. It is shown in
Fig.2 that heterophase fluctuations will be stable only in the case when their
dimensions are large enough, i.e. larger than a certain critical radius, even
in the presence of overcooling. If the dimension of a fluctuation is less than the
critical one, it will dissociate even in the presence of overcooling. If the
dimension of a fluctuation is larger than the critical one, its growth becomes
more expedient (see above). There is a stipulation to be made.
The point is that a fluctuation cannot be growing gradually. By definition, it
must arise at once, on the instant. It was stated above that the probability of
this or that fluctuation decreases exponentially with an increase in its
quantity. Thus, heterophase fluctuations of critical and overcritical
dimensions are highly improbable here. As we see, the theory of heterophase fluctuations
transfers crystallization process from the class of regular phenomena to the
category of accidental, probable ones. This is an essential drawback of the theory, because,
for thousands of years, metallurgy and foundry practice has been demonstrating
the regularity of crystallization. 1.8. Of the combination of the thermodynamic theory of nucleation with the heterophase fluctuations theory Let us trace in detail the connection between the
thermodynamic theory of nucleation with the heterophase fluctuations theory. It follows from the laws of
statistic physics that there is a finite probability I of any system's
transition through the energy barrier ?G by energy fluctuations: I = K exp(-Δ G / kT),
(5) where k is Boltzman constant; K
is the kinetic coefficient depending on the rapidity of the atomic exchange
between the fluctuation and the melt. By inserting the value ΔG into (5) for the critical nucleus from (4), it will be possible to
calculate the probability of the critical nucleus formation by fluctuations
after taking the logarithm: lg I = lg K – 32 s 3 T0 lg e
/ L2 (Δ T)2 k Ya.S.Umansky considers a particular example of
homogeneous crystallization of iron at the discriminate overcooling of 100, 200
and 2950C /68/. The example illustrates the
possibilities of the heterophase fluctuations theory for calculating the
processes of crystallization. So let us take a brief survey of the general data
supplied by Ya.S.Umansky. For iron, specific surface energy along the section
of crystal-melt s = 200 erg/sq.cm, T0 = 1803 K, L = 3.64 kcal/g atom = 2
1010 erg/sq.cm. Hence lgI/K = -32 2003 1800
0.434/4 1020 1.38 10-16 (Δ T)2 The results are presented in the table 1. Table 1 The Probability I of the Appearance of
Heterophase Fluctuations of Critical Dimensions in Liquid Iron at Discriminate
Overcoolings
The table shows that the declining of overcooling
from 295 to 200 K, i.e. 1.5 times on the whole, reduces the probability of
equilibrium nuclear formation in correspondence with the heterophase
fluctuations theory almost 100 000 times as small. At the overcooling of 100
degrees the probability of nucleation, by Ya.S.Umansky’s calculations, comes to
10-35. It is a vanishingly minor quantity. Out of the recorded calculations
Ya.S.Umansky and others arrive at the conclusion that the practically
homogeneous fluctuation does not take place. The probability of forced
crystallization, by this theory, is much higher than the probability of
spontaneous crystallization. In particular, in case when liquid iron wets the
particle surface of some insoluble solid extraneous agent so that the wetting
angle is q = 45 degrees (the case of the average wetting level), Ya.S.Umansky
derives the correlation ΔGheterog / Δ Ghomog
= 0.06. Consequently, heterogeneous nucleation in this case
becomes more or less probable at the overcooling of 100 degrees. Let us note
that in the real processes of casting the overcooling quantities amount to 0.1
- 10 degrees centigrade. At such overcoolings the probability of the formation
of spontaneous, as well as forced, crystallization centers in accordance with
present theory is actually indistinguishable from the zero-point. The supposed improbability of
spontaneous crystallization and the requirements of considerable overcoolings
even for heterogeneous nucleation are regarded as substantial defects of
existent theory, because metal and alloy crystallization goes on unhampered,
with negligible overcoolings. Often overcooling in the process of
crystallization is so small that it can hardly be measured. Thus, the formula (5) connects probabilistic ideas
with the thermodynamic quantity of ΔG, so the heterophase
fluctuations theory starts to laboriously fill in the inconvenient blank of the
thermodynamic theory that is related to the introduction of the critical radius
idea. Ya.I.Frenkel made an assumption that the probability
K of the atomic transition from the melt into a crystalline nucleus is
proportionate to the mobility of atoms in the melt at the temperature of T
/69–70/: K = Kl exp(-U/ RT), where K is the proportionality factor, approximately equal to the
number of atoms in the melt volume viewed (K equals approximately 1023
for one mole of substance); U is the energy of atom activation in the melt; R
being the universal gas constant. Taking into account the three latter formulas, we
arrive at the expression that characterizes the dependence of the rapidity n of
crystallization centers nucleation on the overcooling ΔT of the melt
/74,75/: n = Kl exp(-U/ RT) exp[- Bs3 / T (ΔT)2], (6) where B = 2 (4MT0/ r q) / k is the substance constant.
With an increase in overcooling
there is observed an increase in the rapidity of nucleation; after reaching its
maximum it is again reduced to zero. G.Tamman first formulated a similar
dependence while he was undertaking experimental studies of a series of organic
substances like naphthalene, salol, etc. It was termed Tamman’s curve /76/.
It is assumed that the first exponential multiplier exp[-U/RT] reflects the influence
of the factors that hinder nucleation process, since the lowering of the
temperature provokes the decrease in the rapidity of atomic exchange between
the nuclei and the melt. The second exponential multiplier exp[-Bs3 / T (ΔT)2]
Such a definition presents rather a play upon words,
since the disparity between the vanishingly minor quantity of nucleation, on
the one hand, and the practically measurable one has never yet been possible to
calculate. One more drawback of the theory under analysis
concerns the fact that Tamman’s curve never found its experimental
corroboration for metals. The dependence of the rapidity of crystallization
centers nucleation and the linear rapidity of crystal growth on overcooling for
metals based upon experimental facts is shown in Fig.4 /74/. As we see it in the picture, the real process of
crystal nucleation in metals as dependent on overcooling only increases in
metals. Therefore it is accepted that in application to metals only the
ascending part of Tamman’s curve can be observed. It is assumed that the high
rapidity of atoms in liquid metals causes the latter. So, the heterophase
fluctuations theory enables us to overpass the problem of the critical radius
of crystalline nuclei. Since fluctuations are not growing gradually but appear
on the instant, theoretically large heterophase fluctuations of overcritical
dimensions may in principle arise in a leap. Thus, the thermodynamic problem of
the critical dimension has to be solved by means of probabilistic concepts. The heterophase fluctuations theory, as it follows
from the above-said, interprets liquid as a medium consisting of separate
atoms. 1.9. Of the growth of crystals theory From the viewpoint of present
theory, crystal growth is the result of separate atoms adjunction to the
surface of a crystal. Theoretically, this process is regulated by diffusion
rapidity, whereas in practice the process of growth goes far faster than
diffusion processes and is chiefly determined in reality by the rapidity of
heat abstraction. Yet the existent theory of crystallization ignores the factor
of heat abstraction rapidity and historically uses the overcooling factor. It is assumed that crystal growth depends on the
geometry of growing crystalline planes as well as growth direction /66–67/. For smooth corpuscular surfaces
layer-by-layer crystal growth is thought characteristic by way of formation of
two-dimensional nuclei upon those planes in the form of solid phase monatomic
layer with the ensuing growth of the crystals specified along the whole plane.
Layer transition is realized through the spiral step-by-step mechanism of
growth. However, by contrast with theoretical conceptions, the value of a step
was always several times as large as atom dimensions /66/. On the basis of
experimental facts there originates an idea that the elementary building block
of crystal growth is a certain formation far larger than a separate atom.
Yet crystal growth arises from the already existing
nuclei. Those are different stages of the process, and they are occurring under
dissimilar conditions. In particular, it seems highly important that the
process of growth takes place on the existent surface of the section, whereas
nucleation requires theoretically the forming of a new surface.
v =K2 exp (-U / RT ) exp [-E ( s )2 / T (Δ T)2], (7)
If we take into consideration that
atoms in liquid metals are mobile enough, the first exponential multiplier in
formula (7), as well as in formula (5), may be estimated approximately as one.
Then, formula (7) will be transformed as applied to metals /74/: v = K2 exp [-E ( s’)2 / T (Δ T)]. The dependence of the linear rapidity of
layer-by-layer crystal growth with smooth corpuscular planes on overcooling
will be expressed by the curve reflected in Fig.5. The experimental dependence v on T
in case of liquid gallium crystallization is shown in Fig.6. In a number of
cases, crystal growth goes on without threshold overcooling (Fig.7,8). In the
case given it is premised that growing goes on by the dislocation mechanism. For rough corpuscular planes of crystals
the so-called normal growth by way of chaotic atom joining to any points of
such surfaces is thought characteristic. As a result, the growing crystalline
plane advances far inside the melt, being self-parallel. In this case, the
dependence of the linear rapidity of growing on overcooling is expressed by the
simple formula /74/:
where K4 is the kinetic coefficient characterizing
substance properties; it is premised constant at negligible overcoolings. It is assumed that the normal growth of crystals
occurs at smaller overcoolings. Experimentally, it is esteemed rather hard to
prove. R.Cahn’s theory corroborates that normal growth, on
the contrary, takes place at considerable overcoolings /66/. All the
above-mentioned theories premise that nucleation and the growth of crystals
occur by joining to the solid phase of separate atoms (the monatomic theory).
Such an assumption perceptibly constricts the possibilities of the theory.
Thermodynamics classifies the process of
crystallization alongside with typical processes referring to the dissipation
(dispersion) of energy. Hence there should be several structural levels of
energy abstraction in the system of crystallizable casting, i.e. latent
crystallization heat. Moreover, it is very important to allow for the process
of energy dissipation proper.
1.10. Of the correlation of the structural theory of crystallization with the theory of hardening The terms of crystallization and
hardening denote, in application to castings, the same process though their
semantic load differs essentially. In particular, when we say
‘crystallization’, we mean the structural aspect of the process, i.e. we
understand the process of crystallization as a transition from liquid state to
solid one with the forming of a crystalline structure. When we say ‘hardening’, we imply
the same transition from liquid state to solid one but only as a heat process
without tackling structural problems at all. Such a semantic complexion of the
two of these cognate terms developed historically /77/. The specifically scientific
differentiation under analysis is not so convenient, that is why casting
practice frequently ignores the details of the semantic difference between
these two terms using both of them jointly to define the casting formation
process on the whole. Such a practical usage of the terms also developed
historically. However, the specified distinction
is important enough in scientific works. Thus, apart from the theory of
crystallization that regards hardening as the process of transition from liquid
to solid state with the forming of a crystalline structure, there exists a
theory of hardening which considers the same process without looking into the
crystalline structure of castings and ingots, as a heat redistribution process
exclusively. The latter theory has been elaborated to perfection mathematically
and is steeped in history /74/. A large number of works is
dedicated to the interrelation of the two theories, for the problem of energy
dissipation in the heat theory of hardening is being solved without establishing
the connection between the structure and properties of castings. In its turn,
the structural theory of crystallization does not contemplate energy
dissipation process to the sufficient extent. As a result, the interrelation of
the two theories describing the same process is practically lacking,
notwithstanding considerable efforts to combine them. It is evident that the complete
theory of the forming of castings should fuse both the structural and heat
aspects of crystallization process. 1.11. Statistic theory of crystallization Apart from the two main theories
of crystallization and hardening pointed out above, there exists one more
practically independent trend in the theory of crystallization – the statistic
theory of isothermal crystallization by A.N.Kolmogorov /78/. A.N.Kolmogorov viewed crystallization from the purely
statistic attitude. Such an approach fails to open structural questions nor
does it join the issue of heat abstraction proving once more the correctness of
Pointcarret’s theorem in the sense that any task can be accomplished in a
limitless number of ways. A.N.Kolmogorov’s formula for the solid phase volume V
that is generated in the process of crystallization dependent on time t for the
case of the isothermal crystallization of spherical crystals takes the shape: V = V [1 - exp ( - pnv3 t4 / 3)]
(8) where n is the rapidity of crystalline centers nucleation in a melt
volume unit; v is the linear rapidity of crystal growth. These quantities Kolmogorov accepts as known. I.L.Mirkin applied Kolmogorov’s method to solving the
same problem in case of cubic crystals, N.N.Sirota accomplished the same task
generally for the crystals of arbitrary shapes /74/. Further, using the fact that the
number of crystals N is, as a rule, proportionate to the volume of the melt
crystallizable, there is established a connection between the number of
crystals and the solid phase volume as shown below: V(t) = n (V0 –V)
dt (9) where V is the total volume of the crystallizable metal. If we introduce the v from (8) into (9) at n being
constant and t = 8, we obtain
N = 0.896V0 (n/v)3/4
(10) If we know the volume of crystals
and their quantity, the statistic theory makes it possible to calculate the
average dimensions of the grain d: d = 1.093 (v/n)1/4
(11) 1.12. Of the links between the thermodynamic, fluctuation and probability theories of crystallization Neither the thermodynamic theory of crystallization
nor the heterophase fluctuations theory solves the essential problems of the
crystalline quantity and dimensions of liquid phase in castings. A.N.Kolmogorov’s statistic theory
fills in that breach and is therefore used, as a rule, together with the
fluctuation theory of crystallization that supplies the values of the rapidity
of crystalline centers nucleation as well as the linear rapidity of crystalline
growth for such symbiosis, which cannot be determined by A.N.Kolmogorov’s
theory. In its turn, the fluctuations theory supplements the fundamental
thermodynamic theory of crystallization that is unable to solve the problem of
crystallization on its own because of the thermodynamic barrier presented by
the critical radius of crystallization centers. Thus, the three theories complement one another.
However, the inner unity within such symbiosis of theories unrelated between
them is lacking, which presents one of the main problems in the existent theory
of crystallization that can by right be treated as eclectic. We must observe that A.N.Kolmogorov’s theory allows,
in principle, the stuffing of other quantities of n and v obtained by some other
source. I.e. A.N.Kolmogorov’s theory is not bound up directly with the theory
of fluctuations. A.N.Kolmogorov’s statistic theory, like other ones,
does not respond to the thermodynamic requirement of the two-factorness of
crystallization (as well as melting) process. It cannot step forward with such
a response by nature, for it fails to reveal the causes of melting and
crystallization processes. G.F.Balandin writes: ‘any theory of crystallization
should give answers to the following three questions: how crystals nucleate and
how many of them appear in a unit of time; how these crystals grow and what is
the rapidity of their growth under these or those conditions; what solid phase
quantity appears at any given moment of crystallizing the melt volume specified,
and what is the rapidity of crystallization under the given conditions’ /74/. None of the three above-mentioned basic theories
taken separately gives answers to these questions. Yet their aggregate seems
fraught with contradictions connected with the independence and the lack of
direct ties between the theories named. It is evident from the brief survey that was carried
out that none of the three parts of the modern crystallization theory viewed
above supplies the answer to the first cardinal question broached in the
beginning of this book: what is the cause of melting and crystallization /79/? The clue to the problem of the connection between the
processes of melting and crystallization, the relation between solid and liquid
state is totally lacking in modern theory. Therefore, we arrive at the conclusion that there
exists a pressure to initiate a united theory of the melting and
crystallization of metals and alloys. Chapter 2. General Principles of the Structure of Liquid and Solid Metals as Systems of Interacting Elements of Matter and Space 2.1. General definition of the states of aggregation of matter as different ways of matter-in-space distribution This part includes the most general newest principal
definitions concerning the new approach to the defining of the states of
aggregation of matter. The new approach consists in regarding different states
of aggregation of matter as not the states of matter only and exclusively but
as the states of the systems of interacting elements of matter and space, as
various ways of matter arrangement in physical space, and v.v. /80,81/. Thus, the states of aggregation of
matter are treated as systems comprising two elements: matter and space. Matter
and space are equally important in the structure and properties of the states
of aggregation of matter though they affect this or that specific
characteristic of the system to a different extent. The significance of the
suggested approach is considerably weightier than the problems analyzed in the
book. The novelty of this approach lies in the fact that for the first time it
declares and allows for the equipollent role of the elements of matter and
space in the formation of the states of aggregation of matter, as well as the
forming of any macroscopic physical systems on the whole. The basic premises of the new approach are the
following. The characteristics of the states of aggregation of
matter, e.g. liquid and solid ones, originate by definition at the level of
particle aggregates, i.e. in the presence of a certain large number of
particles per volume. It can never be said about a separately taken particle –
an atom or a molecule – that it is solid, liquid or gaseous. It means that
separate particles act as only the chemical property carriers of this or that substance
but do not bear the characteristics of the states of aggregation of matter
/81–83/. The volume where the properties of the states of
aggregation of matter start becoming apparent is precisely unknown. It is known
only that the volume in question is not large – it approximates the volume of
the smallest drops of liquid. However, any volume is a certain construction that
contains the elements of matter, arranged and granulated in space, and v.v.
Such a construction may be viewed from the angle of geometry, too. Let us
consider various states of aggregation of matter from the mentioned
standpoints, i.e. the points of view of matter-in-space distribution. This
approach has certain advantages. Let us introduce the most general definition of the
states of aggregation of matter. Various states of aggregation of matter present
various ways of matter-in-space distribution inside all real bodies, and v.v. It follows from the definition that any physical
body, any state of aggregation of matter including, comprises two essential
inner components: material and spatial ones /83/. Let us make possible inferences out of the general
definition introduced. Such a way of describing various states, solid and
liquid metallic states including, can be further developed in two ways. These
are the scientifically known substantial and relational ways of describing the
system ‘matter-space’. The substantial approach that originates from Isaac
Newton’s works understands space mainly as a distance, a container inert toward
space, inert atmosphere /116/. Within such an approach the role of space in
forming various states of aggregation of matter is usually ignored as
inessential. Traditionally the role of space in forming various states of
aggregation of matter was by default disregarded, while all the characteristics
of the aggregate states are explained as if they were purely material. The most
significant progress in this sphere developed into the idea of unconfined space
inside this or that substance, which is used but rarely and as a supplementary
concept exclusively. The historically developed negligence of the role of inner
elements of space at macrocosm level is based upon the understanding of space
as the passive part of the environment. The relational approach, arisen
from Einstein’s works, views space as a physical object that is inseparably
united with matter, and presumes the interaction and interrelation of the two
of these components in any states of aggregation of matter. Yet the theory of
relativity contemplates the interaction of material objects only within a
certain space outer towards them without looking into the interaction between
matter and the inner space of the system /86/. So it is accepted in the theory of
relativity and the quantum theory that the interaction and the interference of
matter and space are apparent either at great speeds nearing the velocity of
light or in the vicinity of huge masses of matter. This is true for the
interaction of bodies and particles with the outer space. At macrocosm level the
influence of space on the properties of real physical systems is estimated as
negligible even in this theory. The theory of relativity does not contain any
ideas on various inner elements (particles) of space. The author affirms and proves below by the example of
the states of aggregation of metals and alloys that, apart from the outer space
as regards real space systems, there exists an interior space represented by
discrete spatial elements of various orders inside any physical bodies, which
is inherent in any systems as an integral part of their structure. Active
interaction between the inner elements of matter and space occurs in any
systems, at any dimensional and energy levels; such interaction is a part of
our reality, so we cannot advance multidimensional scientific development but
with the consideration of this factor. Such interaction, its forms and
manifestations are extremely manifold yet subject to theoretic as well as
experimental research. Let us introduce the concept of the relativity of
inner and outer space. Dividing the elements of space into outer and inner is
relative, since the elements of space that are inner toward one system can act
as outer towards some other one. Nevertheless, the relativity under
consideration does not imply the insignificance of such division, for it allows
a better comprehending of the properties of real bodies to describe them with
more precision. Let us also introduce the principle of equivalence
between the inner elements of matter and space on the basis of the above-stated
general geometric definition of the states of aggregation of matter, where the
concepts of matter and space are actually equivalent in the sense that they
cannot exist separately. Out of their interacting elements, matter and space
form combinations and systems of exceeding diversity. Any specific physical
characteristic of any system, liquid and solid metals including, depends, to a
different extent, on the contribution of both the material and spatial parts of
the system specified. Apart from that, the principle of equivalence means
that matter and space perform similar functions of mutual essentiality in the
states of aggregation of matter. It is supposed that the principle of
interaction and equivalence of the elements of matter and space has some
fundamental significance for the forming of various natural systems. Let us consider the role of the inner elements of
matter and space in the formation of the states of aggregation of matter, solid
and liquid states including, from the standpoint of the relational approach and
taking into account the principle of equivalence of the elements of matter and
space formulated above. It must be pointed out that the number of works where
the principles similar to the relational approach are applied to the study of
the connection between solid and liquid states is inconsiderable. In the theory
of liquid and solid states as well as in the theory of crystallization we can
find but separate harbinger elements of this approach. In the previous century, the mentioned approach was
applied to solid metals and other crystals by one of the founders of
crystallography Ye.S.Fyodorov. Ye.S.Fyodorov’s crystalline lattices are
discontinuous-continuous, so space is inhomogeneous within them, there is an
interrelation and interference of the components of the system ‘matter –
space’. It is known that the orientation and arrangement of matter in space
influences the characteristics of crystals greatly. Suffice it to supply the
well-known examples of various forms of carbon presented by graphite, diamond,
and carbyne. These examples demonstrate substances consisting of the same atoms
of carbon that acquire highly different properties due to the various
distributions and interaction of these atoms in space. However, it is premised
by crystallography and chemistry that everything depends on the elements of
matter – atoms –only, whereas the elements of inner space are disregarded as
the passive component of the system. Academician N.S.Kurnakov touched in his works upon
the subject of the interaction of matter and space in various chemical
compounds. He showed that there is a tangible parallel to be traced between the
chemical process and characteristics of space in the phenomena of equilibrium
of chemical reactions between various compounds that are expressed (the
phenomena) by geometric surfaces. Unfortunately, those were nothing
but separate phrases in the works by N.S.Kurnakov, which never found their
evolution /85/. Recently, on the grounds of the analysis of the works
dedicated to the problem given, V.I.Vernadsky’s works in particular /84/, there
appears the following definition of the interrelation between the states of
matter and space: ‘ It is obvious that the same spatial structures cannot
correlate with all the divergent states of matter; on the contrary,
qualitatively different states of matter will inevitably meet their counterpart
among correspondingly different spatial structures, among various states of
space.’ Matter, by V.I.Vernadsky’s term, means substance according to our
terminology. This discovery made by V.I.Vernadsky must have
forestalled its time. It should be marked that V.I.Vernadsky’s work ‘ Space and
Time in Living and Inorganic Matter’ that comprises these ideas, was published
only after the author’s death, in the 70-s. It remained totally unclaimed by
physics and exact science. We treat V.I.Vernadsky’s idea here as one of the
fundamental concepts of the will-be relational theory of
melting-crystallization as well as the connection between the structure of liquid
and solid metals. The theory under analysis may be also termed as the theory of
relativity for metals and other macrosystems. From this point of view, let us consider liquid and
the adjacent solid and gaseous states as the ways of matter-in-space distribution
and v.v. Let us start from gaseous state as the simplest and the most
well-explored. Ya.I.Frenkel was among the first to discover one of
the forms of the inner elements of space in solid metals represented by
vacancies – hollow lattice points. He linked the structure of vacancies to the
vacuum medium in gases. By Ya.I.Frenkel’s definition, ‘…in case of gaseous
state… hollows merge with vacuum, where separate molecules appear to be
ingrained so that vacuum ceases acting as the dispersion phase and becomes the
dispersion medium’ /70/. Such a definition of gas quite agrees with the way of
description accepted in this work. Thus, we can assume that space is continuous in
gaseous state acting as the dispersion medium, whereas matter is discrete and
acts as the dispersion phase, in which connection both of the two components of
gaseous state are at equilibrium with each other. The continuous form of matter represented by a set of
crystalline lattices, as is well known, characterizes solid crystalline state
at the level of aggregation states (at the phase level). If we proceed from the above-accepted definition of
the states of aggregation of matter as various forms of matter-in-space and
v.v. distribution taken together with the principle of equivalence and symmetry
of the elements of matter and space, it ensues that there must be certain
elements of physical space in solid state existing at equilibrium with the
crystalline lattice. The works by the two founders of the point-defects
theory in crystals – Schottky and Ya.I.Frenkel – give us the reason to surmise
that the target elements of space in crystals are represented by vacancies
/70,88/. Thus, Ya.I.Frenkel writes that in liquid state
hollows (vacancies) are caused by ‘…the process that can be termed the
dissolution of the surrounding vacuum in a crystal’. Ibidem further: ‘…the
lattice point left vacant…can be regarded as a hollow that appears to be
absorbed by a crystal from the surrounding space’. Thus, Ya.I.Frenkel considered
vacancies, by their origin as well as by their characteristics, as the elements
of physical space (vacuum) in the crystalline lattice. A similar point of view
is advanced in the works by B.Ya.Piness and Ya.Ye.Guegouzin, where vacancies in
the crystalline lattice of metals are viewed as a parity component having its
volume but deprived of mass /89-90/. Later on the concepts of the parity
metal-vacancy diagrams were worked out on this basis /91/. If we stick to the set-point
approach, we may conclude that matter represented by crystalline lattices is
continuous in solid state whereas space represented by vacancies is discrete;
on the contrary, space is continuous in gaseous state, matter being discrete.
There is a case of dissymmetry of matter and space in the states of aggregation
of matter. Inferences concerning liquid state will be made
below. Yet the following conclusion shapes right away: liquid state also act as
a form of matter-in-space distribution, and v.v. Consequently, there must exist
both material and spatial elements in liquid state that are at a dynamic
equilibrium. Further we shall ascertain what these elements are. 2.2. Premelting Let us consider the processes that precede melting in
metals. It is known that all the characteristics of metals change depending on
temperature. In the broad sense, the total of the changes of all the metal
characteristics with an increase in temperature is premelting, since, one way
or another, all of them reflect the changes in the equilibrium between the
elements of matter and space in solid state, which result in melting. A possibility of transition to
other aggregation states inheres in any state of aggregation of matter. So if
aggregation state is a form of the interacting elements of matter-in-space (and
v.v.) distribution, then, phase transitions are the transitions from one form
of matter-in-space distribution to another. However, such a definition sounds excessively
general. It defines the essence that is shared by all phase transitions, which
is important enough but does not reveal the specificities of the mechanism of
melting and crystallization processes that we are interested in. The mechanism
of every kind of phase transitions has peculiarities of its own, since each
form of matter correlates with its respective form of space. These
specificities are to be brought to light here for the processes of melting and
crystallization. Therefore, let us single out the processes that
relate to the preparation of melting in solid state most directly and are
directly responsible for the mechanism of this process. It was found before that vacancies act as the
characteristic form of the elements of space in the crystalline lattice of
solid metals. Vacancies within the crystalline lattice of metals are in motion,
the motion being similar to that of matter molecules in gases: it is chaotic
and accelerates with an increase in temperature. The behavior of vacancies in
metals is described by the same expressions as the behavior of particles in
gases with the essential distinction lying in the fact that the rapidity of
vacancy motion in solid metals is much lower than particle velocities in gases
while the trajectory of motion structurizes by the crystalline lattice. Still,
those are quantitative differences, whereas in the qualitative aspect vacancies
in solid metals generate the same gas of space-in-matter elements as is
produced by atoms and molecules in gases. So there exists in physics a concept of 'vacancy
gas'. In addition, the concept of vacancy gas pressure is applicable to such
gas, similar to particle gas pressure. Vacancy gas pressure diffuses all through the volume
of solid metal, similar to the pressure of regular gas within the total volume
of such gas. Hence, vacancy gas pressure operates from within upon the
crystalline lattice which functions as the environment and a shell for vacancy
gas at the same time. Similar to regular gas within a rubber bladder, vacancy
gas generates tensions within its shell inside the crystalline lattice and,
should the value of pressure overpower durability, may destroy this shell. Vacancy gas pressure p within the crystalline lattice
can be calculated out of the known relation: pv = nkT, where n is the relative concentration of vacancies; k is Boltzman constant; T is
temperature. The graph of the given function is
shown in Fig.9. The values of vacancy gas pressure
in certain metals at the melting temperature are calculated in Table 2. Table 2. Vacancy Gas Pressure in Solid Metals in
the Vicinity of the Melting Temperature
At low temperatures vacancy gas
pressure is negligible as compared to the durability of solid metals, so the
presence of vacancies does not endanger the latter. However, vacancy
concentration and pressure rise exponentially with an increase in temperature
reaching the values listed in Table 2. These are small but quite measurable
pressure values. It must be noted that vacancy gas
pressure in solid metals cannot be measured experimentally so far; the author
is not even acquainted with such measurement procedures. There exists an
opinion that the formula suggested above allows pressure measurement for gases
only. Vacancy pressure in solid metals must conform to the
same principles as molecule pressure in gases. The difference consists in the
fact that experimental procedures for measuring vacancy pressure are to be
developed yet, followed by the measurement of other forms of spatial elements
at other levels. Those should be the procedures of pressure measurement of the
inner elements of matter and space (not outer ones) represented by vacancies in
case of solid metals. Once such procedures are established, the equation of
state is not excluded to become much more universal than it appears at present.
Still, the given equation should be applied to the evaluation of a system's
inner parameters only. The pressure of the inner elements of matter and space
may diverge from the ambient pressure. Such a phenomenon takes place in case of
vacancies.
Let us mark that the presence of vacancies performs a
crucial function for the majority of the characteristics of solid metals, which
is decisive at times, as it is in case of electron conduction and
superconductivity. Yet vacancy gas pressure leads to melting only with an
increase in temperature and together with the other factor only.
Fig.10, which demonstrates that some elements
(carbon) and compounds (SiC, UC, BN) do not suffer durability loss with a
temperature increase, presents experimental curves of the dependence of the
durability of certain refractory elements and compounds on temperature. It is
important to stress that the same elements and compounds do not melt in the
accepted sense, i.e. they do not form liquid phase. Thus, experimental data
corroborate the importance of durability as one of the factors that determine
melting. With an increase in temperature vacancy gas pressure
in solid metals rises rapidly, whereas the durability of the same metals
declines correspondingly. Let us recall that the same units measure pressure
and durability. Hence, the equality is invariably reached in solid
metals at a certain temperature. pv = σв,
(12) pv is vacancy gas pressure; σв being metal durability
of elongation. The point of the intersection of the curves pv
= f(T) and σв=
f(T) in Fig.10 coincides with the melting temperature. From the position of the
above-said, melting is the process of destroying the continuous crystalline
lattice under the pressure of vacancy gas. Still, it is a general definition.
It does not reveal the specialities of destruction process, whereas if those
are the specialities, or the details, of the process that determine its result,
what liquid metals structure will be like after melting? We are going to look into the
details of melting process exemplified by the elementary act of melting below
with the gradual interpretation of the peculiarities of this process.
|
Element |
Effective coordina-ting number |
DHmelting,
C/mole /97,98/ |
DHvap, C/mole
/97,98/ |
rc, 10-10 m, calculations by (33) |
rc/ a |
nc, cube, calcula-tions by (32) |
nc, sphere |
Cu |
12 |
3.1 |
80.3 |
14.8 |
5.8 |
3300 |
1650 |
Ag |
12 |
2.69 |
60.0 |
17.0 |
6.4 |
4300 |
2160 |
Au |
12 |
3.05 |
82.0 |
19.5 |
6.7 |
4800 |
2400 |
Pt |
12 |
5.2 |
112.0 |
14.4 |
5.2 |
3300 |
1250 |
Pd |
12 |
3.5 |
110.0 |
21.3 |
7.7 |
7800 |
3900 |
Al |
12 |
2.57 |
69.0 |
20.5 |
6.7 |
6300 |
3150 |
Pb |
12 |
1.15 |
42.5 |
26.2 |
9.2 |
12600 |
3150 |
Ni |
12 |
4.22 |
89.4 |
13.2 |
5.3 |
2400 |
1200 |
Co |
12 |
3.75 |
91.4 |
15.0 |
6.0 |
3600 |
1800 |
Ti |
12 |
4.5 |
102.5 |
16.2 |
5.6 |
3000 |
1500 |
Zr |
12 |
4.6 |
128.0 |
22.1 |
6.8 |
5400 |
2700 |
Re |
12 |
8.0 |
169.0 |
14.4 |
5.2 |
2340 |
1170 |
Ce |
12 |
2.12 |
75.0 |
31.7 |
8.7 |
11000 |
5500 |
Zn |
12 |
1.74 |
27.3 |
10.4 |
3.9 |
960 |
480 |
Cd |
12 |
1.53 |
23.9 |
11.5 |
3.9 |
960 |
480 |
Ca |
12 |
2.1 |
39.9 |
18.5 |
4.7 |
|
860 |
Mg |
12 |
2.1 |
30.5 |
11.3 |
3.5 |
|
340 |
Hg |
6 + 6 |
0.549 |
14.13 |
22.0 |
6.4 |
|
2200 |
Fed |
10 |
3.3 |
81.3 |
18.0 |
7.27 |
5400 |
2700 |
V |
10 |
5.05 |
109.6 |
16.8 |
6.4 |
|
1820 |
Cr |
10 |
4.6 |
89.9 |
13.3 |
5.3 |
|
1050 |
W |
10 |
8.4 |
183.0 |
17.5 |
6.4 |
|
1850 |
Mo |
10 |
6.6 |
121.0 |
14.5 |
5.4 |
|
1100 |
Nb |
10 |
6.4 |
166.5 |
22.0 |
7.7 |
|
3160 |
Ta |
10 |
5.9 |
180.0 |
25.8 |
9.0 |
|
5100 |
Sn |
10 |
1.69 |
64.7 |
34.7 |
11.2 |
|
10000 |
Li |
10 |
0.7 |
35.3 |
44.4 |
15.0 |
|
23400 |
Na |
10 |
0.63 |
23.7 |
41.8 |
11.2 |
|
10000 |
K |
10 |
0.57 |
18.9 |
44.6 |
9.7 |
|
6500 |
Cs |
10 |
0.50 |
15.9 |
49.2 |
9.3 |
|
5700 |
Bi |
6 + 1 |
2.6 |
42.8 |
26.7 |
8.0 |
4500 |
2250 |
Ga |
6 |
1.336 |
61.4 |
62.6 |
22.5 |
96000 |
48000 |
Si |
4 |
12.1 |
72.5 |
- |
3.88 |
320 |
160 |
Ge |
4 |
7.7 |
78.3 |
- |
6.5 |
|
765 |
Table 3 demonstrates that cluster
dimensions are calculated for quite a wide range of metals prioritized in
technics and metal science.
If we compare nc for cubic and spherical
clusters, it is possible to observe that spherical clusters contain the number
of atoms twice as small as cubic ones. Such a distinction seems essential
enough; it shows that the choice of the right cluster shape is sufficiently
important. The idea of spherical clusters will be used further on as the basis.
As the table shows, cluster dimensions differ
essentially between themselves in various metals though preserving their order
upon the whole. The minimal number of atoms in a cluster nc for
silicon, magnesium and zinc amounts to 160, 340 and 480 atoms accordingly; the
maximal values of nc for gallium and lithium are 48000 and 23400
correspondingly. However, nc has the order of 103 for the
majority of metals at the melting temperature, which coincides on the whole
with the evaluation of cluster dimensions carried out by other researchers
/99,100/.
The average radius of a cluster at the melting
temperature equals approx.10-9m for the majority of metals. Thus,
clusters are very small formations that are difficult to detect by means of
direct observation. Besides, clusters exist only in motion, only in aggregates
and at interaction with the intercluster splits. Clusters have neither stabile
boundaries habitual to macrocosm nor surface sections but flickering boundaries
or surfaces only. These are quite specific objects with unusual
characteristics, so we need new experimental methods to study them.
The elements of space – flickering intercluster
splits of bonds – form the other equilibrium structural zone in liquid, which,
by interacting with the zone of clusters, constitutes the specific structure of
liquid metals. The basic parameters of the given zone may be calculated
quantitatively. In particular, the average dimensions of a single intercluster
split can be determined, as well as the quota of volume occupied by the
totality of splits in liquid metals.
Since intercluster splits relate to clusters by
definition, their area will be equal to the cluster section area, which is
proportional and closely approximate to the value of rc2.
Let us determine the width of intercluster splits
proceeding from the following considerations. The formation of such splits is
possible only in case when intercluster spacing expands to the value of α
equivalent to the relative theoretic deformation of matter at distension and
will make a (1 + α). Under the conditions of heat oscillations it
corresponds to the situation when a half of all the spacings between clusters
will be less than a (1 + α) = (a + aα). In the meantime,
splits are either lacking or closing. The other half of intercluster spacings
will exceed (a + aα), which corresponds to the split of bonds. The
quota of the element of space proper out of the present quantity will amount to
aα.
It was underlined above in parts 3.1 and 3.2 that no
sooner is the intercluster split formed, than the returning of the cluster into
its original position starts. Therefore, the average quantity of the width of
intercluster split must also approximate the quantity of αa, where
a is the shortest interatomic spacing in solid metal in the vicinity of the
melting temperature. The area of a single intercluster split will be
approximately equivalent to rc2. Admitting that a
cluster performs heat oscillations along the three axes, we obtain that the
area of intercluster splits per cluster equals approx. 3rc2.
Then, the total area of
intercluster splits per gram-atom of any liquid metal will be equivalent to
Scl = 3N0 rc2 / nc
.
Allowing that N0 = 6 1023, rc
= 10-9 m on average, while nc = 103
on average, we get that Scl ≈100
sq.m / g-atom on average. It means that liquid
metals have a gigantic surface area of the inner elements of space –
intercluster splits. So these flickering inner surfaces constitute an essential
part of the structure of any liquid metal and any other liquid, too. The
presence of such surfaces determines many characteristics of liquids, including
such a fundamental characteristic of liquids as fluidity, in particular (see
below).
Returning to the volume of a
single intercluster split in liquid metals, we obtain that the given volume is
equal to the surface area of a single element of space, multiplied by the width
of such an element:
vs ≈rc2
αa. (34)
The quantity of α may be found from the expression cited by
Ya.I.Frenkel /70/:
α = σmax / E,
where σmax is the ultimate theoretic strain
of matter at elongation; E being the modulus of elasticity of matter.
In its turn /101/, the ultimate
theoretic strain of matter can be evaluated from the expression
σmax = (E γ / a) 1/2
and
α = (E γ / a) 1/2 /
E, (35)
where γ is the coefficient of surface tension in liquid metal
at the melting temperature.
Turning back to the volume of a single intercluster split in liquid
metals, we get:
vs ≈rc2
αa. (36)
The number of intercluster splits Ns in a mole of
liquid approximates the molar quantity of clusters Nc:
Nc ≈ N0 / nc.
(37)
If N0 = 6 1023, while nc
averages 103, the number of intercluster splits Nc = Ns
= 6 1020 on average per mole at the melting temperature. This is
quite a large quantity.
The summarized absolute volume of splits in liquid
per mole will be equal to
Vs ≈Nc vs
= (N0 / nc) rc2
αa. (38)
Practice requires the knowledge of
the quota of the total volume occupied by the elements of space rather than the
absolute volume of the zone of the elements of space (which may also be termed
as the zone of unconfined space).
Having evaluated the average spacing between clusters
by the value of a (1 + α) and its expansion as compared with the
non-split state by the quantity of aα, it is quite easy to find the
corresponding change in the system's volume ΔVspl out
of the known expression that relates the change of the length of the object to
the change of its volume /102/.
If the length of a cube-shaped
body is 1, while length increase equals α, the relative
augmentation of the volume of the body will approximate
Δ Vspl = 3α.
Since length for clusters is
l = 2rc,
the relative volume of the zone of intercluster splits will be as
large as
ΔVspl = (3α / 2rc)
100% (39)
By inserting the value of α from (35) into (39), we get
Δ Vspl = 3 (E γ / a) 1/2 / E 2 rc.
Expression (39) is the most suitable for calculations, since the
values of rc are already known there. The values of the
quantities required for calculations are listed in Table 4 below.
Calculating ΔVspl under (39) shows that the
volume occupied by the zone of intercluster splits (the elements of space) in
liquid metals at the melting temperature fluctuates within the limits of 1-6%
for the majority of metals (v. Table 4 below).
Table 4. The Volume of the Zone of Intercluster
Splits in Liquid Metals at the Melting Temperature
Metal |
g, erg/ccm /12,20/ |
E, kg/ccm /101/ |
a, calculations by (35) |
DVspl,
%, calculations by (39) |
Cu |
1133 |
11200 |
0.19 |
4.85 |
Ag |
927 |
7700 |
0.205 |
4.70 |
Au |
1350 |
11000 |
0.226 |
4.95 |
Pt |
1800 |
15400 |
0.205 |
5.7 |
Pd |
1500 |
11900 |
0.214 |
4.08 |
Al |
914 |
5500 |
0.24 |
5.30 |
Pb |
423 |
1820 |
0.26 |
4.15 |
Ni |
1825 |
21000 |
0.183 |
5.10 |
Co |
1890 |
21000 |
0.185 |
4.56 |
Zn |
770 |
13000 |
0.145 |
5.47 |
Feg |
1835 |
20000 |
0.177 |
4.84 |
Fed |
1835 |
13200 |
0.227 |
5.1 |
Sn |
770 |
4150 |
0.248 |
3.3 |
Cs |
68 |
175 |
0.27 |
4.3 |
Ta |
2400 |
19000 |
0.21 |
3.46 |
Mo |
2250 |
35000 |
0.153 |
4.27 |
Nb |
1900 |
16000 |
0.204 |
3.93 |
W |
2300 |
35000 |
0.155 |
3.59 |
Bi |
3900 |
- |
0.207 |
3.7 |
Ga |
735 |
- |
0.20 |
1.33 |
Thus, the elements of space in
liquid metals occupy from 1.33 to 5.7% of the total volume of liquid.
Accordingly, clusters occupy from 94.3 to 98.67% of the total volume of liquid.
The volumes that are occupied by the latent elements of matter and space are
included in the quantities specified.
At melting liquid acquires a large amount of extra
energy as the latent heat of melting, yet the temperature of the liquid does
not change during the process. This is possible only in case when there
originate new degrees of freedom, i.e. new kinds of motion, within the system -
liquid metal in the case given. When analyzing the motion of a cluster at the
point of its formation (the elementary act of melting), it was proved above
that a new kind of motion – heat oscillations of clusters – arises in liquid as
a result of melting.
Let us find the energy of the
oscillations in question, which will enable us to calculate the frequency of
heat cluster oscillations in liquid further on.
In conformity to the theorem of classical statistics
of the uniform distribution of energy according to the degrees of its spareness,
any extra energy within the systems that consist of a large number of particles
is distributed uniformly among all the constituent parts of the given system at
microlevel.
The constituents of liquid at microlevel are clusters
and atoms.
Each particle receives an amount of energy equal to
Ei = Δ Hmelting/ (N0 + Nc),
where N0 is Avogadro Number; Nc
is the number of clusters in a gram-atom of liquid metal.
Since Nc << N0,
the latter expression can be written without any appreciable error as
Ei = Δ Hmelting
/ N0. (40)
On the other hand, we know that the energy of heat
oscillations of one atom makes
Ea = (3/2)
kT (41)
In compliance with the theorem of
the uniform distribution of energy, particle dimensions are not to be taken
into consideration, so the energy of heat oscillations of a cluster that
comprises many atoms will be equal to the same quantity as the energy of
oscillations of a single atom:
Ec = Ea = (3/2) kT (42)
At the melting temperature the quantities of Ec
and Ei must be equal, or
Ec = (3/2) kTmelting
(43)
Comparing the values of Ec and Ei
from (43) and (40) correspondingly gives us the possibility to test the
accepted hypothesis of the equality between the two quantities specified.
To do this, let us calculate the values of Ec and Ei.
The results of calculations are
listed in Table 5.
Table 5. The Energy of Heat Oscillations of
Clusters
Metal |
Ec, J, calculation by (43) |
Ei, J, calculation by (40) |
Ei /Ec |
Na |
0.77 10-20 |
0.43 10-20 |
0.56 |
Pb |
1.24 10-20 |
1.15 10-20 |
0.93 |
Zn |
1.87 10-20 |
1.73 10-20 |
0.92 |
Fe |
3.74 10-20 |
3.08 10-20 |
0.83 |
Cr |
4.50 10-20 |
3.22 10-20 |
0.72 |
Ni |
4.66 10-20 |
2.92 10-20 |
0.62 |
Co |
4.77 10-20 |
2.73 10-20 |
0.57 |
It follows from the data presented in Table 5 that
the suggested hypothesis of the equality between the quantities of Ei
and Ec is corroborated, for the values of these quantities are very
close numerically. A negligible error of determination constitutes approx.± 20 %,
which is rather rare to be observed in calculations of such a kind, if we allow
for the difference in electron structure, as well as the peculiarities of the
structure of crystalline lattices, etc. Presuming that the quantity of Ec
is determined with more precision, we can calculate the average correction
factor to formula (43) on the basis of the data listed in Table 3. The
coefficient in question is 0.707.
By way of inserting the signalized
coefficient into (43), we arrive at the improved formula
Ei = 0.707 Δ Hmelting / N0.
(44)
The most important conclusion to
the given part of the work is the following: the latent heat of melting equals
the energy of heat oscillations of particles at the melting temperature with a
negligible error, hence after melting the specified energy is really spent to
establish new degrees of motion freedom in liquid metals – heat oscillations of
clusters and atoms included into them as a unit.
The data supplied in Table 5
corroborate numerically the correctness of the given inference.
The previous calculations, if they
prove to be correct, allow completing a successive procedure - to calculate the
point of metal melting. It suffices to equate the right sides of expressions
(40) and (43) allowing for the fact that calculations for Table 3 presume that
T = Tmelting.
Thus, we get
Δ Hmelting nc / N0
= (3/2) kTmelting.
Hence we derive the expression for
calculating the melting temperature of metals:
Tmelting = Δ Hmelting / 1.5 N0 k.
(45)
An extraordinarily simple expression (45) is derived
to calculate the melting temperature of metals which relates the given
temperature to the known physical constants: the latent heat of melting,
Avogadro Number and Boltzman
constant.
The results of calculating the
melting temperature of metals under (45) are presented in Table 6.
Table 6. The Melting Temperature of Metals
Metal |
Δ Hmelting, C/mole
-1 |
The Melting Temperature, Тmelting, К |
|
Тmelting, К
by (98) |
Тmelting, К
exper./98/ |
||
Al |
2.58 |
876 |
933 |
V |
5.51 |
1857 |
2190 |
Mn |
3.5 |
1179 |
1517 |
Fe |
4.4 |
1428 |
1811 |
Ni |
4.18 |
1406 |
1728 |
Cu |
3.12 |
1051 |
1357 |
Zn |
1.73 |
583 |
692 |
Sn |
1.72 |
529 |
505 |
Mo |
8.74 |
2945 |
2890 |
As we see it from Table 6, formula
(45) lets obtain only approximate values of the melting temperature accurate
within 2 to 30%.
Although the accuracy under
consideration is not so high for practical application, we should observe that
other methods of calculation the melting temperature with the same or higher
accuracy do not exist so far. Formula (45) ensures the highest accuracy of
calculating the melting temperature of metals at present. In the aggregate with
other calculated data, the data in Table 6 corroborate the applicability of the
developed theory to the description and calculation of a wide range of the
parameters and properties of liquid metals.
A considerable amount of activated atoms in liquid
metals is the next essential peculiarity of their structure. The term of
activated atoms presupposes atoms that have at least one free bond. Such atoms
are represented by surface-located ones as compared with the atoms positioned within
the volume.
Since liquid is saturated with a
large quantity of inner flickering section surfaces, all the atoms that come to
be on such surfaces at a definite moment become activated during the
half-period of flickers, i.e. they acquire extra free energy for the period of
the existence of the given surface.
Such atoms are far more mobile and reactive in
comparison with the atoms that are located within cluster volume both on
account of a higher energy of their own and their position on the surface /30/.
Therefore, we reckon it worthwhile to conditionally single out the zone of
activated atoms taking into consideration their relative concentration in
liquid Ca.
Let us underscore that activated atoms in liquid do
not form any structural zone in liquid. All activated atoms enter into
clusters. There are no other explicit structural units of matter in liquid
except clusters. Activated atoms differ in the sole respect that they come to
be located on the flickering surface for a short time, so they acquire extra
energy and a relative freedom of moving along cluster surface or between
clusters for that short period of time only. The split closes next moment, and
the existent activated atoms lose their supplementary energy. We may say that
activated atoms in liquid metals are flickering, too. Disappearing together
with the split at one site, activated atoms emerge at some other location, so
their average amount in liquid is constant at any moment of time under constant
conditions.
The quantity of Ca may serve as the
measure of the disordering of liquid metals contrasted with solid metals, where
the quantity of Ca is very small being approximately equal to
vacancy concentration inside them (0.001) by the order of their quantity.
Let us determine the concentration of activated atoms
in liquid metals Ca as the relation of the number of free bonds on
the surface of a cluster n to the number of atoms in a cluster nc.
It was shown above that n equals
to a half of all the bonds on the 'surface' of a cluster, i.e.
n = n1/ 2.
Applying the above-used procedure of expressing the
number of bonds through the area of cluster surface S and its volume V,
we get
Ca = n1 /2nc = S / 2V = 4π rc2
/ (4/3) π rc3 = 3/2 rc-1.
(46)
Expressing rc according to (33),
for spherical clusters we have
Ca = (3/2) (ΔHmelting / Δ Hvap)
β-1/3. (47)
The values of Ca calculated under (47) can
be found in Table 7.
Table 7. The Concentration of Activated Atoms in
Liquid Metals at the Melting Temperature
Element |
Cu |
Ag |
Au |
Pt |
Ni |
Co |
Fe |
Zn |
Si |
Cs |
Al |
Pb |
W |
Ca, %, by (47) |
23 |
25 |
21 |
25 |
26 |
23 |
28 |
36 |
28 |
14 |
18 |
15 |
22 |
It is demonstrated that the
concentration of activated atoms in liquid metals is high enough at the
temperature of melting already. A large quantity of activated atoms secures the
high reactivity of liquid metals, as well as the intensive mass exchange
between clusters, and accounts for some other distinctions of liquid metals.
The quantity specified in the
headline is of extreme importance, for it determines the major dynamic
parameters of liquid metal, particularly the characteristics of mass transfer,
impulse, the period of relaxation in liquid and certain other practically
significant quantities.
We are not acquainted with any
other ways of calculating the frequency of cluster heat oscillations in liquid
metals, which imparts a peculiar actuality to our calculation procedure. The
problem of the frequency of intercluster splits flicker is not only unexplored
but it has never been opened to discussion.
It should be stipulated that heat
oscillations of clusters as units do not substitute for atomic heat
oscillations in liquid. Those are two different kinds of motion that exist in
liquid simultaneously. The frequency of flickers of intercluster splits equals
numerically the frequency of cluster heat oscillations, since heat oscillations
of clusters and the flickers of intercluster splits represent the two aspects
(material and spatial) of one and the same process of the interaction of the
elements of matter and space in liquid.
The very existence of clusters is
possible only under the condition of their heat oscillations, since only one
half of the ‘surface’ of a cluster is indicated and separated by intercluster
splits at any given moment, hence a cluster can be singled out only as the
totality of atoms performing simultaneous heat oscillations.
It must be noticed that any motion of matter is
performed in space being reflected there. We may affirm that any kind of motion
of a certain material form is always accompanied by a related kind of motion of
the corresponding elements of space. Matter and space move but simultaneously.
Such an approach is absolutely new and unstudied yet
challenging in many respects, since it enlarges essentially the existent
concepts of motion and suggests investigating as well as allowing for the
previously unknown forms of motion of various spatial elements. The concept of
the motion of spatial elements is quite new on the whole, so it requires
specification by examples. The motion of vacancies inside crystals may be
supplied as an example of motion of the elements of space, which is propagated
in literature.
In case of liquid metals such a previously unknown
form of motion of the elements of space is the oscillatory process of
intercluster splits flickering. The process under consideration can be
expressed through the following formula:
αn + αn →← 2αn
The given scheme reflects the
constant process of cluster flickering when intercluster splits are
periodically opened and closed, while clusters periodically merge and separate.
The same scheme works as applied to melting or crystallization with a shift to
the right (crystallization) or left (melting) but not under oscillatory
operation.
Small dimensions of clusters make
it possible to employ the theorem of the uniform distribution of energy
according to the degrees of its spareness. We substantiated such a possibility
above in Part 3.6.
Let us designate the frequency of heat oscillations
of clusters as φ.
The energy of heat oscillations of clusters can be
determined by (42) as
Ec = (3/2) kT.
The quantity of φ is to be found from the expression
suggested by the oscillations theorem /103/:
φ = (1/2π A) (2Ec/
mc) 1/2, (48)
where A is the amplitude of cluster oscillations; mc is
cluster mass.
It was shown in Part 3.5 that the spacing between
clusters in liquid increases by the quantity of aα, where a is the
shortest interatomic spacing in a crystal at the melting temperature, while
α is the relative maximum deformation of matter at distension.
Hence A = aα. Let us find cluster mass by the expression
mc = M nc / N0, where M is the
atomic weight of matter; nc being the number of atoms in a
cluster according to Table 2, Part 3.4.
By inserting the values of Ec, A and mc
into (48), we arrive at
φ = (1/2πaα) (3kT N0
/ ncM) 1/2.
(49)
The period of heat oscillations of clusters τ will be
equivalent to τ = φ-1, or τ = 2πaα (3kT N0 / ncM)-1/2.
As it follows from (49), the expression for the
frequency of heat oscillations of a cluster differs from the frequency of
atomic heat oscillations by the value of the amplitude of oscillations and the
presence of the nc quantity under the radical.
Nevertheless, the numerical quantities of the
frequency of heat oscillations of clusters are cited in Table 8 below.
Table 8. The Frequency φ and the Period τ
of Heat Oscillations of Clusters and the Frequency of Intercluster Splits
Flickering in Liquid s at the Melting Temperature (Calculation by (49))
Metal |
j, s -1×10-8 |
t, s×108 |
Li |
0.74 |
1.35 |
K |
0.43 |
2.38 |
Cu |
13.5 |
7.40 |
Ag |
21.3 |
4.7 |
Au |
4.20 |
0.24 |
Al |
4.40 |
0.23 |
Pb |
0.7 |
1.43 |
Fe |
10.0 |
0.10 |
Co |
18 |
0.05 |
W |
13 |
0.078 |
It is known that the frequency of heat oscillations
of atoms inside the crystalline lattice by the order of magnitude comes to 1012-1013c-1,
which is on average four orders larger than the corresponding cluster
dimensions. The calculated values of the frequency of heat oscillations of
clusters approximate the values cited in literature and obtained by other
procedures /31/.
The listed data show once more
that at least two independent kinds of heat oscillations of particles coexist
simultaneously in liquid metals, so we should take it into consideration.
The issues of stability of the elements of matter in
the structure of liquid metals and alloys were repeatedly taken in literature
/104,105/. The concept of the elements of space introduced and described in
detail in the present work is not being discussed yet.
The stability in time, or the
period of cluster existence, is important for the understanding of numerous
practical results of metallurgical and casting practice. For instance, it is
useful to account for metallurgical structural heredity /105/.
The initiator of the cybotaxis theory Stewart
considered cybotaxes as rather unstable formations with a short period of
existence correlative to the period of atomic heat oscillations (1 10-12-1
10-13 sec.) /3-4/.
Atomic fluctuations in liquid, heterophase
fluctuations including, exist for a very short period by definition, too – of
the order of 1 10-12sec. V.I.Nikitin and others determine the period
of cluster existence as 10-5 … 10-8sec. /105/.
Undoubtedly, this is insufficient to treat clusters as hereditary information
carriers during the whole period of the existence of liquid state.
Our results show that the period of heat oscillations
of a cluster amounts to the order of 1 10-8sec. However, the period
of cluster existence must by far exceed the given quantity.
In connection with the specific characteristics of
clusters, e.g. the absence of stable surfaces and composition, the flickering
nature of interaction with the elements of space, etc., the period of cluster
existence may be determined only under the premise of their mutability. At the
same time, the changeability of real objects in time is virtually the universal
characteristic, so there is nothing objectionable in that.
To determine the time of cluster existence, we should
recall the definition of clusters as the main structural units of matter in
liquid within the entire temperature-temporal interval of the existence of
liquid aggregation state.
Allowing for all these stipulations, we may assert
that the period of cluster existence is limited by nothing except the interval
of the existence of liquid aggregation state.
In technological processes the duration of the
existence of alloys in liquid state and the period of cluster existence are
evaluated in hours. In natural processes, the period of liquid state existence
as well as that of clusters can take milliards of years.
Similarly to that, the period of crystal existence is
limited by the duration of solid crystalline aggregation state that may also
total milliards of years in natural processes.
It means that clusters are quite stable formations in
liquid state that have nothing in common with fluctuations and other
short-lived formations.
Consequently, clusters with definite changings in dimensions,
composition, etc., exist continuously in liquid alloys from the moment of
fusing up to the moment of crystallization. There are no other limitations to
the period of cluster existence. Or τcl = τliq,
where τcl is the period of cluster existence, τcl
being the duration of the existence of liquid state.
We can refer the above-said to the period of the
existence of the elements of space – intercluster splits - in liquid
considering their specificities.
Such a period of cluster existence
seems quite valid to account for their property to act as the carriers of
certain structural information while interrelating liquid and solid states by
some parameters.
Certain important parameters of the structure of
liquid metals were determined and calculated in Chapter 3. However, the
characteristics of liquid metals change depending on environmental conditions.
A minor dependence of the characteristics of the majority of liquid metals and
alloys on pressure is observed in literature. Still, it is known that this is
the change of temperature that affects the structure and properties of any
metals and alloys strongly enough.
Let us consider the influence of
temperature on the structure of liquid metals.
The quantitative characteristics of the theory under
development concerning the interaction of the elements of matter and space as
applied to the processes of melting and crystallization of metals are closely
interconnected, so if we know the parameters of one of the components, we may
find the corresponding values of other constituent parts. The existence of such
interaction facilitates the accomplishment of the task set in this work.
In particular, Chapter 3 supplies us with the values
of the main structural parameters of the elements of matter as well as the
elements of space in liquid metals at the unique temperature.
At the same time, literature gives the general form
of temperature dependencies of certain quantities used in our theory.
Particularly, /87/ and a series of other sources
quote an expression for the dependency of concentration n of the elements of
space in solid metals – vacancies – on temperature. Viz.:
n = exp (ΔSf / k) exp
( -Ef / kT), (50)
where Δ Sf is the entropy of vacancy forming,
while Ef is the energy of their formation.
On the other hand, the mentioned
source adduces the following expression of statistic thermodynamics for the
dependence of the equilibrium number of activated particles of matter on
temperature:
Ca = B exp (Δ Sf / k) exp (- Ef
/ kT) (51)
where B is the constant depending on the way of distribution of
particles in space.
Let us emphasize that expressions
(50) and (51) are practically identical.
It corroborates once more our thesis that was
advanced above of the equivalence of the elements of matter and space.
Let us find the values of the quantities Ef
and Δ Sf making use of the fact that expression (51) refers to
the same concentration of activated atoms as expression (43) derived above.
At the temperature equal to the melting temperature
expressions (51) and (43) must be equivalent, i.e.
B exp (Δ Sf / k) exp (- Ef / kT) = Δ
Hmelting nc / N0
It was demonstrated above that Δ Hmelting
is required to form intercluster splits which, in their turn, initiate the
formation of activated atoms on the ‘surface’ of such splits. Therefore, having
divided the latent heat of melting by the energy of forming new atoms, we can
determine their molar concentration Ca
Ca = Δ Hmelting / Efz
(52)
Expression (52) supplies us with the absolute value
of the concentration of activated atoms per mole of substance. The relative
quantity of Ca may be obtained from (52) by way of division by
Avogadro number. Thus
Ca =
Δ Hmelting / Efz N0
(53)
On the other hand, we determined the same relative quantity of Ca
earlier by expression (47) as
Ca = γ1 Δ Hmelting /
Δ Hvap,
(54)
where γ1 = (3/2) β, or γ1
= 2 / (3/π)1/3; 3 / (3/π)1/3; 5 / (3/π)1/3;
6 / (3/π)1/3 for a cubic diamond, simple cubic, body-centered cubic and
face-centered cubic types of granulation correspondingly.
Let us equate the right parts of expressions (53) and
(54). We get
Δ Нпл / Ef z
N0 = γ1 Δ Hmelting / Δ Hvap
Hence Ea = Δ Hmelting / γ1 z
N0.
At present, inserting the values of T = Tmelting,
Ca from (54) and Ef from (55) into (51), we find:
γ1 Δ Hmelting / Δ Hvap =
B exp (Δ Sf / k) exp -( Δ Hvap / γ1
z k N0 Tmelting).
Two quantities are unknown here: B and Δ
Sf. Let us recognize B = 1, since we know the distribution
of particles that is reflected in coefficients z and γ1.
In this case, the value exp(ΔSf k) at the melting
temperature will be equal to:
exp (Δ Sf / k) = (γ1 Δ Hmelting
/ Δ Hvap) exp (Δ Hvap / γ1
z R Tmelting), (56)
where R = k N0 – the universal gas constant.
Using (56), we arrive at the final
expression for calculating the dependence of the concentration of activated
atoms in liquid metals on temperature:
Ca = (γ1 Δ Hmelting /
Δ Hvap) exp (Δ Hvap / γ1
z R Tmelting) exp -( Δ Hvap / γ1
z R T) (57)
At T = Tmelting expression (57) transforms
automatically into expression (54).
The dependencies of Ca = f(T) and rc
= f(T) for certain metals under (57) are shown in Fig.12.
As it follows from Fig.12, the concentration
of activated atoms in liquid metals rises rapidly with an increase in
overheating, reaching 100% in the vicinity of the vaporization point of the
given metal.
However, even 100% of activated atoms in liquid do not mean that
there are no clusters in such liquid. Activated atoms are far from being
isolated monatoms independent of one another. The totality of activated atoms
enters into cluster structure. The approximation of activated atoms
concentration in liquid to 100% implies that cluster dimensions in liquid with
an increase in temperature decrease so that the totality of atoms entering into
a cluster emerge on its surface in the vicinity of the vaporization point.
Cluster dimensions in liquid metals and alloys are
modified with an increase in temperature, too. Let us determine the nature of
such modification.
Using the relation of quantities of rc
and Ca from (46), we find
rc = 3 Ca-1 a /
2. (58)
Inserting the value of С from (57) here, we find the dependence of cluster dimensions on
temperature
rc = (2/3ag1) (Δ Hvap / Δ Hmelting)
exp-( Δ Hvap / g1zRTmelting) exp(Δ Hvap / g1zRT) (59)
We can find the number of atoms in
clusters in f(T) allowing for the definite interrelation between the radius of
clusters and the number of atoms inside them:
nc = (4p/3z) rc3 .
By way of inserting here the value of rc
from (58), we get
nc = (4p/3z) (3/2)3 (Ca )-3.
Let us designate d = 9/2z and
introduce the value of Са.
For nc = f (T) we get:
nc = dp [g1 Δ Hmelting /
Δ Hvap) exp(Δ Hvap / g1zRTmelting) exp-( Δ Hvap /g1zRT)]-3 (60)
The subset of formula (60) is represented as
nc = dp (Ca )-3
(61)
The dependence of rc = f (T) is
also represented by Fig.12. The findings show that the rise of temperature
brings about the reduction of cluster dimensions in liquid metals. This
corresponds to the entire current data on the increase of disorder in liquid
metals with temperature rise /2,4,12/.
However, the derived expressions (59) and (60)
predict the existence of clusters in liquid metals up to the temperature of
evaporation. Such a conclusion is at variance with the inferences made in
certain works which state that cluster structure is inherent in liquids near
the melting temperature only, while the monatomic structure with the statistic
distribution of particles takes place at high temperatures /44-46/.
Our theory never employs the concept of ideal,
homogeneous phases. It was maintained above that each of the aggregation states
necessarily includes, except for the basic (predominant) intrinsical paired
elements of matter and space, the equilibrium latent characteristics of the
elements of matter and space that pertain to the states of aggregation adjacent
to the given state by the temperature scale. This inference equally concerns
liquid as well as solid, gaseous and other aggregation states.
Clusters are the equilibrium form of the elements of
matter that is peculiar to liquid state and determines the material aspect of
the characteristics of the present state within the whole temperature range of
its existence.
Some works conclude about the discontinuous nature of
the dimensional modification of the structural units of matter in liquid state
on the basis of measuring the dependency of a series of structure-sensitive
properties of liquid metals on temperature.
The continuous nature of the received dependencies nc
= f (T) and rc = f (T) impels the author to subscribe to
the opinion advanced in /106/, where such fractures of the
characteristic-temperature curves are explained neither by sudden changes in
cluster dimensions nor by the transition from the cluster to monatomic
structure but by the polymorphous transitions in clusters. Since there exists
the neighboring order of atomic granulation inside clusters, its modifications
are quite possible as a result of the interaction of the elements of matter and
space inherent in the specified neighboring order. In their turn, such
modifications may cause the change of cluster dimensions but not their
disappearance.
The reduction of cluster
dimensions with temperature presupposes the increase of their number in a unit
of volume in liquid at the same time. Since clusters are particle aggregations
with an intensive mutual interaction, particle interchange including, such
modification turns out to be quite feasible. As a result of such interplay and
mass transfer, clusters are capable of rapid reorganization; moreover, they get
reorganized constantly.
The motive force of the reduction of cluster
dimensions with an increase in temperature as well as the process of increasing
cluster dimensions at the cooling of melts is the mentioned vacancy gas
pressure. The concentration of vacancies inside clusters increases with the
rise of temperature, which causes their reorganization into clusters with
lesser dimensions. By definition, a cluster may contain not more than one
vacancy. If two or more vacancies arise in a cluster, they generate inner
pressure inside it that leads to its splitting by the mechanism analogous to
the mechanism of melting described above.
If we know the nc, it is easy to
determine the number of clusters per gram-atom of the given metal in liquid
state. Thus
Nc = N0 / nc = N0 Ca-3
/ πδ
(62)
In accordance with (62), the
number of clusters increases rapidly with the rise of temperature.
The reduction of dimensions and the increase in the
number of clusters in liquid metals with the rise of temperature must result in
the expansion of the volume occupied by the zone of intercluster splits (the
elements of space) ΔVspl. The relation between ΔVspl
and Ca can be expressed under (39):
ΔVspl = aα (3/2 rс) 100% = aα Ca 100%.
Inserting here the value of Ca from
(57), we have
Δ Vspl = aα (γ1 Δ Hmelting
/ Δ Hvap) exp (Δ Hvap / g1zRTmelting) exp- (Δ Hvap / g1zRT) 100% (63)
We should note that (63) cannot be
considered as the only contributor to the changing of the volume of liquid at
heating. Similar to any thermodynamic characteristic of a system that is
measured experimentally, the modification of volume is a complex quantity
formed out of the total contribution of both the elements of matter and the
elements of space at all the hierarchical levels of matter and space
interaction that exist in the given system. Still, the contribution of the
upper level of the system always prevails.
Except the quantity of ΔVspl,
at least four more factors must contribute to thermal expansion in the
specified concrete case: 1) the thermal expansion of the residuals of
crystalline lattice inside clusters analogous to the thermal expansion of
solids; 2) the possibility of re-granulation of clusters after their formation
into a compact mutual granulation irrespective of atomic granulation inside
clusters; 3) the possibility of volume modification at polymorphous transitions
in liquid state; 4) the increase in vacancy concentration.
The current evaluations of unconfined space in
liquids do not allow for the contribution of each of the five indicated factors
/2/, therefore, the collation of the obtained quantity with experiment is not
possible so far.
Liquid metals, similar to any physical bodies, are
systems of interacting elements of matter and space. Such interaction directly
affects various thermodynamic and other characteristics of the given system in
the first place. Viz. any property of such systems that is experimentally
determinable will be complex, reflecting the contribution of material as well
as spatial elements at various levels of the system’s hierarchy.
The specific quantity of such
contribution depends on the characteristic in question. There may be properties
determined mainly by the contribution of the material component of the system,
e.g. the mass of liquid and solid metals. There can be properties dependent in
preference on the contribution of the spatial elements of the system, such as
the fluidity of liquid metals, and there are characteristics that depend
equally upon the contribution of both the elements of matter and the elements
of space (density). However, all the characteristics reflect, though in a
different degree, the influence of both the material and spatial elements of
the system under analysis.
We shall determine the totality of liquid metals
characteristics proceeding from the present general conception by specifying
every time the contribution of material and spatial elements into this or that
concrete characteristic at the hierarchical level that corresponds to the level
of aggregation states and the elements of matter and space inherent in this
very state.
Let us mark that we cannot specify the absolute
quantity of the contribution of matter and space to this or that specific
property of a system, yet it is in our power to determine the relative property
modification under the influence of the contribution of this or that specific
element of the system in question.
For instance, we cannot calculate the entire volume
of a system in solid and liquid states being able to do the calculation of the
relative modification of the volume of metals at melting and crystallization.
The same refers to other properties.
The original principle of relativity ensues from the general
theses of the hierarchy of real bodies structure, the presence of a great many
levels of the interacting elements of matter and space inside them.
We cannot yet determine the summarized contribution
of each of such levels, the majority of which are underexplored. Still, we can
evaluate the relative modification of this or that characteristic of a system
while the latter is passing from one aggregation state into another, for
example, at the transition from solid to liquis state and v.v.
At times such modification will be insignificant or
negligible, - occasionally it will be conclusive. Everything depends on the
nature of the property.
Let us start considering the properties of liquid
metals with the property that depends decisively upon the contribution of the
spatial part of a system. This is fluidity.
Let us view the elementary act of fluidity in liquid
metals at the level of clusters and intercluster splits.
Let us symbolically represent two
adjacent clusters as squares A and B in Fig.6. Let us assume that displacement
force F influences cluster A in the direction from left to right. At the moment
1 clusters A and B, being in the state of performing continuous heat
oscillations, approximate so that the flickering split between them is closed
and there occurs no displacement of cluster A towards cluster B, the analyzed
zone of liquid does not flow in such a configuration but behave as a solid.
At moment 2 as a result of the same heat oscillations
clusters A and B separate so a flickering split forms between them for a short
period of time. During the specified time period, clusters A and B are not
connected, and cluster A, under the impact of force F, is easily displaced relative to cluster B by the quantity of δ
termed as the elementary step of the process of flowing.
At moment 3 clusters A and B come together again, and the flickering
split between them closes. However, cluster A is already displaced relative to
cluster B by the quantity of δ. The process under consideration will be
repeated as long as there is the impact of force F without any counteraction.
Totalizing, the elementary acts of flowing lead to
the visual effect of the flowing of liquid metals. Let us remark that the
quantity of displacement δ equals to or is divisible by the width
of a single intercluster split, δ = αa (v. Part 3.5 above).
The very possibility of displacement is caused by the
presence of spare spacings in liquid represented by intercluster splits. In
other words, intercluster splits provide the space for cluster displacement,
increasing the fluidity of liquids by several orders as compared to solid
state.
It follows from the cited
description of the elementary act of fluidity that the process of flowing of
liquid metals and alloys at cluster level is not exactly continuous but it puts
up from minute steps δ following one another.
If force F acts short-term, liquid may respond
to such a short-period impact as a solid body. The mentioned phenomenon exists
and is widely acknowledged, while the short period of time when liquid behaves
as a solid under the influence of force F is termed the relaxation
period.
Let us do the calculation of the
elementary act of fluidity. The speed v of the displacement of cluster A toward
cluster B is:
v = δ / τ
(64)
where τ is the duration of the
elementary displacement act, equal to the period of heat oscillations of a
cluster. The given quantity was determined previously in 3.9:
τ = 2παa (3 kT N0 / M)-1/2
In its turn, the speed of v may be found through the coefficient
of fluidity Te. Thus, for the specified case
v = Te
F (65)
By equating the right sides of (64) and (65), we get:
Te F = δ / τ
(66)
In turn, the force of F can be calculated as
the pressure upon liquid p, multiplied by the area of transverse section of
cluster A, which we shall designate as rc2. Hence
F = p rc2.
(67)
Introducing (67) into (66), we get
Te = δ / τ p rc2.
(68)
Expression (68) correlates
fluidity with such parameters as the width of intercluster split δ,
the radius and frequency of heat oscillations of clusters.
It also follows from (68) that the
fluidity of liquid metals must increase with an increase in temperature, for
cluster dimensions deflate with temperature rise.
Viscosity is traditionally referred to the group of
the basic structure-sensitive properties of liquid metals being used as the
characteristic of internal friction in liquid.
There are numerous theories of
viscosity: the unconfined space theory that leads to Bachinsky’s formula /107/;
Arrenius’ equation derived theoretically by Frenkel and Andrade /69-70/ with
its numerous modifications; the equation suggested by the statistic theory of
liquid plus its modifications /109/. The presence of a large number of theories
concerning the same phenomenon is, on the one hand, a typical scientific
situation, since there always exist multitudinous possibilities to give a
many-sided description to the same phenomenon. Such theories may complement one
another.
On the other hand, the presence of various theories
that are mutually exclusive testifies to the situation of incomplete knowledge.
The latter is the very situation with regard to the viscosity theory. Similar
to diffusion, viscosity description is rather unsatisfactory on the whole,
although we observe some acceptable coincidences between experimental and
calculation data in a series of cases. Such a situation requires a further
theoretic development in order to construct an adequate viscosity theory.
Let us build up the theory of viscosity of liquid
metals allowing for their cluster-vacuum structure.
On the one hand, such a theory can be constructed if
we premise the known correlation between fluidity and viscosity.
η = 1/ Te = τ p rc2 / δ
However, expression (68) and the
latter one contain the variable quantity of p to be rid of, which we regard as
a drawback. In this connection, there arises a necessity to develop a more
convenient theory of liquid metals viscosity taking into account the existence
of both the elements of matter and space inside them.
Such a theory can be grounded on Andrade’s kinetic
equation, derived on the basis of the concept of the monatomic impulse
transmission mechanism /110/, yet neutral in reality with respect to the
dimension of structural units of matter in liquid state.
Andrade supposed that impulse transmission occurs at
the deviation of the structural units of liquid from their layer resulting from
oscillations. Evidently, the term ‘structural unit’ can be equally substituted
here for the concept of ‘atom’ as well as ‘cluster’. In the case given, the
differentiation is quantitative, not qualitative.
Andrade explored two adjacent layers of structural
units of liquid, parallel to the direction of the flowing of liquid. If n is
the number of such particles in 1ccm, then there falls ≈n2/3
of the structural units of matter, clusters in our interpretation, at 1sq.cm.
Let 1/3 of their total number oscillate
perpendicularly to the layer plane. If impulse transmission takes place at the
maximal deviation from the layer plane, the quantity of the transmissed impulse
at a single particle oscillation will make »m n-1/3 (dw/ dy), where m is the mass of a particle (a cluster), while n-1/3
is the average spacing between the layers; y is the coordinate perpendicular to
the layer plane; dw/ dy being the gradient of the tangential speed of flow. The
number of such impulse transmissions per 1sec. reaches »(1/3) j n2/3, where j is the frequency of heat oscillations of a cluster.
Hence, the resultant impulse
equivalent to the force of viscosity and transmitted during 1sec. through a
unit of layer surface area, will be
dP/dt » (4/3) j m n1/3 = h dw/ dy.
Multiplier 4 stipulates here that
a particle transverses the layer plane four times during the period of its heat
oscillations.
Therefrom it follows that
h » (4/3) j mc n1/3,
(69)
where nis the number of clusters in a unit of liquid metal volume.
Let us find the value of j from (49): j = (1/2paа) (3kT N0 / nc M)1/2.
Cluster mass is known, too: mc = М nc / N0 .
Let us determine the number of
clusters in a unit of liquid metal volume by the expression:
n = N0 r / M nc,
(70)
where r is the density of liquid metal.
Inserting the obtained values of j, mc and n into
(69), we get:
h = (2 / 3paа) (3kT N0 / nc M) 1/2 (N0
r / M nc) 1/3 (М
nc / N0) (71)
Expression (71) is correct for Т = Тmelting. To find the dependency of h = f(T), the value of n = f(T)
should be inserted into (71):
nc = dp [g1ΔHmelting /ΔHvap)
exp(ΔHvap /g1zRTmelting) exp-(ΔHvap /g1zRT)]-3
or
h = B exp -(ΔHvap /g1zRT)-1/2 T1/2,
(72)
where B = (2 / 3paа) (3R / nc
M) 1/2 (N0 r / M nc)
1/3 (М nc / N0) (g1ΔHmelting /ΔHvap)exp(ΔHvap /g1zRTmelting) - being constant.
The analysis of the obtained
dependency (72) shows that the expression under consideration is similar to the
well-known Panchenkov formula only /108/ presented as
h =3 (6R) 1/2
(b2/ N) (r4/3 / M5/6) exp (e / RT) T1/2
[1 - exp - (e / RT)] (73)
Here, as well as in (72), we observe the term of Т1/2, while the quantity of e = 2Еvap / z is determined through the energy of vaporization and the
coordinating number of z, which is close to our findings.
Panchenkov’s theory, however, is
based on other assumptions, which accounts for an insufficient degree of its
similarity to the obtained data.
Numerical check (72) demonstrates
a coincidence between calculation and experimental data, which is close enough,
if we consider the proximity of the original Andrade’s expression. The data are
adduced in Table 9.
Table 9. Viscosity of Liquid Metals at the Melting
Temperature
Metal |
h, cps, |
|
calculation by (72) |
exper. by /12/ |
|
Fe |
4.2 |
5.4 |
Co |
5.5 |
4.8 |
Ni |
5.5 |
5.0 |
Cu |
5.0 |
4.1 |
Au |
5.2 |
5.38 |
Al |
1.48 |
1.13 |
Zn |
4.5 |
2.82 |
Cd |
3.9 |
2.3 |
Na |
0.9 |
0.68 |
Temperature dependencies of the viscosity of liquid
metals are shown in Fig.14, 15. The character of the calculation and
experimental dependencies in Fig.14, 15 coincides, their numerical correlation
is quite satisfactory.
Thus, the developed theory of the structure of liquid
metals is quite applicable to the analysis of their viscosity, too.
The traditionally studied properties and processes, such as
diffusion and viscosity in liquid metals, are also regarded as a traditional
object of applying liquid state theories and models with the purpose of adequacy
check of the mentioned theories.
Unfortunately, theoretical skill
created the situation when we have a whole variety of diffusion as well as
melting theories. This deprives the process of the working out of new diffusion
theories of experimentum crucis meaning, of the seemingly essential importance
which similar developments used to have in the past while the number of
diffusion theories was not so great.
Nevertheless, there remains the essential though not
exactly underlying significance of such pursuits. It consists in the fact that
although the building up of the theory of diffusion or any other similar
property of liquid does not play the decisive part in this or that theory of
liquid state, it is one of the necessary steps to check the applicability of
the theory-to-be to the description of a wide range of liquid metals phenomena
and properties - as wide as possible.
In point of fact, under the conditions of the
competition between various theories of one and the same phenomenon, the theory
giving the most exact description to the widest range of phenomena in its
respective field will take the priority.
Besides this, focusing on diffusion is explained by
the practical importance of the specified process for metallurgy and casting.
A large amount of experimental data accumulated on
diffusion makes it possible to test this or that theory on the material that
seems sufficiently extensive /12,17,20/.
The main theoretic expression in the sphere of
diffusion in liquid metals remains the equation, analogous to Arrenius’
equation for viscosity /12/:
D = D0 exp -(ED /
kT), (74)
where D is the coefficient of diffusion; ED
is the energy of diffusion activation (self-diffusion); D0
being the fore-exponential multiplier.
Expression (74) does not always
describe the observed regularities of diffusion satisfactorily, especially
within a wide temperature interval /152/, therefore, attempts at constructing a
diffusion theory on variant bases were and are still being made.
By way of examples, we may cite Zaxton and Sherby’s
empiric correlations /111/, self-diffusion calculations under the hole theory
by Eiring /103/ and Frenkel /69-70/, Andrade’s calculations /110/, Cohen and
Turnball inactivation theory based on the unconfined space model /112/,
Swalin’s fluctuation theory /113/ and a series of modifications of the
mentioned theories.
However, there was no junction between theory and
experiment to be detected in the given works as regards a relatively wide scope
of metals. It is supposed that the coefficients of diffusion, as well as
viscosity, can be calculated on the basis of the consecutive statistic theory
of liquid state /18,109/, under the condition of the exact knowledge of
interatomic potentials /109/, which is lacking so far /14/. The overwhelming
majority of the stated theories employ the ideas of the in-liquid migration of
a separate atom or ion understood as the basic structural unit of liquid.
Thus, the description of diffusion phenomena from the
viewpoint of the set-forth melting and liquid state theory where an atomic
grouping – cluster – is considered to be the main structural unit of matter in
liquid state, merits attention, being of principal interest. Our theory also
premises that each aggregation state, except for the structural units of matter
and space predominant in the state given, bears the latent properties of the
adjacent aggregation states.
It was pointed out above that all the atoms of liquid
enter into clusters, while the atoms that happen to be located on cluster
‘surface’ at the given moment form an aggregate of activated atoms, capable of
migration and acting as the latent elements of gaseous state matter in liquid
state. Apart from that, there are atoms and vacancies inside clusters, which are
bound with one another in the crystal-like structure of neighboring order.
These are the latent elements of solid state in liquid.
Cluster mechanism must be the major mechanism of mass
transfer in liquid metals, for it proves to be the most effective one. However,
according to the principles of synergetics, any dissipative process (diffusion
refers to typically dissipative processes) always occurs at all possible
levels. Therefore, with the exception of the main cluster mechanism,
intercluster diffusion responsible for mass transfer inside clusters will
operate in liquid metals through the mechanism similar to the vacancy mechanism
in solids, and it will be accompanied by the interchange of activated atoms
between clusters by way of separate atoms jumping over the zones of
intercluster splits.
Hence, there are at least three diffusion mechanisms
operating simultaneously in liquid metals: the basic mechanism of mass transfer
through clusters, characteristic of liquid state, and those of latent
aggregation states – solid-like vacancy mechanism inside clusters and gas-like
atomic interchange between clusters.
So our theory presents the process of diffusion as a
composite, aggregative one, whereas the value of diffusion coefficient measured
experimentally becomes the effective, resultant mass transfer coefficient by
the three mechanisms displayed above.
Such an approach meets the accepted conception of the
presence of latent properties belonging to other states of aggregation of
matter in the aggregation state given. In the case under analysis, clusters
should be viewed as the structural elements of the elements of matter form that
dominates in liquid state, while activated atoms are considered as the latent
properties of gaseous aggregation state.
The specified conclusion concerns any other
aggregation states in full measure.
Consequently, the current partial diffusion theory
incorporates into a completer theory that should differentiate between the
contributions of each of the mechanisms into the observed diffusion process.
To create the integrate diffusion theory of such a
complex system is the goal heritable into the future.
We may note here that the first approximation at the
calculation of diffusion coefficient allows neglecting mass transfer inside
clusters, since the contribution of this mechanism into the diffusion
coefficient value under observation seems insignificant. The contribution of
the gas-like mechanism does not appear to be manifest by its quantity. Hence,
we are going to consider two mechanisms of diffusion in liquid state further –
cluster and gas-like, aiming at finding the respective contributions of both of
them /115/.
In conformity with the above-said, we can add
D0 = Dc0 + Da0
Ca, (75)
where D is self-diffusion coefficient; Dc0
is self-diffusion coefficient by the cluster mechanism; Da0
is self-diffusion coefficient by the mechanism of activated atoms; Ca
is the concentration of activated atoms in liquid.
As we see it in (75), the summarized
self-diffusion coefficient is combined of partial coefficients extensively as
contrasted with additive composition.
Let us find the quantities of partial coefficients of
cluster and activated atoms diffusion that enter into (75) on the basis of the
random walk theory. With reference to the given case we have:
Dc0 = k d2
ν (76)
where k = 1/6; d is the space of a single particle
displacement at its transition from the original to some other equilibrium
position; ν is the frequency of such transitions.
Since intercluster spacings are
small in comparison with their dimensions, we may conjecture but a collective
mechanism of their displacements, e.g. the circular mechanism. At such a
mechanism, the adjacent equilibrium positions will be separated by the space
equal to the doubled cluster radius plus the width of one intercluster split
α. Thus,
l = 2 rc + a.
(77)
As usual, let us recognize as a
single diffusion act the displacement when a cluster passes from the original
equilibrium state to the adjacent equilibrium state. Evidently, in this case
d = l = 2 rc + a.
Or, since a << 2 rc at Т = Тmelting, we may admit without any noticeable error that
d = 2 rc
. (78)
To determine the period of
diffusion jumps, let us use the concept of diffusion as an oscillatory process
that is introduced here for the first time. Such an assumption is unacceptable
when analyzing the direction of particles travelling in space. However, if we
consider the process of diffusion in time, abstract from the displacement
direction and allowing for the periodicity of the specified process only, it is
quiet acceptable to regard this process as periodic, i.e. oscillatory. The
frequency of such a process, which is periodic in time, can be found on the
basis of the oscillations energy E equation /116/:
Eс = ( mс A2 w2 )/2, (79)
where mc is the mass of a cluster; Ec
is the energy of cluster oscillations; w is the angular frequency of oscillations; А
is amplitude.
mc = М nc / N0.
Ec = (3/2) kT.
w = 2p n.
А
= d = 2 rc.
n= w /2p
Having accomplished the corresponding substitutions,
out of (79) we derive
w = (2Еc /mc) 1/2
/ d; (80)
and
n = (2Еc /mc) 1/2
/ 2pd (81)
In turn, expressing nc through Сa under the expression (61) derived
earlier, we get
nc = dp (Ca )-3.
Introducing the obtained values of d, Ec
and mc into (81), for t
n = (3 k T N0
/ M dp Ca-3
)1/2/ 4 prc
Let us allow for the previously derived expression for
rc:
rc = (3/2) а Ca-1.
After the introduction of the
concluding value we obtain:
n = (3 R T / M dp Ca-3
) 1/2/6 p a Ca-1
Or
n = Ca5/2
(3 R T / M dp )
1/2/6 p a (82)
The latter expression determines
the frequency of cluster transitions from one to another equilibrium state, or
the frequency of the elementary acts of diffusion process.
By the insertion of the obtained value n from (82) into (76), we
arrive at the final expression for the partial self-diffusion coefficient in
liquid metals by the cluster mechanism:
Dc0 = k d2 n = (a / 4p) Ca1/2
(3 R T / M dp) 1/2
(83)
The found values of Dc0
by equation (83) at Т = Тmelting are listed in Table 8.
This table includes the principal
data on self-diffusion coefficient in liquid metals. The amount of the
published experimental data on self-diffusion is small, their reliability being
unfortunately entirely unknown.
The partial self-diffusion coefficient of activated
atoms can be also evaluated on the basis of equation (76) with the introduction
of the frequency of heat oscillations of clusters from (49) and at d = a.
Under these conditions
Dа0 =(a/12pa) (3RT /dp M) 1/2 Ca3/2
(84)
Calculations under formula (84)
supply the values of the partial self-diffusion coefficient by the mechanism of
activated atoms that are approx. by order of magnitude less than the values of
the partial self-diffusion coefficient by the cluster mechanism. That signifies
that mass transfer in liquid metals at the melting temperature is achieved for
the most part through the displacement of clusters, but not separate atoms,
from the equilibrium positions. The contribution of gas-like diffusion in
liquid metals approximates to 10% of the total value of diffusion coefficient.
It suffices not to neglect the mentioned fact; furthermore, the value of this
diffusion mechanism and its contribution to liquid metals will increase with
the rise of temperature.
The resultant values of the
effective self-diffusion coefficient are derived from (75) with the
introduction of the partial coefficient values from (83) and (84).
Thus
D = (a Ca / 4p) (3RT /dp M) 1/2[(
Ca2 /3pa ) + 1] (85)
The data on
the calculation of self-diffusion coefficient according to equation (85) are to
be found in Table 10, too.
Table 10. Self-Diffusion Coefficients in Liquid
Metals at the Melting Temperature
Metal |
Dc0×105
sq.cm/s, calculation by (83) |
D0×105sq.cm/s, calculation by (85) |
D0×105 sq.cm/s, exp. /2,12,98/ |
Li |
1.9 |
2.1 |
5.6 |
Na |
1.4 |
1.6 |
4.3 |
K |
1.3 |
1.44 |
5.3 |
Cu |
1.4 |
1.5 |
- |
Ag |
0.66 |
0.70 |
2.3 |
Au |
0.75 |
0.80 |
- |
Al |
1.8 |
1.95 |
- |
Pb |
0.47 |
0.50 |
2.0 |
Zn |
1.22 |
1.40 |
1.9 |
Cd |
1.0 |
1.28 |
- |
Fe |
1.04 |
1.1 |
0.17 |
Ni |
1.6 |
1.7 |
- |
Co |
1.55 |
1.65 |
- |
Ti |
2.6 |
2.2 |
- |
W |
1.5 |
1.6 |
- |
Sn |
0.57 |
0.70 |
2.5¸2.0 |
Hg |
0.52 |
0.62 |
1.0 |
Bi |
0.73 |
0.85 |
- |
Ga |
0.47 |
0.65 |
1.7 |
Usually the influence of external
factors that accelerate or decelerate diffusion cannot be eliminated completely
in experiments. Gravity and convection refer to such factors in the first
place. Therefore, the values of experimental self-diffusion coefficients are
higher than the calculated ones, which is to be expected. Self-diffusion data
are rather insufficient for a more accurate evaluation. So far we may state
that the developed diffusion theory registers the acceptable calculated data on
self-diffusion coefficients for a sufficiently wide range of liquid metals.
The process of mass transfer, or
pore diffusion, occurs at any aggregation state. Qualitatively, this is the
same process. However, the rapidity of mass transfer in various aggregation
states differs considerably by quantitative parameters. So, what causes such
distinctions?
To facilitate solving, let us view the distinctions
related to the differentiation between the principal structural units of
various aggregation states.
From the suggested standpoint, such distinctions are
associated with the change of the predominant structural units of matter and
space, the elements of space coming first, in various aggregation states. This
is the presence of unconfined space that operates as the factor determining the
possibility and the rapidity of such a displacement.
Current theory reflects the specified fact through the
known diffusion formulas in gases and liquids.
For gases
D = k u l, (86)
where D is diffusion coefficient; u is the average
rapidity of thermal motion of gas molecules; l is the average length of
a free range of molecule path; k being constant - k = 1/3.
For solids and liquids /101,115/:
D = k d2 n,
(87)
where d is the average space between material particles in
liquid; n is the frequency of heat oscillations of the elements of matter in
liquid; k is the coefficient dependent on particle granulation relative
to one another. Normally k = 1/3 - 1/6.
It was
repeatedly observed that expressions (86) and (87) quite reducible mutually,
since their dimensions and physical implications are similar. The foremost
point is that the quantity of d in expression (87) characterizes the length of
the elementary displacement of material elements during the process of
diffusion prior to their dimensions in liquid state. Consequently, even the
existent incomplete diffusion theory neglects the dimensions of the structural
units of matter in liquid, although taking into consideration the dimensions of
spatial elements as the spacing of the elementary act of diffusion.
I.e.
d = l.
Let us present (86) as
D = k u l = k l (l / t), (88)
where u = l / t.
If we do the substitution (1 / t) = n, (88) assumes the following form:
D = k l2 n, (89)
where n = 1/ t is the frequency of the material element displacement from the
equilibrium state; l being the spacing of such a displacement in space.
Certainly, expression (89)
presupposes the possibility of representing or describing the process of
diffusion as a process oscillatory in time, for (89) employs the idea of the
process frequency n.
This is a new concept in the theory of diffusion.
Still, it was demonstrated while deriving (79) that such an assumption is quite
acceptable when referring to the frequency of displacements in time as
abstracted from displacement direction.
Expression (89) quite coincides with expression (87)
in all its details, allowing for the fact that the constant factor of k can be
different to a certain degree.
It is important to mark that the dimensions of
material particles are absolutely lacking in (86-89) while spacings are present
only, i.e. the parameters of space in the given aggregation state. It
underlines once more the decisive role of the elements of free space in
diffusion process.
Let us presume that expression (89) has a more general
character than expressions (86) and (87) being applicable to any state of
matter.
Hence, if we know the differences in the spacing of
the elementary act of diffusion that are connected with the differentiation
between the predominant elements of matter and space in this or that
aggregation state, we can determine the difference in the coefficients of
diffusion for the adjacent aggregation states (without considering the latent
elements of matter and space and their contribution to diffusion).
Let (89) be
Dl0 = kl ll2 nl
for liquid state, whereas for solid state
Ds0 = ks ls2 ns.
Here ll and ls are
the spacings of the elementary act of the prevalent diffusion mechanism for
liquid and solid states correspondingly; nl and ns represent the frequency of the
elementary acts of diffusion for liquid and solid states correspondingly; kl
and ks are the constants of the granulation of the
predominant elements of matter in liquid and solid states correspondingly.
Then the ratio of diffusion
coefficients for solid and liquid aggregation states will be presented as
Dl0 / Ds0 = kl
ll2 nl / ks ls2
ns
up to a constant.
It is known that ls = а, where а is the shortest interatomic
spacing in a crystal.
We recognized ll as dc
above for liquid state, where dc is the diameter of a
cluster.
It follows from Table 1 that dc » 10 а.
Correspondingly
Dl0 / Ds0 » (10а) 2nl / a2 ns
or
Dl0 / Ds0 » 102 nl / ns
(90)
It is known that diffusion coefficient in liquid metals
at the melting temperature equals approx. to 10-5 sq.cm/s, while in
solid monocrystals at the sate temperature it is 10-7 sq.cm/s /129/.
I.e. the actual correlation at the melting temperature
Dl0 / Ds0 » 100.
That means that the correlation
between the frequencies of diffusion jumps in equation (90) is nl / ns = 1.
Consequently,
the frequencies of diffusion jumps in solid and liquid states at the melting
temperature are approximately equal.
In turn, we know /101/ that the frequency of jumps in
solid metals at the melting temperature equals 108 c-1 at
the frequency of heat oscillations of atoms 1013 c –1.
The frequency of heat oscillations of clusters in
liquid metals at the melting temperature was calculated above (look up Table 8)
and mounts to 108 c-1, which approximates the frequency
of diffusion jumps of atoms in solid metals near the melting temperature.
The curious fact under analysis signifies that a
cluster completes no more than several (less than ten) heat oscillations
between diffusion jumps in liquid state. It is not excluded that clusters in
liquid metals change their equilibrium state at each period of oscillations.
The mentioned peculiarity makes liquid metals cognate
to gases, where the direction of atomic motion changes at a collision with neighbors.
Probably, such affinity is one of the causes of a continuous transition
possibility between liquid and gaseous states.
Apart from this, the fact of the equality between the
frequency of jumps and the frequency of heat oscillations in clusters means
that diffusion process in liquid metals, by contrast with solid metals, has an
inactivated nature, because a cluster does not require any additional energy to
shift its position, except for the energy of heat oscillations. The same
phenomenon takes place in gases, whereas activation energy that exceeds by far
the energy of heat atomic oscillations is needed for the elementary act of
diffusion in solid state.
Thus, there is a fundamental distinction between the
mechanisms of diffusion in solid and liquid metals, while we observe a
fundamental similarity of the diffusion mechanisms in liquid and solid states
by the activation parameters.
In the light of the analysis carried out above, the
difference in the processes of diffusion in liquids and gases becomes mainly
quantitative, so at the equality between the quantities ll = lg
and nl = ng the parameters of mass transfer (and not that only) in solid and
liquid states equalize and grow indistinguishable, which occurs within the
vicinities of the melting temperature.
So the use of the concepts of the
role of spatial elements in the processes of mass transfer gives us the
possibility to calculate the coefficients of diffusion in liquid metals, first,
as well as consider diffusion processes from the unified viewpoint in various
aggregation states, and establish the features of similarity and difference
between the given processes, including the quantitative aspect.
Real liquid metals and alloys always contain a certain
amount of admixtures. It is known that admixture diffusion in liquid metals is
qualitatively subordinate to the same regularities as self-diffusion, yet the
quantitative distinctions in diffusion coefficients of diverse admixtures may
vary considerably enough.
The theory of admixture diffusion,
which is of practical importance for calculating the processes of alloying,
segregation, etc., does not always ensure the sufficient convergence with
experimental data at present.
Thus, one of the latest among the recently developed
fluctuation mechanisms of admixture diffusion in liquid metals /113/ provides
acceptable results for admixture diffusion in liquid alkaline metals but it
turns out to be rather inexact for calculating admixture diffusion in liquid
iron and other refractory metals /117/.
Another admixture diffusion theory – Swalin’s theory –
considers liquid as homogeneous, in connection with which individual properties
of the admixture and the solvent are almost disregarded so the coefficients of
various admixtures diffusion come to be similarized /113/.
In reality, in liquid iron, for instance, the values
of D ranging from 1.7 10-6 sq.cm /s (self-diffusion) to 1 10-3sq.cm
/s (hydrogen diffusion) are observed. The quasipolycrystalline model of liquid
melts was engaged to account for such appreciable discrepancies /31/. As it was
demonstrated, such a model admits that liquid consists of separate clusters
(atomic microgroups) and the surrounding disordered zone, monatomic by its
structure. For such a structure, V.I.Arkharov suggested the following
correlation /31/:
ψcl + ψdis =
1, (91)
where ψcl is the relative cluster contents in
the structure of liquid metal; ψdis being the relative
contents of the disordered (monatomic) zone.
I.e. a
hypothesis of a complex, aggregative nature of liquid metals structure was put
forward within the limits of the quasipolycrystalline model. Therefrom the
authors of the hypothesis in question concluded that the quantity of the
measurable admixture diffusion in liquid metals is complex, composite,
additive, determined by the sum total of the diffusion conduction of various
structural zones in the melt /31/:
D = ψcl Dcl + ψdis Ddis,
(92)
where Dcl and Ddis are partial
admixture diffusion coefficients in various zones.
It was pointed out above that the
existence of two structurally independent zones in liquid state contradicts the
phase rule as well as the quantum mechanics rule of quantum objects
indistinguishability.
In the theory of the material-spatial structure of
liquid metals under our development clusters, and nothing else but clusters,
are thought the prevalent material elements characteristic of liquid state.
Still, the existence of clusters by no means denies the existence of atoms –
these are different levels of matter organization.
It was proved in Part 5.4 above that there are three
mechanisms of mass transfer operating simultaneously in liquid metals: the
basic one being that of mass transfer through clusters; solid-like vacancy mass
transfer through atoms inside clusters; gas-like mass transfer between clusters
by way of interchanging activated atoms located on cluster ‘surface’ with the
participation of intercluster splits (not vacancies). The distinction between
the mentioned process and the solid-like one consists in the following: when
inside a cluster or a solid, migrating atoms preserve their bonds with the
adjacent atoms, whereas the former jump over the intercluster split during the
atomic interchange between clusters. Atoms involved in this process separate
from all their neighbors for a very short time, which is typical of gaseous
aggregation state. Therefore, we term such a process as gas-like.
The gas-like mechanism of mass transfer in liquid
metals acts inside clusters only, being responsible for the redistribution of
atoms at the intercluster level, which is highly important for the homogenizing
of liquid alloys composition, as well as the progress of alloying processes,
but hardly perceptible at measuring diffusion coefficient. So we may neglect
the contribution of the solid-like mechanism to the coefficient of diffusion in
liquid metals as a first approximation. However, the contribution of the other
two diffusion mechanisms must be taken into consideration, though they differ
in the degree of their significance.
Thus, we must allow for two diffusion mechanisms
operating simultaneously in liquid metals: the mechanism of mass transfer
through separate atoms peculiar to gaseous state, the role of which is but
tributary, and the cluster mechanism of diffusion, characteristic of liquid
state proper, to which the principal role in the given process is assigned.
So V.I.Arkharov’s idea of the composite nature of the
observed diffusion coefficient can be applied here by the extensive, contrasted
with the additive, scheme /115/.
D = Dcl + Da Ca,
(93)
where Da is the coefficient of diffusion by the
activated atoms mechanism; Dcl is diffusion coefficient by
the basic cluster mechanism; Ca is the concentration of
activated atoms in liquid metal.
The extensive character (93) is realized through the
fact that diffusion coefficient by the cluster mechanism incorporates into
diffusion as the main one, while the coefficient of Ca acts
as supplementary, proportionate to the concentration of activated atoms. The
given correlation reflects the entire amount of matter enter into clusters in
liquid state, whereas only few atoms inside clusters are activated.
Formula (93) allows for the existence of two
discriminate mechanisms of diffusion in liquid state irrespective of varied
admixture distribution in clusters and among activated atoms.
Let us label admixture diffusion in various structural
zones of liquid as solubility. Different admixture solubility within the
specified structural zones of liquid metals and alloys is explained by the
dissimilarity of the structure of metals.
As it was noted above, the inside cluster structure is
close to that of solid crystals. So there are sufficient reasons for
recognizing the solubility of admixtures in clusters Scl as
equal to the solubility of the same admixture in solid metals Ssol
in the vicinity of the melting temperature, i.e.
Scl = Ssol
(94)
On the other hand, the zone of activated atoms that
adjoins intercluster splits has a structure that is less ordered and less
stabile in time. It was accentuated above that activated atoms on the
‘surface’ of clusters are the latent elements of gaseous state in liquid state.
Disorder, mobility and a considerable volume of unconfined space in this zone
abate dimensional and other constraints for soluted particles.
Therefore, we may expect that the
solubility of the majority of admixtures in the given zone will be higher than
in the solid-like structure of clusters.
In this connection, the observed jump in the
solubility of the majority of admixtures at melting should be associated with
the forming of intercluster splits and the gas-like zone of activated atoms
during the mentioned process.
Thus, admixture solubility within the zone of
activated atoms Saa can be found as the difference in general
admixture solubility in liquid metals Sl and its solubility
in solid metals near the temperature of melting. So
Saa = Sl – Ssol
(95)
or
Saa = Sl – Scl
(96)
Correspondingly
Sl = Scl + Saa
(97)
It seems rather difficult to allow for the
discriminate solubility of admixtures in (93), because the quantities of Dcl
and Da in (93) can differ from partial self-diffusion
coefficients of Dc0 and Da0
in (75) due to the presence of admixtures. So let us apply here another way of
calculating D by omitting the partial coefficients of Dcl and
Da.
Let us employ the well-known
Stokes-Einstein equation. In case of admixture diffusion we have /114/:
D = kT / 6πη
r. (98)
In case of self-diffusion
D0 = kT / 6πη r0,
(99)
where η is liquid metal viscosity; r is the effective
radius of admixture diffusion; r0 being the effective radius
of self-diffusion.
Under the condition that T is equal to η,
dividing (98) by (99) will lead us to
D = D0 r0 /
r (100)
Since the quantity of D0 can be
obtained through (85), there remain the effective values of r0
and r to be found. Let us determine the effective self-diffusion radius r0
as the extensive sum of the radii of material particles that constitute liquid,
i.e. clusters rcl and atoms rme:
r0 = rcl + rме Cа,
(101)
where Cа is the
concentration of activated atoms in liquid metal.
To determine the effective radius
of admixture diffusion r we must take into consideration that the
admixture entering into the composition of clusters (cluster-soluble) moves
together with them, whereas the admixture soluble within the zone of activated
atoms migrates as separate atoms. Thus
r = rcl (Scl / 0,5) + rа Cа (Sаа/0,5), (102)
where rа is the
radius of an admixture atom.
The coefficient of 0.5 here
stipulates that the solubility of any admixture, by formal reasons, cannot
exceed 50% (at the complete mutual solubility), for if the solution content of
admixture B is more than 50%, then, the solution of B in A
transforms into the solution of A in B.
The validity of (102) can be checked by its
application to self-diffusion. Hence Scl = Sаа = 0,5, while rа = rме, so (101) lets arrive
at (102).
Inserting (101) and (102) into
(100), we get:
D = D0 (rcl + rме Cа) / [rcl (Scl
/ 0,5) + rа Cа (Sаа /
0,5)] (103)
All the values of the quantities
constituting (103) are known, in which connection the given formula is quite
applicable to calculations.
Considering the particular practical importance of
iron-based alloys, admixture diffusion in liquid iron and its alloys is the
best explored, which provides ample experimental data for comparison and
checking. Therefore, let us use expression (103) derived previously to
calculate admixture diffusion in liquid iron and compare the obtained
information with experimental data.
Such calculations require the knowledge of the
coefficient of self-diffusion in liquid iron. There are only two values of the
self-diffusion coefficient of liquid iron. These are presented by the only
experimental value of D0Fe = 1.7 10-6
sq.cm/s /138/ and the value of D0Fe=1.1 10-5
sq.cm/s that we obtained earlier.
Because of such an essential discrepancy let us do the
calculations with the use of either of the values of D0Fe
with the purpose of comparison.
We can single out three groups of admixtures depending
on their solubility in liquid iron.
To the first group of admixtures refer the admixtures
with the unrestricted solubility in solid as well as liquid iron.
To the second group of admixtures we refer the
elements with the restricted solubility in solid iron and unrestricted
solubility in liquid iron.
The elements that have the restricted solubility in
solid as well as liquid iron are included into the third group of admixtures.
In this case, the following correlations of the
quantities required for calculations are appropriate:
Scl + Sаа = 1;
Scl = Sаа.
(104)
Hence for the specified group of admixtures
Scl = Sаа =
0.5.
The procedure of calculating the diffusion of such
admixtures in liquid iron may be illustrated by the example of cobalt. Let us
insert the known quantities of Scl, Sаа, rcl, Cа into the
basic expression (103). We shall use the experimental value of D0Fe
in our calculation. Thus, at Т= 1600 0С:
DCo = 1.7 10-6 (2.48 0.387 + 18)/ [1.38 0.387
(0.5/0.5) + 18 (0.5/0.5)] = 1.74 10-6 sq.cm/s.
Using the calculation value of D0Fe
we get DCo = 1.12 10-5 sq.cm/s, which is much
closer to the main experimental data.
Here and further the data on
admixture solubility in liquid iron are cited on the basis of works /119/ and
/120/.
The principal peculiarity of the values of admixture
diffusion coefficients belonging to this group consists in the fact that their
quantities are close being practically equal to the self-diffusion coefficient
of liquid iron. It corroborates that the admixtures of the first group enter
into the melt cluster composition and their prevalent mass migrates within the
melt by the cluster mechanism in general.
The elements that are soluble to this or that extent
both in the clusters of the melt and within the zone of activated atoms form
the second group of admixtures. We can determine the degree and correlation of
the two of these solubility kinds, knowing the solubility of the given
admixture in solid state and recognizing the complete solubility in liquid
state as 1.
The main body of current experimental data /1,17/
indicates that clusters in liquid iron in the vicinity of the melting
temperature have the structure of neighboring order that is similar to the
structure of solid g - iron. In this connection, the quantity of Scl
for liquid iron can be equalized to the maximum solubility of the given
admixture in liquid g - iron (Sg).
For the second group of admixtures in liquid iron
Scl = Sg;
Sаа = 1 - Sg,
i.e. the increase in admixture solubility at the transition of iron
into liquid state takes place wholly due to solubility within the zone of
activated atoms. Thus, in calculations under (103) for this group, the more Saa=1
- Sg value is, the weightier mass transfer by the gas-like mechanism
through atomic interchange between clusters becomes. Since the stated value
fluctuates within relatively wide limits for the elements of this group – from
0.7 for ruthenium up to 1.0 for alkaline and alkali-earth metals and hydrogen –
the values of diffusion coefficients for the given group of admixtures are
remarkable for a wide diversity of values from 1*10-6 for ruthenium
to 1*10-4 sq.cm/s for cerium (v. Table 11 below).
The third group of admixtures includes
the elements with the restricted and low solubility in liquid as well as solid
iron. Hydrogen is a typical representative of this group as regards iron. Since
hydrogen solubility in liquid iron is less than 1, expression (104) for
hydrogen and the entire Group Three can be presented more conveniently as
(Scl / Sl) + (Sаа / Sl) = 1,
where Sl is the maximum effective admixture
solubility in liquid metal at the given temperature.
It is known that hydrogen
dissolves in solid and liquid iron in the quantity of thousands of one percent,
while its solubility in solid iron is approx. three times as little as it is in
liquid state. According to autoradiography data /101/, hydrogen is distributed
non-uniformly in solid iron, accumulating mainly within the areas of disordered
structure (dislocation cores, the surfaces of domain and mosaic blocs sections
inside granules, crystalline borders). There is a low probability of the
presence of such defects inside clusters because of small cluster dimensions.
Consequently, we may assume accurately enough that hydrogen dissolves mainly within
the zone of activated atoms both in solid and liquid iron. So the solubility of
hydrogen in clusters of iron tents to zero:
SclH ®
0. (105)
Hence in correspondence with (104)
for hydrogen and the whole third group of admixtures in liquid iron:
Sаа =
1. (106)
Introducing the values of SclН from (105) here and Sаа from (106) into (102), we obtain that the effective diffusion
radius of admixtures belonging to the third group equals
r = rа Cа / 0.5.
It corresponds to a solubility
that is quite close to monatomic solubility and the monatomic type of diffusion
of these elements, hydrogen in the first place, by the gas-like mechanism
within the zone of activated atoms only. Such a conclusion captures our
attention generating a further inference that the behavior of hydrogen in
liquid iron is similar to its behavior in gas. Hydrogen does not interact with
clusters and migrates through intercluster splits only. Data of Table 11 shown
grafically on the Fig.16
Thus, the properties of latent
states in the prevalent state, the latent properties of gaseous state in liquid
iron represented by the zone of activated atoms, in our case, may suffice for
certain admixtures to concentrate within the zone of such latent elements only.
In this case, the behavior of
admixtures of such a kind becomes the same as it is in gaseous state, but only
within the elements of the latent state area. For hydrogen, liquid iron
constitutes gas with a peculiar configuration of the ramified flickering system
of slits that permeate the entire volume of liquid iron. Otherwise speaking,
for hydrogen liquid iron constitutes a capillary-porous body with flickering
capillaries.
The possibilities of the migration
of the atoms of hydrogen and other elements in such a capillary-porous body are
determined in general by the dimensions of the atoms of the diffusing elements.
The quantities of diffusion coefficients are maximal
for the elements of this group in liquid iron reaching 10-3 sq.cm in
case of hydrogen (v. Table 11).
The totality of the experimental
and calculation data of the values of D of admixtures in iron obtained on the
basis of (103) as compared with the known experimental data is adduced in Table
11. A full line in Fig.15 indicates the distribution of elements by the
quantity of their diffusion coefficient in liquid iron (calculated data).
Experimental data are shown in Fig.15 by separate dots.
Table 11. Calculated and Experimental
Quantities of Diffusion Coefficients of Some Elements in Liquid Iron at 16000C
№ of admixture group |
Metal |
Scl, % at. /119-120/ |
Atomic radius, /66/ |
D, sq.cm /s calculation by exp. D0Fe |
D, sq.cm /s calculation by D0Fe in Table 8 |
D, sq.cm /s experiment |
Source |
1 |
Mn |
50 |
1.428 |
1.74 10-6 |
1.12 10-5 |
- |
- |
1 |
Ni |
50 |
1.385 |
1.74 10-6 |
1.12 10-5 |
- |
- |
1 |
Co |
50 |
1.377 |
1.74 10-6 |
1.12 10-5 |
3.4 10-6 |
/17/ |
1 |
Rh |
50 |
1.487 |
1.74 10-6 |
1.12 10-5 |
- |
- |
1 |
Pd |
50 |
1.500 |
1.74 10-6 |
1.12 10-5 |
- |
- |
1 |
Ir |
50 |
1.534 |
1.74 10-6 |
1.12 10-5 |
- |
- |
2 |
Ru |
29.5 |
1.480 |
2.60 10-6 |
1.82 10-5 |
- |
- |
2 |
Re |
16.7 |
1.520 |
4.70 10-6 |
2.60 10-5 |
- |
- |
2 |
Cr |
12.0 |
1.423 |
6.7010-5 |
4.00 10-5 |
- |
- |
2 |
N |
10.3 |
0.547 |
0.78 10-5 |
4.6 10-5 |
3.78 10-5 |
/70/ |
2 |
- |
- |
- |
- |
- |
5.50 10-5 |
/121/ |
2 |
C |
8.6 |
1.107 |
1.02 10-5 |
5.35 10-5 |
3.28 10-5 |
/70/ |
2 |
Cu |
7.5 |
1.413 |
1.15 10-5 |
6.20 10-5 |
- |
- |
2 |
Zn |
7.0 |
1.538 |
1.10 10-5 |
6.25 10-5 |
- |
- |
2 |
Si |
4.2 |
1.670 |
2.03 10-5 |
7.20 10-5 |
3.10 10-5 |
/122/ |
2 |
- |
- |
- |
- |
- |
1.23 10-5 |
/122/ |
2 |
- |
- |
- |
- |
- |
3.0 10-5 |
/122/ |
2 |
Ge |
4.0 |
1.755 |
2.0 10-5 |
7.20 10-5 |
- |
- |
2 |
Pu |
2.0 |
1.770 |
3.50 10-5 |
9.20 10-5 |
- |
- |
2 |
Mo |
1.6 |
1.550 |
4.30 10-5 |
9.60 10-5 |
- |
- |
2 |
V |
1.6 |
1.491 |
4.40 10-5 |
1.0 10-4 |
- |
- |
2 |
Nb |
1.2-1.9 |
1.625 |
4.00 10-5 |
1.0 10-4 |
- |
- |
2 |
Al |
1.55 |
1.582 |
4.00 10-5 |
1.24 10-4 |
5.0 10-4 |
/122/ |
2 |
- |
- |
- |
- |
- |
5.0 10-4 |
/123-124/ |
2 |
Gd |
2.00 |
1.992 |
6.0 10-5 |
1.0 10-4 |
- |
- |
2 |
Sn |
1.00 |
1.862 |
6.10 10-5 |
1.10 10-4 |
- |
- |
2 |
W |
1.00 |
1.549 |
6.40 10-5 |
1.30 10-4 |
- |
- |
2 |
Ta |
0.95 |
1.626 |
6.50 10-5 |
1.40 10-4 |
- |
- |
2 |
Ti |
0.72 |
1.614 |
8.10 10-5 |
1.40 10-4 |
5.95 10-5 |
/125/ |
2 |
- |
- |
- |
- |
- |
1.38 10-4 |
/126/ |
2 |
Zr |
0.50 |
1.771 |
8.97 10-4 |
1.50 10-4 |
1.18 10-4 |
/125/ |
2 |
O |
0.56 |
0.603 |
1.25 10-4 |
1.50 10-4 |
1.22 10-4 |
/125/ |
2 |
P |
0.25 |
1.582 |
1.40 10-4 |
1.3 10-4 |
- |
- |
2 |
La |
0.20 |
3.335 |
0.85 10-4 |
0.79 10-4 |
0.95 10-4 |
/121/ |
2 |
S |
0.11 |
1.826 |
1.70 10-4 |
1.44 10-4 |
4.94 10-4 |
/122/ |
3 |
Ce |
0.04 |
2.070 |
1.80 10-4 |
1.50 10-4 |
4.4 10-4 |
/123/ |
3 |
Na |
0.0 |
2.110 |
1.80 10-4 |
1.28 10-4 |
- |
- |
3 |
Mg |
0.0 |
2.853 |
1.40 10-4 |
1.20 10-4 |
- |
- |
3 |
Ca |
0.0 |
2.180 |
1.70 10-4 |
1.25 10-4 |
- |
- |
3 |
H |
0.0 |
0.370 |
1.1 10-3 |
7.2 10-3 |
1.32 10-3 |
/31/ |
3 |
- |
- |
- |
- |
- |
3.51 10-3 |
/117/ |
The table shows that the optimal
coordination between calculated and experimental diffusion coefficients of
various admixtures in liquid iron is achieved in case of using the calculated
value of self-diffusion coefficient. In any case, this is the first time when
theory provides a satisfactory congruence with experiment for such a wide range
of data. Diffusion coefficients in liquid iron for a series of elements are
calculated for the first time having never been determined experimentally. It
creates ample possibilities of testing theory through experiment.
As it follows from the above-said,
among the peculiarities of the suggested procedure of calculating diffusion
coefficients we should mention the fact that the coefficient of self-diffusion
of iron (and any other alloy-forming element) is the least possible, whereas
the coefficients of diffusion of any admixtures can equal or exceed the
coefficient of self-diffusion. Such an admission ensues from the premise that
cluster radius in expression (103) is considered constant. In reality, cluster
radius can vary in both the directions with an increase in admixture content.
This may introduce certain corrections into the process of diffusion as well as
its calculation, negligible in the majority of cases at low admixture content
in the alloy specified.
The main factor that determines the quantity of admixture diffusion
coefficient in the given theory is the distribution of admixture between the
structural zones of liquid iron, which, in turn, accounts for the dominance of
this or that diffusion mechanism in liquid iron or the combination of such
mechanisms.
One of the essential structural characteristics of solid and liquid
states of metals is the compactness of their atomic granulation. The latter is
quantitatively evaluated by the coordinating number k that determines
the number of atoms located in the neighborhood equidistantly from one another
or some central atom.
In case of metallic bonds that are not saturated and directional,
atoms in a crystal can be presented with a certain approximation as mutually
attracting incompressible spheres with the radius of R.
The coefficient of compactness q that characterizes the density of
structure granulation is equal to the correlation between the volume of the
particles forming a crystal and the crystalline volume /127/.
From the viewpoint accepted in our work, the coefficient of
compactness characterizes the volume occupied by matter at the intracrystalline
level, where matter is represented by atoms while interatomic spacings inside
the crystalline lattice represent space.
The level of the intracrystalline structure of matter-space systems
differs from the level of aggregation states. This is a finer dimensional
level. However, as it was shown above, any state of any matter-space systems
incorporates latently the ulterior properties of other states and other levels
of such systems. In the meantime, the latent levels, as it was demonstrated by
the example of diffusion, may impart a considerable or even the decisive
contribution to this or that property of the system in a number of cases.
Therefore, it is of theoretic and practical importance to explore
the levels of the matter-space systems structure that are adjacent to the level
of aggregation states, e.g. the level of the intracrystalline structure of
metals.
In case of spherical atomic granulation, the coefficient of compactness
is /127/:
q =
4pnR3 / 3Va,
(107)
where n is the
number of particles in an elementary cell; Va is the volume
of an elementary cell.
For the closest packings, compactness coefficient equals 74%, i.e.
interior elements of space that are peculiar to the given state occupy more
than a quarter of the entire volume even in the most compact crystalline
lattices.
Table 12 lists the coordinating numbers (k) and compactness
coefficient (q) for the main types of crystalline structures.
Table 12. Coordinating Numbers k and Compactness
Coefficient q for Various Structures
Lattice
type |
k |
q,
% |
Face-centered
cubic and hexagonal compact |
12 |
74 |
Tetragonal
body-centered (с = 0.817; n = 2) |
10 |
69.8 |
Body-centered
cubic |
8 |
68.1 |
Simple cubic |
6 |
52 |
Diamond cubic |
4 |
34 |
Tellurium
lattice |
2 |
23 |
Table 12 illustrates that the volume occupied by the elements of
space in crystalline lattices of different types is always appreciable, its
content fluctuating from 26 up to 77% of the total crystalline volume.
Correspondingly, the properties of crystals depend on the contribution of the
elements of space in a highly noticeable way.
The theory of the influence of inner spatial elements upon solid
physical bodies, liquid metals including, is yet to be originated.
Let us note that even such a weighty concept as the atomic radius
can be defined as a half of the shortest interatomic spacing in the crystalline
lattice within the limits of the crystalline lattice only. Such a definition is
inexact, since it also comprises the interatomic space. Still, there exists no
other way of defining the atomic radius, for the radius of an atom taken
separately cannot be determined with accurateness because of the fuzziness of
the electron cloud, i.e. due to the interaction between matter and space, too.
The above-said corroborates once again that matter does not exist
independently of space, that any element of matter has its corresponding
equivalent among the elements of space, so any system can only be described as
a matter-space system. We may isolate neither the elements of matter nor the
elements of space from such a system, neither physically nor theoretically,
since that changes the properties of the analyzed elements.
Returning to the coordinating numbers concept, let us remark in
conformity with the above-said that the real composite structure of any systems
is to be taken into consideration while determining such a number. The concept
of the average coordinating number makes sense only in the case when we allow
for both the spatial and material components and their interaction.
The coordinating number in liquid, as well as in solid, metals is
generally determined by the method of x-rays dispersion or other procedures
/12/.
However, measurement accuracy for solid state still exceeds that for
liquid state, which can be exemplified by the diversity of coordinating numbers
data obtained by various researchers (v. Table 11). The whole array of data
concerning the quantities of k accumulated by present is averaged, i.e. these
data reflect the continued attempts at describing liquids as a homogeneous
monatomic medium.
The simultaneous existence of both the elements of matter and space
in real liquid metals, as well as the existence of the derivative structural
zones generated by the interaction of the mentioned elements - for instance,
the zone of activated atoms - presupposes the existence of different local
coordinating numbers within them.
In particular, the existence of mobile clusters presupposes the
existence of their mutual granulation and the coordinating number of such a
granulation kc = 12, for this granulation will always tend to
the compact one because of the mobility of clusters toward one another,
irrespective of the type of atomic granulation inside clusters.
At the same time, there exists the granulation of atoms inside
clusters with the coordinating number of kd, which may be
equalized (under the condition of the absence of polymorphous transitions in
liquid state) to the coordinating number of the corresponding solid metal at T
= Tmelting. Finally, we should single out the coordinating
number for activated atoms ka. Activated atoms, by
definition, have at least one free bond. In the meanwhile, activated atoms
participate both in the atomic granulation inside clusters and in the mutual
cluster granulation, since activated atoms are located on cluster 'surface'.
In this connection, the coordinating number of activated atoms
equals
ka
= (kc + kd / 2) - 1. (108)
The average (effective) coordinating number k in liquid depends on
the correlation in liquid of the quantities of atoms that participate in this
or that granulation type within this or that structural zone of liquid, at
different dimensional levels inclusive. Such a number is combined of the
coordinating numbers of this or that structural zone additively as a first
approximation, multiplied by the relative number of atoms entering into each
zone. There arise some problems related to the circumstance that the same atoms
can enter different structural zones at different system-dimensional levels.
For example, activated atoms constitute clusters at the same time.
Taking it into consideration, we can derive the expression for
calculating the coordinating number in liquid metals:
k =
[(kc + kd)/ 2 - 1] Ca + kd(1 - Ca).
(109)
The multiplier (1 - Ca) is introduced here in
order not to double our consideration of activated atoms.
There are two variable quantities forming (109): Ca
and kd, which enables us to obtain simplified formulas for
calculating the coordinating numbers of liquid metals with dissimilar original
atomic granulation in solid crystals. Introducing the corresponding values of
the quantities Ca and kd into (109), we
derive
k =
12 - Ca ; k = 10; k = 6 + 2 Ca; k = 4 + 3 Ca, (110)
for
face-centered cubic, body-centered cubic, simple cubic and diamond cubic
granulations.
In accordance with (110), the coordinating number of metals with
close packing in solid state decreases at the melting and heating of liquid;
for b.c.c. metals the coordinating number does not change at melting. Such
constancy is caused by the influence of two factors: the first is that the
effective number of b.c.c. metals equals 10 for solid state, the second being
the mutual compensation of the zonal coordinating numbers of liquid ka
and kd in such metals with the rise of T. It should be
pointed out that kп for such
metals makes 10 only under the stipulations concerning the participation of
atoms of the second coordinating sphere, since the radii of the first and the
second coordinating spheres are very close /1,2,12,127/, so a slight dependency
of the coordinating number on temperature for such metals is yet to be
expected. A slight dependency k of liquid metals with the original
b.c.c. granulation on T is experimentally proved /17/.
The quantities of k for s.c. and c.d. granulations, according to
(110), increase both at melting and at the heating of liquid. The given
quantity rises in the most noticeable way for the c.d. granulation type. Such
is the consequence of the role of the partial factor of the mutual
cluster-in-liquid granulation and the partial coordinating number of kc.
This is the first time we introduce the factor of cluster-in-liquid
re-granulation into a compact mutual granulation. The factor under
consideration also contributes much to the change of volume, density, electric
conductivity and a series of other properties of some metals and non-metals at
melting, which will be shown below.
The calculated values of the coordinating numbers of a series of
liquid metals derived on the basis of (110) are cited in Table 13 as compared
with the present experimental data.
Table 13. The Average Coordinating Numbers in Solid
and Liquid Metals at the Melting Temperature
Element |
Coordinating
number in solid state /17,127/ |
Coordinating
number in liquid state, calculation by (110) |
Coordinating
number in liquid state, experiment /2,17,127/ |
Li |
10 |
10 |
9.5 |
Na |
10 |
10 |
9.5;
9.0; 10.0 |
K |
10 |
10 |
9.0;
10.0 |
Ag |
12 |
11.8 |
10.0 |
Au |
12 |
11.7 |
8.5;
11.5 |
Cu |
12 |
11.8 |
10.9 |
Al |
12 |
11.8 |
10.6;
11.4 |
Pb |
12 |
11.8 |
11.7;
12.1 |
Zn |
12 |
11.6 |
10.8;
11.0 |
Ni |
12 |
11.6 |
12.0 |
Co |
12 |
11.6 |
12.0 |
Mn |
12 |
11.6 |
- |
Feg |
12 |
11.7 |
10.0;
12.0 |
Sn |
10 |
10 |
10 |
Bi |
6 |
6.4 |
7.0;
7.6; 8.0 |
Ge |
4 |
5.0 |
6.0;
8.0 |
The coincidence between calculated and experimental values of liquid
metals coordinating numbers according to the data listed in Table 13 is quite
satisfactory, especially if we observe the considerable diversity of the
existent experimental data. The latter is to be expected, because various
experimental methods can be sensitive to the three dissimilar partial
coordinating numbers in liquid metals – ka, kc and
kd - to a different extent.
Expressions (110) prognosticate a smooth change of the dependencies k
= f(T). Sudden changes of these dependencies and other structure-sensitive
characteristics in liquid metals must be related to the polymorphous transitions
in atomic granulation inside clusters, i.e. the change of the quantity of kd
/17,44-46,128/.
The changes in electrical resistivity at the melting
of metals are very noticeable being of a most diverse character for various
metals. For typical metals with close packing like copper, silver, gold,
titanium, zinc and some other metals electrical resistivity increases more than
twice at melting. Still, the more friable the crystalline lattice in solid
state is, the less electrical resistivity increases at melting, and a decrease
in electrical resistivity at melting is observed in metals with the loosest
lattices like stibium, bismuth, gallium.
The modification of electrical
resistivity of a series of elements cannot be confined to the suggested simple
scheme. For example, for liquid semi-conductors – silicon and germanium – the
rise of electrical resistivity at melting is so considerable that we may
indicate the change of conductivity type in liquid state in comparison with
solid state, particularly, the transition to the metallic conductivity type in
these elements after melting.
In connection with such diversity, the theory that
describes the modifications of electrical resistivity at the melting of at
least the main groups of metals in a more or less satisfactory way is to be
originated.
A.Ubbelode
/1,2/ rightly states that the modification of the electric properties of metals
at melting may be caused by different reasons. Among such causes are quoted the
distant order disappearance and the rise of positional disorder, as well as the
increase of atomic heat oscillations amplitude at melting, which leads to the
increase in the dispersion of the conductivity electrons within atomic oscillations.
A possible change of Fermi energy level and other possible modifications at
corpuscular and electron levels, including the change of conductivity type in
liquid semi-conductors, are supposed.
From the positions of our theory,
all these factors are probable; however, the specificities of liquid
aggregation state proper are lacking among them. All the above-mentioned
factors refer to the corpuscular or even electron level but not to that of
aggregation states.
Notwithstanding the fundamental
importance of these factors, we should note that the level of the description
of this or that phenomenon must be adequate to the phenomenon described. If we
consider the influence of the change of aggregation state upon a certain
phenomenon, we ought to describe such influence at the level of the structural
elements of matter and space, inherent in the given aggregation state.
As it was emphasized above, the
real structure of real systems is quite complex, being distinguished by the
presence of many hierarchical levels embedded one into another, and also the
latent properties of other possible states.
Hence ensues a relativity
principle for the description of the properties of real bodies:
all the measurable properties are
determined by the contribution of both the elements of matter and the elements
of space;
each of the possible levels of the
elements of matter and space can contribute to this or that specified property;
at every change of the system’s
state, its aggregation state including, the elements of matter and space
intrinsic in the given state make a decisive contribution to the change of this
or that property;
it is improbable to find the
summarized absolute value of each property by calculation, we can only
determine the relative contribution of the elements of matter and space at this
or that modification of the system’s aggregation state.
I.e. it is possible to calculate
the value of the relative change of the property only - at melting or
crystallization, for instance. We cannot calculate exactly the absolute value
of liquid or solid metal volume, yet we are able to do the calculation of the
metal volume change at melting and crystallization by the modifications of the
predominant elements of matter and space.
Such an approach has never been
used before.
In particular, the factors of the
influence of the elements of space formed at melting and peculiar to liquid
state proper - i.e. the influence of intercluster splits - upon electrical
resistivity have never been taken into account.
In the meantime, it was demonstrated
above that intercluster splits surround half of the ‘surface’ of any given
cluster at melting. Such intercluster splits have vacuum properties by
definition, i.e. they are impenetrable for the conductivity electrons. If we
isolate the underlined factor that is inherent in such an aspect in liquid
state proper, it (the factor) can reduce the effective conducting section of
any liquid sample doubly sharp, providing the double increase in electric
conductivity at melting. Let us label this factor as fs.
Thus, fs = 2.
Apart from that, the factor of the
increase of the elements of space volume at melting can also be referred to the
factors peculiar to liquid aggregation state proper (see 3.5). Such a factor
reduces the effective conducting section of liquid metals, too, directly
proportional to the volume of the elements of space inside them. Let us
designate this factor as the quantity of fv.
fv = ΔVp,
where ΔVp is
found on the basis of (39) and Table 4.
The two factors in question are
responsible for the rise of electric conductivity at melting by the forming of
intercluster splits - the elements of space characteristic of liquid state -
during melting process.
However, there may occur self-compacting processes in liquid at
melting in comparison with solids. It becomes possible due to cluster mobility.
We touched upon the causes of cluster mobility in the parts dedicated to the
mechanisms of diffusion and self-diffusion in liquids metals above.
The first consequence of such a
cluster re-granulation is a certain compacting of liquid in cases when the
original atomic granulation in solid state and inside clusters is loose with
the coordinating number less than 12. Such compacting equals zero in metals
with the original close packing, then, it is quite negligible in metals with
the b.c.c. granulation of atoms, reaching considerable quantities for metals
with the simple cubic granulation and for the elements with even looser
granulations like the diamond type.
Let us recognize the factor of
compacting due to the re-granulation of clusters as fp.
In accordance with /129/, the
factor of compacting equals
fp = ΔVc = kp Ca,
where kp = 0; 0.06; 0.217; 0.4
- for f.c.c., b.c.c., s.c. and c.d. granulation types correspondingly (see
above). This factor reduces the electrical resistivity of liquids.
The second consequence of cluster
mobility formulates as the change of the number of conducting contacts between
clusters in liquid because of cluster re-granulation into mutual close packing.
Like balls moving inside a box, clusters moving inside liquid always pack into
mutual close packing.
If a certain metal has the close
packing of atoms in the crystalline lattice in solid state, the re-granulation
factor is of no importance for it - no changes of volume, density and the
effective conducting section are observed in this case, since the number of
neighbors as well as the number of conducting contacts is equal here in both
solid and liquid states.
However, if the original atomic
granulation in solid state (and inside clusters) differs from the compact one,
the cluster re-granulation factor will inevitably cause volume reduction in
liquid accompanied by the boost of its density, which was pointed out above,
plus electric conductivity reduction due to the increase of the effective
conducting section of liquid.
Let us designate the influence of
the cluster re-granulation factor on electric conductivity as the quantity of fc.
This quantity can be evaluated on the basis of the correlation between the
coordinating numbers in solid state and the coordinating number of the compact
mutual granulation of clusters in liquid, which makes 12, allowing for the
factor of fs that doubly reduces the number of neighbors for
any given cluster to contact with (electrically) at any given moment. The fs
factor is numerically equivalent to the ratio of the coordinating numbers in
metals in solid state to 12. Or
fs = ks / 12
Let us mark that the fs
factor is not additive with the first three factors by nature but it imparts an
extensive contribution. In connection with the above-said, we obtain the
cumulative influence of the stated four factors presented as
rL / rS = (fs + fv
+ fp) fs (111)
Let us underline once more that
expression (111) by no means pretends to fully describe the electric
conductivity of liquid metals and its mechanism. It considers nothing but the
influence of the structural elements of matter and space upon the electric
conductivity of metals as well as the reorganizations caused by them that occur
at melting.
Calculations under expression
(111) compared with the known experimental data are tabulated below.
Table 14. The Change of Electric Conductivity of
Metals at Melting
Element |
fs |
fv |
fp |
fp |
rL / rS, calculation by (111) |
rL / rS, experiment /1,98/ |
Сu |
2 |
0.045 |
0 |
1 |
2.045 |
2.04 |
Ag |
2 |
0.047 |
0 |
1 |
2.047 |
2.09 |
Au |
2 |
0.05 |
0 |
1 |
2.05 |
2.08 |
Al |
2 |
0.053 |
0 |
1 |
2.053 |
2.20 |
Zn |
2 |
0.055 |
0 |
1 |
2.055 |
2.24 |
Cd |
2 |
0.056 |
0 |
1 |
2.056 |
1.97 |
Ti |
2 |
0.06 |
0 |
1 |
2.06 |
2.06 |
Co |
2 |
0.051 |
0 |
1 |
2.051 |
1.3 |
Ni |
2 |
0.045 |
0 |
1 |
2.045 |
1.05 |
Fe |
2 |
0.05 |
-0.005 |
0.66 |
1.33 |
1.09 |
Li |
2 |
0.043 |
-0.019 |
0.83 |
1.68 |
1.64 |
Na |
2 |
0.038 |
-0.03 |
0.83 |
1.66 |
1.451 |
K |
2 |
0.04 |
-0.015 |
0.83 |
1.67 |
1.56 |
Rb |
2 |
0.041 |
-0.015 |
0.83 |
1.68 |
1.60 |
Cs |
2 |
0.056 |
-0.017 |
0.83 |
1.69 |
2.6 |
Mg |
2 |
0.048 |
-0.01 |
0.83 |
1.69 |
1.78 |
Ba |
2 |
0.05 |
-0.01 |
0.83 |
1.69 |
1.62 |
Ga |
2 |
0.01 |
-0.03 |
0.3 |
0.59 |
0.45-1.46 |
Bi |
2 |
0.037 |
-0.08 |
0.5 |
0.98 |
0.35-0.47 |
Sb |
2 |
0.06 |
-0.12 |
0.5 |
0.97 |
0.61 |
Si |
2 |
0.076 |
-0.08 |
0.3 |
0.57 |
0.034 |
The data listed in Table 14 show
an excellent coincidence between the calculated and experimental values of the
change in metal electric conductivity at melting for typical metals with close
packing, as well as for alkaline and alkali-earth metals with the b.c.c. type
of granulation in solid state. Calculation and experimental values of ρL
/ ρS for such metals coincide with the accuracy of
experiment error.
The roughest coincidence was
observed in case of 3-d transitive metals: iron, nickel, cobalt.
The maximum divergence (one order)
was obtained in case of semi-conductors.
Probably, the maximum divergence
is to be expected in such cases because of the changes of their electron
structure, especially significant for liquid semi-conductors, where, as it was
demonstrated, the transition to the metallic bond type takes place. These are
the changes occurring on the levels distinct from that of aggregation states
although initiated by the aggregation transitions, which stresses once again
the interrelation between various levels of the systems of material and spatial
elements.
In any case, the calculations
carried out above show that electric conductivity is a property that is highly
sensitive to the presence of the spatial elements in liquid. The influence of
such elements of space as intercluster splits turns out to be decisive for a
series of metals to change their electric conductivity at melting. Anyway, this
influence is noticeable enough to be taken into consideration.
Allowing for the influence of the
elements of space upon the electric conductivity of metals and alloys may
acquire some practical relevance for the future control of this fundamental
property.
A great number of widely known facts indicate that
ideal aggregation states, i.e. states taken purely, do not exist. There are no
ideal gases, no ideal crystals, no ideal liquids. We can only approach what we
understand under the ideal state with more or less approximation.
We may say that the idealization of aggregation states
exists only as a way of interpreting such states, as an attempt at specifying
the most significant features in the given phenomenon from the standpoint of
researchers. The author views aggregation states idealization as a lag behind
the scientific progress.
A wide array of data testifies that any aggregation
state comprises quite distinguishable properties of other states in a more or
less latent form. Let us recall the instances of such latent, or ulterior,
presence of one state within some other.
Gases in the vicinity of the melting temperature are
frequently represented by a suspension of the smallest drops of liquid being
termed vapor in this state. The smallest drops of vapor retain the elements of
matter and space inherent in liquid state. However, atom conglomerates are
observed in gases even at the temperatures that far exceed the boiling point
/94/.
It is also easy to evince that at the vaporization of
any liquids these are small atom conglomerates (but not separate atoms) that
pass into gas phase composition, the approximate number of atoms inside them
equaling K/2, where K is the coordinating number in the neighboring order
granulation characteristic of solid state, too.
Thus, real gases retain, though to a varied extent,
the properties of the elements of matter and space peculiar to both liquid and
solid states. It is known that there exists the possibility of the transition
from liquid to gaseous state, and v.v.
The majority of the properties of material and spatial
elements, intrinsic in liquid state, are preserved in solid crystal state.
In particular, the elements of analogy with clusters
are traced explicitly in the presence of such submicroscopic formations in the
structure of solid metals as mosaic domains and blocs, dislocations, twins,
borders of crystals and other formations. Such formations are usually reckoned
as the defects of crystalline structure /1,2,10,37,66,69,70,87,101/. The
borders of these elements and the mobile vacancy complexes possess certain
specificities of the elements of space inherent in liquid state – flickering
intercluster areas of bond splits.
Liquids, as it was corroborated by numerous x-ray and
other types of research, retain the properties of solid state presented by
neighboring order, etc. /10,17,18-24,31-41,44-47/.
We demonstrated above that liquid metals preserve the
properties of material elements that are peculiar to solid crystal state as
local areas of neighboring order inside clusters. Liquid metals also preserve
the elements of space characteristic of solid state – i.e. vacancies, as
intercluster monovacancies. The main difference between the latent elements of
matter and space and the prevalent elements is that the latent elements enter into
the composition of the prevalent ones, being overpowered by the latter.
Therefore, the latent elements of matter and space form neither phases nor
aggregation states but maintain the possibility of the system passing into some
other aggregation state upon the whole. After such a transition the latent
elements of matter and space become predominant, whereas the elements that used
to be predominant before such a transition, pass into the latent state.
There were repeated attempts at proving that defective
crystals thermodynamic stability is higher, for instance, than that of ideal
crystals /1/. The latent properties of aggregation states are often regarded as
the defects of the prevalent aggregation state in question, while we perceive
defects as something objectionable and eliminable.
In a number of cases, there was introduced a
contradictory idea of the equilibrium, i.e. removable, defects for solid metals
vacancies, in the first place.
Here we suggest another way of reasoning. Namely, it
should be admitted on the basis of the ample experimental data quoted above
that each aggregation state, except for the prevalent form of material and
spatial elements, contains more or less latently the properties of the elements
of matter and space that are peculiar to the neighboring aggregation states.
Such properties are equilibrium and unremovable,
acting as the essential constituent part of the hierarchical structure of real
bodies.
The presence of the latent elements of matter and
space pertaining to other states in the state given implies that any
aggregation state reserves the possibility of transition to another aggregation
state.
Gibbs’ phase principle prohibits the existence of more
than one phase at the same temperature and concentration. Consequently, Gibbs’
phase principle refers to the prevalent forms of aggregation states
irrespective of their latent forms. In turn, when speaking about the totality
of both the prevalent and latent aggregation states, we may premise on the
basis of the afore-said that the totality of the predominant and latent
aggregation states is constant for every given system.
Thermodynamics also premises that the stability of
this or that state is determined by way of comparing the free energies of any
two states, for example, liquid state Gl and solid state Gs.
Thus, for solid state /1-2/:
Gs = Hs - T Ss
(112)
and for liquid state
Gl = Hl – T Sl,
(113)
where Hs and Hl represent
enthalpy of solid and liquid states correspondingly; Ss and Sl
being the respective entropy of solid and liquid states.
It is reckoned by right that Gs = Gl
at the temperature of crystallization only. At any other temperature, the
phase, or aggregation state, with the lesser free energy under the given
conditions, will be stable.
However, the application limits of equations (112) and
(113) are thermodynamically unrestricted, which generates the problem of
two-phaseness touched upon above. In theory, thermodynamics allows to compare Gs
and Gl by calculations according to (112) and (113) under any
conditions, at any temperatures, whereas it is actually assumed that the two
aggregation states coexist at the crystallization temperature only, where their
free energies are equal. It is accepted that there exists only liquid state
above the temperature of melting, there being nothing but solid state below the
melting temperature. What comparison can be discussed under such a premise?
We introduced above the concept of the prevalent and
latent elements of matter and space, inherent in different aggregation states,
and that enables us to solve this problem. Actually, the comparison on the
basis of (112) and (113) of the free energies (as well as many other
parameters) of various states is possible under any conditions, if we take into
account the existence of the predominant and latent forms of these states.
Let us note so far that a real and constant comparison
of the stability of the prevalent and latent forms of aggregation states occurs
within any system implying a continuous competition, or extrusion, between the
specified forms. The attempts at changing aggregation states are constantly
taking place within any system, their success or failure being determined by
the inner structure of the system as well as the environmental conditions.
Let us term such a process as the
competition between the prevalent and latent aggregation states.
It means that preparatory processes for the change of
aggregation states are always in progress to a certain degree, and any change
of the aggregation state structure and properties advances or postpones the
transition of the original aggregation state into another. Premelting and
precrystallization always occur in liquid and solid state but they proceed with
a different development degree under dissimilar conditions, with a different
degree of approximating the transition of the predominant aggregation state
form.
Let us view the mechanism of the competition between
the prevalent and latent states exemplified by liquid and solid states of
metals.
It was shown above that melting goes according to the
cluster reactions scheme (19), with the essential addition that scheme (19)
reflects only the material aspect of the processes of melting and
crystallization irrespective of the elements of space participating in these
processes. Let us reproduce expression (19) here in a somewhat modified form:
This scheme equally describes the process of
crystallization, too. The difference lies in the direction of cluster reactions
in scheme (19). In the direction from left to right the scheme describes the
process of crystallization. In the direction from right to left the same scheme
describes melting process.
In liquid state clusters αn
perform heat oscillations, in the process of which the flickering elements of
space – intercluster splits – arise and disappear between the neighboring
clusters. The mechanism and parameters of such oscillations were discussed
above. It is important to remark here that the process of cluster accretion
into elementary crystals in liquid is one of the elements of the existence form
of liquid aggregation state. In the process of such infinitely repeated acts of
cluster accretion and separation there also occurs a repeated transition of the
kinetic energy of heat oscillations of clusters into the potential energy of
bonds between the accrete clusters. This is the repeated evolving of potential
energy that heats up the locality of cluster accretion and makes them separate
anew under the pressure of the vacancies that are contained in clusters.
Such an infinitely repeated
process of consecutive elementary acts of melting and crystallization at the
level of clusters is the mechanism of extrusion between the prevalent liquid
and the latent solid aggregation states inside liquid state.
Liquid does not know about its crystallization, if we
can say so, but the system, due to the continuous competition between the
prevalent and latent states, seems to be constantly testing the environment
through the flickering interaction of the elements of matter and space, as if
adapting for it; the system changes its structure and properties adjusting them
to the environmental conditions. In particular, the dimensions of clusters and
intercluster spacings change in the course of this process, vacancies emerge
and disappear, etc. At the change of the environment, liquid prepares for
crystallization in quite a short time by way of constant extrusion between the
elementary acts of melting and crystallization.
The extrusion of the prevalent and latent material and
spatial elements inside liquid state is, in its wide sense, the mechanism of
the system’s adaptation for the environmental conditions. The possibility of
such adaptation is ensured by constant heat oscillations and other kinds of
heat motion of the elements of matter and space in the aggregate with the
constant flickering of spatial elements and the bonds between the elements of
matter. This is the flickering interaction of the elements of matter and space
that imparts flexibility, mobility to real systems, as well as the ability of
reorganization and reaction to the environmental changes.
A kind of similar process occurs in any other
prevalent aggregation state. In solid state, vacancy gas pressure is constantly
testing crystalline lattices for strength. In gas state, atoms and their small
groupings are constantly colliding, accreting and separating, etc., etc.
Thus, precrystallization is a continuous process of
the interacting, reorganization and extrusion of the elements of matter and
space of the prevalent liquid and the latent solid aggregation state inside the
prevalent liquid state. Such a process is an existence form of any state. It
enables any system to do a quick re-structuring of the total of its intrinsic
parameters and properties in correspondence with the change of the
environmental conditions, the preparation for the process of crystallization
including.
So the coexistence of equations (112) and (113) quite
reflects the real complex structure of aggregation states that turns out to be
distant from idealized concepts. The concepts of ideal simple monatomic liquid,
ideal crystals and ideal gases also prove to be reality-discordant.
Existent theory presents the formation of
crystallization centers as rather a complex and contradictory process. Certain
problems of crystallization centers formation in connection with current theory
were tackled in Part 1.7 above.
Let us consider this problem one
more time in order to suggest its new solution.
The problem of crystallization centers is described in
a great number of works in present theory. Turnball and Hollomon /63/, as well
as W.C.Winegard /68/, give a good account of this problem from the viewpoint of
corpuscular structure of liquid metals. W.C.Winegard writes that when atoms
group so that a nucleating center is formed, the surface of section emerges
between it and liquid. Section surface formation leads to energy consumption,
which brings about a certain increase in the free energy of the system at the
origination of the nucleus. The nucleus, however, can increase only in the case
if the total free energy of the system is decreasing.
The core of the problem of solid phase nucleation in
the existent monatomic theory is formulated here with precision. The
origination of the nucleus within the idealized homogeneous monatomic liquid
inevitably causes the forming of a section surface, which leads to the increase
in free energy, resulting, in turn, in the impossibility of zero growth of such
a nucleus. It is Ya.I.Frenkel’s heterophase fluctuations theory that suggests
rather a controversial way out of this typical circularity.
Mathematically the problem of crystallization centers
formation is presented in the following way /63,66-68/.
The change of the system’s free energy at the forming
of a solid phase crystal in liquid equals
DF = -V DFv + S s,
(114)
where V is the volume of a crystal; S is its surface; DFv is the change of specific volumetric free energy; s is surface tension.
This expression is identical with
the formula cited in Part 1, except for the fact that the latter expression
employs free energy according to Helmholz. Free energies, according to Gibbs
and Helmholz, practically coincide for condensed states.
If we suppose that a microcrystal is spherical, (114)
will be presented as
DF = -(4/3)pr3
DFv
+ 4pr2 s, (115)
where r is the radius of a solid phase nucleus.
The main assumption at formulating
expression (115) is that a certain new surface, for the formation of which work
(energy) must be spent, arises at the crystallization center formation, which
is reflected by the plus in front of the second term on the right in expression
(115). Such an assumption seems quite logical, being the only possible within
the limits of the monatomic theory of liquid metals structure. Still, let us
bring it into focus that such an assumption initiates all the difficulties of
present theory. It was stated above that the introduction of the opposite signs
into the right side of expressions (114-115) causes the insoluble inner
contradictions in existent theory.
In particular, it follows from this very assumption
that the function of
d (DF)/dr = - 4pr2
DFv
+ 8pr s, (116)
has the extremum, while the radius r that corresponds to the bending
point can be obtained under the condition that
4pr2
DFv
+ 8pr s = 0
Hence originates the idea of the
so-called critical radius of the solid phase nucleus:
rc = 2s/ DFv
(117)
Then, with the use of the known value of DFv /66-68/ found under rather debatable premises, the following is
derived:
rc = 2sТmelting / DН DТ
(118)
The physical meaning of the
critical radius of crystallization center is that the growth of all the
crystals with the r > rc is accompanied by the decrease in
the total free energy of the system, so such crystals can grow freely and
unrestrictedly. However, the growth of all the crystals with the r < rc
will be accompanied by the increase in the total free energy of the system, so
such crystals no sooner arise than they must disintegrate. In point of fact, it
should be regarded as a thermodynamic prohibition of crystallization.
Graphically, the relation between DF and r is expressed by curve 1 in Fig.17 and
Fig.2.
According to the
graph, solid phase can set in after having overjumped the interspace of the
prohibited nuclei dimensions from 0 to rc.
Thermodynamics cannot interpret
the possibility of such jumps. Moreover, equations (115) and (116) presuppose a
continuous configuration of the function of DF = f(r), actually prohibiting similar jumps.
So, to overpass the problem of the prohibited interval, there was
initiated a non-thermodynamic theory of heterophase fluctuations that lets
crystals grow stepwise, and not continuously, up to the reaching of
overcritical dimensions. The heterophase fluctuations theory was considered in
Chapter 1 above in a more detailed way.
Such a point looks very unnatural in existent theory,
so nothing but a long-term habit defends it against criticism.
Nevertheless, we shall try to test the assumptions of
present theory for their correspondence to facts.
The major premise breeding all the contradictions of
the mentioned theory is the assumption of the emerging of new section surfaces
at crystallization both at the moment of nucleation and in the process of crystal
growth. Energy must be spent to form such numerous section surfaces. Actually,
it implies that the system must absorb some energy.
However, at crystallization energy is not absorbed but
evolved, moreover, such energy equals the latent heat of melting taken with the
opposite sign to a high degree of accuracy. The lack of differentiation between
the latent heats of melting and crystallization speaks for the complete
dissymmetry of these processes in the sense of work expenditures, including
those for surface formation. On the contrary, theory assumes that there should
always be work expenditure for the forming of the surfaces of solid phase
nuclei section. Since energy evolves in the one case (at crystallization) being
absorbed in the other case, there must be a difference of the latent heats of
melting and crystallization reflected in the quantity of DF = Ss. Yet it does not really exist.
Thus, facts are at variance with existent theory.
In this situation such a discrepancy can either be
neglected, which was established practice for almost 70 years, or some
artificial account of the situation may be suggested (which was also done), or
we are just to accept the facts and search out new explanations. Let us accept
the facts.
Let us accept the fact of energy evolving at
crystallization as the principal one.
It testifies that there are no new surfaces emerging
within the system, as it was thought to be, but, on the contrary, certain
inside surfaces existing in liquid state are closed.
In case of the monatomic theory of the structure of
liquid it is absolutely impossible, since the monatomic liquid is homogeneous
providing no inner surfaces of section.
From the viewpoint of the theory under development,
flickering inner intercluster splits saturate liquid. Let us write the
elementary act of crystallization as the reaction of
an +ds + an - dНcr ® a2n - ds + dНcr
(119)
The given reaction means that at the accretion of two
neighboring clusters into an elementary crystal the elementary surface of
section ds closes between them into a flickering
intercluster split.
It was shown above that this process is accompanied by
the transition of the kinetic energy of heat cluster oscillations into heat, so
the elementary amount of the latent heat of crystallization dНcr is evolved.
If it is true, we have to admit that those are not new
surfaces that emerge at crystallization by the accretion of clusters in liquid,
but the existent flickering intercluster section surfaces that close, which is
accompanied by the evolving of heat to corroborate the facts completely.
Then, we should re-write expression (112) as
DF = -V DFv - S s,
(120)
while expression (120) takes the shape of
DF = -(4/3)pr3
DFv
- 4pr2 s. (121)
Curve 2 in Fig.17. represents the graph of (121). It
is clear that the function of DF = f(r) does not have the extremum in
the given case, decreasing monotone with the rise of r.
Expression (121) differs from
expression (115) only by the sign in front of the second term on the right, but
the physical meaning of expressions (115) and (121) differs fundamentally, and
such a distinction changes all the existent concepts of the mechanism of the
processes of crystallization centers formation and crystal growth.
The minus has a physical significance in the given
case, as well as the sign in front of the first term on the right in (115) and
(121), meaning that at the closing of the intercluster surface S energy evolves
but it is not absorbed. Such a seemingly negligible difference in signs
radically changes our understanding of the problem of nucleation and smoothes
away the contradictions pointed out above.
The point is that crystal growth will be
thermodynamically expedient at any radius of the nucleus r in liquid
cooled down below the melting point temperature.
It also ensues therefrom that the key problem of the
present crystallization theory - that of the critical radius of crystalline
nuclei - is farfetched, it does not exist in reality.
This is a new and essential
conclusion shaking the fundamentals of the current crystallization theory. In
particular, the inference of mass crystalline centers nucleation in cooled
liquid issues among the first consequences of the new theory, which, in turn,
changes our ideas on the mechanism of crystal growth. The new ideas are viewed
in detail within the theory of overcooling and the competition theory of
crystallization below.
Distinct from the artificial problem of the critical
radius of solid phase crystalline nuclei, which used to exist in theory but not
in reality, the phenomenon of the overcooling of liquid before and during the
process of crystallization is an experimental fact.
Present theory closely relates
overcooling to the problem of the critical radius of nucleating centers.
Namely, existent theory considers overcooling as a structure-forming factor
that influences the probability of heterophase fluctuations formation, as well
as the operation of crystalline formation and growth and the dimension of the
critical radius of the solid phase nucleus.
The fabulous nature of the mentioned parameters as
applied to crystallization by no means affects the fact of the existence of
overcooling.
Consequently, overcooling performs some other
functions at crystallization distinct from those that were declared earlier.
To define the role of overcooling at crystallization,
let us consider the heat aspect of this process.
Let us write the equation of the elementary act of
melting-crystallization (119) as
α n + αn = α2n +
δHcr (122)
According to (122), an elementary crystal is formed by
the fusion of any of the two neighboring clusters with the evolving of the hard
amount of heat δHcr.
This is the elementary heat of cluster, or
intercluster split, formation, i.e. the elementary heat of cluster accretion or
the closing of intercluster splits. The quantity of δHcr
can be found through the following expression:
δHcr = DHmeltingnc / N0 (123)
All the quantities included into (123) are known
having been cited previously.
For an elementary crystal to get formed and for
reaction (122) to stop being oscillatory, the heat of δHcr
must be absorbed by the melt without the heating up of the latter above the
melting temperature.
Yet at the temperature of the melt equaling or
exceeding the melting temperature, the melt cannot absorb δHcr
without being heated above the melting temperature, so reaction (123) is
infinitely repeated in the directions both from left to right and from right to
left. The temperature of the melt does not change at that, for the energy of δHcr
is periodically passing from its potential form into kinetic, and v.v.
Crystals cannot form under such conditions. For at
least one elementary crystal to get formed, the heat of δHcr
must finally pass into the form of potential heat energy, so it must be
absorbed by the surrounding melt without the heating of the latter above the
melting temperature.
In turn, it is possible only in
case when the melt is cooled down below the melting temperature.
The stated phenomenon is termed overcooling DT. As it follows from the above-said, overcooling before
crystallization is required for the only purpose – for the melt to absorb the
latent heat of elementary crystal crystallization on its own, without being
heated up above the temperature of crystallization.
It is normally admitted that
crystallization does not go within the overcooling interval. Our theory affirms
that reaction (123) goes on repeatedly with the frequency of 109
acts per second. However, while the heat of δHcr cannot be absorbed by the melt, it passes again and again into the
form of the potential energy of heat oscillations of clusters.
It is obvious that overcooling performs but a purely
heat part in our theory; it is devoid of any structure-forming functions.
Such a purely heat approach to the quantity of DT enables to determine by calculation the quantity of overcooling
necessary for crystallization to set in by the method of heat balance between
the crystallization center and the surrounding melt. The elementary
microcrystal heat balance equation may be presented as
Qm = Qn.c.
(124)
where Qn.c. is the heat of the
elementary crystallization act according to (123), i.e.
Qn.c. = δHcr = DHmeltingnc
/ N0, (125)
while Qm is the heat absorbed by the
melt under the condition of its being heated up to the melting temperature
exactly. This quantity can be obtained by applying familiar heat methods. Thus,
Qm = (Tmelting - Ti) c v
ρ, (126)
where Ti is the maximum temperature of crystallization
start; Tmelting - Ti = DT, where DT is the minimum overcooling of
crystallization start; c is the heat capacity of the melt; ρ
is melt density; v being the melt volume that absorbs the heat of Qn.c.
during the time equal to one period of heat cluster oscillations.
The latter is of immense importance. The heat of Qn.c.
evolves during the time equal to one half-period of heat oscillations of a
cluster. It must be absorbed by the environment during the same or even lesser
time without the heating up of the specified zone of the medium above the
melting temperature, otherwise clusters re-separate and the elementary volume
of liquid gets formed again.
The process of heat absorption cannot be prolonged for
an indefinite period of time. Time factor refers to decisive ones alongside
with structural factors at crystallization.
Hence ensues the second essential conclusion – heat
can only be absorbed by the immediate environment of accreting clusters for
such a short period.
I.e. the volume of v in expression
(126) must be very small, because at the elementary act of crystallization
there is no time to be spent on slow redistribution of heat within the melt
volume and beyond its limits.
We can put forward the following suggestion concerning
the amount of such heat.
The elementary
act of crystallization consists in the accretion of two neighboring clusters.
The elementary heat of crystallization evolves along the accretion border
between the given clusters, so the heat in question, before dispersing in the environment,
will be inevitably absorbed in the main by the accreting clusters themselves,
and elevate their temperature. In case of a successful elementary
crystallization act, the elevation of the temperature of the two clusters under
accretion caused by such heat must not excede the temperature of melting.
To launch our analysis, it is therefore natural to
presume that the volume of v in expression (126) equals the volume of
the elementary crystal itself, i.e. the volume of two clusters.
The volume of two clusters amounts to
v = g Vm nc /N0
(127)
where Vm is the molar (corpuscular) metal volume; g
is the number of clusters participating in the elementary act of
crystallization. In the simplest case g = 2.
By way of inserting the v from (127) into (126), we
obtain
Qm = g DТ c r Vm nc /N0.
Let us stipulate that Vm = M / ρ, where M is the atomic metal mass.
Finally we
obtain:
Qm = g DТ c M nc /N0.
(128)
Now, introducing the value of Qm
from (128) and the value of Qn.c. from (125) into
the original heat balance equation (124), we arrive at
g DТ c M nc /N0 = DНmelting nc / N0
Hence we derive the minimum
overcooling required to start crystallization as the first elementary act at
the absorbing of crystallization heat by two accreting clusters:
DТ = DНmelting / g М с.
(129)
Expression (129) defines
overcooling as a purely heat quantity. Moreover, this expression has a certain
beauty and compactness, which is also important. Expression (129) includes
reference quantities only, in which connection the value of overcooling
necessary for nucleation according to the elementary crystallization act scheme
(122), can be easily calculated.
The data obtained by our
calculation are to be found in Table 15.
Table 15. The Minimum Overcooling Necessary to
Start Crystallization of Pure Liquid Metals at g = 2
Metal |
М, kg/mole /98/ |
DHmelting,
c/mole /98/ |
с, c/mole deg., /98/ |
DТ, deg. by (129) |
Ga |
69.72 |
1335 |
6.24 |
1.53 |
Cu |
63.54 |
3120 |
6.86 |
3.58 |
Sn |
118.7 |
1720 |
7.6 |
0.95 |
Al |
26.98 |
2580 |
7.66 |
6.24 |
Bi |
208.98 |
2730 |
7.43 |
0.88 |
Zn |
65.37 |
1730 |
7.01 |
1.89 |
Fe |
55.85 |
3290 |
10.29 |
2.86 |
Ni |
58.70 |
4180 |
9.20 |
3.87 |
Co |
58.93 |
3900 |
9.60 |
3.44 |
W |
183.85 |
8420 |
26.67 |
0.80 |
Overcooling calculated in Table 15
is quite close to the values of liquid metals overcooling observed
experimentally /1,2,10,68/. It corroborates the inference of the role of overcooling
as mainly the thermal factor of crystallization.
Still, the calculated overcooling
is not limited to the suggested values, for only thermal factors were taken
into account when doing the calculation, while crystallization is also bounded
by positional factors, e.g. the afore-mentioned re-granulation factor, plus the
degree of terrain-contour matching of clusters before accretion, as well as
time factor, external and other factors. Thus, real overcooling before the
start of crystallization may either exceed or be less than its calculated
values under the influence of nonthermal factors.
In this connection, the quantity
of factor g, i.e. cluster quantity participating in the absorption of
the heat of the elementary crystallization act, which we introduced, is of
paramount importance for the experimental research of liquid metals structure.
By measuring the actual overcooling of DТ, we can calculate the
quantity of factor g.
For instance
g = DНmelting / DТ М с.
(130)
Interestingly, g may either exceed or be less than 2. This is
a new and worthy experimental reseach subject.
For example, the maximum known overcooling for liquid iron equals
2950C /68/. Introducing the specified value into (130), we obtain
that g1 = 0.019 in this case. As it is known, crystalline
dimensions are very small in case of crystallization with a considerable
overcooling.
During the process of slow crystalline growth in liquid iron,
overcooling often approximates 0.10C. By introducing the given value
into (130), we find that g2 = 57.14 in this case. The
correlation between g2 and g1 is g2
/ g1 = 3000. Such a correlation characterizes the possible
relation of crystalline dimensions that can be obtained in cases of crystallization
going at either the maximum or the minimum speed for iron.
Thus, the quantity of g turns out to be proportionate to the
crystallization act duration as well as crystalline dimensions in castings, and
it can be used to determine the mentioned quantities.
We should discriminate between the overcooling of nucleation and the
overcooling of crystal growth. The latter is always less than the former, since
heat abstraction conditions are facilitated during crystalline growth.
On the whole, the calculated values of overcooling fit the familiar
experimental values of this quantity. W.C.Winegard allocates
the typical quantity of liquid metals overcooling before the start of
crystallization within the limits of 1-10 deg. /68/. Overcooling values that we
calculated according to (129) and cited in Table 15 are positioned exactly
within the given interval.
The term 'spontaneous' means 'evoked by internal causes' (often
unknown). The term 'forced' in application to crystalline centers nucleation
signifies 'initiated by external causes'.
As it was shown above, crystallization results from the interaction
of both the internal causes, such as the interplay of material and spatial
elements in liquid metals, and external factors, for instance, temperature,
pressure, etc. External and internal causes interreact.
Therefore, the distinction between the processes of nucleation in
liquid metals into spontaneous and forced seems inappropriate.
Nevertheless, such distinction arose, so it is to be taken into
consideration.
W.C.Winegard defines spontaneous crystalline centers nucleation as
nucleation in absolutely homogeneous medium with the presence of overcooling
/68/.
G.F.Balandin /74/ defines spontaneous nucleation as a result of
monatomic heterophase fluctuations formation, also with the sine qua non of
overcooling, which does not contradict W.C.Winegard’s definition.
W.C.Winegard writes that 'in the vicinity of the melting point
critical nucleus dimensions must be infinitely large, because, when overcooling
approximates zero, the decrease in volumetric free energy related to the phase
transition of liquid into solid cannot compensate for the increase in free
surface energy. As overcooling increases, critical nucleus dimensions
decrease…' /68/.
It follows from the given reasoning that overcooling must act as the
measure of spontaneity or forcedness of the process of crystalline centers
nucleation.
The greater overcooling becomes, the closer spontaneous nucleation
is.
Unfortunately, it is impossible to evaluate the degree of
overcooling necessary for spontaneous nucleation on the basis of these
arguments, in contrast to our theory.
It is accepted that spontaneous nucleation is possible only in
liquid metals that are completely purified of any admixtures. The production of
such metals encounters serious experimental difficulties, for no analysis can
secure against the presence of the minimal quantities of foreign admixtures in
the melt.
It is also assumed theoretically that one can overcome the mentioned
difficulties by way of dividing liquid metal into smallest drops. If there be a
little amount of admixture particles within the volume, some drops would not
contain foreign particles by virtue of their own small dimensions, so
homogeneous nucleation could be observed within them. Experiment
overcorroborated such suppositions. As it turned out, overcooling does increase
considerably at the dissection of the melt into drops, - and not for some of
them, as it was to be expected, but practically for all the drops, usually
inversely proportional to their dimensions. Actually it means that at the
crystallization of small drops those are not admixtures that perform the
salient function but a certain, or some, ignored factor(s).
For instance, it may be the factor of time. Small drops cool down
much faster than volume-bounded liquid during the same time period.
Time is also required for the fitting, or terrain-contour matching,
of the accreting cluster structures and for their re-granulation (see above),
it is simply necessary to evolve the latent heat of crystallization and provide
its transition from the kinetic energy of heat cluster oscillations into heat
energy, as well as redistribute the given energy at least within the limits of
two clusters.
Existent theory neglects these factors; it is reckoned that
nucleation in small drops is actually honogeneous.
The values of overcoolings obtained by the small drops method are
listed in Table 16.
Table 16. The Maximum Overcoolings (ΔT)
Obtained by the Small Drops Method /63/
Metal |
ΔT, deg.
C |
Metal |
ΔT, deg.
C |
Mercury |
77 |
Silver |
227 |
Tin |
118 |
Copper |
236 |
Lead |
80 |
Nickel |
319 |
Aluminium |
130 |
Iron |
295 |
As W.C.Winegard notes, such overcoolings are never observed in
practice; overcooling quantity fluctuates from 1 to 10 degrees under real
conditions. Let us add that the calculated quantities of overcooling of the
elementary crystallization act (v. Table 15) fluctuate within the limits of
1-10 degrees.
However, it has been assumed till now on the basis of the data
quoted in Table 16 that heterogeneous, as contrasted with spontaneous,
nucleation takes place under real conditions, i.e. crystals get formed on the
surface of a foreign solid body present in the system.
Thus, current theory regards spontaneous crystallization as an
occurrence that is almost improbable, or practically unobservable, in any case.
For example, V.I.Danilov used to admit that spontaneous
crystallization is hard to observe, too. Liquid metals should be purified of
practically all admixtures for that /1,2,10,63-70/.
The imperfections of such an approach are obvious here having been
commented upon earlier; they result from idealized views of liquid metals
nature as well as the incorrect ideas of the role of overcooling at crystallization.
The interaction between the elements of matter and space theory
developed here presumes that all real bodies consist of the interacting
elements of matter and space, while bodies contain not only the prevalent
material and spatial elements, but also their latent forms from the standpoint
of aggregation states. Thus, liquid metals are non-ideal and inhomogeneous in
principle in their structural aspect, similarly to any other real bodies. So
the premises of the ideal monatomic, monomolecular and any other monoparticle
structure of liquid metals are erroneous in principle, for there are no such
simple liquids in nature.
Evidently, it would be better to define spontaneous nucleation as a
natural elementary crystallization act occurring by way of accretion between
any of two oscillating elements of matter in liquid – i.e. clusters – into a
single elementary crystal accompanied by the evolving of the elementary amount
of crystallization heat under the influence of the totality of external, as
well as internal, factors.
Since clusters within the melt can be of a various chemical
composition, the presence of soluble admixtures does not make the elementary
act of crystallization forced, though influencing it. This is the same natural
spontaneous crystallization, because the essence of the process never changes.
The presence of insoluble insertions or gases in the melt does not
change the core of crystallization process but rather modifies its conditions.
Thus, our approach, as distinct from present theory, establishes
that natural spontaneous nucleation is not a rarity but the major fundamental
and the most prevalent variant of nucleation both in pure metals and alloys.
Spontaneous nucleation may occur within a wide range of
overcoolings. Overcooling does not determine the degree of spontaneity or
forcedness of crystallization process at all. Overcooling is required, first,
for the heat abstraction of the elementary crystallization act, as it was
demonstrated above. Besides, overcooling can be sensitive to metal cooling
speed, as well as structural metal modifications and other factors.
The theory that existed earlier could not calculate the overcooling
necessary for spontaneous crystallization to set in. Such indeterminancy
gradually lead to the fact that spontaneous nucleation came to be considered as
an infrequent, particularly laboratory phenomenon.
The overcooling calculated in Table 14 above is determined for
chemically pure metals.
In essence, this is the overcooling of spontaneous crystallization
yet calculated for the concrete case of the complete two-cluster accretion into
an elementary crystal under the condition that there is time sufficient for the
complete terrain-contour matching of the adjacent cluster structures. Actually,
the specified calculation was done for the conditions of a very slow
overcooling. It is a typical but not the only possible case of spontaneous
crystallization. So the overcooling calculated in Table 15 by expression (129)
is not the only possible spontaneous crystallization overcooling either.
External and internal factors, for example, overcooling speed or
alloy composition change, can strongly affect the conditions of spontaneous
nucleation and the corresponding overcooling.
Thus, spontaneous nucleation may occur under different conditions
and at dissimilar overcoolings. Nucleation is always spontaneous in a sense,
for it is determined by fundamental causes. For instance, nucleation always
goes by reaction (122).
Forced nucleation does not exist as such without spontaneous nucleation.
Consequently, spontaneous nucleation is primary, forced nucleation being
secondary.
Any external action can alter the conditions of reaction (122)
course, but the reaction of the elementary crystalline centers nucleation act
always remains the same.
Therefore, our theory, as distinct from existent views, affirms and
proves that spontaneous crystallization is the main crystallization type,
whereas external action can either hamper or facilitate this process without
changing its essence.
Current theory asserts that the speed of crystalline centers
nucleation is determined by the following expression of the heterophase
fluctuations theory (6):
n = К1 еxp (-U/ RT) exp [- Bs3 / T (DT) 2],
where the quantity
of n is measured by с-1 m-3. I.e. the quantity of n
represents the onset of heterophase fluctuations of critical dimensions
(nucleation centers) frequency per volume unit of liquid.
Let us compare the given approach with the data in our theory.
In accordance with expression (127), every act of heat
oscillations of clusters potentially represents the elementary act of a
crystalline center nucleation. So the frequency of heat cluster oscillations is
the highest possible frequency of crystalline centers nucleation. Out of
expression (49) we derive:
n
= j = (1/2paа)
(3kT N0 / nc M)1/2
To find the frequency of crystalline centers nucleation per certain
volume, (49) is to be multiplied by the number of clusters within the given
volume. It seems most appropiate to determine the unknown quantity per mole of
substance. Thus
N
= n Nc,
where Nc
is the number of clusters in a mole (gram-atom) of liquid metal at the
temperature of melting.
In turn, Nc = N0 / nc.
Finally we obtain the expression for calculating the highest possible frequency
of crystalline centers nucleation per gram-atom of metal:
n
=( N0 / nc)(1/2paа) (3kT N0 / nc M)1/2
(131)
This is an extremely great number, of the order of 1032 с-1.
Thus, the theory of the interaction between the elements of matter
and space that we develop accentuates that the process of spontaneous
crystalline centers nucleation in liquid metals refers to regular but not
random phenomena. Certainly, any cluster pair can form a crystallization
center, but only in case when there occur favorable conditions, the conditions
for the elementary crystallization heat abstraction, in the first place. Such
continuously merging and separating cluster pairs are flickering, or virtual,
crystallization centers.
Flickering crystallization centers nucleate with high frequency in
liquid by (131), and separate again and again with the same frequency. Liquid
seems not to know about its forthcoming crystallization, yet it can prepare for
crystallization with the help of the flicker mechanism as the environment
provides the corresponding conditions for the process.
Crystallization
requires time by various reasons, so the time necessary for crystallization
influences its results. This is familiar from practice. Let us briefly survey
the causes of time influence upon the process of crystallization.
It was stated above that the elementary crystallization act
represents a reaction of fusing two adjacent clusters into a single elementary
crystal. Under the most favorable conditions such a reaction requires the
minimal time equal to one period of heat cluster oscillations
j = 2paа (nc M /3kT N0)1/2
(132)
That is the time equalling approx.10-9 sec. It is
absolutely the minimal time requisite for a single elementary act of
crystallization.
The real minimal crystallization time may exceed the quantity of
(132), yet it cannot be less than the mentioned quantity.
In the first place, the crystallization of metal mass runs
consecutively and not simultaneously. Hence ensues the general rule: the
grosser the casting is, the more time it requires for its crystallization.
Secondly, there exists the above-cited factor of cluster
re-granulation in liquid. It means that in liquid clusters are packed otherwise
than, or not exactly as, the atoms in the crystalline lattice of a solid. At
crystallization, clusters must re-form into a configuration that suits to their
accretion into a single crystal most.
Such reconfiguration occurs by way of consecutive restructurings
until the optimal or at least acceptable configuration of cluster granulation
is reached. Re-granulation requires for the renewing of the same form of
interatomic bonds between neighboring clusters, as is peciliar to a solid
crystal. The closer cluster configuration approaches that of a solid body, the
more thoroughly intercluster bonds get renewed at crystallization, the more
equilibrium the growing crystal is, the more perfect its structure becomes.
However, it is hardly possible to arrive at the complete compatibility between
cluster structures, - this can only be approximated to some degree. The process
under consideration is termed as intercluster bonds matching and it requires a
considerable amount of time for its more or less satisfactory completion.
Still, the matching of clusters and growing crystals need not attain
absolute completion for a successful crystallization course. Clusters can
accrete with a certain mismatch of interatomic bonds. The developing
crystalline lattice will be defective in this case, i.e. far from being
equilibrium, which is usually observed under real casting conditions.
As experience shows, the degree of a possible mismatching of
intercluster bonds is relatively high for metals.
It follows from the experiments of the so-called amorphous metals
production. Even at the speed of overcooling that reaches 106
degrees per second, it is possible to obtain only an extremely fine
microcrystalline structure in metals in the majority of cases. These are but
some specific alloys that let obtain a quasi-amorphous structure.
We should remark that, in connection with the composite real
structure of liquid alloys, the presence of clusters and intercluster splits
inside them, as well as the presence of the neighboring order of atomic
arrangement inside clusters, it is impossible in principle to obtain a
completely amorphous, wholly chaotic structure of metals at their
crystallization from liquid.
All that is to be done in this direction is to obtain a solid metal
structure proximate to the monocluster pattern. Such a structure will contain
an increased amount of the elements of space extrinsic to solid state, the
correspondingly reduced density and immense free energy, becoming, in this
connection, very unstable thermodynamically.
Crystalline growth also requires time – by the same reasons.
In the third place, time is needed to abstract the latent heat of
crystallization away from the growing crystal, as well as crystallization front
and casting upon the whole. The time factor of crystallization heat abstraction
plays the vital or decisive role under regular foundry conditions. Its cause
consists in the sluggishness of the process of heat abstraction by the heat
conductivity mechanism, whereas this is the very mechanism that operates under
heat abstraction conditions in solids, for instance, in a solid mold wall or
within a solid casting zone.
The mass
character of crystalline centers nucleation represents a specific and totally
unexplored problem. The inference that there arises a whole mass of
crystallization centers at the onset of crystallization ensues from the
experimental fact of the instantaneous liquidation of overcooling after
crystallization starts. The nucleation of one or several elementary crystals
cannot almost instantly raise the temperature of the entire metal mass up to
the melting point, which takes place in reality. The growth of several crystals
can elevate metal temperature up to the observed value in principle, but not so
rapidly as it really happens. Nevertheless, temperature rises, which
corresponds to the crystallization of a considerable part of metal volume.
Namely, with the overcooling determined in Table 15, the increase of
the temperature of the entire metal mass up to the melting point means that all
clusters entering into the overcooled metal composition have on average united
pairwise.
Hardly the strict pairwise union is it, in fact, yet the nucleation
of a large number of crystallization centers occurring simultaneously within
the whole volume of the overcooled liquid zone is beyond any doubt.
There are experimental facts corroborating this conclusion. In
particular, it is the familiar formation of a fine-grained disoriented crystals
zone at the surface of castings, or, as it is otherwise termed, the ‘skin of a
casting’ zone. So the more the speed of heat abstraction and the speed of
hardening are, the finer-grained the structure of castings grows.
Certainly, crystals within the zone under analysis are much larger
than elementary crystalline dimensions; however, temperature leap at the onset
of crystallization is but the first stage of the process which leads to the
forming of the ‘skin of a casting’ zone, as well as other structural zones in
castings.
The first inference of the given part lies in the following: a
discontinuous temperature rise within the entire volume of the overcooled metal
at the start of crystallization can be explained, most probably, by the mass
nucleation of a huge amount of elementary crystals, or crystalline centers,
within the zone specified.
Such a supposition seems natural to our theory, since, as it was
pointed out, the elementary crystallization act reaction (122) is continuously
repeated within the whole volume of liquid with the frequency of 10-32
times per second per gram-atom of metal by (131).
As it was shown, it signifies that liquid gets ready for
crystallization as soon as favorable conditions arise.
Overcooling subsumes under such conditions, implying the possibility
of absorbing the elementary heat of crystallization without overheating the
metal. At the reaching of such overcooling, the elementary acts of
crystallization (122) go spontaneously at any point of liquid, so a huge number
of elementary microcrystals (crystalline centers) emerge spontaneously at a
very short time period – approx. 10-9 sec. – independently of one
another. Their ultimate number can reach the quantity of Nmax = N0
/ 2 nc.. The quantity of Nmax for liquid
metals reaches 1020 per gram-atom of metal.
The given amount is much greater than the number of crystals that we
observe in a final casting at the end of crystallization.
Hence, first, the number of crystallization centers in the course of
crystallization does not equal the number of crystals that are obtained by
casting.
Secondly, the number of arising crystallization centers exceeds
multiply the number of crystals that we get through casting.
In the third place, it means that crystalline dimensions increase in
the course of crystallization, whereas their number diminishes.
Such conclusions are novel. The question of changes in crystalline number
in the process of crystallization has never been raised in current theory.
It is supposed by default that the number of crystals in the process
of crystallization does not change, so if a crystal comes into being, it
survives some way or another to be present in a solid casting later. Existent
theory presumes only a mechanical interaction between crystals in the process
of crystallization, for example, crystalline competition and selection in the
direction of their growth. Such interaction does not change the original
crystalline number during crystallization.
Our theory asserts that the number of crystallization centers under
regular casting conditions exceeds multiply the number of crystals obtained in
a final casting.
A question generates how a small number of large crystals result
from the original large number of small elementary crystals. This question is
of paramount importance - both practical and theoretic.
In practice, it is important that we obtain fine-grained castings,
consequently, it is useful to know how to fix such a huge number of
crystallization centers that we have at the beginning of the process in order
not to let them form into large crystals.
As far as theory is concerned, the emerging of a small number of
large crystals from a huge amount of microcrystals means that the process of
crystalline growth goes otherwise than it was previously surmised. Thus, the
theory of crystalline growth from the melt is to be improved.
The competition theory of crystallization that regards the
mechanisms of crystalline growth from the melt /130/ gives answers to the
questions formulated above.
Its major premise is that crystals can grow simultaneously at
different dimensional levels using dissimilar building material.
Correspondingly, there can exist several mechanisms of crystalline
growth in castings.
A certain mechanism of crystalline growth may turn out to be
prevalent under given concrete conditions, yet more often various mechanisms of
growth operate simultaneously complementing one another at different
dimensional levels. These different growth mechanisms are always competing with
one another, which lets obtain crystals with the least free energy.
Among the basic mechanisms we may cite the monatomic mechanism of
crystal growth, when separate atoms act as the main building material for
crystals, the cluster mechanism, when clusters serve as the building material,
and the microcrystalline (or bloc, mosaic, domain) mechanism, when small and
smallest crystals function as the building material for the growth of large
crystals.
The monatomic growth mechanism prevails at the growing of crystals
from gas phase. Still, even in gases there exist, as it was demonstrated above,
the latent elements of matter intrinsic in liquid state – small groupings of
atoms or molecules, and they can also participate in the process of crystalline
growth as the building material. The participation of such complexes is
thermodynamically expedient in the process of crystallization, since it
accelerates crystalline growth. On the other hand, the participation of such
complexes in the growth of crystals increases the probability of the appearance
of the so-called defects inside crystals.
It is the cluster mechanism that dominates at the nucleation and
growth of small crystals from metal and alloy melts. The mechanism under
analysis was viewed in detail earlier when treating the nucleation question.
The basis of the given mechanism is the bicluster reactions scheme as applied
to crystallization:
(133)
where αn
is a cluster within the composition of liquid; 2αn is
the elementary crystal obtained by the accretion of two clusters; iαn
is a crystal formed through the accretion of i clusters.
Reaction (133) can go in both the left and right directions,
dependent on heat absorption or abstraction, while the reaction of the
interaction between the neighboring clusters in liquid at T> Tmelting
goes continuously and reciprocally providing the basis for the flickering
interaction mechanism between material and spatial elements in liquid metals
and alloys:
αn
+ αn→← 2αn.
Cluster mechanism of growth according to scheme (133) is
characteristic of a rapid crystalline growth from the melt, - for instance, at
a high original overcooling, or, on the contrary, for a very slow growth under
the conditions of high temperature gradient within the liquid zone by the front
of crystallization, and also for pure metals.
The mechanism of crystal growth by way of the attachment of
microcrystals to larger crystals is peculiar to slow growth conditions with the
presence of inconsiderable overcooling, as well as the solid-liquid zone in
castings, which is most typical of the alloys that get crystallized under the
conditions of volume hardening.
Let us underline that the basic mechanisms of growth operate most
often simultaneously in different combinations complementing one another. Each
growth mechanism performs its functions and creates certain structural
specificities that can be traced in the structure of solid metals and alloys.
The latter mechanism of crystalline growth has the following
peculiarity: small crystals may unite competing in the process of growth, so
larger crystals may absorb smaller ones.
Let us term the given mechanism of growth as the competitive
mechanism, and let us consider it more extensively due to its appreciable
practical importance for the structure of the overwhelming majority of
castings.
The point is that this is the competitive mechanism of growth that
is responsible for the accretion of crystalline centers under the conditions of
their mass nucleation, typical of regular casting process, and eventually for
the coarsening of the original crystalline casting structure undesirable for
casters.
In principle, there are no insurmountable barriers to the accretion
of crystals with arbitrarily large dimensions in liquid, except for the problem
of their inner structure matching.
Let us mark that structure matching and the accretion of neighboring
microcrystals are thermodynamically expedient, since the system’s free energy
decreases in this case. Thus, the process of mutual fitting between the
structures of the neighboring microcrystals that are at motion in liquid will
not be quite accidental developing from the lesser to a more exact matching.
Consequently, this is a natural process accompanied by the decrease
in the free energy of the system.
So, if such matching exists, crystals can unite without forming
section surfaces, i.e. by way of forming a single large crystal from two or
more smaller crystals.
Stationary, fixed crystals cannot fit in with one another.
Therefore, the process of accretion between small and smallest
crystals is possible until the mentioned crystals retain mobility, i.e. until
they hover within the liquid medium participating in heat motion. It is
possible only with the presence of the solid-liquid zone in a casting.
Such microcrystals, hovering in liquid, are the Brownian motion
objects. While they are able to move, they can fit in with one another in the
course of multiple collisions and accrete into a single larger crystal after
attaining the compatibility of their crystalline lattices.
The dimensions of particles participating in the Brownian motion are
known – they amount to the tenth fractions of a millimeter (0.1mm.). The given
quantity can be considered by convention as the size limit for the hovering
crystals that maintain their accretionability. Larger crystals may accrete,
though, but only in case when the orientation of their crystalline lattices
chances on being compatible.
The process of small crystals accreting into larger ones is
energetically expedient, because the internal grain border area diminishes
during the process, so the free energy of the system decreases.
Similarly to any other dissipative process, the process of
crystallization, in correspondence with I. Prigozhin’s synergetic theses, is to
take place simultaneously at all its possible levels.
The competition crystallization theory asserts the same thesis.
The bicluster reaction mechanism similar to (134) is the basic
mechanism of nucleation and growth of crystals from metal melts. However, the
process of crystalline growth may go with the use of any building material
available in the given medium, including separate small crystals plus separate
activated atoms to fill out hollows.
The major tendency of the competitive crystallization mechanism at
macrolevel is the survival and growth of larger crystals by their absorbing
smaller ones.
This tendency determines the real crystalline structure of castings.
At the same time, the tendency in question reflects the struggle-for-existence
competition between crystals.
The process of competitive crystallization can be represented by the
following scheme of the growth and accretion of three neighboring crystals:
1st
microcrystal αn + αn→ 2αn |
2nd
microcrystal αn + αn→ 2αn 2αn + αn→ 3αn |
3rd
microcrystal αn + αn→ 2αn 2αn + αn→ 3αn ………………………….. ………………………….. iαn + αn→ (i+1)αn |
2nd and 3rd microcrystal accretion:
2αn
+3 αn→ 5αn
(134)
The accretion of the remaining microcrystals:
(i+1)αn
+5 αn→(i+6) αn
It follows from the scheme that crystals nucleate and start their
growth in a parallel way.
Crystalline accretion occurs under definite conditions only.
It is only the degree of crystalline competition development that
determines whether we get a coarse-grained or fine-grained casting structure as
a result. The more intensive competition development is, the farther this
process penetrates, the larger are the crystals that are obtained in the
structure of castings. Therefore, it is of practical importance to know how to
control the competitive crystallization process. To get a fine-grained casting
structure, the competitive mechanism is to be inhibited to hinder crystalline
accretion.
What determines the possibility or impossibility of crystalline
accretion?
Many dissimilar factors, external as well as internal, influence
this process.
However, the possibility of competitive crystalline accretion
process is determined in general by the presence of the solid-liquid zone, as
well as time and crystal contact conditions within the casting zone specified.
In many respects, time factor is the decisive one for the given process. Time
is required for the fitting of the adjacent hovering crystals structures. The
longer the time of crystals hovering within the solid-liquid zone is, the
larger crystals grow, the less their number in a casting is.
On the contrary, if the casting cools down fast, the time period
reserved for the matching of the adjacent crystalline structures diminishes,
they do not have time to accrete, forming independent crystals with the section
border of their own in the structure of the casting. In this case, the casting
has a fine-grained primary crystalline structure.
The operation of competitive crystallization and time factor account
for crystalline dimensions zonality in castings: metal cools down faster within
the ‘skin of a casting’ zone than in the center of the casting, thus, the
competitive process of crystalline accretion within the mentioned zone does not
go up to the end, so we obtain a fine-grained structure.
In the zone of columnar crystals, the width of the two-phase zone by
the crystallization front is small, and microcrystals hovering in this zone do
not have time to augment their dimensions being absorbed by the growing
columnar crystals and acting as the building material for them. The largest
among the hovering crystals cannot fit in with the growing columnar crystals
already, so they are forced toward the center of the casting.
The period of two-phase existence is maximal in the center of the
casting. Correspondingly, this is in the center of the casting that the most
favorable conditions for competition development and crystal accretion are
created, so there we obtain the structure with the coarsest grain.
Thus, the existence of the zone of large disoriented crystals in the
center of castings, typical of alloys and lacking in pure metals, is the
consequence of the competitive crystallization of crystals hovering within the
solid-liquid zone.
Mathematically, the dependence of competitive crystalline growth on
time can be expressed by the following correlation:
r =
k te,
(135)
where r
represents the maximal crystalline dimensions; k is the coefficient; te is the period of two-phase existence in the casting site where the
given crystal is located.
Expression (135) directly relates the dimensions of crystals in
castings to the period of crystallization.
The development of arborescent and other forms of crystalline growth
is well described in literature and therefore left out of consideration here.
The change in the volume of metals at melting and
crystallization is the traditional discussion subject in the theory of metals
in connection with the importance of the volumetric parameter for thermodynamic
constructions.
The change of metal volume at
crystallization is even more important for casting practice. The phenomenon
under analysis leads to the formation of shrinkage cavities and porosity in
castings.
The volume of systems, as well as their entropy,
enters into the main thermodynamic equations. However, if the entropy of metals
always increases at melting, which corresponds to general ideas and the data on
the disordering of matter at melting, the volume of metals, according to
experimental data, may either expand or sink at melting. The given
contradiction complicates the explanation of the change in the volume of metals
at melting.
As a result, none of all existent theories and models
of melting can offer any more or less acceptable theory of volume change at
melting, without touching upon the theories of crystallization.
Apart from general discourse that
the mechanism of melting includes intensive disordering through the formation
of simple, as well as cooperative, positional defects of the corpuscular
structure of matter /1/, there is no other achievement in the field signalized.
The phenomenon under consideration
takes on special practical significance, because the change of volume at
crystallization results in the so-called shrinkage of metals and alloys, the
change in casting and ingot dimensions, as well as the formation of shrinkage
strain, shrinkage cavities and porosity inside them, that affect directly the
quality of casting. If there exist several theories of diffusion and viscosity
of liquid metals, the mechanism and theory of castings shrinkage at
crystallization are totally lacking /74,75/.
So, practice suffers from the lack of theory to some
extent in the case given.
There is a general principle of approaching the change
of these or those properties of systems at the aggregation state transition in
the theory that is set forth. We termed it earlier as the relativity principle
of the forming of real systems properties.
Such an approach has been already applied above at the
founding of diffusion theory, fluidity theory, the theory of the change of
coordinating numbers, the theory of metal electrical resistivity at melting,
and a series of other questions.
The essence of this approach consists in the
following: we can calculate only the relative change of properties at the
aggregation state transition. It is to be done allowing for the changes in the
system’s structure at the level of aggregation states in the first place,
because the level of this or that property description must be adequate to the
property described.
It means that there is no point in finding explanations
to the changes occurring at melting at the levels distinct from that of
aggregation states. For instance, it is senseless to explain atomic structure
without referring to protons, neutrons amd electrons. It is senseless to try to
explain molecule structure without mentioning atoms, although it is an
incomplete description yet. But it is equally pointless to describe the
structure and properties of crystals limiting oneself to the ideas of electrons
and nucleons without considering the existence of atoms. It is pointless or
extremely difficult to describe ingot properties without referring to crystals,
etc.
I.e., for each level of real systems structure or
state, there exists its respective level of structural units that bear the
fundamental properties of the given state.
If we regard the change of properties that is caused
by the aggregation state transition, the adequate description of such a change
is to be carried out with the allowing for the transition from the structural
material and spatial units of one kind, inherent in the original state, to the
structural units of another kind, peculiar to the final state.
Real systems
possess a highly complicated hierarchical structure; they have many levels of
various elements of matter and space. Each hierarchical level imparts its
contribution to any property of the integral system, which the synergetic
science, which means ‘the science of joint action’, takes into consideration.
Unfortunately, the simplified viewpoint on the
structure and properties of real bodies and systems, explaining any properties
and their changes at the atomic-molecular level exclusively, is still widely
disseminated. It seems as if atoms and molecules that took us two millennia to
discover mesmerized the researchers. So, atoms and molecules – the elements of
matter, important yet positioned in a long row of hierarchical structures of
real bodies – come to be considered as the major, if not the sole, elements
that are responsible for all the properties of bodies and all the changes of
these properties. It is an error. Any hierarchical level of the structure of
real bodies is of no less importance than any other level, being major in the
prevalent state.
We should be fully aware of the limitations of the
specified principle. The level correspondence principle does not make it
possible to entirely describe this or that property, to give its absolute
value. It can only determine, including the quantitative aspect, the relative
contribution of the given level to the given property. I.e. such an approach
does not give any absolute values of properties, supplying their relative
values; it gives the quantity of their changes at the transition of the
system’s state, for example, the degree of volume change at the transition from
solid into liquid state. The relative change of volume, not volume proper, is
meant.
That is why the principle under analysis deserved its
label of the relativity principle of the states and properties of real systems.
For theoretical calculation, as well as the determination
of the absolute value of this or that property, we must know every element of
the hierarchical structure of the given system to determine the respective
contribution of each of them. The number of such levels of real bodies
structure is large enough, there being unexplored and unfamiliar ones.
Therefore, the stated synergetic problem was actually formulated here to date.
The absolute values of properties can be experimentally measured today in
certain cases only.
The expressed considerations, the given principle of
describing properties and their changes we shall also apply to the description
of metal volume changes at melting.
At the level of liquid aggregation state, the change
of any properties at the transition to the specified state is related, in the
first place, to the formation of the structural units of matter and space,
intrinsic in liquid state exclusively, i.e. clusters and intercluster spacings,
and their interplay.
The forming of intercluster spacings that have vacuum
properties is mainly responsible for the expansion of metal volume at melting,
and the given expansion will equal the overall volume of the elements of space
in liquid. Let us designate this factor as ΔVspl. The
value of the factor of ΔVspl was determined earlier by
expression (39) in Part 3.5.
ΔVspl = (3α / 2rc) 100%
Allowing for the relation between cluster radius rc
and the concentration of activated atoms on the basis of (45) as Ca
= 3/2 rc-1, we obtain for ΔVspl
ΔVspl = α Ca (136)
The convenience of the given
expression is conditioned by its compactness.
Aside from the factor of the
expansion of the volume of liquid at melting due to the formation of new
elements of space – intercluster spacings, – other factors operate in liquid,
which are related to clusters and able to cause self-compacting processes.
The forming of flickering
intercluster splits makes intercluster bonds unstable and flickering, too,
giving the opportunity of cluster displacing in liquid relative to one another.
As it was shown above, the possibility of such
displacements and the existence of splits account for the phenomena of fluidity
and mass transfer – diffusion in liquid state. The same cause influences the
change of volume concerning both its expansion and its sinking. As it was
demonstrated, the elements of space are responsible for the increase in volume,
and the volume that they occupy corresponds to the increase in the volume of
liquid. Those are clusters that account for the diminishing of the volume of
liquid.
The loosening of bonds between clusters provides the
possibility of their mutual displacement and re-granulation /129/. Under the
condition of free migration of compactly-shaped bodies relative to one another,
as we know, the granulation closest to the most compact one with the
coordinating number of 12 is reached. A compact packing of balls within any
capacitance at vibration serves as a familiar example to this.
Hence, two kinds of material elements granulation seem
to arise in liquid at different levels: 1. there remains inside clusters the
original atomic granulation peculiar to solid bodies; 2. there reappears mutual
cluster granulation. The in-cluster granulation of atoms seems to be enclosed
within the mutual granulation of clusters.
It is a good example of the hierarchy of real bodies
structure at different levels, which was discussed earlier.
If atomic granulation inside clusters has the same value
of its coordinating number as the compact mutual cluster granulation, then, the
re-granulation of clusters does not affect the volume of liquid. However, if
the in-cluster atomic granulation differs from the compact one, the
re-granulation of clusters will promote the compacting of liquid, so we are to
take it into consideration.
Let us designate the quantity of cluster
re-granulation factor as ΔVc. The quantity of ΔVc
can be determined on the basis of the following concepts. At the
re-granulation of clusters it is only their mutual position that changes, but
the in-cluster atomic granulation remains the same. We may reckon that only the
atoms located on the ‘surface’ of clusters participate in the forming of the
new, cluster granulation. They also take part in the granulation of atoms
inside clusters. Consequently, the quantity of ΔVc must
be proportionate to the relation of the number of atoms located on cluster
‘surface’ n, to the aggregate number of atoms in a cluster nc,
or
DVc = - kcompn’
/ 2 nc, (137)
where the coefficient of 2 allows for the fact that atoms located on
cluster ‘surface’ equally participate in the two mentioned types of
granulation inside liquid; kcomp is the coefficient of
compactness characterizing the change of volume at the transition from a
certain given granulation to a more compact one.
Out of expression (137), considering the value of Са from (46), we derive:
DVc = - kcomp
Са
(138)
On the basis of /101/ we obtain kcomp=
0.0; 0.06; 0.217; 0.4 for face-centered cubic, body-centered cubic, simple
cubic and cubic diamond granulation types correspondingly.
The aggregate relative change of
metal volume at melting and crystallization is found by the algebraic summation
of expressions (136) and (138). We obtain
DV = DVspl + DVc = a Са – kcomp Са = Са (a - kcomp)
If we express it as a percentage, as it is accepted,
we arrive at
DV = Са (a - kcomp)
100% (139)
Expression (139) relates the change in the volume of
metals at melting to the parameters of the elements of matter and space in
liquid – the width of spatial elements a and the factor of cluster re-granulation kcomp.
The values of DV found on the basis of (141) in
comparison with experimental data are listed in Table 17.
Table 17 The Change in the Volume of Metals at
Melting and Crystallization (Shrinkage)
Metal |
g, erg/sq.cm. /2,15, 20/ |
Е, kg/sq.mm. /10, 101/ |
a, calculation by (35) |
DVcomp,
%, calculation by (39) |
DVc, % calculation by (139) |
DV, % calculation by (139) |
DV, % experim. data /1,2,98/ |
Cu |
1133 |
11200 |
0.19 |
4.85 |
0 |
4.85 |
4.33-5.30 |
Ag |
927 |
7700 |
0.205 |
4.70 |
0 |
4.70 |
3.8-5.40 |
Au |
1350 |
11000 |
0.226 |
4.95 |
0 |
4.95 |
5.1-5.47 |
Pt |
1800 |
15400 |
0.205 |
5.7 |
0 |
5.7 |
no data |
Pd |
1500 |
11900 |
0.214 |
4.08 |
0 |
4.08 |
no data |
Al |
914 |
5500 |
0.24 |
5.30 |
0 |
5.30 |
6.0-7.14 |
Pb |
423 |
1820 |
0.26 |
4.15 |
0 |
4.15 |
3.5-3.56 |
Ni |
1825 |
21000 |
0.183 |
5.10 |
0 |
5.10 |
4.5-6.34 |
Co |
1890 |
21000 |
0.185 |
4.56 |
0 |
4.56 |
3.5-5.69 |
Zn |
770 |
13000 |
0.145 |
5.47 |
0 |
5.47 |
4.08-4.20 |
Feg |
1835 |
20000 |
0.177 |
4.84 |
0 |
4.84 |
2.8-3.58 |
Fed |
1835 |
13200 |
0.227 |
5.1 |
-1.64 |
3.46 |
2.8-3.58 |
Sn |
770 |
4150 |
0.248 |
3.3 |
-0.78 |
2.52 |
2.6-3.0 |
Cs |
68 |
175 |
0.27 |
4.3 |
-1.9 |
2.4 |
2.60 |
Ta |
2400 |
19000 |
0.21 |
3.46 |
-1.18 |
2.28 |
- |
Mo |
2250 |
35000 |
0.153 |
4.27 |
-2.04 |
2.23 |
- |
Nb |
1900 |
16000 |
0.204 |
3.93 |
-1.38 |
2.55 |
- |
W |
2300 |
35000 |
0.155 |
3.59 |
-1.59 |
2.0 |
- |
Bi |
3900 |
2550 |
0.207 |
3.7 |
-7.9 |
-4.2 |
-3.35 |
Ga |
735 |
- |
0.20 |
1.33 |
-2.82 |
-1.49 |
-3.2 |
Li |
377 |
1120 |
0.30 |
4.2 |
-2.1 |
2.1 |
1.65 |
Na |
171 |
530 |
0.45 |
4.6 |
-2.0 |
2.6 |
2.5 |
K |
91 |
460 |
0.38 |
4.0 |
-1.8 |
2.2 |
2.55 |
Rb |
754 |
235 |
0.31 |
4.7 |
-2.1 |
2.6 |
2.5 |
Mg |
728 |
2650 |
0.25 |
3.9 |
- |
3.9 |
3.05 |
As we see, the convergence of calculation and
experimental data is quite precise for a wide range of metals. We should note
that the entire data were obtained on the basis of theoretical assumptions for
the first time. It is for the first time, too, that the quantities of volume
change for ‘regular’, as well as the so-called anomalous metals – gallium and
bismuth – were calculated on a single basis. It was shown that their seemingly
anomalous behavior does not in the least differ from the behavior of all the
other metals in the aspect of volume change, obeying the same regularities.
In particular, the factor of
cluster re-granulation contributes much to the change of volume of anomalous
metals – bismuth, gallium, stibium, as well as silicon, water and some other
substances – at melting. In the indicated substances, the given factor prevails
over the factor of the forming of intercluster splits, which causes a visible
volume decrease at melting.
In practice, the change of metal volume at
crystallization that was calculated above leads to the forming of shrinkage
cavities and blisters in castings.
The process of forming shrinkage
cavities and blisters consists in the following: at crystallization, separate
submicroscopic intercluster splits – the elements of space in liquid state –
unite by the same cluster reactions scheme (19) as clusters accrete at melting.
The complete crystallization scheme if we allow for
the elements of space participating in it at the corresponding level is
presented as
(an + b) + (an + b) ® 2an + 2b;
(2an + 2b) + (an + b) ® 3an + 3b;
......................................................
(ian + ib) + (an + b) ® (i + 1)an + (i + 1)b,
(140)
where b is a single intercluster split (a single spatial element in liquid
metals); (i + 1)b is a shrinkage cavity or pore formed by the merger of (i + 1)
single elements of space.
At crystallization, the elements
of space can partially escape to the ambient space. It lowers the level of
liquid metal in a casting, yet no cavity formation takes place inside the
casting.
However, after the forming of a hard skin on the
surface of the casting, the evolving of the elements of space inside the
remaining amount of liquid metal results in the formation of hollows presented
as shrinkage cavities and porosity.
Since shrinkage cavities are formed by way of spatial
elements (vacuum) merger, they possess the characteristics of vacuum, too. It
can be proved by the well-known experimental fact that the pressure inside such
hollows equals zero at the moment of their formation.
Certainly, gas or air may fill such hollows, which
does not change the vacuum nature of the latter.
It is important that shrinkage cavities can be located
in castings both in the vicinity of their formation site and at a distance from
it.
The distribution of shrinkage cavities inside a
casting is the result of joint (synergetic) action of subsidiary factors, such
as the competitive crystallization character, redistribution of the remaining
liquid inside the casting under the influence of pressure differential,
capillary forces and gravity.
The competitive nature of crystallization leads to the
distribution of shrinkage cavities on the casting section surface, so that it
replicates the distribution of crystals to some extent. Namely, pores, similar
to crystals, have the minimal dimensions in the casting surface vicinity. The
dimensions of shrinkage cavities, similar to crystalline dimensions, increase
toward the center of a casting.
The same dependency as is employed to evaluate the
dimensions of crystals can be applied to the evaluation of the average
dimensions of shrinkage cavities on the casting section surface, i.e.:
rcav = kcav U-1,
(141)
where U is crystallization rate.
Under the influence of gravity the
last remaining portions of liquid lower down, whereas hollows are
correspondingly displaced upwards. So shrinkage cavities acquire their maximum
dimensions in the central part of castings.
Thus, we infer that the major cause of shrinkage
cavities formation in the process of crystallization is the process of the
merging of single intercluster elements of space in liquid.
The mentioned process as such pertains to natural laws
and cannot be eliminated.
Therefore, practical measures directed at increasing
the density of castings should perforce have a compensatory or displacement
character – if shrinkage cannot be eliminated as such, it may be compensated at
one location and displaced to some other safer site – which is the basis of
applying risers, coolers and a series of other techniques to casting
technology.
The issues of alloy formation are usually related to
the diagrams of state. Really, the diagrams of binary alloys state give a
considerable amount of information on the structure of solid alloys.
However, the given book brings the
unexplored problems of structure and crystallization of liquid metals and
alloys into its preferential focus.
Therefore, we shall apply here a somewhat different
approach to the problems of alloy formation proceeding from the structure of
liquid alloys and the mechanisms of melting and crystallization processes.
Melting and crystallization refer to everyday repeated
processes of foundry practice. At the same time, these are the basic structure-
and property-forming foundry processes. Cast alloys, their structure and
properties are formed during each smelting. The supplement of finishing
additions, alloying elements, ligatures and modifiers relate to everyday
routine foundry procedures.
Still, notwithstanding the mentioned ordinariness, the
mechanism of admixture dissolution, as well as the formation and structure of
alloys in liquid state and the order of the process specified, its adequate
description are lacking in literature, except for the most general
thermodynamic description. Thermodynamics, though giving a general
(phenomenological) description to this or that phenomenon at macrolevel, is not
to describe – and does not describe, by nature – the mechanism of the given
phenomenon at the level adequate to this process. Structure is no concern of
thermodynamics.
At the same time, it is extremely important to foundry
practice to know the structural mechanism of cast alloys formation process – to
effectively control these processes, uppermost.
Let us analyze the process of alloy formation
beginning from the dissolution of alloying and other elements proceeding from
the concepts of liquid metals structure stated above.
The process of alloy formation is rather complicated;
it includes several dissimilar mechanisms, or formation stages.
The process of the dissolution of
ligatures and other admixtures within the liquid alloy base marks the first
stage of alloy formation. By its nature, the given process is identical with
that of melting, being described by the same cluster reactions scheme.
However, the process of alloy formation is essentially
different from melting in relation to the solubility temperature in this or
that medium, in the first place. As a rule, the specified temperature is
considerably lower than the melting temperature of the given admixture in its
pure form. Apart from this, the so-called contact phenomena, as well as the
processes of mass transfer, perform a significant function in the processes of
admixture dissolution and alloy formation.
In particular, these are contact phenomena that cause
a change in the solubility temperature in comparison with the melting
temperature of the given substance. For the first time, the role of contact
phenomena in the process of eutectic formation was scrutinized in V.M. Zalkin’s
book /131/. A different conception of contact melting that allows for the
interaction of material and spatial elements in the process of melting is
suggested there.
These are contact phenomena that differentiate the
melting and crystallization of many alloys from the melting and crystallization
of pure metals, to a certain degree. Moreover, contact phenomena participate in
the melting of alloys by sequencing prior to the processes of mass transfer.
So, let us view first the mechanism of contact
phenomena influence upon the melting of alloys. The mechanisms of the influence
of mass transfer of various kinds upon alloy formation will be analyzed later.
Let us recall that there exist two intersecting
processes taking place at the rise of temperature in solid metals, and not only
there, that result in melting. On the one hand, it is the familiar phenomenon
of the decrease of durability of all metals with a temperature rise. On the
other hand, it is the increase in concentration and pressure of vacancy gas in
solid metals. The operation of this mechanism was described above.
To investigate the mechanism of alloy formation, it is
important to consider that both the factor of durability and the factor of
vacancy gas pressure change at the contact between two metals, first within the
contact zone of various materials and phases.
At the developing of alloys, there are two or several
various metals, alloys or ligatures that are melting together.
I.e. the contact between various metals in the process
of their formation and dissolution characterizes the process of alloy
development.
At the beginning, let us regard how vacancy gas
pressure changes at the borderline between two different metals, and trace
its influence on the temperature of melting.
Let us note that metals may exchange atoms at a close
contact under diffusion laws. This is a well-known phenomenon. In a similar
way, various metals exchange vacancies at a close contact.
Each metal has its own intrinsic vacancy concentration
at a given temperature. Let us admit that two metals - A and B –
are contacting. One of them has an equilibrium vacancy concentration of CA,
the other one having the concentration of CB. Let us presume
that CA exceeds CB.
As a result, a difference in the density of vacancy
gas dC generates along the border of their
section, which acts as the motive force for the onset of the diffusion vacancy
exchange process. Vacancies will flow from the metal with the greater density
of vacancies to the metal with the lower density. The rate of this process
equals the rate of corpuscular diffusion. As a result, vacancy concentrations
within the boundary zone come to be equalized.
However, the number of vacancies becomes less than the
equilibrium amount in metal A within the contact zone, whereas it
exceeds the equilibrium amount in metal B.
Evidently, the melting temperature
of metal B within the zone of contact decreases under such conditions
proportionate to the difference of vacancy concentration
dC = CA
- CB
Still, let us add that the temperature of the melting
of the second metal within the contact zone must be rising due to vacancy
redistribution, but it does not always occur in reality. There often decrease
the melting temperatures of both the contacting metals. The cause of this
phenomenon will be considered later when discussing the role of Rebinder’s effect in
contact melting.
The contact zone depth is not
great measuring several mcm or even less, - yet the given zone may get renewed
owing to the mass exchange within the contact zone.
Or, for instance, metals A and B forming
a binary alloy could form a fine byturn structure with the thickness of layers A
and B approximating the thickness in the contact zone of each of the
mentioned metals A and B. Such a structure could melt by the
contact mechanism within the entire melt volume.
The hypothesis of such a structure seems too
far-fetched at first sight. However, we know that this is the very structure to
be typical of many eutectic alloys – a fine microstructure with the alternation
of layers or the zones of other form of metals A and B, or solid
solutions a and b.
In point of fact, all alloys are inhomogeneous by
their microstructure to a certain extent, though the ideal conditions for
contact melting are created only in eutectic fine structures with the
alterntion of the zones of two different phases at microlevel.
The rate of metal dissolution in the process of
melting deserves our special attention. It is commonly known, having been
corroborated by hundreds of researches, that the mutual dissolution of metals
goes at the rate of corpuscular diffusion. One of the well-known methods of
measuring diffusion coefficient - the rotating disk method - is based on this
phenomenon.
These widely known facts are frequently used as the
proof of the corpuscular mechanism of the process of metal dissolution and
melting.
However, the alternative is neglected here - melting
by cluster mechanism requires vacancies. The latter move within the metal at
the corpuscular diffusion rate. So the diffusion rates of dissolution processes
do not in the least withhold these processes from going according to cluster
mechanism.
This is an extra example how synergetic principles
work in the processes of alloy formation.
Namely, the given example demonstrates once again that
dissipative processes actually go according to the suggested scheme
simultaneously at all the possible levels of the given system with the use of
all possible mechanisms.
In the specified case, the cluster process of metal
dissolution goes by the corpuscular mechanism of vacancy diffusion.
Unfortunately, the excessive generality of synergetic
principles hampers their direct application. For instance, we cannot exactly
determine the respective level of the course of this or that process for the
phenomenon under our consideration.
We have to admit that every particular phenomenon
requires the correspondingly particular study, - the finding of the structural
or any other kind of hierarchy of the given phenomenon, foremost, after the
completion of which synergetic principles are to be applied to the found
hierarchy.
Thus, synergetic principles can be generally applied
to practice to explain the already-found phenomena, which is also very
important, though.
At present let us analyze the influence of the durability
decrease factor within the zone of various metals contact upon the
formation of alloys and contact melting.
The so-called Rebinder’s effect is widely
acknowledged in physics. It consists in multiple decrease of solid metals
durability while they are contacting with liquid metals and some other liquids.
The same situation arises at the introduction of any
other additions into the crucible with liquid metal, which noticeably
facilitates mulling and the dissolution of any additions in the process of
alloy melting later on.
The mechanism of Rebinder’s effect is not unveiled so
far, but it is quite probable that it relates to moistening and solubility. We
may suppose that Rebinder’s
effect is connected with the diffusion of spatial
elements from liquid contacting substance into solid metal. The elements of
space of all kinds can diffuse in the same way as the elements of matter. Such
a diffusion of microhollows has been explored long since by the example of
vacancy diffusion.
Incorporating by way of diffusion into the surface
zone of solid metal, intercluster splits sharply depreciate the durability of
this zone, generating numerous flickering intercluster splits within it, which
act as microcracks reducing the durability of solid substance contact layer to
the durability of liquid, i.e. almost to zero. The sum durability of solid
metal decreases in this connection.
Consequently, in correspondence with Rebinder’s effect, the
durability of solid admixtures sharply decreases in foundry melting furnaces.
Whereas, in accordance with modern theory of melting, durability reduction
causes, in turn, the inevitable decrease in the melting temperature of the
given solid metal.
Thus, if the contact redistribution of vacancies can
lower the melting temperature of the given metal while increasing the melting
temperature of the other contacting metal, Rebinder’s effect reduces
to zero the possible melting temperature rise of the other metal. As a result,
the melting temperature of both the contacting metals may lower down within the
zone of their contact.
Now it is time to survey the process of contact
melting successively upon the whole.
Contact melting starts from vacancy redistribution and
the lowering of the melting temperature of the metal the vacancy concentration
of which increases as a result of such redistribution. The condition of such
vacancy redistribution is the initial perceptible difference in vacancy
concentration within contacting bodies.
However, after the forming of liquid phase the
Rebinder’s effect mechanism may start acting toward the metal that remains
solid. As a result, the durability of the given metal reduces on the contact
surface. Durability decrease in this metal, while vacancy gas pressure remains
the same, lowers its melting temperature within the contact zone according to
Rebinder’s effect /132/.
Finally, the melting temperature of such an alloy may
be less than the melting temperature of both its components.
We are familiar with such alloys - these are eutectic
alloys.
In alloys of other types - in monophase alloys, for
example - contact mechanism does not operate fully, so their melting
temperature is always higher than the melting temperature of the
low-melting-point component.
It is time to relegate the simplified and incorrect
concept of liquid alloys as a homogeneous atom mixture to the past. We know
that the structure of liquid and solid alloys is hereditarily bounded. We also
know that liquid metals have a microinhomogeneous cluster-vacuum structure,
where clusters are atomic microgroups with the proximate order similar to that
of solid state, whereas the elements of space are represented by intercluster
bond splits possessing the characteristics of vacuum.
At the same time, clusters are not
microcrystals, not the remainders of solid phase in liquid – these are the
structural elements of matter in liquid state.
It means that liquid alloys consist of clusters within
the entire temperature range of the existence of liquid – starting from the
melting temperature and ending in the temperature of evaporation. Except for
clusters, flickering intercluster splits perform the function tantamount to
that of clusters but qualitatively distinct from it.
Medley eclectic ideas on liquid metals and alloys
consisting of clusters and separate atoms at the same time are propagated in
scientific literature. In such a ‘raisin pudding’ structure, as A.Ubbelode
termed it, separate clusters seem to flow within a homogeneous mix of separate
atoms /1,2/.
Other authors assume that liquid metals and alloys
consist of clusters before the liquid reaches a certain temperature, after
which the given liquid passes into a purely corpuscular structure.
These approaches are erroneous, since they contradict
the quantum theory postulate of the indistinguishability of quantum objects,
atoms including. In application to liquid metals the specified postulate
signifies that all atoms must either enter into the composition of clusters or
the entire liquid must be monatomic. The simultaneous existence of various
atomic states is prohibited. The transition of liquid into monatomic state is impossible
in our theory, for clusters and only clusters are the prevalent elements of
matter in liquid.
The latent elements of the adjacent aggregation states
in liquid or any other aggregation state, as it was demonstrated above, do not
segregate from the predominant structural elements. For example, the latent
elements of matter in gaseous state – activated atoms – necessarily enter into
the composition of clusters. The latent elements of crystalline structure
represented by the proximate order also enter into cluster composition, as well
as vacancies.
I.e. the latent elements of matter and space do not
form phases of their own in any state.
Similar to that, iron in solid state may exist either
as ferrite or austenite, but it cannot exist in both the forms simultaneously
within a wide temperature range. There is the same prohibition acting here.
Thermodynamics furnishes an analogue of this
prohibition as the familiar Gibbs’ phase principle.
Since there exists proximate order in clusters, there
may occur its changes analogous to polymorphous transitions, if we touch upon
polymorphous transitions in liquid metals, but with the conservation of
clusters.
In the process of their formation, all alloys go through their
mixing stage that supersedes melting.
If a homogeneous substance is melting, there are formed clusters of
the same type.
If two or more substances or a composite monophase alloy are melting
or dissolving, clusters of various types get formed at melting. In liquid
state, clusters of various types are exposed to opposite forces.
Some of them aim at dividing heterogeneous clusters – these are
gravity and the forces of interaction between like clusters.
Other forces tend to uniformly intermix all clusters with the
forming of a homogeneous cluster mix. These are intermixing forces, including
the processes of natural and artificial convection, as well as corpuscular and
cluster diffusion.
Each of these forces has its specificities, its particular sphere of
influencing alloy formation.
Corpuscular diffusion prevails at short distances, for instance, at
the atomic exchange between clusters and the redistribution of atoms inside
clusters. The peculiarity of the process of corpuscular diffusion lies in the
relative slowness of this process. The typical value of the corpuscular
diffusion coefficient in liquid metals amounts to 10-7sq.cm / sec.
approximately (see the above part dealing with diffusion). This is quite
sufficient for the exchanging of atoms between neighboring clusters and inside
them.
However, it was shown above that a much faster cluster diffusion
mechanism that provides cluster intermixing operates in liquid metals, too. The
coefficient of cluster diffusion in liquid metals in the vicinity of the
melting point comes to 10-5 – 10-3 sq.cm /sec.
Cluster diffusion ensures the mixing and transfer of clusters within
relatively thin layers of liquid, for example, in contact layers and the narrow
zone directly by the crystallization front. Yet calculations show that cluster
diffusion cannot provide the homogenizing of alloy composition within the
entire volume of the melting crucible during a short melting period.
Macroscopic melts intermixing occurs due to natural and artificial
convection.
Alloys with the reciprocal solubility of their components go through
an extra formation stage – that of atomic diffusive mixing. At this stage, the
activated atoms of substance A penetrate into the clusters of substance B
and v.v. with the forming of mixed composition clusters AmBn.
At the formation of chemical compounds the alloy actually dissociates into two.
In contrast to the stages of melting, cluster and convection mixing,
the stage of atomic diffusive mixing does not affect all alloys.
For example, the given stage is atypical of the eutectic type
alloys, yet it is of extreme importance to alloys with the complete or partial
solubility of their components in solid state, or for alloys which incorporate
chemical compounds. For instance, this is the intercluster diffusion stage that
determines the possibility or impossibility of the removal of certain
admixtures present within clusters. It concerns many practical occurrences,
e.g. the case of extracting iron admixtures from aluminum alloys. Entering into
the cluster composition of a chemical intermetallic compound – ferrous
aluminide, the atoms of iron react with admixtures but sluggishly, so iron is
hard to extract from such alloys. It is necessary to decompose iron-containing
clusters to extract it, which is possible, for instance, at a considerable
overheating of alloys.
The forming of alloys with the maximum homogeneity should not be
carried out unless we allow for the stage of intercluster diffusion. It must be
taken into consideration that this is the slowest stage of alloy formation
process in liquid state that requires high temperatures.
Thus, the mechanism of alloy formation is complex enough, so the
structure of liquid alloys can be very complicated and multiform similarly to
that of solid alloys.
Upon the whole, we do not examine the stage of atomic diffusive
mixing here in detail, since the stage specified is reflected in well-known
works on alloy formation most fully and thoroughly.
The splitting of alloy formation process into four stages is of
course conventional in the sense that these processes can occur and do occur
simultaneously in the actual process of melting – but at dissimilar dimensional
levels. Nevertheless, such a distinction proves useful for the analysis and
understanding of cast alloys formation processes, their heredity, different
types of alloys and different types of diagrams of their state, as well as the
distinction between their characteristics, etc.
Natural and artificial convection provides the
homogenizing of alloy composition on a macroscale, on the scale of the smelting
furnace crucible, for example. This is a very powerful alloy formation
mechanism underestimated thus far.
The degree of the development of
regular gravity convection is proportionate to Rayleigh criterion and the cube
of the size of a sample, so convection works within the entire volume of
melting crucibles, ensuring the homogenization of alloy composition at
macrolevel. The larger the melting crucible is, the greater are convection
forces that operate within it. Thus, it is convection that ensures alloying in
production quantities.
However, convection may ensure alloy homogeneity at
macrolevel only. The homogeneity of alloy composition at microlevel is provided
by cluster diffusion, while corpuscular diffusion secures homogeneity at
lattice, inertcluster and, correspondingly, intercrystal level. Therefore, only
the cumulative simultaneous operation of various mechanisms provides the
forming of high-grade alloys. It also serves as an example of synergetic and
metallurgy laws working.
At the same time, special experiments have shown that
gravity effect essentially prevails over diffusion when convection is lacking.
It signifies that thin alloy samples under earth conditions tend to
stratification by the density of their components: heavy clusters lower down,
whereas those with the lesser density rise to the surface /82/.
So, if it was not for convection and intermixing, we
could never obtain more or less homogeneous alloys under earth conditions at
all. It also means that the decisive role in alloy formation processes belongs
to natural and artificial convection neglected till now as far as alloy formation
is concerned, rather than diffusion, as it is usually assumed.
Thus, under the influence of cluster diffusion and
convection of various kinds heterogeneous clusters intermix in liquid alloys,
while gravity hampers this process as much as possible.
No sooner does convection stop, than any alloy starts
segregating density-wise fast or slowly within gravitational field. In a series
of alloys with a considerable difference of the densities of their components
gravity effect is noticeable even when convection is present. For instance,
leaded bronzes segregate actively in liquid state in ordinary smelting
furnaces, which is well known to all practicists. Some other alloys behave
similarly to that.
We affirm here that all alloys without any exception
behave in the same way, yet segregation processes go very slowly sometimes so
natural convection successfully hinders them under regular melting conditions.
Therefore, we introduced the concept of thermal
kinetic processing of melts to be distinguished from thermal temporal and high
speed thermal treatment, which are based upon the using of time delay at a
definite temperature as well as the definite rate of the cooling of the melt
correspondingly.
Thermal kinetic processing includes the mentioned
factors, but the core of this process consists in using the preset modes of the
mixing of the melt in the process of its formation. Thermal kinetic processing
allows for both the intermixing degree and the degree of turbulence, as well as
the scope of turbulence at different dimensional levels of the melt. Thermal
kinetic processing lets obtain very homogeneous alloys the characteristics of
which prove to be highly stable.
As it was mentioned, gravity hampers the forming of homogeneous
alloys upon the whole and promotes their complete or partial segregation by the
density of their components.
Interestingly, the processes of alloy gravity or centrifugal forces
segregation may be normally applied to practice for the forming of castings
with the preset inhomogeneous structure. For example, there may be created a
smooth transition from the zone of gray iron to that of steel within one
casting at centrifuging in the process of slow hardening.
We got castings repeatedly from different alloys, silumins
including, with a stepless set of various structures and compositions according
to ingot height from hypoeutectic to deeply hypereutectic alloys by way of a
continuous holding of samples of originally homogeneous alloys within gravitational
field while convection is being specially suppressed.
It follows from the aforesaid, too, that the diagrams of alloys
state constitute quite a relative picture, so we are to accustom ourselves to
the fact that the whole set of all possible compositions and structures may be
simultaneously present in a casting under definite conditions.
Let us label such castings as variable structure castings. Apropos,
there are no physical prohibitions of getting variable structure castings. The
forming of castings with a controlled variable structure is possible even now,
yet the know-how of their production is to include the holding of a casting in
liquid state within gravitational field or the centrifugal force during the
period necessary for alloy segregation under the conditions of suppressed
convection.
The specified time period is always individual depending on a series
of conditions. Nevertheless, this is a real time amounting to minutes – or tens
of minutes at the utmost.
We should note that the production of insufficiently homogeneous
alloys is quite feasible if convection is underdeveloped. It is possible in
small-sized samples or in large furnaces, the inside temperature being too low
and the time of melting insufficient. Under such conditions the melt gets
inhomogeneous to a varying extent, ‘spotty’ by its composition and
submicrostructure.
In principle, the extent of such inhomogeneity spots within alloy
structure can be arbitrary. In practice, cast alloys and castings get ‘spotty’
very often – rather almost always – with the locally inhomogeneous structure
and properties. As a rule, it adversely affects casting characteristics.
For example, in cupola heat cast irons there may occur zones with
both their composition and structure varying because of the low temperature of
the process within the same casting. The process of segregation, as well as the
inhomogeneity of hardening, is usually listed among the causes of this
phenomenon.
Such homogeneity frequently acts in reality as a result of an
incomplete intermixing of alloy components in liquid state at cluster stage.
Such an alloy may be considered undersmelted.
The uncontrollable ‘spotty’ structure is one of the main causes of
inexplicable fluctuations of casting characteristics from melting to melting or
even within the same melting, which is familiar to practical experts. This
cannot be detected by regular methods of chemical or microstructure analysis.
Only the procedure of the micro-X-ray spectrometry analysis can be applied to
detect it at the corresponding dimensional level.
However, ‘spottiness’ may sometimes result not only in the worsening
but also in the betterment of certain service properties of metal in castings.
Therefore, the study of the conditions for the forming of local
inhomogeneity, or ‘spottiness’, of metal structure at different levels is one
of new trends in alloy research.
However, ‘spottiness’ may sometimes result not only in worsening but
also in the betterment of certain service properties of metal in castings.
Therefore, the study of the conditions for the forming of local
inhomogeneity, or ‘spottiness’, of metal structure at different levels is one
of new trends of alloy research.
The obtaining of controlled local inhomogeneity in alloy structure
is also possible even at present. In this connection we worked out the method
of the forming of cast alloys by cold emulsification. This method does not make
use of the regular ‘hot’ alloying while applying the emulsification of alloying
elements in the melt at low temperatures or in solid-liquid state. As a result,
there can be obtained some alloys with the controlled ‘spotty’ structure and
high mechanical qualities.
This is also a new trend of cast alloys production, which is probable
to find its expanded application in the coming century.
The above-said underlines the special significance of the observance
of technological thermal temporal and, particularly, thermal kinetic melting
mode in practice for the getting of stable homogeneous alloys, imparting the
physical meaning of a means of attaining such homogeneity into the mode
specified.
Alloy theory includes issues of practical importance
that abound in unsolved questions.
One of them concerns the causes of
alloys hardening within a temperature interval.
By way of illustration, let us consider an alloy with
the unrestricted solubility of its components in liquid and solid state, e.g.
copper-nickel. If such an alloy consists of a homogeneous mix of atoms in
liquid state and in solid state, why then these atoms separate intensively at
crystallization to later form a homogeneous solution anew?
The key to the problem analyzed relates to alloy
inhomogeneity in liquid state. There is a set of clusters with various
compositions present in the alloy. So, when crystallization sets in, clusters
containing a large amount of refractory element atoms get crystallized in the
first place, and v.v.
I.e. the existence of alloy hardening interval is a corollary
to the inhomogeneity of liquid alloy structure at cluster level, namely, the
simultaneous existence of clusters with dissimilar compositions in liquid
alloys. Clusters separate at crystallization, dividing on the basis of their
similarity or difference. Clusters that are similar by composition crystallize
together, under similar conditions, at the same temperature, in particular.
Thus, this is the existence of a set of variable
composition clusters within the temperature interval in liquid alloys that is
the primary cause and the motive force of selective crystallization process.
The continuous changing of cluster composition creates the effect of continuous
crystallization within the temperature range of liquidus-solidus. However, the
given process is discontinuous step-like at cluster level.
Under the condition if updated precision measurers are
applied, the given gradation of crystallization within the interval of
hardening can be detected. Even the measuring instruments that are currently in
use are able to register a spotty solid phase separation within the
crystallization interval of certain alloys. It denotes the possibility of the
existence of a discontinuous character of cluster composition change within the
liquid melt - up to its becoming step-like.
Alloys with the unrestricted solubility of elements in solid and
liquid state form a liquid structure consisting of a set of variable
composition clusters that represent the elements of matter, with a continuous
change of this composition in accordance with the diagram of state.
Correspondingly, such alloys have a definite set of transition elements both of
matter and space.
The structural formula of such
alloys structure is:
S = S a(am + bn),
(142)
where S is alloy composition; a and b are alloy components; m
and n are the variable portions of the atoms of a and b in clusters. The
quantities of m and n enter into the following correlation:
m + n = 1.
Corpuscular diffusion is highly
important in the formation of such alloys, along with cluster diffusion and
convection. It is corpuscular diffusion that provides the atomic interchange
between clusters which differ by their composition, as well as the penetration
of atoms inside clusters if there exist sufficient power of bonds between
heterogeneous atoms.
The existence of a stepless
cluster composition change and a smooth change in heat and solid phase evolving
within the interval of hardening is prognosticated in such melts.
On the one hand, small cluster dimensions facilitate
the penetration of admixture atoms into clusters; on the other hand,
neighboring order distortions and the corresponding submicrotensions always
arise in small clusters at the penetration of the atoms of other types, which
extrudes admixture atoms. Small cluster dimensions also promote a fast cluster
clearing of admixture atoms that create tensions inside clusters. We may say
that clusters are capable of self-purification from admixture atoms.
Therefore, as it was affirmed
above, admixture atoms can penetrate into clusters with a different composition
in liquid metals only with the presence of the sufficient physical affinity, or
at the forming of chemical bonds that are more durable than one-type atomic
bonds A-A or B-B.
In many cases admixture solubility in clusters is lower
than in a solid crystal. The increase of admixture solubility in liquid state
is achieved by admixture forming its own clusters, or through the locating of
admixture atoms on cluster ‘surface’, within the zone of activated atoms. The
given question was discussed earlier, too, in connection with admixture
diffusion.
Thus, admixture solubility in clusters cannot exceed
that in a solid crystal. Really, the solubility in solid state is rather a
complicated concept, too. It was demonstrated by many authors that admixtures
frequently concentrate along the boundaries of grains in solids, as well as at
dislocations and other defects. So the change of the average admixture
concentration in a solid crystal does not at all signify that the figure of
this change coincides with admixture solubility in the ideal crystalline
lattice of the given type.
Similarly to that, the aggregate solubility in liquid
state is also a composite quantity compounding of several parameters. The
presence of intercluster splits of vacuum nature in liquid alloys facilitates
corpuscular diffusion by the mechanism of activated atoms migration inside
them, while the huge internal area of the surface of spatial elements zone lets
a far greater amount of admixture atoms occupy cluster boundaries. It is one of
the causes of the increase of admixture solubility in liquid metals.
Thus, alloys with the restricted solubility of
elements in solid and liquid state possess a complex composition, where the
following three elements are necessarily present:
1.
clusters with the mixed inner composition of the
solid solution type a(am + bn);
2.
clusters with the composition of pure components
(or one of the components) aa and ab;
3.
clusters, ‘covered’ or ‘separated’ by individual
admixture atoms or monatomic admixture layers. For example, v. the clusters of
one of the original components aa, ‘covered’ with individual activated admixture atoms b: b(aa)b.
Let us conditionally label such clusters as clad.
Let us emphasize that admixture atoms b also enter
into cluster composition aa in the latter case, but with a
peculiar location on cluster ‘surface’, analogously to the arrangement of
certain admixtures along the boundaries of mosaic blocs and other structural
defects in solid metals. In liquid metals, in connection with a huge amount of
spatial elements in their structure, clusters formed in such a way may
constitute a considerable part of their total quantity.
The structural formula of the alloys of such a type is
presented as:
S = S { aa + ab + a(am +bn) + b(aa )b} (143)
Such alloys are the most
complicated by the composition of their elements of matter and space. Both the
step-like and continuous change of cluster composition is possible within them.
Correspondingly, the zones of more or less continuous heat evolving and solid
phase alternate with the areas of singularity - the departure form the
continuous course of hardening – within the hardening interval of such alloys.
The problems of the forming of
liquid eutectic alloys were broached earlier in Part 7.2. It was demonstrated
that contact phenomena play a significant part in their formation. Still,
contact phenomena affect but some of the parameters of alloy melting and crystallization
without essentially changing their structure.
Liquid eutectics refer to alloys
that do not possess mutual solubility in solid state, or have the restricted
solubility in solid state and the unrestricted solubility in liquid state.
They are characterized by the eutectic melting
temperature below the temperature of melting of both the basic components of a
binary alloy. In solid state, dispersed microstructure is inherent in
eutectics, where the dispersed elements of metals A and B alternate according
to a definite order.
In this connection, solid eutectics have been
classified long since as peculiar mixes. K.P.Bunin and his followers refer
liquid eutectics to mixes, too /59-61/.
From the viewpoint of the theory under development,
liquid eutectics differ from other alloys only by the extreme degree of cluster
composition inhomogeneity in liquid state. If we observe a continuous change of
cluster composition in other liquid alloys or a multi-step fractional change of
such a composition, liquid eutectics may have but two steps of cluster
composition at the utmost.
Liquid eutectics represent cluster mixes of the
elements of A and B or their solid solutions, where the interaction between the
like clusters AA and BB is higher than the interelement interaction
AB. It means that clusters A and B mix reluctantly, not
spontaneously.
The latest experiments have also shown that many, or
possibly even all the liquid eutectics, are unstable when convection is lacking
and tend to segregation by the original elements within gravitational field
(see below).
For the production of liquid eutectic mixes of
heterogeneous clusters that interact reluctantly, a certain amount of energy is
to be spent.
Usually natural or artificial convection suffices for
the forming of such mixes. Corpuscular diffusion does not actually participate
in the forming of liquid eutectics, since they are lacking in the atomic
exchange between clusters A and B.
The structural formula of liquid eutectics for the
case of the complete lack of mutual solubility in liquid state is
S = S {aa + ab}, (144)
where aa and ab are the respective clusters consisting of the atoms of A or B
elements only.
Such a formula of structure is
incomplete in the sense that it does not reflect the bonds between the clusters
of different types. Such bonds must be flickering by liquid state nature, i.e.
they must arise at the approximation of the neighboring dissimilar clusters,
splitting at their separation in the process of heat oscillations.
Such bonds must be of intermetallic nature without the
forming of permanent-type intermetallic compounds.
Evidently, more detailed researches of liquid
eutectics can supply more extensive data on the flickering intermetallic bonds
of such a type. Similar bonds exist in solid eutectics, too, as stable bonds
between the elements of their microstructure, yet these bonds remain
underexplored so far.
At the same time, the presence of the flickering
AB-type bonds in liquid eutectics acts as the factor securing their relative
stability and technical utilization possibility. At the complete lack or
weakness of such bonds the alloy simply segregates into two liquids.
The existence of such relatively weak bonds in liquid
eutectics between dissimilar clusters facilitates the displacement or shift of
these dissimilar clusters relative to one another. It accounts for the widely
known fact of a higher fluidity of eutectic alloys in comparison with other
alloy types.
So liquid eutectic alloys have a high fluidity due to
the heterogeneous cluster bonds that they contain being weaker than the bonds
between similar clusters.
We may say that this is the instability of liquid
eutectics that imparts a higher fluidity to them.
Liquid alloys with a peritectic structure have a
structure similar to eutectics. The structural formula of the former coincides
with formula (145) with the distinction that there can be several peritectics
in the same system. Correspondingly, there can be several cluster types in liquid
peritectics. In this case, the structural formula of liquid peritectics may be
presented as:
S = S {aa + aab + ab}.
(145)
The dissimilarity between
eutectics and peritectics also consists in the different influence of contact
phenomena at the melting and crystallization of these two alloy types. For
instance, the difference in vacancy concentration between the components of
peritectics is considerably lower than it is between the constituents of eutectics,
and Rebinder’s effect works for one component only. As a result, the effect of
the melting temperature lowering is not so appreciable to the components of
peritectics as it is in eutectics, and it acts relative to only one alloy
component.
In eutectics, contact phenomena
perform the leading role, which is probably caused by the highest possible
difference in vacancy concentration at the melting points of metals
constituting the given eutectic, as well as the considerable quantity of
Rebinder’s effect in these metal vapors.
Eutectics differ by far from the source metals by a
series of parameters. In particular, the paramount distinction of eutectics is
the lowering of their melting temperature as compared with that of the alloy
components.
The improvement of important foundry properties such
as shrinkage diminution and fluidity increase is also characteristic of many
eutectics. I.e. the properties of eutectics change nonadditively to the content
of the elements of A and B within them. Eutectics, i.e. the mixes of two
different substances, behave as a certain new substance by a series of basic
parameters. It is caused by various factors.
Among the structural causes of these significant
changes of eutectic alloys properties at the level of the elements of liquid
state we may and are to single out the factor of cluster re-granulation in
liquid eutectics after their formation. The mentioned factor was viewed above
when analyzing the mechanism of metal volume change at melting and
crystallization.
Let us consider the role of the factor of cluster
re-granulation at the forming of eutectics. Eutectics represent a mix of two
types of clusters different by their composition. In turn, it means that the
dimensions and shape of the two given cluster types, as well as the dimensions
and shape of the spatial elements of the original liquid metals A and B are
different, too.
It is known that the mixes consisting of particles of
different dimensions can fill space more compactly than particles, e.g. balls,
of similar dimensions. To achieve this, the balls of dissimilar dimensions are
to occupy definite positions in space, by way of alternating according to a
definite pattern, for instance. Or smaller particles may fill the spacings
between larger ones.
We also know that such a distribution results, for
instance, from intermixing. In this case, a certain mutual configuration in
space is attained with the minimal volume of spatial elements, which explains
shrinkage diminution in eutectics at subsequent hardening.
Certainly, this is not a rigid construction, so it may
decompose under certain conditions, for example, at segregating within
gravitational field. We can state that the structure of liquid eutectics does
not only get formed but is also sustained owing to convection to a considerable
extent.
At the convective mixing of heterogeneous clusters A
and B the shape and dimensions of clusters do not change. It is only the
shape and dimensions of the spatial elements of liquid state that are subject
to changes, i.e. the shape and dimensions of intercluster splits.
Such a seemingly negligible change turns out to be
sufficient for a relatively unstable mix of two different substances to acquire
the properties of a certain third substance.
All that was stated above concerning the role of
cluster re-granulation and the role of the change of spatial elements –
intercluster splits, – relates, though to a different extent, to the forming of
alloys of other types, but the mentioned factors affect eutectics most.
The alloys of the iron-carbon system - steels and
cast irons - refer to prevailing industrial alloys. Cast irons are most widely
applied to foundry.
Thus, the study of liquid cast
irons structure in connection with the processes of crystallization and
structure formation seems worth making.
Iron-carbon alloys pertain to the alloys with the
restricted solubility of their elements in solid state and not quite definite
mutual solubility of their components in liquid state. Such indefiniteness is
caused by the fact that there were no successful attempts at deriving the alloy
of iron with carbon with the content of carbon higher than 25% at. in
connection with the necessity of obtaining a stable and precisely measurable
temperature for researches carried out at temperatures above 20000C.
However, this is not the sole speciality of such
alloys.
Iron-carbon alloys are distinguished by the feature
that one of their components - carbon - does not melt at all in its free form
and does not form liquid phase, thus presenting quite a rare, though not the
only one, exception among the elements of the periodic system /119-120/.
From the viewpoint of the melting theory that was
developed above, carbon in its most stable graphite form does not melt, because
its durability does not decrease with the rise of temperature, as it does in
case of the overwhelming majority of elements, but even increases to a certain
degree. The decrease of durability, as it was noted earlier, is one of the
requisite factors pre-starting melting.
Apart from this, solid-state carbon does not dissolve
within itself the elements of any other kind, practically. Rather a restricted
number of elements that form limitary carbon solutions, including iron, are
known.
At the same time, carbon readily reacts with many
elements, which results in carbide forming.
Carbon does not completely dissolve iron within itself
either, yet iron has a limited dissolvability area with carbon and also forms a
series of carbides with it, among which cementite usually occurs in cast iron.
The system of iron-carbon also refers to the alloys of
the eutectic type, i.e. it is characterized by a high degree of cluster
composition inhomogeneity in liquid state and a high degree of the irregularity
of heat and crystallizable phases evolving within the interval of hardening.
In the processes of the melting and crystallization of
cast iron, as well as it is in other alloys of the eutectic type, a significant
role belongs to contact phenomena – the diffusive vacancy redistribution at the
contact between iron and carbon and Rebinder’s effect, in the first place.
Let us consider the process of the
contact melting of iron in detail.
We shall proceed from the familiar fact that carbon
does not dissolve iron within itself. Consequently, the exchange of substance
between iron and carbon is one-way during the contact – it is only carbon that
can penetrate into iron, while iron cannot penetrate into solid carbon.
However, such prohibition does not work in case of the
exchange between the elements of space. Vacancies, or intercluster splits,
being flickering and having neither a stable shape nor stable dimensions, are
very plastic and easily adjustable to any substance. So the exchange of spatial
elements is also possible for the elements that do not exchange their elements
of matter.
Liquid iron, or liquid austenite, contacting with
solid graphite inclusions, serves as the source of all the elements of space
existing within carbon – vacancies, as well as intercluster splits.
Vacancies generate the inner vacancy gas pressure
within the contact zone of carbon, while intercluster splits diminish the
durability of the given layer. As a result, a typical cluster-melting situation
arises in the contact layer of carbon.
Carbon in the contact layer (within it only) melts
forcedly under the stated conditions with the forming of clusters and
intercluster splits of its own, which mix with the clusters of iron, or
austenite. Intercluster splits that exist between carbon clusters differ from
intercluster splits in liquid austenite. As a result, a counter exchange
between material and spatial elements takes place, now in the direction from
carbon to iron. Rebinder’s effect starts working relative to iron, too. Its
melting temperature also lowers down.
The re-granulation of clusters into a mutual mix
system occurs simultaneously, so the volume of the mix decreases. As a result,
the liquid eutectic mix shrinkage diminishes at subsequent crystallization.
There arise flickering bonds between the clusters of
iron and carbon that turn out to be weaker than iron-iron and carbon-carbon
bonds.
The decrease in the power of bonds within the system
leads to the reduction of internal friction (viscosity) and the increase in
cast iron fluidity in comparison with that of liquid iron.
The suggested description of the melting of cast iron
and the forming of its structure in liquid state, as well as any other description,
neither claims for exhaustiveness nor for the consideration of all the factors
that are possible in this connection. It qualitatively reflects only the
relative contribution of the material and spatial elements of liquid state to
some properties of liquid cast iron. Such an approach meets the accepted
relativity principle in the description of the changes concerning the
characteristics of metals and alloys at the level of material and spatial
elements interaction at aggregation state transitions.
The structure of liquid cast iron possesses all the properties of
liquid eutectics structure, having peculiarities of its own, though.
In particular, numerous experimental results of X-ray as well as
sedimentation tests bring the authors to the conclusion that carbon occurs in
liquid iron not only in the solution of clusters with the neighboring order
structure similar to that of austenite, but also as clusters with the
dimensions of 2.7…4.9nm at the temperatures approximating the temperature of
cast iron liquidus /16,17,30,37,55, 59,133,134/.
X-ray tests corroborate the presence of the compound Fe3C
in liquid cast iron, too.
There is a contradiction consisting in Fe3C being
unstable at high temperatures: at any sufficiently prolonged holding at
elevated temperatures in solid state it will inevitably disintegrate into
ferrite and graphite or austenite and graphite.
In principle, such disintegration must go considerably faster and
more completely than in solid state owing to the accelerated mass exchange
processes, but it does not take place.
It was demonstrated earlier that the flickering bonds between
heterogeneous clusters inevitably generate in liquid eutectics, and they are
weaker than the bonds between like clusters.
Therefore, we may assert that the compound Fe3C is
present within liquid cast iron as the flickering interatomic bonds between
austenite and graphite clusters. Graphite and cementite coexist simultaneously
in liquid state, yet graphite exists in cluster form that is stable for liquid
state, whereas Fe3C is present only as the flickering bonds between
graphite and austenite clusters, constantly arising and disappearing /140/.
It was stated above that such flickering bonds between heterogeneous
clusters are in principle characteristic of all liquid eutectics, as well as of
any alloys generally. The specificity of iron-carbon alloys consists in the
relative durability of the compound of such a type and the possibility of the
growth of such bonds at fast crystallization or at the presence of
carbide-stabilizing elements in the alloy.
It seems relevant to underline for further research that a close
contact between the clusters of austenite and graphite promotes the forming of
such bonds, while the formation of splits or the separation of dissimilar
clusters, on the contrary, hinders the formation of the given type of bonds.
The structural formula of liquid alloys of the iron-carbon system
may be presented as the follows /140/:
S =
S
{naa + mag}, (146)
where aa denotes austenite-like clusters; ag are graphite clusters; n is the number of austenite clusters
per unit of volume or gram-atom of liquid alloy; m is the number of graphite
clusters within the same volume.
The given expression relates both to liquid cast iron and liquid
steel. The difference lies only in the quantitative correlation between the
clusters of the two types.
Formula (146), like other similar formulas, does not allow for the
existence of spatial elements in liquid alloys alongside with the elements of
matter. It also ignores the presence of flickering bonds between similar and
dissimilar clusters.
Considering the special importance of the bonds of Fe3C
type in liquid cast irons, we can supplement formula (146) with the scheme of
cluster interaction:
S =
S {naa ®Fe3C ¬mag},
(147)
We are to take the instability of Fe3C bonds into our
consideration, - such bonds are unstable, have a flickering nature and
alternate with intercluster splits in time. This is reflected by the following
scheme:
aa ®Fe3C ¬ag + t/2,
aa « ag + t,
aa ®Fe3C ¬ag + 3t/2, (148)
......................... etc.,
where the symbol
of « represents
an intercluster split, t is the duration of one period of heat oscillations of a cluster.
Actually the time period of t, as it was already demonstrated, means the duration of the
existence of flickering intercluster bonds and flickering elements of space –
intercluster splits «.
The succession of cluster reactions (148) does not reflect the
structural formula of liquid cast iron upon the whole but the sequence of the
flickering bonds of the Fe3C type alternate between the clusters of
austenite and graphite and the flickering intercluster splits «.
Various alloying additions (Si, Mn, etc.) and undesirable admixtures
(S, P, etc.), as well as a series of uncontrollable admixtures, occur in the
structure of real cast irons.
In the structural aspect, some of them do not form clusters of their
own in liquid cast iron (Si, Mn and other elements) but enter into the
composition of austenite clusters. Other elements exist in the form of special
clusters (iron sulphide, phosphide eutectic and others), then, the third group
of elements may exist as activated atoms on the ‘surface’ of ‘clad’ clusters of
austenite or graphite (the same sulphur and phosphorus in small
concentrations).
Melting and
crystallization, as it was shown above, are partially reversible processes, so
these are mainly the processes inverse to melting that constitute
crystallization.
Contact processes perform a significant role at the crystallization
of liquid cast iron, the same as it is at melting.
Austenite clusters perform the function of the leading phase at the
crystallization of hypoeutectic cast irons and steels. They are the first to
form solid phase by the regular cluster scheme of crystallization. The
crystalline surface of solid austenite serves as the vacancy sink area for
graphite clusters contacting with this surface.
As a result of selective crystallization, graphite clusters are
forced back piling up at the boundaries of growing austenite crystals with the
forming of agglomerations. When vacancy concentration in graphite clusters
becomes lower than critical as a result of vacancy sink from graphite to
austenite clusters, graphite clusters start crystallizing, too, within the interdendritic
austenite spacings, as a rule, which reflects the weighty part of graphite
crystallization in the given system.
The accretion between graphite clusters is also accompanied by the
accretion of the elements of space characteristic of liquid state –
intercluster splits. While intergrowing, intercluster splits form shrinkage
microhollows at the boundaries of the growing crystals of graphite.
Such a scheme is peculiar to the regular relatively slow cast iron
crystallization with the forming of the structure of gray cast iron. Its
distinctive feature is the separation of clusters and intercluster spacings of
austenite and carbon in space in time.
Separation occurs on the basis of ‘like to like’ principle, for the
energy of flickering bonds between like clusters is higher than that between
dissimilar ones. Correspondingly, the flickering bonds of the Fe3C
type are replaced at the separation of clusters by iron-iron or carbon-carbon
bonds. However, the specified process requires time. If there is enough time,
graphite clusters have time to meet, accrete and separate from the surrounding
austenite by a vacuum layer that generates from the joined intercluster splits.
Therefore, the opinion that graphite performs the role of vacuum in
cast iron is incorrect. The bonds of iron-carbon actually disappear and get
replaced by microhollows at the forming of graphite insertions in cast iron at
crystallization.
That is the way of forming gray iron microstructure.
If there is not enough time to cluster separation or selective
crystallization, if crystallization goes too fast for that, then cluster mix
get crystallized as mix proper. There occurs no cluster separation. Such is the
basic distinction of the formation of white cast iron at the level of the
elements of matter and space (clusters and intercluster splits).
At crystallization, as it was pointed out earlier, clusters accrete
and their heat oscillations stop. If separation is lacking, both similar and
dissimilar clusters accrete forcedly, the clusters of austenite and graphite,
in particular. In this case, the flickering bonds between austenite and
graphite clusters of the Fe3C type become stable, since flickers
stop at crystallization.
As a result, an extremely nonequilibrium structure of alternating
graphite and austenite microzones is formed. The nonequilibrium of the derived
structure partially withdraws due to corpuscular diffusion, the redistribution
of carbon atoms.
Considering the extreme smallness of cluster dimensions – 1-10nm,
the process of diffusive redistribution of carbon goes during an extremely
short period of time (fractions of a second), so it can be caught by special
hardening experiments only with the cooling rate of millions of degrees per
second within the temperature interval from the temperature of melting to room
temperature. Such experiments are known, and there was registered the presence
of carbon microzones within the microstructure of iron hardened from its liquid
state.
Thus, white cast iron is originally crystallized as a mix of
austenite and graphite clusters. The original cementite generates right after
fast crystallization and not from liquid state but in solid state already due
to the fast redistribution of carbon atoms from the clusters of graphite into
the surrounding austenite.
Intercluster bonds of the Fe3C type registered in the
process of fast crystallization function as the nuclei of a new phase
–cementite - in this process, they accelerate and organize new phase growth
according to their pattern.
This is how the microstructure of white cast iron arises.
As we see, the same original structure of a liquid iron-carbon alloy
can generate structures differing in a drastic way as a result of
crystallization going at different rates.
The natural
crystalline structure of castings is distinguished by the pronounced
inhomogeneity of the dimensions of primary crystals along the section of a
casting. If no special measures are taken, there arise within the majority of
castings crystals that are not only dissimilar, but also too large by their
dimensions.
It promotes the generating of other kinds of inhomogeneities -
physical and chemical - in castings represented by shrinkage, segregation, etc.
The properties of castings differ within the zones of dissimilar
structures, too. The highest properties, homogeneous along the section of
castings, are normally obtained at the forming of a homogeneous and
fine-grained structure.
Moreover, we know that the smaller the dimensions of primary
crystals in castings are, the higher are a series of important service and
technological casting properties.
Therefore, most often casters aim at the forming of the
finest-grained and the most homogeneous casting structure.
Modifying is one among the most widespread means of attaining this
object. Casters understand modifying as the insertion of small quantities of
various additions into liquid metal before crystallization to achieve a
fine-grained structure of castings /65,74,75,135/.
What are the given additions requisite for?
As it was shown above, there is always a large number of nucleation
centers in castings and ingots – much more than the amount of crystals in a
final casting – owing to the mass nature of crystalline centers nucleation by a
spontaneous mechanism.
However, the structure of castings turns out to be coarse-grained
and inhomogeneous along the section of castings.
The cause of the zoning of crystalline castings structure, as it was
demonstrated, consists in the competitive character of crystal growing from the
melt with the two-phase zone being present.
If there is enough time for structure correlation, grosser crystals
absorb smaller ones, so the structure of a casting is gradually becoming more
and more coarse-grained as a result, while crystallization rate is decreasing.
Crystallization rate regularly decelerates in the direction from the surface of
the casting to its center, which brings about the zoning of the crystalline
structure of castings.
In accordance with current theory, spontaneous crystallization is
actually impossible, crystalline centers nucleation entails much difficulty,
the number of these centers is always insufficient, and so the leading role in
the crystallization of castings belongs to special additions – modifiers,
requisite for the multiplying of nucleation centers.
Our theory asserts that the process of spontaneous crystallization
in liquid metals goes naturally and without extra difficulties. The number of
nucleation centers is always redundant, exceeding by far the amount of crystals
in a casting. Thus, the role of modifiers is different, according to our
theory.
First, let us consider the issues that are general to both the new
and old modifying theory.
We also regard modifying as the introducing of additions that refine
grain dimensions in castings. Such additions are termed modifiers.
Correspondingly, the additions that reduce the number of grains in castings and
augment their dimensions are called demodifiers.
Modifiers are divided into modifiers of the first type dissoluble to
a different extent in the metallic base of the liquid alloy of the addition.
Modifiers of the second type are represented by the particles of refractory
substances insoluble in liquid alloy (at least during the process of
crystallization). The given theses remain constant in this theory.
New modifying theory provisions are stated below with the
consideration of the real material-spatial structure of liquid metals.
Let us analyze the operation of first-type modifiers from the
standpoint of thermodynamics.
The existent theory of modifying is based upon the thesis of work
expenditure necessity for the nucleation and growth of crystals. The
incorrectness of the suggested thesis was proved earlier in Part 6.
It was shown in Part 6.2 that flickering inner intercluster surfaces
saturate liquid. At the elementary act of crystallization by the reaction of an + an®
a2n two neighboring clusters accrete
into an elementary crystal and the section surface represented as a flickering
intercluster split closes between them.
Consequently, at crystallization going by cluster accretion these
are not only new surfaces that arise in liquid, but also the existent
flickering intercluster section surfaces that close, which is accompanied by
the evolving of crystallization heat, and not its expenditure, in complete
conformity with facts.
Then the change in the free energy of the system at the forming of
an elementary crystal (nucleation center) at spontaneous crystallization
according to (126) amounts to:
DF = -(4/3)pr3 DFv - 4pr2 s,
(149)
Graph (149) is visualized by curve 2 in Fig.17.
The latter expression signifies that energy is evolved but not
consumed at the forming of a nucleation center. Correspondingly, it also means
that the formation of nucleation centers does not require any work to be done
but, on the contrary, crystallization is thermodynamically expedient at any
crystalline dimensions.
As a result, fundamental changes in crystallization theory imply the
changes in the theory of modifying.
It is known that first- type modifiers refer to surface-active
substances that lower the surface tension s of liquid melt. The lowering of s is an experimental fact
/65,74,75,135/.
Let us denote the surface tension of liquid metal or alloy without
modifiers as s. Let us designate the surface tension of alloy with modifiers as sм.
By definition, s
>sм by the
absolute quantity.
Expression (126) in case if we introduce some modifier will be
presented as
DFм = -(4/3)pr3 DFv - 4pr2sм,
(150)
Subtracting (150) from (148), we obtain DFм - DF = - 4pr2 sм +4pr2 s
or DF - DFм = 4pr2
(s-sм) > 0 by the absolute quantity.
Consequently, the decrease of the free energy of the system lowers
by the absolute quantity at the crystallization of metals that contain
modifiers.
It ensues that the growth of crystals with first- type modifiers is
less thermodynamically expedient than spontaneous crystal growth.
Next, it follows that first- type modifiers hinder and retard the
nucleation and growth of crystals in comparison with spontaneous nucleation and
growth.
On the contrary, demodifiers increase the quantity of s, which results in the
facilitating and accelerating of crystalline growth.
Graphically, the influence of modifiers and demodifiers of the first
type upon the change of free energy at crystallization can be represented by
the three curves in Fig.18.
Curve 1 corresponds to the process of spontaneous crystallization;
curve 2 conforms to the process of crystallization with modifiers, curve 3
reflecting the nucleation and growth of crystals with the presence of
demodifiers.
What is the mechanism of the influence of modifiers upon crystalline
dimensions in the light of the above-said?
Modifiers, by hindering crystalline growth, hamper the process of
competitive crystallization, too, i.e. the process of small crystals accreting
with larger ones. A large number of small crystals that nucleated by the scheme
of spontaneous crystallization accrete with more difficulty, grow slower and
get a chance to survive in competitive activity with larger crystals. As a result,
a fine-grained primary crystalline structure is registered in a casting at the
same crystallization rate.
It also means that forced crystallization with modifiers does not
replace spontaneous crystallization, as it is assumed now. On the contrary, forced
crystallization with modifiers of the first type is less thermodynamically
expedient, less equilibrium than spontaneous crystallization. First- type
modifiers do not in the least facilitate the formation and growth of crystals,
as it is accepted. They hamper these processes, on the contrary. Still, by
hampering spontaneous crystallization processes, first- type modifiers provide
the refining of the primary crystalline structure of castings.
Such is the general thermodynamic mechanism of the influence of
first- type modifiers upon the dimensions of primary crystals in castings. We
shall underline that the specified general considerations do not reflect the
entire diversity of modifying. Therefore, the thermodynamic theory of modifying
is to be supplemented by other means at other levels.
The fundamentals
of the electron theory of modifying were laid by G.V.Samsonov,
V.K.Grigorovitch, Khoudokormov, Tiller and Takahashi, as well as others
/136,137/.
G.V.Samsonov worked out the concept of the donor-acceptor mechanism
of modifier and matrix interaction. Khoudokormov and Grigorovitch /138/
developed the ideas of the role of the bond type and the electron structure of
matter in the aspects of interaction and modifying. The given concepts are
widely employed and developed at present.
We shall assume after G.V.Samsonov that good modifiers are to be
free electron donors for liquid metal.
The ability of this or that substance to act as a free electron
donor in alloys is always relative, i.e. it is determined by the comparison
with the metal of a casting.
Work function /136/, electronegativity, after Gordy, or the relative
ionization potential, after V.M.Vozdvizhensky /139/, may characterize the
ability of the given substance to donate free electrons.
Our research showed that the two latter parameters characterize the
modifying ability in an approximately equal degree; however, the application of
the effective ionization potential proves more convenient in practice, after
V.M.Vozdvizhensky /139/.
All substances having a lesser quantity of electronegativity, or the
effective ionization potential Uef, than the metallic base of
the given alloy, will have a more or less modifying influence at
crystallization, i.e. they will deflate crystalline dimensions.
All substances having the quantity of Uef that
exceeds that of the metallic base of the alloy, will have a demodifying
influence at crystallization, i.e. they will promote the enlarging of the
primary crystalline structure.
It relates to the following specificity: the lower the ionization
potential quantity is, the easier it is for the substance to donate its valence
electrons, and v.v.
The degree of the modifying influence of this or that element can be
evaluated by the sign of the difference between the effective ionization matrix
and modifier potentials:Ume – Umod
If the given difference is above zero, i.e. positive, then the
specified element can act as a modifier. If this difference is below zero, the
element under consideration will be a demodifier of the first type. I.e.
Ume
– Umod > 0 – a modifier,
Ume
– Umod < 0 – a demodifier.
The second factor that characterizes the ability of some substance
to affect nucleation and the growth of crystals is the factor of admixture
solubility in the given matrix. A good modifier must locate along the
boundaries of crystals and clusters without entering into their composition.
I.e. a modifier or demodifier is to form clad clusters, where modifier atoms
are distributed between clusters.
A modifier is not to form clusters of its own, because a certain
modifier amount will not be located along cluster boundaries of the melt in
this case.
Correspondingly, the element possessing modifier characteristics
must have a low solubility in solid metal and a restricted solubility in liquid
metal.
Let us denote the factor of solubility as CS. We
shall conditionally assume that modifiers have the solubility in a hard matrix
of this or that alloy that does not exceed one percent: CS<1%.
Both the noted factors can be united in the following semi-empirical
formula for the calculation of the modifying activity of modifiers
(demodifiers) of the first type:
m = (Ume –
Umod)/ СS, (151)
m being the coefficient of modifying activity.
Expression (151) is very simple and convenient for calculations. The
quantities of CS are listed in reference books concerning diagrams
of state, the quantities of U are cited in literature, too /139/.
Expression (151) is also convenient for the reason that it allows to
clearly divide all elements into first- type modifiers or demodifiers. Namely, m < 0 for
demodifiers in accordance with (151), i.e. their modifying coefficient value is
subzero, while modifiers will have a plus value of the modifying coefficient.
The quantity of m has but a relative value according to (151) and serves for the
comparing of the modifying coefficients of various elements exclusively.
The values of the coefficient of m for various first- type modifiers and
demodifiers for liquid alloys on the basis of iron and aluminium are to be
found in Tables 18 and 19.
Table 18. The Coefficient of Modifying Activity for
Various Elements in Liquid Iron-Based Alloys (Modifiers of the First Type)
Element |
CS, % at. /119,120/ |
Umod, /139/ |
m, calculation by (150) |
First-Type
Demodifiers |
|||
Fe |
- |
3.00 |
0 |
Co |
50 |
3.11 |
-2.2 10-1
|
Ni |
50 |
3.20 |
-4.0 10-1
|
Ir |
50 |
3.26 |
-5.2 10-1
|
Pt |
50 |
3.34 |
-6.8 10-1 |
Pd |
50 |
3.45 |
-9.0 10-1 |
Mn |
50 |
3.66 |
-1.3 |
Ru |
29.5 |
3.45 |
-1.5 |
Zn |
7.00 |
3.17 |
-2.4 |
Re |
16.7 |
3.57 |
-3.4 |
Cr |
12 |
3.47 |
-3.9 |
Al |
1.55 |
3.14 |
-9.0 |
Mo |
1.60 |
3.29 |
-1.8 101
|
Ge |
4.00 |
3.27 |
-1.9 101 |
Si |
4.20 |
3.84 |
-2.0 101 |
C |
8.60 |
4.86 |
-2.1 101 |
Nb |
1.90 |
3.42 |
-2.2 101 |
Sn |
1.00 |
3.31 |
-3.1 101 |
V |
1.60 |
3.71 |
-4.4 101 |
Ta |
0.95 |
3.44 |
-4.5 101 |
W |
1.00 |
3.81 |
-8.1 101 |
P |
0.25 |
4.30 |
-5.2 102 |
O |
0.56 |
5.0-6.0 |
-5.0-8.0 102 |
S |
0.11 |
4.76 |
-1.6 103 |
F |
<1.0 10-4 |
>5.0 |
-1.0 (104-
105) |
Cl |
<1.0 10-4 |
>5.0 |
-1.0 (104-
105) |
Br |
<1.0 10-4 |
>4.0 |
-1.0 (103-
104) |
J |
<1.0 10-4 |
>4.0 |
-1.0 (103-
104) |
|
|
|
|
First-Type
Modifiers |
|||
|
|
|
|
Fe |
- |
3.0 |
0 |
Rh |
50 |
2.91 |
0.9 |
Cu |
7.5 |
2.56 |
5.9 |
Ti |
0.72 |
2.85 |
0.21 |
Zr |
0.5 |
2.87 |
26 |
Gd |
2.0 |
2.38 |
31 |
La |
0.2 |
2.15 |
420 |
Ce |
4.0 10-2 |
2.25 |
1900 |
Mg |
~0.01 |
2.42 |
~3000 |
Ca |
<0.02 |
1.86 |
>2000 |
Na |
<0.001 |
1.34 |
>10000 |
B |
~0.001 |
1.44 |
~1000 |
Sr |
~0.001 |
1.64 |
~1000 |
Y |
~0.001 |
2.30 |
~700 |
Pr |
~0.001 |
2.24 |
~700 |
Sc |
~0.001 |
2.57 |
~400 |
The data listed in Table 18 coincide upon the whole with available
practical data on the modifying (demodifying) ability of these or those
admixtures in iron-based alloys.
Thus, alkali-earth and rare-earth metals have been rightly used long
since as modificators at the casting of steel and cast iron.
According to the data supplied by Table 18, all elements may be
divided into three groups by the degree of their modifying activity in
iron-based alloys /147/.
1.
The elements that do not practically affect
crystallization have the following coefficient of m: m = 0-10.
2.
The elements that influence crystallization to a
minor degree possess the coefficient m = 10-100.
4.
Strong modifiers have the coefficient m >100.
The following elements refer to strong modifiers by the order of the
increase of their modifying ability:
Sc, La, Y, Pr, Sr, Ba, Ce, Ca, Mg, Na. The data on the modifying activity
of metals and elements in iron-based alloys are presented in Fig.19.
Out of this series it is only sodium that is not applied to the
modifying of steel because of the extreme volatility of the former. At its
introduction into liquid steel or liquid iron, sodium evaporates almost
instantly, so the amount of sodium atoms in the structure of the melt is not
enough to detect the effect of sodium at crystallization.
The data on the demodifying activity of elements are little used in
practice. The demodifier elements are practically used only in the cases when a
single-crystal or a coarse-grained directional structure is to be formed. Thus,
sulphur is specially introduced into the composition of magnetohard alloys
while growing cast single-crystal magnets. Phosphorus is used to obtain the
maximum overcooling at the forming of amorphous metals.
It is of high practical importance that
modifiers and demodifiers have a different sign of the coefficient of m.
It means that modifiers and demodifiers counteract in alloys as far
as their influence on crystallization is concerned. We know it from practice
that steel and cast iron contaminated by sulphur, phosphorus and oxygen
actually resist modifying.
However, the majority of modifiers forms compounds with sulphur and
oxygen that are insoluble in liquid steel. Therefore, the larger part of
modifiers at their introduction into the melt is spent on the neutralization of
the demodifying effect of detrimental and other impurities, including the means
of bounding the given impurities into insoluble compounds, and not on attaining
the modifying effect.
However, deoxidation and desulphurizing concern only the first part
of modifier-demodifier interaction. Unremovable demodifiers, like phosphorus,
remain in the alloy. Some weak demodificators, like carbon and silicon, are the
essential components of iron alloys. They are not to be removed, - their
influence can be but neutralized. So the second part of modifier-demodifier
interaction consists in the neutralization of the influence of demodifiers
simply due to the quantitative dominance of the modifying effect of modifiers.
Modifying proper becomes possible only after the neutralization of
demodifiers.
Therefore, a considerably larger amount of modifiers than is
requisite for modifying proper must be introduced into melts.
Table 19. The Coefficient of Modifying Activity of
Various Elements in Aluminum-Based Alloys (Modifiers and Demodifiers of the
First Type)
Element |
CS,
% at. /119,120/ |
Umod,
/139/ |
m, calculation by (150) |
First-Type
Demodifiers |
|||
Al |
- |
3.14 |
0 |
Zn |
50.0 |
3.17 |
-0.06 |
Ge |
2.80 |
3.77 |
-22 |
Si |
1.65 |
3.84 |
-42 |
Ti |
0.28 |
3.27 |
-46 |
Rh |
0.29 |
3.37 |
-79 |
Bi |
0.20 |
3.42 |
-140 |
Re |
0.26 |
3.57 |
-160 |
Sn |
0.10 |
3.31 |
-170 |
Mo |
0.07 |
3.29 |
-210 |
Be |
0.10 |
3.40 |
-260 |
B |
0.44 |
4.47 |
-300 |
C |
0.08 |
4.86 |
-2100 |
Sb |
0.05 |
3.87 |
-3600 |
|
|
|
|
First-Type
Modifiers |
|||
|
|
|
|
Ga |
9.50 |
3.12 |
0.21 |
Cr |
0.44 |
3.04 |
23 |
Cu |
2.5 |
2.56 |
23 |
Mg |
18.9 |
2.42 |
38 |
Mn |
1.46 |
3.06 |
55 |
Y |
0.80 |
2.30 |
100 |
Co |
0.02 |
3.11 |
150 |
Hf |
0.18 |
2.78 |
200 |
Au |
0.70 |
1.00 |
220 |
Cd |
0.11 |
2.89 |
230 |
Ca |
0.40 |
1.86 |
320 |
Ba |
0.40 |
1.44 |
420 |
Na |
0.10 |
1.34 |
1700 |
Ce |
0.05 |
2.23 |
1800 |
La |
0.05 |
2.15 |
2000 |
Nd |
0.04 |
2.27 |
2200 |
In |
0.04 |
2.05 |
2250 |
Sr |
<0.01 |
1.64 |
15000 |
According to the data in Table 19,
sodium, cerium, lanthanum, neodymium, indium, and strontium refer to the
strongest modifiers for aluminium.
Sodium and strontium are most frequently applied in practice. Fig.20
graphically presents the data on the properties of modifiers in aluminium.
The point is that in practice we are to consider not only the
modifying ability of this or that substance, but also its cost, accessibility,
as well as the availability of its forms convenient for the introduction into
the melt.
Practice shows that there exists a certain optimal amount of each
modifier, at the introduction of which into the melt the given modifier affects
the process of crystallization to the maximum extent. It merits our attention
that this amount usually approximates 0.1% mas. of the acting substance for the
greater part of the most widely-used modifiers of the first type.
The larger modifier quantity, as it was demonstrated, is spent on
the oxidation and neutralization of the influence of demodifiers. The amount of
modifiers approximately equal to their residual content within the melt is
spent for the attaining of the modifying effect proper, i.e. for the refinement
of the dimensions of primary crystals.
Taking into consideration that clusters in liquid metals at the
temperature of melting contain 1000 atoms on the average, we have to state that
one modifier atom falls at one to ten clusters on the average in liquid metal
before crystallization. Such admixture amount under the condition of its
regular distribution within the volume of metal can hardly influence the
process of crystallization.
On the other hand, we know that first-type modifiers are not located
regularly within the volume of metal but concentrate on any section surfaces,
inner ones including. Not only do they concentrate on these surfaces, but
stabilize them, sometimes even bringing about the increase of the area of the
surface of the liquid as foam.
Since there are enough inner section surfaces represented by the
elements of space in liquid metals, the atoms of the majority of modifiers
migrate along these section surfaces, like particles of gas, changing and
stabilizing the spatial constituent of the melt to a certain extent, augmenting
the volume of the given part of the system.
In the meantime, the density of liquid alloy decreases, which may be
proved experimentally.
Undoubtedly, these are not all the modifiers of the first type that
behave similarly to gas molecules in alloys. Still, such a modifying mechanism,
let us term it as gas-like, is characteristic of many most frequently used
modifiers. Such is the mechanism of the modifying effect of sodium in silumins,
magnesium and rare-earth metals in steel and cast iron. The specified mechanism
is possible owing to the comparatively low vaporization temperature of such
modifiers and their low solubility in cast iron and steel.
This side of the modifying mechanism of a series of main modifiers
should also be allowed for in practice together with the thermodynamic and
electron factors that we touched upon earlier.
It is the mentioned property of modifiers which, together with their
electron characteristics, enables such small quantities of modifiers to affect
the process of crystallization to such an appreciable degree.
These are the expanded interlayers of the elements of space along
the boundaries of growing crystals that hinder the joining of new clusters or
other crystals to them, thus retarding crystalline growth.
However, the same modifier property to stabilize and cause the
expansion of the elements of space in melts may bring about a totally different
effect at the increase of modifier amount over its optimal quantity.
Namely, the redundant amount of modifier may even lead to foam
formation in melts, as we know. It means that the volume of the elements of
space increases excessively. The melt becomes gas-like, frothing easily.
In this case, the modifying effect is disguised with unfavorable
after-effects of modifier redundancy. Such an effect is termed overmodifying.
In turn, the lack of modifier results in the fact that the entire
modifier amount introduced is wholly spent on the suppression of the activity
of demodifiers, deoxidation, desulphurizing and other chemical reactions,
removing the modifier out of the given solution, so we observe the lack of
substance for modifying proper. Such an effect is termed undermodifying.
Therefore, the optimal amount of modifiers is usually determined at
the level of 0.1% of the acting substance content to be adjusted in the process
of operation.
Since any melt contains a combination of controllable and
uncontrollable demodifiers of its own, it seems very hard, and often
impossible, to suppress the demodifying effect of the whole set of the known
and unknown admixtures with the help of only one modifier.
It is caused by
the circumstance that dissimilar substances have a different degree of their
chemical affinity, they interact in a different way or turn out to be inert
toward each other. In this connection, the so-called complex modifying is
widely used during the last decades, when two or several types of substances –
not the sole modifier – that possess a certain modifying activity are
introduced into the melt. Such a complex provides a much more complete blocking
of the negative influence of the demodifiers present in the melt.
As a result, the effect achieved at modifying comes to be stronger
and more stable.
It was stated in Part 9.1 that the introduction of modifiers causes
the increase of the free energy of the system and the decrease of its
thermodynamic stability.
The system tends to re-establish its equilibrium through the
removing of modifiers.
After the introduction of modifiers into
the melt their quantity always diminishes in the course of time and dependent
on temperature.
Correspondingly, the modifying effect is unstable by nature.
Practically, the specified effect reaches its maximum right after modifier
introduction and subsides in the course of time.
There are several reasons for the decrease of the concentration of
modifiers in the melt with time. One of the major causes is modifier
vaporization, since the larger part of modifiers is represented by substances
that boil relatively easily and exist in the thermodynamically unstable gaseous
state in the melt.
Secondly, there come modifier losses caused by chemical reactions
with alloy components and atmospheric gases, oxygen in the first place.
The higher the temperature of the melt is, the faster modifier
vaporization runs.
The typical
dependency of the modifying effect on time for liquid steel is shown in Fig.21.
The quantity of the modifying effect was measured here by the number of primary
crystals in steel per 1ccm of section area. Titanium nitride particles were
used as modifier. Time keeping started from the moment of the introduction of
the nitride-forming element into liquid steel. The modifying of steel by
titanium nitrides has the most appreciable effect on the refining of the
primary crystalline grain in steel.
Any familiar modifiers preserve a more or less noticeable effect on
the refining of the primary crystalline structure in liquid steel during
approx. 10 minutes from the moment of introduction up to the onset of
solidification of the casting. It is therefore assumed that steel is hard to
modify. In-mold process, or modifying in a mold, is considered practically
effective for steel castings.
For liquid cast iron, the effect of modifying lasts longer, up to 30
min. Thus, both in-ladle modifying and inmold process are possible as regards
cast iron.
Each of these processes has its advantages and disadvantages.
In-ladle modifying requires a greater modifier consumption, approx. 10-20%
more. However, this process allows holding the metal after modifier
introduction and lets emerge the products of the reaction between modifiers and
alloy components. As a result, the metal gets purer. On the other hand, the
modifying effect lowers to a certain extent during the period of in-ladle
holding (10-15 minutes), as well as the content of the modifier in the melt.
In-mold process allows saving the modifier attaining the maximum
modifying effect, but all the products of side reactions of modifiers with
alloy components remain within the casting. The metal becomes contaminated. It
does not matter at times – while forming low sort ductile cast iron. However,
the in-mold process effect may turn out to be unfavorable for the obtaining of
high-quality cast iron, or the source cast iron of high purity concerning
admixtures should be used.
In case of the modifying of aluminum alloys by sodium and strontium,
the effect of modifying is considerably more stable and longer lasting than it
is at the modifying of steel and cast iron by any known modifiers.
It can be explained by the low melting temperature of aluminum
alloys and a close aluminium affinity toward oxygen, which partially protects
modifiers in liquid aluminum alloys from oxidation. For the mentioned reasons,
the effect of modifying in aluminum alloys lasts for many hours and can be
noticed even after re-melting.
In steel and cast iron the modifying effect after the re-melting of
modified castings is not observed.
If this or that alloy forms two or more phases at crystallization,
the modifying effect may be attained selectively, in principle, by the refining
of this or that phase. Even at complex modifying different phases react to
modifying in a different way.
There are enough examples of selective modifying in practice.
For instance, the most widely used gray cast iron modifying by
ferrosilicon is a typical example of selective modifying. The point is that
silicon refines the dimensions of graphite crystals in the main having
practically no influence upon the structure of the metallic matrix of cast
iron.
It is of special interest that cast iron contains the redundant
amount of silicon without modifying – 2% approx.
Consequently, it is not silicon proper that renders the modifying
effect but its form in the melt.
At dissolution, ferrosilicon goes through all the stages of alloy
formation that were mentioned before. It forms clusters of its own, which are
first diffused within the volume of cast iron, and it is only then, during the
process of corpuscular diffusion, that silicon passes into the composition of
austenite clusters.
Silicon clusters, as long as they exist, act as a good substrate for
the formation and growth of graphite crystals, because silicon and graphite are
close analogues in Mendeleev’s periodic law. Their properties and structure are
affined enough to realize the principle of structural-dimensional
correspondence.
I.e. silicon is actually the selective
modifier of the second type for graphite in cast iron.
It merits our particular attention that ferrosilicon acts
short-time, although silicon does not burn out in liquid cast iron. It also
stresses that the particles of silicon, capable of acting as crystallization
centers for graphite, are short-lived, disappearing gradually.
There exist other examples of selective modifying.
For instance, we found a strong selective modifying effect of
silumin (11% Si) by the particles of titanium carbide TiC.
Silumin microstructures with different quantities of TiC particles
are shown in Fig.22.
The dark particles of titanium carbide cause the singling out of the
eutectic and the active dendrite growth of primary aluminum crystals (the white
a-phase
in the picture). Eutectic decomposition also leads to the growth of large
silicon crystals (light-gray particles of crystalline cut).
As a result of such modifier influence, the eutectic may be actually
destroyed, which is clearly observed in the second picture. It is interesting
that alloy microstructure simultaneously combines the features of both the
hypoeutectic (a-phase) and hypereutectic (free silicon) alloy.
As a rule, the problem of the nucleation of solid phase on the
surface of solid materials that prove to be insoluble in the melt is solved by
considering a special case of nucleation upon a flat substrate of unlimited
dimensions. In practice, the given case corresponds to nucleation on the
surface of contacting with the mold. This is quite an important particular
case.
However, solid phase in modifying practice must nucleate on the
surface of refractory dispersion particles that hover in liquid alloy. It is
the means of influencing the process of crystallization within the entire
casting volume.
So let us view the process of nucleation and the growth of solid phase
crystals on the surface of dispersion particles, wholly dipped in liquid.
The statement of the problem runs as follows.
Let us assume that there is a solid particle in the melt. The
specified particle possesses the characteristics of the modifier of the second
type, i.e. the nucleation of solid phase crystals occurs on its surface and the
particle itself may be regarded as a crystallization center.
In order to simplify the problem, let us presume that the particle
is spherical with the radius of rp. Solid phase existing as
shell 2 with the external radius of r (v. Fig.23) is formed on the
surface of particle 1 with the radius of rp under certain
conditions.
If a particle does not have the characteristics of a second-type
modifier, the shell of solid phase is not formed under the same conditions.
Let us consider a thermodynamic problem of the interaction between
the particle and the melt. Depending on the properties of the particle and its
interaction with the melt, there forms solid phase on the inclusion surface (r>rp)
– or it does not (r=rp).
The larger the quantity of r is, other parameters being equal, the
more effective the given modifier proves.
It is obvious that the quantity of r depends on both the dimensions
and characteristics of the particle and the properties and temperature of the
melt, as well as the correlation between the parameters of the particle and the
melt.
As a first approximation, let us denominate the given correlation as
the function of
r=f
(rp). (152)
V. the solution to the problem.
Such a formation (particle-solid shell-melt) will be stable only in
the case when the free energy of the system decreases monotone or when the
dependency of DF = f (r) has the minimum.
Thus, let us find and explore the extremum area for the function of
the change in the system’s free energy DF = f (r) at the forming or melting of the solid phase shell on the surface
of a solid foreign particle in the melt near the melting point, aiming at the
determining the fundamental possibility of solid phase formation on the surface
of the particles – second-type modifiers in liquid alloys.
In its general form, the change of the system’s free energy while
forming a solid shell on the particle surface will be the same as it is at the
formation of a new phase center at spontaneous crystallization:
DF = S DFv
+ S DFs,
(153)
where DFv is the change of the
volumetric free energy of the system; DFs is the change of the surface free energy of the system.
We shall presume that the particle and the solid phase shell on its
surface are spherical.
The change in volumetric free energy at the forming or melting of
the shell equals
S DFv
= (4/3)p(r3 – rp3) DFv.
(154)
The change of the system’s free surface energy at the formation of
solid phase on the solid inclusion surface is
S DFv
= Sp sp – S s,
(155)
where Sp
is the particle surface area; S is the area of the external surface of a solid
shell; sp and s represent the specific
interphase energy on the section surface of particle-liquid phase and solid
phase-liquid phase correspondingly.
The minus in front of the second term in equation (155), the same as
is observed earlier, has a physical meaning and signifies that there are not
new surfaces that get formed at the formation of solid phase in microinhomogeneous
liquid, but intercluster surfaces – the elements of space existent in liquid –
that are closed. There is no work expenditure for the formation of surface S;
on the contrary, during its formation there evolves energy in the system as the
latent crystallization heat.
Thus, as it was demonstrated earlier when analyzing the nucleation
of crystallization centers, equation (155) allows for the real structure of
liquid that consists of the elements of matter and space.
Taking into consideration the spherical shape of particles and solid
phase, out of (155) we derive:
S DFs
= 4pr2 sp – 4pr2 s.
(156)
By inserting the values of S DFv and S DFs from (153) and (155) into
(152), we obtain:
DF = (4/3)p(r3
– rp3) DFv + 4pr2
sp – 4pr2
s. (157)
Expression (157) is a linear dependency of the type DF = f
(r).
Such dependencies may be monotone, or they can have bending points.
We are interested in the corroboration of the existence or the absence of the
minimum on the curve DF=f(r). If the mentioned minimum exists,
then, the formation of solid phase is thermodynamically expedient, and v.v.
To find the minimum, it is necessary to test function (154) for the
existence of extremum and determine the nature of the given extremum further
on, if it does exist.
Let us take into consideration that the quantity of DF, according to (157), is the function of two variables DF = f (r,
rp). However, this is but the particle
radius of rp that acts as the independent argument in a
physical sense, since the radius of the solid phase shell on the particle
surface depends on rp in its turn, or r = j (rp).
If we consider that, let us find the first derivative DF = f (r,
rp) to determine the extremum existence,
and equate it with zero. When the function of one variable (the function of DF on our case) is defined as U = f (x, y), where y = rp
= j (x), the chain rule of differentiation
as applied to our case lets derive the following formula:
DF¢ = DF¢rp + DF¢r r¢rp = 0 (158)
Let us differentiate (157) by the chain rule scheme (158) and equate
the first derivative with zero. Thus
4pr2DFv
(dr/drp) - 4pr2DFv - 8prs (dr/drp)
+ 8prpsp = 0.
Having completed the requisite cancellations, we obtain
DFv r2
(dr/drp) - DFv rp2 - 2rs (dr/drp)
+ 2sp rp =
0. (159)
For further analysis, we are to know the mode of the function r =
j (rp). As a first
approximation, the correlation between the quantities of r and rp
can be expressed by a linear dependency presented as r = A (rp), where
A is a certain constant.
It is obvious that only the values of A>1 have a
physical sense.
It is known that a small section of any even curve can be approximated
by a line segment. So the expression of r = A (rp) is quite
acceptable.
In this case dr/drp = А, and equation (159) assumes the form
А r2 DFv - 2А s r - DFv rp2 + 2sp rp =
0 (160)
This is an equation of the second order relative to r. It is
known that if the order of the first derivative is even, then, the function
under analysis has the extremum.
Now it is time to test the nature of the extremum for the
availability of the minimum.
Let us express s and sp through the independent argument
of rp.
Referring to B.Chalmers /65/, we obtain
sp = rp DFvp
/2
(161)
s = rp DFv
/2 (162)
r = rp is used in expression (161) to simplify the problem, because the
shell radius may assume any values r ³ rp under the conditions of the problem.
Apart from that, the usage of the variable quantity of r is
inexpedient, for (160) does not have any solution relative to r in this case.
By inserting the values of sp and s from (161) and
(162) into (160), we obtain:
А DFv r2
- А DFv r rp - DFv rp2
+ DFvp rp2 = 0.
or
А DFv r2
- А DFv r rp + rp2 (DFvp - DFv) =
0. (163)
As modifiers of the first type, dispersion particles of various
refractory particles, the melting temperature of which is considerably higher
than the temperature of modifying. In this case, the following inequality DFvp >> DFv takes place.
Consequently, we can assume without any considerable error that DFvp - DFv @ DFvp.
Then expression (163) will assume the form of
А DFv r2
- А DFv rp r + rp2 DFvp =
0 (164)
Equation (164) as related to r is a regular quadratic
equation of the type ax2+bx+c=0, where а = А DFv; b = - А DFv rp; c = rp2
DFvp .
The solution of the given expression is presented as:
r1,2 = rp
{(1/2) + [(1/4) - (DFvp /А DFv)]1/2} (165)
Now we can determine the nature of the extremum of the function of DF = f(r). Its flexion is
DF² = 2А DFv r - А DFv rp . (166)
Since there exists a flexion, it is relevant to continue the
research concerning the existence of the maximum or the minimum of the
function.
To achieve this aim, we are to determine the sign of the flexion DF². To do so, let us introduce the value of r1 from
(165) into (166). Thus
DF² = 2А DFv rp {(1/2) + [(1/4) - (DFvp
/ А DFv)]1/2} - А DFv rp (167)
Since DFvp < 0 for refractory substances under
modifying conditions, then we have the sum and the value in braces in
expression (167) positive and >1. Consequently, the first term in the right side of (167)
2А DFv rp {(1/2)
+ [(1/4) - (DFvp / А DFv)]1/2}
is more than the
second one А DFv rp .
Consequently, DF² > 0.
Since the value of the flexion is
positive, then the function of DF = f(r) has the minimum if r = r1.
The given conclusion is of fundamental importance. It means that
there exists the possibility of forming the solid phase shell of limited
dimensions on the particles of refractory modifiers of the second type in melts
at the temperature that is somewhat higher than the melting temperature. The
radius of such a solid phase zone id determined by expression (165).
This is an unexpected and convincing conclusion. It was accepted
earlier that solid phase cannot exist at the temperature that is higher than
the temperature of melting. The analysis that we carried out shows that solid
phase can be formed within limited zones in the vicinity of the surfaces of
strong modifiers of the second type and exist at the temperature that exceeds
the melting temperature of the given metal or alloy. Such zones cannot grow if
the dimensions exceed the quantity of r. The graph of function (157) under the
specified conditions has the form represented in Fig.24.
The picture shows that the nuclei of solid phase that were formed in
liquid metal or alloy on the surface of modifiers of the second type, remain in
a sort of potential well and have strictly specified dimensions. Such
formations are lacking in either growth or decomposition tendency under
constant conditions. However, if the conditions change, the dimensions of such
formations change, too. For example, at the cooling of the melt such
microcrystals will be boundedly growing to a new equilibium value of r,
diminishing till zero at heating. The temperature of the zero value of r can be
calculated on the basis of the listed expressions.
This is the first time when we predict the possibility of the
existence of equilibrium though small crystals of solid phase in liquid metal
at the temperature that is higher than the temperature of melting. Such an
inference does not contradict the general theses of thermodynamics about the
impossibility of coexistence of solid and liquid phases within the same
temperature interval, since we have got a more complicated case here when three
phases – solid, liquid and that of modifier particles – coexist. In the
particular case viewed above, the interaction between the three given phases
may create conditions for a specific form of the coexistence of solid and
liquid phases within the limited temperature interval.
In case if the melt cools down to the temperature of
crystallization, the microcrystals arisen as a result of the interaction
between the three phases will be growing without bound.
Certainly, such solid phase areas that are limited by dimensions get
the advantage over smaller spontaneously nucleating crystals in their
competitive activity at the cooling of the given liquid down to the temperature
of melting. On the other hand, the formations under consideration are too large
to absorb one another.
As a result, crystalline dimensions in a casting with the modifiers
of the second type are determined by the number of particles of the modifier
specified: the larger their number is, the smaller crystals become.
In general, such is the mechanism of the influence of second-type
modifiers upon the dimensions of primary crystals in the structure of castings.
Comparing the modifying activity of various modifiers of the second
type to correspondingly opt between them is possible through the application
and analysis of expression (164). This expression allows calculating the
dimensions of the shell of solid phase that is formed on the surface of this or
that modifier and comparing the effectiveness of various second-type modifiers
by the value of r from (164).
According to (164), the value of the shell of solid phase radius r
depends rather on the correlation between the thermodynamic properties of the
substance of modifier particles and the melt presented as DFvp and DFv than on rp. By B.Chalmers /65/, we obtain:
DFvp = DНp DТp/ ТLp;
DFv
= DН DТ/ ТLm,
(168)
where DНp is the enthalpy of the forming of modifier substance; DТp = ТLp - Т; ТLp is the melting temperature of
the substance of a second-type modifier.
DН, DТ and ТLm represent the same for the
metal of the melt; Т being the current
temperature.
By
introducing (168) into (165), we obtain:
r1,2
= rp {(1/2) + [(1/4) - (DНp DТp ТLm / А DН DТ
ТLp )]1/2} (169)
Expression (169 determines the shell radius as the sphere of the
direct particle influence upon the surrounding melt; in this connexion, the
forming of solid shells on modifier particles at the temperatures that exceed
the melting temperature of the melt can also be referred to the rank of contact
phenomena. We observe that substances may act otherwise than lowering their
respective temperatures of melting within the bounded contact zone, as it takes
place at contact melting. Within the contact zone of certain substances, the
given substance may increase the temperature of melting (crystallization) of
the other substance under certain conditions.
Let us term this phenomenon, unknown till present, as contact
crystallization.
We can deduce some conditions of successful modifying of the second
type on the basis of (169.
In the first place, contact crystallization area enlarges with the
decrease of the temperature of the melt.
Secondly, the higher the modifier thermodynamic stability presented
by the value of DНp is, the more r increases.
In the third place, the greater the difference of the melting
temperatures of the modifier and the given alloy (DТp) is, the higher is the modifier
effectiveness.
Hence we can make several inferences about the characteristics of
second-type modifiers and the conditions of their application.
Modifiers of the second type must be refractory and
thermodynamically stable under the condition of being modified by substances.
It also follows from Part 9.3 that there should be some electron
affinity between modifiers of the second type and the alloy. It implies that
the substance of the modifier of the second type should possess the metallic
type of conductivity.
Finally, it is desirable that the substance of the modifier should
be insoluble in the given melt. Soluble modifiers of the second type are
possible (similarly to the presence of silicon in cast iron), yet they act up
to the moment of their complete dissolution in the melt only, i.e. their effect
is a pronounced short-term one. For instance, steel powder immixed in liquid
steel can act as a second-type modifier refining the primary crystalline
structure of steel.
The effect described is used in suspension casting. However, this
effect is only observed until the powder particles melt completely. Therefore,
metallic powders are introduced into liquid steel at suspension casting only in
the process of pouring steel into the mold. The in-ladle introduction of the
same powders does not produce any modifying effect.
Thus, we formulated the four factors of the choice of second-type
modifiers:
1.
Second-type modifiers must possess a high
temperature of melting that proves to be considerably higher than the melting
temperature of the alloy they are introduced into (‘considerably’ means
‘hundreds of degrees higher’ in the given context).
2.
Second-type modifiers must have the enthalpy of
forming that exceeds considerably the enthalpy of the forming of the melt they
are to be introduced into.
3.
Second-type modifiers must have the metallic
type of conductivity.
4.
It is desirable that second-type modifiers
should be melt-insoluble.
Are the mentioned requirements sufficient to realize the choice of
particular substances acting as modifiers?
As an example, let us make the choice of modifiers of the second type
for steel and cast iron in accordance with the stated requirements (v. Table
20).
The requirement of refractoriness alone tangibly restricts the range
of possible candidates for being second-type modifiers.
If we consider substances the melting temperature of which exceeds
2500 K, the number of such substances is but 43. These are refractory metals,
oxides, carbides, nitrides and borides.
Out of the mentioned 43 substances, these are only 10 that strictly
conform to the requirement of being insoluble in liquid steel and cast iron.
At last, the number of substances that possess the metallic type of
conductivity and satisfy other conditions apart from that, is restricted to but
three compounds in this case: titanium and zirconium nitrides and zirconium
diboride. The modifying ability of a series of substances remains undecided
because of certain data missing.
We must admit that this is quite a concrete set of a highly
restricted range of substances.
Practical application of these substances as modifiers of the second
type for steel propagates poorly affecting but titanium nitrides in the main.
Table 20. The Evaluation of the Suitability of
Various Refractory Substances as Modifiers of the Second Type for Steel and
Cast Iron
Substance |
Temperature
of melting, degr.K, /92/ |
Solubility
in liquid steel and cast iron, /92/ |
Free
energy of forming, kJ/mole, at 2000 К, /92/ |
Conductivity
nature at 2000К, /92/ |
Suitability
as a second-type modifier |
Elements |
|||||
C |
4020 |
s |
- |
metallic |
no |
Мо |
2890 |
s |
- |
metallic |
no |
Nb |
2740 |
s |
- |
metallic |
no |
Os |
3320 |
s |
- |
metallic |
no |
Re |
3450 |
s |
- |
metallic |
no |
Ta |
3269 |
s |
- |
metallic |
no |
W |
3680 |
s |
- |
metallic |
no |
Borides |
|||||
Hf B2 |
3520 |
? |
310 |
metallic |
? |
LaB6 |
2800 |
? |
? |
metallic |
? |
NbB2 |
3270 |
? |
155 |
metallic |
? |
ThB4 |
2775 |
? |
188 |
metallic |
? |
TaB2 |
3370 |
s |
? |
metallic |
no |
TiB2 |
3190 |
s
(0.5 %) |
238 |
metallic |
? |
UB2 |
2700 |
? |
? |
metallic |
? |
W2B |
3040 |
? |
? |
metallic |
? |
ZrB2 |
3310 |
no |
268 |
metallic |
suitable |
Carbides |
|||||
HfC |
4220 |
? |
209 |
metallic |
? |
NbC |
3870 |
1
% |
134 |
metallic |
no |
SiC |
3100 |
? |
44 |
metallic |
? |
Ta2 C |
3770 |
0.5
% |
188 |
metallic |
no |
TaC |
4270 |
0.5
% |
144 |
metallic |
no |
ThC |
2900 |
? |
17 |
metallic |
? |
Th2C |
2930 |
? |
190 |
metallic |
? |
TiC |
3340 |
0.5
% |
158 |
metallic |
no |
UC |
2670 |
20
% |
75 |
metallic |
no |
UC2 |
2770 |
20
% |
104 |
metallic |
no |
VC |
2970 |
3
% |
103 |
metallic |
no |
WC |
3058 |
7
% |
60 |
metallic |
no |
ZrC |
3690 |
? |
182 |
metallic |
? |
Nitrides |
|||||
BN |
3240 |
no |
80 |
semicond. |
no |
HfN |
3580 |
? |
184 |
metallic |
? |
TaN |
3360 |
chem.
react. |
99 |
metallic |
no |
ThN |
3060 |
? |
137 |
metallic |
? |
TiN |
3220 |
no |
149 |
metallic |
suitable |
UN |
3120 |
? |
118 |
metallic |
? |
ZrN |
3250 |
no |
178 |
metallic |
suitable |
Oxides |
|||||
BeO |
2820 |
no |
401 |
ionic |
no |
CeO2 |
3070 |
no |
600 |
semicond. |
no |
HfO2 |
3170 |
no |
753 |
semicond. |
no |
MgO |
3098 |
no |
321 |
ionic |
no |
ThO2 |
3540 |
no |
840 |
semicond. |
no |
UO2 |
3130 |
no |
738 |
semicond. |
no |
ZrO2 |
2973 |
no |
721 |
semicond. |
no |
We tested the inferences of Table 20
concerning the modifying activity of titanium and zirconium nitrides in
practice. At the introduction of approx. 0.1% of the given nitride particles
per entire volume into steel with 0.3% C content, we obtain a thorough primary
crystalline grain refinement (v. the photograph in Fig.25) in case of both
titanium and zirconium nitrides application /148/. The largest crystalline
dimensions in modified castings with the mass of 10 kg and wall thickness of
100 mm within the zone of a shrinkage cavity equaled approx. 1 mm. Judging from
the cited results, titanium and zirconium nitrides are the strongest
second-type modifiers for steel at present.
Nitride inclusions surrounded by a dark mantle are found in the
microstructure of cast samples extracted from nitride-modified steel hardened
from its liquid state (v. the photograph in Fig.26).
We suppose that these areas correspond to the shells of solid phase
that have been already formed around modifier inclusions in liquid state before
the onset of total crystallization.
Such solid phase formation around
inclusions is to be reflected by the curves of the cooling of castings and be
noticeable when the amount of inclusions comes to be large enough.
Special experiments were carried out on the basis of aluminum alloys
reinforced with a considerable particle quantity. Liquidus and solidus
temperatures were measured by the method of differential thermal analysis
concerning alloys reinforced with titanium carbides of any given quantities.
Experiments have shown the rising of liquidus temperature of such
composite alloys with the increase in the content of carbide particles, the
temperature of solidus being constant (v. Fig.27).
Thus, experimental data corroborate the validity of theoretic
inferences made above concerning the mechanism of the influence of second-type
modifiers upon the process of castings crystallization.
Apart from refractory inclusions, there are fusible, liquid, gaseous
inclusions in liquid alloys. Do the former affect nucleation and the growth of
crystals in castings?
Let us consider the case when the melting temperature of the given
particle and the enthalpy of its formation approximate the temperature of
melting and the latent heat of metal melting, i.e. the following equality takes
place:
ТLp @ ТLm and DНp = D Н.
Then, on the basis of (168) we obtain:
DFvp = DFv and DFvp - DFv = 0.
In this case, equation (163) assumes the form:
А DFv r2 - А DFv r rp =
0 (169)
Hence we derive
А DFv r(r
- rp) = 0 (170)
Equation (171) can be solved in two ways:
r1
= 0; r2 = rp (171)
Solvation (171) implies that a stable solid phase shell cannot be
formed in liquid around fusible particles. However, the specified conclusion
applies but to the temperatures exceeding the temperature of liquidus. Certain
fusible inclusions may function as modifiers of the second type within the
interval of crystallization. This inference is proved by the practical examples
of applying microchills as modifiers. The application of ferrosilicon to the
modifying of cast iron serves as an analogous example. The generality
integrating the given cases is that relatively fusible substances can act as
second-type modifiers for a short time period only until they melt or dissolve
completely in the surrounding liquid metal.
Liquid
inclusions.
Let us examine
the case when liquid inclusions are present, i.e. their melting temperature
proves to be considerably lower than the melting temperature of metal.
TLp
<< TLm .
Then DFvp << DFv and DFvp - DFv @ - DFv
In this case, equation (162) will assume the form of
А DFv r2 - А DFv r rp - rpDFv =
0 (173)
Equation (173) has the following roots
r1,2 = rp
{(1/2) + [(1/4) - (1/ А)]1/2} (174)
Since А ³ 1, equation (174) either has no
solution within the interval of А = 1 ¸ 4, or results in r1 < rp.
Consequently, the shell of solid phase cannot be formed on the
surface of liquid and all the more – on the surface of gaseous inclusions
within melts. Consequently, liquid and gaseous inclusions in liquid cast alloys
cannot function as modifiers of the second type.
Moreover, such inclusions hinder the nucleation of solid phase in
the vicinity of its surface, i.e. they are second-type demodifiers in melts.
The dimensions of the elements of matter in liquid alloys were
repeatedly measured by various procedures. X-ray diffraction researches,
unfortunately, furnish ambiguous results, which allow interpretation both from
the standpoint of cluster existence and the monatomic structure of liquid
metals and alloys.
In this connection, much more univalent researches of sedimentation
processes in melts seem to be of particular interest – for instance, at the
centrifuging or finer modern sedimentation methods within gravity field.
K.P.Bounin, on the basis of V.I.Danilov’s works /10/, as well as the
liquid eutectic alloys research of his own, was the first to put forward a
hypothesis of the possibility of melts microstratification by structural areas,
similar to the structure of pure components /59/. Broaching the question why
spontaneous stratification of such melts does not take place, K.P.Bounin wrote:
‘… thermal motion ensures the kinetic stability of eutectic melt, and,
notwithstanding the melt being microheterogeneous, there is no stratification
at microlevel.’ In this connection, K.P.Bounin substantiated the possibility of
applying centrifuging to the investigation of liquid eutectics structure.
Profound researches of liquid eutectics, carried out by Yu.N.Taran
and V.I.Mazur /60-61/, as well as a series of other investigations
/49-55,57-58/, corroborate the microheterogeneity of liquid eutectics in a
tenable way nowadays.
Outstanding pioneer experimental works on centrifuging by
A.A.Vertman and A.M.Samarin proved the existence of sedimentation phenomena and
disclosed the first symptoms of liquid eutectics stratification start /16-17/.
Unfortunately, though the suggested ideas proved right, the stated
experiments were not quite mastered methodically. They failed to mark the
difference between sedimentation at crystallization and liquid state
sedimentation with a considerable degree of methodical reliability. For the
mentioned methods shortcoming, as well as other ones, the interpretation of the
results of centrifuging experiments was subjected to severe criticism on the
part of the followers of the monatomic theory of liquid alloys structure
/20,26/. Being fastidious to the methodical imperfections of the experiments
that were carried out and the accepted way of calculations, the critics of
centrifuging rejected the microheterogeneity idea as the basis of the given
experiments uppermost.
The theory of liquid metals and alloys structure stated in our work
evolved even farther than the model of the microinhomogeneous structure of
eutectics. The existence of both the elements of matter in liquid state –
clusters – and the elements of space in any liquid metals and alloys including
eutectics is grounded here.
In this connection, a theoretical substantiation and the carrying
out of sedimentation experiments, considering the newest data /30,142-143/ and
methods /142-144,149-150/, seems to be of an appreciable interest.
The distribution of any particles in liquids goes under the
influence of gravity forces, on the one hand, and the forces of thermal and
convective mixing, on the other hand.
The only theory that views such a distribution at present is the
Brownian motion theory.
Einstein, Smoloukhovsky, Jean Perren were engaged in the Brownian
motion theory research. The latter scientist investigated the Brownian motion
experimentally (as applied to water) for determining Avogadro Number. The given
trend has been physically validated and elaborated to perfection, resulting in
the Nobel Prize award to Jean Perren for the Brownian motion researches.
The Brownian motion was not investigated in liquid metals, therefore
the application of the Brownian motion theory and experiments to the research
of liquid metals and alloys is of no little interest, yet it is requisite for
the measuring of the dimensions of material elements in liquid melts rather
than determining Avogadro Number.
Let us consider the Brownian motion theory with the purpose of
checking for the possibility of finding the dimensions of the elements of
matter in liquids.
Fokker-Plank equation /142/ serves as the basis for the theoretical
solution of the problem concerning the particle Brownian motion in liquid if we
take gravity factor into consideration:
¶C/¶t = D ¶2C/ ¶h2 + k ¶C/¶h,
where С is the relative Brownian particles concentration; С = f(h0, h, t); t is time; h is the sample height in our case; D is the
coefficient of diffusion; k being the gravity constant.
We must note that the given equation was composed regardless either
of convection or particle interaction. Its general solution was obtained by
Smoloukhovsky. Still, Smoloukhovsky's solution is convenient to use if particle
dimensions and mass are known a priori. Such a way proves unsuitable for our
purpose.
In order to arrive at the requisite solution, let us consider the
case of equilibrium in the Brownian motion, when particle redistribution under
the influence of forces is completed, so the concentration of particles does
not change in time, i.e.
¶C/¶t = 0.
In this case the displacement of Brownian particles will be zero,
i.e. diffusion and gravity influence will be equalized.
For the given stationary case we obtain:
D ¶C/¶h + CV =
0, (175)
where V
is the gravity speed of sedimentation or floating.
Expression (175) is true if heat and gravity particle energy
approximate, i.e. if the particles are sufficiently small. Solution (175)
assumes the form of
С = С exp (-
Vh/D) (176)
We do not know the quantity of gravity rate V for the expression
specified. To find it, let us consider the known mechanical problem of the
floating (sink) of small particles at Stokes’ approximation. Let us set up a
motion equation (without viewing the interaction of particles).
The force necessary to displace the particles of the mass of m and
the radius of r is determined by the regular correlation accepted in mechanics
Fv
= m dV/dt
There being no auxiliary effects, the given force equates with
gravity:
Fg
= mg = 4pr3Dg/3.
We should also take into consideration the force of internal
friction (viscosity):
Fh = 6 phrV.
Archimedean force resists gravity: Fd = 4 pdgr3/3
If we sum up the interaction of the mentioned forces, the equation
of a particle motion in liquid will be presented as
Fv
= Fg - Fh - Fd , (177)
or
m
dV/dt = 4pr3Dg/3 - 6 phrV - 4 pdgr3/3.
Cancellations completed, we obtain
dV/dt = (D - d)g/D - 9hV/ r2D.
(178)
Equation (178) can be written as
dV/dt + AV + B
= 0, (179)
where
А = 9h/2Vr2; B = - (D - d)g/D.
Equation (179)
belongs to linear differential ones. Its complete solution is
V
= (C - 1) e-At - B/A,
where С is a constant.
It follows from the initial conditions that V = 0 at t = 0, thus
V
= B (e-At - 1)/A.
By introducing the values of the constants of A and B,
we find:
V
= - [2 (D - d)g r2 / 9h] [exp (-9ht/2r2) -
1]
or
V =
[2 (1 - d/D) g r2 D/ 9h] [1 - exp (-9ht/2r2)] (180)
The analysis of expression (180) shows that the sink rate of Brownian
particles is directly proportionate to the square of the radii of these
particles. Particle sedimentation rate accelerates to a certain degree with the
increase of time, asymptotically approaching the equilibrium value of Ve.
The given value can be determined if we assume that е-Аt®0 at t®¥, so according to (180)
Ve
= 2 (1 - d/D) g r2 D/ 9h (181)
It is easy to calculate that the mentioned limit is actually
instantly reached for small particles with r £ 10-8 m.
Let us
introduce the V from (181) into (176) arriving at
С/С0 =
exp [- 2 (1 - d/D) g r2 D h/ 9hD]
(182)
A similar solution for the stationary distribution of Brownian
particles was obtained by J.Perren /151/. He focused on the equilibrium of two
forces only: gravity
Fg
= 4pr3 (D - d)g dh/3,
and the Brownian
motion force
Fb
= -kT dC.
Solving the equilibrium equation of the specified forces, J. Perren
derived the expression /151/:
ln
C/C0 = 4pr3 (D - d)g h/3kT. (183)
If we substitute the D in expression (182) for Stokes-Einstein
correlation
D
= kT/ 6phr,
we shall derive
the expression that is wholly identical with Perren’s equation (183).
Namely:
C/C0
= exp [- 4pr3 (D - d)g
h/3kT] (184)
Thus, two variant solutions to the problem result in similar
solutions for the stationary distribution of Brownian particles (184).
It is of extreme importance to note that the distribution of Brownian
particles in liquid, according to (184), is proportionate to the cube of the
radius of Brownian particles.
I.e. particle dimensions strongly affect their in-melt distribution.
Consequently, the experiments concerning the study of in-melt Brownian particles
distribution should be rendered extremely sensitive to the dimensions of
sedimentator particles. This is very significant, since it allows carrying out
the experiment the result of which will differ essentially depending on the
nature of particles – the structural elements of matter in liquid: atoms or
clusters. The difference in the order of the given two types of material
elements, according to (184), should affect the distribution of Brownian
particles by the sample height in a deciding way.
It enables us to employ (184) for the calculation of the dimensions
of Brownian particles in melts with a considerable certainty. It follows from
(184) that
r =
[(3kT ln C/C0)/ 4p (D - d)g h]1/3 , (185)
where d is the average
melt density; D being the density of Brownian particles.
Expression (183) is quite useful for the study of the dimensions of
the elements of matter in liquid alloys on the basis of the results of
sedimentation experiments.
Actually, if any more or less noticeable difference of element
concentration is obtained by the sample height of several cm as a result of
holding liquid cast alloys within gravity field, that will mean that the
in-melt dimensions of the elements of matter exceed the dimensions of separate
atoms.
Calculations show that with the corpuscular structure of liquids no
measurable difference of element concentration by the sample height with the h
= 50-100 mm is to be observed in such experiments.
Jean Perren was awarded the Nobel Prize for determining Avogadro
Number in the experiments dedicated to the study of the Brownian motion in
water /151/. As a matter of fact, Jean Perren estimated the total number of
particles moving in liquid on the basis of the measurement of the motion energy
of some of them, namely the particles that were specially introduced into
transparent liquid (water), - the particles visible by microscope with their
dimensions and density known a priori. The researcher made use of the fact
that the heat motion energy of any particles moving in liquid is identical.
By Perren /151/, the motion energy of one Brownian particle equals
W =
2pr3 (D - d)g h/ ln C/C0
, (186)
where h is the
height of a sample.
Jean Perren determined the quantities entering into (186) from
experience to further calculate the quantity of W on their basis.
The quantity of W being known, Jean Perren found the number
of the structural units of liquid per mole N:
N =
3RT/2W (187)
According to his data, the number under analysis was approximately
equal to Avogadro Number.
From the viewpoint of present theory, any energy in liquid is
distributed uniformly between all the particles constituting the liquid, be it
atoms or clusters or other particles that are sufficiently small. The stated
fact is reflected in the known energy equidistribution theorem.
It was demonstrated in Part 3.6 above that the heat energy of
clusters does not differ from the heat energy of separate atoms in consequence
of energy equidistribution by the degrees of freedom.
Besides, the number of clusters per mole of liquid amounts to
approx. 0.1% of the number of atoms, i.e. Avogadro Number. Thus, the aggregate
amount of the elements of matter in liquid among which heat energy is
distributed equals N0 + Nc + Nb » N0
,since Nc + Nb << N0
, where Nc is the number of
clusters per mole of liquid; Nb is the number of Brownian
particles introduced.
Consequently, the total number of atoms and clusters per mole of
liquid differs very little from Avogadro Number (Np = N0
+ 0.1%).
Therefore, the result obtained by J. Perren gives the amount of
particles per mole of water that hardly exceeds the theoretical Avogadro Number
of N0.
However, Perren conducted experiments with visible particles that
were introduced into water in advance and the dimensions and density of which
were set-point.
In case of liquid alloys research allowing for the existence of clusters
cluster dimensions are unknown a priori. Still, the Brownian motion theory lets
employ the difference in the density of the particles constituting the alloy to
determine the dimensions of these particles, even if the latter are invisible.
For instance, at the detection of a concentration difference in the composition
of elements by the height of a liquid alloy sample while holding it within
gravity field J. Perren’s methods can be successfully applied to calculate
cluster dimensions by the formula (185) derived above.
It is possible in the connection with the fact that matter in liquid
alloys is represented by atomic groupings-clusters that possess the value of r
by order of magnitude greater in comparison with atoms. Since r enters into
(184) in the cube, the existence of clusters must affect the distribution of
the elements of matter in liquid alloys within gravity field most decisively,
so the quantity of C/C0 in case of the monatomic structure of
liquid metals must be three orders as small as it is in case of the cluster
structure of melts.
Thus, theoretical analysis shows that there exists a quantity that
is highly sensitive to the dimensions of the elements of matter in liquid
alloys – this is the change of the concentration of elements in alloy samples
by their height while holding the samples within gravity field (convection
lacking).
In case of the monatomic structure of liquid alloys there have to be
no visible change of concentration for the time of holding reaching tens of
minutes or hours.
In case of the cluster structure of liquid alloys such a change
should be quite apparent, amounting to the tenth fractions of one percent in
samples that measure approx. 10cm by height.
The given arguments and calculations were assumed as a basis for
creating a new procedure of liquid alloy structure research.
There exists a diversity of opinions with respect to component
sedimentation in liquid alloys.
For example, B.Chalmers supposes that in alloys with the
unrestricted solubility of their components in liquid state ‘… there must be no
segregation in liquid until the latter starts hardening’ /67/.
Well-known monographs by other authors treat but segregation at
crystallization, too /64,66,68,74,75,135/.
At the same time, there is K.P.Bounin’s opinion on the possibility
of such a phenomenon in liquid eutectic alloys /59/.
There are centrifuging experiments results obtained by A.A.Vertman
and A.M.Samarin, the results of V.P.Tchernobrovkin’s observations concerning
segregation in liquid cast iron /16,17/, as well as a whole series of other
data.
We also have objection and discussion data that deny segregation at
centrifuging /20/.
Thus, there exist opposite opinions on liquid state segregation,
even under the conditions of simulated gravity at centrifuging. It testifies to
the insufficient study of the question.
The development of the procedure that will provide reliable and
unambiguous results is of paramount importance for a secure record of an
unknown phenomenon and its mechanisms.
The results of cosmic metal research demonstrated that convection in
samples increases inevitably with an increase in gravity g, for Rayleigh Number
is on the rise /102/:
Rа = (gh3/n)[(btDT/a) + (bcDC/D)],
where h is the
height of the metal layer; DT is the in-layer temperature drop; DC is the difference in concentrations; bc is the metal volume expansion coefficient; n is the
kinematic viscosity coefficient.
It is clear from the formula that Rayleigh Number speedily increases
with an increase in g and h reaching the critical value of Ra = 1700.
Hence, the influence of gravity in sedimentation experiments does
not only promote sedimentation but inhibits it.
Therefore, centrifuging experiments should not be considered
effective by their nature, since the quantity of g is too large there,
so convection cannot be eliminated because of various mechanical interference
types like unavoidable vibrations, rotation speed fluctuation, Coriolis forces,
etc.
The given procedure may be used only as a preliminary, qualitative
one. It can hardly be of help for the finding of precise quantitative
information.
We developed different variants of experimental mass transfer
research methods in liquid alloys within gravity field, there being no initial
concentration gradient in samples.
The basics of these methods consist in the following.
Specified composition alloys were produced out of pure components
(of extra pure brands and brands tested pure for analysis). Experiments were performed
on the following alloys: Pb-Sn, Pb-Bi, Zn-Al, Al-Si, Al-Fe-Cr, Cu-Pb, Cu-Sn,
Fe-C, Cu-Pb-Sn. For the obtaining of a homogeneous parent composition, alloys
were overheated 200-300 K above the temperature of liquidus to be held 1 hour
at the given temperature, and were stirred thoroughly with an alundum or quartz
stick.
No sooner was the stirring completed than alloy samples were taken
into quartz or alundum capillaries or tubes by way of vacuum suction. Capillary
or tube diameters went from 0.3 to 50mm, their height being changed from 40mm
to 500mm, since the way in which the diameter and height of samples, as well as
their material, affected the result was our subject matter, too (Fig.28).
For the most part, samples of 1-3mm in diameter were used. Such a
choice allows a practically full suppression of the onset of in-sample
convection.
The samples were brought to crystallization by way of air cooling.
Then the initial distribution of components by sample height was studied on one
of the samples by way of chemical and metallographic analyses.
Other samples were tested by sight for continuity. Such a test is of
fundamental importance, for samples where experiment discontinuity is observed
show a sharp distortion in results.
After the sorting through, the samples were sealed hermetically.
Quartz capillaries were sealed through vacuum soldering. Alundum capillaries
were plugged at the ends with a pulverized alundum-based stopper to be sealed
in with quartz further. Practice showed that a similar sealing shuts the
samples off from air and guards against volatile elements evaporation much
better than, for example, smelting in inert gas atmosphere.
To hold the metal thread fixed inside a capillary to guard the
former against bias at turning, an asbestos layer 2-3mm thick was put and
packed on top of metal before soldering.
The small diameter samples prepared in such a way were placed into
the holes of a graphite casing – a cylinder with 8-10 openings along its
external perimeter extent - in groups of 8-10 identical items, so all the
samples remained under the same thermal conditions. The casing with the samples
was put in an experimental cell that had been heated up to the specified
temperature.
The cell constituted the isothermal zone of resistance shaft and
resistance multipurpose furnaces for fusible alloys or a similar zone of
Tamman’s furnace for refractory alloys.
For temperature equalization, as well as the screening of
electromagnetic fields, two cylindrical coaxial graphite and molybdenum screens
shielded the graphite casing with the samples.
Under such conditions, fluctuation in temperature by the length of
the isothermal zone in resistance shaft and resistance multipurpose furnaces
totaled + 2K at most, amounting to + 8K in Tamman’s furnace.
The time of heating the samples up to the specified temperature made
1-5min. for small diameter samples (< 3mm). The temperature was
registered by thermocouples and potentiometers.
The experiment consisted in the holding of capillaries at the
specified temperature during the preset time at a definite position of the
sample relative to the vertical /141/. The working cell of furnaces revolved
around the horizontal axis, which let station the samples in vertical,
horizontal and reverse positions. It enabled us to hold the sample vertically
in liquid state, for instance, and place it horizontally at crystallization.
That was the way we eliminated the influence of segregation upon the obtained
distribution during the crystallization period.
To this end, we compared the distribution of elements in the samples
that had been held in horizontal and vertical positions correspondingly.
To study the influence of time factor, identical samples were
withdrawn out of the casing 5, 15, 30, 60, 90, 120, 180, 240min after the
beginning of the experiment.
Crystallization was effected in varied ways, too, to study the
influence of crystallization processes on the distribution of elements by the
length of samples.
Some samples were crystallized within the casing by the air blowing
of the experimental cell. A certain share of samples was withdrawn out of the
furnace and cooled by water in vertical or horizontal positions.
At the water hardening of samples 3mm in diameter the time of
hardening amounted to 3-4s. A considerable value of overcooling – from two to
three tens of degrees - was observed in small diameter samples.
The obtained samples were taken out of capillaries. Samples with
ruptures and holes were rejected. The selected ones were cut into parts and
tested for the distribution of components by height in solid state by the
methods of chemical and metallographic analyses.
The negative effect of negligible external noises, e.g. those coming
from the vibrations of machinery operating in the neighborhood, was noticed in
the course of the experiment. Such vibrations may cause convection thus utterly
distorting the result. In this connection, furnaces were rigged out with
vibration pads and the experiments were performed mainly at night to minimize
the noise.
Segregation effect at melting was also neglected in the experiments,
which is normally disregarded altogether. Melting was conducted with
differently positioned samples for that purpose.
Mass transfer at melting was not observed in our experiments.
Mass transfer at crystallization amounted from 0 to 8% of the
registered concentrations depending on crystallization conditions. Zero effect
of segregation at crystallization on the distribution of elements by sample
height was found in the experiments concerning the crystallization of samples
in horizontal position. Zero effect of segregation at crystallization was also
observed at the water hardening of samples out of liquid state.
The application of the described method allows detecting and
eliminating the influence of external factors preserving the effect of elements
redistribution by sample height under the influence of gravity in its pure
form, practically.
A possible Coriolis acceleration influence, the influence of a
possible difference in alloy density by sample height, as well as a series of
other probable disturbance interference types, were also taken into account
while carrying out the experiment.
The above-said gives us sufficient grounds to state that the
developed method lets study the redistribution (mass transfer) of alloy components
just in liquid state, whereas our experiments allow for the influence of
melting-crystallization processes reducing it to the minimum.
The temperature and composition of the
samples experimented with by the above-stated method are listed in Fig.29. The
results of the chemical composition research of the obtained samples by their
height under varied conditions of holding are presented in Fig.30 /143/.
The data demonstrated in the given picture show that the
redistribution of alloy elements by sample height undoubtedly occurs in the
liquid alloy of Pb-Sn in the process of holding in liquid state (convection
lacking).
The results of chemical inhomogeneity
development at the holding in liquid state in the course of time are to be
clearly seen in Fig.31.
At the holding of the same samples horizontally in liquid state the
change of lengthwise in-sample concentration was not detected (Fig.32). In this
case, to eliminate Coriolis acceleration influence, the samples were orientated
north south.
It is established that the process of the
redistribution of components in Pb-Sn liquid alloy decelerates but does not
stop with the increase of the time of holding up to three hours.
Thus, the equilibrium distribution of components within the given
alloy was not obtained under the given conditions. Consequently, an even
further alloy segregation into components is possible if the time of holding
increases.
The distribution of components by sample height in Bi-Cd liquid
alloy in capillaries 2-3mm in diameter and with the h of h=100mm, the
composition of the alloy being eutectic and the overcooling exceeding the
liquidus by 500C, is presented in Fig.32.
The mechanism of the transition from the initial homogeneous
distribution to inhomogeneous one is shown in Pict.33.
The nature and mechanisms of the redistribution of elements by
sample height in Bi-Cd liquid alloy prove to be qualitatively similar to the
same mechanisms in the melt of Pb-Sn.
A faster redistribution of components in Bi-Cd alloy in time can be
marked as the distinctive feature of the mentioned alloy.
It is interesting to point out that the degree of inhomogeneity
achieved in samples decreases with the rise in temperature. It may be explained
by the growth of convection and the acceleration of other types of mass
transfer with temperature rise.
Experiments show that eutectic melts are unstable under the
conditions of suppressed convection and tend to segregate by density into the
original components.
Alloys pertaining to Zn-Al and Zn-Al-Cu group are industrial cast
alloys utilized in pressure die casting.
Alloys with the content of aluminium from 3 to 11% are used more
frequently – for instance, Russian standard ZAM 4-1 and ZAM 10-5 alloys
(zink-based alloys with 4% aluminium and 1% magnesium vs. Al 10% and Mg 5%
content respectively). Therefore, alloys with aluminium content from 1 to 11%
were given the most consideration in our experiment /150/.
The change in the concentration of aluminium by the height of a
3.7%Al alloy sample (ZAM 4-2 alloy) at different holdings is to be seen in
Fig.34 and Fig.35. As it is clear from the picture, the degree of inhomogeneity
increases in samples with an increase in the time of holding in liquid state.
Among the peculiarities of ZAM 4-1 alloy we should mention the
character of the dependency of concentration on the sample height – the pattern
tending to linear one, - which indicates non-approximation to equilibrium.
Secondly, the process of the forming of chemical inhomogeneity in ZAM 4-1 melt
is decelerated, though the time order of the forming of inhomogeneity remains
the same.
Kinetic behavior of the transition from homogeneous to inhomogeneous
distribution of components in Zn-Al liquid alloys is shown in Fig.36.
The use of alloys with variant original
content of elements allows visually comparing the stated behavior
specificities. It follows from Fig.36 that the obtained degree of inhomogeneity
increases in its absolute value with an increase in the original average
admixture content.
If we determine the relative degree of sedimentation development as
the relation of the absolute difference in concentrations DC of one of the alloy components to the average content of the latter
in the given alloy C (DC/C), there emerge regularities that
merit our attention, - they are presented in Fig.37.
The data demonstrated in this picture suggest that there exists a
well-defined connection between the degrees of inhomogeneity achieved per
specified time and the diagram of state of the alloy under consideration.
Namely, there is an inflection of the dependency of the absolute inhomogeneity
quantity DAl = f (Al%) in the eutectic
concentration area, whereas the maximum is observed in the dependency curve of
the relative inhomogeneity DAl/Al = f (Al%) in the same area. Fig.35
also reflects the influence of alloy overheating or the time of holding in
liquid state on the degree of inhomogeneity obtained at the time of holding up
to 3hrs. The lower line in the picture corresponds to the overheating of 1500C,
the middle one – to the overheating of 1000C, the top line
representing the overheating of 500C. It is clear that the degree of
inhomogeneity achieved per specified time decreases with an increase in
overheating.
To prove whether the data obtained actually reflect the process of
segregation in liquid state, experiments that allow determining the position of
a sample while holding it in liquid state were carried out.
The results are shown in Fig.37. The middle horizontal line
corresponds to the distribution of elements by sample height at a horizontal
holding.
These data corroborate the trustworthiness of the developed method
and demonstrate that the segregation that is observed really progresses during
liquid state holding.
The obtained segregation regularities in
Zn-Al liquid alloys are substantially congenial to those that are typical of
Pb-Sn and Bi-Cd alloys. The difference lies in the fact that we have noticed no
approaching of the equilibrium in experiments on Zn-Al alloys during three
hours of holding at all. There is a pronounced tendency to further development
of inhomogeneity.
Al-Si liquid alloys are widely used in
industry, in which connection their study becomes a subject of particular
interest.
Sedimentation studies in liquid state in the alloys of Al-Si system
were effected in alundum and graphite capillaries, since liquid aluminium while
contacting with quartz reduces the latter to silicon, which rather distorts the
results.
Kinetic behavior of the transition from the original homogeneous to
inhomogeneous distribution of elements for Al-Si alloys are demonstrated in
Fig.38 as applied to Al-12%Si alloy. As we see, the same regularities of the
transition to inhomogeneous distribution as are observed in other eutectic
alloys occur in Al-12%Si alloy. Quantitatively, the process of redistribution
in Al-12%Si alloy goes faster than in Zn-Al alloys.
The character of the distribution of silicon by sample height does
not differ from linear one, practically, which also indicates that equilibrium
is not established and the system tends to further segregation.
The connection between the degree of the development of
inhomogeneity and the diagram of state is shown in Fig.39. As is obvious, the
observed regularities are close to those discovered in Zn-Al alloys, though the
extremum in the vicinity of the eutectic point turns out more distinct in Al-Si
alloys.
Sedimentation in liquid tin casting bronze with approx. 10% tin and
2% zink content was analyzed. The bronze under consideration also refers to
industrial alloys. Moreover, it belongs to alloys with a peritectic structure,
too. There is only one solid solution a on the basis of copper in the areas of the
indicated tin and zink concentrations in a solid alloy in the diagrams of state
of binary alloys.
However, the mentioned liquid tin casting bronze does not have a
monophase composition. Two solutions are formed in solid bronze on account of
the presence of tin and zink: the solution of tin in copper and the solution
of copper in tin and zink. These solutions have a variable composition. Thus,
the triple system of Cu-Sn-Zn differs from binary Cu-Sn and Cu-Zn systems.
There was no eutectic in the signalized area of concentrations in the triple
alloy.
The experiments aimed at studying sedimentation in bronze were
carried out in Tamman’s furnace equipped with graphite heaters.
The change of phase composition, namely the quantity of the solid
solution of a in microstructure, was the sole subject matter of our study of
samples. The distribution of a-solid solution by sample height is demonstrated in Fig.40. The
kinetics of the process of sedimentation is to be found in Fig.41.
One can see that the results obtained as far as liquid bronze is
concerned do not differ qualitatively from those achieved in eutectic alloys.
This gives us reasons to assert that sedimentation in liquid state is also
characteristic of alloys with a peritectic structure.
We studied carbon sedimentation in liquid
Russian standard cast iron LK-4 with 10% carbon content.
Experiments were performed in quartz capillaries in Tamman’s
furnace. Carbon and sulphur content was analyzed in the upper and lower part of
a sample. After holding the samples for 3 hours in liquid state at the
overheating of 50 degrees above the point of liquidus the difference in carbon
concentrations between the upper and lower sample points averaged 0.3%. The
difference in concentrations reached 0.8% in one of the samples. Naturally, the
concentration of carbon in the upper part of the sample was more saturated than
that in its lower part.
In connection with the fact that equation (184) prognosticates a strong
dependency of the achieved degree of inhomogeneity on the height h of the
sample, experiments were conducted in order to prove if such a dependency
exists.
Experiments were carried out on Zn-4% and Al-12%Si alloys by the
same method. The only distinction consisted in the height of quartz capillaries
for Zn-4% alloy being assumed equal to 50,100 and 200mm respectively, the
diameter equaling 1-3mm.
For Al-12%Si alloy, the height of the alundum capillary was
recognized as 50 and 100mm, its diameter being 1mm.
Sealed samples were held for 3 hours in liquid state at the
overheating of 500C above the liquidus temperature of the given
alloy.
Then the samples were water hardened and subjected to analysis.
The results are tabulated below (Table 21).
Table 21. The Degree of Inhomogeneity in Liquid
Alloys Depending on the Sample Height
Sample
height, mm |
Alloy
composition |
Cupper,
%, experiment |
Clower,
%, experiment |
Cupper,
%, calculation
by (183) |
Clower,
%, calculation by (183) |
50 |
Zn-4%Al |
4.1 |
3.8 |
4.1 |
3.8 |
100 |
¸ |
4.75 |
2.8 |
4.2 |
3.8 |
200 |
¸ |
5.0 |
2.8 |
4.4 |
3.6 |
50 |
Al-12%Si |
12.4 |
11.3 |
12.6 |
11.5 |
100 |
¸ |
13.2 |
11.1 |
12.5 |
11.4 |
Note:
calculation values are derived for equilibrium theoretic distribution.
As is seen from the table, the experiment corroborates a definite
increase in the degree of inhomogeneity obtained in liquid alloy with the rise
of the height of the sample. However, experimental data by far exceed
calculation data, though our calculation was done for equilibrium distribution.
Granting that the majority of the conducted experiments as regards
sedimentation in liquid alloys demonstrated that equilibrium was not
established, we carried out experiments with a prolonged time of holding of the
following alloys: Zn -5%Al; Zn - 10%Al; Zn - 15%Al in liquid state for 24, 48,
72, 96 hours.
The principal result of the experiments in question was that
equilibrium was not established even after 96 hours of holding, so the
redistribution of elements continued.
Aluminium content in the upper and lower parts of samples after such
a long-term holding in liquid state is represented in Table22.
Table 22. Aluminium Content in the Upper and Lower
Parts of Zn-Al Alloy Samples 100mm Tall Subjected to a Prolonged Holding
Alloy
composition |
The time of
holding in liquid state, hrs |
Aluminium
concentration in the upper part of the sample, % |
Aluminium
concentration in the lower part of the sample, % |
Zn-5%Al |
24 |
14.0 |
1.1 |
48 |
16.4 |
0.8 |
|
72 |
17.7 |
0.8 |
|
96 |
19.3 |
0.7 |
|
Zn -10%Al |
24 |
22.2 |
5.9 |
48 |
23.9 |
5.2 |
|
72 |
26.0 |
4.4 |
|
96 |
27.9 |
4.0 |
|
Zn-15%Al |
24 |
26.6 |
7.1 |
48 |
28.7 |
6.4 |
|
72 |
29.7 |
5.9 |
|
96 |
33.8 |
5.1 |
As it is clear from Table 22, the redistribution of aluminium
continues even after a 96-hour holding with quite a high intensity. It brings
us to predict the complete segregation of the given liquid alloy into its
original components at a sufficiently prolonged holding in liquid state.
The observed passage of chemical inhomogeneity into structural one
with the forming of conglomerates proved to be a new fact of utmost importance
in our experiments.
Fig.42 shows the microphotographs of the structure of the original
solid samples (a), as well as the samples that have been held for 24, 72 and 96
hours correspondingly (the respective photographs b, c and d).
Considerable changes in the microstructure of the upper zone of the
samples should be noted while examining these photographs.
After the expiry of 24 hours of holding in liquid state we observed
the origination of drop-shaped formations – we termed them as conglomerates -
of the phase rich in aluminium (Fig.42, b).
The appearance of cut crystals was observed along with the formation
of round conglomerates on the expiry of 72 hours of holding (Fig.42, c).
Having been held for 96 hours, crystals enlarge (Fig.42, c). The
composition of crystals corresponds to the intermetallic compound of Zn-Al type
that is lacking in the diagram of state of the given alloy.
Thus, the possibility of the transition of the microinhomogeneous
structure of liquid alloys into the macroinhomogeneous structure of solid
alloys was experimentally proved.
Sedimentation experiments let us
calculate the dimensions of Brownian particles in liquid alloys by equation
(185). The Brownian motion theory applies to any particles whose heat and
gravity energies are comparable. The mentioned equation is inactive toward the
dimensions of sedimentator particles and therefore quite applicable to our
calculations, - to the initial stages of the process of sedimentation, at
least.
Let us remark that when deriving (185) we make an assumption
concerning the difference between the average melt density and the density of
sedimentator particles (clusters and their conglomerates).
Assuming the density of particles equal to that of the pure
component of the eutectic, we cause certain indeterminancy, which is
unavoidable at the current level of knowledge. By prior estimation, the
indeterminancy of the value Δ imparts a relative error of 3-10% into the
calculation of r by (184).
By assuming that cluster conglomerates enriched b+
By one of the components do exist in melts, which results from our
experimental data, we get the following dimensions of cluster conglomerates in
the examined alloys on the basis of (184) (v. Table 23).
Table 23. The Dimensions of Conglomerates with the
Predominance of One of the Components in Liquid Alloys after Holding in Liquid
State within Gravity Field at Suppressed Convection for 3 Hours
Alloy |
Second
element content, % |
Temperature,
C |
Conglomerate
radius, calculation, nm |
Conglomerate radius, calculation, nm |
Bi-Cd |
50 |
180 |
rBi
= 9.05 |
|
|
|
|
|
|
Pb-Sn |
60 |
200 |
rBi
= 4.0-4.9 |
rSn
= 5.6 |
|
|
250 |
rBi
= 3.6-4.2 |
|
|
|
350 |
rBi
= 2.1-3.7 |
|
|
|
|
|
|
Al-Si |
6.0 |
650 |
rSi
= 11.2 |
|
|
6.9 |
650 |
rSi
= 11.5 |
|
|
8.0 |
650 |
rSi
= 11.5 |
|
|
9.2 |
650 |
rSi
= 12.5 |
|
|
10.8 |
650 |
rSi
= 12.1 |
|
|
12.1 |
650 |
rSi
= 12.1 |
|
|
13.3 |
650 |
rSi
= 11.3 |
|
|
13.8 |
650 |
rSi
= 11.4 |
|
|
|
|
|
|
Cu-Sn |
5.0 |
1100 |
rSn
= 4.2-2.1 |
|
|
5.0 |
1050 |
rSn
= 8.1-7.0 |
|
|
|
|
|
|
Cu-Sn-Pb |
5.0%Sn+4.9%Pb |
1050 |
rSn
= 3.8-5.6 |
|
|
|
1100 |
rSn
= 1.8-2.7 |
|
|
|
|
|
|
Fe-C |
4.2 |
1200 |
rC
= 2.7-4.9 |
|
If we take into account the relatively large effective dimensions of
Brownian particles, according to Table 23, that exceed cluster dimensions by
order of magnitude approximately, we may conclude that the dimensions of
Brownian particles in alloys considerably exceed those of separate clusters as
early as on the expiry of three hours of holding.
In case of the monatomic structure of liquid alloys no noticeable
inhomogeneity can arise in liquid samples 10-100mm tall, which we demonstrated
earlier /142, 150, 153/.
The presence of microgroups with the radius of 1-10nm, on the
contrary, must cause a certain slight chemical inhomogeneity in samples 100mm
tall at the level of 0.01-0.1%, by calculation. However, our experiments
indicated a much higher degree of inhomogeneity present. Therefore, the results
of our experiments overpassed the limits of the simplified theoretical
alternative: clusters or separate atoms.
Concentration inhomogeneity obtained in the samples of various
alloys in capillaries 50-100mm in height under the conditions of holding in
liquid state within gravity field at suppressed convection reaches tens of
percents. It also leads to the forming of structural inhomogeneity represented
by drop-shaped conglomerates. Moreover, even after holding in liquid state up
to 96 hours the equilibrium distribution is not achieved, so the process of alloys
segregating into the original components continues.
The achieved results are unprecedented by the obtained inhomogeneity
in liquid state. Such a considerable inhomogeneity was not achieved even while
conducting centrifuging experiments.
There are no precedents to the discovered tendency of the
continuation of liquid alloys segregation into the original components. Both
sedimentation theory and the theory of centrifuging prognosticate quite a rapid
establishment of equilibrium. In reality, equilibrium was not struck. This fact
cannot be interpreted either.
What processes can lead to such a substantial segregation of alloys
in liquid state?
The influence of oxidation and admixtures was eliminated by way of
testing. The influence of capillary material was also eliminated through the
varying of the materials.
We excluded the possibility of thermal diffusion and barometric
diffusion effect, too, by carrying out special experiments. The effect of
shallow diffusion was easy to eliminate, too. A series of other less probable
processes, such as Marangony surface convection, was discussed and eliminated
/102/.
As a result of such elimination, we can name two basic causes of
alloys segregation into the original components:
1.
The lack of convection in samples.
2.
Structural inhomogeneity of melts in liquid
state.
The conclusion concerning the influence of the lack of convection
points out that the given factor is primary in the formation of liquid alloys
under earth conditions. To all appearances, if it were not for convection,
there would not be such diversity in alloys, diagrams of state and structures.
Some of the diagrams of state of binary alloys would become unrecognizable.
This is convection, and not diffusion, that sustains many of the existent
alloys in liquid state as a macroscopic homogeneous mix.
As for structural inhomogeneity, the presence of the elements of
matter - clusters - and the elements of space in liquid metals provides the
incensive for the process of segregation to set on. Yet if there were nothing
but clusters in liquid alloys, segregation process would have established
equilibrium at a very low inhomogeneity degree with the difference in
concentrations in a 100mm tall sample that does not exceed 0.1% and stopped,
because the process of diffusion, according to the theory, must balance the
subsequent process of the redistribution of single clusters.
Still, unexpectedly for us, the process of segregation in
experiments developed very extensively, so the process continued even on the
expiry of 96 hours of holding in liquid state, although the difference in
concentrations 10-20 times as large (1000 – 2000%) was reached in samples and
considerable structural changes were observed.
The existence of clusters only cannot explain such a result.
Besides, the given result does not keep within the limits of the monatomic
alloy structure theory. Only the formations that are larger than clusters can
ensure such a speed and depth of the processes of spontaneous segregation of
liquid alloys.
We suppose that here we handle a synergetic process going
simultaneously at several levels and developing in time. This process causes
the formation of a whole hierarchy of liquid state structures. The term of the
hierarchy of structures of liquid state was introduced by the author in works
/143, 152, 154, 155, 141/ on the basis of sedimentation researches of our own
in liquid alloys.
A vertical displacement of like clusters under the influence of
gravity and Archimedean force triggers the process. I.e. the process starts
according to the Brownian motion theory.
However, the Brownian motion theory does not allow for the
interaction between Brownian particles. Judjing from experimental results, such
an interaction occurs and determines a further development of segregation
process in liquid alloys.
A slow displacement of like clusters in one direction cannot but
result in a phenomenon similar to orthokinetic coagulation. The frequency of
encounters of like clusters and the time of their side-by-side stay increase in
the process of such a motion.
Under such conditions, even an inconsiderable prevalence of
interaction forces between like particles (A-A and B-B forces symbolically, the
composition of a binary alloy being AB) over the forces of interaction between
unlike particles AB will inevitably lead to the forming of agglomerations of
like clusters, or cluster conglomerates, inside liquid. The above-mentioned
hierarchy of structures of liuqid state arises in this way.
Such processes of conglomeration of like clusters go at any moment
of time within any liquid alloy but unstable cluster conglomerates get easily
decomposed through convection. This is convection that sustains liquid alloys
in an as-homogeneous mixes of heterogeneous clusters condition.
The given data is corroborated by the already ample data of orbital
experiments /102/, where sedimentation processes in melts turned out to be
unexpectedly significant under the conditions of the lack of natural
convection.
Diffusion, which used to be considered as the principal motive force
of alloy forming, participates in the mentioned process to a certain extent,
yet it can resist neither gravity nor Archimedean force, to say nothing about
the forces of interaction between like clusters.
The dimensions of the elements of matter in liquid alloys – i.e.
clusters – that numerous authors, the author of the present work included,
calculated by a variety of methods, fluctuate within the limits of 1-10nm,
whereas the dimensions of drop-shaped conglomerates formed as a result of
sedimentation amounts from 0.1 to 1.0mm. Consequently, there can exist three or
four hierarchical dimensional levels of the agglomerations of like clusters
between clusters and visible conglomerates in liquid alloys within the limits
of 10nm-0.01mm. These levels correspond to the dimensions of colloidal
particles.
Thus, there may arise the following hierarchical levels of cluster
conglomerates in the process of sedimentation in liquid alloys.
Separate clusters – 1-10nm dimensional level
Cluster conglomerates - 10-100nm dimensional level
Colloidal cluster conglomerates - 100-1000nm dimensional level
Colloidal microdrops - 1000-10000nm (1-10mkm) dimensional level
Drop-shaped macroconglomerates - 10-1000mkm dimensional level.
Colloidal microdrops and drop-shaped conglomerates are the only two
largest dimensional levels of conglomerates registered in the course of the
experiment. Smaller cluster aggregations act as a constituent part of solid
alloys in the process of crystallization and remain undisclosed by existent
methods. We should underline that cluster conglomerates are unstable, labile
formations at any dimensional levels so they can exist and be developing only
if convection is lacking. However, under certain conditions these formations
may become stabilized and even form new phases. And, of course, conglomerates
can directly affect the structure of cast metal, which was demonstrated
earlier. Still, the basic result of the forming of conglomerates at all
dimensional levels is the acceleration of the process of segregating this or
that liquid alloy into its original components or solutions.
The growth of cluster conglomerates, as well as the competitive
growth of crystals, is possible at different levels: conglomerates can grow
both by way of separate clusters adjunction and the coalescence of neighboring
conglomerates.
Therefore, cluster conglomerates are able to dimensionally evolve
very far from the original clusters while remaining nothing but a constituent
of liquid.
In the sequel, conglomerates can evolve into new phases.
In the process of sedimentation conglomerates behave as indivisible
formations of far larger dimensions than clusters. Correspondingly, the larger
conglomerates are, the faster they emerge, since the speed of the floating of
particles, according to Stokes’ equation, is proportionate to the square of
their radius. So sedimentation process, developing synergetically, i.e. going
simultaneously at different dimensional levels, accelerates and develops
towards the complete segregation of the melt into its original components.
Such complete melt segregation was not achieved in our experiments,
yet we noted a definite tendency to the development of the process toward the
segregation of melts. The obtained twentyfold segregation is quite a quantity,
though, that seems to be comparable to the degree of segregation reached after
approx. ten passings when purifying metals of admixtures by the zone refining
method /68/.
Consequently, the phenomenon that we disclosed – that of liquid
alloys segregation developing within gravity field – can be applied both
scientifically and practically as the real alternative to the method of zone
refining of metals and alloys, as well as a way of segregating certain alloys
into their original components.
A wide spectrum of the phenomena of metallurgical heredity
corroborates the inference concerning the complex hierarchical structure of
liquid alloys. The above-cited data of Ch.10 on sedimentation experiments
lasting many hours confirm both the stated conclusion and the inference that we
made earlier: the basic structural elements of liquid alloys are extremely
stable in time.
It was shown above that the period of cluster existence equals the
period of the existence of liquid state. Sedimentation experiments lasting
24-96 hours corroborate the latter conclusion within the limits of the
experiment– up to 96 hours, at any rate.
Metallurgical heredity is a highly complicated phenomenon. Most
often heredity is meant when we touch upon the inheritance of a crystalline
structure. However, the phenomena of metallurgical heredity have a wider
implication than a structural factor as such.
Mechanical properties, as well as the tendency to cracking,
shrinkage, and other technologically significant properties, are inherited
under certain conditions, too.
Therefore, it would be more correct to speak about the spectre of
metallurgical heredity phenomena. The common feature of the given
manifestations, widely different one from another, concerning the connection
between the structure and properties of liquid and solid metals is that these
or those properties or alloy parameters are passed from charge materials to
final castings through liquid state.
Consequently, when broaching the phenomena of metallurgical
heredity, we deal with a certain mechanism, or mechanisms, of data transfer
from the original solid charge through melting, liquid state and
crystallization to a final casting.
The feature that integrates the phenomena of metallurgical heredity
is their wide prevalence and practical importance, on the one hand, and their
being unstable, labile, on the other hand. In particular, it is well known that
some manifestations of metallurgical heredity can be eliminated by the
overheating of liquid metal, for instance, and its thorough mixing. It refers
to structural heredity in the first place.
The problem of heredity carriers in liquid metals, as well as the
hereditary information transfer mechanism, is of practical importance. The
problem mentioned has been given insufficient scrutiny so far.
Traditionally, solid insoluble particles of various inclusions that
are strictly indeterminable are reckoned as information carriers in liquid
metals /10/.
V.I.Nikitin systematized the ideas on the carriers of hereditary
information adding clusters to admixture particles, as well as cluster
conglomerations and other elements of the structure of melts /105/.
Unfortunately, the mechanism of passing hereditary information remains unknown.
The zone of the ‘responsibility’ of this or that hereditary information carrier
for this or that type of hereditary phenomena has not been determined either.
V.I.Nikitin also introduced the concept of gene engineering for
melts and castings that is based upon the control of the structure and
properties of castings with the application of metallurgical heredity phenomena
and modifying; yet first it is necessary to study the mechanisms of
transferring variant hereditary features from liquid to solid state in order to
practically apply gene engineering to melts.
Our research shows that the particles of modifiers of the second
type can preserve the shell of solid phase on their surface at temperatures
higher than the melting temperature of the alloy. So a considerable overcooling
is required to liquidate these particles, as well as time to deactivate the
surface of such inclusions. Thus, the operation mechanism of second-type
modifiers that was disclosed here is one of the mechanisms of structural
heredity.
Undoubtedly, clusters and intercluster splits fuction as the
carriers of hereditary information, too. This follows from the melting and
crystallization theories initiated by the author that reveal the connection
between liquid and solid state structures. However, clusters carry but the
principal information of the crystalline lattice type and the way it is built.
Clusters cannot carry information concerning the number of crystals, their
dimensions, etc. It is another hierarchical level of metal structure – and
another informational level.
More specific information may be carried by different cluster
conglomerates and conglomerates of intercluster splits that are often to be
found in liquid alloys, which was proved by our sedimentation experiments in
melts. Possible variants of such conglomerates are actually unlimited both by
their composition and their dimensions, structure and other parameters.
Therefore, the possible variants of their hereditary influence are also
extremely diverse and unpredictable as yet. Unlike clusters proper, their conglomerates
are far less stable. The overheating of the melt, as well as natural convection
or artificial intermixing of the melt, etc. may cause their decomposition.
Thus, many types of metallurgical heredity are highly sensitive to
the overheating and intermixing of melts.
Metallurgical heredity researches closely relate to the studies of
the structure of melts and appear to be quite challenging as far as the forming
of castings with controllable high nonequilibrium properties is concerned.
Proceeding from
the most general considerations of the structure of real bodies that include
both the elements of matter and the elements of space, we succeeded in
developing a new unified non-contradictory theory of the melting and
crystallization of metals and alloys.
The given theory differs fundamentally from the existent theory and
turns out to be incompatible with it.
A surprising persistence of science, accumulated for more than a
hundred-year period of the existence of crystal nucleation theory, with respect
to this field keeps the new theory from gaining ground. Prevailing ideas will
obviously offer a strong and long-term resistance.
However, the drawbacks of existent theory seem to be so substantial
that only the lack of a more or less reasonable alternative can account for the
existence of the former for such a long time period. Moreover, the lack of
ideas on a far more complex, or even a fundamentally different, structure of
aggregation states and real bodies in general aggravates the situation.
Still, on the expiry of a certain time, with the accumulation of new
data, the former theory will inevitably be relegated to the past.
We ought not to blame anyone for the founding of the wrong crystal
nucleation theory, since the structure of the states of aggregation of matter
used to be presumed monatomic until recently, while the concept of the inner
elements of space was never introduced. The ideas of the flickering nature of
the interplay of material and spatial elements inside real systems were
non-existent all the more. Those are the mentioned ideas that prove to be basic
ones for the new theory of melting, crystallization and the liquid state of
metals.
The given concepts will take the longest time to engraft, for the
store of knowledge in this sphere is too insufficient being confined to the
material of the present book. Still, the road is clear, and anyone can take it.
The methods of experimental research of the elements of space,
vacancies excepting, are to be established yet.
We may hypothetically propose to study subtle oscillations of
electrical resistivity in capillary samples of liquid metals and alloys. The
flickering nature of the interaction between the elements of matter and space
in liquid metals at the level of clusters and intercluster splits must result
in the flickering character of electrons passing through the melt, too.
Certainly, the samples should be very small; otherwise subtle oscillations of
electrical resistivity with the order period of a billionth fraction of a
second in large samples will level because of mutual multiple superposition of
such oscillations.
The discrete character of the elementary acts of melting and
crystallization can also become a subiect of experimental study, as well as the
discrete mechanism of liquid metals fluidity. The measurement of the elementary
amount of the latent heat of crystallization or melting could generate a lot of
new actual information on the specified processes.
These are precision experiments requiring precision instrumentation
that the author was unable to get on account of science subsistence conditions
at this time in this country. Somebody may be luckier, though.
Certainly, the importance of the above-stated methods of precision
sedimentation experiments oriented on the study of liquid alloys structure,
cannot be denied.
In fact, this is the first direct method of measuring the dimensions
of sedimentator units of matter in liquid alloys that was specially dedicated
to serve the given purpose and proved highly sensitive to the dimensions of
sedimentator particles in liquid alloys.
The tendency displayed by many liquid alloys to segregate into their
original components – we discovered it in the course of our experiments –
within gravity field is an essentially new experimental result that can also be
applied to industrially purify metals of admixtures instead of the method of
zone melting of metals.
The results that we obtained in this field have been repeatedly
published including the publications in magazines issued by the USSR Academy of
Sciences and Russian Academy of Sciences. The comment was but favorable.
In conclusion, we would like to make a hypothesis concerning the
applicability of the equation of state to condensed aggregation states – liquid
and solid.
The attempts at adapting the equation of state to liquids and solids
were, and are, continuously being made. On the one hand, there are no
theoretical bars to it – which brings no practical results either, on the other
hand.
When calculating vacancy gas pressure in solids earlier, it was
stated and corroborated by quantitative data that the equation under analysis
is applicable to vacancy gas pressure, at least, at the point of melting, with
high accuracy.
In this connection, there arises a hypothesis that the equation of
state PV =RT should be applied if we allow for the existence of
different types of the elements of matter and space at many levels.
For gases, the measured pressure and volume turned out to
incidentally coincide with the pressure and volume of material and spatial elements
that are characteristic of the given state. Namely, the intrinsic pressure of
the elements of matter and space in gases coincided with that measured by
customary equipment.
Our hypothesis of the applicability of the equation of state
consists in the following: this is the intrinsic – and not ambient – pressure
of the elements of matter and space in the state specified that must be taken
into account while applying this equation to condensed states.
The concept of the intrinsic pressure of the elements of space in
solids and liquids, and at other levels of the structure of real bodies, is a
new one.
To know how it can be measured is a new experimental trend in the
research of matter-space real systems, too. To be more exact, a series of new
trends should propagate, since specific elements of matter and space are
peculiar to every level of matter-space systems, as well as their specific
internal interaction and its parameters, intrinsic pressure including.
Certainly, various forms of the elements of space characteristic of
every form of the elements of matter are yet to be disclosed to build up a
system similar to the periodic law of the elements of space at corpuscular
level and at other levels, too. There is a demand for the periodic law of
matter-space element complexes.
An extra hypothesis refers to the flickering nature of the
interaction between the elements of matter and space. Evidently, flickers in
the diversity of their forms, oscillatory, rotary and other flickers including,
are typical of many levels of real systems’ structure. Interatomic interaction
in solids and liquids and molecules is very likely to have a flickering nature,
which determines all practically important properties of solids - for instance,
durability, plasticity, electric conductivity, density, etc. etc. The
parameters of spatial elements, including the characteristics of their
flickers, are specified in the present work for the liquid state of aggregation
of matter only. Solid state has parameters of its own. They are to be determined
as yet. We are to focus on the contribution of the latter to the real
properties of solids, which will surely turn out to be as significant as the
contribution of material elements – atoms - within solid crystals.
Concluding the book by a series of hypotheses, the author suggests
that all interested people – both the supporters and adversaries of the
proposed new concepts - should volunteer and test the new potentialities. New
ways always yield new results.
Good luck!
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Igor V. Gavrilin
Melting and Crystallization of Metals
and Alloys
Table of
Contents
Chapter 1.
Melting and Crystallization. State of Issue
1.1. Thermodynamics of melting and crystallization
1.2. Structural theories of melting and crystallization and the
structure of liquid metals
1.3. Statistic and monatomic theories of liquid state of metals
1.4. Models of microinhomogeneous structure of liquid metals
1.6. The elements of the thermodynamic theory of crystalline
centers nucleation
1.7. The elements of the heterophase fluctuations theory as applied
to the theory of crystallization
1.9. Of the growth of crystals theory
1.10. Of the correlation of the structural theory of
crystallization with the theory of hardening
1.11. Statistic theory of crystallization
2.4. The correlation of melting centers with the centers of
crystallization
Chapter 3.
The Mechanism of Metals and Alloys Melting Process
3.2. Calculating the elementary act of melting
3.3. The structure of liquid metals at the point of their formation
3.4. Calculating cluster dimensions in liquid metals at the
melting temperature
3.5. Calculating the dimensions of spatial elements in liquid
metals at the melting temperature
3.6. Calculating the energy of clusters and intercluster splits
motion in liquid
3.7. Calculating the point of metal melting
3.8. Calculating the content of activated atoms in liquid
3.10. The period of cluster existence
Chapter 4.
The Change of Liquid Metals Structure at Heating and Cooling
4.2. The modification of cluster dimensions with the change of
temperature
5.1. Of the connection between the structure and characteristics of
liquid metals
5.2. The mechanism of fluidity in liquid metals
5.3. Viscosity in liquid metals
5.4. Self-diffusion in liquid metals
5.5. Comparing the effects of mass transfer in various states of
aggregation of matter in metals.
5.6. Admixture diffusion in liquid metals and alloys
5.7. Admixture diffusion in liquid iron
5.8. Of the change of coordinating numbers at melting
5.9. Of the change of the electrical resistivity of metals at
melting
Chapter 6.
Of the Mechanism of Crystallization of Metals and Alloys
6.2. The formation of crystallization centers
6.3. The overcooling problem at crystallization
6.4. Spontaneous and forced crystalline centers nucleation in
liquid metals
6.5. The frequency of crystalline centers nucleation
6.6. Time factor at crystallization
6.7. The problem of mass crystalline centers nucleation
6.8. The competition theory of crystallization
6.9. Of the change in the volume of metals at melting and
crystallization
6.10. The formation of shrinkage cavities and blisters in metals
and alloys
Chapter 7.
Alloy Formation and the Structure of Liquid Metals
7.1. Of the mechanism of the formation of liquid alloys
7.2. The point of metal dissolution and contact phenomena
7.3. The formation of alloy structure in liquid state
7.4. Cluster mixing stage in the formation of liquid alloys
7.5. The stage of atomic diffusive mixing
7.6. The stage of convective mixing in the formation of liquid
metals
7.7. Of the function of gravity in liquid metals formation
7.8. The production of locally inhomogeneous alloys
7.9. The formation of alloy hardening interval
7.10. The structure of liquid metals with the unrestricted
solubility of elements
7.12. The structure of liquid eutectic alloys
Chapter 8.
The Structure and Crystallization of Liquid Cast Iron
8.1. General data on iron-carbon alloys
8.2. The formation of liquid cast iron structure
8.3. The structure of liquid cast iron
8.4. The peculiarities of cast iron crystallization. The formation
of gray cast iron
8.5. The formation of white cast iron
9.1. General data on modifying
9.4. The choice of the amount of modifiers of the first type.
Gas-like modifying mechanism
9.9. Of the characteristics and choice of second-type modifiers for
alloys
Chapter 10
Experimental Research of Liquid Alloy Structure
10.3. Of the Brownian motion experiments
10.4. Sedimentation research procedures of the structure of liquid
alloys
10.5. Sedimentation in Pb-Sn liquid alloy
10.6. Sedimentation of the elements of matter in Bi-Cd liquid alloy
10.7. Sedimentation in Zn-Al liquid alloy
10.8. Sedimentation in Al-Si liquid alloys /149/
10.9. Sedimentation in liquid casting bronze /152/
10.10. Sedimentation in liquid cast iron /144/
10.11. The influence of the sample height upon the degree of
inhomogeneity obtained in liquid alloys
10.12. Conglomeration in liquid alloys as a result of sedimentation
10.15. Of the mechanisms of metallurgical heredity
A new general theory of the crystallization of metals
and alloys was worked out. It is based on a principle - never yet discovered -
of the structure of real physical bodies, metals as well as any others.
According to the new principle, metals and alloys, as
well as any other bodies, contain not only the elements of matter, i.e. atoms,
but also the interior elements of space (unknown up to the present moment),
that have the characteristics of vacuum. The elements of matter and space
inside physical bodies are closely interconnected, being at a continuous
interplay. The features of both the matter and space elements determine any
characteristics of physical bodies. Each configuration of the elements of
matter conforms to the corresponding structure of the elements of space that is
inseparably linked with the former. The interior elements of matter and space
are indivisible in the sense that they do not exist separately. The division
between the interior and exterior spatial elements is relative.
There exists the hierarchy of the
levels of real bodies structure and the hierarchy of their states, e.g. the
states of aggregation of matter. Each of these levels can be distinguished by
having its own pairs of elements of matter and space, interconnected and
interacting.
The main type of interaction between spatial and
material elements at macrolevel is the flicker of various elements of space
inside real bodies during the process of heat oscillations of the elements of
matter and space and the respective flicker of connections between the elements
of matter. The flickering type of interplay between the elements of matter and
space essentially determines all the properties of real physical bodies,
including durability, density, plasticity, etc. of solids, fluidity and
viscosity etc. of liquids.
Any state, except for its dominant form of the
elements of matter and space, includes latently the attributes of other
configurations of the elements of matter and space, inherent in other possible
states of the physical body.
Dominant and latent elements of matter and space are
constantly interactive, mutually expulsive. Such extrusion provides the inner
rearrangement of a system in conformity to the environmental conditions,
secures the preparedness of a system for the change of the dominant
configuration of spatial and material elements, i.e. the change of the states
of aggregation of matter.
The principles mentioned above enable the author to
initiate a new quantitative theory of the melting and crystallization of
metals.
It was demonstrated that the process of melting
results from the interaction of the elements of matter and space in solid state
and, in its turn, is the process of the formation of new moving clusters,
dominant only in liquid state of the elements of matter - and flickering F
splits, dominant in liquid state of the elements of space only. A definition
and equation of the elementary act of melting were put forward. All the
parameters of the elements of matter and space in liquid metals were calculated
- their dimensions, the energy of oscillations, parameters of interaction, etc.
A theory of the structure of liquid metals was established on this basis.
Considering the presence of paired interactive
elements of matter (clusters) and space (intercluster splits) in the structure
of liquid metals, the author formulates a new theory of crystallization, where
the elementary act of crystallization represents the act of accretion of any of
the two adjoining clusters and the shutting of the isolated element of space -
the intercluster split. Calculations show that the total area of similar space
elements in a mole of any liquid metal or alloy mounts from 100 to 500 sq.m.
So, crystallization is not accompanied by the forming of new section surfaces,
as it is usually accepted, but, on the contrary, by the closing of a huge area
of the interior surfaces of space elements.
This observation helps solve the problem of the
critical radii of new-phase nuclei within the theory of crystallization. In its
turn, the elimination of the artificial problem of the critical radius
annihilates the necessity of the heterophase fluctuations theory.
As a result, the new theory of crystallization is
entirely different from the already existent one with respect to its conformity
to the facts. In particular, current theory admits that the spontaneous
formation of crystals is actually impossible, since it requires a cooling of
several hundred degrees[1],
which can hardly be put into practice. The new theory states that
crystallization occurs easily and without hindrance; spontaneous
crystallization is always the main type of crystallization, and the influence
of external factors and admixtures accelerates or hampers spontaneous
crystallization, yet by no means replaces it. Overcooling, as the new theory
specifies, fulfills but a thermal function of a factor necessary for absorbing
latent heat of the elementary act of crystallization within the volume of
clusters crystallized. The process demands a cooling of but a few degrees,
sometimes - even 1/10 of a degree, which is observed in reality.
Another theory closely associated with the theory of
crystallization - the theory of accretion of spatial elements in the process of
crystallization – was elaborated. It is founded on the new quantitative theory
of shrinkage. The author points out that during the process of crystallization
shrinkage is wholly determined by the presence of spatial elements in liquid
metals.
A competition theory of crystal growth proposes that
the process of crystal growing involves not only separate atoms nor clusters as
the building material, but also microcrystals of various dimensions.
Corresponding calculations are made. It is shown that the structure of casting
closely relates to the factor of hardening rate; formulas are derived
respectively.
There was suggested a new theory of liquid alloys
modifying. It allows for the actual complicated structure of liquid alloys and
the real crystallization mechanism. The methods of calculating the processes of
modifying, as well as the choice of modifiers, are supplied.
Melting and crystallization
concern regular technological processes in the sphere of metallurgy and foundry
practice. The cycles of melting - alloying - thermal-temporal processing -
refining - teeming - crystallization are daily repeated time and again in each
of the thousands of foundry and metallurgic shops all over the world. These are
adjusted processes that seem to be explored to the minute detail. Everything
measurable referring to the processes indicated was measured long ago,
everything left to study underwent exploration and was entered into monographs,
manuals and textbooks. So what is the aim of writing this book?
Strange as it might seem, with all the knowledge of many practical
details, there is no answer to the simplest though cardinal questions about the
essence of the processes of melting and crystallization. The questions are the
following:
1.
What is the cause of melting and
crystallization?
2.
What is the way they act?
3.
How are melting and crystallization connected,
what is the relation between liquid and solid metals?
4.
How do the mechanisms of melting and
crystallization influence the structure and characteristics of castings and
ingots?
The book is meant to supply
answers to these questions.
Indeed, it has not been found out
yet why solid metals start to melt while heating at a certain temperature.
Why with the cooling of almost the
same temperature liquid metals start crystallizing?
What causes these processes?
How do they develop, in other
words, what is their mechanism?
What is the structure of liquid
metals?
And how is all that connected with
the structure and properties of solid metals?
The knowledge of answers to these
questions is extremely important in scientific-theoretical respect, since it is
time to fill in these absolute blanks of science. It is useful with respect to
practice, too, as soon as the knowledge of answers to such questions enables us
to control the processes of melting and crystallization of metals and alloys
with more comprehension and efficiency.
The research on characteristics
and structure of liquid metallic melts, the study of their relation to the structure
and properties of solid metals and alloys, the exploration of liquid-solid and
v. v. transition processes - these closely interconnected fields traditionally
attract a large number of researchers. A distinctive feature of the scientific
development of the sphere in question can be formulated as the extreme
unevenness of the research of its parts. I.e. solid metals are explored much
better than liquid ones.
The lack of one synthetic idea integrating both
liquid and solid states, both melting and crystallization may be considered as another bar to
melting-crystallization research. Really, the processes of melting and
crystallization must be interrelated, if we follow simple logic. Moreover, they
are reversible at any moment and a priori must be, in general, the mirror image
of each other. Correspondingly, the theory of crystallization ought to be the
theory of melting at the same time, and v. v.
Yet there is no unified general
theory of melting-crystallization with the exception of the remotest thermodynamic
approach which by no means allows looking into the details of these processes.
We have a whole bunch of
crystallization theories whereas the incomparably lesser amount of theories of
melting fails to counterbalance them, as if those were completely
non-correlative processes. In this connection it is apparent that this is an
inadmissible thesis. A widely acknowledged phenomenon of metallurgical heredity
manifests itself in various parameters, i.e. in the alloy structure succession
before and after melting and crystallization. Such a phenomenon indicates the
existence of a certain structural parallelism of liquid and solid states. So a
thorough, well-balanced theory of melting-crystallization ought to explain the
mechanism of this congeniality.
Notwithstanding the obviousness of
these points to every scientist and practical experts engaged in
melting-crystallization research, the response to them are almost a century
overdue, if the reading is taken from the initiation of Tamman’s theory, and
even more, counting from Gibbs’ works on thermodynamics of the nucleation of
new-phase centers. During the last century, numerous were the researchers who
channeled their energy into the field of melting-crystallization; numerous
issues were joined except for the problems raised above.
The book furnishes clues to all of
them.
In the author’s opinion, the existence of such
questions and problems underlines the priority of present difficulties. These
problems are not to be solved by ordinary tools. There is a demand for
fundamentally new approaches to these objectives, new ideas, new concepts and
new solutions. Alongside with the other provisions, the book is dedicated to
the working out of new conceptions, ideas and solutions in the sphere of
melting and crystallization. Necessity arising, the newfound data can outspring
the limits of the book.
The book weds theory with practice everywhere - with
the application of the newborn theory to metallurgical practice and casting
production, in particular. Yet metallurgical and casting practice seems to be
overloaded with the core difficulties listed above. For instance, it is of
practical importance to know how structural ties inside solid and liquid metals
become established, how the primary casting characteristics like fluidity and
shrinkage get formed. Rather a troublesome circumstance for the foundry
science, for example, is the total lack of any theory of shrinkage at
crystallization to be coupled with the problem of shrinkage cavities formation
and that of porosity.
A theory like that deploys below, supported by
numerical calculations. It will be shown that the wholly applied, to all
appearance, phenomenon of shrinkage, as many other processes, is inseparably
linked with the mechanism of melting and crystallization, the structure of
liquid and solid metals, the structure of physical bodies at large.
Under the general theory of melting and
crystallization the theories of diffusion, viscosity, the change of
coordinating numbers and electrical resistivity at melting, then, the points of
melting, cooling, the structure of castings and ingots, the theory and
calculations of modifying processes were created and mastered as far as
experimental data can prove.
A new theory of liquid cast iron structure and
crystallization was worked out.
There is a description to a new set of experimental procedures for
the research of liquid alloy structure by means of capillary sedimentation
experiment; the results of testing a series of alloys by such means are
presented. Data concerning the possibility of alloy refining with the help of
the new set of methods are supplied. It is demonstrated that potentially the
new procedures are no inferior to the method of zone refining of metals.
Some additional possibilities and
perspectives of the new approach are outlined in the conclusion.
The most general and undoubtedly correct description
of melting and crystallization processes originates from thermodynamics.
Thermodynamics, by force of its specificity as a
phenomenological science, can give most generalized descriptions to phenomena
through simple ‘exterior’ parameters such as pressure, volume and temperature
without delving into the structure of the substance given and the mechanism of
this or that process. Both the strong and weak points of thermodynamics lie
there. Thermodynamics is unable to furnish data on the structure and mechanism
of the processes under consideration, which shapes the major blank gap in the
sphere of melting-crystallization.
What can thermodynamics supply, at
any rate? It gives the most general description to processes and points out
their basic limitations, which is of utter importance. All morphological or
structural or other types of theories have to comply with thermodynamic
limitations, i.e. thermodynamics creates the most general criteria of truth, if
we can say so. So, if any theory fails to meet thermodynamic requirements, it
is erroneous in principle. However, if a theory conforms to the aforementioned
criteria, it does not always follow that it is automatically true. Compliance
with thermodynamic criteria is the immutable but rather an insufficient
condition of truth of any structural theory. Other criteria of truth refer to
truthfulness-to-facts level.
Still, it is accepted to start
from thermodynamics in order to establish the generalized truth criteria
touched upon before.
The first application of
thermodynamics to melting dates from the middle of 19th century. ‘Krystallizieren
und Schmelzen’ (1903) by Gustav Tamman may be regarded as the first classical
work on melting and crystallization. The crystalline structure of solids had
not been studied yet by that period so Tamman’s theory was by necessity
phenomenological.
Thermodynamics of phase fluctuations, melting and
crystallization in particular, has developed considerably by present, which is
vividly corroborated by A.Ubbbelode’s works /1,2/.
Taking into account the specific features of the work
given, let us survey the most general thermodynamic theses only that relate to
the core part of the research undertaken in this book.
Balance between phases /1,2/.
It follows from the theorems of classical
thermodynamics that at the equilibrium between any two states of any material
systems the free energies of the matter mass units in both of these states must
be equal. Mathematical correlation between the parameters of the two phases, e.
g. solid and liquid, coexisting in equilibrium, is determined by the condition
GS =GL (1)
where GS is Gibbs’ free energy index for solid phase; GL is Gibbs’ free
energy index for liquid phase.
The term of phase here and below as applied to
liquids and solid bodies will denote, for the sake of abbreviation, solid or
liquid states of aggregation of matter.
All the numerous thermodynamic theorems that relate
to melting and crystallization are derived, in any case, with the use of the
fundamental equality (1).
According to Gibbs’ phase principle, if we take a
homogeneous substance forming a single-component system, solid, liquid and
gaseous phases can coexist at the sole combination of temperature and pressure
- at the so-called triple point. The majority of metals, and not only metals,
have such a low pressure of steam at the triple point that the temperature of
the triple point coincides, practically, with the melting temperature at the
ambient pressure of 1 atm. For example, the melting temperature of water and
ice is TM= 0.0000C (by definition) whereas the triple
point temperature is TTR=0.0100C.
At higher pressures than that which corresponds to
the triple point, gaseous phase in a single-component system practically
disappears, while the melting temperature depends on the aggregate pressure.
The change of Gibbs’ free energy of solid and liquid
states of aggregation of matter with temperature (at constant pressure) is
shown in Picture 1. It is clear that, in conformity with the aforesaid, these
two characteristic curves cross at a single point that corresponds to the
melting temperature. Within the area of temperatures lower than the melting
temperature the free energy of solid phase is minimal and there exists
stability in solid state. On the contrary, within the area of temperatures
higher than the melting temperature the free energy of liquid phase is of less
importance than the free energy of solid phase. Within this area liquid phase
is stable. Generally, in conformity to the laws of thermodynamics, the state
the free energy of which under any particular conditions is the least possible
will be stable under the conditions given.
Equality (1) and Fig.1 supply the
principal conclusion of thermodynamics concerning melting and crystallization.
Namely, a certain principle ensues from them; let us label it as the two-phase
principle, or the principle of comparison.
Otherwise speaking, it follows from thermodynamics that under any
conditions, at any temperature in our case, there really exist at least two
quantities of free energy that compete with each other and may be compared.
Such dualism is of great interest - being totally unexplored. The difficulties
in the study of the question given consist in the fact that in reality, at
temperatures lower than the melting temperature, there exists solid phase only
in the open, manifest aspect. On the other hand, at temperatures higher than
the melting temperature there exists only liquids phase in the manifest aspect.
Therefore, the problem of a constant comparison between the two quantities of
free energy of liquid and solid phases remains open to discussion.
We should admit that in literature there is no
unified opinion on the fundamental point under consideration, yet
contradictions are apparent. In particular, there circulates an opinion that
any ‘single-phase’ models and theories of melting and crystallization are
erroneous by definition, because they contradict the principle of
thermodynamics noted above, which we agreed to label as the two-phase
principle, or the principle of structural dualism, also pointed out by
A.R.Ubbelode /1,2/. Other authors construct their theories and models ignoring
this principle.
Still, the principle of dualism, or, to be precise,
the principle of the constant coexistence of two certain complexes of
quantities that determine the states of aggregation of matter acts at least as
a fact of negation of ‘single-phase’ models and theories of melting and
crystallization. Nobody knows for sure what the adequate
melting-crystallization theory should be like, yet it follows from the
principle stated above that it ought to be a ‘two-phase’, or ‘dualistic’,
theory. To be more exact, it should be a ‘two-factor’ theory representing the
processes of existence and change of the states of aggregation of matter as the
corollary of the interaction between two certain factors or two complexes of
factors.
Until present, it has been impossible to find any
convincing proofs of the real existence of such dualism.
The argument is to broach the question whether these
quantities exist in reality, or they are just conceptual. Should thermodynamics
be understood directly, or roundabout ways of comprehension be scouted?
As we see, even the simplest approach to the problems
of melting and crystallization, even within the stabilized area of
thermodynamics, turns out to transcend its seeming simplicity and clarity.
At the minimum, the above-mentioned theses give rise
to one unsolved problem: are there any two factors, existing constantly, the
competition of which leads to the change of the states of aggregation of matter
under certain conditions?
If it is so, what are those factors, unknown to
science?
Questions generating...
There ensues a general idea out of the comparison of
the thermodynamic parameters of a large number of various substances in solid
and liquid states. The attempts at using the volume measurement criterion as
the sole one are still being made, yet they look self-defeating if we recall
the existence of the so-called anomalous metals which do not only increase
their volume while melting but reduce it. That is quite a stumbling block, a
real hindrance to the application of many thermodynamic conceptions, i.e. the
principle of corresponding states, to melting.
However, there remains the major thermodynamic
parameter that changes equally for all the known cases of melting. It is
entropy. For all the known substances in liquid state entropy is higher than in
solid state. Let us use this experimental fact for our purposes. The change of
entropy at melting can be represented by Boltzman equation /1/:
Δ Sf =R ln Wl /Ws,
(2)
where Wl is the number of independent ways of substance
realization in liquid state, Ws - the same for solid state.
It was demonstrated by (2) that
the transition from solid to liquid state is accompanied by the increase in a
certain kind of disorderedness in a substance. Thermodynamics cannot answer the
question about the nature of such disorderedness. Nevertheless, this inference
is sufficiently important to be allowed for further.
Proceeding from the problem specified, let us analyze
some of the most prominent theories of melting and crystallization for the
purpose of satisfying the condition of ‘two-factorness’.
Like any other science, the
science of melting, crystallization and liquid state of metals developed by
accumulating, analyzing and systematizing the experimental data. Such a way is
characteristic of the stable period in the development of any science. With the
exception of the periods of stability, at times science undergoes periods of
discoveries, i.e. unpredictable qualitative leaps in the understanding of this
or that issue or even just the finding of new methods never known before, and
research areas. Scores of years have passed without bringing great changes for
the science of melting, crystallization and the liquid state of metals.
The last substantial experimental discovery in this
sphere was made in the early 20th century together with the
development of X-rays- and further on - neutron diffraction.
The transposition of substance structure research
with the help of penetrating radiation on particle floods from solid to liquid
state was undertaken in the 30-ies. Valuable works were accomplished by Stewart
/3,4/, Kirkwood /5,6/, Bernal /7,8/, Gingrich /9/ and others. However, the
first researcher to carry out systematic exploration of liquid metals structure
with the use of the method of x-rays dispersion by the surface of liquids, as
well as other methods, was V.I.Danilov /10,11/. In the works listed, the
likeness between the atomic structures of neighboring order in solid and liquid
metals in the vicinity of the melting temperature was emphasized with all definiteness,
as well as the gradual diffusion of the crystal-like structure of neighboring
order in liquids within the limits of overheating process.
The result of such a discovery was the retreat from
the formerly predominant notions of the similarity of liquid metals structure
to the chaotic gas structure, that originate in Van-der-Vaals’ works /12/.
Thus, the turn marking the transition from the
conception of the gas-like structure of neighboring order in liquids to the
ideas of their crystal-like structure took place.
As we see, the historic logic of science development
deprived (for reasons unknown) liquid state of the right to the independence of
structure, though thermodynamics affirms unambiguously that the liquid state of
aggregation of matter, as any other state, is sufficiently independent and must
have a certain independent, specific structure. Yet this unsophisticated
inference never evolved, and liquids were looked upon further viewed as
something that has a dependent, transitional structure. Such an approach caused
much damage and hindered the development of the science of liquid metals
structure, melting and crystallization for a long period.
In the course of time the development of the science
of liquid metals structure was distinguished by the rivalry, or eclecticism, of
the two approaches described above: the quasi-gaseous and quasi-crystalline
ones. Either of them was accumulating, and not without success, facts in its
favor, while the structural independence of liquid state was never prospected.
For instance, there was analyzed a number of
experimental facts which testify to the proximity of liquid and solid metals
structure near the melting temperature. This is rather a negligible quantity of
enthalpies and entropies of fusion as compared with the same quantities for
evaporation /13/; a negligible change in the volume of metals at melting; a
slight change in heat capacity, heat conduction and electrical resistivity; a
qualitative similarity in the position of the first maximums and minimums on
the curves of reflected X-rays intensity in liquids with the position of lines
on X-rays photographs of powders of respective solid metals, etc. This
information is supported by the data on Hall’s coefficient, magnetic
characteristics, coordinating numbers, etc.
However, the supporters of the
gaseousness of liquid metals structure did have a considerable amount of data
in their favor.
In the first place, it is almost a
statistic distribution of atoms in liquid metals at heightened temperatures; a
possibility of continued transition of liquids into gas; a huge difference in
the mass transfer coefficients in solid and liquid metals, of fluidity, etc.
Out of the entire array of
controversial facts there can be drawn a reliable conclusion: by nature, liquid
metals possess the features of both the semblance and dissimilarity to solid as
well as gaseous state. Yet no inferences about the structural independence of
liquid state were made, there was not even formulated the conception of the
structural independence of liquid state.
From our point of view, such
ambivalence of liquid state characteristics is really one of the forms of its
structural independence and a means of manifesting the dualism that we touched
upon earlier. So any acceptable theory of melting and liquid state should
interpret the entire mass data available on liquid state, melting and
crystallization, and also the dual nature of liquid state structure and
properties. Such a theory must be able to explain the structural independence
of liquid state, too and its bonds with the structure of the neighboring states
of aggregation of matter.
So far, some researchers are still trying to see into
the nature of liquid state, proceeding from its similarity to solid state,
while others stress the liquid and gaseous states affinity. Both the
approaches, as it follows from the fore-going material, may be appropriate,
because the same liquid reaches, by structure and properties, the indices of
solid state near the melting temperature and of gases near the point of
vaporization, or critical point.
Both the approaches are one-sided, for they cannot
cover the whole range of liquid metals characteristics within the whole
temperature range of their existence.
To define the structural peculiarities of different
states of aggregation of matter, liquid state in particular, the author
introduces the concept of a structural unit (an element) of liquid state.
It is quite a convenient notion, since it lets us
single out the smallest part of liquid state, on the one hand, and classify the
existing structural theories on the basis of structural units or structural
elements these theories operate with, on the other hand.
To bring up the problem of the structural elements of
liquid state is to the point, because any science of a system’s structure is
based on the concept of the smallest structural element of the system
specified, the element carrying the basic attributes of the system. For
instance, an atom (a molecule) is the smallest structural unit of matter in
chemistry.
From the viewpoints of differentiation between
structural units of matter in liquid state there exist two large groups of
theories of melting and liquid state that occupy, directly or indirectly,
unlike positions with regard to this issue. The most numerous group of theories
premise the definition of a structural unit of matter that is accepted in
chemistry and treats an atom (a molecule) as the structural unit of any state
of aggregation of matter. The ideal model of liquid from this viewpoint is a
monatomic liquid. In reality, the most closely related to the model given are
liquids with weak interaction between their particles, i.e. liquid inert gases
near the critical point /14,15/.
The other group of theories and liquid state models
proceeds from the fact that liquids have a complex structure which may consist
of particle groupings of various kinds /1,3,4,16,17/. It leads us to assume
that not only separate atoms but also some of their groupings act as structural
units (elements) of liquid state.
In the conceptual aspect, it can be regarded as a
step forward in comparison with simplified monatomic models and theories of
liquid state. Why? The point is that the monatomic approach equalizes all the
states of aggregation of matter in the sense that they turn out to be
indistinguishable by their structural elements, which is false a priori,
because the properties of the states of aggregation of matter become apparent
at the level of certain particle aggregations but not that of separate
particles.
The role of separate atoms and molecules in the
structure of liquids is not denied but one more level, one more state to the
understanding of the structure of the states of aggregation of matter is
introduced into this group of models and theories of liquid state.
Since any groupings consist of
similar atoms and molecules, there arises a question whether the distinction
between the definitions of a structural unit of liquid state is essential.
We can suggest the positive answer, because the
description level should be adequate to the subject described. Thus, atoms
include protons, neutrons and electrons; however, if we limited ourselves to
these concepts ignoring the idea of the atom, it would be extremely hard or
almost impossible to describe, for instance, the structure of molecules or
crystals. The fact is that any agglomeration of particles is more than a mere
agglomeration for the reason that it has a certain structure, which, as a rule,
is not a mere sum of particles. What is more, the particles under consideration
are tied between them and organized in space in a certain way.
That is why the question about the structural
elements of liquid state seems to be the question of paramount importance, so
using it as the basis for the theories and models of liquid state classification
is quite defensible.
Returning to the thermodynamic requirement of the
‘two-factorness’ of a system, let us note that the structural elements of
liquid as well as any other state should meet this requirement, i.e. the
structural element in question should consist, at the minimum, of two parts of
differing nature. Such duality possesses a totally general character; still, it
is undisclosed as yet. The nature of structural dualism is to be viewed below.
Modern statistic theories of liquid state refer to
the first group of the classification on offer, since they make use,
principally, of the pair interparticle interaction concepts /12-15/. The data
on such interaction come out to be extracted from the information on X-rays-
and neutron diffraction of liquid metals. The assumption function is the
dependence of the diffused X-radiation intensity on the radiation angle, which
allows determining the statistic structural factor. The knowledge of the
structural factor, in its turn, lets us find, with the application of
Furie-transformation procedure, the relative function of radial distribution
and the aggregate correlative function, as well as calculate the coefficients
of self-diffusion, viscosity and some other properties of liquid metals if we
imply their monatomic structure.
With the help of the well-known methods of Born-Green
/18/ and Percus-Yewick /19/ we can calculate pair potentials of interatomic
interaction out of the registered quantities. It may be admitted that there is
no reliable information on the nature of interatomic potential, diverse kinds
of constructed functions of pair potential are used. It is clear that the
results obtained differ greatly between themselves, as well as from the
experimental data, which diminishes the practical value of the indicated
methods /20/.
The connection of the statistic theory of liquid
metals with melting is not established directly, displaying the weakness of the
theory under analysis.
Frequently we come through an assertion that the
statistic theory does not serve as the model one to the extent that it uses the
experimental data on neutron or X-rays diffusion /20/. However, out of these
data there can be inferred nothing but the information about the pair function
of distribution, i.e. the pair interaction between the particles. It is
supposed implicitly that liquids consist of dispersive particles (atoms) only.
Consequently, the monatomic model of liquids structure starts functioning as
the basic approach in the statistic theory though in the most general, implicit
aspect.
Concretizing the specificities of the first group
theories, i.e. those which adhere to the thesis that an atom (a molecule) is
the structural unit of liquid state, leads to the formation of the so-called
corpuscular models /12,14,21/.
Among the widespread models there can be mentioned
the so-called hard sphere model /22,23/, the working out of which amasses a
large number of publications. In accordance with this model, atoms in liquid
metals impute the properties of hard spheres. The method in question as applied
to certain X-rays photographs looks effective enough; still, no-one succeeded
in achieving the correspondence between the experimental and calculated data
for the wide variety of liquid metals /23-25/.
Because of computer expansion,
calculation procedures such as the method of molecular dynamics enabling to
experiment with the modeling of liquids using the statistic theory results and
methods, became popular /26, 27/. These methods are not applied to the sphere
of practical metallurgy and foundry works, their importance is so far purely
theoretic.
To the group of monatomic models we should also refer
the widely known yet rarely used model of J.Bernal /8,28/, where the structure
of liquid is rated as a result of the disordered allocation of separate atoms
with the possible realization in the neighboring order of the symmetry of the
fifth order (nontranslatable) that cannot be realized in solid state. The model
under consideration reveals rather the author’s rich imagination than the real
state of things, yet we cannot deny it the right to existence. While the real
structure of liquid metals remains unexplored, any models and theories have a
right to existence.
It must be noted that neither the model of hard
spheres nor that of J.Bernal allows modeling explicitly the process of melting
and crystallization, though such attempts are occasionally being made. There
was advanced an opinion that if computer memory volume amplified a bit
increasing the number of modeling steps, the problem solution would ensue by
itself /26/. It looks fairly improbable.
Let us emphasize as the common drawback of all these
theories and models that none of them conforms to the principle of dualism or
two-phaseness analyzed above.
A special position belongs to the models of melting
the most widely known of which is Lindeman’s model. Lindeman put forward a
hypothesis that melting sets in when atomic oscillation frequency and amplitude
in crystalline lattice points increase so much that the atomic bonds start
breaking. Further research did not corroborate Lindeman’s suppositions, since
calculations made under his theory issue the points of melting that approach
the vaporization point /1,2/. Still, if we take into account the extreme
scarcity of logical theories of melting, Lindeman’s model continues being
mentioned in reviews for scores of years, which we are in order to do by
tradition /29/.
Let us observe that Lindeman’s model fails to meet
the thermodynamic requirement of dualism of melting and liquid state, because,
according to this model, there is only one reason for melting – it is an
increase in amplitude and frequency of atomic oscillations at metal heating,
and the cause in question does not compete, or interact, with any other factor.
Among the second group of theoretical conceptions
that present complex views upon the structural unit of liquid substance we
should place diverse variants of models of microinhomogeneous structure of
liquid metals.
This trend is intensely developing
during the last years /30-36/. The first works representing such a type belong
to Stewart /3-4/. Stewart was the first to introduce the cybotaxis concept that
turned out to be highly viable. Stewart’s cybotaxes are, in their own way,
short-lived crystals forming liquid.
A divergent but analogous idea of liquid metals is
observed in works /31,36,37/. Liquid is viewed here as a double-structure
system which consists of relatively long-lived atomic microgroups with the
ordered structure of the neighboring order similar to solid state, and the
disordered zone with the chaotic arrangement of atoms.
The eclectic nature of such views cannot but manifest
itself: monatomic and cluster models are forced together. It might not even be
considered an error if Gibbs’ phase principle was not disregarded here. One and
the same substance within a wide range of the single state of aggregation of
matter temperatures appears to form two different phases, which conflicts with
reality. The quantum mechanics rule of quantum objects indistinguishability
comes to be violated, too.
It is evident that atoms in microgroups and within
the disordered zone must differ in certain characteristics, whereas quantum
mechanics states that under the same conditions atoms of the same kind must be
similar by their properties (i.e. indistinguishable). It is possible that there
is a way out of this disparity which remains, however, undisclosed.
Hypothetical atomic microgroups in liquid metals
structure are termed differently in various works: cybotaxes, microcrystals,
microgroups, clusters, blocs, etc. Still, the meditation upon these works
enables us to infer that similar objects divergent by details only are meant.
That is why we shall use the term of ‘cluster’ to label the objects of such a
sort further, concretizing or enlarging it as required.
The conclusions about the existence of clusters in
liquid metals are made on the basis of precision analysis of thin structure
curves of X-rays dispersion /34,40/, where the ordered areas dimensions
(approx.1 nm) as well as the atomic granulation type, are estimated on the
basis of the first maximum splitting, its intensity and width. In a number of
works, on the contrary, quite acceptable curves of X-rays dispersion are
diagrammed by means of calculations, proceeding from the microcrystal or
paracrystal models /41,42/.
There was given a high-grade description to the
process of metal crystallization on the basis of the quasipolycrystalline model
of liquid metals structure by Arkharov-Novokhatsky /43,44/.
An original description of the mechanism of melting
and the structural transitions in liquid metals can be found in Ye.S.Filippov’s
works /44,46/. The presence of such transitions is permitted and substantiated
by the repeated observation (carried out by numerous authors) of anomalies in
temperature and concentration dependencies of various structure-sensitive
properties of liquid metals and alloys /46/.
Still, the quantities of the anomalies observed are
relatively low at times and located within the error limits of standard
measuring. Nevertheless, the data mass on the anomalies in structure-sensitive
properties of liquid metals and alloys is so impressive at present that on the
whole the conclusion about the existence of such anomalies seems sound. Their
presence is especially indubitable in relation to some alloys, yet it is not
proved by a series of works on pure metals /47/. Upon the whole, the existence
of certain analogues of phase transitions of the second order in liquid metals
and alloys appears to be logical in the presence of microgroups that have the
features of neighboring order in the arrangement of atoms inside such a group,
in their structure. If there is order, a transition from one of its forms to
others is possible. Clusters lacking, we consider phase transitions impossible,
on the contrary. So the problem of the existence of similar transitions serves,
for tens of years already, as a battleground between the supporters of
different approaches to the description of liquid metals and alloys nature.
This discussion going on for almost a century took
place because of the possibility of a polysemantic interpretation of the
majority of the results obtained in most of the experiments with liquid metals,
which, in its turn, reflects the objective complexity of the liquid state
nature and its ambiguity.
Till latest, for example, there were no direct
experiments that could definitely corroborate or disprove the hypothesis of
cluster, microinhomogeneous structure of liquid metals. Its postulational
character of the hypothesis under analysis, as well as the ambiguity of
suppositions advanced concerning the structure of liquids, can be regarded as
its main shortcomings /25/.
In this connection, the prevailing position in the
theory of pure single-component liquid metals occupies the statistic theory
that denies cluster existence and the microgroups similar to them in liquid
metals /48/. A series of computer models of liquid metals structure, adherent
to the hard sphere model, sides with the theory mentioned. There were created
several successful computer methods of calculating certain characteristics of
some liquid metals, which have as their basis the curves of X-rays dispersion
and the supposition about the monatomic structure of liquid metals, or diverse
forms of interatomic interaction potential. Yet we do not have a right to
attribute the achieved results to the whole range of liquid metals.
The situation in the alloy theory field is developing
for the most part in favor of the cluster approach, especially in the area of
studying the systems Fe – C, Fe – Si, Cr – C etc. that are of practical
importance /49–51/, and liquid eutectics /50,52–55/. It is explained by the presence
in this area of a far greater mass of experimental facts testifying to the
advantage of the cluster theory of liquid metals structure.
Above all, there should be placed the well-known
experiments on the melts centrifuging that lead to the enrichment of the sample
remote from the rotation axis by the heavier component in all the cases. It
allows determining the dimensions of the areas enriched by this component,
proceeding from the experimental conditions.
Experiments of such a kind were carried out systematically
for the first time in Russia by A.A.Vertman and A.M.Samarin /16–17/ to be
repeated periodically later by a series of researchers with certain
modifications but always with the same result / 55,57,58/. The definite
significance of these experiments consists in asserting that during the
experiment and under its conditions no appreciable change of the alloy
concentration along the sample was to take place in the centrifugal force
direction with the monatomic structure of melts.
However, the change was continually occurring, so it
was possible to calculate the floating zone dimensions by the simple Stokes’
formula on the basis of this change.
Since such experiments were abolishing the monatomic
hypothesis, their interpretation underwent a whirlwind of criticism by the
monatomic hypothesis advocates from the ground of the theory of regular
solutions /20,56/. Unfortunately, the criticism came out to be valid in a
number of items, because Vertman–Samarin experimental procedures were
imperfect. To our disappointment, a comprehensive response to the criticism in
question never followed, so the centrifuging trend was stigmatized doubtful and
the research in this sphere was suspended for a long time. Still, scientific
development proved that the trend referred to was right upon the whole,
notwithstanding the incompleteness of its methods, and is being gradually
reestablished at present as a reliable way of liquid alloys study /57,60–61/.
The inferences in favor of cluster existence are
generally made on the grounds of the congeniality between the whole range of
liquids and solids listed above. It merits our attention to notice that a large
mass of data on liquid metals structure that is accepted as naturally
fundamental may be viewed ambivalently.
These are the X-ray diffraction analysis data, in the
first place. They are successfully used in calculations and conclusions,
testifying to the cluster, as well as monatomic, structure of liquid metals. On
the basis of this we may state, to our regret, that so weighty a tool of X-ray
diffraction analysis, cogent in other cases, was unable to supply any precise
information upon the nature of liquid metals.
The conclusion sequent to the aforesaid formulates
like this: the nature of liquid metals, in all probability, is not so plain and
definite as it happens to be presented in the already existing models, that
there must be some factors, unfound till now, which determine the nature under
research. Simpler speaking, there is something in liquid metals beside clusters
or monatoms. What is it?
Two main groups of theoretical concepts of liquid
metals and alloys structure that were accentuated above almost fail to have
common ground at present. Several statistic theory supporters see it clearly
enough. I.Z.Fisher argued convincingly /48/ that if a monatom is to be considered as the
structural unit of liquid state, the microgroup idea looks absolutely redundant.
Let us observe, for justice, that the premises of the afore-mentioned work by I.Z.Fisher
contain this conclusion already.
In any case, the significance of Fisher’s work cannot
be assessed by the fact that it proves the absence of microgroups (clusters) in
real liquid yet it demonstrates the antagonism and incompatibility of the
monatomic and cluster approaches to the description of liquid metals and alloys
structure. The antagonism under analysis is not always accepted or even
understood in the works on liquid metals theory. There are some attempts at its
negation, the bringing together of the two approaches /62/, or assuming that
structural units may differ in the same liquid /43,44/. These works mark
nothing but the quantitative difference of the two approaches; the trends are
forced to reconciliation or joining.
Still, the hypothesis of monatomic structure of
liquid metals does not require the cluster concept by its inner logic, which
was shown by I.Z.Fisher.
In its turn, the cluster hypothesis implies an
insolvency of the monatomic idea for explaining the phase characteristics,
though it may be applied to interpret cluster structure.
The acknowledgment of the antagonism of these
approaches could be more advisable at present stage than the attempts at linking
fundamentally disparate ideas.
The cardinal incompatibility of these approaches, as
it was pointed out above, is that similar atoms cannot be at two different
states under constant conditions. It is quite obvious that atoms in clusters
and atoms in their free state (monatoms) are at divergent states. Therefore,
clusters and monatoms in liquid metals cannot exist simultaneously as
structural elements. Either must be held to. The alloy of these states is
totally unacceptable, if we proceed from the quantum principle of
indistinguishability of similar atoms.
Hence, it is clear that the question of the choice of
the liquid state structural unit is sufficiently important. It is also evident
that this problem does not have any satisfactory solution so far. That is why
the analysis of the areas where this or that approach leads to the optimal
results plays a vital part in the model of liquid state choice.
The previous analysis shows that the monatomic
approach (statistic theory, hard sphere model, molecular dynamics method, etc.)
agrees with experiment most thoroughly in the critical point area, or generally
at high temperatures remote from the melting temperature, for liquids with weak
interaction between their particles like liquid argon, and pure metals.
The maximum conformity with the experimental data for
the cluster approach is achieved in the low-temperature area near the melting
temperature or liquidus point, for alloys, systems with strong interaction
between the particles, eutectic alloys.
Undoubtedly, only the preferred spheres of relative
achievement are listed here. In certain cases these areas intersect, yet on the
whole the state of things is as presented. If we investigate the situation
given, the models of cluster structure of liquid metals and alloys are
preferably suited for metallurgists and castors in their practical work, since
these models describe the properties of liquid metals and alloys in the optimal
way within the practically important temperature interval.
Unfortunately, the applied advances of the models in
question and their use for the practically important theories of
crystallization, hardening, the theory and practice of modifying and doping of
alloys have an exclusively qualitative nature and can be applied to practice to
the least degree possible.
In the sphere of the statistic theory, practical
applications are also scanty. Though the heterophase fluctuations theory allows
to qualitatively describe the new phase nucleation process, it is
correspondingly weak in handling this area having no practical applications,
notwithstanding the more than a semi-centennial history of its development.
Finally, both the major approaches to the description
of liquid metals and alloys structure – monatomic and cluster ones – are
subject to the lack of correspondence with the thermodynamic requirement of
two-factorness. Both of these approaches premise that the concept of monatoms,
or clusters, suffices to describe liquid state and the states of aggregation of
matter in general.
Thermodynamics claims that it does not sound
exhaustive, that there must be at least one factor essential to liquid state.
The detection of this attendant factor is of paramount importance to the theory
of liquid metals and alloys structure at present.
In the structural aspect,
crystallization process starts from the formation of elementary microcrystals
that are termed crystalline centers. Since W.Gibbs, it has been accepted to
assume in the existent thermodynamic theory that the process indicated requires
energy consumption for the formation of solid and liquid phase section surface
/63,64/.
Such a hypothesis seems valid from the angle of
common sense, yet it leads to a rather drastic assertion of the existence of a
certain critical radius of crystalline nuclei, which had distressing
consequences for the theory of crystallization.
Let us consider the problem how the concept of the
critical dimensions of crystalline centers arose in present theory.
It is surmised that the aggregate change in the free
energy of a system while forming a solid phase zone in liquid, amounts to the
sum total of the changes in the volumetric and surface energies of the system’s
zone specified /64–68/:
ΔG = -ΔGv + ΔGs,
where ΔGv is specific volumetric free energy; ΔGs is specific surface energy.
The signs in the equation given constitute the main
theoretic problem. These signs have a physical significance and are introduced
for the reasons of common sense. The minus in front of the first term on the
right denotes that energy at solid phase formation at T = T0 evolves in correspondence with
the experimental fact of latent heat crystallization emittance. The plus in
front of the second term on the right signifies that, on the contrary, there
must be energy consumption for the formation of phase section surface.
The latest conclusion in the stated case is based not
upon the experimental facts but common sense considerations that there must be
work (energy) consumption for the formation of phase section surface. This
consideration is true in many cases, so an inference that it is right in all
the cases was formerly made.
As a result, it follows from the theory that the
process of solid phase center formation at the crystallization of metals has
some features of inherent contradiction.
In particular, solid phase center formation is
recognized thermodynamically expedient whereas the surface formation of the
same phase is reckoned thermodynamically inexpedient. We are going to expand on
the present contradiction in Chapter 6. So far let us mark the fact of its
existence, - it gives rise to the conclusion of the existence of solid phase
nucleus critical dimensions further.
Let us make a conjecture that a solid phase nucleus
is spherical. Then, the latest expression for ΔG will shape into
ΔG = - ΔGv 4/3 pr3 + s 4pr2
The expression is graphically shown by the curve in
Fig.2.
It follows from Fig.2 that the dependence
ΔG = f (r) has a maximum. The nuclear
radius where the function attains its maximum is termed critical. As a
corollary to the existence of this maximum at the primary period of the new
phase center nucleation, the process of the formation of solid phase demands
energy consumption, so solid phase nucleation becomes expedient only after
amounting to a certain critical dimension.
Factually, such a character of the curve in Fig.2
stands for a thermodynamic prohibition of the nucleation of microcrystals with
the radius less than the critical one. In liquids, such crystals must
dissociate.
Consequently, it is necessary for crystallization process to set in
that crystals should nucleate gross enough having the dimensions larger than
the critical ones. How can it be? Thermodynamics does not answer this question.
Let us consider the existent solutions to the problem
of the critical radius of crystallization centers. Willard Gibbs demonstrated
for the first time that, to form a critical dimension nucleus, it is necessary
to expend work (energy) A = Δ Gs, which equals to one third of the spare surface energy of the
nucleus mentioned:
Δ Gs = 1/3 S Si si,
where Si is
the specific surface of the ith zone of an equilibrium crystal at
nucleation; si is the
specific surface energy for the zone given.
If a cube is the equilibrium form of a crystal, then
Δ G = 8s r2 c,
(3)
where rc is the radius of the sphere inscribed
into a cube of the critical dimension.
To calculate the quantity of rc of a spherical
nucleus the following formula is used /65,67,68/:
rc = 2s T0/ L Δ T.
(4)
Graphically, the last expression corresponds to the
minimum in Pict.2.
The concluding formula shows that the critical dimension
of a solid phase nucleus deflates with an increase in the melt overcooling ΔT = (T0 – T). The nucleus formation work ?G
decreases at the same time as the overcooling increase is taking place.
For a separate nucleus in the form of a cube with
the edge a = 2 rc there is the following quantity of work that is to be expended for
the formation of the section surface of the nucleus given in accordance with
the present idea:
Δ G = 1/3 Ss = 32 s 3 T0 / L2 (Δ
T)2
This expression confirms that, in correspondence with
present theory, the formation of the solid phase nucleus of the critical
dimension demands the expenditure of work (energy) and is therefore
thermodynamically inexpedient.
Thus, the existent thermodynamic theory of solid
phase nucleation in liquids cannot surmount the theoretical bar it constructed
concerning the idea of the critical radius of the crystalline nucleus as well
as the thermodynamic inexpedience of the process of crystal growth with the
crystalline dimensions less than the critical ones. To overpass the
contradiction under analysis, the thermodynamic theory required either the
shift of the conception of the energy expenditure necessity for nucleation in
liquid, or extraneous help. It happened that the scientific development in the signalized
sphere chose the latter, i.e. the support from non-thermodynamic theories. This
support was rendered by the theory of heterophase fluctuations nucleation.
The noted contradictions in the thermodynamic theory
of nucleation remained a problem for quite a time and were formally mastered
only with the help of the non-thermodynamic, probabilistic by nature, theory of
heterophase fluctuations.
The aforesaid theory was being created for scores of
years by the efforts of scientists innumerable, so it is termed by the names of
different authors. Among the most frequently-mentioned pioneers of this theory
we can enumerate Frenkel, Volmer, Weber, Bekker, During,
Eiring and several other authors occasionally /69–72/.
In physics, fluctuations are any contingent
deviations from the average state and distribution of particles in any large
systems, which are determined by the chaotic thermal motion of the system's
particles. The measure of fluctuations is the average square of the difference
in any local value of any physical quantity in this system L' and the average
value of the same quantity for the whole system L.
(Δ L)2 = (L – L')2
As a rule, fluctuations are small and the probability
of any fluctuation given decreases exponentially with an increase in its
quantity.
If a system consists of N
independent parts, the relative fluctuation of any additive function of the
state L of the system specified is inversely proportionate to the square root
of the number of its (the system's) parts.
The theory of heterophase fluctuations employs the
fact there can be fluctuations of any type, heterophase including
/69,70,73–75/. The latter means that as a result of contingent chaotic thermal
motion of atoms there may occasionally appear some zones in the melt that have
the atomic distribution similar to that of a crystal.
It is supposed that within the limits of such
fluctuations neighboring order is casually realized, the order which is
characteristic of a crystal, so the surface of the section with the surrounding
melt is established. Theoretically, this supposition is quite possible. In the
theory in question, such fluctuations are identified with microcrystals.
It means that the theory under consideration implies
rather groundlessly that the instantaneous contingent organization of a certain
atomic configuration in space suffices for this zone to acquire the structure
and properties of some other phase. Such a statement cannot be assessed as
potent or convincing.
Fluctuations are unstable and transient by nature.
The period of their existence correlates with the duration of heat oscillations
of particles forming up liquid. In case of atomic fluctuations, the noted
period equals to 10-12 of a second. Fluctuations set in and fade
right away. Only under this condition the average state of the system remains
constant.
The theory of heterophase fluctuations admits that
under certain conditions heterophase fluctuations may turn from the unstable
state into the stable one acting as crystallization centers. These conditions
are adopted from thermodynamics.
The first similar condition in the heterophase
fluctuations theory reads as overcooling, since a stable existence of zones
with solid crystal structure is thermodynamically possible only in the melt
overcooled below the melting temperature. The greater overcooling gets, the
less the critical radius of nucleation is, in accordance with (4).
Correspondingly, the less must be the dimension of a heterophase fluctuation
and the greater the probability of such a fluctuation to set in.
The second condition is
that of the critical dimension of fluctuations under analysis. It is shown in
Fig.2 that heterophase fluctuations will be stable only in the case when their
dimensions are large enough, i.e. larger than a certain critical radius, even
in the presence of overcooling.
If the dimension of a fluctuation is less than the
critical one, it will dissociate even in the presence of overcooling. If the
dimension of a fluctuation is larger than the critical one, its growth becomes
more expedient (see above).
There is a stipulation to be made.
The point is that a fluctuation cannot be growing gradually. By definition, it
must arise at once, on the instant. It was stated above that the probability of
this or that fluctuation decreases exponentially with an increase in its
quantity. Thus, heterophase fluctuations of critical and overcritical
dimensions are highly improbable here.
As we see, the theory of heterophase fluctuations
transfers crystallization process from the class of regular phenomena to the
category of accidental, probable ones.
This is an essential drawback of the theory, because,
for thousands of years, metallurgy and foundry practice has been demonstrating
the regularity of crystallization.
Let us trace in detail the connection between the
thermodynamic theory of nucleation with the heterophase fluctuations theory.
It follows from the laws of
statistic physics that there is a finite probability I of any system's
transition through the energy barrier ?G by energy fluctuations:
I = K exp(-Δ G / kT),
(5)
where k is Boltzman constant; K
is the kinetic coefficient depending on the rapidity of the atomic exchange
between the fluctuation and the melt.
By inserting the value ΔG into (5) for the critical nucleus from (4), it will be possible to
calculate the probability of the critical nucleus formation by fluctuations
after taking the logarithm:
lg I = lg K – 32 s 3 T0 lg e
/ L2 (Δ T)2 k
Ya.S.Umansky considers a particular example of
homogeneous crystallization of iron at the discriminate overcooling of 100, 200
and 2950C /68/.
The example illustrates the
possibilities of the heterophase fluctuations theory for calculating the
processes of crystallization. So let us take a brief survey of the general data
supplied by Ya.S.Umansky.
For iron, specific surface energy along the section
of crystal-melt s = 200 erg/sq.cm, T0 = 1803 K, L = 3.64 kcal/g atom = 2
1010 erg/sq.cm.
Hence
lgI/K = -32 2003 1800
0.434/4 1020 1.38 10-16 (Δ T)2
The results are presented in the table 1.
Table 1 The Probability I of the Appearance of
Heterophase Fluctuations of Critical Dimensions in Liquid Iron at Discriminate
Overcoolings
ΔT, K |
100 |
200 |
295 |
I/K |
10-35 |
10-8.8 |
10-4 |
The table shows that the declining of overcooling
from 295 to 200 K, i.e. 1.5 times on the whole, reduces the probability of
equilibrium nuclear formation in correspondence with the heterophase
fluctuations theory almost 100 000 times as small. At the overcooling of 100
degrees the probability of nucleation, by Ya.S.Umansky’s calculations, comes to
10-35. It is a vanishingly minor quantity.
Out of the recorded calculations
Ya.S.Umansky and others arrive at the conclusion that the practically
homogeneous fluctuation does not take place. The probability of forced
crystallization, by this theory, is much higher than the probability of
spontaneous crystallization.
In particular, in case when liquid iron wets the
particle surface of some insoluble solid extraneous agent so that the wetting
angle is q = 45 degrees (the case of the average wetting level), Ya.S.Umansky
derives the correlation
ΔGheterog / Δ Ghomog
= 0.06.
Consequently, heterogeneous nucleation in this case
becomes more or less probable at the overcooling of 100 degrees. Let us note
that in the real processes of casting the overcooling quantities amount to 0.1
- 10 degrees centigrade. At such overcoolings the probability of the formation
of spontaneous, as well as forced, crystallization centers in accordance with
present theory is actually indistinguishable from the zero-point.
The supposed improbability of
spontaneous crystallization and the requirements of considerable overcoolings
even for heterogeneous nucleation are regarded as substantial defects of
existent theory, because metal and alloy crystallization goes on unhampered,
with negligible overcoolings. Often overcooling in the process of
crystallization is so small that it can hardly be measured.
Thus, the formula (5) connects probabilistic ideas
with the thermodynamic quantity of ΔG, so the heterophase
fluctuations theory starts to laboriously fill in the inconvenient blank of the
thermodynamic theory that is related to the introduction of the critical radius
idea.
Ya.I.Frenkel made an assumption that the probability
K of the atomic transition from the melt into a crystalline nucleus is
proportionate to the mobility of atoms in the melt at the temperature of T
/69–70/:
K = Kl exp(-U/ RT),
where K is the proportionality factor, approximately equal to the
number of atoms in the melt volume viewed (K equals approximately 1023
for one mole of substance); U is the energy of atom activation in the melt; R
being the universal gas constant.
Taking into account the three latter formulas, we
arrive at the expression that characterizes the dependence of the rapidity n of
crystallization centers nucleation on the overcooling ΔT of the melt
/74,75/:
n = Kl exp(-U/ RT) exp[- Bs3 / T (ΔT)2], (6)
where B = 2 (4MT0/ r q) / k is the substance constant.
Fig.3 shows the dependence of the
rapidity of crystallization centers nucleation on the degree of overcooling.
With an increase in overcooling
there is observed an increase in the rapidity of nucleation; after reaching its
maximum it is again reduced to zero. G.Tamman first formulated a similar
dependence while he was undertaking experimental studies of a series of organic
substances like naphthalene, salol, etc. It was termed Tamman’s curve /76/.
There exists a certain divergence between the experimental and
theoretic curves. Tamman’s curve does not start from the zero-point. I.e. there
is an area in the immediate vicinity of the melting temperature, where the
rapidity of nucleation equals zero. The theory does not prognosticate the
existence of such an area explicitly; neither does it give any convincing
explanation to this phenomenon. According to the theory, the probability of
nucleation at any temperature lower than the melting temperature is otherwise
than zero.
It is assumed that the first exponential multiplier exp[-U/RT] reflects the influence
of the factors that hinder nucleation process, since the lowering of the
temperature provokes the decrease in the rapidity of atomic exchange between
the nuclei and the melt.
The second exponential multiplier exp[-Bs3 / T (ΔT)2]
accounts for increasing the rapidity of
nucleation at slight overcoolings, which relates to the deflating of the
critical dimension of a crystalline nucleus at a decrease in the temperature of
the melt and, correspondingly, to the reduction of energy (work) expenditures
for its formation.
The interval of slight overcoolings, i.e. the temperatures under
which crystals cannot nucleate as yet, in the area near and lower than the
melting temperature, is called theoretically the interval of the melt
metastability before the onset of crystallization. The heterophase fluctuations
theory, as it was stated, does not prognosticate the existence of such an
interval explicitly. It is assumed that the interval of metastability is,
theoretically, the overcooling of the melt in question, when the probability of
nucleation passes from the vanishingly minor quantity to a definite,
practically measurable one.
Such a definition presents rather a play upon words,
since the disparity between the vanishingly minor quantity of nucleation, on
the one hand, and the practically measurable one has never yet been possible to
calculate.
One more drawback of the theory under analysis
concerns the fact that Tamman’s curve never found its experimental
corroboration for metals. The dependence of the rapidity of crystallization
centers nucleation and the linear rapidity of crystal growth on overcooling for
metals based upon experimental facts is shown in Fig.4 /74/.
As we see it in the picture, the real process of
crystal nucleation in metals as dependent on overcooling only increases in
metals. Therefore it is accepted that in application to metals only the
ascending part of Tamman’s curve can be observed. It is assumed that the high
rapidity of atoms in liquid metals causes the latter. So, the heterophase
fluctuations theory enables us to overpass the problem of the critical radius
of crystalline nuclei. Since fluctuations are not growing gradually but appear
on the instant, theoretically large heterophase fluctuations of overcritical
dimensions may in principle arise in a leap. Thus, the thermodynamic problem of
the critical dimension has to be solved by means of probabilistic concepts.
The heterophase fluctuations theory, as it follows
from the above-said, interprets liquid as a medium consisting of separate
atoms.
From the viewpoint of present
theory, crystal growth is the result of separate atoms adjunction to the
surface of a crystal. Theoretically, this process is regulated by diffusion
rapidity, whereas in practice the process of growth goes far faster than
diffusion processes and is chiefly determined in reality by the rapidity of
heat abstraction. Yet the existent theory of crystallization ignores the factor
of heat abstraction rapidity and historically uses the overcooling factor.
It is assumed that crystal growth depends on the
geometry of growing crystalline planes as well as growth direction /66–67/.
For smooth corpuscular surfaces
layer-by-layer crystal growth is thought characteristic by way of formation of
two-dimensional nuclei upon those planes in the form of solid phase monatomic
layer with the ensuing growth of the crystals specified along the whole plane.
Layer transition is realized through the spiral step-by-step mechanism of
growth. However, by contrast with theoretical conceptions, the value of a step
was always several times as large as atom dimensions /66/. On the basis of
experimental facts there originates an idea that the elementary building block
of crystal growth is a certain formation far larger than a separate atom.
Still, if we accept the hypothesis of
two-dimensional nuclei as the right one, the linear rapidity of crystal growth
will be determined by the probability of the formation of such two-dimensional
nuclei. It is interesting that the theory reduces the problem of crystal growth
to the problem of nuclear formation.
Yet crystal growth arises from the already existing
nuclei. Those are different stages of the process, and they are occurring under
dissimilar conditions. In particular, it seems highly important that the
process of growth takes place on the existent surface of the section, whereas
nucleation requires theoretically the forming of a new surface.
Returning to present theory, let us point out that the probability
of two-dimensional nuclei formation, be it real or hypothetical, is expressed
in existent theory by the formula analogous to (5):
v =K2 exp (-U / RT ) exp [-E ( s )2 / T (Δ T)2], (7)
where K2 is the
substance constant; U is the energy of activation, analogous to U
in formula (6); E is the substance constant, analogous in its essence to
the value B in formula (6); s being the surface tension of the melt along the border of a
two-dimensional nucleus.
The curve graph (7) is absolutely analogous to the curve of the
rapidity of crystallization centers nucleation shown in Fig.4. It means that
crystal growth starts and continues at a definite overcooling only. Hence, in
compliance with present theory, nucleation and the growth of crystals unfold
according to the same core patterns.
If we take into consideration that
atoms in liquid metals are mobile enough, the first exponential multiplier in
formula (7), as well as in formula (5), may be estimated approximately as one.
Then, formula (7) will be transformed as applied to metals /74/:
v = K2 exp [-E ( s’)2 / T (Δ T)].
The dependence of the linear rapidity of
layer-by-layer crystal growth with smooth corpuscular planes on overcooling
will be expressed by the curve reflected in Fig.5.
The experimental dependence v on T
in case of liquid gallium crystallization is shown in Fig.6. In a number of
cases, crystal growth goes on without threshold overcooling (Fig.7,8). In the
case given it is premised that growing goes on by the dislocation mechanism.
For rough corpuscular planes of crystals
the so-called normal growth by way of chaotic atom joining to any points of
such surfaces is thought characteristic. As a result, the growing crystalline
plane advances far inside the melt, being self-parallel. In this case, the
dependence of the linear rapidity of growing on overcooling is expressed by the
simple formula /74/:
v = K4 Δ T,
where K4 is the kinetic coefficient characterizing
substance properties; it is premised constant at negligible overcoolings.
It is assumed that the normal growth of crystals
occurs at smaller overcoolings. Experimentally, it is esteemed rather hard to
prove.
R.Cahn’s theory corroborates that normal growth, on
the contrary, takes place at considerable overcoolings /66/. All the
above-mentioned theories premise that nucleation and the growth of crystals
occur by joining to the solid phase of separate atoms (the monatomic theory).
Such an assumption perceptibly constricts the possibilities of the theory.
In conformity with one of the synergetic
theses of I. Prigozhin’s theory, any processes connected with the
redistribution of energy and its dissipation, in particular, always occur at
several, including all the possible, levels. The total of these levels
constitutes the structural or any other hierarchy of the structural levels of
the system in question. The provision that crystal growth happens at the
corpuscular level only, contradicts the stated synergetic theses.
Thermodynamics classifies the process of
crystallization alongside with typical processes referring to the dissipation
(dispersion) of energy. Hence there should be several structural levels of
energy abstraction in the system of crystallizable casting, i.e. latent
crystallization heat. Moreover, it is very important to allow for the process
of energy dissipation proper.
The corpuscular mechanism of nucleation and crystal growth that laid
the foundation of present theory does not provide for other levels and growth
possibilities except the corpuscular one. Without denying the obviousness of
atomic participation in the process of crystallization, it must be said that
this is only one of the possible structural levels of the realization of the
process signified; besides, in accordance with synergetic theses, there must
exist several other levels of crystal growth. Therefore, the fluctuation theory
should be regarded as nothing but a step in the development of the science of
crystallization processes.
The terms of crystallization and
hardening denote, in application to castings, the same process though their
semantic load differs essentially. In particular, when we say
‘crystallization’, we mean the structural aspect of the process, i.e. we
understand the process of crystallization as a transition from liquid state to
solid one with the forming of a crystalline structure.
When we say ‘hardening’, we imply
the same transition from liquid state to solid one but only as a heat process
without tackling structural problems at all. Such a semantic complexion of the
two of these cognate terms developed historically /77/.
The specifically scientific
differentiation under analysis is not so convenient, that is why casting
practice frequently ignores the details of the semantic difference between
these two terms using both of them jointly to define the casting formation
process on the whole. Such a practical usage of the terms also developed
historically.
However, the specified distinction
is important enough in scientific works. Thus, apart from the theory of
crystallization that regards hardening as the process of transition from liquid
to solid state with the forming of a crystalline structure, there exists a
theory of hardening which considers the same process without looking into the
crystalline structure of castings and ingots, as a heat redistribution process
exclusively. The latter theory has been elaborated to perfection mathematically
and is steeped in history /74/.
A large number of works is
dedicated to the interrelation of the two theories, for the problem of energy
dissipation in the heat theory of hardening is being solved without establishing
the connection between the structure and properties of castings. In its turn,
the structural theory of crystallization does not contemplate energy
dissipation process to the sufficient extent. As a result, the interrelation of
the two theories describing the same process is practically lacking,
notwithstanding considerable efforts to combine them.
It is evident that the complete
theory of the forming of castings should fuse both the structural and heat
aspects of crystallization process.
Apart from the two main theories
of crystallization and hardening pointed out above, there exists one more
practically independent trend in the theory of crystallization – the statistic
theory of isothermal crystallization by A.N.Kolmogorov /78/.
A.N.Kolmogorov viewed crystallization from the purely
statistic attitude. Such an approach fails to open structural questions nor
does it join the issue of heat abstraction proving once more the correctness of
Pointcarret’s theorem in the sense that any task can be accomplished in a
limitless number of ways.
A.N.Kolmogorov’s formula for the solid phase volume V
that is generated in the process of crystallization dependent on time t for the
case of the isothermal crystallization of spherical crystals takes the shape:
V = V [1 - exp ( - pnv3 t4 / 3)]
(8)
where n is the rapidity of crystalline centers nucleation in a melt
volume unit; v is the linear rapidity of crystal growth.
These quantities Kolmogorov accepts as known.
I.L.Mirkin applied Kolmogorov’s method to solving the
same problem in case of cubic crystals, N.N.Sirota accomplished the same task
generally for the crystals of arbitrary shapes /74/.
Further, using the fact that the
number of crystals N is, as a rule, proportionate to the volume of the melt
crystallizable, there is established a connection between the number of
crystals and the solid phase volume as shown below:
V(t) = n (V0 –V)
dt (9)
where V is the total volume of the crystallizable metal.
If we introduce the v from (8) into (9) at n being
constant and t = 8, we obtain
N = 0.896V0 (n/v)3/4
(10)
If we know the volume of crystals
and their quantity, the statistic theory makes it possible to calculate the
average dimensions of the grain d:
d = 1.093 (v/n)1/4
(11)
Neither the thermodynamic theory of crystallization
nor the heterophase fluctuations theory solves the essential problems of the
crystalline quantity and dimensions of liquid phase in castings.
A.N.Kolmogorov’s statistic theory
fills in that breach and is therefore used, as a rule, together with the
fluctuation theory of crystallization that supplies the values of the rapidity
of crystalline centers nucleation as well as the linear rapidity of crystalline
growth for such symbiosis, which cannot be determined by A.N.Kolmogorov’s
theory. In its turn, the fluctuations theory supplements the fundamental
thermodynamic theory of crystallization that is unable to solve the problem of
crystallization on its own because of the thermodynamic barrier presented by
the critical radius of crystallization centers.
Thus, the three theories complement one another.
However, the inner unity within such symbiosis of theories unrelated between
them is lacking, which presents one of the main problems in the existent theory
of crystallization that can by right be treated as eclectic.
We must observe that A.N.Kolmogorov’s theory allows,
in principle, the stuffing of other quantities of n and v obtained by some other
source. I.e. A.N.Kolmogorov’s theory is not bound up directly with the theory
of fluctuations.
A.N.Kolmogorov’s statistic theory, like other ones,
does not respond to the thermodynamic requirement of the two-factorness of
crystallization (as well as melting) process. It cannot step forward with such
a response by nature, for it fails to reveal the causes of melting and
crystallization processes.
G.F.Balandin writes: ‘any theory of crystallization
should give answers to the following three questions: how crystals nucleate and
how many of them appear in a unit of time; how these crystals grow and what is
the rapidity of their growth under these or those conditions; what solid phase
quantity appears at any given moment of crystallizing the melt volume specified,
and what is the rapidity of crystallization under the given conditions’ /74/.
None of the three above-mentioned basic theories
taken separately gives answers to these questions. Yet their aggregate seems
fraught with contradictions connected with the independence and the lack of
direct ties between the theories named.
It is evident from the brief survey that was carried
out that none of the three parts of the modern crystallization theory viewed
above supplies the answer to the first cardinal question broached in the
beginning of this book: what is the cause of melting and crystallization /79/?
The clue to the problem of the connection between the
processes of melting and crystallization, the relation between solid and liquid
state is totally lacking in modern theory.
Therefore, we arrive at the conclusion that there
exists a pressure to initiate a united theory of the melting and
crystallization of metals and alloys.
This part includes the most general newest principal
definitions concerning the new approach to the defining of the states of
aggregation of matter. The new approach consists in regarding different states
of aggregation of matter as not the states of matter only and exclusively but
as the states of the systems of interacting elements of matter and space, as
various ways of matter arrangement in physical space, and v.v. /80,81/.
Thus, the states of aggregation of
matter are treated as systems comprising two elements: matter and space. Matter
and space are equally important in the structure and properties of the states
of aggregation of matter though they affect this or that specific
characteristic of the system to a different extent. The significance of the
suggested approach is considerably weightier than the problems analyzed in the
book. The novelty of this approach lies in the fact that for the first time it
declares and allows for the equipollent role of the elements of matter and
space in the formation of the states of aggregation of matter, as well as the
forming of any macroscopic physical systems on the whole.
The basic premises of the new approach are the
following.
The characteristics of the states of aggregation of
matter, e.g. liquid and solid ones, originate by definition at the level of
particle aggregates, i.e. in the presence of a certain large number of
particles per volume. It can never be said about a separately taken particle –
an atom or a molecule – that it is solid, liquid or gaseous. It means that
separate particles act as only the chemical property carriers of this or that substance
but do not bear the characteristics of the states of aggregation of matter
/81–83/.
The volume where the properties of the states of
aggregation of matter start becoming apparent is precisely unknown. It is known
only that the volume in question is not large – it approximates the volume of
the smallest drops of liquid.
However, any volume is a certain construction that
contains the elements of matter, arranged and granulated in space, and v.v.
Such a construction may be viewed from the angle of geometry, too. Let us
consider various states of aggregation of matter from the mentioned
standpoints, i.e. the points of view of matter-in-space distribution. This
approach has certain advantages.
Let us introduce the most general definition of the
states of aggregation of matter.
Various states of aggregation of matter present
various ways of matter-in-space distribution inside all real bodies, and v.v.
It follows from the definition that any physical
body, any state of aggregation of matter including, comprises two essential
inner components: material and spatial ones /83/.
Let us make possible inferences out of the general
definition introduced. Such a way of describing various states, solid and
liquid metallic states including, can be further developed in two ways. These
are the scientifically known substantial and relational ways of describing the
system ‘matter-space’.
The substantial approach that originates from Isaac
Newton’s works understands space mainly as a distance, a container inert toward
space, inert atmosphere /116/. Within such an approach the role of space in
forming various states of aggregation of matter is usually ignored as
inessential. Traditionally the role of space in forming various states of
aggregation of matter was by default disregarded, while all the characteristics
of the aggregate states are explained as if they were purely material. The most
significant progress in this sphere developed into the idea of unconfined space
inside this or that substance, which is used but rarely and as a supplementary
concept exclusively. The historically developed negligence of the role of inner
elements of space at macrocosm level is based upon the understanding of space
as the passive part of the environment.
The relational approach, arisen
from Einstein’s works, views space as a physical object that is inseparably
united with matter, and presumes the interaction and interrelation of the two
of these components in any states of aggregation of matter. Yet the theory of
relativity contemplates the interaction of material objects only within a
certain space outer towards them without looking into the interaction between
matter and the inner space of the system /86/.
So it is accepted in the theory of
relativity and the quantum theory that the interaction and the interference of
matter and space are apparent either at great speeds nearing the velocity of
light or in the vicinity of huge masses of matter. This is true for the
interaction of bodies and particles with the outer space. At macrocosm level the
influence of space on the properties of real physical systems is estimated as
negligible even in this theory. The theory of relativity does not contain any
ideas on various inner elements (particles) of space.
The author affirms and proves below by the example of
the states of aggregation of metals and alloys that, apart from the outer space
as regards real space systems, there exists an interior space represented by
discrete spatial elements of various orders inside any physical bodies, which
is inherent in any systems as an integral part of their structure. Active
interaction between the inner elements of matter and space occurs in any
systems, at any dimensional and energy levels; such interaction is a part of
our reality, so we cannot advance multidimensional scientific development but
with the consideration of this factor. Such interaction, its forms and
manifestations are extremely manifold yet subject to theoretic as well as
experimental research.
Let us introduce the concept of the relativity of
inner and outer space. Dividing the elements of space into outer and inner is
relative, since the elements of space that are inner toward one system can act
as outer towards some other one. Nevertheless, the relativity under
consideration does not imply the insignificance of such division, for it allows
a better comprehending of the properties of real bodies to describe them with
more precision.
Let us also introduce the principle of equivalence
between the inner elements of matter and space on the basis of the above-stated
general geometric definition of the states of aggregation of matter, where the
concepts of matter and space are actually equivalent in the sense that they
cannot exist separately. Out of their interacting elements, matter and space
form combinations and systems of exceeding diversity. Any specific physical
characteristic of any system, liquid and solid metals including, depends, to a
different extent, on the contribution of both the material and spatial parts of
the system specified.
Apart from that, the principle of equivalence means
that matter and space perform similar functions of mutual essentiality in the
states of aggregation of matter. It is supposed that the principle of
interaction and equivalence of the elements of matter and space has some
fundamental significance for the forming of various natural systems.
Let us consider the role of the inner elements of
matter and space in the formation of the states of aggregation of matter, solid
and liquid states including, from the standpoint of the relational approach and
taking into account the principle of equivalence of the elements of matter and
space formulated above.
It must be pointed out that the number of works where
the principles similar to the relational approach are applied to the study of
the connection between solid and liquid states is inconsiderable. In the theory
of liquid and solid states as well as in the theory of crystallization we can
find but separate harbinger elements of this approach.
In the previous century, the mentioned approach was
applied to solid metals and other crystals by one of the founders of
crystallography Ye.S.Fyodorov.
Ye.S.Fyodorov’s crystalline lattices are
discontinuous-continuous, so space is inhomogeneous within them, there is an
interrelation and interference of the components of the system ‘matter –
space’. It is known that the orientation and arrangement of matter in space
influences the characteristics of crystals greatly. Suffice it to supply the
well-known examples of various forms of carbon presented by graphite, diamond,
and carbyne. These examples demonstrate substances consisting of the same atoms
of carbon that acquire highly different properties due to the various
distributions and interaction of these atoms in space. However, it is premised
by crystallography and chemistry that everything depends on the elements of
matter – atoms –only, whereas the elements of inner space are disregarded as
the passive component of the system.
Academician N.S.Kurnakov touched in his works upon
the subject of the interaction of matter and space in various chemical
compounds. He showed that there is a tangible parallel to be traced between the
chemical process and characteristics of space in the phenomena of equilibrium
of chemical reactions between various compounds that are expressed (the
phenomena) by geometric surfaces.
Unfortunately, those were nothing
but separate phrases in the works by N.S.Kurnakov, which never found their
evolution /85/.
Recently, on the grounds of the analysis of the works
dedicated to the problem given, V.I.Vernadsky’s works in particular /84/, there
appears the following definition of the interrelation between the states of
matter and space: ‘ It is obvious that the same spatial structures cannot
correlate with all the divergent states of matter; on the contrary,
qualitatively different states of matter will inevitably meet their counterpart
among correspondingly different spatial structures, among various states of
space.’ Matter, by V.I.Vernadsky’s term, means substance according to our
terminology.
This discovery made by V.I.Vernadsky must have
forestalled its time. It should be marked that V.I.Vernadsky’s work ‘ Space and
Time in Living and Inorganic Matter’ that comprises these ideas, was published
only after the author’s death, in the 70-s. It remained totally unclaimed by
physics and exact science.
We treat V.I.Vernadsky’s idea here as one of the
fundamental concepts of the will-be relational theory of
melting-crystallization as well as the connection between the structure of liquid
and solid metals. The theory under analysis may be also termed as the theory of
relativity for metals and other macrosystems.
From this point of view, let us consider liquid and
the adjacent solid and gaseous states as the ways of matter-in-space distribution
and v.v. Let us start from gaseous state as the simplest and the most
well-explored.
Ya.I.Frenkel was among the first to discover one of
the forms of the inner elements of space in solid metals represented by
vacancies – hollow lattice points. He linked the structure of vacancies to the
vacuum medium in gases. By Ya.I.Frenkel’s definition, ‘…in case of gaseous
state… hollows merge with vacuum, where separate molecules appear to be
ingrained so that vacuum ceases acting as the dispersion phase and becomes the
dispersion medium’ /70/.
Such a definition of gas quite agrees with the way of
description accepted in this work.
Thus, we can assume that space is continuous in
gaseous state acting as the dispersion medium, whereas matter is discrete and
acts as the dispersion phase, in which connection both of the two components of
gaseous state are at equilibrium with each other.
The continuous form of matter represented by a set of
crystalline lattices, as is well known, characterizes solid crystalline state
at the level of aggregation states (at the phase level).
If we proceed from the above-accepted definition of
the states of aggregation of matter as various forms of matter-in-space and
v.v. distribution taken together with the principle of equivalence and symmetry
of the elements of matter and space, it ensues that there must be certain
elements of physical space in solid state existing at equilibrium with the
crystalline lattice.
The works by the two founders of the point-defects
theory in crystals – Schottky and Ya.I.Frenkel – give us the reason to surmise
that the target elements of space in crystals are represented by vacancies
/70,88/.
Thus, Ya.I.Frenkel writes that in liquid state
hollows (vacancies) are caused by ‘…the process that can be termed the
dissolution of the surrounding vacuum in a crystal’. Ibidem further: ‘…the
lattice point left vacant…can be regarded as a hollow that appears to be
absorbed by a crystal from the surrounding space’.
Thus, Ya.I.Frenkel considered
vacancies, by their origin as well as by their characteristics, as the elements
of physical space (vacuum) in the crystalline lattice. A similar point of view
is advanced in the works by B.Ya.Piness and Ya.Ye.Guegouzin, where vacancies in
the crystalline lattice of metals are viewed as a parity component having its
volume but deprived of mass /89-90/. Later on the concepts of the parity
metal-vacancy diagrams were worked out on this basis /91/.
If we stick to the set-point
approach, we may conclude that matter represented by crystalline lattices is
continuous in solid state whereas space represented by vacancies is discrete;
on the contrary, space is continuous in gaseous state, matter being discrete.
There is a case of dissymmetry of matter and space in the states of aggregation
of matter.
Inferences concerning liquid state will be made
below. Yet the following conclusion shapes right away: liquid state also act as
a form of matter-in-space distribution, and v.v. Consequently, there must exist
both material and spatial elements in liquid state that are at a dynamic
equilibrium. Further we shall ascertain what these elements are.
Let us consider the processes that precede melting in
metals. It is known that all the characteristics of metals change depending on
temperature. In the broad sense, the total of the changes of all the metal
characteristics with an increase in temperature is premelting, since, one way
or another, all of them reflect the changes in the equilibrium between the
elements of matter and space in solid state, which result in melting.
A possibility of transition to
other aggregation states inheres in any state of aggregation of matter. So if
aggregation state is a form of the interacting elements of matter-in-space (and
v.v.) distribution, then, phase transitions are the transitions from one form
of matter-in-space distribution to another.
However, such a definition sounds excessively
general. It defines the essence that is shared by all phase transitions, which
is important enough but does not reveal the specificities of the mechanism of
melting and crystallization processes that we are interested in. The mechanism
of every kind of phase transitions has peculiarities of its own, since each
form of matter correlates with its respective form of space. These
specificities are to be brought to light here for the processes of melting and
crystallization.
Therefore, let us single out the processes that
relate to the preparation of melting in solid state most directly and are
directly responsible for the mechanism of this process.
It was found before that vacancies act as the
characteristic form of the elements of space in the crystalline lattice of
solid metals. Vacancies within the crystalline lattice of metals are in motion,
the motion being similar to that of matter molecules in gases: it is chaotic
and accelerates with an increase in temperature. The behavior of vacancies in
metals is described by the same expressions as the behavior of particles in
gases with the essential distinction lying in the fact that the rapidity of
vacancy motion in solid metals is much lower than particle velocities in gases
while the trajectory of motion structurizes by the crystalline lattice. Still,
those are quantitative differences, whereas in the qualitative aspect vacancies
in solid metals generate the same gas of space-in-matter elements as is
produced by atoms and molecules in gases.
So there exists in physics a concept of 'vacancy
gas'. In addition, the concept of vacancy gas pressure is applicable to such
gas, similar to particle gas pressure.
Vacancy gas pressure diffuses all through the volume
of solid metal, similar to the pressure of regular gas within the total volume
of such gas. Hence, vacancy gas pressure operates from within upon the
crystalline lattice which functions as the environment and a shell for vacancy
gas at the same time. Similar to regular gas within a rubber bladder, vacancy
gas generates tensions within its shell inside the crystalline lattice and,
should the value of pressure overpower durability, may destroy this shell.
Vacancy gas pressure p within the crystalline lattice
can be calculated out of the known relation:
pv = nkT,
where n is the relative concentration of vacancies; k is Boltzman constant; T is
temperature.
The graph of the given function is
shown in Fig.9.
The values of vacancy gas pressure
in certain metals at the melting temperature are calculated in Table 2.
Table 2. Vacancy Gas Pressure in Solid Metals in
the Vicinity of the Melting Temperature
Metal |
n |
Vacancy gas pressure, pv, kg/sq.cm |
Al |
0.3 10-3 |
2.4 |
Cu |
0.6 10-3 |
9.0 |
Pb |
0.16 10-3 |
0.4 |
Fe |
0.37 10-3 |
7.0 |
W |
0.5 10-3 |
16.0 |
At low temperatures vacancy gas
pressure is negligible as compared to the durability of solid metals, so the
presence of vacancies does not endanger the latter. However, vacancy
concentration and pressure rise exponentially with an increase in temperature
reaching the values listed in Table 2. These are small but quite measurable
pressure values.
It must be noted that vacancy gas
pressure in solid metals cannot be measured experimentally so far; the author
is not even acquainted with such measurement procedures. There exists an
opinion that the formula suggested above allows pressure measurement for gases
only.
Vacancy pressure in solid metals must conform to the
same principles as molecule pressure in gases. The difference consists in the
fact that experimental procedures for measuring vacancy pressure are to be
developed yet, followed by the measurement of other forms of spatial elements
at other levels. Those should be the procedures of pressure measurement of the
inner elements of matter and space (not outer ones) represented by vacancies in
case of solid metals. Once such procedures are established, the equation of
state is not excluded to become much more universal than it appears at present.
Still, the given equation should be applied to the evaluation of a system's
inner parameters only. The pressure of the inner elements of matter and space
may diverge from the ambient pressure. Such a phenomenon takes place in case of
vacancies.
The congruence of the intrinsic pressure
with the measurable pressure in the gas experiment is but an accidental effect
of our being enveloped by the gas medium.
Let us mark that the presence of vacancies performs a
crucial function for the majority of the characteristics of solid metals, which
is decisive at times, as it is in case of electron conduction and
superconductivity. Yet vacancy gas pressure leads to melting only with an
increase in temperature and together with the other factor only.
Crystalline lattice durability acts as such a factor. It is
generally known that the durability of all metals reduces multiply with an
increase in temperature. This is an experimental fact registered in reference books
/92/. A typical dependence of metal durability on temperature is also shown in
Fig.9.
Fig.10, which demonstrates that some elements
(carbon) and compounds (SiC, UC, BN) do not suffer durability loss with a
temperature increase, presents experimental curves of the dependence of the
durability of certain refractory elements and compounds on temperature. It is
important to stress that the same elements and compounds do not melt in the
accepted sense, i.e. they do not form liquid phase. Thus, experimental data
corroborate the importance of durability as one of the factors that determine
melting.
With an increase in temperature vacancy gas pressure
in solid metals rises rapidly, whereas the durability of the same metals
declines correspondingly. Let us recall that the same units measure pressure
and durability.
Hence, the equality is invariably reached in solid
metals at a certain temperature.
pv = σв,
(12)
pv is vacancy gas pressure; σв being metal durability
of elongation.
The point of the intersection of the curves pv
= f(T) and σв=
f(T) in Fig.10 coincides with the melting temperature.
From the position of the
above-said, melting is the process of destroying the continuous crystalline
lattice under the pressure of vacancy gas.
Still, it is a general definition.
It does not reveal the specialities of destruction process, whereas if those
are the specialities, or the details, of the process that determine its result,
what liquid metals structure will be like after melting?
We are going to look into the
details of melting process exemplified by the elementary act of melting below
with the gradual interpretation of the peculiarities of this process.
Melting, as well as crystallization, starts at
certain points of a solid body. We shall term such points as melting centers.
Practice shows that outside surfaces,
especially acute angles, melt first of all. The role of surfaces and angles
becomes particularly apparent at the melting of fine powders where a certain
melting temperature decrease was even observed occasionally.
After outside surfaces the inside interfaces of all
kinds break into melting, namely: crystalline borders, enriched by admixtures
in particular, the boundaries of grains inside primary crystals, the borders of
mosaic structure blocks inside the grains, planar, linear and point defects of
crystalline structure also participate in the process of melting.
Integrating the information, it should be inferred
that zones with the locally increased free energy start functioning as melting
centers in solid metals.
This can easily be paralleled with the above-marked role of
vacancies and durability in melting process. As for vacancies, nothing but the
mentioned zones are the sources of vacancies in crystals /87/. Such is the
literary term of the vacancy origination sites in crystals. Out of these sites
vacancies disperse all over the volume of crystals through diffusion.
Accordingly, vacancy concentration at such sites is the highest possible at the
heating of crystals so the sites in question build up the conditions for
implementing equation (12) in the first place.
In its turn, solid metals durability is unequal at
different zones, too. So nothing except the above-named sites are the weaker
spots of a solid body, the durability of solid metals is minimal there.
Thus, experimental data on melting centers coincide
with the inferences about the role of vacancies and durability in the melting
of solid metals.
The sites of melting and crystallization centers
location do not coincide; moreover, they are polar in a certain sense. Namely,
crystallization centers turn out to occupy the position inside crystalline axes
in the zone crystallizable in the first place. Whereas melting centers are
spread over the surface of crystals along the boundaries of grains, at the
areas of fusible admixture congestion, etc. (see above).
Melting starts where
crystallization ends, and v.v.
Thus, notwithstanding the reversibility of the
processes of melting and crystallization, there are some distinctions, or a
peculiar dissymmetry, in their mechanism.
Returning to the role of vacancies and durability in
the processes of melting and crystallization, it should be observed that if
melting starts at the sites acting as vacancy sources, crystallization sets in
at the areas of vacancy sink. If melting starts at the zones of minimal
durability, crystallization, on the contrary, begins at the areas that will
have the maximum durability in solid metal structure later.
These are the essential distinctions in the location
of the centers of melting and crystallization, so they are to be taken into
consideration at the initial stages of the processes specified while studying
the connection and differences between the processes of melting and
crystallization, the connection and differences between solid and liquid state.
The question of the elementary act of melting and the
structural units of matter in liquid is raised here for the first time.
As a rule, the process of melting
is viewed as continuous where the elementary act of melting is hard or
impossible to detect.
Or melting process is considered corpuscular where
the process of the transition of a separate solid body atom into liquid is the
elementary act while separate atoms are the structural units of matter in
liquid state, similar to solid and gaseous states.
However, such an assumption is incorrect, since
aggregation states characteristics like the property of a body to be liquid or
solid are not inherent in separate atoms. It was stated above that a separate
atom (a molecule) couldn’t be solid, liquid or gaseous. The characteristics of
the states of aggregation of matter become apparent only at the level of
certain aggregations of matter particles and space elements.
The formation of such a minimal aggregation of matter
particles and space elements that bears the properties of the given state we
term here as the elementary act of the formation of this or that aggregation
state, liquid state in the case specified.
It was emphasized above that the two main factors
leading to melting are the increase in vacancy gas pressure, on the one hand,
and the declining of metal and alloy durability with a temperature increase, on
the other hand.
However, vacancy concentration at which melting
occurs is rather small constituting one vacancy per several thousands of atoms
on average.
It is quite important that the presence of vacancies
distorts the crystalline lattice within the area around the vacancy. Vacancy
interaction potential in the crystalline lattice takes the form of damped
periodic function /93/. It means that at the approximation of vacancies to a
definite spacing they seem to feel mutual presence so at further approximation
the zones of repulsion between vacancies supersede the areas of attraction.
Certain vacancies can overcome repulsion zones, the
activation energy being sufficient. Such vacancies merge and are able to form
vacancy disks, micropores and other vacancy complexes of various configurations
/66,87,88/.
However, calculations show that such vacancies are in
the absolute minority – from 1 to 7% of their general number at the melting
temperature.
The absolute majority of vacancies does not
acquire the sufficient activation energy and cannot overcome the repulsion
barrier. I.e. we may assert that the overwhelming majority of vacancies within
the crystalline lattice of metals and alloys repel one another at
approximation. Such repulsion generates vacancy gas pressure.
With an increase in temperature vacancy concentration
in metals increases, - they collide more frequently, so vacancy gas pressure
rises with the temperature increase.
Still, the concentration of
vacancies can increase only within the limit determined by vacancy interaction
potential. It signifies that vacancies that repel one another cannot come
closer than the determined spacing permits – they do not dispose of the
sufficient energy of activation to approximate closer.
Thus, vacancy concentration
reaches a certain critical limit. Simultaneously, the durability of metals and
alloys diminishes to a value equal to vacancy gas pressure. This is the very
point where melting starts.
The elementary act of melting
consists in the following: the crystalline zone surrounding a vacancy (each
vacancy) partially separates from the crystal under vacancy gas pressure.
The diagram of the elementary act
of melting is represented in Fig.11 a, b.
Fig.11a shows the original state
of a crystalline lattice zone with vacancies before the elementary act of
melting, Fig.11b displays the state of the same zone after the partial
separation of the zone that surrounds the single vacancy, from the crystal. The
active force of the elementary act of melting process is vacancy gas pressure,
the direction of which is indicated in Fig.11a by an arrow.
However, hardly does the specified
crystalline zone separate, when there originate slit-like hollows - the areas of
interatomic bond breaking marked by the hatched space in Fig.11b - between the
zone and the areas shown at the bottom of Fig.11b. At that the ejecting
pressure of vacancy gas upon the given area vanishes after the formation of
breaking. At the same time, as it can be seen in the picture, reactive forces
act upon the separated area - the forces of ambient pressure (if any), viscous
friction and surface tension - it exists always.
The forces in question restore the separated area to
the original position.
After that vacancy gas pressure arises
anew in the original position, the area separates partially once again to
return again under the influence of reactive forces, so the process goes into
the oscillatory one.
Such a partially separated crystalline area that
keeps at a constant oscillation we shall further term 'cluster', since such a
term circulates in scientific literature already.
A cluster is the structural unit of space in liquid
state, characteristic of the given state exclusively.
The intercluster split is the structural unit of
space inherent in liquid state of aggregation of matter.
Let us equate the elementary act of melting that
implicates the forming of a single cluster and a single intercluster split. Let
us remark that the conditions of melting at the origin of this process and at
its completion differ to a certain extent, different may be the external
melting conditions, which imposes its constraints on the process. It is
experimentally established that external factors, e.g. pressure or environment,
may affect melting considerably. The external factors under analysis influence
melting in combination with the intrinsic causes of this process. For instance,
the environment can affect, by the mechanism of P.Rebinder’s effect in
particular, the durability of metals and alloys, whereas ambient pressure can
interact with vacancy gas pressure under definite conditions. As a result, it
is the temperature of melting that changes first.
Considering all the stipulations,
let us remark that below there is adduced only one among the totality of the
possible variants of the equation of the elementary act of melting. This is the
variant for the prevailing case of melting at the midpoint of the process, in the
presence of both the liquid and solid phase layers, i.e. at the two-phase
state.
Under these constraints the
equation of the elementary act of melting process may be put down like this:
Fv = Fη + Fσ,
(13)
where Fv is the motive (active) force of the process of
melting represented by vacancy gas pressure that passes after the forming of a
cluster into the kinetic energy of heat oscillations of the latter; i.e. Fv
= mc d2x/ dt2; Fη is the reactive force of viscous friction within the
existing liquid; Fσ are surface forces.
In this connection
Fη
= 6 p η
rc dx/dt ;
Fσ = 2p rc σ x;
where x is the displacement of a crystalline lattice zone, which
comprises one vacancy, from the equilibrium position; mc is the mass
of such a zone (after the separation of a cluster); rc is cluster
radius; η is
liquid metal viscosity at T = Tmelting; σ is the
interphase tension coefficient; t being time.
Let us input
the given quantities of the components into (13). Then, the elementary act of
melting or the equation of the cluster motion at the point of its formation
will be recorded in the following way:
mc d2x/ dt2 = 6pηrc dx/dt + 2prcσx
(14)
Let us introduce the designations
6pηrc = -β; 2prcσ = -α2.
In this case equation (14) will be
similar to the classical oscillation equation in the presence of resistivity:
m d2x/d t2 + β dx/dt + α2x
= 0 (15)
Equation (15) is solved by Eiler substitutions and
hyperbola functions. In the specified case Eiler substitutions assume the
following aspect:
x = exp(kt); dx/dt = k exp(kt); d2x/dt2 = k2
exp(kt).
After introducing Eiler substitutions into (15) we
get:
exp(kt) (mc k2 + β k + α2)
= 0.
Having solved the given equation,
we derive the expression for the motion of a cluster at the point of its
formation:
x =v0 exp( - βt / 2 mc ) [exp(ωt)
– exp( - ωt)]/ ω (16),
where v0 is the original speed of cluster motion;
ω is the conditional frequency of cluster oscillations.
The final expression (16) is an oscillatory motion
equation. At a one-pass impact of the active force of Fv this
equation turns into the damped oscillations equation. At a repeated multiple
impact of the active force the equation shifts to the equation of undamped
oscillations. It was stated above that after the resetting of a cluster under
the influence of reactive forces the active force of vacancy gas pressure
generates anew so the process repeats again and again. Consequently, the
cluster at the point of its formation acquires an oscillatory motion of its
own. It accounts for the fact that metals and alloys absorb a large amount of
heat at melting without changing the temperature of the body. It is known that
this is possible only in the case when the system takes on new degrees of
spareness, i.e. new kinds of motion. Such a new kind of motion, or new degrees
of spareness for metals in liquid state, is the oscillatory motion of clusters
and atoms inside them. Such a phenomenon will be viewed in detail below.
The importance of equation (16)
lies in its showing how the new kind of motion that is lacking in solid state –
cluster oscillations –arises.
Out of the latter expression (16),
in its turn, there may be found some significant quantitative parameters of
cluster oscillations, particularly the maximum deviation of a cluster from the
original position, i.e. the amplitude of cluster heat oscillations.
Simultaneously, the value in question will characterize the width of the
flickering intercluster slit-like hollows - the areas of the elements of space
peculiar to liquid state.
Determining the given value is quite important due to
the fact that it helps us answer the following questions: 1) whether the
cluster detaches wholly from the solid body at melting; 2) if the liquid
intermixes right after its formation. The latter is significant for determining
the mechanism of the well-known phenomenon of metallurgical heredity.
If we specify the maximum of function x from (16), we
get /83/:
x = ( v0 mc /6pηrc )[1 -
exp( - p)]
(17)
The latter expression should be solved separately for
each given metal. The author accomplished this procedure as applied to iron,
mercury, lead, zinc. It was obtained that the maximum deviation of clusters from
the original position at the point of their formation has the order of 0.1-1.0
angstroem units. This is considerably less than the dimensions of an atom in
the metals specified and is approximately equal or more than the ultimate
theoretic strain of matter at elongation (Frenkel constant) /70/.
Hence, clusters at the point of
their formation do not detach fully from the remaining metal mass, solid phase
including. They deviate 1 angstroem from the original position at the most,
after which they return to the original position, and further such oscillations
are continuously repeated.
A distension of 0.1-1.0 angstroem suffices for a
short-time breaking of interatomic cluster bonds to the remaining solid metal
mass on the plane perpendicular to the cluster motion direction. In its turn,
the direction of the motive force of melting process - i.e. vacancy gas
pressure upon the crystalline zone specified - defines the cluster motion
direction, as shown in Pict.11. The former direction is always perpendicular to
the surface of the specified zone of liquid and solid phase section surface and
is oriented
outward solid phase.
It follows from the figures obtained that at the
point of their formation clusters remain attached to their respective sites so
the spontaneous intermixing of liquid at the point of its formation does not
occur, because cluster dimensions far exceed their deviation value. It was
demonstrated above that approx. a thousand of atoms, the dimension of which
averages 1-10 nm, enter into a cluster at melting. (Cluster dimensions will be
calculated precisely enough below.) Thus, liquid metals and alloys have the
same distribution of matter in their volume at the point of their formation as
it is in solid state. Time passing, there gradually ensues an intermixing and
homogenizing of the liquid alloy composition due to convection at macrolevel
and cluster diffusion at microlevel. Such an inference totally conforms to the
well-known facts of metallurgical structural and chemical heredity and its
connection with the time of holding, overcooling and the intermixing of melts.
A short-time split of bonds suffices for the removal
of vacancy gas pressure in this direction, whereas the contact of a cluster
with the environment remains at other planes, - there is only a displacement.
As it is to be expected during such a displacement, interatomic bonds are
destroyed but partially at the shifting planes, for their complete rupture
occurs at splitting.
As it follows from the analysis carried out above,
the conclusions concerning such phenomena as cluster and intercluster splits
formation, as well as their oscillatory motion, do not act as postulates. They
present a mathematical consequence to the analysis of the elementary act of
melting /83/.
Hence, right after melting liquid
metals and alloys consist of rather small (a thousand atoms approx.) atomic
groupings - clusters that were formed at melting and are performing continuous
heat oscillations. The totality of clusters constitutes the material part of
liquid aggregation state of metals and alloys.
The spatial component is formed at melting, too,
representing the areas of slit-like flickering splits of interatomic bonds
between clusters. These are quite narrow slits not more than 1 angstroem wide
arising and vanishing (flickering) in consequence of the separation and
approximation of clusters in the process of their heat oscillations. While
oscillating in such a way, any cluster approaches one half of its immediate
neighbors on average at any moment given moving away from the other half.
As a result, only a half of the 'surface' of every
cluster is marked and separated by splits at any given moment. The other half
of the 'surface' serves to bond a cluster to the whole material mass in liquid
at any given moment. The term of 'surface' is applicable to clusters with
certain stipulations only. Such surfaces are flickering, i.e. they arise and
vanish periodically. The flicker of intercluster splits is not a postulate
either but a corollary to the mathematical analysis of the elementary act of
melting.
The concept of flickering surfaces, or flickering
intercluster splits, as a form of space inherent in liquid state of aggregation
of matter, is introduced here for the first time. Such forms of spatial
elements have never been known before. Certainly, the phenomenon under
consideration is to be studied further. The existence of such elements of space
shows that the forms of the elements of space equilibrated by these or those
material forms can really be varied, new, unknown yet subject to study.
The process of flickering is highly important by
itself, since flickers are the main form of interaction between the elements of
matter and space at the level of solid and liquid states of aggregation of
matter. It is the flickering form of the elements of matter and space that
determines the basic metal characteristics to a certain extent, as it will be
shown below.
The causes of the insufficient exploration of the
inner elements of matter and space lie in the sphere of psychology but not
physics: no-one studied the given aspect of reality so far because of the
physical invisibility of the inner elements of space, first, and the universal
negligence of the researchers toward the role of the inner elements of space in
the structure of real systems, in the second place.
Our approach consists in premising that the role of
spatial elements in the forming of the structure and properties of liquid
metals and other physical bodies is none the less important than that of the
material component (atoms, molecules and elements of other levels of matter).
However, the role of the elements of space in the formation of the properties
of various systems differs essentially from the role of matter.
Matter and space represent, in their own way,
different poles of the characteristics of all real systems. Still, let us
underline it once more that any property of any material system is determined
by the summarized contribution of the two of these inseparable elements of
matter and space.
The particular contribution of
material and spatial elements differs in application to every given system or
property. Still, such contribution is always clearly distinguishable and always
present, so it must be improper to neglect the contribution of both the
components even if we are forced to do so. The given conclusion will be
illustrated by the example of calculating a whole series of the characteristics
of liquid metals.
Liquid metals preserve the neighboring order of
atomic distribution inside clusters, which is inherent in solid state. The
heritage of solid state in liquid one is represented by monovacancies - one
vacancy per cluster on average. Namely, these are vacancies that make clusters
repel one another at approximation.
Thus, liquid has some specific
prevalent elements of structure and motion kinds of its own as well as those
inherited from solid state.
Flickering clusters are the predominant specific
elements of matter in liquid metals. We term them predominant, for nothing but
clusters are the smallest structural units of matter in liquid state.
The prevalent elements - the structural units of
space in liquid metals - are flickering intercluster splits.
The totality of the elements of matter and space in
liquid state - clusters and intercluster splits - bears the basic properties of
liquid state taken as a whole.
The properties of crystalline structure presented by
the neighboring order of atomic granulation in clusters are the elements of
matter inherited from solid state in liquid metals. The elements of space
inherited from solid state in the structure of liquid metals are monovacancies
located inside clusters.
The ulterior characteristics of gaseous aggregation
state in liquid state can be singled out in a similar way.
The simultaneous presence of such characteristics or
premises of one aggregation state within others has, to all appearance, quite a
general character, i.e. it occurs at any state though under different aspects.
Let us term the ulterior properties of other states
within the state given as latent properties to be distinguished from the
characteristics of the prevalent state of a system.
Thus, there is observed a clear structural tie
between liquid and solid metal states. Each of these aggregation states bears
the latent properties of the other state, which promotes melting and
crystallization processes as well as phase transitions upon the whole.
The aforesaid may be treated as the structural
substantiation of the thermodynamic principle of dualism, or two-factorness,
deployed in Ch.1.
In particular, it follows that there exist at least
two factors the interplay of which causes melting: the factor of the increase
in vacancy gas pressure with the rise of temperature and the factor of the
decrease of metal durability with a temperature rise.
On the other hand, ascertaining the fact of the
presence of one aggregation state within the other may also be regarded as the
basis of the thermodynamic dualism of the processes of melting and crystallization.
The free energies of the prevalent and latent states
can be calculated and compared for any temperatures. Yet it seems far more
important that prevalent and latent characteristics are constantly changeable
while coexisting simultaneously and extrude each other with the change of the
environmental conditions. Such extrusion between the latent and predominant
elements of matter and space ensures the constant readiness of a system for the
transition of aggregation states with the corresponding change of external
conditions.
The extrusion under consideration is the motive force
of melting, crystallization and other forms of state transition of systems.
Such dualism is to be viewed in detail below.
Let us calculate cluster dimensions in liquid metals
at the melting temperature premising that the latent heat of vaporization of
metals, similar to the latent heat of melting, is known.
We shall proceed from the bimolecular reactions
scheme known from the vaporization (condensation) theory /66,94/. This scheme
presents as follows:
(18)
where α1 is a separate atom (a
molecule); αm is a complex consisting of m atoms.
The scheme under consideration is reliable enough to
describe melting and crystallization processes, where phase transition from one
aggregation state into the other one is effected atom by atom, though there
might be some stipulations even in this case.
However, such a scheme is
unacceptable for the universal description of melting and crystallization
processes, since these are whole atomic complexes - clusters - but not separate
atoms that enter liquid at melting. Some authors assume that at melting
separate atoms pass from solid into liquid state to further assemble into
clusters again, while surrounded by liquid.
Nevertheless, such a gradation contradicts the
well-known minimum principle, annihilating the hereditary interrelation between
liquid and solid states which, as experiments show, really exists represented
by the phenomena of structural heredity and its other types. Therefore scheme
(18), acceptable for describing subliming processes, must be substituted for a
certain other scheme for melting, that would evince more conformity to the
mechanism of melting, the minimum principle and the facts of the hereditary
bonds between solid and liquid states.
We suggest the following scheme for the cluster
mechanism of the process of melting /30/:
(19)
where αn is a cluster including n atoms; α
in being a crystal consolidating i clusters.
The two latter schemes, except their structural
distinctions, require dissimilar energy consumption for their realization. In
absolute accordance with experimental data, the bimolecular reactions scheme
(18) demands far greater energy expenditure than the scheme of cluster
reactions (19). The fact in question enables us to determine cluster
dimensions. In conformity to the facts available, to realize vaporization
according to the bimolecular reactions scheme, there must be energy consumption
equal to the latent heat of vaporization ΔHvap. Such
energy is required, as we know, for the splitting of interatomic bonds.
For the realization of melting through the cluster
reactions scheme (19), energy expenditure that equals the latent heat of
melting ΔHmelting must take place. The heat
under analysis is spent to split the bonds between separate clusters, though
the phrase 'a separate cluster' lacks accuracy. Clusters do not exist
separately but only in agglomerations.
The cluster scheme of
melting-crystallization reflects the participation of only the elements of
matter – clusters and crystals - in liquid and solid states in the processes
specified. It does not allow for the part introduced by the elements of space.
This will be accomplished further.
It is known that approx. one half of interatomic
bonds but not all of them split at vaporization /94/. Almost the same happens
at melting when about a half of all the bonds between clusters but not all of
them are split.
The cited data of the distribution of quantities of ΔHvap and ΔHmelting
enable us to calculate the average cluster dimensions in liquid metals at
the melting temperature.
To simplify our calculations, let us premise that the
source solid has a simple cubic granulation (s.c.) the coordinating number of
which is K = 6, while clusters are in the form of the cube. It can be
easily demonstrated that the number of interatomic bonds n1 on the
surface of the cube with such a granulation relates to the number of atoms
inside it n by a simple ratio /30,95/:
n1 = 6n2/3. (20)
The suggested formula was derived by the author proceeding from the
following obvious considerations. The number of atoms along the edge of the
cube with a simple cubic granulation of atoms inside it amounts to n1/3.
The number of atoms (interatomic bonds including) on the surface of one plane
of the cube will be equivalent to the square of the latter quantity, i.e. n2/3.
The number of cubic planes is 6. Hence originates formula (20).
Let us designate the energy of one interatomic bond
as U1. Then, the specific heat of melting per cluster δHmelting
will be equal to the energy of one half-bond U1/2, multiplied by the
number of bonds that are split at melting on the 'surface' of a cluster, i.e. n1/2.
Or
δHmelting = U1 n1/ 4 = 6n2/3
U1 / 4 = 3U1 n2/3 /
2. (21)
The total energy of one-atom bond with the
coordinating number of the immediate neighbors K = 6 will make 6 U1/
2 = 3 U1. Divisor 2 is introduced here because at the splitting of
one-atom bond its energy is divided between two atoms.
The energy that is calculated by
formula (21) is the energy of the elementary act of melting.
At present let us consider the constituents of the
latent heat of vaporization of the same number of atoms n. As it is stated in
literature /94/, the latent heat of vaporization constitutes approx. a half of
the aggregate energy of interatomic bonds of matter. Taking this into
consideration, we get
2δHvap = 3 U1 n,
or
δHvap = 3/2 U1 n. (22)
To determine the quantity of n, let us find the
correlation between the specific quantities of δHvap and
δHmelting from (22) and (21). We get
δHvap / δHmelting = (3/2 U1
n) / (3/2 U1 n2/3),
therefrom
n = (δHvap / δHmelting)3,
for a simple cubic granulation of atoms inside a cluster.
However, the application of the latter formula has
certain constraints, for the specific quantities of δHvap
and δHmelting are unknown.
Let us make use of the circumstance that in the given
case it suffices to know but the correlation of the quantities under analysis.
Since (21) and (22) relate by definition to the equal number of atoms n, it
will be quite correct to substitute the specific quantities of δHvap
and δHmelting for the molar ones ΔHvap
and ΔHmelting, because their proportion will be
identical. Thus,
nsc = (ΔHvap
/ ΔHmelting)3,
(23)
where nsc is the number of atoms in a cluster with
a simple cubic granulation of atoms inside it.
Since the granulation of atoms
inside a cluster is known, we can find the radius of the cluster insphere. For
a cluster with a simple cubic granulation of atoms we have:
rsc = a (n1/3) /
2, (24)
where n is determined by (23); a is the
shortest interatomic spacing within the crystalline lattice of the given type.
Using simple geometric transforms, it is easy to turn
from the simple cubic type of granulation to the calculating of the number of
atoms in clusters for the b.c.c. (body-centered cubic granulation) and f.c.c.
(face-centered cubic granulation) granulation types. We derive the following
/30,96/:
nbcc = (9/16) (ΔHvap
/ ΔHmelting)3
; (25)
r = (a/2) (3nbcc /4)1/3;
(26)
nfcc = (1/4) (ΔHvap
/ ΔHmelting)3
; (27)
rfcc = (a/2) (nfcc /2)1/3
; (28)
Allowing for a negligible error,
we may use equations (27) and (28) to calculate the cluster structure of
liquid metals that have a compact hexagonal granulation in solid state near the
melting temperature. It is evident that all the latter formulas are derived
under the hypothesis that neighboring order remains in clusters during melting
process as it was in the source solid metal or alloy.
Moreover, the obtained equations
are derived under the premise of a cubic cluster shape. A spherical cluster
shape is closer to reality because of its minimal surface. Therefore, we are to
elaborate the equations for calculating the dimensions of spherical clusters.
Analogously to deriving (23 - 28), let us arrive at a
solution for spherical clusters with a simple cubic granulation of atoms inside
it. We express cluster volume through the number of atoms inside it getting
Vc = n a3.
In its turn, the volume may be
expressed through cluster radius, too /81/:
Vc = (4/3) π r3.
Equating the right sides of the two latter equations,
we can find
rsc = (3n/4π)1/3
a. (29)
Generalizing the latter expression for any
granulation types, we get
rc = (3z n /4 π)1/3
a, (30)
where z = ksc / kc ; ksc = 6
- is the coordinating number of a simple cubic granulation; kc is
the coordinating number of granulation in a cluster assumed equivalent to the
coordinating number in solid metal at the melting temperature.
Let us admit further that the
total energy of cluster bonding is proportionate to its volume V, the surface
energy of bonding being proportionate to the area of its surface S. Out of the
correlation of the given quantities we obtain:
V/S = 2 ΔHvap / 2 ΔHmelting
= 3U1 n / (4 πr2
U1)/2a2
By inserting here the value of r from (29),
for a simple cubic granulation we obtain:
ΔHvap / ΔHmelting = 3U1 n
/ 3πU1 n2/3 (3/4π )2/3≈ n1/3 /
(1/2)1/3.
Hence, for spherical clusters with
a simple cubic granulation of atoms in neighboring order inside clusters we get
nsc = 1/2 (ΔHvap
/ Δ Hmelting)3.
(31)
Using the correlation V/S is quite
correct here, for its usage in case of cubic clusters allows to arrive at
equations (23-28), i.e. it is adequate to the use of the means of deriving the
expressions for calculating cluster dimensions that was accepted earlier.
If we generalize equation (31) for
all the types of granulation by the previously accepted procedure, we get the
correlation for calculating the number of atoms in spherical clusters with any
types of atomic granulation in neighboring order:
nc = ac (ΔHvap / ΔHmelting)3,
(32)
where ac is the geometric coefficient
that depends on the shape of a cluster as well as atomic granulation in it. For
cubic clusters with a simple cubic, body-centered cubic and face-centered cubic
granulation of neighboring order ac = 1; 9/16 and 1/4
correspondingly.
Similar to (32) we derive the generalized expression
for the calculation of cluster radius rc at the melting
temperature:
rc = ( ΔHvap
/ ΔHmelting) β1/3 a, (33)
where β = 3z ac / 4π for spheric clusters.
The results of calculating cluster
dimensions in liquid metals at the melting temperature are tabulated below.
Table 3. Cluster Dimensions in Liquid Metals at the
Melting Temperature
Element |
Effective coordina-ting number |
DHmelting,
C/mole /97,98/ |
DHvap, C/mole
/97,98/ |
rc, 10-10 m, calculations by (33) |
rc/ a |
nc, cube, calcula-tions by (32) |
nc, sphere |
Cu |
12 |
3.1 |
80.3 |
14.8 |
5.8 |
3300 |
1650 |
Ag |
12 |
2.69 |
60.0 |
17.0 |
6.4 |
4300 |
2160 |
Au |
12 |
3.05 |
82.0 |
19.5 |
6.7 |
4800 |
2400 |
Pt |
12 |
5.2 |
112.0 |
14.4 |
5.2 |
3300 |
1250 |
Pd |
12 |
3.5 |
110.0 |
21.3 |
7.7 |
7800 |
3900 |
Al |
12 |
2.57 |
69.0 |
20.5 |
6.7 |
6300 |
3150 |
Pb |
12 |
1.15 |
42.5 |
26.2 |
9.2 |
12600 |
3150 |
Ni |
12 |
4.22 |
89.4 |
13.2 |
5.3 |
2400 |
1200 |
Co |
12 |
3.75 |
91.4 |
15.0 |
6.0 |
3600 |
1800 |
Ti |
12 |
4.5 |
102.5 |
16.2 |
5.6 |
3000 |
1500 |
Zr |
12 |
4.6 |
128.0 |
22.1 |
6.8 |
5400 |
2700 |
Re |
12 |
8.0 |
169.0 |
14.4 |
5.2 |
2340 |
1170 |
Ce |
12 |
2.12 |
75.0 |
31.7 |
8.7 |
11000 |
5500 |
Zn |
12 |
1.74 |
27.3 |
10.4 |
3.9 |
960 |
480 |
Cd |
12 |
1.53 |
23.9 |
11.5 |
3.9 |
960 |
480 |
Ca |
12 |
2.1 |
39.9 |
18.5 |
4.7 |
|
860 |
Mg |
12 |
2.1 |
30.5 |
11.3 |
3.5 |
|
340 |
Hg |
6 + 6 |
0.549 |
14.13 |
22.0 |
6.4 |
|
2200 |
Fed |
10 |
3.3 |
81.3 |
18.0 |
7.27 |
5400 |
2700 |
V |
10 |
5.05 |
109.6 |
16.8 |
6.4 |
|
1820 |
Cr |
10 |
4.6 |
89.9 |
13.3 |
5.3 |
|
1050 |
W |
10 |
8.4 |
183.0 |
17.5 |
6.4 |
|
1850 |
Mo |
10 |
6.6 |
121.0 |
14.5 |
5.4 |
|
1100 |
Nb |
10 |
6.4 |
166.5 |
22.0 |
7.7 |
|
3160 |
Ta |
10 |
5.9 |
180.0 |
25.8 |
9.0 |
|
5100 |
Sn |
10 |
1.69 |
64.7 |
34.7 |
11.2 |
|
10000 |
Li |
10 |
0.7 |
35.3 |
44.4 |
15.0 |
|
23400 |
Na |
10 |
0.63 |
23.7 |
41.8 |
11.2 |
|
10000 |
K |
10 |
0.57 |
18.9 |
44.6 |
9.7 |
|
6500 |
Cs |
10 |
0.50 |
15.9 |
49.2 |
9.3 |
|
5700 |
Bi |
6 + 1 |
2.6 |
42.8 |
26.7 |
8.0 |
4500 |
2250 |
Ga |
6 |
1.336 |
61.4 |
62.6 |
22.5 |
96000 |
48000 |
Si |
4 |
12.1 |
72.5 |
- |
3.88 |
320 |
160 |
Ge |
4 |
7.7 |
78.3 |
- |
6.5 |
|
765 |
Table 3 demonstrates that cluster
dimensions are calculated for quite a wide range of metals prioritized in
technics and metal science.
If we compare nc for cubic and spherical
clusters, it is possible to observe that spherical clusters contain the number
of atoms twice as small as cubic ones. Such a distinction seems essential
enough; it shows that the choice of the right cluster shape is sufficiently
important. The idea of spherical clusters will be used further on as the basis.
As the table shows, cluster dimensions differ
essentially between themselves in various metals though preserving their order
upon the whole. The minimal number of atoms in a cluster nc for
silicon, magnesium and zinc amounts to 160, 340 and 480 atoms accordingly; the
maximal values of nc for gallium and lithium are 48000 and 23400
correspondingly. However, nc has the order of 103 for the
majority of metals at the melting temperature, which coincides on the whole
with the evaluation of cluster dimensions carried out by other researchers
/99,100/.
The average radius of a cluster at the melting
temperature equals approx.10-9m for the majority of metals. Thus,
clusters are very small formations that are difficult to detect by means of
direct observation. Besides, clusters exist only in motion, only in aggregates
and at interaction with the intercluster splits. Clusters have neither stabile
boundaries habitual to macrocosm nor surface sections but flickering boundaries
or surfaces only. These are quite specific objects with unusual
characteristics, so we need new experimental methods to study them.
The elements of space – flickering intercluster
splits of bonds – form the other equilibrium structural zone in liquid, which,
by interacting with the zone of clusters, constitutes the specific structure of
liquid metals. The basic parameters of the given zone may be calculated
quantitatively. In particular, the average dimensions of a single intercluster
split can be determined, as well as the quota of volume occupied by the
totality of splits in liquid metals.
Since intercluster splits relate to clusters by
definition, their area will be equal to the cluster section area, which is
proportional and closely approximate to the value of rc2.
Let us determine the width of intercluster splits
proceeding from the following considerations. The formation of such splits is
possible only in case when intercluster spacing expands to the value of α
equivalent to the relative theoretic deformation of matter at distension and
will make a (1 + α). Under the conditions of heat oscillations it
corresponds to the situation when a half of all the spacings between clusters
will be less than a (1 + α) = (a + aα). In the meantime,
splits are either lacking or closing. The other half of intercluster spacings
will exceed (a + aα), which corresponds to the split of bonds. The
quota of the element of space proper out of the present quantity will amount to
aα.
It was underlined above in parts 3.1 and 3.2 that no
sooner is the intercluster split formed, than the returning of the cluster into
its original position starts. Therefore, the average quantity of the width of
intercluster split must also approximate the quantity of αa, where
a is the shortest interatomic spacing in solid metal in the vicinity of the
melting temperature. The area of a single intercluster split will be
approximately equivalent to rc2. Admitting that a
cluster performs heat oscillations along the three axes, we obtain that the
area of intercluster splits per cluster equals approx. 3rc2.
Then, the total area of
intercluster splits per gram-atom of any liquid metal will be equivalent to
Scl = 3N0 rc2 / nc
.
Allowing that N0 = 6 1023, rc
= 10-9 m on average, while nc = 103
on average, we get that Scl ≈100
sq.m / g-atom on average. It means that liquid
metals have a gigantic surface area of the inner elements of space –
intercluster splits. So these flickering inner surfaces constitute an essential
part of the structure of any liquid metal and any other liquid, too. The
presence of such surfaces determines many characteristics of liquids, including
such a fundamental characteristic of liquids as fluidity, in particular (see
below).
Returning to the volume of a
single intercluster split in liquid metals, we obtain that the given volume is
equal to the surface area of a single element of space, multiplied by the width
of such an element:
vs ≈rc2
αa. (34)
The quantity of α may be found from the expression cited by
Ya.I.Frenkel /70/:
α = σmax / E,
where σmax is the ultimate theoretic strain
of matter at elongation; E being the modulus of elasticity of matter.
In its turn /101/, the ultimate
theoretic strain of matter can be evaluated from the expression
σmax = (E γ / a) 1/2
and
α = (E γ / a) 1/2 /
E, (35)
where γ is the coefficient of surface tension in liquid metal
at the melting temperature.
Turning back to the volume of a single intercluster split in liquid
metals, we get:
vs ≈rc2
αa. (36)
The number of intercluster splits Ns in a mole of
liquid approximates the molar quantity of clusters Nc:
Nc ≈ N0 / nc.
(37)
If N0 = 6 1023, while nc
averages 103, the number of intercluster splits Nc = Ns
= 6 1020 on average per mole at the melting temperature. This is
quite a large quantity.
The summarized absolute volume of splits in liquid
per mole will be equal to
Vs ≈Nc vs
= (N0 / nc) rc2
αa. (38)
Practice requires the knowledge of
the quota of the total volume occupied by the elements of space rather than the
absolute volume of the zone of the elements of space (which may also be termed
as the zone of unconfined space).
Having evaluated the average spacing between clusters
by the value of a (1 + α) and its expansion as compared with the
non-split state by the quantity of aα, it is quite easy to find the
corresponding change in the system's volume ΔVspl out
of the known expression that relates the change of the length of the object to
the change of its volume /102/.
If the length of a cube-shaped
body is 1, while length increase equals α, the relative
augmentation of the volume of the body will approximate
Δ Vspl = 3α.
Since length for clusters is
l = 2rc,
the relative volume of the zone of intercluster splits will be as
large as
ΔVspl = (3α / 2rc)
100% (39)
By inserting the value of α from (35) into (39), we get
Δ Vspl = 3 (E γ / a) 1/2 / E 2 rc.
Expression (39) is the most suitable for calculations, since the
values of rc are already known there. The values of the
quantities required for calculations are listed in Table 4 below.
Calculating ΔVspl under (39) shows that the
volume occupied by the zone of intercluster splits (the elements of space) in
liquid metals at the melting temperature fluctuates within the limits of 1-6%
for the majority of metals (v. Table 4 below).
Table 4. The Volume of the Zone of Intercluster
Splits in Liquid Metals at the Melting Temperature
Metal |
g, erg/ccm /12,20/ |
E, kg/ccm /101/ |
a, calculations by (35) |
DVspl,
%, calculations by (39) |
Cu |
1133 |
11200 |
0.19 |
4.85 |
Ag |
927 |
7700 |
0.205 |
4.70 |
Au |
1350 |
11000 |
0.226 |
4.95 |
Pt |
1800 |
15400 |
0.205 |
5.7 |
Pd |
1500 |
11900 |
0.214 |
4.08 |
Al |
914 |
5500 |
0.24 |
5.30 |
Pb |
423 |
1820 |
0.26 |
4.15 |
Ni |
1825 |
21000 |
0.183 |
5.10 |
Co |
1890 |
21000 |
0.185 |
4.56 |
Zn |
770 |
13000 |
0.145 |
5.47 |
Feg |
1835 |
20000 |
0.177 |
4.84 |
Fed |
1835 |
13200 |
0.227 |
5.1 |
Sn |
770 |
4150 |
0.248 |
3.3 |
Cs |
68 |
175 |
0.27 |
4.3 |
Ta |
2400 |
19000 |
0.21 |
3.46 |
Mo |
2250 |
35000 |
0.153 |
4.27 |
Nb |
1900 |
16000 |
0.204 |
3.93 |
W |
2300 |
35000 |
0.155 |
3.59 |
Bi |
3900 |
- |
0.207 |
3.7 |
Ga |
735 |
- |
0.20 |
1.33 |
Thus, the elements of space in
liquid metals occupy from 1.33 to 5.7% of the total volume of liquid.
Accordingly, clusters occupy from 94.3 to 98.67% of the total volume of liquid.
The volumes that are occupied by the latent elements of matter and space are
included in the quantities specified.
At melting liquid acquires a large amount of extra
energy as the latent heat of melting, yet the temperature of the liquid does
not change during the process. This is possible only in case when there
originate new degrees of freedom, i.e. new kinds of motion, within the system -
liquid metal in the case given. When analyzing the motion of a cluster at the
point of its formation (the elementary act of melting), it was proved above
that a new kind of motion – heat oscillations of clusters – arises in liquid as
a result of melting.
Let us find the energy of the
oscillations in question, which will enable us to calculate the frequency of
heat cluster oscillations in liquid further on.
In conformity to the theorem of classical statistics
of the uniform distribution of energy according to the degrees of its spareness,
any extra energy within the systems that consist of a large number of particles
is distributed uniformly among all the constituent parts of the given system at
microlevel.
The constituents of liquid at microlevel are clusters
and atoms.
Each particle receives an amount of energy equal to
Ei = Δ Hmelting/ (N0 + Nc),
where N0 is Avogadro Number; Nc
is the number of clusters in a gram-atom of liquid metal.
Since Nc << N0,
the latter expression can be written without any appreciable error as
Ei = Δ Hmelting
/ N0. (40)
On the other hand, we know that the energy of heat
oscillations of one atom makes
Ea = (3/2)
kT (41)
In compliance with the theorem of
the uniform distribution of energy, particle dimensions are not to be taken
into consideration, so the energy of heat oscillations of a cluster that
comprises many atoms will be equal to the same quantity as the energy of
oscillations of a single atom:
Ec = Ea = (3/2) kT (42)
At the melting temperature the quantities of Ec
and Ei must be equal, or
Ec = (3/2) kTmelting
(43)
Comparing the values of Ec and Ei
from (43) and (40) correspondingly gives us the possibility to test the
accepted hypothesis of the equality between the two quantities specified.
To do this, let us calculate the values of Ec and Ei.
The results of calculations are
listed in Table 5.
Table 5. The Energy of Heat Oscillations of
Clusters
Metal |
Ec, J, calculation by (43) |
Ei, J, calculation by (40) |
Ei /Ec |
Na |
0.77 10-20 |
0.43 10-20 |
0.56 |
Pb |
1.24 10-20 |
1.15 10-20 |
0.93 |
Zn |
1.87 10-20 |
1.73 10-20 |
0.92 |
Fe |
3.74 10-20 |
3.08 10-20 |
0.83 |
Cr |
4.50 10-20 |
3.22 10-20 |
0.72 |
Ni |
4.66 10-20 |
2.92 10-20 |
0.62 |
Co |
4.77 10-20 |
2.73 10-20 |
0.57 |
It follows from the data presented in Table 5 that
the suggested hypothesis of the equality between the quantities of Ei
and Ec is corroborated, for the values of these quantities are very
close numerically. A negligible error of determination constitutes approx.± 20 %,
which is rather rare to be observed in calculations of such a kind, if we allow
for the difference in electron structure, as well as the peculiarities of the
structure of crystalline lattices, etc. Presuming that the quantity of Ec
is determined with more precision, we can calculate the average correction
factor to formula (43) on the basis of the data listed in Table 3. The
coefficient in question is 0.707.
By way of inserting the signalized
coefficient into (43), we arrive at the improved formula
Ei = 0.707 Δ Hmelting / N0.
(44)
The most important conclusion to
the given part of the work is the following: the latent heat of melting equals
the energy of heat oscillations of particles at the melting temperature with a
negligible error, hence after melting the specified energy is really spent to
establish new degrees of motion freedom in liquid metals – heat oscillations of
clusters and atoms included into them as a unit.
The data supplied in Table 5
corroborate numerically the correctness of the given inference.
The previous calculations, if they
prove to be correct, allow completing a successive procedure - to calculate the
point of metal melting. It suffices to equate the right sides of expressions
(40) and (43) allowing for the fact that calculations for Table 3 presume that
T = Tmelting.
Thus, we get
Δ Hmelting nc / N0
= (3/2) kTmelting.
Hence we derive the expression for
calculating the melting temperature of metals:
Tmelting = Δ Hmelting / 1.5 N0 k.
(45)
An extraordinarily simple expression (45) is derived
to calculate the melting temperature of metals which relates the given
temperature to the known physical constants: the latent heat of melting,
Avogadro Number and Boltzman
constant.
The results of calculating the
melting temperature of metals under (45) are presented in Table 6.
Table 6. The Melting Temperature of Metals
Metal |
Δ Hmelting, C/mole
-1 |
The Melting Temperature, Тmelting, К |
|
Тmelting, К
by (98) |
Тmelting, К
exper./98/ |
||
Al |
2.58 |
876 |
933 |
V |
5.51 |
1857 |
2190 |
Mn |
3.5 |
1179 |
1517 |
Fe |
4.4 |
1428 |
1811 |
Ni |
4.18 |
1406 |
1728 |
Cu |
3.12 |
1051 |
1357 |
Zn |
1.73 |
583 |
692 |
Sn |
1.72 |
529 |
505 |
Mo |
8.74 |
2945 |
2890 |
As we see it from Table 6, formula
(45) lets obtain only approximate values of the melting temperature accurate
within 2 to 30%.
Although the accuracy under
consideration is not so high for practical application, we should observe that
other methods of calculation the melting temperature with the same or higher
accuracy do not exist so far. Formula (45) ensures the highest accuracy of
calculating the melting temperature of metals at present. In the aggregate with
other calculated data, the data in Table 6 corroborate the applicability of the
developed theory to the description and calculation of a wide range of the
parameters and properties of liquid metals.
A considerable amount of activated atoms in liquid
metals is the next essential peculiarity of their structure. The term of
activated atoms presupposes atoms that have at least one free bond. Such atoms
are represented by surface-located ones as compared with the atoms positioned within
the volume.
Since liquid is saturated with a
large quantity of inner flickering section surfaces, all the atoms that come to
be on such surfaces at a definite moment become activated during the
half-period of flickers, i.e. they acquire extra free energy for the period of
the existence of the given surface.
Such atoms are far more mobile and reactive in
comparison with the atoms that are located within cluster volume both on
account of a higher energy of their own and their position on the surface /30/.
Therefore, we reckon it worthwhile to conditionally single out the zone of
activated atoms taking into consideration their relative concentration in
liquid Ca.
Let us underscore that activated atoms in liquid do
not form any structural zone in liquid. All activated atoms enter into
clusters. There are no other explicit structural units of matter in liquid
except clusters. Activated atoms differ in the sole respect that they come to
be located on the flickering surface for a short time, so they acquire extra
energy and a relative freedom of moving along cluster surface or between
clusters for that short period of time only. The split closes next moment, and
the existent activated atoms lose their supplementary energy. We may say that
activated atoms in liquid metals are flickering, too. Disappearing together
with the split at one site, activated atoms emerge at some other location, so
their average amount in liquid is constant at any moment of time under constant
conditions.
The quantity of Ca may serve as the
measure of the disordering of liquid metals contrasted with solid metals, where
the quantity of Ca is very small being approximately equal to
vacancy concentration inside them (0.001) by the order of their quantity.
Let us determine the concentration of activated atoms
in liquid metals Ca as the relation of the number of free bonds on
the surface of a cluster n to the number of atoms in a cluster nc.
It was shown above that n equals
to a half of all the bonds on the 'surface' of a cluster, i.e.
n = n1/ 2.
Applying the above-used procedure of expressing the
number of bonds through the area of cluster surface S and its volume V,
we get
Ca = n1 /2nc = S / 2V = 4π rc2
/ (4/3) π rc3 = 3/2 rc-1.
(46)
Expressing rc according to (33),
for spherical clusters we have
Ca = (3/2) (ΔHmelting / Δ Hvap)
β-1/3. (47)
The values of Ca calculated under (47) can
be found in Table 7.
Table 7. The Concentration of Activated Atoms in
Liquid Metals at the Melting Temperature
Element |
Cu |
Ag |
Au |
Pt |
Ni |
Co |
Fe |
Zn |
Si |
Cs |
Al |
Pb |
W |
Ca, %, by (47) |
23 |
25 |
21 |
25 |
26 |
23 |
28 |
36 |
28 |
14 |
18 |
15 |
22 |
It is demonstrated that the
concentration of activated atoms in liquid metals is high enough at the
temperature of melting already. A large quantity of activated atoms secures the
high reactivity of liquid metals, as well as the intensive mass exchange
between clusters, and accounts for some other distinctions of liquid metals.
The quantity specified in the
headline is of extreme importance, for it determines the major dynamic
parameters of liquid metal, particularly the characteristics of mass transfer,
impulse, the period of relaxation in liquid and certain other practically
significant quantities.
We are not acquainted with any
other ways of calculating the frequency of cluster heat oscillations in liquid
metals, which imparts a peculiar actuality to our calculation procedure. The
problem of the frequency of intercluster splits flicker is not only unexplored
but it has never been opened to discussion.
It should be stipulated that heat
oscillations of clusters as units do not substitute for atomic heat
oscillations in liquid. Those are two different kinds of motion that exist in
liquid simultaneously. The frequency of flickers of intercluster splits equals
numerically the frequency of cluster heat oscillations, since heat oscillations
of clusters and the flickers of intercluster splits represent the two aspects
(material and spatial) of one and the same process of the interaction of the
elements of matter and space in liquid.
The very existence of clusters is
possible only under the condition of their heat oscillations, since only one
half of the ‘surface’ of a cluster is indicated and separated by intercluster
splits at any given moment, hence a cluster can be singled out only as the
totality of atoms performing simultaneous heat oscillations.
It must be noticed that any motion of matter is
performed in space being reflected there. We may affirm that any kind of motion
of a certain material form is always accompanied by a related kind of motion of
the corresponding elements of space. Matter and space move but simultaneously.
Such an approach is absolutely new and unstudied yet
challenging in many respects, since it enlarges essentially the existent
concepts of motion and suggests investigating as well as allowing for the
previously unknown forms of motion of various spatial elements. The concept of
the motion of spatial elements is quite new on the whole, so it requires
specification by examples. The motion of vacancies inside crystals may be
supplied as an example of motion of the elements of space, which is propagated
in literature.
In case of liquid metals such a previously unknown
form of motion of the elements of space is the oscillatory process of
intercluster splits flickering. The process under consideration can be
expressed through the following formula:
αn + αn →← 2αn
The given scheme reflects the
constant process of cluster flickering when intercluster splits are
periodically opened and closed, while clusters periodically merge and separate.
The same scheme works as applied to melting or crystallization with a shift to
the right (crystallization) or left (melting) but not under oscillatory
operation.
Small dimensions of clusters make
it possible to employ the theorem of the uniform distribution of energy
according to the degrees of its spareness. We substantiated such a possibility
above in Part 3.6.
Let us designate the frequency of heat oscillations
of clusters as φ.
The energy of heat oscillations of clusters can be
determined by (42) as
Ec = (3/2) kT.
The quantity of φ is to be found from the expression
suggested by the oscillations theorem /103/:
φ = (1/2π A) (2Ec/
mc) 1/2, (48)
where A is the amplitude of cluster oscillations; mc is
cluster mass.
It was shown in Part 3.5 that the spacing between
clusters in liquid increases by the quantity of aα, where a is the
shortest interatomic spacing in a crystal at the melting temperature, while
α is the relative maximum deformation of matter at distension.
Hence A = aα. Let us find cluster mass by the expression
mc = M nc / N0, where M is the
atomic weight of matter; nc being the number of atoms in a
cluster according to Table 2, Part 3.4.
By inserting the values of Ec, A and mc
into (48), we arrive at
φ = (1/2πaα) (3kT N0
/ ncM) 1/2.
(49)
The period of heat oscillations of clusters τ will be
equivalent to τ = φ-1, or τ = 2πaα (3kT N0 / ncM)-1/2.
As it follows from (49), the expression for the
frequency of heat oscillations of a cluster differs from the frequency of
atomic heat oscillations by the value of the amplitude of oscillations and the
presence of the nc quantity under the radical.
Nevertheless, the numerical quantities of the
frequency of heat oscillations of clusters are cited in Table 8 below.
Table 8. The Frequency φ and the Period τ
of Heat Oscillations of Clusters and the Frequency of Intercluster Splits
Flickering in Liquid s at the Melting Temperature (Calculation by (49))
Metal |
j, s -1×10-8 |
t, s×108 |
Li |
0.74 |
1.35 |
K |
0.43 |
2.38 |
Cu |
13.5 |
7.40 |
Ag |
21.3 |
4.7 |
Au |
4.20 |
0.24 |
Al |
4.40 |
0.23 |
Pb |
0.7 |
1.43 |
Fe |
10.0 |
0.10 |
Co |
18 |
0.05 |
W |
13 |
0.078 |
It is known that the frequency of heat oscillations
of atoms inside the crystalline lattice by the order of magnitude comes to 1012-1013c-1,
which is on average four orders larger than the corresponding cluster
dimensions. The calculated values of the frequency of heat oscillations of
clusters approximate the values cited in literature and obtained by other
procedures /31/.
The listed data show once more
that at least two independent kinds of heat oscillations of particles coexist
simultaneously in liquid metals, so we should take it into consideration.
The issues of stability of the elements of matter in
the structure of liquid metals and alloys were repeatedly taken in literature
/104,105/. The concept of the elements of space introduced and described in
detail in the present work is not being discussed yet.
The stability in time, or the
period of cluster existence, is important for the understanding of numerous
practical results of metallurgical and casting practice. For instance, it is
useful to account for metallurgical structural heredity /105/.
The initiator of the cybotaxis theory Stewart
considered cybotaxes as rather unstable formations with a short period of
existence correlative to the period of atomic heat oscillations (1 10-12-1
10-13 sec.) /3-4/.
Atomic fluctuations in liquid, heterophase
fluctuations including, exist for a very short period by definition, too – of
the order of 1 10-12sec. V.I.Nikitin and others determine the period
of cluster existence as 10-5 … 10-8sec. /105/.
Undoubtedly, this is insufficient to treat clusters as hereditary information
carriers during the whole period of the existence of liquid state.
Our results show that the period of heat oscillations
of a cluster amounts to the order of 1 10-8sec. However, the period
of cluster existence must by far exceed the given quantity.
In connection with the specific characteristics of
clusters, e.g. the absence of stable surfaces and composition, the flickering
nature of interaction with the elements of space, etc., the period of cluster
existence may be determined only under the premise of their mutability. At the
same time, the changeability of real objects in time is virtually the universal
characteristic, so there is nothing objectionable in that.
To determine the time of cluster existence, we should
recall the definition of clusters as the main structural units of matter in
liquid within the entire temperature-temporal interval of the existence of
liquid aggregation state.
Allowing for all these stipulations, we may assert
that the period of cluster existence is limited by nothing except the interval
of the existence of liquid aggregation state.
In technological processes the duration of the
existence of alloys in liquid state and the period of cluster existence are
evaluated in hours. In natural processes, the period of liquid state existence
as well as that of clusters can take milliards of years.
Similarly to that, the period of crystal existence is
limited by the duration of solid crystalline aggregation state that may also
total milliards of years in natural processes.
It means that clusters are quite stable formations in
liquid state that have nothing in common with fluctuations and other
short-lived formations.
Consequently, clusters with definite changings in dimensions,
composition, etc., exist continuously in liquid alloys from the moment of
fusing up to the moment of crystallization. There are no other limitations to
the period of cluster existence. Or τcl = τliq,
where τcl is the period of cluster existence, τcl
being the duration of the existence of liquid state.
We can refer the above-said to the period of the
existence of the elements of space – intercluster splits - in liquid
considering their specificities.
Such a period of cluster existence
seems quite valid to account for their property to act as the carriers of
certain structural information while interrelating liquid and solid states by
some parameters.
Certain important parameters of the structure of
liquid metals were determined and calculated in Chapter 3. However, the
characteristics of liquid metals change depending on environmental conditions.
A minor dependence of the characteristics of the majority of liquid metals and
alloys on pressure is observed in literature. Still, it is known that this is
the change of temperature that affects the structure and properties of any
metals and alloys strongly enough.
Let us consider the influence of
temperature on the structure of liquid metals.
The quantitative characteristics of the theory under
development concerning the interaction of the elements of matter and space as
applied to the processes of melting and crystallization of metals are closely
interconnected, so if we know the parameters of one of the components, we may
find the corresponding values of other constituent parts. The existence of such
interaction facilitates the accomplishment of the task set in this work.
In particular, Chapter 3 supplies us with the values
of the main structural parameters of the elements of matter as well as the
elements of space in liquid metals at the unique temperature.
At the same time, literature gives the general form
of temperature dependencies of certain quantities used in our theory.
Particularly, /87/ and a series of other sources
quote an expression for the dependency of concentration n of the elements of
space in solid metals – vacancies – on temperature. Viz.:
n = exp (ΔSf / k) exp
( -Ef / kT), (50)
where Δ Sf is the entropy of vacancy forming,
while Ef is the energy of their formation.
On the other hand, the mentioned
source adduces the following expression of statistic thermodynamics for the
dependence of the equilibrium number of activated particles of matter on
temperature:
Ca = B exp (Δ Sf / k) exp (- Ef
/ kT) (51)
where B is the constant depending on the way of distribution of
particles in space.
Let us emphasize that expressions
(50) and (51) are practically identical.
It corroborates once more our thesis that was
advanced above of the equivalence of the elements of matter and space.
Let us find the values of the quantities Ef
and Δ Sf making use of the fact that expression (51) refers to
the same concentration of activated atoms as expression (43) derived above.
At the temperature equal to the melting temperature
expressions (51) and (43) must be equivalent, i.e.
B exp (Δ Sf / k) exp (- Ef / kT) = Δ
Hmelting nc / N0
It was demonstrated above that Δ Hmelting
is required to form intercluster splits which, in their turn, initiate the
formation of activated atoms on the ‘surface’ of such splits. Therefore, having
divided the latent heat of melting by the energy of forming new atoms, we can
determine their molar concentration Ca
Ca = Δ Hmelting / Efz
(52)
Expression (52) supplies us with the absolute value
of the concentration of activated atoms per mole of substance. The relative
quantity of Ca may be obtained from (52) by way of division by
Avogadro number. Thus
Ca =
Δ Hmelting / Efz N0
(53)
On the other hand, we determined the same relative quantity of Ca
earlier by expression (47) as
Ca = γ1 Δ Hmelting /
Δ Hvap,
(54)
where γ1 = (3/2) β, or γ1
= 2 / (3/π)1/3; 3 / (3/π)1/3; 5 / (3/π)1/3;
6 / (3/π)1/3 for a cubic diamond, simple cubic, body-centered cubic and
face-centered cubic types of granulation correspondingly.
Let us equate the right parts of expressions (53) and
(54). We get
Δ Нпл / Ef z
N0 = γ1 Δ Hmelting / Δ Hvap
Hence Ea = Δ Hmelting / γ1 z
N0.
At present, inserting the values of T = Tmelting,
Ca from (54) and Ef from (55) into (51), we find:
γ1 Δ Hmelting / Δ Hvap =
B exp (Δ Sf / k) exp -( Δ Hvap / γ1
z k N0 Tmelting).
Two quantities are unknown here: B and Δ
Sf. Let us recognize B = 1, since we know the distribution
of particles that is reflected in coefficients z and γ1.
In this case, the value exp(ΔSf k) at the melting
temperature will be equal to:
exp (Δ Sf / k) = (γ1 Δ Hmelting
/ Δ Hvap) exp (Δ Hvap / γ1
z R Tmelting), (56)
where R = k N0 – the universal gas constant.
Using (56), we arrive at the final
expression for calculating the dependence of the concentration of activated
atoms in liquid metals on temperature:
Ca = (γ1 Δ Hmelting /
Δ Hvap) exp (Δ Hvap / γ1
z R Tmelting) exp -( Δ Hvap / γ1
z R T) (57)
At T = Tmelting expression (57) transforms
automatically into expression (54).
The dependencies of Ca = f(T) and rc
= f(T) for certain metals under (57) are shown in Fig.12.
As it follows from Fig.12, the concentration
of activated atoms in liquid metals rises rapidly with an increase in
overheating, reaching 100% in the vicinity of the vaporization point of the
given metal.
However, even 100% of activated atoms in liquid do not mean that
there are no clusters in such liquid. Activated atoms are far from being
isolated monatoms independent of one another. The totality of activated atoms
enters into cluster structure. The approximation of activated atoms
concentration in liquid to 100% implies that cluster dimensions in liquid with
an increase in temperature decrease so that the totality of atoms entering into
a cluster emerge on its surface in the vicinity of the vaporization point.
Cluster dimensions in liquid metals and alloys are
modified with an increase in temperature, too. Let us determine the nature of
such modification.
Using the relation of quantities of rc
and Ca from (46), we find
rc = 3 Ca-1 a /
2. (58)
Inserting the value of С from (57) here, we find the dependence of cluster dimensions on
temperature
rc = (2/3ag1) (Δ Hvap / Δ Hmelting)
exp-( Δ Hvap / g1zRTmelting) exp(Δ Hvap / g1zRT) (59)
We can find the number of atoms in
clusters in f(T) allowing for the definite interrelation between the radius of
clusters and the number of atoms inside them:
nc = (4p/3z) rc3 .
By way of inserting here the value of rc
from (58), we get
nc = (4p/3z) (3/2)3 (Ca )-3.
Let us designate d = 9/2z and
introduce the value of Са.
For nc = f (T) we get:
nc = dp [g1 Δ Hmelting /
Δ Hvap) exp(Δ Hvap / g1zRTmelting) exp-( Δ Hvap /g1zRT)]-3 (60)
The subset of formula (60) is represented as
nc = dp (Ca )-3
(61)
The dependence of rc = f (T) is
also represented by Fig.12. The findings show that the rise of temperature
brings about the reduction of cluster dimensions in liquid metals. This
corresponds to the entire current data on the increase of disorder in liquid
metals with temperature rise /2,4,12/.
However, the derived expressions (59) and (60)
predict the existence of clusters in liquid metals up to the temperature of
evaporation. Such a conclusion is at variance with the inferences made in
certain works which state that cluster structure is inherent in liquids near
the melting temperature only, while the monatomic structure with the statistic
distribution of particles takes place at high temperatures /44-46/.
Our theory never employs the concept of ideal,
homogeneous phases. It was maintained above that each of the aggregation states
necessarily includes, except for the basic (predominant) intrinsical paired
elements of matter and space, the equilibrium latent characteristics of the
elements of matter and space that pertain to the states of aggregation adjacent
to the given state by the temperature scale. This inference equally concerns
liquid as well as solid, gaseous and other aggregation states.
Clusters are the equilibrium form of the elements of
matter that is peculiar to liquid state and determines the material aspect of
the characteristics of the present state within the whole temperature range of
its existence.
Some works conclude about the discontinuous nature of
the dimensional modification of the structural units of matter in liquid state
on the basis of measuring the dependency of a series of structure-sensitive
properties of liquid metals on temperature.
The continuous nature of the received dependencies nc
= f (T) and rc = f (T) impels the author to subscribe to
the opinion advanced in /106/, where such fractures of the
characteristic-temperature curves are explained neither by sudden changes in
cluster dimensions nor by the transition from the cluster to monatomic
structure but by the polymorphous transitions in clusters. Since there exists
the neighboring order of atomic granulation inside clusters, its modifications
are quite possible as a result of the interaction of the elements of matter and
space inherent in the specified neighboring order. In their turn, such
modifications may cause the change of cluster dimensions but not their
disappearance.
The reduction of cluster
dimensions with temperature presupposes the increase of their number in a unit
of volume in liquid at the same time. Since clusters are particle aggregations
with an intensive mutual interaction, particle interchange including, such
modification turns out to be quite feasible. As a result of such interplay and
mass transfer, clusters are capable of rapid reorganization; moreover, they get
reorganized constantly.
The motive force of the reduction of cluster
dimensions with an increase in temperature as well as the process of increasing
cluster dimensions at the cooling of melts is the mentioned vacancy gas
pressure. The concentration of vacancies inside clusters increases with the
rise of temperature, which causes their reorganization into clusters with
lesser dimensions. By definition, a cluster may contain not more than one
vacancy. If two or more vacancies arise in a cluster, they generate inner
pressure inside it that leads to its splitting by the mechanism analogous to
the mechanism of melting described above.
If we know the nc, it is easy to
determine the number of clusters per gram-atom of the given metal in liquid
state. Thus
Nc = N0 / nc = N0 Ca-3
/ πδ
(62)
In accordance with (62), the
number of clusters increases rapidly with the rise of temperature.
The reduction of dimensions and the increase in the
number of clusters in liquid metals with the rise of temperature must result in
the expansion of the volume occupied by the zone of intercluster splits (the
elements of space) ΔVspl. The relation between ΔVspl
and Ca can be expressed under (39):
ΔVspl = aα (3/2 rс) 100% = aα Ca 100%.
Inserting here the value of Ca from
(57), we have
Δ Vspl = aα (γ1 Δ Hmelting
/ Δ Hvap) exp (Δ Hvap / g1zRTmelting) exp- (Δ Hvap / g1zRT) 100% (63)
We should note that (63) cannot be
considered as the only contributor to the changing of the volume of liquid at
heating. Similar to any thermodynamic characteristic of a system that is
measured experimentally, the modification of volume is a complex quantity
formed out of the total contribution of both the elements of matter and the
elements of space at all the hierarchical levels of matter and space
interaction that exist in the given system. Still, the contribution of the
upper level of the system always prevails.
Except the quantity of ΔVspl,
at least four more factors must contribute to thermal expansion in the
specified concrete case: 1) the thermal expansion of the residuals of
crystalline lattice inside clusters analogous to the thermal expansion of
solids; 2) the possibility of re-granulation of clusters after their formation
into a compact mutual granulation irrespective of atomic granulation inside
clusters; 3) the possibility of volume modification at polymorphous transitions
in liquid state; 4) the increase in vacancy concentration.
The current evaluations of unconfined space in
liquids do not allow for the contribution of each of the five indicated factors
/2/, therefore, the collation of the obtained quantity with experiment is not
possible so far.
Liquid metals, similar to any physical bodies, are
systems of interacting elements of matter and space. Such interaction directly
affects various thermodynamic and other characteristics of the given system in
the first place. Viz. any property of such systems that is experimentally
determinable will be complex, reflecting the contribution of material as well
as spatial elements at various levels of the system’s hierarchy.
The specific quantity of such
contribution depends on the characteristic in question. There may be properties
determined mainly by the contribution of the material component of the system,
e.g. the mass of liquid and solid metals. There can be properties dependent in
preference on the contribution of the spatial elements of the system, such as
the fluidity of liquid metals, and there are characteristics that depend
equally upon the contribution of both the elements of matter and the elements
of space (density). However, all the characteristics reflect, though in a
different degree, the influence of both the material and spatial elements of
the system under analysis.
We shall determine the totality of liquid metals
characteristics proceeding from the present general conception by specifying
every time the contribution of material and spatial elements into this or that
concrete characteristic at the hierarchical level that corresponds to the level
of aggregation states and the elements of matter and space inherent in this
very state.
Let us mark that we cannot specify the absolute
quantity of the contribution of matter and space to this or that specific
property of a system, yet it is in our power to determine the relative property
modification under the influence of the contribution of this or that specific
element of the system in question.
For instance, we cannot calculate the entire volume
of a system in solid and liquid states being able to do the calculation of the
relative modification of the volume of metals at melting and crystallization.
The same refers to other properties.
The original principle of relativity ensues from the general
theses of the hierarchy of real bodies structure, the presence of a great many
levels of the interacting elements of matter and space inside them.
We cannot yet determine the summarized contribution
of each of such levels, the majority of which are underexplored. Still, we can
evaluate the relative modification of this or that characteristic of a system
while the latter is passing from one aggregation state into another, for
example, at the transition from solid to liquis state and v.v.
At times such modification will be insignificant or
negligible, - occasionally it will be conclusive. Everything depends on the
nature of the property.
Let us start considering the properties of liquid
metals with the property that depends decisively upon the contribution of the
spatial part of a system. This is fluidity.
Let us view the elementary act of fluidity in liquid
metals at the level of clusters and intercluster splits.
Let us symbolically represent two
adjacent clusters as squares A and B in Fig.6. Let us assume that displacement
force F influences cluster A in the direction from left to right. At the moment
1 clusters A and B, being in the state of performing continuous heat
oscillations, approximate so that the flickering split between them is closed
and there occurs no displacement of cluster A towards cluster B, the analyzed
zone of liquid does not flow in such a configuration but behave as a solid.
At moment 2 as a result of the same heat oscillations
clusters A and B separate so a flickering split forms between them for a short
period of time. During the specified time period, clusters A and B are not
connected, and cluster A, under the impact of force F, is easily displaced relative to cluster B by the quantity of δ
termed as the elementary step of the process of flowing.
At moment 3 clusters A and B come together again, and the flickering
split between them closes. However, cluster A is already displaced relative to
cluster B by the quantity of δ. The process under consideration will be
repeated as long as there is the impact of force F without any counteraction.
Totalizing, the elementary acts of flowing lead to
the visual effect of the flowing of liquid metals. Let us remark that the
quantity of displacement δ equals to or is divisible by the width
of a single intercluster split, δ = αa (v. Part 3.5 above).
The very possibility of displacement is caused by the
presence of spare spacings in liquid represented by intercluster splits. In
other words, intercluster splits provide the space for cluster displacement,
increasing the fluidity of liquids by several orders as compared to solid
state.
It follows from the cited
description of the elementary act of fluidity that the process of flowing of
liquid metals and alloys at cluster level is not exactly continuous but it puts
up from minute steps δ following one another.
If force F acts short-term, liquid may respond
to such a short-period impact as a solid body. The mentioned phenomenon exists
and is widely acknowledged, while the short period of time when liquid behaves
as a solid under the influence of force F is termed the relaxation
period.
Let us do the calculation of the
elementary act of fluidity. The speed v of the displacement of cluster A toward
cluster B is:
v = δ / τ
(64)
where τ is the duration of the
elementary displacement act, equal to the period of heat oscillations of a
cluster. The given quantity was determined previously in 3.9:
τ = 2παa (3 kT N0 / M)-1/2
In its turn, the speed of v may be found through the coefficient
of fluidity Te. Thus, for the specified case
v = Te
F (65)
By equating the right sides of (64) and (65), we get:
Te F = δ / τ
(66)
In turn, the force of F can be calculated as
the pressure upon liquid p, multiplied by the area of transverse section of
cluster A, which we shall designate as rc2. Hence
F = p rc2.
(67)
Introducing (67) into (66), we get
Te = δ / τ p rc2.
(68)
Expression (68) correlates
fluidity with such parameters as the width of intercluster split δ,
the radius and frequency of heat oscillations of clusters.
It also follows from (68) that the
fluidity of liquid metals must increase with an increase in temperature, for
cluster dimensions deflate with temperature rise.
Viscosity is traditionally referred to the group of
the basic structure-sensitive properties of liquid metals being used as the
characteristic of internal friction in liquid.
There are numerous theories of
viscosity: the unconfined space theory that leads to Bachinsky’s formula /107/;
Arrenius’ equation derived theoretically by Frenkel and Andrade /69-70/ with
its numerous modifications; the equation suggested by the statistic theory of
liquid plus its modifications /109/. The presence of a large number of theories
concerning the same phenomenon is, on the one hand, a typical scientific
situation, since there always exist multitudinous possibilities to give a
many-sided description to the same phenomenon. Such theories may complement one
another.
On the other hand, the presence of various theories
that are mutually exclusive testifies to the situation of incomplete knowledge.
The latter is the very situation with regard to the viscosity theory. Similar
to diffusion, viscosity description is rather unsatisfactory on the whole,
although we observe some acceptable coincidences between experimental and
calculation data in a series of cases. Such a situation requires a further
theoretic development in order to construct an adequate viscosity theory.
Let us build up the theory of viscosity of liquid
metals allowing for their cluster-vacuum structure.
On the one hand, such a theory can be constructed if
we premise the known correlation between fluidity and viscosity.
η = 1/ Te = τ p rc2 / δ
However, expression (68) and the
latter one contain the variable quantity of p to be rid of, which we regard as
a drawback. In this connection, there arises a necessity to develop a more
convenient theory of liquid metals viscosity taking into account the existence
of both the elements of matter and space inside them.
Such a theory can be grounded on Andrade’s kinetic
equation, derived on the basis of the concept of the monatomic impulse
transmission mechanism /110/, yet neutral in reality with respect to the
dimension of structural units of matter in liquid state.
Andrade supposed that impulse transmission occurs at
the deviation of the structural units of liquid from their layer resulting from
oscillations. Evidently, the term ‘structural unit’ can be equally substituted
here for the concept of ‘atom’ as well as ‘cluster’. In the case given, the
differentiation is quantitative, not qualitative.
Andrade explored two adjacent layers of structural
units of liquid, parallel to the direction of the flowing of liquid. If n is
the number of such particles in 1ccm, then there falls ≈n2/3
of the structural units of matter, clusters in our interpretation, at 1sq.cm.
Let 1/3 of their total number oscillate
perpendicularly to the layer plane. If impulse transmission takes place at the
maximal deviation from the layer plane, the quantity of the transmissed impulse
at a single particle oscillation will make »m n-1/3 (dw/ dy), where m is the mass of a particle (a cluster), while n-1/3
is the average spacing between the layers; y is the coordinate perpendicular to
the layer plane; dw/ dy being the gradient of the tangential speed of flow. The
number of such impulse transmissions per 1sec. reaches »(1/3) j n2/3, where j is the frequency of heat oscillations of a cluster.
Hence, the resultant impulse
equivalent to the force of viscosity and transmitted during 1sec. through a
unit of layer surface area, will be
dP/dt » (4/3) j m n1/3 = h dw/ dy.
Multiplier 4 stipulates here that
a particle transverses the layer plane four times during the period of its heat
oscillations.
Therefrom it follows that
h » (4/3) j mc n1/3,
(69)
where nis the number of clusters in a unit of liquid metal volume.
Let us find the value of j from (49): j = (1/2paа) (3kT N0 / nc M)1/2.
Cluster mass is known, too: mc = М nc / N0 .
Let us determine the number of
clusters in a unit of liquid metal volume by the expression:
n = N0 r / M nc,
(70)
where r is the density of liquid metal.
Inserting the obtained values of j, mc and n into
(69), we get:
h = (2 / 3paа) (3kT N0 / nc M) 1/2 (N0
r / M nc) 1/3 (М
nc / N0) (71)
Expression (71) is correct for Т = Тmelting. To find the dependency of h = f(T), the value of n = f(T)
should be inserted into (71):
nc = dp [g1ΔHmelting /ΔHvap)
exp(ΔHvap /g1zRTmelting) exp-(ΔHvap /g1zRT)]-3
or
h = B exp -(ΔHvap /g1zRT)-1/2 T1/2,
(72)
where B = (2 / 3paа) (3R / nc
M) 1/2 (N0 r / M nc)
1/3 (М nc / N0) (g1ΔHmelting /ΔHvap)exp(ΔHvap /g1zRTmelting) - being constant.
The analysis of the obtained
dependency (72) shows that the expression under consideration is similar to the
well-known Panchenkov formula only /108/ presented as
h =3 (6R) 1/2
(b2/ N) (r4/3 / M5/6) exp (e / RT) T1/2
[1 - exp - (e / RT)] (73)
Here, as well as in (72), we observe the term of Т1/2, while the quantity of e = 2Еvap / z is determined through the energy of vaporization and the
coordinating number of z, which is close to our findings.
Panchenkov’s theory, however, is
based on other assumptions, which accounts for an insufficient degree of its
similarity to the obtained data.
Numerical check (72) demonstrates
a coincidence between calculation and experimental data, which is close enough,
if we consider the proximity of the original Andrade’s expression. The data are
adduced in Table 9.
Table 9. Viscosity of Liquid Metals at the Melting
Temperature
Metal |
h, cps, |
|
calculation by (72) |
exper. by /12/ |
|
Fe |
4.2 |
5.4 |
Co |
5.5 |
4.8 |
Ni |
5.5 |
5.0 |
Cu |
5.0 |
4.1 |
Au |
5.2 |
5.38 |
Al |
1.48 |
1.13 |
Zn |
4.5 |
2.82 |
Cd |
3.9 |
2.3 |
Na |
0.9 |
0.68 |
Temperature dependencies of the viscosity of liquid
metals are shown in Fig.14, 15. The character of the calculation and
experimental dependencies in Fig.14, 15 coincides, their numerical correlation
is quite satisfactory.
Thus, the developed theory of the structure of liquid
metals is quite applicable to the analysis of their viscosity, too.
The traditionally studied properties and processes, such as
diffusion and viscosity in liquid metals, are also regarded as a traditional
object of applying liquid state theories and models with the purpose of adequacy
check of the mentioned theories.
Unfortunately, theoretical skill
created the situation when we have a whole variety of diffusion as well as
melting theories. This deprives the process of the working out of new diffusion
theories of experimentum crucis meaning, of the seemingly essential importance
which similar developments used to have in the past while the number of
diffusion theories was not so great.
Nevertheless, there remains the essential though not
exactly underlying significance of such pursuits. It consists in the fact that
although the building up of the theory of diffusion or any other similar
property of liquid does not play the decisive part in this or that theory of
liquid state, it is one of the necessary steps to check the applicability of
the theory-to-be to the description of a wide range of liquid metals phenomena
and properties - as wide as possible.
In point of fact, under the conditions of the
competition between various theories of one and the same phenomenon, the theory
giving the most exact description to the widest range of phenomena in its
respective field will take the priority.
Besides this, focusing on diffusion is explained by
the practical importance of the specified process for metallurgy and casting.
A large amount of experimental data accumulated on
diffusion makes it possible to test this or that theory on the material that
seems sufficiently extensive /12,17,20/.
The main theoretic expression in the sphere of
diffusion in liquid metals remains the equation, analogous to Arrenius’
equation for viscosity /12/:
D = D0 exp -(ED /
kT), (74)
where D is the coefficient of diffusion; ED
is the energy of diffusion activation (self-diffusion); D0
being the fore-exponential multiplier.
Expression (74) does not always
describe the observed regularities of diffusion satisfactorily, especially
within a wide temperature interval /152/, therefore, attempts at constructing a
diffusion theory on variant bases were and are still being made.
By way of examples, we may cite Zaxton and Sherby’s
empiric correlations /111/, self-diffusion calculations under the hole theory
by Eiring /103/ and Frenkel /69-70/, Andrade’s calculations /110/, Cohen and
Turnball inactivation theory based on the unconfined space model /112/,
Swalin’s fluctuation theory /113/ and a series of modifications of the
mentioned theories.
However, there was no junction between theory and
experiment to be detected in the given works as regards a relatively wide scope
of metals. It is supposed that the coefficients of diffusion, as well as
viscosity, can be calculated on the basis of the consecutive statistic theory
of liquid state /18,109/, under the condition of the exact knowledge of
interatomic potentials /109/, which is lacking so far /14/. The overwhelming
majority of the stated theories employ the ideas of the in-liquid migration of
a separate atom or ion understood as the basic structural unit of liquid.
Thus, the description of diffusion phenomena from the
viewpoint of the set-forth melting and liquid state theory where an atomic
grouping – cluster – is considered to be the main structural unit of matter in
liquid state, merits attention, being of principal interest. Our theory also
premises that each aggregation state, except for the structural units of matter
and space predominant in the state given, bears the latent properties of the
adjacent aggregation states.
It was pointed out above that all the atoms of liquid
enter into clusters, while the atoms that happen to be located on cluster
‘surface’ at the given moment form an aggregate of activated atoms, capable of
migration and acting as the latent elements of gaseous state matter in liquid
state. Apart from that, there are atoms and vacancies inside clusters, which are
bound with one another in the crystal-like structure of neighboring order.
These are the latent elements of solid state in liquid.
Cluster mechanism must be the major mechanism of mass
transfer in liquid metals, for it proves to be the most effective one. However,
according to the principles of synergetics, any dissipative process (diffusion
refers to typically dissipative processes) always occurs at all possible
levels. Therefore, with the exception of the main cluster mechanism,
intercluster diffusion responsible for mass transfer inside clusters will
operate in liquid metals through the mechanism similar to the vacancy mechanism
in solids, and it will be accompanied by the interchange of activated atoms
between clusters by way of separate atoms jumping over the zones of
intercluster splits.
Hence, there are at least three diffusion mechanisms
operating simultaneously in liquid metals: the basic mechanism of mass transfer
through clusters, characteristic of liquid state, and those of latent
aggregation states – solid-like vacancy mechanism inside clusters and gas-like
atomic interchange between clusters.
So our theory presents the process of diffusion as a
composite, aggregative one, whereas the value of diffusion coefficient measured
experimentally becomes the effective, resultant mass transfer coefficient by
the three mechanisms displayed above.
Such an approach meets the accepted conception of the
presence of latent properties belonging to other states of aggregation of
matter in the aggregation state given. In the case under analysis, clusters
should be viewed as the structural elements of the elements of matter form that
dominates in liquid state, while activated atoms are considered as the latent
properties of gaseous aggregation state.
The specified conclusion concerns any other
aggregation states in full measure.
Consequently, the current partial diffusion theory
incorporates into a completer theory that should differentiate between the
contributions of each of the mechanisms into the observed diffusion process.
To create the integrate diffusion theory of such a
complex system is the goal heritable into the future.
We may note here that the first approximation at the
calculation of diffusion coefficient allows neglecting mass transfer inside
clusters, since the contribution of this mechanism into the diffusion
coefficient value under observation seems insignificant. The contribution of
the gas-like mechanism does not appear to be manifest by its quantity. Hence,
we are going to consider two mechanisms of diffusion in liquid state further –
cluster and gas-like, aiming at finding the respective contributions of both of
them /115/.
In conformity with the above-said, we can add
D0 = Dc0 + Da0
Ca, (75)
where D is self-diffusion coefficient; Dc0
is self-diffusion coefficient by the cluster mechanism; Da0
is self-diffusion coefficient by the mechanism of activated atoms; Ca
is the concentration of activated atoms in liquid.
As we see it in (75), the summarized
self-diffusion coefficient is combined of partial coefficients extensively as
contrasted with additive composition.
Let us find the quantities of partial coefficients of
cluster and activated atoms diffusion that enter into (75) on the basis of the
random walk theory. With reference to the given case we have:
Dc0 = k d2
ν (76)
where k = 1/6; d is the space of a single particle
displacement at its transition from the original to some other equilibrium
position; ν is the frequency of such transitions.
Since intercluster spacings are
small in comparison with their dimensions, we may conjecture but a collective
mechanism of their displacements, e.g. the circular mechanism. At such a
mechanism, the adjacent equilibrium positions will be separated by the space
equal to the doubled cluster radius plus the width of one intercluster split
α. Thus,
l = 2 rc + a.
(77)
As usual, let us recognize as a
single diffusion act the displacement when a cluster passes from the original
equilibrium state to the adjacent equilibrium state. Evidently, in this case
d = l = 2 rc + a.
Or, since a << 2 rc at Т = Тmelting, we may admit without any noticeable error that
d = 2 rc
. (78)
To determine the period of
diffusion jumps, let us use the concept of diffusion as an oscillatory process
that is introduced here for the first time. Such an assumption is unacceptable
when analyzing the direction of particles travelling in space. However, if we
consider the process of diffusion in time, abstract from the displacement
direction and allowing for the periodicity of the specified process only, it is
quiet acceptable to regard this process as periodic, i.e. oscillatory. The
frequency of such a process, which is periodic in time, can be found on the
basis of the oscillations energy E equation /116/:
Eс = ( mс A2 w2 )/2, (79)
where mc is the mass of a cluster; Ec
is the energy of cluster oscillations; w is the angular frequency of oscillations; А
is amplitude.
mc = М nc / N0.
Ec = (3/2) kT.
w = 2p n.
А
= d = 2 rc.
n= w /2p
Having accomplished the corresponding substitutions,
out of (79) we derive
w = (2Еc /mc) 1/2
/ d; (80)
and
n = (2Еc /mc) 1/2
/ 2pd (81)
In turn, expressing nc through Сa under the expression (61) derived
earlier, we get
nc = dp (Ca )-3.
Introducing the obtained values of d, Ec
and mc into (81), for t
n = (3 k T N0
/ M dp Ca-3
)1/2/ 4 prc
Let us allow for the previously derived expression for
rc:
rc = (3/2) а Ca-1.
After the introduction of the
concluding value we obtain:
n = (3 R T / M dp Ca-3
) 1/2/6 p a Ca-1
Or
n = Ca5/2
(3 R T / M dp )
1/2/6 p a (82)
The latter expression determines
the frequency of cluster transitions from one to another equilibrium state, or
the frequency of the elementary acts of diffusion process.
By the insertion of the obtained value n from (82) into (76), we
arrive at the final expression for the partial self-diffusion coefficient in
liquid metals by the cluster mechanism:
Dc0 = k d2 n = (a / 4p) Ca1/2
(3 R T / M dp) 1/2
(83)
The found values of Dc0
by equation (83) at Т = Тmelting are listed in Table 8.
This table includes the principal
data on self-diffusion coefficient in liquid metals. The amount of the
published experimental data on self-diffusion is small, their reliability being
unfortunately entirely unknown.
The partial self-diffusion coefficient of activated
atoms can be also evaluated on the basis of equation (76) with the introduction
of the frequency of heat oscillations of clusters from (49) and at d = a.
Under these conditions
Dа0 =(a/12pa) (3RT /dp M) 1/2 Ca3/2
(84)
Calculations under formula (84)
supply the values of the partial self-diffusion coefficient by the mechanism of
activated atoms that are approx. by order of magnitude less than the values of
the partial self-diffusion coefficient by the cluster mechanism. That signifies
that mass transfer in liquid metals at the melting temperature is achieved for
the most part through the displacement of clusters, but not separate atoms,
from the equilibrium positions. The contribution of gas-like diffusion in
liquid metals approximates to 10% of the total value of diffusion coefficient.
It suffices not to neglect the mentioned fact; furthermore, the value of this
diffusion mechanism and its contribution to liquid metals will increase with
the rise of temperature.
The resultant values of the
effective self-diffusion coefficient are derived from (75) with the
introduction of the partial coefficient values from (83) and (84).
Thus
D = (a Ca / 4p) (3RT /dp M) 1/2[(
Ca2 /3pa ) + 1] (85)
The data on
the calculation of self-diffusion coefficient according to equation (85) are to
be found in Table 10, too.
Table 10. Self-Diffusion Coefficients in Liquid
Metals at the Melting Temperature
Metal |
Dc0×105
sq.cm/s, calculation by (83) |
D0×105sq.cm/s, calculation by (85) |
D0×105 sq.cm/s, exp. /2,12,98/ |
Li |
1.9 |
2.1 |
5.6 |
Na |
1.4 |
1.6 |
4.3 |
K |
1.3 |
1.44 |
5.3 |
Cu |
1.4 |
1.5 |
- |
Ag |
0.66 |
0.70 |
2.3 |
Au |
0.75 |
0.80 |
- |
Al |
1.8 |
1.95 |
- |
Pb |
0.47 |
0.50 |
2.0 |
Zn |
1.22 |
1.40 |
1.9 |
Cd |
1.0 |
1.28 |
- |
Fe |
1.04 |
1.1 |
0.17 |
Ni |
1.6 |
1.7 |
- |
Co |
1.55 |
1.65 |
- |
Ti |
2.6 |
2.2 |
- |
W |
1.5 |
1.6 |
- |
Sn |
0.57 |
0.70 |
2.5¸2.0 |
Hg |
0.52 |
0.62 |
1.0 |
Bi |
0.73 |
0.85 |
- |
Ga |
0.47 |
0.65 |
1.7 |
Usually the influence of external
factors that accelerate or decelerate diffusion cannot be eliminated completely
in experiments. Gravity and convection refer to such factors in the first
place. Therefore, the values of experimental self-diffusion coefficients are
higher than the calculated ones, which is to be expected. Self-diffusion data
are rather insufficient for a more accurate evaluation. So far we may state
that the developed diffusion theory registers the acceptable calculated data on
self-diffusion coefficients for a sufficiently wide range of liquid metals.
The process of mass transfer, or
pore diffusion, occurs at any aggregation state. Qualitatively, this is the
same process. However, the rapidity of mass transfer in various aggregation
states differs considerably by quantitative parameters. So, what causes such
distinctions?
To facilitate solving, let us view the distinctions
related to the differentiation between the principal structural units of
various aggregation states.
From the suggested standpoint, such distinctions are
associated with the change of the predominant structural units of matter and
space, the elements of space coming first, in various aggregation states. This
is the presence of unconfined space that operates as the factor determining the
possibility and the rapidity of such a displacement.
Current theory reflects the specified fact through the
known diffusion formulas in gases and liquids.
For gases
D = k u l, (86)
where D is diffusion coefficient; u is the average
rapidity of thermal motion of gas molecules; l is the average length of
a free range of molecule path; k being constant - k = 1/3.
For solids and liquids /101,115/:
D = k d2 n,
(87)
where d is the average space between material particles in
liquid; n is the frequency of heat oscillations of the elements of matter in
liquid; k is the coefficient dependent on particle granulation relative
to one another. Normally k = 1/3 - 1/6.
It was
repeatedly observed that expressions (86) and (87) quite reducible mutually,
since their dimensions and physical implications are similar. The foremost
point is that the quantity of d in expression (87) characterizes the length of
the elementary displacement of material elements during the process of
diffusion prior to their dimensions in liquid state. Consequently, even the
existent incomplete diffusion theory neglects the dimensions of the structural
units of matter in liquid, although taking into consideration the dimensions of
spatial elements as the spacing of the elementary act of diffusion.
I.e.
d = l.
Let us present (86) as
D = k u l = k l (l / t), (88)
where u = l / t.
If we do the substitution (1 / t) = n, (88) assumes the following form:
D = k l2 n, (89)
where n = 1/ t is the frequency of the material element displacement from the
equilibrium state; l being the spacing of such a displacement in space.
Certainly, expression (89)
presupposes the possibility of representing or describing the process of
diffusion as a process oscillatory in time, for (89) employs the idea of the
process frequency n.
This is a new concept in the theory of diffusion.
Still, it was demonstrated while deriving (79) that such an assumption is quite
acceptable when referring to the frequency of displacements in time as
abstracted from displacement direction.
Expression (89) quite coincides with expression (87)
in all its details, allowing for the fact that the constant factor of k can be
different to a certain degree.
It is important to mark that the dimensions of
material particles are absolutely lacking in (86-89) while spacings are present
only, i.e. the parameters of space in the given aggregation state. It
underlines once more the decisive role of the elements of free space in
diffusion process.
Let us presume that expression (89) has a more general
character than expressions (86) and (87) being applicable to any state of
matter.
Hence, if we know the differences in the spacing of
the elementary act of diffusion that are connected with the differentiation
between the predominant elements of matter and space in this or that
aggregation state, we can determine the difference in the coefficients of
diffusion for the adjacent aggregation states (without considering the latent
elements of matter and space and their contribution to diffusion).
Let (89) be
Dl0 = kl ll2 nl
for liquid state, whereas for solid state
Ds0 = ks ls2 ns.
Here ll and ls are
the spacings of the elementary act of the prevalent diffusion mechanism for
liquid and solid states correspondingly; nl and ns represent the frequency of the
elementary acts of diffusion for liquid and solid states correspondingly; kl
and ks are the constants of the granulation of the
predominant elements of matter in liquid and solid states correspondingly.
Then the ratio of diffusion
coefficients for solid and liquid aggregation states will be presented as
Dl0 / Ds0 = kl
ll2 nl / ks ls2
ns
up to a constant.
It is known that ls = а, where а is the shortest interatomic
spacing in a crystal.
We recognized ll as dc
above for liquid state, where dc is the diameter of a
cluster.
It follows from Table 1 that dc » 10 а.
Correspondingly
Dl0 / Ds0 » (10а) 2nl / a2 ns
or
Dl0 / Ds0 » 102 nl / ns
(90)
It is known that diffusion coefficient in liquid metals
at the melting temperature equals approx. to 10-5 sq.cm/s, while in
solid monocrystals at the sate temperature it is 10-7 sq.cm/s /129/.
I.e. the actual correlation at the melting temperature
Dl0 / Ds0 » 100.
That means that the correlation
between the frequencies of diffusion jumps in equation (90) is nl / ns = 1.
Consequently,
the frequencies of diffusion jumps in solid and liquid states at the melting
temperature are approximately equal.
In turn, we know /101/ that the frequency of jumps in
solid metals at the melting temperature equals 108 c-1 at
the frequency of heat oscillations of atoms 1013 c –1.
The frequency of heat oscillations of clusters in
liquid metals at the melting temperature was calculated above (look up Table 8)
and mounts to 108 c-1, which approximates the frequency
of diffusion jumps of atoms in solid metals near the melting temperature.
The curious fact under analysis signifies that a
cluster completes no more than several (less than ten) heat oscillations
between diffusion jumps in liquid state. It is not excluded that clusters in
liquid metals change their equilibrium state at each period of oscillations.
The mentioned peculiarity makes liquid metals cognate
to gases, where the direction of atomic motion changes at a collision with neighbors.
Probably, such affinity is one of the causes of a continuous transition
possibility between liquid and gaseous states.
Apart from this, the fact of the equality between the
frequency of jumps and the frequency of heat oscillations in clusters means
that diffusion process in liquid metals, by contrast with solid metals, has an
inactivated nature, because a cluster does not require any additional energy to
shift its position, except for the energy of heat oscillations. The same
phenomenon takes place in gases, whereas activation energy that exceeds by far
the energy of heat atomic oscillations is needed for the elementary act of
diffusion in solid state.
Thus, there is a fundamental distinction between the
mechanisms of diffusion in solid and liquid metals, while we observe a
fundamental similarity of the diffusion mechanisms in liquid and solid states
by the activation parameters.
In the light of the analysis carried out above, the
difference in the processes of diffusion in liquids and gases becomes mainly
quantitative, so at the equality between the quantities ll = lg
and nl = ng the parameters of mass transfer (and not that only) in solid and
liquid states equalize and grow indistinguishable, which occurs within the
vicinities of the melting temperature.
So the use of the concepts of the
role of spatial elements in the processes of mass transfer gives us the
possibility to calculate the coefficients of diffusion in liquid metals, first,
as well as consider diffusion processes from the unified viewpoint in various
aggregation states, and establish the features of similarity and difference
between the given processes, including the quantitative aspect.
Real liquid metals and alloys always contain a certain
amount of admixtures. It is known that admixture diffusion in liquid metals is
qualitatively subordinate to the same regularities as self-diffusion, yet the
quantitative distinctions in diffusion coefficients of diverse admixtures may
vary considerably enough.
The theory of admixture diffusion,
which is of practical importance for calculating the processes of alloying,
segregation, etc., does not always ensure the sufficient convergence with
experimental data at present.
Thus, one of the latest among the recently developed
fluctuation mechanisms of admixture diffusion in liquid metals /113/ provides
acceptable results for admixture diffusion in liquid alkaline metals but it
turns out to be rather inexact for calculating admixture diffusion in liquid
iron and other refractory metals /117/.
Another admixture diffusion theory – Swalin’s theory –
considers liquid as homogeneous, in connection with which individual properties
of the admixture and the solvent are almost disregarded so the coefficients of
various admixtures diffusion come to be similarized /113/.
In reality, in liquid iron, for instance, the values
of D ranging from 1.7 10-6 sq.cm /s (self-diffusion) to 1 10-3sq.cm
/s (hydrogen diffusion) are observed. The quasipolycrystalline model of liquid
melts was engaged to account for such appreciable discrepancies /31/. As it was
demonstrated, such a model admits that liquid consists of separate clusters
(atomic microgroups) and the surrounding disordered zone, monatomic by its
structure. For such a structure, V.I.Arkharov suggested the following
correlation /31/:
ψcl + ψdis =
1, (91)
where ψcl is the relative cluster contents in
the structure of liquid metal; ψdis being the relative
contents of the disordered (monatomic) zone.
I.e. a
hypothesis of a complex, aggregative nature of liquid metals structure was put
forward within the limits of the quasipolycrystalline model. Therefrom the
authors of the hypothesis in question concluded that the quantity of the
measurable admixture diffusion in liquid metals is complex, composite,
additive, determined by the sum total of the diffusion conduction of various
structural zones in the melt /31/:
D = ψcl Dcl + ψdis Ddis,
(92)
where Dcl and Ddis are partial
admixture diffusion coefficients in various zones.
It was pointed out above that the
existence of two structurally independent zones in liquid state contradicts the
phase rule as well as the quantum mechanics rule of quantum objects
indistinguishability.
In the theory of the material-spatial structure of
liquid metals under our development clusters, and nothing else but clusters,
are thought the prevalent material elements characteristic of liquid state.
Still, the existence of clusters by no means denies the existence of atoms –
these are different levels of matter organization.
It was proved in Part 5.4 above that there are three
mechanisms of mass transfer operating simultaneously in liquid metals: the
basic one being that of mass transfer through clusters; solid-like vacancy mass
transfer through atoms inside clusters; gas-like mass transfer between clusters
by way of interchanging activated atoms located on cluster ‘surface’ with the
participation of intercluster splits (not vacancies). The distinction between
the mentioned process and the solid-like one consists in the following: when
inside a cluster or a solid, migrating atoms preserve their bonds with the
adjacent atoms, whereas the former jump over the intercluster split during the
atomic interchange between clusters. Atoms involved in this process separate
from all their neighbors for a very short time, which is typical of gaseous
aggregation state. Therefore, we term such a process as gas-like.
The gas-like mechanism of mass transfer in liquid
metals acts inside clusters only, being responsible for the redistribution of
atoms at the intercluster level, which is highly important for the homogenizing
of liquid alloys composition, as well as the progress of alloying processes,
but hardly perceptible at measuring diffusion coefficient. So we may neglect
the contribution of the solid-like mechanism to the coefficient of diffusion in
liquid metals as a first approximation. However, the contribution of the other
two diffusion mechanisms must be taken into consideration, though they differ
in the degree of their significance.
Thus, we must allow for two diffusion mechanisms
operating simultaneously in liquid metals: the mechanism of mass transfer
through separate atoms peculiar to gaseous state, the role of which is but
tributary, and the cluster mechanism of diffusion, characteristic of liquid
state proper, to which the principal role in the given process is assigned.
So V.I.Arkharov’s idea of the composite nature of the
observed diffusion coefficient can be applied here by the extensive, contrasted
with the additive, scheme /115/.
D = Dcl + Da Ca,
(93)
where Da is the coefficient of diffusion by the
activated atoms mechanism; Dcl is diffusion coefficient by
the basic cluster mechanism; Ca is the concentration of
activated atoms in liquid metal.
The extensive character (93) is realized through the
fact that diffusion coefficient by the cluster mechanism incorporates into
diffusion as the main one, while the coefficient of Ca acts
as supplementary, proportionate to the concentration of activated atoms. The
given correlation reflects the entire amount of matter enter into clusters in
liquid state, whereas only few atoms inside clusters are activated.
Formula (93) allows for the existence of two
discriminate mechanisms of diffusion in liquid state irrespective of varied
admixture distribution in clusters and among activated atoms.
Let us label admixture diffusion in various structural
zones of liquid as solubility. Different admixture solubility within the
specified structural zones of liquid metals and alloys is explained by the
dissimilarity of the structure of metals.
As it was noted above, the inside cluster structure is
close to that of solid crystals. So there are sufficient reasons for
recognizing the solubility of admixtures in clusters Scl as
equal to the solubility of the same admixture in solid metals Ssol
in the vicinity of the melting temperature, i.e.
Scl = Ssol
(94)
On the other hand, the zone of activated atoms that
adjoins intercluster splits has a structure that is less ordered and less
stabile in time. It was accentuated above that activated atoms on the
‘surface’ of clusters are the latent elements of gaseous state in liquid state.
Disorder, mobility and a considerable volume of unconfined space in this zone
abate dimensional and other constraints for soluted particles.
Therefore, we may expect that the
solubility of the majority of admixtures in the given zone will be higher than
in the solid-like structure of clusters.
In this connection, the observed jump in the
solubility of the majority of admixtures at melting should be associated with
the forming of intercluster splits and the gas-like zone of activated atoms
during the mentioned process.
Thus, admixture solubility within the zone of
activated atoms Saa can be found as the difference in general
admixture solubility in liquid metals Sl and its solubility
in solid metals near the temperature of melting. So
Saa = Sl – Ssol
(95)
or
Saa = Sl – Scl
(96)
Correspondingly
Sl = Scl + Saa
(97)
It seems rather difficult to allow for the
discriminate solubility of admixtures in (93), because the quantities of Dcl
and Da in (93) can differ from partial self-diffusion
coefficients of Dc0 and Da0
in (75) due to the presence of admixtures. So let us apply here another way of
calculating D by omitting the partial coefficients of Dcl and
Da.
Let us employ the well-known
Stokes-Einstein equation. In case of admixture diffusion we have /114/:
D = kT / 6πη
r. (98)
In case of self-diffusion
D0 = kT / 6πη r0,
(99)
where η is liquid metal viscosity; r is the effective
radius of admixture diffusion; r0 being the effective radius
of self-diffusion.
Under the condition that T is equal to η,
dividing (98) by (99) will lead us to
D = D0 r0 /
r (100)
Since the quantity of D0 can be
obtained through (85), there remain the effective values of r0
and r to be found. Let us determine the effective self-diffusion radius r0
as the extensive sum of the radii of material particles that constitute liquid,
i.e. clusters rcl and atoms rme:
r0 = rcl + rме Cа,
(101)
where Cа is the
concentration of activated atoms in liquid metal.
To determine the effective radius
of admixture diffusion r we must take into consideration that the
admixture entering into the composition of clusters (cluster-soluble) moves
together with them, whereas the admixture soluble within the zone of activated
atoms migrates as separate atoms. Thus
r = rcl (Scl / 0,5) + rа Cа (Sаа/0,5), (102)
where rа is the
radius of an admixture atom.
The coefficient of 0.5 here
stipulates that the solubility of any admixture, by formal reasons, cannot
exceed 50% (at the complete mutual solubility), for if the solution content of
admixture B is more than 50%, then, the solution of B in A
transforms into the solution of A in B.
The validity of (102) can be checked by its
application to self-diffusion. Hence Scl = Sаа = 0,5, while rа = rме, so (101) lets arrive
at (102).
Inserting (101) and (102) into
(100), we get:
D = D0 (rcl + rме Cа) / [rcl (Scl
/ 0,5) + rа Cа (Sаа /
0,5)] (103)
All the values of the quantities
constituting (103) are known, in which connection the given formula is quite
applicable to calculations.
Considering the particular practical importance of
iron-based alloys, admixture diffusion in liquid iron and its alloys is the
best explored, which provides ample experimental data for comparison and
checking. Therefore, let us use expression (103) derived previously to
calculate admixture diffusion in liquid iron and compare the obtained
information with experimental data.
Such calculations require the knowledge of the
coefficient of self-diffusion in liquid iron. There are only two values of the
self-diffusion coefficient of liquid iron. These are presented by the only
experimental value of D0Fe = 1.7 10-6
sq.cm/s /138/ and the value of D0Fe=1.1 10-5
sq.cm/s that we obtained earlier.
Because of such an essential discrepancy let us do the
calculations with the use of either of the values of D0Fe
with the purpose of comparison.
We can single out three groups of admixtures depending
on their solubility in liquid iron.
To the first group of admixtures refer the admixtures
with the unrestricted solubility in solid as well as liquid iron.
To the second group of admixtures we refer the
elements with the restricted solubility in solid iron and unrestricted
solubility in liquid iron.
The elements that have the restricted solubility in
solid as well as liquid iron are included into the third group of admixtures.
In this case, the following correlations of the
quantities required for calculations are appropriate:
Scl + Sаа = 1;
Scl = Sаа.
(104)
Hence for the specified group of admixtures
Scl = Sаа =
0.5.
The procedure of calculating the diffusion of such
admixtures in liquid iron may be illustrated by the example of cobalt. Let us
insert the known quantities of Scl, Sаа, rcl, Cа into the
basic expression (103). We shall use the experimental value of D0Fe
in our calculation. Thus, at Т= 1600 0С:
DCo = 1.7 10-6 (2.48 0.387 + 18)/ [1.38 0.387
(0.5/0.5) + 18 (0.5/0.5)] = 1.74 10-6 sq.cm/s.
Using the calculation value of D0Fe
we get DCo = 1.12 10-5 sq.cm/s, which is much
closer to the main experimental data.
Here and further the data on
admixture solubility in liquid iron are cited on the basis of works /119/ and
/120/.
The principal peculiarity of the values of admixture
diffusion coefficients belonging to this group consists in the fact that their
quantities are close being practically equal to the self-diffusion coefficient
of liquid iron. It corroborates that the admixtures of the first group enter
into the melt cluster composition and their prevalent mass migrates within the
melt by the cluster mechanism in general.
The elements that are soluble to this or that extent
both in the clusters of the melt and within the zone of activated atoms form
the second group of admixtures. We can determine the degree and correlation of
the two of these solubility kinds, knowing the solubility of the given
admixture in solid state and recognizing the complete solubility in liquid
state as 1.
The main body of current experimental data /1,17/
indicates that clusters in liquid iron in the vicinity of the melting
temperature have the structure of neighboring order that is similar to the
structure of solid g - iron. In this connection, the quantity of Scl
for liquid iron can be equalized to the maximum solubility of the given
admixture in liquid g - iron (Sg).
For the second group of admixtures in liquid iron
Scl = Sg;
Sаа = 1 - Sg,
i.e. the increase in admixture solubility at the transition of iron
into liquid state takes place wholly due to solubility within the zone of
activated atoms. Thus, in calculations under (103) for this group, the more Saa=1
- Sg value is, the weightier mass transfer by the gas-like mechanism
through atomic interchange between clusters becomes. Since the stated value
fluctuates within relatively wide limits for the elements of this group – from
0.7 for ruthenium up to 1.0 for alkaline and alkali-earth metals and hydrogen –
the values of diffusion coefficients for the given group of admixtures are
remarkable for a wide diversity of values from 1*10-6 for ruthenium
to 1*10-4 sq.cm/s for cerium (v. Table 11 below).
The third group of admixtures includes
the elements with the restricted and low solubility in liquid as well as solid
iron. Hydrogen is a typical representative of this group as regards iron. Since
hydrogen solubility in liquid iron is less than 1, expression (104) for
hydrogen and the entire Group Three can be presented more conveniently as
(Scl / Sl) + (Sаа / Sl) = 1,
where Sl is the maximum effective admixture
solubility in liquid metal at the given temperature.
It is known that hydrogen
dissolves in solid and liquid iron in the quantity of thousands of one percent,
while its solubility in solid iron is approx. three times as little as it is in
liquid state. According to autoradiography data /101/, hydrogen is distributed
non-uniformly in solid iron, accumulating mainly within the areas of disordered
structure (dislocation cores, the surfaces of domain and mosaic blocs sections
inside granules, crystalline borders). There is a low probability of the
presence of such defects inside clusters because of small cluster dimensions.
Consequently, we may assume accurately enough that hydrogen dissolves mainly within
the zone of activated atoms both in solid and liquid iron. So the solubility of
hydrogen in clusters of iron tents to zero:
SclH ®
0. (105)
Hence in correspondence with (104)
for hydrogen and the whole third group of admixtures in liquid iron:
Sаа =
1. (106)
Introducing the values of SclН from (105) here and Sаа from (106) into (102), we obtain that the effective diffusion
radius of admixtures belonging to the third group equals
r = rа Cа / 0.5.
It corresponds to a solubility
that is quite close to monatomic solubility and the monatomic type of diffusion
of these elements, hydrogen in the first place, by the gas-like mechanism
within the zone of activated atoms only. Such a conclusion captures our
attention generating a further inference that the behavior of hydrogen in
liquid iron is similar to its behavior in gas. Hydrogen does not interact with
clusters and migrates through intercluster splits only. Data of Table 11 shown
grafically on the Fig.16
Thus, the properties of latent
states in the prevalent state, the latent properties of gaseous state in liquid
iron represented by the zone of activated atoms, in our case, may suffice for
certain admixtures to concentrate within the zone of such latent elements only.
In this case, the behavior of
admixtures of such a kind becomes the same as it is in gaseous state, but only
within the elements of the latent state area. For hydrogen, liquid iron
constitutes gas with a peculiar configuration of the ramified flickering system
of slits that permeate the entire volume of liquid iron. Otherwise speaking,
for hydrogen liquid iron constitutes a capillary-porous body with flickering
capillaries.
The possibilities of the migration
of the atoms of hydrogen and other elements in such a capillary-porous body are
determined in general by the dimensions of the atoms of the diffusing elements.
The quantities of diffusion coefficients are maximal
for the elements of this group in liquid iron reaching 10-3 sq.cm in
case of hydrogen (v. Table 11).
The totality of the experimental
and calculation data of the values of D of admixtures in iron obtained on the
basis of (103) as compared with the known experimental data is adduced in Table
11. A full line in Fig.15 indicates the distribution of elements by the
quantity of their diffusion coefficient in liquid iron (calculated data).
Experimental data are shown in Fig.15 by separate dots.
Table 11. Calculated and Experimental
Quantities of Diffusion Coefficients of Some Elements in Liquid Iron at 16000C
№ of admixture group |
Metal |
Scl, % at. /119-120/ |
Atomic radius, /66/ |
D, sq.cm /s calculation by exp. D0Fe |
D, sq.cm /s calculation by D0Fe in Table 8 |
D, sq.cm /s experiment |
Source |
1 |
Mn |
50 |
1.428 |
1.74 10-6 |
1.12 10-5 |
- |
- |
1 |
Ni |
50 |
1.385 |
1.74 10-6 |
1.12 10-5 |
- |
- |
1 |
Co |
50 |
1.377 |
1.74 10-6 |
1.12 10-5 |
3.4 10-6 |
/17/ |
1 |
Rh |
50 |
1.487 |
1.74 10-6 |
1.12 10-5 |
- |
- |
1 |
Pd |
50 |
1.500 |
1.74 10-6 |
1.12 10-5 |
- |
- |
1 |
Ir |
50 |
1.534 |
1.74 10-6 |
1.12 10-5 |
- |
- |
2 |
Ru |
29.5 |
1.480 |
2.60 10-6 |
1.82 10-5 |
- |
- |
2 |
Re |
16.7 |
1.520 |
4.70 10-6 |
2.60 10-5 |
- |
- |
2 |
Cr |
12.0 |
1.423 |
6.7010-5 |
4.00 10-5 |
- |
- |
2 |
N |
10.3 |
0.547 |
0.78 10-5 |
4.6 10-5 |
3.78 10-5 |
/70/ |
2 |
- |
- |
- |
- |
- |
5.50 10-5 |
/121/ |
2 |
C |
8.6 |
1.107 |
1.02 10-5 |
5.35 10-5 |
3.28 10-5 |
/70/ |
2 |
Cu |
7.5 |
1.413 |
1.15 10-5 |
6.20 10-5 |
- |
- |
2 |
Zn |
7.0 |
1.538 |
1.10 10-5 |
6.25 10-5 |
- |
- |
2 |
Si |
4.2 |
1.670 |
2.03 10-5 |
7.20 10-5 |
3.10 10-5 |
/122/ |
2 |
- |
- |
- |
- |
- |
1.23 10-5 |
/122/ |
2 |
- |
- |
- |
- |
- |
3.0 10-5 |
/122/ |
2 |
Ge |
4.0 |
1.755 |
2.0 10-5 |
7.20 10-5 |
- |
- |
2 |
Pu |
2.0 |
1.770 |
3.50 10-5 |
9.20 10-5 |
- |
- |
2 |
Mo |
1.6 |
1.550 |
4.30 10-5 |
9.60 10-5 |
- |
- |
2 |
V |
1.6 |
1.491 |
4.40 10-5 |
1.0 10-4 |
- |
- |
2 |
Nb |
1.2-1.9 |
1.625 |
4.00 10-5 |
1.0 10-4 |
- |
- |
2 |
Al |
1.55 |
1.582 |
4.00 10-5 |
1.24 10-4 |
5.0 10-4 |
/122/ |
2 |
- |
- |
- |
- |
- |
5.0 10-4 |
/123-124/ |
2 |
Gd |
2.00 |
1.992 |
6.0 10-5 |
1.0 10-4 |
- |
- |
2 |
Sn |
1.00 |
1.862 |
6.10 10-5 |
1.10 10-4 |
- |
- |
2 |
W |
1.00 |
1.549 |
6.40 10-5 |
1.30 10-4 |
- |
- |
2 |
Ta |
0.95 |
1.626 |
6.50 10-5 |
1.40 10-4 |
- |
- |
2 |
Ti |
0.72 |
1.614 |
8.10 10-5 |
1.40 10-4 |
5.95 10-5 |
/125/ |
2 |
- |
- |
- |
- |
- |
1.38 10-4 |
/126/ |
2 |
Zr |
0.50 |
1.771 |
8.97 10-4 |
1.50 10-4 |
1.18 10-4 |
/125/ |
2 |
O |
0.56 |
0.603 |
1.25 10-4 |
1.50 10-4 |
1.22 10-4 |
/125/ |
2 |
P |
0.25 |
1.582 |
1.40 10-4 |
1.3 10-4 |
- |
- |
2 |
La |
0.20 |
3.335 |
0.85 10-4 |
0.79 10-4 |
0.95 10-4 |
/121/ |
2 |
S |
0.11 |
1.826 |
1.70 10-4 |
1.44 10-4 |
4.94 10-4 |
/122/ |
3 |
Ce |
0.04 |
2.070 |
1.80 10-4 |
1.50 10-4 |
4.4 10-4 |
/123/ |
3 |
Na |
0.0 |
2.110 |
1.80 10-4 |
1.28 10-4 |
- |
- |
3 |
Mg |
0.0 |
2.853 |
1.40 10-4 |
1.20 10-4 |
- |
- |
3 |
Ca |
0.0 |
2.180 |
1.70 10-4 |
1.25 10-4 |
- |
- |
3 |
H |
0.0 |
0.370 |
1.1 10-3 |
7.2 10-3 |
1.32 10-3 |
/31/ |
3 |
- |
- |
- |
- |
- |
3.51 10-3 |
/117/ |
The table shows that the optimal
coordination between calculated and experimental diffusion coefficients of
various admixtures in liquid iron is achieved in case of using the calculated
value of self-diffusion coefficient. In any case, this is the first time when
theory provides a satisfactory congruence with experiment for such a wide range
of data. Diffusion coefficients in liquid iron for a series of elements are
calculated for the first time having never been determined experimentally. It
creates ample possibilities of testing theory through experiment.
As it follows from the above-said,
among the peculiarities of the suggested procedure of calculating diffusion
coefficients we should mention the fact that the coefficient of self-diffusion
of iron (and any other alloy-forming element) is the least possible, whereas
the coefficients of diffusion of any admixtures can equal or exceed the
coefficient of self-diffusion. Such an admission ensues from the premise that
cluster radius in expression (103) is considered constant. In reality, cluster
radius can vary in both the directions with an increase in admixture content.
This may introduce certain corrections into the process of diffusion as well as
its calculation, negligible in the majority of cases at low admixture content
in the alloy specified.
The main factor that determines the quantity of admixture diffusion
coefficient in the given theory is the distribution of admixture between the
structural zones of liquid iron, which, in turn, accounts for the dominance of
this or that diffusion mechanism in liquid iron or the combination of such
mechanisms.
One of the essential structural characteristics of solid and liquid
states of metals is the compactness of their atomic granulation. The latter is
quantitatively evaluated by the coordinating number k that determines
the number of atoms located in the neighborhood equidistantly from one another
or some central atom.
In case of metallic bonds that are not saturated and directional,
atoms in a crystal can be presented with a certain approximation as mutually
attracting incompressible spheres with the radius of R.
The coefficient of compactness q that characterizes the density of
structure granulation is equal to the correlation between the volume of the
particles forming a crystal and the crystalline volume /127/.
From the viewpoint accepted in our work, the coefficient of
compactness characterizes the volume occupied by matter at the intracrystalline
level, where matter is represented by atoms while interatomic spacings inside
the crystalline lattice represent space.
The level of the intracrystalline structure of matter-space systems
differs from the level of aggregation states. This is a finer dimensional
level. However, as it was shown above, any state of any matter-space systems
incorporates latently the ulterior properties of other states and other levels
of such systems. In the meantime, the latent levels, as it was demonstrated by
the example of diffusion, may impart a considerable or even the decisive
contribution to this or that property of the system in a number of cases.
Therefore, it is of theoretic and practical importance to explore
the levels of the matter-space systems structure that are adjacent to the level
of aggregation states, e.g. the level of the intracrystalline structure of
metals.
In case of spherical atomic granulation, the coefficient of compactness
is /127/:
q =
4pnR3 / 3Va,
(107)
where n is the
number of particles in an elementary cell; Va is the volume
of an elementary cell.
For the closest packings, compactness coefficient equals 74%, i.e.
interior elements of space that are peculiar to the given state occupy more
than a quarter of the entire volume even in the most compact crystalline
lattices.
Table 12 lists the coordinating numbers (k) and compactness
coefficient (q) for the main types of crystalline structures.
Table 12. Coordinating Numbers k and Compactness
Coefficient q for Various Structures
Lattice
type |
k |
q,
% |
Face-centered
cubic and hexagonal compact |
12 |
74 |
Tetragonal
body-centered (с = 0.817; n = 2) |
10 |
69.8 |
Body-centered
cubic |
8 |
68.1 |
Simple cubic |
6 |
52 |
Diamond cubic |
4 |
34 |
Tellurium
lattice |
2 |
23 |
Table 12 illustrates that the volume occupied by the elements of
space in crystalline lattices of different types is always appreciable, its
content fluctuating from 26 up to 77% of the total crystalline volume.
Correspondingly, the properties of crystals depend on the contribution of the
elements of space in a highly noticeable way.
The theory of the influence of inner spatial elements upon solid
physical bodies, liquid metals including, is yet to be originated.
Let us note that even such a weighty concept as the atomic radius
can be defined as a half of the shortest interatomic spacing in the crystalline
lattice within the limits of the crystalline lattice only. Such a definition is
inexact, since it also comprises the interatomic space. Still, there exists no
other way of defining the atomic radius, for the radius of an atom taken
separately cannot be determined with accurateness because of the fuzziness of
the electron cloud, i.e. due to the interaction between matter and space, too.
The above-said corroborates once again that matter does not exist
independently of space, that any element of matter has its corresponding
equivalent among the elements of space, so any system can only be described as
a matter-space system. We may isolate neither the elements of matter nor the
elements of space from such a system, neither physically nor theoretically,
since that changes the properties of the analyzed elements.
Returning to the coordinating numbers concept, let us remark in
conformity with the above-said that the real composite structure of any systems
is to be taken into consideration while determining such a number. The concept
of the average coordinating number makes sense only in the case when we allow
for both the spatial and material components and their interaction.
The coordinating number in liquid, as well as in solid, metals is
generally determined by the method of x-rays dispersion or other procedures
/12/.
However, measurement accuracy for solid state still exceeds that for
liquid state, which can be exemplified by the diversity of coordinating numbers
data obtained by various researchers (v. Table 11). The whole array of data
concerning the quantities of k accumulated by present is averaged, i.e. these
data reflect the continued attempts at describing liquids as a homogeneous
monatomic medium.
The simultaneous existence of both the elements of matter and space
in real liquid metals, as well as the existence of the derivative structural
zones generated by the interaction of the mentioned elements - for instance,
the zone of activated atoms - presupposes the existence of different local
coordinating numbers within them.
In particular, the existence of mobile clusters presupposes the
existence of their mutual granulation and the coordinating number of such a
granulation kc = 12, for this granulation will always tend to
the compact one because of the mobility of clusters toward one another,
irrespective of the type of atomic granulation inside clusters.
At the same time, there exists the granulation of atoms inside
clusters with the coordinating number of kd, which may be
equalized (under the condition of the absence of polymorphous transitions in
liquid state) to the coordinating number of the corresponding solid metal at T
= Tmelting. Finally, we should single out the coordinating
number for activated atoms ka. Activated atoms, by
definition, have at least one free bond. In the meanwhile, activated atoms
participate both in the atomic granulation inside clusters and in the mutual
cluster granulation, since activated atoms are located on cluster 'surface'.
In this connection, the coordinating number of activated atoms
equals
ka
= (kc + kd / 2) - 1. (108)
The average (effective) coordinating number k in liquid depends on
the correlation in liquid of the quantities of atoms that participate in this
or that granulation type within this or that structural zone of liquid, at
different dimensional levels inclusive. Such a number is combined of the
coordinating numbers of this or that structural zone additively as a first
approximation, multiplied by the relative number of atoms entering into each
zone. There arise some problems related to the circumstance that the same atoms
can enter different structural zones at different system-dimensional levels.
For example, activated atoms constitute clusters at the same time.
Taking it into consideration, we can derive the expression for
calculating the coordinating number in liquid metals:
k =
[(kc + kd)/ 2 - 1] Ca + kd(1 - Ca).
(109)
The multiplier (1 - Ca) is introduced here in
order not to double our consideration of activated atoms.
There are two variable quantities forming (109): Ca
and kd, which enables us to obtain simplified formulas for
calculating the coordinating numbers of liquid metals with dissimilar original
atomic granulation in solid crystals. Introducing the corresponding values of
the quantities Ca and kd into (109), we
derive
k =
12 - Ca ; k = 10; k = 6 + 2 Ca; k = 4 + 3 Ca, (110)
for
face-centered cubic, body-centered cubic, simple cubic and diamond cubic
granulations.
In accordance with (110), the coordinating number of metals with
close packing in solid state decreases at the melting and heating of liquid;
for b.c.c. metals the coordinating number does not change at melting. Such
constancy is caused by the influence of two factors: the first is that the
effective number of b.c.c. metals equals 10 for solid state, the second being
the mutual compensation of the zonal coordinating numbers of liquid ka
and kd in such metals with the rise of T. It should be
pointed out that kп for such
metals makes 10 only under the stipulations concerning the participation of
atoms of the second coordinating sphere, since the radii of the first and the
second coordinating spheres are very close /1,2,12,127/, so a slight dependency
of the coordinating number on temperature for such metals is yet to be
expected. A slight dependency k of liquid metals with the original
b.c.c. granulation on T is experimentally proved /17/.
The quantities of k for s.c. and c.d. granulations, according to
(110), increase both at melting and at the heating of liquid. The given
quantity rises in the most noticeable way for the c.d. granulation type. Such
is the consequence of the role of the partial factor of the mutual
cluster-in-liquid granulation and the partial coordinating number of kc.
This is the first time we introduce the factor of cluster-in-liquid
re-granulation into a compact mutual granulation. The factor under
consideration also contributes much to the change of volume, density, electric
conductivity and a series of other properties of some metals and non-metals at
melting, which will be shown below.
The calculated values of the coordinating numbers of a series of
liquid metals derived on the basis of (110) are cited in Table 13 as compared
with the present experimental data.
Table 13. The Average Coordinating Numbers in Solid
and Liquid Metals at the Melting Temperature
Element |
Coordinating
number in solid state /17,127/ |
Coordinating
number in liquid state, calculation by (110) |
Coordinating
number in liquid state, experiment /2,17,127/ |
Li |
10 |
10 |
9.5 |
Na |
10 |
10 |
9.5;
9.0; 10.0 |
K |
10 |
10 |
9.0;
10.0 |
Ag |
12 |
11.8 |
10.0 |
Au |
12 |
11.7 |
8.5;
11.5 |
Cu |
12 |
11.8 |
10.9 |
Al |
12 |
11.8 |
10.6;
11.4 |
Pb |
12 |
11.8 |
11.7;
12.1 |
Zn |
12 |
11.6 |
10.8;
11.0 |
Ni |
12 |
11.6 |
12.0 |
Co |
12 |
11.6 |
12.0 |
Mn |
12 |
11.6 |
- |
Feg |
12 |
11.7 |
10.0;
12.0 |
Sn |
10 |
10 |
10 |
Bi |
6 |
6.4 |
7.0;
7.6; 8.0 |
Ge |
4 |
5.0 |
6.0;
8.0 |
The coincidence between calculated and experimental values of liquid
metals coordinating numbers according to the data listed in Table 13 is quite
satisfactory, especially if we observe the considerable diversity of the
existent experimental data. The latter is to be expected, because various
experimental methods can be sensitive to the three dissimilar partial
coordinating numbers in liquid metals – ka, kc and
kd - to a different extent.
Expressions (110) prognosticate a smooth change of the dependencies k
= f(T). Sudden changes of these dependencies and other structure-sensitive
characteristics in liquid metals must be related to the polymorphous transitions
in atomic granulation inside clusters, i.e. the change of the quantity of kd
/17,44-46,128/.
The changes in electrical resistivity at the melting
of metals are very noticeable being of a most diverse character for various
metals. For typical metals with close packing like copper, silver, gold,
titanium, zinc and some other metals electrical resistivity increases more than
twice at melting. Still, the more friable the crystalline lattice in solid
state is, the less electrical resistivity increases at melting, and a decrease
in electrical resistivity at melting is observed in metals with the loosest
lattices like stibium, bismuth, gallium.
The modification of electrical
resistivity of a series of elements cannot be confined to the suggested simple
scheme. For example, for liquid semi-conductors – silicon and germanium – the
rise of electrical resistivity at melting is so considerable that we may
indicate the change of conductivity type in liquid state in comparison with
solid state, particularly, the transition to the metallic conductivity type in
these elements after melting.
In connection with such diversity, the theory that
describes the modifications of electrical resistivity at the melting of at
least the main groups of metals in a more or less satisfactory way is to be
originated.
A.Ubbelode
/1,2/ rightly states that the modification of the electric properties of metals
at melting may be caused by different reasons. Among such causes are quoted the
distant order disappearance and the rise of positional disorder, as well as the
increase of atomic heat oscillations amplitude at melting, which leads to the
increase in the dispersion of the conductivity electrons within atomic oscillations.
A possible change of Fermi energy level and other possible modifications at
corpuscular and electron levels, including the change of conductivity type in
liquid semi-conductors, are supposed.
From the positions of our theory,
all these factors are probable; however, the specificities of liquid
aggregation state proper are lacking among them. All the above-mentioned
factors refer to the corpuscular or even electron level but not to that of
aggregation states.
Notwithstanding the fundamental
importance of these factors, we should note that the level of the description
of this or that phenomenon must be adequate to the phenomenon described. If we
consider the influence of the change of aggregation state upon a certain
phenomenon, we ought to describe such influence at the level of the structural
elements of matter and space, inherent in the given aggregation state.
As it was emphasized above, the
real structure of real systems is quite complex, being distinguished by the
presence of many hierarchical levels embedded one into another, and also the
latent properties of other possible states.
Hence ensues a relativity
principle for the description of the properties of real bodies:
all the measurable properties are
determined by the contribution of both the elements of matter and the elements
of space;
each of the possible levels of the
elements of matter and space can contribute to this or that specified property;
at every change of the system’s
state, its aggregation state including, the elements of matter and space
intrinsic in the given state make a decisive contribution to the change of this
or that property;
it is improbable to find the
summarized absolute value of each property by calculation, we can only
determine the relative contribution of the elements of matter and space at this
or that modification of the system’s aggregation state.
I.e. it is possible to calculate
the value of the relative change of the property only - at melting or
crystallization, for instance. We cannot calculate exactly the absolute value
of liquid or solid metal volume, yet we are able to do the calculation of the
metal volume change at melting and crystallization by the modifications of the
predominant elements of matter and space.
Such an approach has never been
used before.
In particular, the factors of the
influence of the elements of space formed at melting and peculiar to liquid
state proper - i.e. the influence of intercluster splits - upon electrical
resistivity have never been taken into account.
In the meantime, it was demonstrated
above that intercluster splits surround half of the ‘surface’ of any given
cluster at melting. Such intercluster splits have vacuum properties by
definition, i.e. they are impenetrable for the conductivity electrons. If we
isolate the underlined factor that is inherent in such an aspect in liquid
state proper, it (the factor) can reduce the effective conducting section of
any liquid sample doubly sharp, providing the double increase in electric
conductivity at melting. Let us label this factor as fs.
Thus, fs = 2.
Apart from that, the factor of the
increase of the elements of space volume at melting can also be referred to the
factors peculiar to liquid aggregation state proper (see 3.5). Such a factor
reduces the effective conducting section of liquid metals, too, directly
proportional to the volume of the elements of space inside them. Let us
designate this factor as the quantity of fv.
fv = ΔVp,
where ΔVp is
found on the basis of (39) and Table 4.
The two factors in question are
responsible for the rise of electric conductivity at melting by the forming of
intercluster splits - the elements of space characteristic of liquid state -
during melting process.
However, there may occur self-compacting processes in liquid at
melting in comparison with solids. It becomes possible due to cluster mobility.
We touched upon the causes of cluster mobility in the parts dedicated to the
mechanisms of diffusion and self-diffusion in liquids metals above.
The first consequence of such a
cluster re-granulation is a certain compacting of liquid in cases when the
original atomic granulation in solid state and inside clusters is loose with
the coordinating number less than 12. Such compacting equals zero in metals
with the original close packing, then, it is quite negligible in metals with
the b.c.c. granulation of atoms, reaching considerable quantities for metals
with the simple cubic granulation and for the elements with even looser
granulations like the diamond type.
Let us recognize the factor of
compacting due to the re-granulation of clusters as fp.
In accordance with /129/, the
factor of compacting equals
fp = ΔVc = kp Ca,
where kp = 0; 0.06; 0.217; 0.4
- for f.c.c., b.c.c., s.c. and c.d. granulation types correspondingly (see
above). This factor reduces the electrical resistivity of liquids.
The second consequence of cluster
mobility formulates as the change of the number of conducting contacts between
clusters in liquid because of cluster re-granulation into mutual close packing.
Like balls moving inside a box, clusters moving inside liquid always pack into
mutual close packing.
If a certain metal has the close
packing of atoms in the crystalline lattice in solid state, the re-granulation
factor is of no importance for it - no changes of volume, density and the
effective conducting section are observed in this case, since the number of
neighbors as well as the number of conducting contacts is equal here in both
solid and liquid states.
However, if the original atomic
granulation in solid state (and inside clusters) differs from the compact one,
the cluster re-granulation factor will inevitably cause volume reduction in
liquid accompanied by the boost of its density, which was pointed out above,
plus electric conductivity reduction due to the increase of the effective
conducting section of liquid.
Let us designate the influence of
the cluster re-granulation factor on electric conductivity as the quantity of fc.
This quantity can be evaluated on the basis of the correlation between the
coordinating numbers in solid state and the coordinating number of the compact
mutual granulation of clusters in liquid, which makes 12, allowing for the
factor of fs that doubly reduces the number of neighbors for
any given cluster to contact with (electrically) at any given moment. The fs
factor is numerically equivalent to the ratio of the coordinating numbers in
metals in solid state to 12. Or
fs = ks / 12
Let us mark that the fs
factor is not additive with the first three factors by nature but it imparts an
extensive contribution. In connection with the above-said, we obtain the
cumulative influence of the stated four factors presented as
rL / rS = (fs + fv
+ fp) fs (111)
Let us underline once more that
expression (111) by no means pretends to fully describe the electric
conductivity of liquid metals and its mechanism. It considers nothing but the
influence of the structural elements of matter and space upon the electric
conductivity of metals as well as the reorganizations caused by them that occur
at melting.
Calculations under expression
(111) compared with the known experimental data are tabulated below.
Table 14. The Change of Electric Conductivity of
Metals at Melting
Element |
fs |
fv |
fp |
fp |
rL / rS, calculation by (111) |
rL / rS, experiment /1,98/ |
Сu |
2 |
0.045 |
0 |
1 |
2.045 |
2.04 |
Ag |
2 |
0.047 |
0 |
1 |
2.047 |
2.09 |
Au |
2 |
0.05 |
0 |
1 |
2.05 |
2.08 |
Al |
2 |
0.053 |
0 |
1 |
2.053 |
2.20 |
Zn |
2 |
0.055 |
0 |
1 |
2.055 |
2.24 |
Cd |
2 |
0.056 |
0 |
1 |
2.056 |
1.97 |
Ti |
2 |
0.06 |
0 |
1 |
2.06 |
2.06 |
Co |
2 |
0.051 |
0 |
1 |
2.051 |
1.3 |
Ni |
2 |
0.045 |
0 |
1 |
2.045 |
1.05 |
Fe |
2 |
0.05 |
-0.005 |
0.66 |
1.33 |
1.09 |
Li |
2 |
0.043 |
-0.019 |
0.83 |
1.68 |
1.64 |
Na |
2 |
0.038 |
-0.03 |
0.83 |
1.66 |
1.451 |
K |
2 |
0.04 |
-0.015 |
0.83 |
1.67 |
1.56 |
Rb |
2 |
0.041 |
-0.015 |
0.83 |
1.68 |
1.60 |
Cs |
2 |
0.056 |
-0.017 |
0.83 |
1.69 |
2.6 |
Mg |
2 |
0.048 |
-0.01 |
0.83 |
1.69 |
1.78 |
Ba |
2 |
0.05 |
-0.01 |
0.83 |
1.69 |
1.62 |
Ga |
2 |
0.01 |
-0.03 |
0.3 |
0.59 |
0.45-1.46 |
Bi |
2 |
0.037 |
-0.08 |
0.5 |
0.98 |
0.35-0.47 |
Sb |
2 |
0.06 |
-0.12 |
0.5 |
0.97 |
0.61 |
Si |
2 |
0.076 |
-0.08 |
0.3 |
0.57 |
0.034 |
The data listed in Table 14 show
an excellent coincidence between the calculated and experimental values of the
change in metal electric conductivity at melting for typical metals with close
packing, as well as for alkaline and alkali-earth metals with the b.c.c. type
of granulation in solid state. Calculation and experimental values of ρL
/ ρS for such metals coincide with the accuracy of
experiment error.
The roughest coincidence was
observed in case of 3-d transitive metals: iron, nickel, cobalt.
The maximum divergence (one order)
was obtained in case of semi-conductors.
Probably, the maximum divergence
is to be expected in such cases because of the changes of their electron
structure, especially significant for liquid semi-conductors, where, as it was
demonstrated, the transition to the metallic bond type takes place. These are
the changes occurring on the levels distinct from that of aggregation states
although initiated by the aggregation transitions, which stresses once again
the interrelation between various levels of the systems of material and spatial
elements.
In any case, the calculations
carried out above show that electric conductivity is a property that is highly
sensitive to the presence of the spatial elements in liquid. The influence of
such elements of space as intercluster splits turns out to be decisive for a
series of metals to change their electric conductivity at melting. Anyway, this
influence is noticeable enough to be taken into consideration.
Allowing for the influence of the
elements of space upon the electric conductivity of metals and alloys may
acquire some practical relevance for the future control of this fundamental
property.
A great number of widely known facts indicate that
ideal aggregation states, i.e. states taken purely, do not exist. There are no
ideal gases, no ideal crystals, no ideal liquids. We can only approach what we
understand under the ideal state with more or less approximation.
We may say that the idealization of aggregation states
exists only as a way of interpreting such states, as an attempt at specifying
the most significant features in the given phenomenon from the standpoint of
researchers. The author views aggregation states idealization as a lag behind
the scientific progress.
A wide array of data testifies that any aggregation
state comprises quite distinguishable properties of other states in a more or
less latent form. Let us recall the instances of such latent, or ulterior,
presence of one state within some other.
Gases in the vicinity of the melting temperature are
frequently represented by a suspension of the smallest drops of liquid being
termed vapor in this state. The smallest drops of vapor retain the elements of
matter and space inherent in liquid state. However, atom conglomerates are
observed in gases even at the temperatures that far exceed the boiling point
/94/.
It is also easy to evince that at the vaporization of
any liquids these are small atom conglomerates (but not separate atoms) that
pass into gas phase composition, the approximate number of atoms inside them
equaling K/2, where K is the coordinating number in the neighboring order
granulation characteristic of solid state, too.
Thus, real gases retain, though to a varied extent,
the properties of the elements of matter and space peculiar to both liquid and
solid states. It is known that there exists the possibility of the transition
from liquid to gaseous state, and v.v.
The majority of the properties of material and spatial
elements, intrinsic in liquid state, are preserved in solid crystal state.
In particular, the elements of analogy with clusters
are traced explicitly in the presence of such submicroscopic formations in the
structure of solid metals as mosaic domains and blocs, dislocations, twins,
borders of crystals and other formations. Such formations are usually reckoned
as the defects of crystalline structure /1,2,10,37,66,69,70,87,101/. The
borders of these elements and the mobile vacancy complexes possess certain
specificities of the elements of space inherent in liquid state – flickering
intercluster areas of bond splits.
Liquids, as it was corroborated by numerous x-ray and
other types of research, retain the properties of solid state presented by
neighboring order, etc. /10,17,18-24,31-41,44-47/.
We demonstrated above that liquid metals preserve the
properties of material elements that are peculiar to solid crystal state as
local areas of neighboring order inside clusters. Liquid metals also preserve
the elements of space characteristic of solid state – i.e. vacancies, as
intercluster monovacancies. The main difference between the latent elements of
matter and space and the prevalent elements is that the latent elements enter into
the composition of the prevalent ones, being overpowered by the latter.
Therefore, the latent elements of matter and space form neither phases nor
aggregation states but maintain the possibility of the system passing into some
other aggregation state upon the whole. After such a transition the latent
elements of matter and space become predominant, whereas the elements that used
to be predominant before such a transition, pass into the latent state.
There were repeated attempts at proving that defective
crystals thermodynamic stability is higher, for instance, than that of ideal
crystals /1/. The latent properties of aggregation states are often regarded as
the defects of the prevalent aggregation state in question, while we perceive
defects as something objectionable and eliminable.
In a number of cases, there was introduced a
contradictory idea of the equilibrium, i.e. removable, defects for solid metals
vacancies, in the first place.
Here we suggest another way of reasoning. Namely, it
should be admitted on the basis of the ample experimental data quoted above
that each aggregation state, except for the prevalent form of material and
spatial elements, contains more or less latently the properties of the elements
of matter and space that are peculiar to the neighboring aggregation states.
Such properties are equilibrium and unremovable,
acting as the essential constituent part of the hierarchical structure of real
bodies.
The presence of the latent elements of matter and
space pertaining to other states in the state given implies that any
aggregation state reserves the possibility of transition to another aggregation
state.
Gibbs’ phase principle prohibits the existence of more
than one phase at the same temperature and concentration. Consequently, Gibbs’
phase principle refers to the prevalent forms of aggregation states
irrespective of their latent forms. In turn, when speaking about the totality
of both the prevalent and latent aggregation states, we may premise on the
basis of the afore-said that the totality of the predominant and latent
aggregation states is constant for every given system.
Thermodynamics also premises that the stability of
this or that state is determined by way of comparing the free energies of any
two states, for example, liquid state Gl and solid state Gs.
Thus, for solid state /1-2/:
Gs = Hs - T Ss
(112)
and for liquid state
Gl = Hl – T Sl,
(113)
where Hs and Hl represent
enthalpy of solid and liquid states correspondingly; Ss and Sl
being the respective entropy of solid and liquid states.
It is reckoned by right that Gs = Gl
at the temperature of crystallization only. At any other temperature, the
phase, or aggregation state, with the lesser free energy under the given
conditions, will be stable.
However, the application limits of equations (112) and
(113) are thermodynamically unrestricted, which generates the problem of
two-phaseness touched upon above. In theory, thermodynamics allows to compare Gs
and Gl by calculations according to (112) and (113) under any
conditions, at any temperatures, whereas it is actually assumed that the two
aggregation states coexist at the crystallization temperature only, where their
free energies are equal. It is accepted that there exists only liquid state
above the temperature of melting, there being nothing but solid state below the
melting temperature. What comparison can be discussed under such a premise?
We introduced above the concept of the prevalent and
latent elements of matter and space, inherent in different aggregation states,
and that enables us to solve this problem. Actually, the comparison on the
basis of (112) and (113) of the free energies (as well as many other
parameters) of various states is possible under any conditions, if we take into
account the existence of the predominant and latent forms of these states.
Let us note so far that a real and constant comparison
of the stability of the prevalent and latent forms of aggregation states occurs
within any system implying a continuous competition, or extrusion, between the
specified forms. The attempts at changing aggregation states are constantly
taking place within any system, their success or failure being determined by
the inner structure of the system as well as the environmental conditions.
Let us term such a process as the
competition between the prevalent and latent aggregation states.
It means that preparatory processes for the change of
aggregation states are always in progress to a certain degree, and any change
of the aggregation state structure and properties advances or postpones the
transition of the original aggregation state into another. Premelting and
precrystallization always occur in liquid and solid state but they proceed with
a different development degree under dissimilar conditions, with a different
degree of approximating the transition of the predominant aggregation state
form.
Let us view the mechanism of the competition between
the prevalent and latent states exemplified by liquid and solid states of
metals.
It was shown above that melting goes according to the
cluster reactions scheme (19), with the essential addition that scheme (19)
reflects only the material aspect of the processes of melting and
crystallization irrespective of the elements of space participating in these
processes. Let us reproduce expression (19) here in a somewhat modified form:
This scheme equally describes the process of
crystallization, too. The difference lies in the direction of cluster reactions
in scheme (19). In the direction from left to right the scheme describes the
process of crystallization. In the direction from right to left the same scheme
describes melting process.
In liquid state clusters αn
perform heat oscillations, in the process of which the flickering elements of
space – intercluster splits – arise and disappear between the neighboring
clusters. The mechanism and parameters of such oscillations were discussed
above. It is important to remark here that the process of cluster accretion
into elementary crystals in liquid is one of the elements of the existence form
of liquid aggregation state. In the process of such infinitely repeated acts of
cluster accretion and separation there also occurs a repeated transition of the
kinetic energy of heat oscillations of clusters into the potential energy of
bonds between the accrete clusters. This is the repeated evolving of potential
energy that heats up the locality of cluster accretion and makes them separate
anew under the pressure of the vacancies that are contained in clusters.
Such an infinitely repeated
process of consecutive elementary acts of melting and crystallization at the
level of clusters is the mechanism of extrusion between the prevalent liquid
and the latent solid aggregation states inside liquid state.
Liquid does not know about its crystallization, if we
can say so, but the system, due to the continuous competition between the
prevalent and latent states, seems to be constantly testing the environment
through the flickering interaction of the elements of matter and space, as if
adapting for it; the system changes its structure and properties adjusting them
to the environmental conditions. In particular, the dimensions of clusters and
intercluster spacings change in the course of this process, vacancies emerge
and disappear, etc. At the change of the environment, liquid prepares for
crystallization in quite a short time by way of constant extrusion between the
elementary acts of melting and crystallization.
The extrusion of the prevalent and latent material and
spatial elements inside liquid state is, in its wide sense, the mechanism of
the system’s adaptation for the environmental conditions. The possibility of
such adaptation is ensured by constant heat oscillations and other kinds of
heat motion of the elements of matter and space in the aggregate with the
constant flickering of spatial elements and the bonds between the elements of
matter. This is the flickering interaction of the elements of matter and space
that imparts flexibility, mobility to real systems, as well as the ability of
reorganization and reaction to the environmental changes.
A kind of similar process occurs in any other
prevalent aggregation state. In solid state, vacancy gas pressure is constantly
testing crystalline lattices for strength. In gas state, atoms and their small
groupings are constantly colliding, accreting and separating, etc., etc.
Thus, precrystallization is a continuous process of
the interacting, reorganization and extrusion of the elements of matter and
space of the prevalent liquid and the latent solid aggregation state inside the
prevalent liquid state. Such a process is an existence form of any state. It
enables any system to do a quick re-structuring of the total of its intrinsic
parameters and properties in correspondence with the change of the
environmental conditions, the preparation for the process of crystallization
including.
So the coexistence of equations (112) and (113) quite
reflects the real complex structure of aggregation states that turns out to be
distant from idealized concepts. The concepts of ideal simple monatomic liquid,
ideal crystals and ideal gases also prove to be reality-discordant.
Existent theory presents the formation of
crystallization centers as rather a complex and contradictory process. Certain
problems of crystallization centers formation in connection with current theory
were tackled in Part 1.7 above.
Let us consider this problem one
more time in order to suggest its new solution.
The problem of crystallization centers is described in
a great number of works in present theory. Turnball and Hollomon /63/, as well
as W.C.Winegard /68/, give a good account of this problem from the viewpoint of
corpuscular structure of liquid metals. W.C.Winegard writes that when atoms
group so that a nucleating center is formed, the surface of section emerges
between it and liquid. Section surface formation leads to energy consumption,
which brings about a certain increase in the free energy of the system at the
origination of the nucleus. The nucleus, however, can increase only in the case
if the total free energy of the system is decreasing.
The core of the problem of solid phase nucleation in
the existent monatomic theory is formulated here with precision. The
origination of the nucleus within the idealized homogeneous monatomic liquid
inevitably causes the forming of a section surface, which leads to the increase
in free energy, resulting, in turn, in the impossibility of zero growth of such
a nucleus. It is Ya.I.Frenkel’s heterophase fluctuations theory that suggests
rather a controversial way out of this typical circularity.
Mathematically the problem of crystallization centers
formation is presented in the following way /63,66-68/.
The change of the system’s free energy at the forming
of a solid phase crystal in liquid equals
DF = -V DFv + S s,
(114)
where V is the volume of a crystal; S is its surface; DFv is the change of specific volumetric free energy; s is surface tension.
This expression is identical with
the formula cited in Part 1, except for the fact that the latter expression
employs free energy according to Helmholz. Free energies, according to Gibbs
and Helmholz, practically coincide for condensed states.
If we suppose that a microcrystal is spherical, (114)
will be presented as
DF = -(4/3)pr3
DFv
+ 4pr2 s, (115)
where r is the radius of a solid phase nucleus.
The main assumption at formulating
expression (115) is that a certain new surface, for the formation of which work
(energy) must be spent, arises at the crystallization center formation, which
is reflected by the plus in front of the second term on the right in expression
(115). Such an assumption seems quite logical, being the only possible within
the limits of the monatomic theory of liquid metals structure. Still, let us
bring it into focus that such an assumption initiates all the difficulties of
present theory. It was stated above that the introduction of the opposite signs
into the right side of expressions (114-115) causes the insoluble inner
contradictions in existent theory.
In particular, it follows from this very assumption
that the function of
d (DF)/dr = - 4pr2
DFv
+ 8pr s, (116)
has the extremum, while the radius r that corresponds to the bending
point can be obtained under the condition that
4pr2
DFv
+ 8pr s = 0
Hence originates the idea of the
so-called critical radius of the solid phase nucleus:
rc = 2s/ DFv
(117)
Then, with the use of the known value of DFv /66-68/ found under rather debatable premises, the following is
derived:
rc = 2sТmelting / DН DТ
(118)
The physical meaning of the
critical radius of crystallization center is that the growth of all the
crystals with the r > rc is accompanied by the decrease in
the total free energy of the system, so such crystals can grow freely and
unrestrictedly. However, the growth of all the crystals with the r < rc
will be accompanied by the increase in the total free energy of the system, so
such crystals no sooner arise than they must disintegrate. In point of fact, it
should be regarded as a thermodynamic prohibition of crystallization.
Graphically, the relation between DF and r is expressed by curve 1 in Fig.17 and
Fig.2.
According to the
graph, solid phase can set in after having overjumped the interspace of the
prohibited nuclei dimensions from 0 to rc.
Thermodynamics cannot interpret
the possibility of such jumps. Moreover, equations (115) and (116) presuppose a
continuous configuration of the function of DF = f(r), actually prohibiting similar jumps.
So, to overpass the problem of the prohibited interval, there was
initiated a non-thermodynamic theory of heterophase fluctuations that lets
crystals grow stepwise, and not continuously, up to the reaching of
overcritical dimensions. The heterophase fluctuations theory was considered in
Chapter 1 above in a more detailed way.
Such a point looks very unnatural in existent theory,
so nothing but a long-term habit defends it against criticism.
Nevertheless, we shall try to test the assumptions of
present theory for their correspondence to facts.
The major premise breeding all the contradictions of
the mentioned theory is the assumption of the emerging of new section surfaces
at crystallization both at the moment of nucleation and in the process of crystal
growth. Energy must be spent to form such numerous section surfaces. Actually,
it implies that the system must absorb some energy.
However, at crystallization energy is not absorbed but
evolved, moreover, such energy equals the latent heat of melting taken with the
opposite sign to a high degree of accuracy. The lack of differentiation between
the latent heats of melting and crystallization speaks for the complete
dissymmetry of these processes in the sense of work expenditures, including
those for surface formation. On the contrary, theory assumes that there should
always be work expenditure for the forming of the surfaces of solid phase
nuclei section. Since energy evolves in the one case (at crystallization) being
absorbed in the other case, there must be a difference of the latent heats of
melting and crystallization reflected in the quantity of DF = Ss. Yet it does not really exist.
Thus, facts are at variance with existent theory.
In this situation such a discrepancy can either be
neglected, which was established practice for almost 70 years, or some
artificial account of the situation may be suggested (which was also done), or
we are just to accept the facts and search out new explanations. Let us accept
the facts.
Let us accept the fact of energy evolving at
crystallization as the principal one.
It testifies that there are no new surfaces emerging
within the system, as it was thought to be, but, on the contrary, certain
inside surfaces existing in liquid state are closed.
In case of the monatomic theory of the structure of
liquid it is absolutely impossible, since the monatomic liquid is homogeneous
providing no inner surfaces of section.
From the viewpoint of the theory under development,
flickering inner intercluster splits saturate liquid. Let us write the
elementary act of crystallization as the reaction of
an +ds + an - dНcr ® a2n - ds + dНcr
(119)
The given reaction means that at the accretion of two
neighboring clusters into an elementary crystal the elementary surface of
section ds closes between them into a flickering
intercluster split.
It was shown above that this process is accompanied by
the transition of the kinetic energy of heat cluster oscillations into heat, so
the elementary amount of the latent heat of crystallization dНcr is evolved.
If it is true, we have to admit that those are not new
surfaces that emerge at crystallization by the accretion of clusters in liquid,
but the existent flickering intercluster section surfaces that close, which is
accompanied by the evolving of heat to corroborate the facts completely.
Then, we should re-write expression (112) as
DF = -V DFv - S s,
(120)
while expression (120) takes the shape of
DF = -(4/3)pr3
DFv
- 4pr2 s. (121)
Curve 2 in Fig.17. represents the graph of (121). It
is clear that the function of DF = f(r) does not have the extremum in
the given case, decreasing monotone with the rise of r.
Expression (121) differs from
expression (115) only by the sign in front of the second term on the right, but
the physical meaning of expressions (115) and (121) differs fundamentally, and
such a distinction changes all the existent concepts of the mechanism of the
processes of crystallization centers formation and crystal growth.
The minus has a physical significance in the given
case, as well as the sign in front of the first term on the right in (115) and
(121), meaning that at the closing of the intercluster surface S energy evolves
but it is not absorbed. Such a seemingly negligible difference in signs
radically changes our understanding of the problem of nucleation and smoothes
away the contradictions pointed out above.
The point is that crystal growth will be
thermodynamically expedient at any radius of the nucleus r in liquid
cooled down below the melting point temperature.
It also ensues therefrom that the key problem of the
present crystallization theory - that of the critical radius of crystalline
nuclei - is farfetched, it does not exist in reality.
This is a new and essential
conclusion shaking the fundamentals of the current crystallization theory. In
particular, the inference of mass crystalline centers nucleation in cooled
liquid issues among the first consequences of the new theory, which, in turn,
changes our ideas on the mechanism of crystal growth. The new ideas are viewed
in detail within the theory of overcooling and the competition theory of
crystallization below.
Distinct from the artificial problem of the critical
radius of solid phase crystalline nuclei, which used to exist in theory but not
in reality, the phenomenon of the overcooling of liquid before and during the
process of crystallization is an experimental fact.
Present theory closely relates
overcooling to the problem of the critical radius of nucleating centers.
Namely, existent theory considers overcooling as a structure-forming factor
that influences the probability of heterophase fluctuations formation, as well
as the operation of crystalline formation and growth and the dimension of the
critical radius of the solid phase nucleus.
The fabulous nature of the mentioned parameters as
applied to crystallization by no means affects the fact of the existence of
overcooling.
Consequently, overcooling performs some other
functions at crystallization distinct from those that were declared earlier.
To define the role of overcooling at crystallization,
let us consider the heat aspect of this process.
Let us write the equation of the elementary act of
melting-crystallization (119) as
α n + αn = α2n +
δHcr (122)
According to (122), an elementary crystal is formed by
the fusion of any of the two neighboring clusters with the evolving of the hard
amount of heat δHcr.
This is the elementary heat of cluster, or
intercluster split, formation, i.e. the elementary heat of cluster accretion or
the closing of intercluster splits. The quantity of δHcr
can be found through the following expression:
δHcr = DHmeltingnc / N0 (123)
All the quantities included into (123) are known
having been cited previously.
For an elementary crystal to get formed and for
reaction (122) to stop being oscillatory, the heat of δHcr
must be absorbed by the melt without the heating up of the latter above the
melting temperature.
Yet at the temperature of the melt equaling or
exceeding the melting temperature, the melt cannot absorb δHcr
without being heated above the melting temperature, so reaction (123) is
infinitely repeated in the directions both from left to right and from right to
left. The temperature of the melt does not change at that, for the energy of δHcr
is periodically passing from its potential form into kinetic, and v.v.
Crystals cannot form under such conditions. For at
least one elementary crystal to get formed, the heat of δHcr
must finally pass into the form of potential heat energy, so it must be
absorbed by the surrounding melt without the heating of the latter above the
melting temperature.
In turn, it is possible only in
case when the melt is cooled down below the melting temperature.
The stated phenomenon is termed overcooling DT. As it follows from the above-said, overcooling before
crystallization is required for the only purpose – for the melt to absorb the
latent heat of elementary crystal crystallization on its own, without being
heated up above the temperature of crystallization.
It is normally admitted that
crystallization does not go within the overcooling interval. Our theory affirms
that reaction (123) goes on repeatedly with the frequency of 109
acts per second. However, while the heat of δHcr cannot be absorbed by the melt, it passes again and again into the
form of the potential energy of heat oscillations of clusters.
It is obvious that overcooling performs but a purely
heat part in our theory; it is devoid of any structure-forming functions.
Such a purely heat approach to the quantity of DT enables to determine by calculation the quantity of overcooling
necessary for crystallization to set in by the method of heat balance between
the crystallization center and the surrounding melt. The elementary
microcrystal heat balance equation may be presented as
Qm = Qn.c.
(124)
where Qn.c. is the heat of the
elementary crystallization act according to (123), i.e.
Qn.c. = δHcr = DHmeltingnc
/ N0, (125)
while Qm is the heat absorbed by the
melt under the condition of its being heated up to the melting temperature
exactly. This quantity can be obtained by applying familiar heat methods. Thus,
Qm = (Tmelting - Ti) c v
ρ, (126)
where Ti is the maximum temperature of crystallization
start; Tmelting - Ti = DT, where DT is the minimum overcooling of
crystallization start; c is the heat capacity of the melt; ρ
is melt density; v being the melt volume that absorbs the heat of Qn.c.
during the time equal to one period of heat cluster oscillations.
The latter is of immense importance. The heat of Qn.c.
evolves during the time equal to one half-period of heat oscillations of a
cluster. It must be absorbed by the environment during the same or even lesser
time without the heating up of the specified zone of the medium above the
melting temperature, otherwise clusters re-separate and the elementary volume
of liquid gets formed again.
The process of heat absorption cannot be prolonged for
an indefinite period of time. Time factor refers to decisive ones alongside
with structural factors at crystallization.
Hence ensues the second essential conclusion – heat
can only be absorbed by the immediate environment of accreting clusters for
such a short period.
I.e. the volume of v in expression
(126) must be very small, because at the elementary act of crystallization
there is no time to be spent on slow redistribution of heat within the melt
volume and beyond its limits.
We can put forward the following suggestion concerning
the amount of such heat.
The elementary
act of crystallization consists in the accretion of two neighboring clusters.
The elementary heat of crystallization evolves along the accretion border
between the given clusters, so the heat in question, before dispersing in the environment,
will be inevitably absorbed in the main by the accreting clusters themselves,
and elevate their temperature. In case of a successful elementary
crystallization act, the elevation of the temperature of the two clusters under
accretion caused by such heat must not excede the temperature of melting.
To launch our analysis, it is therefore natural to
presume that the volume of v in expression (126) equals the volume of
the elementary crystal itself, i.e. the volume of two clusters.
The volume of two clusters amounts to
v = g Vm nc /N0
(127)
where Vm is the molar (corpuscular) metal volume; g
is the number of clusters participating in the elementary act of
crystallization. In the simplest case g = 2.
By way of inserting the v from (127) into (126), we
obtain
Qm = g DТ c r Vm nc /N0.
Let us stipulate that Vm = M / ρ, where M is the atomic metal mass.
Finally we
obtain:
Qm = g DТ c M nc /N0.
(128)
Now, introducing the value of Qm
from (128) and the value of Qn.c. from (125) into
the original heat balance equation (124), we arrive at
g DТ c M nc /N0 = DНmelting nc / N0
Hence we derive the minimum
overcooling required to start crystallization as the first elementary act at
the absorbing of crystallization heat by two accreting clusters:
DТ = DНmelting / g М с.
(129)
Expression (129) defines
overcooling as a purely heat quantity. Moreover, this expression has a certain
beauty and compactness, which is also important. Expression (129) includes
reference quantities only, in which connection the value of overcooling
necessary for nucleation according to the elementary crystallization act scheme
(122), can be easily calculated.
The data obtained by our
calculation are to be found in Table 15.
Table 15. The Minimum Overcooling Necessary to
Start Crystallization of Pure Liquid Metals at g = 2
Metal |
М, kg/mole /98/ |
DHmelting,
c/mole /98/ |
с, c/mole deg., /98/ |
DТ, deg. by (129) |
Ga |
69.72 |
1335 |
6.24 |
1.53 |
Cu |
63.54 |
3120 |
6.86 |
3.58 |
Sn |
118.7 |
1720 |
7.6 |
0.95 |
Al |
26.98 |
2580 |
7.66 |
6.24 |
Bi |
208.98 |
2730 |
7.43 |
0.88 |
Zn |
65.37 |
1730 |
7.01 |
1.89 |
Fe |
55.85 |
3290 |
10.29 |
2.86 |
Ni |
58.70 |
4180 |
9.20 |
3.87 |
Co |
58.93 |
3900 |
9.60 |
3.44 |
W |
183.85 |
8420 |
26.67 |
0.80 |
Overcooling calculated in Table 15
is quite close to the values of liquid metals overcooling observed
experimentally /1,2,10,68/. It corroborates the inference of the role of overcooling
as mainly the thermal factor of crystallization.
Still, the calculated overcooling
is not limited to the suggested values, for only thermal factors were taken
into account when doing the calculation, while crystallization is also bounded
by positional factors, e.g. the afore-mentioned re-granulation factor, plus the
degree of terrain-contour matching of clusters before accretion, as well as
time factor, external and other factors. Thus, real overcooling before the
start of crystallization may either exceed or be less than its calculated
values under the influence of nonthermal factors.
In this connection, the quantity
of factor g, i.e. cluster quantity participating in the absorption of
the heat of the elementary crystallization act, which we introduced, is of
paramount importance for the experimental research of liquid metals structure.
By measuring the actual overcooling of DТ, we can calculate the
quantity of factor g.
For instance
g = DНmelting / DТ М с.
(130)
Interestingly, g may either exceed or be less than 2. This is
a new and worthy experimental reseach subject.
For example, the maximum known overcooling for liquid iron equals
2950C /68/. Introducing the specified value into (130), we obtain
that g1 = 0.019 in this case. As it is known, crystalline
dimensions are very small in case of crystallization with a considerable
overcooling.
During the process of slow crystalline growth in liquid iron,
overcooling often approximates 0.10C. By introducing the given value
into (130), we find that g2 = 57.14 in this case. The
correlation between g2 and g1 is g2
/ g1 = 3000. Such a correlation characterizes the possible
relation of crystalline dimensions that can be obtained in cases of crystallization
going at either the maximum or the minimum speed for iron.
Thus, the quantity of g turns out to be proportionate to the
crystallization act duration as well as crystalline dimensions in castings, and
it can be used to determine the mentioned quantities.
We should discriminate between the overcooling of nucleation and the
overcooling of crystal growth. The latter is always less than the former, since
heat abstraction conditions are facilitated during crystalline growth.
On the whole, the calculated values of overcooling fit the familiar
experimental values of this quantity. W.C.Winegard allocates
the typical quantity of liquid metals overcooling before the start of
crystallization within the limits of 1-10 deg. /68/. Overcooling values that we
calculated according to (129) and cited in Table 15 are positioned exactly
within the given interval.
The term 'spontaneous' means 'evoked by internal causes' (often
unknown). The term 'forced' in application to crystalline centers nucleation
signifies 'initiated by external causes'.
As it was shown above, crystallization results from the interaction
of both the internal causes, such as the interplay of material and spatial
elements in liquid metals, and external factors, for instance, temperature,
pressure, etc. External and internal causes interreact.
Therefore, the distinction between the processes of nucleation in
liquid metals into spontaneous and forced seems inappropriate.
Nevertheless, such distinction arose, so it is to be taken into
consideration.
W.C.Winegard defines spontaneous crystalline centers nucleation as
nucleation in absolutely homogeneous medium with the presence of overcooling
/68/.
G.F.Balandin /74/ defines spontaneous nucleation as a result of
monatomic heterophase fluctuations formation, also with the sine qua non of
overcooling, which does not contradict W.C.Winegard’s definition.
W.C.Winegard writes that 'in the vicinity of the melting point
critical nucleus dimensions must be infinitely large, because, when overcooling
approximates zero, the decrease in volumetric free energy related to the phase
transition of liquid into solid cannot compensate for the increase in free
surface energy. As overcooling increases, critical nucleus dimensions
decrease…' /68/.
It follows from the given reasoning that overcooling must act as the
measure of spontaneity or forcedness of the process of crystalline centers
nucleation.
The greater overcooling becomes, the closer spontaneous nucleation
is.
Unfortunately, it is impossible to evaluate the degree of
overcooling necessary for spontaneous nucleation on the basis of these
arguments, in contrast to our theory.
It is accepted that spontaneous nucleation is possible only in
liquid metals that are completely purified of any admixtures. The production of
such metals encounters serious experimental difficulties, for no analysis can
secure against the presence of the minimal quantities of foreign admixtures in
the melt.
It is also assumed theoretically that one can overcome the mentioned
difficulties by way of dividing liquid metal into smallest drops. If there be a
little amount of admixture particles within the volume, some drops would not
contain foreign particles by virtue of their own small dimensions, so
homogeneous nucleation could be observed within them. Experiment
overcorroborated such suppositions. As it turned out, overcooling does increase
considerably at the dissection of the melt into drops, - and not for some of
them, as it was to be expected, but practically for all the drops, usually
inversely proportional to their dimensions. Actually it means that at the
crystallization of small drops those are not admixtures that perform the
salient function but a certain, or some, ignored factor(s).
For instance, it may be the factor of time. Small drops cool down
much faster than volume-bounded liquid during the same time period.
Time is also required for the fitting, or terrain-contour matching,
of the accreting cluster structures and for their re-granulation (see above),
it is simply necessary to evolve the latent heat of crystallization and provide
its transition from the kinetic energy of heat cluster oscillations into heat
energy, as well as redistribute the given energy at least within the limits of
two clusters.
Existent theory neglects these factors; it is reckoned that
nucleation in small drops is actually honogeneous.
The values of overcoolings obtained by the small drops method are
listed in Table 16.
Table 16. The Maximum Overcoolings (ΔT)
Obtained by the Small Drops Method /63/
Metal |
ΔT, deg.
C |
Metal |
ΔT, deg.
C |
Mercury |
77 |
Silver |
227 |
Tin |
118 |
Copper |
236 |
Lead |
80 |
Nickel |
319 |
Aluminium |
130 |
Iron |
295 |
As W.C.Winegard notes, such overcoolings are never observed in
practice; overcooling quantity fluctuates from 1 to 10 degrees under real
conditions. Let us add that the calculated quantities of overcooling of the
elementary crystallization act (v. Table 15) fluctuate within the limits of
1-10 degrees.
However, it has been assumed till now on the basis of the data
quoted in Table 16 that heterogeneous, as contrasted with spontaneous,
nucleation takes place under real conditions, i.e. crystals get formed on the
surface of a foreign solid body present in the system.
Thus, current theory regards spontaneous crystallization as an
occurrence that is almost improbable, or practically unobservable, in any case.
For example, V.I.Danilov used to admit that spontaneous
crystallization is hard to observe, too. Liquid metals should be purified of
practically all admixtures for that /1,2,10,63-70/.
The imperfections of such an approach are obvious here having been
commented upon earlier; they result from idealized views of liquid metals
nature as well as the incorrect ideas of the role of overcooling at crystallization.
The interaction between the elements of matter and space theory
developed here presumes that all real bodies consist of the interacting
elements of matter and space, while bodies contain not only the prevalent
material and spatial elements, but also their latent forms from the standpoint
of aggregation states. Thus, liquid metals are non-ideal and inhomogeneous in
principle in their structural aspect, similarly to any other real bodies. So
the premises of the ideal monatomic, monomolecular and any other monoparticle
structure of liquid metals are erroneous in principle, for there are no such
simple liquids in nature.
Evidently, it would be better to define spontaneous nucleation as a
natural elementary crystallization act occurring by way of accretion between
any of two oscillating elements of matter in liquid – i.e. clusters – into a
single elementary crystal accompanied by the evolving of the elementary amount
of crystallization heat under the influence of the totality of external, as
well as internal, factors.
Since clusters within the melt can be of a various chemical
composition, the presence of soluble admixtures does not make the elementary
act of crystallization forced, though influencing it. This is the same natural
spontaneous crystallization, because the essence of the process never changes.
The presence of insoluble insertions or gases in the melt does not
change the core of crystallization process but rather modifies its conditions.
Thus, our approach, as distinct from present theory, establishes
that natural spontaneous nucleation is not a rarity but the major fundamental
and the most prevalent variant of nucleation both in pure metals and alloys.
Spontaneous nucleation may occur within a wide range of
overcoolings. Overcooling does not determine the degree of spontaneity or
forcedness of crystallization process at all. Overcooling is required, first,
for the heat abstraction of the elementary crystallization act, as it was
demonstrated above. Besides, overcooling can be sensitive to metal cooling
speed, as well as structural metal modifications and other factors.
The theory that existed earlier could not calculate the overcooling
necessary for spontaneous crystallization to set in. Such indeterminancy
gradually lead to the fact that spontaneous nucleation came to be considered as
an infrequent, particularly laboratory phenomenon.
The overcooling calculated in Table 14 above is determined for
chemically pure metals.
In essence, this is the overcooling of spontaneous crystallization
yet calculated for the concrete case of the complete two-cluster accretion into
an elementary crystal under the condition that there is time sufficient for the
complete terrain-contour matching of the adjacent cluster structures. Actually,
the specified calculation was done for the conditions of a very slow
overcooling. It is a typical but not the only possible case of spontaneous
crystallization. So the overcooling calculated in Table 15 by expression (129)
is not the only possible spontaneous crystallization overcooling either.
External and internal factors, for example, overcooling speed or
alloy composition change, can strongly affect the conditions of spontaneous
nucleation and the corresponding overcooling.
Thus, spontaneous nucleation may occur under different conditions
and at dissimilar overcoolings. Nucleation is always spontaneous in a sense,
for it is determined by fundamental causes. For instance, nucleation always
goes by reaction (122).
Forced nucleation does not exist as such without spontaneous nucleation.
Consequently, spontaneous nucleation is primary, forced nucleation being
secondary.
Any external action can alter the conditions of reaction (122)
course, but the reaction of the elementary crystalline centers nucleation act
always remains the same.
Therefore, our theory, as distinct from existent views, affirms and
proves that spontaneous crystallization is the main crystallization type,
whereas external action can either hamper or facilitate this process without
changing its essence.
Current theory asserts that the speed of crystalline centers
nucleation is determined by the following expression of the heterophase
fluctuations theory (6):
n = К1 еxp (-U/ RT) exp [- Bs3 / T (DT) 2],
where the quantity
of n is measured by с-1 m-3. I.e. the quantity of n
represents the onset of heterophase fluctuations of critical dimensions
(nucleation centers) frequency per volume unit of liquid.
Let us compare the given approach with the data in our theory.
In accordance with expression (127), every act of heat
oscillations of clusters potentially represents the elementary act of a
crystalline center nucleation. So the frequency of heat cluster oscillations is
the highest possible frequency of crystalline centers nucleation. Out of
expression (49) we derive:
n
= j = (1/2paа)
(3kT N0 / nc M)1/2
To find the frequency of crystalline centers nucleation per certain
volume, (49) is to be multiplied by the number of clusters within the given
volume. It seems most appropiate to determine the unknown quantity per mole of
substance. Thus
N
= n Nc,
where Nc
is the number of clusters in a mole (gram-atom) of liquid metal at the
temperature of melting.
In turn, Nc = N0 / nc.
Finally we obtain the expression for calculating the highest possible frequency
of crystalline centers nucleation per gram-atom of metal:
n
=( N0 / nc)(1/2paа) (3kT N0 / nc M)1/2
(131)
This is an extremely great number, of the order of 1032 с-1.
Thus, the theory of the interaction between the elements of matter
and space that we develop accentuates that the process of spontaneous
crystalline centers nucleation in liquid metals refers to regular but not
random phenomena. Certainly, any cluster pair can form a crystallization
center, but only in case when there occur favorable conditions, the conditions
for the elementary crystallization heat abstraction, in the first place. Such
continuously merging and separating cluster pairs are flickering, or virtual,
crystallization centers.
Flickering crystallization centers nucleate with high frequency in
liquid by (131), and separate again and again with the same frequency. Liquid
seems not to know about its forthcoming crystallization, yet it can prepare for
crystallization with the help of the flicker mechanism as the environment
provides the corresponding conditions for the process.
Crystallization
requires time by various reasons, so the time necessary for crystallization
influences its results. This is familiar from practice. Let us briefly survey
the causes of time influence upon the process of crystallization.
It was stated above that the elementary crystallization act
represents a reaction of fusing two adjacent clusters into a single elementary
crystal. Under the most favorable conditions such a reaction requires the
minimal time equal to one period of heat cluster oscillations
j = 2paа (nc M /3kT N0)1/2
(132)
That is the time equalling approx.10-9 sec. It is
absolutely the minimal time requisite for a single elementary act of
crystallization.
The real minimal crystallization time may exceed the quantity of
(132), yet it cannot be less than the mentioned quantity.
In the first place, the crystallization of metal mass runs
consecutively and not simultaneously. Hence ensues the general rule: the
grosser the casting is, the more time it requires for its crystallization.
Secondly, there exists the above-cited factor of cluster
re-granulation in liquid. It means that in liquid clusters are packed otherwise
than, or not exactly as, the atoms in the crystalline lattice of a solid. At
crystallization, clusters must re-form into a configuration that suits to their
accretion into a single crystal most.
Such reconfiguration occurs by way of consecutive restructurings
until the optimal or at least acceptable configuration of cluster granulation
is reached. Re-granulation requires for the renewing of the same form of
interatomic bonds between neighboring clusters, as is peciliar to a solid
crystal. The closer cluster configuration approaches that of a solid body, the
more thoroughly intercluster bonds get renewed at crystallization, the more
equilibrium the growing crystal is, the more perfect its structure becomes.
However, it is hardly possible to arrive at the complete compatibility between
cluster structures, - this can only be approximated to some degree. The process
under consideration is termed as intercluster bonds matching and it requires a
considerable amount of time for its more or less satisfactory completion.
Still, the matching of clusters and growing crystals need not attain
absolute completion for a successful crystallization course. Clusters can
accrete with a certain mismatch of interatomic bonds. The developing
crystalline lattice will be defective in this case, i.e. far from being
equilibrium, which is usually observed under real casting conditions.
As experience shows, the degree of a possible mismatching of
intercluster bonds is relatively high for metals.
It follows from the experiments of the so-called amorphous metals
production. Even at the speed of overcooling that reaches 106
degrees per second, it is possible to obtain only an extremely fine
microcrystalline structure in metals in the majority of cases. These are but
some specific alloys that let obtain a quasi-amorphous structure.
We should remark that, in connection with the composite real
structure of liquid alloys, the presence of clusters and intercluster splits
inside them, as well as the presence of the neighboring order of atomic
arrangement inside clusters, it is impossible in principle to obtain a
completely amorphous, wholly chaotic structure of metals at their
crystallization from liquid.
All that is to be done in this direction is to obtain a solid metal
structure proximate to the monocluster pattern. Such a structure will contain
an increased amount of the elements of space extrinsic to solid state, the
correspondingly reduced density and immense free energy, becoming, in this
connection, very unstable thermodynamically.
Crystalline growth also requires time – by the same reasons.
In the third place, time is needed to abstract the latent heat of
crystallization away from the growing crystal, as well as crystallization front
and casting upon the whole. The time factor of crystallization heat abstraction
plays the vital or decisive role under regular foundry conditions. Its cause
consists in the sluggishness of the process of heat abstraction by the heat
conductivity mechanism, whereas this is the very mechanism that operates under
heat abstraction conditions in solids, for instance, in a solid mold wall or
within a solid casting zone.
The mass
character of crystalline centers nucleation represents a specific and totally
unexplored problem. The inference that there arises a whole mass of
crystallization centers at the onset of crystallization ensues from the
experimental fact of the instantaneous liquidation of overcooling after
crystallization starts. The nucleation of one or several elementary crystals
cannot almost instantly raise the temperature of the entire metal mass up to
the melting point, which takes place in reality. The growth of several crystals
can elevate metal temperature up to the observed value in principle, but not so
rapidly as it really happens. Nevertheless, temperature rises, which
corresponds to the crystallization of a considerable part of metal volume.
Namely, with the overcooling determined in Table 15, the increase of
the temperature of the entire metal mass up to the melting point means that all
clusters entering into the overcooled metal composition have on average united
pairwise.
Hardly the strict pairwise union is it, in fact, yet the nucleation
of a large number of crystallization centers occurring simultaneously within
the whole volume of the overcooled liquid zone is beyond any doubt.
There are experimental facts corroborating this conclusion. In
particular, it is the familiar formation of a fine-grained disoriented crystals
zone at the surface of castings, or, as it is otherwise termed, the ‘skin of a
casting’ zone. So the more the speed of heat abstraction and the speed of
hardening are, the finer-grained the structure of castings grows.
Certainly, crystals within the zone under analysis are much larger
than elementary crystalline dimensions; however, temperature leap at the onset
of crystallization is but the first stage of the process which leads to the
forming of the ‘skin of a casting’ zone, as well as other structural zones in
castings.
The first inference of the given part lies in the following: a
discontinuous temperature rise within the entire volume of the overcooled metal
at the start of crystallization can be explained, most probably, by the mass
nucleation of a huge amount of elementary crystals, or crystalline centers,
within the zone specified.
Such a supposition seems natural to our theory, since, as it was
pointed out, the elementary crystallization act reaction (122) is continuously
repeated within the whole volume of liquid with the frequency of 10-32
times per second per gram-atom of metal by (131).
As it was shown, it signifies that liquid gets ready for
crystallization as soon as favorable conditions arise.
Overcooling subsumes under such conditions, implying the possibility
of absorbing the elementary heat of crystallization without overheating the
metal. At the reaching of such overcooling, the elementary acts of
crystallization (122) go spontaneously at any point of liquid, so a huge number
of elementary microcrystals (crystalline centers) emerge spontaneously at a
very short time period – approx. 10-9 sec. – independently of one
another. Their ultimate number can reach the quantity of Nmax = N0
/ 2 nc.. The quantity of Nmax for liquid
metals reaches 1020 per gram-atom of metal.
The given amount is much greater than the number of crystals that we
observe in a final casting at the end of crystallization.
Hence, first, the number of crystallization centers in the course of
crystallization does not equal the number of crystals that are obtained by
casting.
Secondly, the number of arising crystallization centers exceeds
multiply the number of crystals that we get through casting.
In the third place, it means that crystalline dimensions increase in
the course of crystallization, whereas their number diminishes.
Such conclusions are novel. The question of changes in crystalline number
in the process of crystallization has never been raised in current theory.
It is supposed by default that the number of crystals in the process
of crystallization does not change, so if a crystal comes into being, it
survives some way or another to be present in a solid casting later. Existent
theory presumes only a mechanical interaction between crystals in the process
of crystallization, for example, crystalline competition and selection in the
direction of their growth. Such interaction does not change the original
crystalline number during crystallization.
Our theory asserts that the number of crystallization centers under
regular casting conditions exceeds multiply the number of crystals obtained in
a final casting.
A question generates how a small number of large crystals result
from the original large number of small elementary crystals. This question is
of paramount importance - both practical and theoretic.
In practice, it is important that we obtain fine-grained castings,
consequently, it is useful to know how to fix such a huge number of
crystallization centers that we have at the beginning of the process in order
not to let them form into large crystals.
As far as theory is concerned, the emerging of a small number of
large crystals from a huge amount of microcrystals means that the process of
crystalline growth goes otherwise than it was previously surmised. Thus, the
theory of crystalline growth from the melt is to be improved.
The competition theory of crystallization that regards the
mechanisms of crystalline growth from the melt /130/ gives answers to the
questions formulated above.
Its major premise is that crystals can grow simultaneously at
different dimensional levels using dissimilar building material.
Correspondingly, there can exist several mechanisms of crystalline
growth in castings.
A certain mechanism of crystalline growth may turn out to be
prevalent under given concrete conditions, yet more often various mechanisms of
growth operate simultaneously complementing one another at different
dimensional levels. These different growth mechanisms are always competing with
one another, which lets obtain crystals with the least free energy.
Among the basic mechanisms we may cite the monatomic mechanism of
crystal growth, when separate atoms act as the main building material for
crystals, the cluster mechanism, when clusters serve as the building material,
and the microcrystalline (or bloc, mosaic, domain) mechanism, when small and
smallest crystals function as the building material for the growth of large
crystals.
The monatomic growth mechanism prevails at the growing of crystals
from gas phase. Still, even in gases there exist, as it was demonstrated above,
the latent elements of matter intrinsic in liquid state – small groupings of
atoms or molecules, and they can also participate in the process of crystalline
growth as the building material. The participation of such complexes is
thermodynamically expedient in the process of crystallization, since it
accelerates crystalline growth. On the other hand, the participation of such
complexes in the growth of crystals increases the probability of the appearance
of the so-called defects inside crystals.
It is the cluster mechanism that dominates at the nucleation and
growth of small crystals from metal and alloy melts. The mechanism under
analysis was viewed in detail earlier when treating the nucleation question.
The basis of the given mechanism is the bicluster reactions scheme as applied
to crystallization:
(133)
where αn
is a cluster within the composition of liquid; 2αn is
the elementary crystal obtained by the accretion of two clusters; iαn
is a crystal formed through the accretion of i clusters.
Reaction (133) can go in both the left and right directions,
dependent on heat absorption or abstraction, while the reaction of the
interaction between the neighboring clusters in liquid at T> Tmelting
goes continuously and reciprocally providing the basis for the flickering
interaction mechanism between material and spatial elements in liquid metals
and alloys:
αn
+ αn→← 2αn.
Cluster mechanism of growth according to scheme (133) is
characteristic of a rapid crystalline growth from the melt, - for instance, at
a high original overcooling, or, on the contrary, for a very slow growth under
the conditions of high temperature gradient within the liquid zone by the front
of crystallization, and also for pure metals.
The mechanism of crystal growth by way of the attachment of
microcrystals to larger crystals is peculiar to slow growth conditions with the
presence of inconsiderable overcooling, as well as the solid-liquid zone in
castings, which is most typical of the alloys that get crystallized under the
conditions of volume hardening.
Let us underline that the basic mechanisms of growth operate most
often simultaneously in different combinations complementing one another. Each
growth mechanism performs its functions and creates certain structural
specificities that can be traced in the structure of solid metals and alloys.
The latter mechanism of crystalline growth has the following
peculiarity: small crystals may unite competing in the process of growth, so
larger crystals may absorb smaller ones.
Let us term the given mechanism of growth as the competitive
mechanism, and let us consider it more extensively due to its appreciable
practical importance for the structure of the overwhelming majority of
castings.
The point is that this is the competitive mechanism of growth that
is responsible for the accretion of crystalline centers under the conditions of
their mass nucleation, typical of regular casting process, and eventually for
the coarsening of the original crystalline casting structure undesirable for
casters.
In principle, there are no insurmountable barriers to the accretion
of crystals with arbitrarily large dimensions in liquid, except for the problem
of their inner structure matching.
Let us mark that structure matching and the accretion of neighboring
microcrystals are thermodynamically expedient, since the system’s free energy
decreases in this case. Thus, the process of mutual fitting between the
structures of the neighboring microcrystals that are at motion in liquid will
not be quite accidental developing from the lesser to a more exact matching.
Consequently, this is a natural process accompanied by the decrease
in the free energy of the system.
So, if such matching exists, crystals can unite without forming
section surfaces, i.e. by way of forming a single large crystal from two or
more smaller crystals.
Stationary, fixed crystals cannot fit in with one another.
Therefore, the process of accretion between small and smallest
crystals is possible until the mentioned crystals retain mobility, i.e. until
they hover within the liquid medium participating in heat motion. It is
possible only with the presence of the solid-liquid zone in a casting.
Such microcrystals, hovering in liquid, are the Brownian motion
objects. While they are able to move, they can fit in with one another in the
course of multiple collisions and accrete into a single larger crystal after
attaining the compatibility of their crystalline lattices.
The dimensions of particles participating in the Brownian motion are
known – they amount to the tenth fractions of a millimeter (0.1mm.). The given
quantity can be considered by convention as the size limit for the hovering
crystals that maintain their accretionability. Larger crystals may accrete,
though, but only in case when the orientation of their crystalline lattices
chances on being compatible.
The process of small crystals accreting into larger ones is
energetically expedient, because the internal grain border area diminishes
during the process, so the free energy of the system decreases.
Similarly to any other dissipative process, the process of
crystallization, in correspondence with I. Prigozhin’s synergetic theses, is to
take place simultaneously at all its possible levels.
The competition crystallization theory asserts the same thesis.
The bicluster reaction mechanism similar to (134) is the basic
mechanism of nucleation and growth of crystals from metal melts. However, the
process of crystalline growth may go with the use of any building material
available in the given medium, including separate small crystals plus separate
activated atoms to fill out hollows.
The major tendency of the competitive crystallization mechanism at
macrolevel is the survival and growth of larger crystals by their absorbing
smaller ones.
This tendency determines the real crystalline structure of castings.
At the same time, the tendency in question reflects the struggle-for-existence
competition between crystals.
The process of competitive crystallization can be represented by the
following scheme of the growth and accretion of three neighboring crystals:
1st
microcrystal αn + αn→ 2αn |
2nd
microcrystal αn + αn→ 2αn 2αn + αn→ 3αn |
3rd
microcrystal αn + αn→ 2αn 2αn + αn→ 3αn ………………………….. ………………………….. iαn + αn→ (i+1)αn |
2nd and 3rd microcrystal accretion:
2αn
+3 αn→ 5αn
(134)
The accretion of the remaining microcrystals:
(i+1)αn
+5 αn→(i+6) αn
It follows from the scheme that crystals nucleate and start their
growth in a parallel way.
Crystalline accretion occurs under definite conditions only.
It is only the degree of crystalline competition development that
determines whether we get a coarse-grained or fine-grained casting structure as
a result. The more intensive competition development is, the farther this
process penetrates, the larger are the crystals that are obtained in the
structure of castings. Therefore, it is of practical importance to know how to
control the competitive crystallization process. To get a fine-grained casting
structure, the competitive mechanism is to be inhibited to hinder crystalline
accretion.
What determines the possibility or impossibility of crystalline
accretion?
Many dissimilar factors, external as well as internal, influence
this process.
However, the possibility of competitive crystalline accretion
process is determined in general by the presence of the solid-liquid zone, as
well as time and crystal contact conditions within the casting zone specified.
In many respects, time factor is the decisive one for the given process. Time
is required for the fitting of the adjacent hovering crystals structures. The
longer the time of crystals hovering within the solid-liquid zone is, the
larger crystals grow, the less their number in a casting is.
On the contrary, if the casting cools down fast, the time period
reserved for the matching of the adjacent crystalline structures diminishes,
they do not have time to accrete, forming independent crystals with the section
border of their own in the structure of the casting. In this case, the casting
has a fine-grained primary crystalline structure.
The operation of competitive crystallization and time factor account
for crystalline dimensions zonality in castings: metal cools down faster within
the ‘skin of a casting’ zone than in the center of the casting, thus, the
competitive process of crystalline accretion within the mentioned zone does not
go up to the end, so we obtain a fine-grained structure.
In the zone of columnar crystals, the width of the two-phase zone by
the crystallization front is small, and microcrystals hovering in this zone do
not have time to augment their dimensions being absorbed by the growing
columnar crystals and acting as the building material for them. The largest
among the hovering crystals cannot fit in with the growing columnar crystals
already, so they are forced toward the center of the casting.
The period of two-phase existence is maximal in the center of the
casting. Correspondingly, this is in the center of the casting that the most
favorable conditions for competition development and crystal accretion are
created, so there we obtain the structure with the coarsest grain.
Thus, the existence of the zone of large disoriented crystals in the
center of castings, typical of alloys and lacking in pure metals, is the
consequence of the competitive crystallization of crystals hovering within the
solid-liquid zone.
Mathematically, the dependence of competitive crystalline growth on
time can be expressed by the following correlation:
r =
k te,
(135)
where r
represents the maximal crystalline dimensions; k is the coefficient; te is the period of two-phase existence in the casting site where the
given crystal is located.
Expression (135) directly relates the dimensions of crystals in
castings to the period of crystallization.
The development of arborescent and other forms of crystalline growth
is well described in literature and therefore left out of consideration here.
The change in the volume of metals at melting and
crystallization is the traditional discussion subject in the theory of metals
in connection with the importance of the volumetric parameter for thermodynamic
constructions.
The change of metal volume at
crystallization is even more important for casting practice. The phenomenon
under analysis leads to the formation of shrinkage cavities and porosity in
castings.
The volume of systems, as well as their entropy,
enters into the main thermodynamic equations. However, if the entropy of metals
always increases at melting, which corresponds to general ideas and the data on
the disordering of matter at melting, the volume of metals, according to
experimental data, may either expand or sink at melting. The given
contradiction complicates the explanation of the change in the volume of metals
at melting.
As a result, none of all existent theories and models
of melting can offer any more or less acceptable theory of volume change at
melting, without touching upon the theories of crystallization.
Apart from general discourse that
the mechanism of melting includes intensive disordering through the formation
of simple, as well as cooperative, positional defects of the corpuscular
structure of matter /1/, there is no other achievement in the field signalized.
The phenomenon under consideration
takes on special practical significance, because the change of volume at
crystallization results in the so-called shrinkage of metals and alloys, the
change in casting and ingot dimensions, as well as the formation of shrinkage
strain, shrinkage cavities and porosity inside them, that affect directly the
quality of casting. If there exist several theories of diffusion and viscosity
of liquid metals, the mechanism and theory of castings shrinkage at
crystallization are totally lacking /74,75/.
So, practice suffers from the lack of theory to some
extent in the case given.
There is a general principle of approaching the change
of these or those properties of systems at the aggregation state transition in
the theory that is set forth. We termed it earlier as the relativity principle
of the forming of real systems properties.
Such an approach has been already applied above at the
founding of diffusion theory, fluidity theory, the theory of the change of
coordinating numbers, the theory of metal electrical resistivity at melting,
and a series of other questions.
The essence of this approach consists in the
following: we can calculate only the relative change of properties at the
aggregation state transition. It is to be done allowing for the changes in the
system’s structure at the level of aggregation states in the first place,
because the level of this or that property description must be adequate to the
property described.
It means that there is no point in finding explanations
to the changes occurring at melting at the levels distinct from that of
aggregation states. For instance, it is senseless to explain atomic structure
without referring to protons, neutrons amd electrons. It is senseless to try to
explain molecule structure without mentioning atoms, although it is an
incomplete description yet. But it is equally pointless to describe the
structure and properties of crystals limiting oneself to the ideas of electrons
and nucleons without considering the existence of atoms. It is pointless or
extremely difficult to describe ingot properties without referring to crystals,
etc.
I.e., for each level of real systems structure or
state, there exists its respective level of structural units that bear the
fundamental properties of the given state.
If we regard the change of properties that is caused
by the aggregation state transition, the adequate description of such a change
is to be carried out with the allowing for the transition from the structural
material and spatial units of one kind, inherent in the original state, to the
structural units of another kind, peculiar to the final state.
Real systems
possess a highly complicated hierarchical structure; they have many levels of
various elements of matter and space. Each hierarchical level imparts its
contribution to any property of the integral system, which the synergetic
science, which means ‘the science of joint action’, takes into consideration.
Unfortunately, the simplified viewpoint on the
structure and properties of real bodies and systems, explaining any properties
and their changes at the atomic-molecular level exclusively, is still widely
disseminated. It seems as if atoms and molecules that took us two millennia to
discover mesmerized the researchers. So, atoms and molecules – the elements of
matter, important yet positioned in a long row of hierarchical structures of
real bodies – come to be considered as the major, if not the sole, elements
that are responsible for all the properties of bodies and all the changes of
these properties. It is an error. Any hierarchical level of the structure of
real bodies is of no less importance than any other level, being major in the
prevalent state.
We should be fully aware of the limitations of the
specified principle. The level correspondence principle does not make it
possible to entirely describe this or that property, to give its absolute
value. It can only determine, including the quantitative aspect, the relative
contribution of the given level to the given property. I.e. such an approach
does not give any absolute values of properties, supplying their relative
values; it gives the quantity of their changes at the transition of the
system’s state, for example, the degree of volume change at the transition from
solid into liquid state. The relative change of volume, not volume proper, is
meant.
That is why the principle under analysis deserved its
label of the relativity principle of the states and properties of real systems.
For theoretical calculation, as well as the determination
of the absolute value of this or that property, we must know every element of
the hierarchical structure of the given system to determine the respective
contribution of each of them. The number of such levels of real bodies
structure is large enough, there being unexplored and unfamiliar ones.
Therefore, the stated synergetic problem was actually formulated here to date.
The absolute values of properties can be experimentally measured today in
certain cases only.
The expressed considerations, the given principle of
describing properties and their changes we shall also apply to the description
of metal volume changes at melting.
At the level of liquid aggregation state, the change
of any properties at the transition to the specified state is related, in the
first place, to the formation of the structural units of matter and space,
intrinsic in liquid state exclusively, i.e. clusters and intercluster spacings,
and their interplay.
The forming of intercluster spacings that have vacuum
properties is mainly responsible for the expansion of metal volume at melting,
and the given expansion will equal the overall volume of the elements of space
in liquid. Let us designate this factor as ΔVspl. The
value of the factor of ΔVspl was determined earlier by
expression (39) in Part 3.5.
ΔVspl = (3α / 2rc) 100%
Allowing for the relation between cluster radius rc
and the concentration of activated atoms on the basis of (45) as Ca
= 3/2 rc-1, we obtain for ΔVspl
ΔVspl = α Ca (136)
The convenience of the given
expression is conditioned by its compactness.
Aside from the factor of the
expansion of the volume of liquid at melting due to the formation of new
elements of space – intercluster spacings, – other factors operate in liquid,
which are related to clusters and able to cause self-compacting processes.
The forming of flickering
intercluster splits makes intercluster bonds unstable and flickering, too,
giving the opportunity of cluster displacing in liquid relative to one another.
As it was shown above, the possibility of such
displacements and the existence of splits account for the phenomena of fluidity
and mass transfer – diffusion in liquid state. The same cause influences the
change of volume concerning both its expansion and its sinking. As it was
demonstrated, the elements of space are responsible for the increase in volume,
and the volume that they occupy corresponds to the increase in the volume of
liquid. Those are clusters that account for the diminishing of the volume of
liquid.
The loosening of bonds between clusters provides the
possibility of their mutual displacement and re-granulation /129/. Under the
condition of free migration of compactly-shaped bodies relative to one another,
as we know, the granulation closest to the most compact one with the
coordinating number of 12 is reached. A compact packing of balls within any
capacitance at vibration serves as a familiar example to this.
Hence, two kinds of material elements granulation seem
to arise in liquid at different levels: 1. there remains inside clusters the
original atomic granulation peculiar to solid bodies; 2. there reappears mutual
cluster granulation. The in-cluster granulation of atoms seems to be enclosed
within the mutual granulation of clusters.
It is a good example of the hierarchy of real bodies
structure at different levels, which was discussed earlier.
If atomic granulation inside clusters has the same value
of its coordinating number as the compact mutual cluster granulation, then, the
re-granulation of clusters does not affect the volume of liquid. However, if
the in-cluster atomic granulation differs from the compact one, the
re-granulation of clusters will promote the compacting of liquid, so we are to
take it into consideration.
Let us designate the quantity of cluster
re-granulation factor as ΔVc. The quantity of ΔVc
can be determined on the basis of the following concepts. At the
re-granulation of clusters it is only their mutual position that changes, but
the in-cluster atomic granulation remains the same. We may reckon that only the
atoms located on the ‘surface’ of clusters participate in the forming of the
new, cluster granulation. They also take part in the granulation of atoms
inside clusters. Consequently, the quantity of ΔVc must
be proportionate to the relation of the number of atoms located on cluster
‘surface’ n, to the aggregate number of atoms in a cluster nc,
or
DVc = - kcompn’
/ 2 nc, (137)
where the coefficient of 2 allows for the fact that atoms located on
cluster ‘surface’ equally participate in the two mentioned types of
granulation inside liquid; kcomp is the coefficient of
compactness characterizing the change of volume at the transition from a
certain given granulation to a more compact one.
Out of expression (137), considering the value of Са from (46), we derive:
DVc = - kcomp
Са
(138)
On the basis of /101/ we obtain kcomp=
0.0; 0.06; 0.217; 0.4 for face-centered cubic, body-centered cubic, simple
cubic and cubic diamond granulation types correspondingly.
The aggregate relative change of
metal volume at melting and crystallization is found by the algebraic summation
of expressions (136) and (138). We obtain
DV = DVspl + DVc = a Са – kcomp Са = Са (a - kcomp)
If we express it as a percentage, as it is accepted,
we arrive at
DV = Са (a - kcomp)
100% (139)
Expression (139) relates the change in the volume of
metals at melting to the parameters of the elements of matter and space in
liquid – the width of spatial elements a and the factor of cluster re-granulation kcomp.
The values of DV found on the basis of (141) in
comparison with experimental data are listed in Table 17.
Table 17 The Change in the Volume of Metals at
Melting and Crystallization (Shrinkage)
Metal |
g, erg/sq.cm. /2,15, 20/ |
Е, kg/sq.mm. /10, 101/ |
a, calculation by (35) |
DVcomp,
%, calculation by (39) |
DVc, % calculation by (139) |
DV, % calculation by (139) |
DV, % experim. data /1,2,98/ |
Cu |
1133 |
11200 |
0.19 |
4.85 |
0 |
4.85 |
4.33-5.30 |
Ag |
927 |
7700 |
0.205 |
4.70 |
0 |
4.70 |
3.8-5.40 |
Au |
1350 |
11000 |
0.226 |
4.95 |
0 |
4.95 |
5.1-5.47 |
Pt |
1800 |
15400 |
0.205 |
5.7 |
0 |
5.7 |
no data |
Pd |
1500 |
11900 |
0.214 |
4.08 |
0 |
4.08 |
no data |
Al |
914 |
5500 |
0.24 |
5.30 |
0 |
5.30 |
6.0-7.14 |
Pb |
423 |
1820 |
0.26 |
4.15 |
0 |
4.15 |
3.5-3.56 |
Ni |
1825 |
21000 |
0.183 |
5.10 |
0 |
5.10 |
4.5-6.34 |
Co |
1890 |
21000 |
0.185 |
4.56 |
0 |
4.56 |
3.5-5.69 |
Zn |
770 |
13000 |
0.145 |
5.47 |
0 |
5.47 |
4.08-4.20 |
Feg |
1835 |
20000 |
0.177 |
4.84 |
0 |
4.84 |
2.8-3.58 |
Fed |
1835 |
13200 |
0.227 |
5.1 |
-1.64 |
3.46 |
2.8-3.58 |
Sn |
770 |
4150 |
0.248 |
3.3 |
-0.78 |
2.52 |
2.6-3.0 |
Cs |
68 |
175 |
0.27 |
4.3 |
-1.9 |
2.4 |
2.60 |
Ta |
2400 |
19000 |
0.21 |
3.46 |
-1.18 |
2.28 |
- |
Mo |
2250 |
35000 |
0.153 |
4.27 |
-2.04 |
2.23 |
- |
Nb |
1900 |
16000 |
0.204 |
3.93 |
-1.38 |
2.55 |
- |
W |
2300 |
35000 |
0.155 |
3.59 |
-1.59 |
2.0 |
- |
Bi |
3900 |
2550 |
0.207 |
3.7 |
-7.9 |
-4.2 |
-3.35 |
Ga |
735 |
- |
0.20 |
1.33 |
-2.82 |
-1.49 |
-3.2 |
Li |
377 |
1120 |
0.30 |
4.2 |
-2.1 |
2.1 |
1.65 |
Na |
171 |
530 |
0.45 |
4.6 |
-2.0 |
2.6 |
2.5 |
K |
91 |
460 |
0.38 |
4.0 |
-1.8 |
2.2 |
2.55 |
Rb |
754 |
235 |
0.31 |
4.7 |
-2.1 |
2.6 |
2.5 |
Mg |
728 |
2650 |
0.25 |
3.9 |
- |
3.9 |
3.05 |
As we see, the convergence of calculation and
experimental data is quite precise for a wide range of metals. We should note
that the entire data were obtained on the basis of theoretical assumptions for
the first time. It is for the first time, too, that the quantities of volume
change for ‘regular’, as well as the so-called anomalous metals – gallium and
bismuth – were calculated on a single basis. It was shown that their seemingly
anomalous behavior does not in the least differ from the behavior of all the
other metals in the aspect of volume change, obeying the same regularities.
In particular, the factor of
cluster re-granulation contributes much to the change of volume of anomalous
metals – bismuth, gallium, stibium, as well as silicon, water and some other
substances – at melting. In the indicated substances, the given factor prevails
over the factor of the forming of intercluster splits, which causes a visible
volume decrease at melting.
In practice, the change of metal volume at
crystallization that was calculated above leads to the forming of shrinkage
cavities and blisters in castings.
The process of forming shrinkage
cavities and blisters consists in the following: at crystallization, separate
submicroscopic intercluster splits – the elements of space in liquid state –
unite by the same cluster reactions scheme (19) as clusters accrete at melting.
The complete crystallization scheme if we allow for
the elements of space participating in it at the corresponding level is
presented as
(an + b) + (an + b) ® 2an + 2b;
(2an + 2b) + (an + b) ® 3an + 3b;
......................................................
(ian + ib) + (an + b) ® (i + 1)an + (i + 1)b,
(140)
where b is a single intercluster split (a single spatial element in liquid
metals); (i + 1)b is a shrinkage cavity or pore formed by the merger of (i + 1)
single elements of space.
At crystallization, the elements
of space can partially escape to the ambient space. It lowers the level of
liquid metal in a casting, yet no cavity formation takes place inside the
casting.
However, after the forming of a hard skin on the
surface of the casting, the evolving of the elements of space inside the
remaining amount of liquid metal results in the formation of hollows presented
as shrinkage cavities and porosity.
Since shrinkage cavities are formed by way of spatial
elements (vacuum) merger, they possess the characteristics of vacuum, too. It
can be proved by the well-known experimental fact that the pressure inside such
hollows equals zero at the moment of their formation.
Certainly, gas or air may fill such hollows, which
does not change the vacuum nature of the latter.
It is important that shrinkage cavities can be located
in castings both in the vicinity of their formation site and at a distance from
it.
The distribution of shrinkage cavities inside a
casting is the result of joint (synergetic) action of subsidiary factors, such
as the competitive crystallization character, redistribution of the remaining
liquid inside the casting under the influence of pressure differential,
capillary forces and gravity.
The competitive nature of crystallization leads to the
distribution of shrinkage cavities on the casting section surface, so that it
replicates the distribution of crystals to some extent. Namely, pores, similar
to crystals, have the minimal dimensions in the casting surface vicinity. The
dimensions of shrinkage cavities, similar to crystalline dimensions, increase
toward the center of a casting.
The same dependency as is employed to evaluate the
dimensions of crystals can be applied to the evaluation of the average
dimensions of shrinkage cavities on the casting section surface, i.e.:
rcav = kcav U-1,
(141)
where U is crystallization rate.
Under the influence of gravity the
last remaining portions of liquid lower down, whereas hollows are
correspondingly displaced upwards. So shrinkage cavities acquire their maximum
dimensions in the central part of castings.
Thus, we infer that the major cause of shrinkage
cavities formation in the process of crystallization is the process of the
merging of single intercluster elements of space in liquid.
The mentioned process as such pertains to natural laws
and cannot be eliminated.
Therefore, practical measures directed at increasing
the density of castings should perforce have a compensatory or displacement
character – if shrinkage cannot be eliminated as such, it may be compensated at
one location and displaced to some other safer site – which is the basis of
applying risers, coolers and a series of other techniques to casting
technology.
The issues of alloy formation are usually related to
the diagrams of state. Really, the diagrams of binary alloys state give a
considerable amount of information on the structure of solid alloys.
However, the given book brings the
unexplored problems of structure and crystallization of liquid metals and
alloys into its preferential focus.
Therefore, we shall apply here a somewhat different
approach to the problems of alloy formation proceeding from the structure of
liquid alloys and the mechanisms of melting and crystallization processes.
Melting and crystallization refer to everyday repeated
processes of foundry practice. At the same time, these are the basic structure-
and property-forming foundry processes. Cast alloys, their structure and
properties are formed during each smelting. The supplement of finishing
additions, alloying elements, ligatures and modifiers relate to everyday
routine foundry procedures.
Still, notwithstanding the mentioned ordinariness, the
mechanism of admixture dissolution, as well as the formation and structure of
alloys in liquid state and the order of the process specified, its adequate
description are lacking in literature, except for the most general
thermodynamic description. Thermodynamics, though giving a general
(phenomenological) description to this or that phenomenon at macrolevel, is not
to describe – and does not describe, by nature – the mechanism of the given
phenomenon at the level adequate to this process. Structure is no concern of
thermodynamics.
At the same time, it is extremely important to foundry
practice to know the structural mechanism of cast alloys formation process – to
effectively control these processes, uppermost.
Let us analyze the process of alloy formation
beginning from the dissolution of alloying and other elements proceeding from
the concepts of liquid metals structure stated above.
The process of alloy formation is rather complicated;
it includes several dissimilar mechanisms, or formation stages.
The process of the dissolution of
ligatures and other admixtures within the liquid alloy base marks the first
stage of alloy formation. By its nature, the given process is identical with
that of melting, being described by the same cluster reactions scheme.
However, the process of alloy formation is essentially
different from melting in relation to the solubility temperature in this or
that medium, in the first place. As a rule, the specified temperature is
considerably lower than the melting temperature of the given admixture in its
pure form. Apart from this, the so-called contact phenomena, as well as the
processes of mass transfer, perform a significant function in the processes of
admixture dissolution and alloy formation.
In particular, these are contact phenomena that cause
a change in the solubility temperature in comparison with the melting
temperature of the given substance. For the first time, the role of contact
phenomena in the process of eutectic formation was scrutinized in V.M. Zalkin’s
book /131/. A different conception of contact melting that allows for the
interaction of material and spatial elements in the process of melting is
suggested there.
These are contact phenomena that differentiate the
melting and crystallization of many alloys from the melting and crystallization
of pure metals, to a certain degree. Moreover, contact phenomena participate in
the melting of alloys by sequencing prior to the processes of mass transfer.
So, let us view first the mechanism of contact
phenomena influence upon the melting of alloys. The mechanisms of the influence
of mass transfer of various kinds upon alloy formation will be analyzed later.
Let us recall that there exist two intersecting
processes taking place at the rise of temperature in solid metals, and not only
there, that result in melting. On the one hand, it is the familiar phenomenon
of the decrease of durability of all metals with a temperature rise. On the
other hand, it is the increase in concentration and pressure of vacancy gas in
solid metals. The operation of this mechanism was described above.
To investigate the mechanism of alloy formation, it is
important to consider that both the factor of durability and the factor of
vacancy gas pressure change at the contact between two metals, first within the
contact zone of various materials and phases.
At the developing of alloys, there are two or several
various metals, alloys or ligatures that are melting together.
I.e. the contact between various metals in the process
of their formation and dissolution characterizes the process of alloy
development.
At the beginning, let us regard how vacancy gas
pressure changes at the borderline between two different metals, and trace
its influence on the temperature of melting.
Let us note that metals may exchange atoms at a close
contact under diffusion laws. This is a well-known phenomenon. In a similar
way, various metals exchange vacancies at a close contact.
Each metal has its own intrinsic vacancy concentration
at a given temperature. Let us admit that two metals - A and B –
are contacting. One of them has an equilibrium vacancy concentration of CA,
the other one having the concentration of CB. Let us presume
that CA exceeds CB.
As a result, a difference in the density of vacancy
gas dC generates along the border of their
section, which acts as the motive force for the onset of the diffusion vacancy
exchange process. Vacancies will flow from the metal with the greater density
of vacancies to the metal with the lower density. The rate of this process
equals the rate of corpuscular diffusion. As a result, vacancy concentrations
within the boundary zone come to be equalized.
However, the number of vacancies becomes less than the
equilibrium amount in metal A within the contact zone, whereas it
exceeds the equilibrium amount in metal B.
Evidently, the melting temperature
of metal B within the zone of contact decreases under such conditions
proportionate to the difference of vacancy concentration
dC = CA
- CB
Still, let us add that the temperature of the melting
of the second metal within the contact zone must be rising due to vacancy
redistribution, but it does not always occur in reality. There often decrease
the melting temperatures of both the contacting metals. The cause of this
phenomenon will be considered later when discussing the role of Rebinder’s effect in
contact melting.
The contact zone depth is not
great measuring several mcm or even less, - yet the given zone may get renewed
owing to the mass exchange within the contact zone.
Or, for instance, metals A and B forming
a binary alloy could form a fine byturn structure with the thickness of layers A
and B approximating the thickness in the contact zone of each of the
mentioned metals A and B. Such a structure could melt by the
contact mechanism within the entire melt volume.
The hypothesis of such a structure seems too
far-fetched at first sight. However, we know that this is the very structure to
be typical of many eutectic alloys – a fine microstructure with the alternation
of layers or the zones of other form of metals A and B, or solid
solutions a and b.
In point of fact, all alloys are inhomogeneous by
their microstructure to a certain extent, though the ideal conditions for
contact melting are created only in eutectic fine structures with the
alterntion of the zones of two different phases at microlevel.
The rate of metal dissolution in the process of
melting deserves our special attention. It is commonly known, having been
corroborated by hundreds of researches, that the mutual dissolution of metals
goes at the rate of corpuscular diffusion. One of the well-known methods of
measuring diffusion coefficient - the rotating disk method - is based on this
phenomenon.
These widely known facts are frequently used as the
proof of the corpuscular mechanism of the process of metal dissolution and
melting.
However, the alternative is neglected here - melting
by cluster mechanism requires vacancies. The latter move within the metal at
the corpuscular diffusion rate. So the diffusion rates of dissolution processes
do not in the least withhold these processes from going according to cluster
mechanism.
This is an extra example how synergetic principles
work in the processes of alloy formation.
Namely, the given example demonstrates once again that
dissipative processes actually go according to the suggested scheme
simultaneously at all the possible levels of the given system with the use of
all possible mechanisms.
In the specified case, the cluster process of metal
dissolution goes by the corpuscular mechanism of vacancy diffusion.
Unfortunately, the excessive generality of synergetic
principles hampers their direct application. For instance, we cannot exactly
determine the respective level of the course of this or that process for the
phenomenon under our consideration.
We have to admit that every particular phenomenon
requires the correspondingly particular study, - the finding of the structural
or any other kind of hierarchy of the given phenomenon, foremost, after the
completion of which synergetic principles are to be applied to the found
hierarchy.
Thus, synergetic principles can be generally applied
to practice to explain the already-found phenomena, which is also very
important, though.
At present let us analyze the influence of the durability
decrease factor within the zone of various metals contact upon the
formation of alloys and contact melting.
The so-called Rebinder’s effect is widely
acknowledged in physics. It consists in multiple decrease of solid metals
durability while they are contacting with liquid metals and some other liquids.
The same situation arises at the introduction of any
other additions into the crucible with liquid metal, which noticeably
facilitates mulling and the dissolution of any additions in the process of
alloy melting later on.
The mechanism of Rebinder’s effect is not unveiled so
far, but it is quite probable that it relates to moistening and solubility. We
may suppose that Rebinder’s
effect is connected with the diffusion of spatial
elements from liquid contacting substance into solid metal. The elements of
space of all kinds can diffuse in the same way as the elements of matter. Such
a diffusion of microhollows has been explored long since by the example of
vacancy diffusion.
Incorporating by way of diffusion into the surface
zone of solid metal, intercluster splits sharply depreciate the durability of
this zone, generating numerous flickering intercluster splits within it, which
act as microcracks reducing the durability of solid substance contact layer to
the durability of liquid, i.e. almost to zero. The sum durability of solid
metal decreases in this connection.
Consequently, in correspondence with Rebinder’s effect, the
durability of solid admixtures sharply decreases in foundry melting furnaces.
Whereas, in accordance with modern theory of melting, durability reduction
causes, in turn, the inevitable decrease in the melting temperature of the
given solid metal.
Thus, if the contact redistribution of vacancies can
lower the melting temperature of the given metal while increasing the melting
temperature of the other contacting metal, Rebinder’s effect reduces
to zero the possible melting temperature rise of the other metal. As a result,
the melting temperature of both the contacting metals may lower down within the
zone of their contact.
Now it is time to survey the process of contact
melting successively upon the whole.
Contact melting starts from vacancy redistribution and
the lowering of the melting temperature of the metal the vacancy concentration
of which increases as a result of such redistribution. The condition of such
vacancy redistribution is the initial perceptible difference in vacancy
concentration within contacting bodies.
However, after the forming of liquid phase the
Rebinder’s effect mechanism may start acting toward the metal that remains
solid. As a result, the durability of the given metal reduces on the contact
surface. Durability decrease in this metal, while vacancy gas pressure remains
the same, lowers its melting temperature within the contact zone according to
Rebinder’s effect /132/.
Finally, the melting temperature of such an alloy may
be less than the melting temperature of both its components.
We are familiar with such alloys - these are eutectic
alloys.
In alloys of other types - in monophase alloys, for
example - contact mechanism does not operate fully, so their melting
temperature is always higher than the melting temperature of the
low-melting-point component.
It is time to relegate the simplified and incorrect
concept of liquid alloys as a homogeneous atom mixture to the past. We know
that the structure of liquid and solid alloys is hereditarily bounded. We also
know that liquid metals have a microinhomogeneous cluster-vacuum structure,
where clusters are atomic microgroups with the proximate order similar to that
of solid state, whereas the elements of space are represented by intercluster
bond splits possessing the characteristics of vacuum.
At the same time, clusters are not
microcrystals, not the remainders of solid phase in liquid – these are the
structural elements of matter in liquid state.
It means that liquid alloys consist of clusters within
the entire temperature range of the existence of liquid – starting from the
melting temperature and ending in the temperature of evaporation. Except for
clusters, flickering intercluster splits perform the function tantamount to
that of clusters but qualitatively distinct from it.
Medley eclectic ideas on liquid metals and alloys
consisting of clusters and separate atoms at the same time are propagated in
scientific literature. In such a ‘raisin pudding’ structure, as A.Ubbelode
termed it, separate clusters seem to flow within a homogeneous mix of separate
atoms /1,2/.
Other authors assume that liquid metals and alloys
consist of clusters before the liquid reaches a certain temperature, after
which the given liquid passes into a purely corpuscular structure.
These approaches are erroneous, since they contradict
the quantum theory postulate of the indistinguishability of quantum objects,
atoms including. In application to liquid metals the specified postulate
signifies that all atoms must either enter into the composition of clusters or
the entire liquid must be monatomic. The simultaneous existence of various
atomic states is prohibited. The transition of liquid into monatomic state is impossible
in our theory, for clusters and only clusters are the prevalent elements of
matter in liquid.
The latent elements of the adjacent aggregation states
in liquid or any other aggregation state, as it was demonstrated above, do not
segregate from the predominant structural elements. For example, the latent
elements of matter in gaseous state – activated atoms – necessarily enter into
the composition of clusters. The latent elements of crystalline structure
represented by the proximate order also enter into cluster composition, as well
as vacancies.
I.e. the latent elements of matter and space do not
form phases of their own in any state.
Similar to that, iron in solid state may exist either
as ferrite or austenite, but it cannot exist in both the forms simultaneously
within a wide temperature range. There is the same prohibition acting here.
Thermodynamics furnishes an analogue of this
prohibition as the familiar Gibbs’ phase principle.
Since there exists proximate order in clusters, there
may occur its changes analogous to polymorphous transitions, if we touch upon
polymorphous transitions in liquid metals, but with the conservation of
clusters.
In the process of their formation, all alloys go through their
mixing stage that supersedes melting.
If a homogeneous substance is melting, there are formed clusters of
the same type.
If two or more substances or a composite monophase alloy are melting
or dissolving, clusters of various types get formed at melting. In liquid
state, clusters of various types are exposed to opposite forces.
Some of them aim at dividing heterogeneous clusters – these are
gravity and the forces of interaction between like clusters.
Other forces tend to uniformly intermix all clusters with the
forming of a homogeneous cluster mix. These are intermixing forces, including
the processes of natural and artificial convection, as well as corpuscular and
cluster diffusion.
Each of these forces has its specificities, its particular sphere of
influencing alloy formation.
Corpuscular diffusion prevails at short distances, for instance, at
the atomic exchange between clusters and the redistribution of atoms inside
clusters. The peculiarity of the process of corpuscular diffusion lies in the
relative slowness of this process. The typical value of the corpuscular
diffusion coefficient in liquid metals amounts to 10-7sq.cm / sec.
approximately (see the above part dealing with diffusion). This is quite
sufficient for the exchanging of atoms between neighboring clusters and inside
them.
However, it was shown above that a much faster cluster diffusion
mechanism that provides cluster intermixing operates in liquid metals, too. The
coefficient of cluster diffusion in liquid metals in the vicinity of the
melting point comes to 10-5 – 10-3 sq.cm /sec.
Cluster diffusion ensures the mixing and transfer of clusters within
relatively thin layers of liquid, for example, in contact layers and the narrow
zone directly by the crystallization front. Yet calculations show that cluster
diffusion cannot provide the homogenizing of alloy composition within the
entire volume of the melting crucible during a short melting period.
Macroscopic melts intermixing occurs due to natural and artificial
convection.
Alloys with the reciprocal solubility of their components go through
an extra formation stage – that of atomic diffusive mixing. At this stage, the
activated atoms of substance A penetrate into the clusters of substance B
and v.v. with the forming of mixed composition clusters AmBn.
At the formation of chemical compounds the alloy actually dissociates into two.
In contrast to the stages of melting, cluster and convection mixing,
the stage of atomic diffusive mixing does not affect all alloys.
For example, the given stage is atypical of the eutectic type
alloys, yet it is of extreme importance to alloys with the complete or partial
solubility of their components in solid state, or for alloys which incorporate
chemical compounds. For instance, this is the intercluster diffusion stage that
determines the possibility or impossibility of the removal of certain
admixtures present within clusters. It concerns many practical occurrences,
e.g. the case of extracting iron admixtures from aluminum alloys. Entering into
the cluster composition of a chemical intermetallic compound – ferrous
aluminide, the atoms of iron react with admixtures but sluggishly, so iron is
hard to extract from such alloys. It is necessary to decompose iron-containing
clusters to extract it, which is possible, for instance, at a considerable
overheating of alloys.
The forming of alloys with the maximum homogeneity should not be
carried out unless we allow for the stage of intercluster diffusion. It must be
taken into consideration that this is the slowest stage of alloy formation
process in liquid state that requires high temperatures.
Thus, the mechanism of alloy formation is complex enough, so the
structure of liquid alloys can be very complicated and multiform similarly to
that of solid alloys.
Upon the whole, we do not examine the stage of atomic diffusive
mixing here in detail, since the stage specified is reflected in well-known
works on alloy formation most fully and thoroughly.
The splitting of alloy formation process into four stages is of
course conventional in the sense that these processes can occur and do occur
simultaneously in the actual process of melting – but at dissimilar dimensional
levels. Nevertheless, such a distinction proves useful for the analysis and
understanding of cast alloys formation processes, their heredity, different
types of alloys and different types of diagrams of their state, as well as the
distinction between their characteristics, etc.
Natural and artificial convection provides the
homogenizing of alloy composition on a macroscale, on the scale of the smelting
furnace crucible, for example. This is a very powerful alloy formation
mechanism underestimated thus far.
The degree of the development of
regular gravity convection is proportionate to Rayleigh criterion and the cube
of the size of a sample, so convection works within the entire volume of
melting crucibles, ensuring the homogenization of alloy composition at
macrolevel. The larger the melting crucible is, the greater are convection
forces that operate within it. Thus, it is convection that ensures alloying in
production quantities.
However, convection may ensure alloy homogeneity at
macrolevel only. The homogeneity of alloy composition at microlevel is provided
by cluster diffusion, while corpuscular diffusion secures homogeneity at
lattice, inertcluster and, correspondingly, intercrystal level. Therefore, only
the cumulative simultaneous operation of various mechanisms provides the
forming of high-grade alloys. It also serves as an example of synergetic and
metallurgy laws working.
At the same time, special experiments have shown that
gravity effect essentially prevails over diffusion when convection is lacking.
It signifies that thin alloy samples under earth conditions tend to
stratification by the density of their components: heavy clusters lower down,
whereas those with the lesser density rise to the surface /82/.
So, if it was not for convection and intermixing, we
could never obtain more or less homogeneous alloys under earth conditions at
all. It also means that the decisive role in alloy formation processes belongs
to natural and artificial convection neglected till now as far as alloy formation
is concerned, rather than diffusion, as it is usually assumed.
Thus, under the influence of cluster diffusion and
convection of various kinds heterogeneous clusters intermix in liquid alloys,
while gravity hampers this process as much as possible.
No sooner does convection stop, than any alloy starts
segregating density-wise fast or slowly within gravitational field. In a series
of alloys with a considerable difference of the densities of their components
gravity effect is noticeable even when convection is present. For instance,
leaded bronzes segregate actively in liquid state in ordinary smelting
furnaces, which is well known to all practicists. Some other alloys behave
similarly to that.
We affirm here that all alloys without any exception
behave in the same way, yet segregation processes go very slowly sometimes so
natural convection successfully hinders them under regular melting conditions.
Therefore, we introduced the concept of thermal
kinetic processing of melts to be distinguished from thermal temporal and high
speed thermal treatment, which are based upon the using of time delay at a
definite temperature as well as the definite rate of the cooling of the melt
correspondingly.
Thermal kinetic processing includes the mentioned
factors, but the core of this process consists in using the preset modes of the
mixing of the melt in the process of its formation. Thermal kinetic processing
allows for both the intermixing degree and the degree of turbulence, as well as
the scope of turbulence at different dimensional levels of the melt. Thermal
kinetic processing lets obtain very homogeneous alloys the characteristics of
which prove to be highly stable.
As it was mentioned, gravity hampers the forming of homogeneous
alloys upon the whole and promotes their complete or partial segregation by the
density of their components.
Interestingly, the processes of alloy gravity or centrifugal forces
segregation may be normally applied to practice for the forming of castings
with the preset inhomogeneous structure. For example, there may be created a
smooth transition from the zone of gray iron to that of steel within one
casting at centrifuging in the process of slow hardening.
We got castings repeatedly from different alloys, silumins
including, with a stepless set of various structures and compositions according
to ingot height from hypoeutectic to deeply hypereutectic alloys by way of a
continuous holding of samples of originally homogeneous alloys within gravitational
field while convection is being specially suppressed.
It follows from the aforesaid, too, that the diagrams of alloys
state constitute quite a relative picture, so we are to accustom ourselves to
the fact that the whole set of all possible compositions and structures may be
simultaneously present in a casting under definite conditions.
Let us label such castings as variable structure castings. Apropos,
there are no physical prohibitions of getting variable structure castings. The
forming of castings with a controlled variable structure is possible even now,
yet the know-how of their production is to include the holding of a casting in
liquid state within gravitational field or the centrifugal force during the
period necessary for alloy segregation under the conditions of suppressed
convection.
The specified time period is always individual depending on a series
of conditions. Nevertheless, this is a real time amounting to minutes – or tens
of minutes at the utmost.
We should note that the production of insufficiently homogeneous
alloys is quite feasible if convection is underdeveloped. It is possible in
small-sized samples or in large furnaces, the inside temperature being too low
and the time of melting insufficient. Under such conditions the melt gets
inhomogeneous to a varying extent, ‘spotty’ by its composition and
submicrostructure.
In principle, the extent of such inhomogeneity spots within alloy
structure can be arbitrary. In practice, cast alloys and castings get ‘spotty’
very often – rather almost always – with the locally inhomogeneous structure
and properties. As a rule, it adversely affects casting characteristics.
For example, in cupola heat cast irons there may occur zones with
both their composition and structure varying because of the low temperature of
the process within the same casting. The process of segregation, as well as the
inhomogeneity of hardening, is usually listed among the causes of this
phenomenon.
Such homogeneity frequently acts in reality as a result of an
incomplete intermixing of alloy components in liquid state at cluster stage.
Such an alloy may be considered undersmelted.
The uncontrollable ‘spotty’ structure is one of the main causes of
inexplicable fluctuations of casting characteristics from melting to melting or
even within the same melting, which is familiar to practical experts. This
cannot be detected by regular methods of chemical or microstructure analysis.
Only the procedure of the micro-X-ray spectrometry analysis can be applied to
detect it at the corresponding dimensional level.
However, ‘spottiness’ may sometimes result not only in the worsening
but also in the betterment of certain service properties of metal in castings.
Therefore, the study of the conditions for the forming of local
inhomogeneity, or ‘spottiness’, of metal structure at different levels is one
of new trends in alloy research.
However, ‘spottiness’ may sometimes result not only in worsening but
also in the betterment of certain service properties of metal in castings.
Therefore, the study of the conditions for the forming of local
inhomogeneity, or ‘spottiness’, of metal structure at different levels is one
of new trends of alloy research.
The obtaining of controlled local inhomogeneity in alloy structure
is also possible even at present. In this connection we worked out the method
of the forming of cast alloys by cold emulsification. This method does not make
use of the regular ‘hot’ alloying while applying the emulsification of alloying
elements in the melt at low temperatures or in solid-liquid state. As a result,
there can be obtained some alloys with the controlled ‘spotty’ structure and
high mechanical qualities.
This is also a new trend of cast alloys production, which is probable
to find its expanded application in the coming century.
The above-said underlines the special significance of the observance
of technological thermal temporal and, particularly, thermal kinetic melting
mode in practice for the getting of stable homogeneous alloys, imparting the
physical meaning of a means of attaining such homogeneity into the mode
specified.
Alloy theory includes issues of practical importance
that abound in unsolved questions.
One of them concerns the causes of
alloys hardening within a temperature interval.
By way of illustration, let us consider an alloy with
the unrestricted solubility of its components in liquid and solid state, e.g.
copper-nickel. If such an alloy consists of a homogeneous mix of atoms in
liquid state and in solid state, why then these atoms separate intensively at
crystallization to later form a homogeneous solution anew?
The key to the problem analyzed relates to alloy
inhomogeneity in liquid state. There is a set of clusters with various
compositions present in the alloy. So, when crystallization sets in, clusters
containing a large amount of refractory element atoms get crystallized in the
first place, and v.v.
I.e. the existence of alloy hardening interval is a corollary
to the inhomogeneity of liquid alloy structure at cluster level, namely, the
simultaneous existence of clusters with dissimilar compositions in liquid
alloys. Clusters separate at crystallization, dividing on the basis of their
similarity or difference. Clusters that are similar by composition crystallize
together, under similar conditions, at the same temperature, in particular.
Thus, this is the existence of a set of variable
composition clusters within the temperature interval in liquid alloys that is
the primary cause and the motive force of selective crystallization process.
The continuous changing of cluster composition creates the effect of continuous
crystallization within the temperature range of liquidus-solidus. However, the
given process is discontinuous step-like at cluster level.
Under the condition if updated precision measurers are
applied, the given gradation of crystallization within the interval of
hardening can be detected. Even the measuring instruments that are currently in
use are able to register a spotty solid phase separation within the
crystallization interval of certain alloys. It denotes the possibility of the
existence of a discontinuous character of cluster composition change within the
liquid melt - up to its becoming step-like.
Alloys with the unrestricted solubility of elements in solid and
liquid state form a liquid structure consisting of a set of variable
composition clusters that represent the elements of matter, with a continuous
change of this composition in accordance with the diagram of state.
Correspondingly, such alloys have a definite set of transition elements both of
matter and space.
The structural formula of such
alloys structure is:
S = S a(am + bn),
(142)
where S is alloy composition; a and b are alloy components; m
and n are the variable portions of the atoms of a and b in clusters. The
quantities of m and n enter into the following correlation:
m + n = 1.
Corpuscular diffusion is highly
important in the formation of such alloys, along with cluster diffusion and
convection. It is corpuscular diffusion that provides the atomic interchange
between clusters which differ by their composition, as well as the penetration
of atoms inside clusters if there exist sufficient power of bonds between
heterogeneous atoms.
The existence of a stepless
cluster composition change and a smooth change in heat and solid phase evolving
within the interval of hardening is prognosticated in such melts.
On the one hand, small cluster dimensions facilitate
the penetration of admixture atoms into clusters; on the other hand,
neighboring order distortions and the corresponding submicrotensions always
arise in small clusters at the penetration of the atoms of other types, which
extrudes admixture atoms. Small cluster dimensions also promote a fast cluster
clearing of admixture atoms that create tensions inside clusters. We may say
that clusters are capable of self-purification from admixture atoms.
Therefore, as it was affirmed
above, admixture atoms can penetrate into clusters with a different composition
in liquid metals only with the presence of the sufficient physical affinity, or
at the forming of chemical bonds that are more durable than one-type atomic
bonds A-A or B-B.
In many cases admixture solubility in clusters is lower
than in a solid crystal. The increase of admixture solubility in liquid state
is achieved by admixture forming its own clusters, or through the locating of
admixture atoms on cluster ‘surface’, within the zone of activated atoms. The
given question was discussed earlier, too, in connection with admixture
diffusion.
Thus, admixture solubility in clusters cannot exceed
that in a solid crystal. Really, the solubility in solid state is rather a
complicated concept, too. It was demonstrated by many authors that admixtures
frequently concentrate along the boundaries of grains in solids, as well as at
dislocations and other defects. So the change of the average admixture
concentration in a solid crystal does not at all signify that the figure of
this change coincides with admixture solubility in the ideal crystalline
lattice of the given type.
Similarly to that, the aggregate solubility in liquid
state is also a composite quantity compounding of several parameters. The
presence of intercluster splits of vacuum nature in liquid alloys facilitates
corpuscular diffusion by the mechanism of activated atoms migration inside
them, while the huge internal area of the surface of spatial elements zone lets
a far greater amount of admixture atoms occupy cluster boundaries. It is one of
the causes of the increase of admixture solubility in liquid metals.
Thus, alloys with the restricted solubility of
elements in solid and liquid state possess a complex composition, where the
following three elements are necessarily present:
1.
clusters with the mixed inner composition of the
solid solution type a(am + bn);
2.
clusters with the composition of pure components
(or one of the components) aa and ab;
3.
clusters, ‘covered’ or ‘separated’ by individual
admixture atoms or monatomic admixture layers. For example, v. the clusters of
one of the original components aa, ‘covered’ with individual activated admixture atoms b: b(aa)b.
Let us conditionally label such clusters as clad.
Let us emphasize that admixture atoms b also enter
into cluster composition aa in the latter case, but with a
peculiar location on cluster ‘surface’, analogously to the arrangement of
certain admixtures along the boundaries of mosaic blocs and other structural
defects in solid metals. In liquid metals, in connection with a huge amount of
spatial elements in their structure, clusters formed in such a way may
constitute a considerable part of their total quantity.
The structural formula of the alloys of such a type is
presented as:
S = S { aa + ab + a(am +bn) + b(aa )b} (143)
Such alloys are the most
complicated by the composition of their elements of matter and space. Both the
step-like and continuous change of cluster composition is possible within them.
Correspondingly, the zones of more or less continuous heat evolving and solid
phase alternate with the areas of singularity - the departure form the
continuous course of hardening – within the hardening interval of such alloys.
The problems of the forming of
liquid eutectic alloys were broached earlier in Part 7.2. It was demonstrated
that contact phenomena play a significant part in their formation. Still,
contact phenomena affect but some of the parameters of alloy melting and crystallization
without essentially changing their structure.
Liquid eutectics refer to alloys
that do not possess mutual solubility in solid state, or have the restricted
solubility in solid state and the unrestricted solubility in liquid state.
They are characterized by the eutectic melting
temperature below the temperature of melting of both the basic components of a
binary alloy. In solid state, dispersed microstructure is inherent in
eutectics, where the dispersed elements of metals A and B alternate according
to a definite order.
In this connection, solid eutectics have been
classified long since as peculiar mixes. K.P.Bunin and his followers refer
liquid eutectics to mixes, too /59-61/.
From the viewpoint of the theory under development,
liquid eutectics differ from other alloys only by the extreme degree of cluster
composition inhomogeneity in liquid state. If we observe a continuous change of
cluster composition in other liquid alloys or a multi-step fractional change of
such a composition, liquid eutectics may have but two steps of cluster
composition at the utmost.
Liquid eutectics represent cluster mixes of the
elements of A and B or their solid solutions, where the interaction between the
like clusters AA and BB is higher than the interelement interaction
AB. It means that clusters A and B mix reluctantly, not
spontaneously.
The latest experiments have also shown that many, or
possibly even all the liquid eutectics, are unstable when convection is lacking
and tend to segregation by the original elements within gravitational field
(see below).
For the production of liquid eutectic mixes of
heterogeneous clusters that interact reluctantly, a certain amount of energy is
to be spent.
Usually natural or artificial convection suffices for
the forming of such mixes. Corpuscular diffusion does not actually participate
in the forming of liquid eutectics, since they are lacking in the atomic
exchange between clusters A and B.
The structural formula of liquid eutectics for the
case of the complete lack of mutual solubility in liquid state is
S = S {aa + ab}, (144)
where aa and ab are the respective clusters consisting of the atoms of A or B
elements only.
Such a formula of structure is
incomplete in the sense that it does not reflect the bonds between the clusters
of different types. Such bonds must be flickering by liquid state nature, i.e.
they must arise at the approximation of the neighboring dissimilar clusters,
splitting at their separation in the process of heat oscillations.
Such bonds must be of intermetallic nature without the
forming of permanent-type intermetallic compounds.
Evidently, more detailed researches of liquid
eutectics can supply more extensive data on the flickering intermetallic bonds
of such a type. Similar bonds exist in solid eutectics, too, as stable bonds
between the elements of their microstructure, yet these bonds remain
underexplored so far.
At the same time, the presence of the flickering
AB-type bonds in liquid eutectics acts as the factor securing their relative
stability and technical utilization possibility. At the complete lack or
weakness of such bonds the alloy simply segregates into two liquids.
The existence of such relatively weak bonds in liquid
eutectics between dissimilar clusters facilitates the displacement or shift of
these dissimilar clusters relative to one another. It accounts for the widely
known fact of a higher fluidity of eutectic alloys in comparison with other
alloy types.
So liquid eutectic alloys have a high fluidity due to
the heterogeneous cluster bonds that they contain being weaker than the bonds
between similar clusters.
We may say that this is the instability of liquid
eutectics that imparts a higher fluidity to them.
Liquid alloys with a peritectic structure have a
structure similar to eutectics. The structural formula of the former coincides
with formula (145) with the distinction that there can be several peritectics
in the same system. Correspondingly, there can be several cluster types in liquid
peritectics. In this case, the structural formula of liquid peritectics may be
presented as:
S = S {aa + aab + ab}.
(145)
The dissimilarity between
eutectics and peritectics also consists in the different influence of contact
phenomena at the melting and crystallization of these two alloy types. For
instance, the difference in vacancy concentration between the components of
peritectics is considerably lower than it is between the constituents of eutectics,
and Rebinder’s effect works for one component only. As a result, the effect of
the melting temperature lowering is not so appreciable to the components of
peritectics as it is in eutectics, and it acts relative to only one alloy
component.
In eutectics, contact phenomena
perform the leading role, which is probably caused by the highest possible
difference in vacancy concentration at the melting points of metals
constituting the given eutectic, as well as the considerable quantity of
Rebinder’s effect in these metal vapors.
Eutectics differ by far from the source metals by a
series of parameters. In particular, the paramount distinction of eutectics is
the lowering of their melting temperature as compared with that of the alloy
components.
The improvement of important foundry properties such
as shrinkage diminution and fluidity increase is also characteristic of many
eutectics. I.e. the properties of eutectics change nonadditively to the content
of the elements of A and B within them. Eutectics, i.e. the mixes of two
different substances, behave as a certain new substance by a series of basic
parameters. It is caused by various factors.
Among the structural causes of these significant
changes of eutectic alloys properties at the level of the elements of liquid
state we may and are to single out the factor of cluster re-granulation in
liquid eutectics after their formation. The mentioned factor was viewed above
when analyzing the mechanism of metal volume change at melting and
crystallization.
Let us consider the role of the factor of cluster
re-granulation at the forming of eutectics. Eutectics represent a mix of two
types of clusters different by their composition. In turn, it means that the
dimensions and shape of the two given cluster types, as well as the dimensions
and shape of the spatial elements of the original liquid metals A and B are
different, too.
It is known that the mixes consisting of particles of
different dimensions can fill space more compactly than particles, e.g. balls,
of similar dimensions. To achieve this, the balls of dissimilar dimensions are
to occupy definite positions in space, by way of alternating according to a
definite pattern, for instance. Or smaller particles may fill the spacings
between larger ones.
We also know that such a distribution results, for
instance, from intermixing. In this case, a certain mutual configuration in
space is attained with the minimal volume of spatial elements, which explains
shrinkage diminution in eutectics at subsequent hardening.
Certainly, this is not a rigid construction, so it may
decompose under certain conditions, for example, at segregating within
gravitational field. We can state that the structure of liquid eutectics does
not only get formed but is also sustained owing to convection to a considerable
extent.
At the convective mixing of heterogeneous clusters A
and B the shape and dimensions of clusters do not change. It is only the
shape and dimensions of the spatial elements of liquid state that are subject
to changes, i.e. the shape and dimensions of intercluster splits.
Such a seemingly negligible change turns out to be
sufficient for a relatively unstable mix of two different substances to acquire
the properties of a certain third substance.
All that was stated above concerning the role of
cluster re-granulation and the role of the change of spatial elements –
intercluster splits, – relates, though to a different extent, to the forming of
alloys of other types, but the mentioned factors affect eutectics most.
The alloys of the iron-carbon system - steels and
cast irons - refer to prevailing industrial alloys. Cast irons are most widely
applied to foundry.
Thus, the study of liquid cast
irons structure in connection with the processes of crystallization and
structure formation seems worth making.
Iron-carbon alloys pertain to the alloys with the
restricted solubility of their elements in solid state and not quite definite
mutual solubility of their components in liquid state. Such indefiniteness is
caused by the fact that there were no successful attempts at deriving the alloy
of iron with carbon with the content of carbon higher than 25% at. in
connection with the necessity of obtaining a stable and precisely measurable
temperature for researches carried out at temperatures above 20000C.
However, this is not the sole speciality of such
alloys.
Iron-carbon alloys are distinguished by the feature
that one of their components - carbon - does not melt at all in its free form
and does not form liquid phase, thus presenting quite a rare, though not the
only one, exception among the elements of the periodic system /119-120/.
From the viewpoint of the melting theory that was
developed above, carbon in its most stable graphite form does not melt, because
its durability does not decrease with the rise of temperature, as it does in
case of the overwhelming majority of elements, but even increases to a certain
degree. The decrease of durability, as it was noted earlier, is one of the
requisite factors pre-starting melting.
Apart from this, solid-state carbon does not dissolve
within itself the elements of any other kind, practically. Rather a restricted
number of elements that form limitary carbon solutions, including iron, are
known.
At the same time, carbon readily reacts with many
elements, which results in carbide forming.
Carbon does not completely dissolve iron within itself
either, yet iron has a limited dissolvability area with carbon and also forms a
series of carbides with it, among which cementite usually occurs in cast iron.
The system of iron-carbon also refers to the alloys of
the eutectic type, i.e. it is characterized by a high degree of cluster
composition inhomogeneity in liquid state and a high degree of the irregularity
of heat and crystallizable phases evolving within the interval of hardening.
In the processes of the melting and crystallization of
cast iron, as well as it is in other alloys of the eutectic type, a significant
role belongs to contact phenomena – the diffusive vacancy redistribution at the
contact between iron and carbon and Rebinder’s effect, in the first place.
Let us consider the process of the
contact melting of iron in detail.
We shall proceed from the familiar fact that carbon
does not dissolve iron within itself. Consequently, the exchange of substance
between iron and carbon is one-way during the contact – it is only carbon that
can penetrate into iron, while iron cannot penetrate into solid carbon.
However, such prohibition does not work in case of the
exchange between the elements of space. Vacancies, or intercluster splits,
being flickering and having neither a stable shape nor stable dimensions, are
very plastic and easily adjustable to any substance. So the exchange of spatial
elements is also possible for the elements that do not exchange their elements
of matter.
Liquid iron, or liquid austenite, contacting with
solid graphite inclusions, serves as the source of all the elements of space
existing within carbon – vacancies, as well as intercluster splits.
Vacancies generate the inner vacancy gas pressure
within the contact zone of carbon, while intercluster splits diminish the
durability of the given layer. As a result, a typical cluster-melting situation
arises in the contact layer of carbon.
Carbon in the contact layer (within it only) melts
forcedly under the stated conditions with the forming of clusters and
intercluster splits of its own, which mix with the clusters of iron, or
austenite. Intercluster splits that exist between carbon clusters differ from
intercluster splits in liquid austenite. As a result, a counter exchange
between material and spatial elements takes place, now in the direction from
carbon to iron. Rebinder’s effect starts working relative to iron, too. Its
melting temperature also lowers down.
The re-granulation of clusters into a mutual mix
system occurs simultaneously, so the volume of the mix decreases. As a result,
the liquid eutectic mix shrinkage diminishes at subsequent crystallization.
There arise flickering bonds between the clusters of
iron and carbon that turn out to be weaker than iron-iron and carbon-carbon
bonds.
The decrease in the power of bonds within the system
leads to the reduction of internal friction (viscosity) and the increase in
cast iron fluidity in comparison with that of liquid iron.
The suggested description of the melting of cast iron
and the forming of its structure in liquid state, as well as any other description,
neither claims for exhaustiveness nor for the consideration of all the factors
that are possible in this connection. It qualitatively reflects only the
relative contribution of the material and spatial elements of liquid state to
some properties of liquid cast iron. Such an approach meets the accepted
relativity principle in the description of the changes concerning the
characteristics of metals and alloys at the level of material and spatial
elements interaction at aggregation state transitions.
The structure of liquid cast iron possesses all the properties of
liquid eutectics structure, having peculiarities of its own, though.
In particular, numerous experimental results of X-ray as well as
sedimentation tests bring the authors to the conclusion that carbon occurs in
liquid iron not only in the solution of clusters with the neighboring order
structure similar to that of austenite, but also as clusters with the
dimensions of 2.7…4.9nm at the temperatures approximating the temperature of
cast iron liquidus /16,17,30,37,55, 59,133,134/.
X-ray tests corroborate the presence of the compound Fe3C
in liquid cast iron, too.
There is a contradiction consisting in Fe3C being
unstable at high temperatures: at any sufficiently prolonged holding at
elevated temperatures in solid state it will inevitably disintegrate into
ferrite and graphite or austenite and graphite.
In principle, such disintegration must go considerably faster and
more completely than in solid state owing to the accelerated mass exchange
processes, but it does not take place.
It was demonstrated earlier that the flickering bonds between
heterogeneous clusters inevitably generate in liquid eutectics, and they are
weaker than the bonds between like clusters.
Therefore, we may assert that the compound Fe3C is
present within liquid cast iron as the flickering interatomic bonds between
austenite and graphite clusters. Graphite and cementite coexist simultaneously
in liquid state, yet graphite exists in cluster form that is stable for liquid
state, whereas Fe3C is present only as the flickering bonds between
graphite and austenite clusters, constantly arising and disappearing /140/.
It was stated above that such flickering bonds between heterogeneous
clusters are in principle characteristic of all liquid eutectics, as well as of
any alloys generally. The specificity of iron-carbon alloys consists in the
relative durability of the compound of such a type and the possibility of the
growth of such bonds at fast crystallization or at the presence of
carbide-stabilizing elements in the alloy.
It seems relevant to underline for further research that a close
contact between the clusters of austenite and graphite promotes the forming of
such bonds, while the formation of splits or the separation of dissimilar
clusters, on the contrary, hinders the formation of the given type of bonds.
The structural formula of liquid alloys of the iron-carbon system
may be presented as the follows /140/:
S =
S
{naa + mag}, (146)
where aa denotes austenite-like clusters; ag are graphite clusters; n is the number of austenite clusters
per unit of volume or gram-atom of liquid alloy; m is the number of graphite
clusters within the same volume.
The given expression relates both to liquid cast iron and liquid
steel. The difference lies only in the quantitative correlation between the
clusters of the two types.
Formula (146), like other similar formulas, does not allow for the
existence of spatial elements in liquid alloys alongside with the elements of
matter. It also ignores the presence of flickering bonds between similar and
dissimilar clusters.
Considering the special importance of the bonds of Fe3C
type in liquid cast irons, we can supplement formula (146) with the scheme of
cluster interaction:
S =
S {naa ®Fe3C ¬mag},
(147)
We are to take the instability of Fe3C bonds into our
consideration, - such bonds are unstable, have a flickering nature and
alternate with intercluster splits in time. This is reflected by the following
scheme:
aa ®Fe3C ¬ag + t/2,
aa « ag + t,
aa ®Fe3C ¬ag + 3t/2, (148)
......................... etc.,
where the symbol
of « represents
an intercluster split, t is the duration of one period of heat oscillations of a cluster.
Actually the time period of t, as it was already demonstrated, means the duration of the
existence of flickering intercluster bonds and flickering elements of space –
intercluster splits «.
The succession of cluster reactions (148) does not reflect the
structural formula of liquid cast iron upon the whole but the sequence of the
flickering bonds of the Fe3C type alternate between the clusters of
austenite and graphite and the flickering intercluster splits «.
Various alloying additions (Si, Mn, etc.) and undesirable admixtures
(S, P, etc.), as well as a series of uncontrollable admixtures, occur in the
structure of real cast irons.
In the structural aspect, some of them do not form clusters of their
own in liquid cast iron (Si, Mn and other elements) but enter into the
composition of austenite clusters. Other elements exist in the form of special
clusters (iron sulphide, phosphide eutectic and others), then, the third group
of elements may exist as activated atoms on the ‘surface’ of ‘clad’ clusters of
austenite or graphite (the same sulphur and phosphorus in small
concentrations).
Melting and
crystallization, as it was shown above, are partially reversible processes, so
these are mainly the processes inverse to melting that constitute
crystallization.
Contact processes perform a significant role at the crystallization
of liquid cast iron, the same as it is at melting.
Austenite clusters perform the function of the leading phase at the
crystallization of hypoeutectic cast irons and steels. They are the first to
form solid phase by the regular cluster scheme of crystallization. The
crystalline surface of solid austenite serves as the vacancy sink area for
graphite clusters contacting with this surface.
As a result of selective crystallization, graphite clusters are
forced back piling up at the boundaries of growing austenite crystals with the
forming of agglomerations. When vacancy concentration in graphite clusters
becomes lower than critical as a result of vacancy sink from graphite to
austenite clusters, graphite clusters start crystallizing, too, within the interdendritic
austenite spacings, as a rule, which reflects the weighty part of graphite
crystallization in the given system.
The accretion between graphite clusters is also accompanied by the
accretion of the elements of space characteristic of liquid state –
intercluster splits. While intergrowing, intercluster splits form shrinkage
microhollows at the boundaries of the growing crystals of graphite.
Such a scheme is peculiar to the regular relatively slow cast iron
crystallization with the forming of the structure of gray cast iron. Its
distinctive feature is the separation of clusters and intercluster spacings of
austenite and carbon in space in time.
Separation occurs on the basis of ‘like to like’ principle, for the
energy of flickering bonds between like clusters is higher than that between
dissimilar ones. Correspondingly, the flickering bonds of the Fe3C
type are replaced at the separation of clusters by iron-iron or carbon-carbon
bonds. However, the specified process requires time. If there is enough time,
graphite clusters have time to meet, accrete and separate from the surrounding
austenite by a vacuum layer that generates from the joined intercluster splits.
Therefore, the opinion that graphite performs the role of vacuum in
cast iron is incorrect. The bonds of iron-carbon actually disappear and get
replaced by microhollows at the forming of graphite insertions in cast iron at
crystallization.
That is the way of forming gray iron microstructure.
If there is not enough time to cluster separation or selective
crystallization, if crystallization goes too fast for that, then cluster mix
get crystallized as mix proper. There occurs no cluster separation. Such is the
basic distinction of the formation of white cast iron at the level of the
elements of matter and space (clusters and intercluster splits).
At crystallization, as it was pointed out earlier, clusters accrete
and their heat oscillations stop. If separation is lacking, both similar and
dissimilar clusters accrete forcedly, the clusters of austenite and graphite,
in particular. In this case, the flickering bonds between austenite and
graphite clusters of the Fe3C type become stable, since flickers
stop at crystallization.
As a result, an extremely nonequilibrium structure of alternating
graphite and austenite microzones is formed. The nonequilibrium of the derived
structure partially withdraws due to corpuscular diffusion, the redistribution
of carbon atoms.
Considering the extreme smallness of cluster dimensions – 1-10nm,
the process of diffusive redistribution of carbon goes during an extremely
short period of time (fractions of a second), so it can be caught by special
hardening experiments only with the cooling rate of millions of degrees per
second within the temperature interval from the temperature of melting to room
temperature. Such experiments are known, and there was registered the presence
of carbon microzones within the microstructure of iron hardened from its liquid
state.
Thus, white cast iron is originally crystallized as a mix of
austenite and graphite clusters. The original cementite generates right after
fast crystallization and not from liquid state but in solid state already due
to the fast redistribution of carbon atoms from the clusters of graphite into
the surrounding austenite.
Intercluster bonds of the Fe3C type registered in the
process of fast crystallization function as the nuclei of a new phase
–cementite - in this process, they accelerate and organize new phase growth
according to their pattern.
This is how the microstructure of white cast iron arises.
As we see, the same original structure of a liquid iron-carbon alloy
can generate structures differing in a drastic way as a result of
crystallization going at different rates.
The natural
crystalline structure of castings is distinguished by the pronounced
inhomogeneity of the dimensions of primary crystals along the section of a
casting. If no special measures are taken, there arise within the majority of
castings crystals that are not only dissimilar, but also too large by their
dimensions.
It promotes the generating of other kinds of inhomogeneities -
physical and chemical - in castings represented by shrinkage, segregation, etc.
The properties of castings differ within the zones of dissimilar
structures, too. The highest properties, homogeneous along the section of
castings, are normally obtained at the forming of a homogeneous and
fine-grained structure.
Moreover, we know that the smaller the dimensions of primary
crystals in castings are, the higher are a series of important service and
technological casting properties.
Therefore, most often casters aim at the forming of the
finest-grained and the most homogeneous casting structure.
Modifying is one among the most widespread means of attaining this
object. Casters understand modifying as the insertion of small quantities of
various additions into liquid metal before crystallization to achieve a
fine-grained structure of castings /65,74,75,135/.
What are the given additions requisite for?
As it was shown above, there is always a large number of nucleation
centers in castings and ingots – much more than the amount of crystals in a
final casting – owing to the mass nature of crystalline centers nucleation by a
spontaneous mechanism.
However, the structure of castings turns out to be coarse-grained
and inhomogeneous along the section of castings.
The cause of the zoning of crystalline castings structure, as it was
demonstrated, consists in the competitive character of crystal growing from the
melt with the two-phase zone being present.
If there is enough time for structure correlation, grosser crystals
absorb smaller ones, so the structure of a casting is gradually becoming more
and more coarse-grained as a result, while crystallization rate is decreasing.
Crystallization rate regularly decelerates in the direction from the surface of
the casting to its center, which brings about the zoning of the crystalline
structure of castings.
In accordance with current theory, spontaneous crystallization is
actually impossible, crystalline centers nucleation entails much difficulty,
the number of these centers is always insufficient, and so the leading role in
the crystallization of castings belongs to special additions – modifiers,
requisite for the multiplying of nucleation centers.
Our theory asserts that the process of spontaneous crystallization
in liquid metals goes naturally and without extra difficulties. The number of
nucleation centers is always redundant, exceeding by far the amount of crystals
in a casting. Thus, the role of modifiers is different, according to our
theory.
First, let us consider the issues that are general to both the new
and old modifying theory.
We also regard modifying as the introducing of additions that refine
grain dimensions in castings. Such additions are termed modifiers.
Correspondingly, the additions that reduce the number of grains in castings and
augment their dimensions are called demodifiers.
Modifiers are divided into modifiers of the first type dissoluble to
a different extent in the metallic base of the liquid alloy of the addition.
Modifiers of the second type are represented by the particles of refractory
substances insoluble in liquid alloy (at least during the process of
crystallization). The given theses remain constant in this theory.
New modifying theory provisions are stated below with the
consideration of the real material-spatial structure of liquid metals.
Let us analyze the operation of first-type modifiers from the
standpoint of thermodynamics.
The existent theory of modifying is based upon the thesis of work
expenditure necessity for the nucleation and growth of crystals. The
incorrectness of the suggested thesis was proved earlier in Part 6.
It was shown in Part 6.2 that flickering inner intercluster surfaces
saturate liquid. At the elementary act of crystallization by the reaction of an + an®
a2n two neighboring clusters accrete
into an elementary crystal and the section surface represented as a flickering
intercluster split closes between them.
Consequently, at crystallization going by cluster accretion these
are not only new surfaces that arise in liquid, but also the existent
flickering intercluster section surfaces that close, which is accompanied by
the evolving of crystallization heat, and not its expenditure, in complete
conformity with facts.
Then the change in the free energy of the system at the forming of
an elementary crystal (nucleation center) at spontaneous crystallization
according to (126) amounts to:
DF = -(4/3)pr3 DFv - 4pr2 s,
(149)
Graph (149) is visualized by curve 2 in Fig.17.
The latter expression signifies that energy is evolved but not
consumed at the forming of a nucleation center. Correspondingly, it also means
that the formation of nucleation centers does not require any work to be done
but, on the contrary, crystallization is thermodynamically expedient at any
crystalline dimensions.
As a result, fundamental changes in crystallization theory imply the
changes in the theory of modifying.
It is known that first- type modifiers refer to surface-active
substances that lower the surface tension s of liquid melt. The lowering of s is an experimental fact
/65,74,75,135/.
Let us denote the surface tension of liquid metal or alloy without
modifiers as s. Let us designate the surface tension of alloy with modifiers as sм.
By definition, s
>sм by the
absolute quantity.
Expression (126) in case if we introduce some modifier will be
presented as
DFм = -(4/3)pr3 DFv - 4pr2sм,
(150)
Subtracting (150) from (148), we obtain DFм - DF = - 4pr2 sм +4pr2 s
or DF - DFм = 4pr2
(s-sм) > 0 by the absolute quantity.
Consequently, the decrease of the free energy of the system lowers
by the absolute quantity at the crystallization of metals that contain
modifiers.
It ensues that the growth of crystals with first- type modifiers is
less thermodynamically expedient than spontaneous crystal growth.
Next, it follows that first- type modifiers hinder and retard the
nucleation and growth of crystals in comparison with spontaneous nucleation and
growth.
On the contrary, demodifiers increase the quantity of s, which results in the
facilitating and accelerating of crystalline growth.
Graphically, the influence of modifiers and demodifiers of the first
type upon the change of free energy at crystallization can be represented by
the three curves in Fig.18.
Curve 1 corresponds to the process of spontaneous crystallization;
curve 2 conforms to the process of crystallization with modifiers, curve 3
reflecting the nucleation and growth of crystals with the presence of
demodifiers.
What is the mechanism of the influence of modifiers upon crystalline
dimensions in the light of the above-said?
Modifiers, by hindering crystalline growth, hamper the process of
competitive crystallization, too, i.e. the process of small crystals accreting
with larger ones. A large number of small crystals that nucleated by the scheme
of spontaneous crystallization accrete with more difficulty, grow slower and
get a chance to survive in competitive activity with larger crystals. As a result,
a fine-grained primary crystalline structure is registered in a casting at the
same crystallization rate.
It also means that forced crystallization with modifiers does not
replace spontaneous crystallization, as it is assumed now. On the contrary, forced
crystallization with modifiers of the first type is less thermodynamically
expedient, less equilibrium than spontaneous crystallization. First- type
modifiers do not in the least facilitate the formation and growth of crystals,
as it is accepted. They hamper these processes, on the contrary. Still, by
hampering spontaneous crystallization processes, first- type modifiers provide
the refining of the primary crystalline structure of castings.
Such is the general thermodynamic mechanism of the influence of
first- type modifiers upon the dimensions of primary crystals in castings. We
shall underline that the specified general considerations do not reflect the
entire diversity of modifying. Therefore, the thermodynamic theory of modifying
is to be supplemented by other means at other levels.
The fundamentals
of the electron theory of modifying were laid by G.V.Samsonov,
V.K.Grigorovitch, Khoudokormov, Tiller and Takahashi, as well as others
/136,137/.
G.V.Samsonov worked out the concept of the donor-acceptor mechanism
of modifier and matrix interaction. Khoudokormov and Grigorovitch /138/
developed the ideas of the role of the bond type and the electron structure of
matter in the aspects of interaction and modifying. The given concepts are
widely employed and developed at present.
We shall assume after G.V.Samsonov that good modifiers are to be
free electron donors for liquid metal.
The ability of this or that substance to act as a free electron
donor in alloys is always relative, i.e. it is determined by the comparison
with the metal of a casting.
Work function /136/, electronegativity, after Gordy, or the relative
ionization potential, after V.M.Vozdvizhensky /139/, may characterize the
ability of the given substance to donate free electrons.
Our research showed that the two latter parameters characterize the
modifying ability in an approximately equal degree; however, the application of
the effective ionization potential proves more convenient in practice, after
V.M.Vozdvizhensky /139/.
All substances having a lesser quantity of electronegativity, or the
effective ionization potential Uef, than the metallic base of
the given alloy, will have a more or less modifying influence at
crystallization, i.e. they will deflate crystalline dimensions.
All substances having the quantity of Uef that
exceeds that of the metallic base of the alloy, will have a demodifying
influence at crystallization, i.e. they will promote the enlarging of the
primary crystalline structure.
It relates to the following specificity: the lower the ionization
potential quantity is, the easier it is for the substance to donate its valence
electrons, and v.v.
The degree of the modifying influence of this or that element can be
evaluated by the sign of the difference between the effective ionization matrix
and modifier potentials:Ume – Umod
If the given difference is above zero, i.e. positive, then the
specified element can act as a modifier. If this difference is below zero, the
element under consideration will be a demodifier of the first type. I.e.
Ume
– Umod > 0 – a modifier,
Ume
– Umod < 0 – a demodifier.
The second factor that characterizes the ability of some substance
to affect nucleation and the growth of crystals is the factor of admixture
solubility in the given matrix. A good modifier must locate along the
boundaries of crystals and clusters without entering into their composition.
I.e. a modifier or demodifier is to form clad clusters, where modifier atoms
are distributed between clusters.
A modifier is not to form clusters of its own, because a certain
modifier amount will not be located along cluster boundaries of the melt in
this case.
Correspondingly, the element possessing modifier characteristics
must have a low solubility in solid metal and a restricted solubility in liquid
metal.
Let us denote the factor of solubility as CS. We
shall conditionally assume that modifiers have the solubility in a hard matrix
of this or that alloy that does not exceed one percent: CS<1%.
Both the noted factors can be united in the following semi-empirical
formula for the calculation of the modifying activity of modifiers
(demodifiers) of the first type:
m = (Ume –
Umod)/ СS, (151)
m being the coefficient of modifying activity.
Expression (151) is very simple and convenient for calculations. The
quantities of CS are listed in reference books concerning diagrams
of state, the quantities of U are cited in literature, too /139/.
Expression (151) is also convenient for the reason that it allows to
clearly divide all elements into first- type modifiers or demodifiers. Namely, m < 0 for
demodifiers in accordance with (151), i.e. their modifying coefficient value is
subzero, while modifiers will have a plus value of the modifying coefficient.
The quantity of m has but a relative value according to (151) and serves for the
comparing of the modifying coefficients of various elements exclusively.
The values of the coefficient of m for various first- type modifiers and
demodifiers for liquid alloys on the basis of iron and aluminium are to be
found in Tables 18 and 19.
Table 18. The Coefficient of Modifying Activity for
Various Elements in Liquid Iron-Based Alloys (Modifiers of the First Type)
Element |
CS, % at. /119,120/ |
Umod, /139/ |
m, calculation by (150) |
First-Type
Demodifiers |
|||
Fe |
- |
3.00 |
0 |
Co |
50 |
3.11 |
-2.2 10-1
|
Ni |
50 |
3.20 |
-4.0 10-1
|
Ir |
50 |
3.26 |
-5.2 10-1
|
Pt |
50 |
3.34 |
-6.8 10-1 |
Pd |
50 |
3.45 |
-9.0 10-1 |
Mn |
50 |
3.66 |
-1.3 |
Ru |
29.5 |
3.45 |
-1.5 |
Zn |
7.00 |
3.17 |
-2.4 |
Re |
16.7 |
3.57 |
-3.4 |
Cr |
12 |
3.47 |
-3.9 |
Al |
1.55 |
3.14 |
-9.0 |
Mo |
1.60 |
3.29 |
-1.8 101
|
Ge |
4.00 |
3.27 |
-1.9 101 |
Si |
4.20 |
3.84 |
-2.0 101 |
C |
8.60 |
4.86 |
-2.1 101 |
Nb |
1.90 |
3.42 |
-2.2 101 |
Sn |
1.00 |
3.31 |
-3.1 101 |
V |
1.60 |
3.71 |
-4.4 101 |
Ta |
0.95 |
3.44 |
-4.5 101 |
W |
1.00 |
3.81 |
-8.1 101 |
P |
0.25 |
4.30 |
-5.2 102 |
O |
0.56 |
5.0-6.0 |
-5.0-8.0 102 |
S |
0.11 |
4.76 |
-1.6 103 |
F |
<1.0 10-4 |
>5.0 |
-1.0 (104-
105) |
Cl |
<1.0 10-4 |
>5.0 |
-1.0 (104-
105) |
Br |
<1.0 10-4 |
>4.0 |
-1.0 (103-
104) |
J |
<1.0 10-4 |
>4.0 |
-1.0 (103-
104) |
|
|
|
|
First-Type
Modifiers |
|||
|
|
|
|
Fe |
- |
3.0 |
0 |
Rh |
50 |
2.91 |
0.9 |
Cu |
7.5 |
2.56 |
5.9 |
Ti |
0.72 |
2.85 |
0.21 |
Zr |
0.5 |
2.87 |
26 |
Gd |
2.0 |
2.38 |
31 |
La |
0.2 |
2.15 |
420 |
Ce |
4.0 10-2 |
2.25 |
1900 |
Mg |
~0.01 |
2.42 |
~3000 |
Ca |
<0.02 |
1.86 |
>2000 |
Na |
<0.001 |
1.34 |
>10000 |
B |
~0.001 |
1.44 |
~1000 |
Sr |
~0.001 |
1.64 |
~1000 |
Y |
~0.001 |
2.30 |
~700 |
Pr |
~0.001 |
2.24 |
~700 |
Sc |
~0.001 |
2.57 |
~400 |
The data listed in Table 18 coincide upon the whole with available
practical data on the modifying (demodifying) ability of these or those
admixtures in iron-based alloys.
Thus, alkali-earth and rare-earth metals have been rightly used long
since as modificators at the casting of steel and cast iron.
According to the data supplied by Table 18, all elements may be
divided into three groups by the degree of their modifying activity in
iron-based alloys /147/.
1.
The elements that do not practically affect
crystallization have the following coefficient of m: m = 0-10.
2.
The elements that influence crystallization to a
minor degree possess the coefficient m = 10-100.
4.
Strong modifiers have the coefficient m >100.
The following elements refer to strong modifiers by the order of the
increase of their modifying ability:
Sc, La, Y, Pr, Sr, Ba, Ce, Ca, Mg, Na. The data on the modifying activity
of metals and elements in iron-based alloys are presented in Fig.19.
Out of this series it is only sodium that is not applied to the
modifying of steel because of the extreme volatility of the former. At its
introduction into liquid steel or liquid iron, sodium evaporates almost
instantly, so the amount of sodium atoms in the structure of the melt is not
enough to detect the effect of sodium at crystallization.
The data on the demodifying activity of elements are little used in
practice. The demodifier elements are practically used only in the cases when a
single-crystal or a coarse-grained directional structure is to be formed. Thus,
sulphur is specially introduced into the composition of magnetohard alloys
while growing cast single-crystal magnets. Phosphorus is used to obtain the
maximum overcooling at the forming of amorphous metals.
It is of high practical importance that
modifiers and demodifiers have a different sign of the coefficient of m.
It means that modifiers and demodifiers counteract in alloys as far
as their influence on crystallization is concerned. We know it from practice
that steel and cast iron contaminated by sulphur, phosphorus and oxygen
actually resist modifying.
However, the majority of modifiers forms compounds with sulphur and
oxygen that are insoluble in liquid steel. Therefore, the larger part of
modifiers at their introduction into the melt is spent on the neutralization of
the demodifying effect of detrimental and other impurities, including the means
of bounding the given impurities into insoluble compounds, and not on attaining
the modifying effect.
However, deoxidation and desulphurizing concern only the first part
of modifier-demodifier interaction. Unremovable demodifiers, like phosphorus,
remain in the alloy. Some weak demodificators, like carbon and silicon, are the
essential components of iron alloys. They are not to be removed, - their
influence can be but neutralized. So the second part of modifier-demodifier
interaction consists in the neutralization of the influence of demodifiers
simply due to the quantitative dominance of the modifying effect of modifiers.
Modifying proper becomes possible only after the neutralization of
demodifiers.
Therefore, a considerably larger amount of modifiers than is
requisite for modifying proper must be introduced into melts.
Table 19. The Coefficient of Modifying Activity of
Various Elements in Aluminum-Based Alloys (Modifiers and Demodifiers of the
First Type)
Element |
CS,
% at. /119,120/ |
Umod,
/139/ |
m, calculation by (150) |
First-Type
Demodifiers |
|||
Al |
- |
3.14 |
0 |
Zn |
50.0 |
3.17 |
-0.06 |
Ge |
2.80 |
3.77 |
-22 |
Si |
1.65 |
3.84 |
-42 |
Ti |
0.28 |
3.27 |
-46 |
Rh |
0.29 |
3.37 |
-79 |
Bi |
0.20 |
3.42 |
-140 |
Re |
0.26 |
3.57 |
-160 |
Sn |
0.10 |
3.31 |
-170 |
Mo |
0.07 |
3.29 |
-210 |
Be |
0.10 |
3.40 |
-260 |
B |
0.44 |
4.47 |
-300 |
C |
0.08 |
4.86 |
-2100 |
Sb |
0.05 |
3.87 |
-3600 |
|
|
|
|
First-Type
Modifiers |
|||
|
|
|
|
Ga |
9.50 |
3.12 |
0.21 |
Cr |
0.44 |
3.04 |
23 |
Cu |
2.5 |
2.56 |
23 |
Mg |
18.9 |
2.42 |
38 |
Mn |
1.46 |
3.06 |
55 |
Y |
0.80 |
2.30 |
100 |
Co |
0.02 |
3.11 |
150 |
Hf |
0.18 |
2.78 |
200 |
Au |
0.70 |
1.00 |
220 |
Cd |
0.11 |
2.89 |
230 |
Ca |
0.40 |
1.86 |
320 |
Ba |
0.40 |
1.44 |
420 |
Na |
0.10 |
1.34 |
1700 |
Ce |
0.05 |
2.23 |
1800 |
La |
0.05 |
2.15 |
2000 |
Nd |
0.04 |
2.27 |
2200 |
In |
0.04 |
2.05 |
2250 |
Sr |
<0.01 |
1.64 |
15000 |
According to the data in Table 19,
sodium, cerium, lanthanum, neodymium, indium, and strontium refer to the
strongest modifiers for aluminium.
Sodium and strontium are most frequently applied in practice. Fig.20
graphically presents the data on the properties of modifiers in aluminium.
The point is that in practice we are to consider not only the
modifying ability of this or that substance, but also its cost, accessibility,
as well as the availability of its forms convenient for the introduction into
the melt.
Practice shows that there exists a certain optimal amount of each
modifier, at the introduction of which into the melt the given modifier affects
the process of crystallization to the maximum extent. It merits our attention
that this amount usually approximates 0.1% mas. of the acting substance for the
greater part of the most widely-used modifiers of the first type.
The larger modifier quantity, as it was demonstrated, is spent on
the oxidation and neutralization of the influence of demodifiers. The amount of
modifiers approximately equal to their residual content within the melt is
spent for the attaining of the modifying effect proper, i.e. for the refinement
of the dimensions of primary crystals.
Taking into consideration that clusters in liquid metals at the
temperature of melting contain 1000 atoms on the average, we have to state that
one modifier atom falls at one to ten clusters on the average in liquid metal
before crystallization. Such admixture amount under the condition of its
regular distribution within the volume of metal can hardly influence the
process of crystallization.
On the other hand, we know that first-type modifiers are not located
regularly within the volume of metal but concentrate on any section surfaces,
inner ones including. Not only do they concentrate on these surfaces, but
stabilize them, sometimes even bringing about the increase of the area of the
surface of the liquid as foam.
Since there are enough inner section surfaces represented by the
elements of space in liquid metals, the atoms of the majority of modifiers
migrate along these section surfaces, like particles of gas, changing and
stabilizing the spatial constituent of the melt to a certain extent, augmenting
the volume of the given part of the system.
In the meantime, the density of liquid alloy decreases, which may be
proved experimentally.
Undoubtedly, these are not all the modifiers of the first type that
behave similarly to gas molecules in alloys. Still, such a modifying mechanism,
let us term it as gas-like, is characteristic of many most frequently used
modifiers. Such is the mechanism of the modifying effect of sodium in silumins,
magnesium and rare-earth metals in steel and cast iron. The specified mechanism
is possible owing to the comparatively low vaporization temperature of such
modifiers and their low solubility in cast iron and steel.
This side of the modifying mechanism of a series of main modifiers
should also be allowed for in practice together with the thermodynamic and
electron factors that we touched upon earlier.
It is the mentioned property of modifiers which, together with their
electron characteristics, enables such small quantities of modifiers to affect
the process of crystallization to such an appreciable degree.
These are the expanded interlayers of the elements of space along
the boundaries of growing crystals that hinder the joining of new clusters or
other crystals to them, thus retarding crystalline growth.
However, the same modifier property to stabilize and cause the
expansion of the elements of space in melts may bring about a totally different
effect at the increase of modifier amount over its optimal quantity.
Namely, the redundant amount of modifier may even lead to foam
formation in melts, as we know. It means that the volume of the elements of
space increases excessively. The melt becomes gas-like, frothing easily.
In this case, the modifying effect is disguised with unfavorable
after-effects of modifier redundancy. Such an effect is termed overmodifying.
In turn, the lack of modifier results in the fact that the entire
modifier amount introduced is wholly spent on the suppression of the activity
of demodifiers, deoxidation, desulphurizing and other chemical reactions,
removing the modifier out of the given solution, so we observe the lack of
substance for modifying proper. Such an effect is termed undermodifying.
Therefore, the optimal amount of modifiers is usually determined at
the level of 0.1% of the acting substance content to be adjusted in the process
of operation.
Since any melt contains a combination of controllable and
uncontrollable demodifiers of its own, it seems very hard, and often
impossible, to suppress the demodifying effect of the whole set of the known
and unknown admixtures with the help of only one modifier.
It is caused by
the circumstance that dissimilar substances have a different degree of their
chemical affinity, they interact in a different way or turn out to be inert
toward each other. In this connection, the so-called complex modifying is
widely used during the last decades, when two or several types of substances –
not the sole modifier – that possess a certain modifying activity are
introduced into the melt. Such a complex provides a much more complete blocking
of the negative influence of the demodifiers present in the melt.
As a result, the effect achieved at modifying comes to be stronger
and more stable.
It was stated in Part 9.1 that the introduction of modifiers causes
the increase of the free energy of the system and the decrease of its
thermodynamic stability.
The system tends to re-establish its equilibrium through the
removing of modifiers.
After the introduction of modifiers into
the melt their quantity always diminishes in the course of time and dependent
on temperature.
Correspondingly, the modifying effect is unstable by nature.
Practically, the specified effect reaches its maximum right after modifier
introduction and subsides in the course of time.
There are several reasons for the decrease of the concentration of
modifiers in the melt with time. One of the major causes is modifier
vaporization, since the larger part of modifiers is represented by substances
that boil relatively easily and exist in the thermodynamically unstable gaseous
state in the melt.
Secondly, there come modifier losses caused by chemical reactions
with alloy components and atmospheric gases, oxygen in the first place.
The higher the temperature of the melt is, the faster modifier
vaporization runs.
The typical
dependency of the modifying effect on time for liquid steel is shown in Fig.21.
The quantity of the modifying effect was measured here by the number of primary
crystals in steel per 1ccm of section area. Titanium nitride particles were
used as modifier. Time keeping started from the moment of the introduction of
the nitride-forming element into liquid steel. The modifying of steel by
titanium nitrides has the most appreciable effect on the refining of the
primary crystalline grain in steel.
Any familiar modifiers preserve a more or less noticeable effect on
the refining of the primary crystalline structure in liquid steel during
approx. 10 minutes from the moment of introduction up to the onset of
solidification of the casting. It is therefore assumed that steel is hard to
modify. In-mold process, or modifying in a mold, is considered practically
effective for steel castings.
For liquid cast iron, the effect of modifying lasts longer, up to 30
min. Thus, both in-ladle modifying and inmold process are possible as regards
cast iron.
Each of these processes has its advantages and disadvantages.
In-ladle modifying requires a greater modifier consumption, approx. 10-20%
more. However, this process allows holding the metal after modifier
introduction and lets emerge the products of the reaction between modifiers and
alloy components. As a result, the metal gets purer. On the other hand, the
modifying effect lowers to a certain extent during the period of in-ladle
holding (10-15 minutes), as well as the content of the modifier in the melt.
In-mold process allows saving the modifier attaining the maximum
modifying effect, but all the products of side reactions of modifiers with
alloy components remain within the casting. The metal becomes contaminated. It
does not matter at times – while forming low sort ductile cast iron. However,
the in-mold process effect may turn out to be unfavorable for the obtaining of
high-quality cast iron, or the source cast iron of high purity concerning
admixtures should be used.
In case of the modifying of aluminum alloys by sodium and strontium,
the effect of modifying is considerably more stable and longer lasting than it
is at the modifying of steel and cast iron by any known modifiers.
It can be explained by the low melting temperature of aluminum
alloys and a close aluminium affinity toward oxygen, which partially protects
modifiers in liquid aluminum alloys from oxidation. For the mentioned reasons,
the effect of modifying in aluminum alloys lasts for many hours and can be
noticed even after re-melting.
In steel and cast iron the modifying effect after the re-melting of
modified castings is not observed.
If this or that alloy forms two or more phases at crystallization,
the modifying effect may be attained selectively, in principle, by the refining
of this or that phase. Even at complex modifying different phases react to
modifying in a different way.
There are enough examples of selective modifying in practice.
For instance, the most widely used gray cast iron modifying by
ferrosilicon is a typical example of selective modifying. The point is that
silicon refines the dimensions of graphite crystals in the main having
practically no influence upon the structure of the metallic matrix of cast
iron.
It is of special interest that cast iron contains the redundant
amount of silicon without modifying – 2% approx.
Consequently, it is not silicon proper that renders the modifying
effect but its form in the melt.
At dissolution, ferrosilicon goes through all the stages of alloy
formation that were mentioned before. It forms clusters of its own, which are
first diffused within the volume of cast iron, and it is only then, during the
process of corpuscular diffusion, that silicon passes into the composition of
austenite clusters.
Silicon clusters, as long as they exist, act as a good substrate for
the formation and growth of graphite crystals, because silicon and graphite are
close analogues in Mendeleev’s periodic law. Their properties and structure are
affined enough to realize the principle of structural-dimensional
correspondence.
I.e. silicon is actually the selective
modifier of the second type for graphite in cast iron.
It merits our particular attention that ferrosilicon acts
short-time, although silicon does not burn out in liquid cast iron. It also
stresses that the particles of silicon, capable of acting as crystallization
centers for graphite, are short-lived, disappearing gradually.
There exist other examples of selective modifying.
For instance, we found a strong selective modifying effect of
silumin (11% Si) by the particles of titanium carbide TiC.
Silumin microstructures with different quantities of TiC particles
are shown in Fig.22.
The dark particles of titanium carbide cause the singling out of the
eutectic and the active dendrite growth of primary aluminum crystals (the white
a-phase
in the picture). Eutectic decomposition also leads to the growth of large
silicon crystals (light-gray particles of crystalline cut).
As a result of such modifier influence, the eutectic may be actually
destroyed, which is clearly observed in the second picture. It is interesting
that alloy microstructure simultaneously combines the features of both the
hypoeutectic (a-phase) and hypereutectic (free silicon) alloy.
As a rule, the problem of the nucleation of solid phase on the
surface of solid materials that prove to be insoluble in the melt is solved by
considering a special case of nucleation upon a flat substrate of unlimited
dimensions. In practice, the given case corresponds to nucleation on the
surface of contacting with the mold. This is quite an important particular
case.
However, solid phase in modifying practice must nucleate on the
surface of refractory dispersion particles that hover in liquid alloy. It is
the means of influencing the process of crystallization within the entire
casting volume.
So let us view the process of nucleation and the growth of solid phase
crystals on the surface of dispersion particles, wholly dipped in liquid.
The statement of the problem runs as follows.
Let us assume that there is a solid particle in the melt. The
specified particle possesses the characteristics of the modifier of the second
type, i.e. the nucleation of solid phase crystals occurs on its surface and the
particle itself may be regarded as a crystallization center.
In order to simplify the problem, let us presume that the particle
is spherical with the radius of rp. Solid phase existing as
shell 2 with the external radius of r (v. Fig.23) is formed on the
surface of particle 1 with the radius of rp under certain
conditions.
If a particle does not have the characteristics of a second-type
modifier, the shell of solid phase is not formed under the same conditions.
Let us consider a thermodynamic problem of the interaction between
the particle and the melt. Depending on the properties of the particle and its
interaction with the melt, there forms solid phase on the inclusion surface (r>rp)
– or it does not (r=rp).
The larger the quantity of r is, other parameters being equal, the
more effective the given modifier proves.
It is obvious that the quantity of r depends on both the dimensions
and characteristics of the particle and the properties and temperature of the
melt, as well as the correlation between the parameters of the particle and the
melt.
As a first approximation, let us denominate the given correlation as
the function of
r=f
(rp). (152)
V. the solution to the problem.
Such a formation (particle-solid shell-melt) will be stable only in
the case when the free energy of the system decreases monotone or when the
dependency of DF = f (r) has the minimum.
Thus, let us find and explore the extremum area for the function of
the change in the system’s free energy DF = f (r) at the forming or melting of the solid phase shell on the surface
of a solid foreign particle in the melt near the melting point, aiming at the
determining the fundamental possibility of solid phase formation on the surface
of the particles – second-type modifiers in liquid alloys.
In its general form, the change of the system’s free energy while
forming a solid shell on the particle surface will be the same as it is at the
formation of a new phase center at spontaneous crystallization:
DF = S DFv
+ S DFs,
(153)
where DFv is the change of the
volumetric free energy of the system; DFs is the change of the surface free energy of the system.
We shall presume that the particle and the solid phase shell on its
surface are spherical.
The change in volumetric free energy at the forming or melting of
the shell equals
S DFv
= (4/3)p(r3 – rp3) DFv.
(154)
The change of the system’s free surface energy at the formation of
solid phase on the solid inclusion surface is
S DFv
= Sp sp – S s,
(155)
where Sp
is the particle surface area; S is the area of the external surface of a solid
shell; sp and s represent the specific
interphase energy on the section surface of particle-liquid phase and solid
phase-liquid phase correspondingly.
The minus in front of the second term in equation (155), the same as
is observed earlier, has a physical meaning and signifies that there are not
new surfaces that get formed at the formation of solid phase in microinhomogeneous
liquid, but intercluster surfaces – the elements of space existent in liquid –
that are closed. There is no work expenditure for the formation of surface S;
on the contrary, during its formation there evolves energy in the system as the
latent crystallization heat.
Thus, as it was demonstrated earlier when analyzing the nucleation
of crystallization centers, equation (155) allows for the real structure of
liquid that consists of the elements of matter and space.
Taking into consideration the spherical shape of particles and solid
phase, out of (155) we derive:
S DFs
= 4pr2 sp – 4pr2 s.
(156)
By inserting the values of S DFv and S DFs from (153) and (155) into
(152), we obtain:
DF = (4/3)p(r3
– rp3) DFv + 4pr2
sp – 4pr2
s. (157)
Expression (157) is a linear dependency of the type DF = f
(r).
Such dependencies may be monotone, or they can have bending points.
We are interested in the corroboration of the existence or the absence of the
minimum on the curve DF=f(r). If the mentioned minimum exists,
then, the formation of solid phase is thermodynamically expedient, and v.v.
To find the minimum, it is necessary to test function (154) for the
existence of extremum and determine the nature of the given extremum further
on, if it does exist.
Let us take into consideration that the quantity of DF, according to (157), is the function of two variables DF = f (r,
rp). However, this is but the particle
radius of rp that acts as the independent argument in a
physical sense, since the radius of the solid phase shell on the particle
surface depends on rp in its turn, or r = j (rp).
If we consider that, let us find the first derivative DF = f (r,
rp) to determine the extremum existence,
and equate it with zero. When the function of one variable (the function of DF on our case) is defined as U = f (x, y), where y = rp
= j (x), the chain rule of differentiation
as applied to our case lets derive the following formula:
DF¢ = DF¢rp + DF¢r r¢rp = 0 (158)
Let us differentiate (157) by the chain rule scheme (158) and equate
the first derivative with zero. Thus
4pr2DFv
(dr/drp) - 4pr2DFv - 8prs (dr/drp)
+ 8prpsp = 0.
Having completed the requisite cancellations, we obtain
DFv r2
(dr/drp) - DFv rp2 - 2rs (dr/drp)
+ 2sp rp =
0. (159)
For further analysis, we are to know the mode of the function r =
j (rp). As a first
approximation, the correlation between the quantities of r and rp
can be expressed by a linear dependency presented as r = A (rp), where
A is a certain constant.
It is obvious that only the values of A>1 have a
physical sense.
It is known that a small section of any even curve can be approximated
by a line segment. So the expression of r = A (rp) is quite
acceptable.
In this case dr/drp = А, and equation (159) assumes the form
А r2 DFv - 2А s r - DFv rp2 + 2sp rp =
0 (160)
This is an equation of the second order relative to r. It is
known that if the order of the first derivative is even, then, the function
under analysis has the extremum.
Now it is time to test the nature of the extremum for the
availability of the minimum.
Let us express s and sp through the independent argument
of rp.
Referring to B.Chalmers /65/, we obtain
sp = rp DFvp
/2
(161)
s = rp DFv
/2 (162)
r = rp is used in expression (161) to simplify the problem, because the
shell radius may assume any values r ³ rp under the conditions of the problem.
Apart from that, the usage of the variable quantity of r is
inexpedient, for (160) does not have any solution relative to r in this case.
By inserting the values of sp and s from (161) and
(162) into (160), we obtain:
А DFv r2
- А DFv r rp - DFv rp2
+ DFvp rp2 = 0.
or
А DFv r2
- А DFv r rp + rp2 (DFvp - DFv) =
0. (163)
As modifiers of the first type, dispersion particles of various
refractory particles, the melting temperature of which is considerably higher
than the temperature of modifying. In this case, the following inequality DFvp >> DFv takes place.
Consequently, we can assume without any considerable error that DFvp - DFv @ DFvp.
Then expression (163) will assume the form of
А DFv r2
- А DFv rp r + rp2 DFvp =
0 (164)
Equation (164) as related to r is a regular quadratic
equation of the type ax2+bx+c=0, where а = А DFv; b = - А DFv rp; c = rp2
DFvp .
The solution of the given expression is presented as:
r1,2 = rp
{(1/2) + [(1/4) - (DFvp /А DFv)]1/2} (165)
Now we can determine the nature of the extremum of the function of DF = f(r). Its flexion is
DF² = 2А DFv r - А DFv rp . (166)
Since there exists a flexion, it is relevant to continue the
research concerning the existence of the maximum or the minimum of the
function.
To achieve this aim, we are to determine the sign of the flexion DF². To do so, let us introduce the value of r1 from
(165) into (166). Thus
DF² = 2А DFv rp {(1/2) + [(1/4) - (DFvp
/ А DFv)]1/2} - А DFv rp (167)
Since DFvp < 0 for refractory substances under
modifying conditions, then we have the sum and the value in braces in
expression (167) positive and >1. Consequently, the first term in the right side of (167)
2А DFv rp {(1/2)
+ [(1/4) - (DFvp / А DFv)]1/2}
is more than the
second one А DFv rp .
Consequently, DF² > 0.
Since the value of the flexion is
positive, then the function of DF = f(r) has the minimum if r = r1.
The given conclusion is of fundamental importance. It means that
there exists the possibility of forming the solid phase shell of limited
dimensions on the particles of refractory modifiers of the second type in melts
at the temperature that is somewhat higher than the melting temperature. The
radius of such a solid phase zone id determined by expression (165).
This is an unexpected and convincing conclusion. It was accepted
earlier that solid phase cannot exist at the temperature that is higher than
the temperature of melting. The analysis that we carried out shows that solid
phase can be formed within limited zones in the vicinity of the surfaces of
strong modifiers of the second type and exist at the temperature that exceeds
the melting temperature of the given metal or alloy. Such zones cannot grow if
the dimensions exceed the quantity of r. The graph of function (157) under the
specified conditions has the form represented in Fig.24.
The picture shows that the nuclei of solid phase that were formed in
liquid metal or alloy on the surface of modifiers of the second type, remain in
a sort of potential well and have strictly specified dimensions. Such
formations are lacking in either growth or decomposition tendency under
constant conditions. However, if the conditions change, the dimensions of such
formations change, too. For example, at the cooling of the melt such
microcrystals will be boundedly growing to a new equilibium value of r,
diminishing till zero at heating. The temperature of the zero value of r can be
calculated on the basis of the listed expressions.
This is the first time when we predict the possibility of the
existence of equilibrium though small crystals of solid phase in liquid metal
at the temperature that is higher than the temperature of melting. Such an
inference does not contradict the general theses of thermodynamics about the
impossibility of coexistence of solid and liquid phases within the same
temperature interval, since we have got a more complicated case here when three
phases – solid, liquid and that of modifier particles – coexist. In the
particular case viewed above, the interaction between the three given phases
may create conditions for a specific form of the coexistence of solid and
liquid phases within the limited temperature interval.
In case if the melt cools down to the temperature of
crystallization, the microcrystals arisen as a result of the interaction
between the three phases will be growing without bound.
Certainly, such solid phase areas that are limited by dimensions get
the advantage over smaller spontaneously nucleating crystals in their
competitive activity at the cooling of the given liquid down to the temperature
of melting. On the other hand, the formations under consideration are too large
to absorb one another.
As a result, crystalline dimensions in a casting with the modifiers
of the second type are determined by the number of particles of the modifier
specified: the larger their number is, the smaller crystals become.
In general, such is the mechanism of the influence of second-type
modifiers upon the dimensions of primary crystals in the structure of castings.
Comparing the modifying activity of various modifiers of the second
type to correspondingly opt between them is possible through the application
and analysis of expression (164). This expression allows calculating the
dimensions of the shell of solid phase that is formed on the surface of this or
that modifier and comparing the effectiveness of various second-type modifiers
by the value of r from (164).
According to (164), the value of the shell of solid phase radius r
depends rather on the correlation between the thermodynamic properties of the
substance of modifier particles and the melt presented as DFvp and DFv than on rp. By B.Chalmers /65/, we obtain:
DFvp = DНp DТp/ ТLp;
DFv
= DН DТ/ ТLm,
(168)
where DНp is the enthalpy of the forming of modifier substance; DТp = ТLp - Т; ТLp is the melting temperature of
the substance of a second-type modifier.
DН, DТ and ТLm represent the same for the
metal of the melt; Т being the current
temperature.
By
introducing (168) into (165), we obtain:
r1,2
= rp {(1/2) + [(1/4) - (DНp DТp ТLm / А DН DТ
ТLp )]1/2} (169)
Expression (169 determines the shell radius as the sphere of the
direct particle influence upon the surrounding melt; in this connexion, the
forming of solid shells on modifier particles at the temperatures that exceed
the melting temperature of the melt can also be referred to the rank of contact
phenomena. We observe that substances may act otherwise than lowering their
respective temperatures of melting within the bounded contact zone, as it takes
place at contact melting. Within the contact zone of certain substances, the
given substance may increase the temperature of melting (crystallization) of
the other substance under certain conditions.
Let us term this phenomenon, unknown till present, as contact
crystallization.
We can deduce some conditions of successful modifying of the second
type on the basis of (169.
In the first place, contact crystallization area enlarges with the
decrease of the temperature of the melt.
Secondly, the higher the modifier thermodynamic stability presented
by the value of DНp is, the more r increases.
In the third place, the greater the difference of the melting
temperatures of the modifier and the given alloy (DТp) is, the higher is the modifier
effectiveness.
Hence we can make several inferences about the characteristics of
second-type modifiers and the conditions of their application.
Modifiers of the second type must be refractory and
thermodynamically stable under the condition of being modified by substances.
It also follows from Part 9.3 that there should be some electron
affinity between modifiers of the second type and the alloy. It implies that
the substance of the modifier of the second type should possess the metallic
type of conductivity.
Finally, it is desirable that the substance of the modifier should
be insoluble in the given melt. Soluble modifiers of the second type are
possible (similarly to the presence of silicon in cast iron), yet they act up
to the moment of their complete dissolution in the melt only, i.e. their effect
is a pronounced short-term one. For instance, steel powder immixed in liquid
steel can act as a second-type modifier refining the primary crystalline
structure of steel.
The effect described is used in suspension casting. However, this
effect is only observed until the powder particles melt completely. Therefore,
metallic powders are introduced into liquid steel at suspension casting only in
the process of pouring steel into the mold. The in-ladle introduction of the
same powders does not produce any modifying effect.
Thus, we formulated the four factors of the choice of second-type
modifiers:
1.
Second-type modifiers must possess a high
temperature of melting that proves to be considerably higher than the melting
temperature of the alloy they are introduced into (‘considerably’ means
‘hundreds of degrees higher’ in the given context).
2.
Second-type modifiers must have the enthalpy of
forming that exceeds considerably the enthalpy of the forming of the melt they
are to be introduced into.
3.
Second-type modifiers must have the metallic
type of conductivity.
4.
It is desirable that second-type modifiers
should be melt-insoluble.
Are the mentioned requirements sufficient to realize the choice of
particular substances acting as modifiers?
As an example, let us make the choice of modifiers of the second type
for steel and cast iron in accordance with the stated requirements (v. Table
20).
The requirement of refractoriness alone tangibly restricts the range
of possible candidates for being second-type modifiers.
If we consider substances the melting temperature of which exceeds
2500 K, the number of such substances is but 43. These are refractory metals,
oxides, carbides, nitrides and borides.
Out of the mentioned 43 substances, these are only 10 that strictly
conform to the requirement of being insoluble in liquid steel and cast iron.
At last, the number of substances that possess the metallic type of
conductivity and satisfy other conditions apart from that, is restricted to but
three compounds in this case: titanium and zirconium nitrides and zirconium
diboride. The modifying ability of a series of substances remains undecided
because of certain data missing.
We must admit that this is quite a concrete set of a highly
restricted range of substances.
Practical application of these substances as modifiers of the second
type for steel propagates poorly affecting but titanium nitrides in the main.
Table 20. The Evaluation of the Suitability of
Various Refractory Substances as Modifiers of the Second Type for Steel and
Cast Iron
Substance |
Temperature
of melting, degr.K, /92/ |
Solubility
in liquid steel and cast iron, /92/ |
Free
energy of forming, kJ/mole, at 2000 К, /92/ |
Conductivity
nature at 2000К, /92/ |
Suitability
as a second-type modifier |
Elements |
|||||
C |
4020 |
s |
- |
metallic |
no |
Мо |
2890 |
s |
- |
metallic |
no |
Nb |
2740 |
s |
- |
metallic |
no |
Os |
3320 |
s |
- |
metallic |
no |
Re |
3450 |
s |
- |
metallic |
no |
Ta |
3269 |
s |
- |
metallic |
no |
W |
3680 |
s |
- |
metallic |
no |
Borides |
|||||
Hf B2 |
3520 |
? |
310 |
metallic |
? |
LaB6 |
2800 |
? |
? |
metallic |
? |
NbB2 |
3270 |
? |
155 |
metallic |
? |
ThB4 |
2775 |
? |
188 |
metallic |
? |
TaB2 |
3370 |
s |
? |
metallic |
no |
TiB2 |
3190 |
s
(0.5 %) |
238 |
metallic |
? |
UB2 |
2700 |
? |
? |
metallic |
? |
W2B |
3040 |
? |
? |
metallic |
? |
ZrB2 |
3310 |
no |
268 |
metallic |
suitable |
Carbides |
|||||
HfC |
4220 |
? |
209 |
metallic |
? |
NbC |
3870 |
1
% |
134 |
metallic |
no |
SiC |
3100 |
? |
44 |
metallic |
? |
Ta2 C |
3770 |
0.5
% |
188 |
metallic |
no |
TaC |
4270 |
0.5
% |
144 |
metallic |
no |
ThC |
2900 |
? |
17 |
metallic |
? |
Th2C |
2930 |
? |
190 |
metallic |
? |
TiC |
3340 |
0.5
% |
158 |
metallic |
no |
UC |
2670 |
20
% |
75 |
metallic |
no |
UC2 |
2770 |
20
% |
104 |
metallic |
no |
VC |
2970 |
3
% |
103 |
metallic |
no |
WC |
3058 |
7
% |
60 |
metallic |
no |
ZrC |
3690 |
? |
182 |
metallic |
? |
Nitrides |
|||||
BN |
3240 |
no |
80 |
semicond. |
no |
HfN |
3580 |
? |
184 |
metallic |
? |
TaN |
3360 |
chem.
react. |
99 |
metallic |
no |
ThN |
3060 |
? |
137 |
metallic |
? |
TiN |
3220 |
no |
149 |
metallic |
suitable |
UN |
3120 |
? |
118 |
metallic |
? |
ZrN |
3250 |
no |
178 |
metallic |
suitable |
Oxides |
|||||
BeO |
2820 |
no |
401 |
ionic |
no |
CeO2 |
3070 |
no |
600 |
semicond. |
no |
HfO2 |
3170 |
no |
753 |
semicond. |
no |
MgO |
3098 |
no |
321 |
ionic |
no |
ThO2 |
3540 |
no |
840 |
semicond. |
no |
UO2 |
3130 |
no |
738 |
semicond. |
no |
ZrO2 |
2973 |
no |
721 |
semicond. |
no |
We tested the inferences of Table 20
concerning the modifying activity of titanium and zirconium nitrides in
practice. At the introduction of approx. 0.1% of the given nitride particles
per entire volume into steel with 0.3% C content, we obtain a thorough primary
crystalline grain refinement (v. the photograph in Fig.25) in case of both
titanium and zirconium nitrides application /148/. The largest crystalline
dimensions in modified castings with the mass of 10 kg and wall thickness of
100 mm within the zone of a shrinkage cavity equaled approx. 1 mm. Judging from
the cited results, titanium and zirconium nitrides are the strongest
second-type modifiers for steel at present.
Nitride inclusions surrounded by a dark mantle are found in the
microstructure of cast samples extracted from nitride-modified steel hardened
from its liquid state (v. the photograph in Fig.26).
We suppose that these areas correspond to the shells of solid phase
that have been already formed around modifier inclusions in liquid state before
the onset of total crystallization.
Such solid phase formation around
inclusions is to be reflected by the curves of the cooling of castings and be
noticeable when the amount of inclusions comes to be large enough.
Special experiments were carried out on the basis of aluminum alloys
reinforced with a considerable particle quantity. Liquidus and solidus
temperatures were measured by the method of differential thermal analysis
concerning alloys reinforced with titanium carbides of any given quantities.
Experiments have shown the rising of liquidus temperature of such
composite alloys with the increase in the content of carbide particles, the
temperature of solidus being constant (v. Fig.27).
Thus, experimental data corroborate the validity of theoretic
inferences made above concerning the mechanism of the influence of second-type
modifiers upon the process of castings crystallization.
Apart from refractory inclusions, there are fusible, liquid, gaseous
inclusions in liquid alloys. Do the former affect nucleation and the growth of
crystals in castings?
Let us consider the case when the melting temperature of the given
particle and the enthalpy of its formation approximate the temperature of
melting and the latent heat of metal melting, i.e. the following equality takes
place:
ТLp @ ТLm and DНp = D Н.
Then, on the basis of (168) we obtain:
DFvp = DFv and DFvp - DFv = 0.
In this case, equation (163) assumes the form:
А DFv r2 - А DFv r rp =
0 (169)
Hence we derive
А DFv r(r
- rp) = 0 (170)
Equation (171) can be solved in two ways:
r1
= 0; r2 = rp (171)
Solvation (171) implies that a stable solid phase shell cannot be
formed in liquid around fusible particles. However, the specified conclusion
applies but to the temperatures exceeding the temperature of liquidus. Certain
fusible inclusions may function as modifiers of the second type within the
interval of crystallization. This inference is proved by the practical examples
of applying microchills as modifiers. The application of ferrosilicon to the
modifying of cast iron serves as an analogous example. The generality
integrating the given cases is that relatively fusible substances can act as
second-type modifiers for a short time period only until they melt or dissolve
completely in the surrounding liquid metal.
Liquid
inclusions.
Let us examine
the case when liquid inclusions are present, i.e. their melting temperature
proves to be considerably lower than the melting temperature of metal.
TLp
<< TLm .
Then DFvp << DFv and DFvp - DFv @ - DFv
In this case, equation (162) will assume the form of
А DFv r2 - А DFv r rp - rpDFv =
0 (173)
Equation (173) has the following roots
r1,2 = rp
{(1/2) + [(1/4) - (1/ А)]1/2} (174)
Since А ³ 1, equation (174) either has no
solution within the interval of А = 1 ¸ 4, or results in r1 < rp.
Consequently, the shell of solid phase cannot be formed on the
surface of liquid and all the more – on the surface of gaseous inclusions
within melts. Consequently, liquid and gaseous inclusions in liquid cast alloys
cannot function as modifiers of the second type.
Moreover, such inclusions hinder the nucleation of solid phase in
the vicinity of its surface, i.e. they are second-type demodifiers in melts.
The dimensions of the elements of matter in liquid alloys were
repeatedly measured by various procedures. X-ray diffraction researches,
unfortunately, furnish ambiguous results, which allow interpretation both from
the standpoint of cluster existence and the monatomic structure of liquid
metals and alloys.
In this connection, much more univalent researches of sedimentation
processes in melts seem to be of particular interest – for instance, at the
centrifuging or finer modern sedimentation methods within gravity field.
K.P.Bounin, on the basis of V.I.Danilov’s works /10/, as well as the
liquid eutectic alloys research of his own, was the first to put forward a
hypothesis of the possibility of melts microstratification by structural areas,
similar to the structure of pure components /59/. Broaching the question why
spontaneous stratification of such melts does not take place, K.P.Bounin wrote:
‘… thermal motion ensures the kinetic stability of eutectic melt, and,
notwithstanding the melt being microheterogeneous, there is no stratification
at microlevel.’ In this connection, K.P.Bounin substantiated the possibility of
applying centrifuging to the investigation of liquid eutectics structure.
Profound researches of liquid eutectics, carried out by Yu.N.Taran
and V.I.Mazur /60-61/, as well as a series of other investigations
/49-55,57-58/, corroborate the microheterogeneity of liquid eutectics in a
tenable way nowadays.
Outstanding pioneer experimental works on centrifuging by
A.A.Vertman and A.M.Samarin proved the existence of sedimentation phenomena and
disclosed the first symptoms of liquid eutectics stratification start /16-17/.
Unfortunately, though the suggested ideas proved right, the stated
experiments were not quite mastered methodically. They failed to mark the
difference between sedimentation at crystallization and liquid state
sedimentation with a considerable degree of methodical reliability. For the
mentioned methods shortcoming, as well as other ones, the interpretation of the
results of centrifuging experiments was subjected to severe criticism on the
part of the followers of the monatomic theory of liquid alloys structure
/20,26/. Being fastidious to the methodical imperfections of the experiments
that were carried out and the accepted way of calculations, the critics of
centrifuging rejected the microheterogeneity idea as the basis of the given
experiments uppermost.
The theory of liquid metals and alloys structure stated in our work
evolved even farther than the model of the microinhomogeneous structure of
eutectics. The existence of both the elements of matter in liquid state –
clusters – and the elements of space in any liquid metals and alloys including
eutectics is grounded here.
In this connection, a theoretical substantiation and the carrying
out of sedimentation experiments, considering the newest data /30,142-143/ and
methods /142-144,149-150/, seems to be of an appreciable interest.
The distribution of any particles in liquids goes under the
influence of gravity forces, on the one hand, and the forces of thermal and
convective mixing, on the other hand.
The only theory that views such a distribution at present is the
Brownian motion theory.
Einstein, Smoloukhovsky, Jean Perren were engaged in the Brownian
motion theory research. The latter scientist investigated the Brownian motion
experimentally (as applied to water) for determining Avogadro Number. The given
trend has been physically validated and elaborated to perfection, resulting in
the Nobel Prize award to Jean Perren for the Brownian motion researches.
The Brownian motion was not investigated in liquid metals, therefore
the application of the Brownian motion theory and experiments to the research
of liquid metals and alloys is of no little interest, yet it is requisite for
the measuring of the dimensions of material elements in liquid melts rather
than determining Avogadro Number.
Let us consider the Brownian motion theory with the purpose of
checking for the possibility of finding the dimensions of the elements of
matter in liquids.
Fokker-Plank equation /142/ serves as the basis for the theoretical
solution of the problem concerning the particle Brownian motion in liquid if we
take gravity factor into consideration:
¶C/¶t = D ¶2C/ ¶h2 + k ¶C/¶h,
where С is the relative Brownian particles concentration; С = f(h0, h, t); t is time; h is the sample height in our case; D is the
coefficient of diffusion; k being the gravity constant.
We must note that the given equation was composed regardless either
of convection or particle interaction. Its general solution was obtained by
Smoloukhovsky. Still, Smoloukhovsky's solution is convenient to use if particle
dimensions and mass are known a priori. Such a way proves unsuitable for our
purpose.
In order to arrive at the requisite solution, let us consider the
case of equilibrium in the Brownian motion, when particle redistribution under
the influence of forces is completed, so the concentration of particles does
not change in time, i.e.
¶C/¶t = 0.
In this case the displacement of Brownian particles will be zero,
i.e. diffusion and gravity influence will be equalized.
For the given stationary case we obtain:
D ¶C/¶h + CV =
0, (175)
where V
is the gravity speed of sedimentation or floating.
Expression (175) is true if heat and gravity particle energy
approximate, i.e. if the particles are sufficiently small. Solution (175)
assumes the form of
С = С exp (-
Vh/D) (176)
We do not know the quantity of gravity rate V for the expression
specified. To find it, let us consider the known mechanical problem of the
floating (sink) of small particles at Stokes’ approximation. Let us set up a
motion equation (without viewing the interaction of particles).
The force necessary to displace the particles of the mass of m and
the radius of r is determined by the regular correlation accepted in mechanics
Fv
= m dV/dt
There being no auxiliary effects, the given force equates with
gravity:
Fg
= mg = 4pr3Dg/3.
We should also take into consideration the force of internal
friction (viscosity):
Fh = 6 phrV.
Archimedean force resists gravity: Fd = 4 pdgr3/3
If we sum up the interaction of the mentioned forces, the equation
of a particle motion in liquid will be presented as
Fv
= Fg - Fh - Fd , (177)
or
m
dV/dt = 4pr3Dg/3 - 6 phrV - 4 pdgr3/3.
Cancellations completed, we obtain
dV/dt = (D - d)g/D - 9hV/ r2D.
(178)
Equation (178) can be written as
dV/dt + AV + B
= 0, (179)
where
А = 9h/2Vr2; B = - (D - d)g/D.
Equation (179)
belongs to linear differential ones. Its complete solution is
V
= (C - 1) e-At - B/A,
where С is a constant.
It follows from the initial conditions that V = 0 at t = 0, thus
V
= B (e-At - 1)/A.
By introducing the values of the constants of A and B,
we find:
V
= - [2 (D - d)g r2 / 9h] [exp (-9ht/2r2) -
1]
or
V =
[2 (1 - d/D) g r2 D/ 9h] [1 - exp (-9ht/2r2)] (180)
The analysis of expression (180) shows that the sink rate of Brownian
particles is directly proportionate to the square of the radii of these
particles. Particle sedimentation rate accelerates to a certain degree with the
increase of time, asymptotically approaching the equilibrium value of Ve.
The given value can be determined if we assume that е-Аt®0 at t®¥, so according to (180)
Ve
= 2 (1 - d/D) g r2 D/ 9h (181)
It is easy to calculate that the mentioned limit is actually
instantly reached for small particles with r £ 10-8 m.
Let us
introduce the V from (181) into (176) arriving at
С/С0 =
exp [- 2 (1 - d/D) g r2 D h/ 9hD]
(182)
A similar solution for the stationary distribution of Brownian
particles was obtained by J.Perren /151/. He focused on the equilibrium of two
forces only: gravity
Fg
= 4pr3 (D - d)g dh/3,
and the Brownian
motion force
Fb
= -kT dC.
Solving the equilibrium equation of the specified forces, J. Perren
derived the expression /151/:
ln
C/C0 = 4pr3 (D - d)g h/3kT. (183)
If we substitute the D in expression (182) for Stokes-Einstein
correlation
D
= kT/ 6phr,
we shall derive
the expression that is wholly identical with Perren’s equation (183).
Namely:
C/C0
= exp [- 4pr3 (D - d)g
h/3kT] (184)
Thus, two variant solutions to the problem result in similar
solutions for the stationary distribution of Brownian particles (184).
It is of extreme importance to note that the distribution of Brownian
particles in liquid, according to (184), is proportionate to the cube of the
radius of Brownian particles.
I.e. particle dimensions strongly affect their in-melt distribution.
Consequently, the experiments concerning the study of in-melt Brownian particles
distribution should be rendered extremely sensitive to the dimensions of
sedimentator particles. This is very significant, since it allows carrying out
the experiment the result of which will differ essentially depending on the
nature of particles – the structural elements of matter in liquid: atoms or
clusters. The difference in the order of the given two types of material
elements, according to (184), should affect the distribution of Brownian
particles by the sample height in a deciding way.
It enables us to employ (184) for the calculation of the dimensions
of Brownian particles in melts with a considerable certainty. It follows from
(184) that
r =
[(3kT ln C/C0)/ 4p (D - d)g h]1/3 , (185)
where d is the average
melt density; D being the density of Brownian particles.
Expression (183) is quite useful for the study of the dimensions of
the elements of matter in liquid alloys on the basis of the results of
sedimentation experiments.
Actually, if any more or less noticeable difference of element
concentration is obtained by the sample height of several cm as a result of
holding liquid cast alloys within gravity field, that will mean that the
in-melt dimensions of the elements of matter exceed the dimensions of separate
atoms.
Calculations show that with the corpuscular structure of liquids no
measurable difference of element concentration by the sample height with the h
= 50-100 mm is to be observed in such experiments.
Jean Perren was awarded the Nobel Prize for determining Avogadro
Number in the experiments dedicated to the study of the Brownian motion in
water /151/. As a matter of fact, Jean Perren estimated the total number of
particles moving in liquid on the basis of the measurement of the motion energy
of some of them, namely the particles that were specially introduced into
transparent liquid (water), - the particles visible by microscope with their
dimensions and density known a priori. The researcher made use of the fact
that the heat motion energy of any particles moving in liquid is identical.
By Perren /151/, the motion energy of one Brownian particle equals
W =
2pr3 (D - d)g h/ ln C/C0
, (186)
where h is the
height of a sample.
Jean Perren determined the quantities entering into (186) from
experience to further calculate the quantity of W on their basis.
The quantity of W being known, Jean Perren found the number
of the structural units of liquid per mole N:
N =
3RT/2W (187)
According to his data, the number under analysis was approximately
equal to Avogadro Number.
From the viewpoint of present theory, any energy in liquid is
distributed uniformly between all the particles constituting the liquid, be it
atoms or clusters or other particles that are sufficiently small. The stated
fact is reflected in the known energy equidistribution theorem.
It was demonstrated in Part 3.6 above that the heat energy of
clusters does not differ from the heat energy of separate atoms in consequence
of energy equidistribution by the degrees of freedom.
Besides, the number of clusters per mole of liquid amounts to
approx. 0.1% of the number of atoms, i.e. Avogadro Number. Thus, the aggregate
amount of the elements of matter in liquid among which heat energy is
distributed equals N0 + Nc + Nb » N0
,since Nc + Nb << N0
, where Nc is the number of
clusters per mole of liquid; Nb is the number of Brownian
particles introduced.
Consequently, the total number of atoms and clusters per mole of
liquid differs very little from Avogadro Number (Np = N0
+ 0.1%).
Therefore, the result obtained by J. Perren gives the amount of
particles per mole of water that hardly exceeds the theoretical Avogadro Number
of N0.
However, Perren conducted experiments with visible particles that
were introduced into water in advance and the dimensions and density of which
were set-point.
In case of liquid alloys research allowing for the existence of clusters
cluster dimensions are unknown a priori. Still, the Brownian motion theory lets
employ the difference in the density of the particles constituting the alloy to
determine the dimensions of these particles, even if the latter are invisible.
For instance, at the detection of a concentration difference in the composition
of elements by the height of a liquid alloy sample while holding it within
gravity field J. Perren’s methods can be successfully applied to calculate
cluster dimensions by the formula (185) derived above.
It is possible in the connection with the fact that matter in liquid
alloys is represented by atomic groupings-clusters that possess the value of r
by order of magnitude greater in comparison with atoms. Since r enters into
(184) in the cube, the existence of clusters must affect the distribution of
the elements of matter in liquid alloys within gravity field most decisively,
so the quantity of C/C0 in case of the monatomic structure of
liquid metals must be three orders as small as it is in case of the cluster
structure of melts.
Thus, theoretical analysis shows that there exists a quantity that
is highly sensitive to the dimensions of the elements of matter in liquid
alloys – this is the change of the concentration of elements in alloy samples
by their height while holding the samples within gravity field (convection
lacking).
In case of the monatomic structure of liquid alloys there have to be
no visible change of concentration for the time of holding reaching tens of
minutes or hours.
In case of the cluster structure of liquid alloys such a change
should be quite apparent, amounting to the tenth fractions of one percent in
samples that measure approx. 10cm by height.
The given arguments and calculations were assumed as a basis for
creating a new procedure of liquid alloy structure research.
There exists a diversity of opinions with respect to component
sedimentation in liquid alloys.
For example, B.Chalmers supposes that in alloys with the
unrestricted solubility of their components in liquid state ‘… there must be no
segregation in liquid until the latter starts hardening’ /67/.
Well-known monographs by other authors treat but segregation at
crystallization, too /64,66,68,74,75,135/.
At the same time, there is K.P.Bounin’s opinion on the possibility
of such a phenomenon in liquid eutectic alloys /59/.
There are centrifuging experiments results obtained by A.A.Vertman
and A.M.Samarin, the results of V.P.Tchernobrovkin’s observations concerning
segregation in liquid cast iron /16,17/, as well as a whole series of other
data.
We also have objection and discussion data that deny segregation at
centrifuging /20/.
Thus, there exist opposite opinions on liquid state segregation,
even under the conditions of simulated gravity at centrifuging. It testifies to
the insufficient study of the question.
The development of the procedure that will provide reliable and
unambiguous results is of paramount importance for a secure record of an
unknown phenomenon and its mechanisms.
The results of cosmic metal research demonstrated that convection in
samples increases inevitably with an increase in gravity g, for Rayleigh Number
is on the rise /102/:
Rа = (gh3/n)[(btDT/a) + (bcDC/D)],
where h is the
height of the metal layer; DT is the in-layer temperature drop; DC is the difference in concentrations; bc is the metal volume expansion coefficient; n is the
kinematic viscosity coefficient.
It is clear from the formula that Rayleigh Number speedily increases
with an increase in g and h reaching the critical value of Ra = 1700.
Hence, the influence of gravity in sedimentation experiments does
not only promote sedimentation but inhibits it.
Therefore, centrifuging experiments should not be considered
effective by their nature, since the quantity of g is too large there,
so convection cannot be eliminated because of various mechanical interference
types like unavoidable vibrations, rotation speed fluctuation, Coriolis forces,
etc.
The given procedure may be used only as a preliminary, qualitative
one. It can hardly be of help for the finding of precise quantitative
information.
We developed different variants of experimental mass transfer
research methods in liquid alloys within gravity field, there being no initial
concentration gradient in samples.
The basics of these methods consist in the following.
Specified composition alloys were produced out of pure components
(of extra pure brands and brands tested pure for analysis). Experiments were performed
on the following alloys: Pb-Sn, Pb-Bi, Zn-Al, Al-Si, Al-Fe-Cr, Cu-Pb, Cu-Sn,
Fe-C, Cu-Pb-Sn. For the obtaining of a homogeneous parent composition, alloys
were overheated 200-300 K above the temperature of liquidus to be held 1 hour
at the given temperature, and were stirred thoroughly with an alundum or quartz
stick.
No sooner was the stirring completed than alloy samples were taken
into quartz or alundum capillaries or tubes by way of vacuum suction. Capillary
or tube diameters went from 0.3 to 50mm, their height being changed from 40mm
to 500mm, since the way in which the diameter and height of samples, as well as
their material, affected the result was our subject matter, too (Fig.28).
For the most part, samples of 1-3mm in diameter were used. Such a
choice allows a practically full suppression of the onset of in-sample
convection.
The samples were brought to crystallization by way of air cooling.
Then the initial distribution of components by sample height was studied on one
of the samples by way of chemical and metallographic analyses.
Other samples were tested by sight for continuity. Such a test is of
fundamental importance, for samples where experiment discontinuity is observed
show a sharp distortion in results.
After the sorting through, the samples were sealed hermetically.
Quartz capillaries were sealed through vacuum soldering. Alundum capillaries
were plugged at the ends with a pulverized alundum-based stopper to be sealed
in with quartz further. Practice showed that a similar sealing shuts the
samples off from air and guards against volatile elements evaporation much
better than, for example, smelting in inert gas atmosphere.
To hold the metal thread fixed inside a capillary to guard the
former against bias at turning, an asbestos layer 2-3mm thick was put and
packed on top of metal before soldering.
The small diameter samples prepared in such a way were placed into
the holes of a graphite casing – a cylinder with 8-10 openings along its
external perimeter extent - in groups of 8-10 identical items, so all the
samples remained under the same thermal conditions. The casing with the samples
was put in an experimental cell that had been heated up to the specified
temperature.
The cell constituted the isothermal zone of resistance shaft and
resistance multipurpose furnaces for fusible alloys or a similar zone of
Tamman’s furnace for refractory alloys.
For temperature equalization, as well as the screening of
electromagnetic fields, two cylindrical coaxial graphite and molybdenum screens
shielded the graphite casing with the samples.
Under such conditions, fluctuation in temperature by the length of
the isothermal zone in resistance shaft and resistance multipurpose furnaces
totaled + 2K at most, amounting to + 8K in Tamman’s furnace.
The time of heating the samples up to the specified temperature made
1-5min. for small diameter samples (< 3mm). The temperature was
registered by thermocouples and potentiometers.
The experiment consisted in the holding of capillaries at the
specified temperature during the preset time at a definite position of the
sample relative to the vertical /141/. The working cell of furnaces revolved
around the horizontal axis, which let station the samples in vertical,
horizontal and reverse positions. It enabled us to hold the sample vertically
in liquid state, for instance, and place it horizontally at crystallization.
That was the way we eliminated the influence of segregation upon the obtained
distribution during the crystallization period.
To this end, we compared the distribution of elements in the samples
that had been held in horizontal and vertical positions correspondingly.
To study the influence of time factor, identical samples were
withdrawn out of the casing 5, 15, 30, 60, 90, 120, 180, 240min after the
beginning of the experiment.
Crystallization was effected in varied ways, too, to study the
influence of crystallization processes on the distribution of elements by the
length of samples.
Some samples were crystallized within the casing by the air blowing
of the experimental cell. A certain share of samples was withdrawn out of the
furnace and cooled by water in vertical or horizontal positions.
At the water hardening of samples 3mm in diameter the time of
hardening amounted to 3-4s. A considerable value of overcooling – from two to
three tens of degrees - was observed in small diameter samples.
The obtained samples were taken out of capillaries. Samples with
ruptures and holes were rejected. The selected ones were cut into parts and
tested for the distribution of components by height in solid state by the
methods of chemical and metallographic analyses.
The negative effect of negligible external noises, e.g. those coming
from the vibrations of machinery operating in the neighborhood, was noticed in
the course of the experiment. Such vibrations may cause convection thus utterly
distorting the result. In this connection, furnaces were rigged out with
vibration pads and the experiments were performed mainly at night to minimize
the noise.
Segregation effect at melting was also neglected in the experiments,
which is normally disregarded altogether. Melting was conducted with
differently positioned samples for that purpose.
Mass transfer at melting was not observed in our experiments.
Mass transfer at crystallization amounted from 0 to 8% of the
registered concentrations depending on crystallization conditions. Zero effect
of segregation at crystallization on the distribution of elements by sample
height was found in the experiments concerning the crystallization of samples
in horizontal position. Zero effect of segregation at crystallization was also
observed at the water hardening of samples out of liquid state.
The application of the described method allows detecting and
eliminating the influence of external factors preserving the effect of elements
redistribution by sample height under the influence of gravity in its pure
form, practically.
A possible Coriolis acceleration influence, the influence of a
possible difference in alloy density by sample height, as well as a series of
other probable disturbance interference types, were also taken into account
while carrying out the experiment.
The above-said gives us sufficient grounds to state that the
developed method lets study the redistribution (mass transfer) of alloy components
just in liquid state, whereas our experiments allow for the influence of
melting-crystallization processes reducing it to the minimum.
The temperature and composition of the
samples experimented with by the above-stated method are listed in Fig.29. The
results of the chemical composition research of the obtained samples by their
height under varied conditions of holding are presented in Fig.30 /143/.
The data demonstrated in the given picture show that the
redistribution of alloy elements by sample height undoubtedly occurs in the
liquid alloy of Pb-Sn in the process of holding in liquid state (convection
lacking).
The results of chemical inhomogeneity
development at the holding in liquid state in the course of time are to be
clearly seen in Fig.31.
At the holding of the same samples horizontally in liquid state the
change of lengthwise in-sample concentration was not detected (Fig.32). In this
case, to eliminate Coriolis acceleration influence, the samples were orientated
north south.
It is established that the process of the
redistribution of components in Pb-Sn liquid alloy decelerates but does not
stop with the increase of the time of holding up to three hours.
Thus, the equilibrium distribution of components within the given
alloy was not obtained under the given conditions. Consequently, an even
further alloy segregation into components is possible if the time of holding
increases.
The distribution of components by sample height in Bi-Cd liquid
alloy in capillaries 2-3mm in diameter and with the h of h=100mm, the
composition of the alloy being eutectic and the overcooling exceeding the
liquidus by 500C, is presented in Fig.32.
The mechanism of the transition from the initial homogeneous
distribution to inhomogeneous one is shown in Pict.33.
The nature and mechanisms of the redistribution of elements by
sample height in Bi-Cd liquid alloy prove to be qualitatively similar to the
same mechanisms in the melt of Pb-Sn.
A faster redistribution of components in Bi-Cd alloy in time can be
marked as the distinctive feature of the mentioned alloy.
It is interesting to point out that the degree of inhomogeneity
achieved in samples decreases with the rise in temperature. It may be explained
by the growth of convection and the acceleration of other types of mass
transfer with temperature rise.
Experiments show that eutectic melts are unstable under the
conditions of suppressed convection and tend to segregate by density into the
original components.
Alloys pertaining to Zn-Al and Zn-Al-Cu group are industrial cast
alloys utilized in pressure die casting.
Alloys with the content of aluminium from 3 to 11% are used more
frequently – for instance, Russian standard ZAM 4-1 and ZAM 10-5 alloys
(zink-based alloys with 4% aluminium and 1% magnesium vs. Al 10% and Mg 5%
content respectively). Therefore, alloys with aluminium content from 1 to 11%
were given the most consideration in our experiment /150/.
The change in the concentration of aluminium by the height of a
3.7%Al alloy sample (ZAM 4-2 alloy) at different holdings is to be seen in
Fig.34 and Fig.35. As it is clear from the picture, the degree of inhomogeneity
increases in samples with an increase in the time of holding in liquid state.
Among the peculiarities of ZAM 4-1 alloy we should mention the
character of the dependency of concentration on the sample height – the pattern
tending to linear one, - which indicates non-approximation to equilibrium.
Secondly, the process of the forming of chemical inhomogeneity in ZAM 4-1 melt
is decelerated, though the time order of the forming of inhomogeneity remains
the same.
Kinetic behavior of the transition from homogeneous to inhomogeneous
distribution of components in Zn-Al liquid alloys is shown in Fig.36.
The use of alloys with variant original
content of elements allows visually comparing the stated behavior
specificities. It follows from Fig.36 that the obtained degree of inhomogeneity
increases in its absolute value with an increase in the original average
admixture content.
If we determine the relative degree of sedimentation development as
the relation of the absolute difference in concentrations DC of one of the alloy components to the average content of the latter
in the given alloy C (DC/C), there emerge regularities that
merit our attention, - they are presented in Fig.37.
The data demonstrated in this picture suggest that there exists a
well-defined connection between the degrees of inhomogeneity achieved per
specified time and the diagram of state of the alloy under consideration.
Namely, there is an inflection of the dependency of the absolute inhomogeneity
quantity DAl = f (Al%) in the eutectic
concentration area, whereas the maximum is observed in the dependency curve of
the relative inhomogeneity DAl/Al = f (Al%) in the same area. Fig.35
also reflects the influence of alloy overheating or the time of holding in
liquid state on the degree of inhomogeneity obtained at the time of holding up
to 3hrs. The lower line in the picture corresponds to the overheating of 1500C,
the middle one – to the overheating of 1000C, the top line
representing the overheating of 500C. It is clear that the degree of
inhomogeneity achieved per specified time decreases with an increase in
overheating.
To prove whether the data obtained actually reflect the process of
segregation in liquid state, experiments that allow determining the position of
a sample while holding it in liquid state were carried out.
The results are shown in Fig.37. The middle horizontal line
corresponds to the distribution of elements by sample height at a horizontal
holding.
These data corroborate the trustworthiness of the developed method
and demonstrate that the segregation that is observed really progresses during
liquid state holding.
The obtained segregation regularities in
Zn-Al liquid alloys are substantially congenial to those that are typical of
Pb-Sn and Bi-Cd alloys. The difference lies in the fact that we have noticed no
approaching of the equilibrium in experiments on Zn-Al alloys during three
hours of holding at all. There is a pronounced tendency to further development
of inhomogeneity.
Al-Si liquid alloys are widely used in
industry, in which connection their study becomes a subject of particular
interest.
Sedimentation studies in liquid state in the alloys of Al-Si system
were effected in alundum and graphite capillaries, since liquid aluminium while
contacting with quartz reduces the latter to silicon, which rather distorts the
results.
Kinetic behavior of the transition from the original homogeneous to
inhomogeneous distribution of elements for Al-Si alloys are demonstrated in
Fig.38 as applied to Al-12%Si alloy. As we see, the same regularities of the
transition to inhomogeneous distribution as are observed in other eutectic
alloys occur in Al-12%Si alloy. Quantitatively, the process of redistribution
in Al-12%Si alloy goes faster than in Zn-Al alloys.
The character of the distribution of silicon by sample height does
not differ from linear one, practically, which also indicates that equilibrium
is not established and the system tends to further segregation.
The connection between the degree of the development of
inhomogeneity and the diagram of state is shown in Fig.39. As is obvious, the
observed regularities are close to those discovered in Zn-Al alloys, though the
extremum in the vicinity of the eutectic point turns out more distinct in Al-Si
alloys.
Sedimentation in liquid tin casting bronze with approx. 10% tin and
2% zink content was analyzed. The bronze under consideration also refers to
industrial alloys. Moreover, it belongs to alloys with a peritectic structure,
too. There is only one solid solution a on the basis of copper in the areas of the
indicated tin and zink concentrations in a solid alloy in the diagrams of state
of binary alloys.
However, the mentioned liquid tin casting bronze does not have a
monophase composition. Two solutions are formed in solid bronze on account of
the presence of tin and zink: the solution of tin in copper and the solution
of copper in tin and zink. These solutions have a variable composition. Thus,
the triple system of Cu-Sn-Zn differs from binary Cu-Sn and Cu-Zn systems.
There was no eutectic in the signalized area of concentrations in the triple
alloy.
The experiments aimed at studying sedimentation in bronze were
carried out in Tamman’s furnace equipped with graphite heaters.
The change of phase composition, namely the quantity of the solid
solution of a in microstructure, was the sole subject matter of our study of
samples. The distribution of a-solid solution by sample height is demonstrated in Fig.40. The
kinetics of the process of sedimentation is to be found in Fig.41.
One can see that the results obtained as far as liquid bronze is
concerned do not differ qualitatively from those achieved in eutectic alloys.
This gives us reasons to assert that sedimentation in liquid state is also
characteristic of alloys with a peritectic structure.
We studied carbon sedimentation in liquid
Russian standard cast iron LK-4 with 10% carbon content.
Experiments were performed in quartz capillaries in Tamman’s
furnace. Carbon and sulphur content was analyzed in the upper and lower part of
a sample. After holding the samples for 3 hours in liquid state at the
overheating of 50 degrees above the point of liquidus the difference in carbon
concentrations between the upper and lower sample points averaged 0.3%. The
difference in concentrations reached 0.8% in one of the samples. Naturally, the
concentration of carbon in the upper part of the sample was more saturated than
that in its lower part.
In connection with the fact that equation (184) prognosticates a strong
dependency of the achieved degree of inhomogeneity on the height h of the
sample, experiments were conducted in order to prove if such a dependency
exists.
Experiments were carried out on Zn-4% and Al-12%Si alloys by the
same method. The only distinction consisted in the height of quartz capillaries
for Zn-4% alloy being assumed equal to 50,100 and 200mm respectively, the
diameter equaling 1-3mm.
For Al-12%Si alloy, the height of the alundum capillary was
recognized as 50 and 100mm, its diameter being 1mm.
Sealed samples were held for 3 hours in liquid state at the
overheating of 500C above the liquidus temperature of the given
alloy.
Then the samples were water hardened and subjected to analysis.
The results are tabulated below (Table 21).
Table 21. The Degree of Inhomogeneity in Liquid
Alloys Depending on the Sample Height
Sample
height, mm |
Alloy
composition |
Cupper,
%, experiment |
Clower,
%, experiment |
Cupper,
%, calculation
by (183) |
Clower,
%, calculation by (183) |
50 |
Zn-4%Al |
4.1 |
3.8 |
4.1 |
3.8 |
100 |
¸ |
4.75 |
2.8 |
4.2 |
3.8 |
200 |
¸ |
5.0 |
2.8 |
4.4 |
3.6 |
50 |
Al-12%Si |
12.4 |
11.3 |
12.6 |
11.5 |
100 |
¸ |
13.2 |
11.1 |
12.5 |
11.4 |
Note:
calculation values are derived for equilibrium theoretic distribution.
As is seen from the table, the experiment corroborates a definite
increase in the degree of inhomogeneity obtained in liquid alloy with the rise
of the height of the sample. However, experimental data by far exceed
calculation data, though our calculation was done for equilibrium distribution.
Granting that the majority of the conducted experiments as regards
sedimentation in liquid alloys demonstrated that equilibrium was not
established, we carried out experiments with a prolonged time of holding of the
following alloys: Zn -5%Al; Zn - 10%Al; Zn - 15%Al in liquid state for 24, 48,
72, 96 hours.
The principal result of the experiments in question was that
equilibrium was not established even after 96 hours of holding, so the
redistribution of elements continued.
Aluminium content in the upper and lower parts of samples after such
a long-term holding in liquid state is represented in Table22.
Table 22. Aluminium Content in the Upper and Lower
Parts of Zn-Al Alloy Samples 100mm Tall Subjected to a Prolonged Holding
Alloy
composition |
The time of
holding in liquid state, hrs |
Aluminium
concentration in the upper part of the sample, % |
Aluminium
concentration in the lower part of the sample, % |
Zn-5%Al |
24 |
14.0 |
1.1 |
48 |
16.4 |
0.8 |
|
72 |
17.7 |
0.8 |
|
96 |
19.3 |
0.7 |
|
Zn -10%Al |
24 |
22.2 |
5.9 |
48 |
23.9 |
5.2 |
|
72 |
26.0 |
4.4 |
|
96 |
27.9 |
4.0 |
|
Zn-15%Al |
24 |
26.6 |
7.1 |
48 |
28.7 |
6.4 |
|
72 |
29.7 |
5.9 |
|
96 |
33.8 |
5.1 |
As it is clear from Table 22, the redistribution of aluminium
continues even after a 96-hour holding with quite a high intensity. It brings
us to predict the complete segregation of the given liquid alloy into its
original components at a sufficiently prolonged holding in liquid state.
The observed passage of chemical inhomogeneity into structural one
with the forming of conglomerates proved to be a new fact of utmost importance
in our experiments.
Fig.42 shows the microphotographs of the structure of the original
solid samples (a), as well as the samples that have been held for 24, 72 and 96
hours correspondingly (the respective photographs b, c and d).
Considerable changes in the microstructure of the upper zone of the
samples should be noted while examining these photographs.
After the expiry of 24 hours of holding in liquid state we observed
the origination of drop-shaped formations – we termed them as conglomerates -
of the phase rich in aluminium (Fig.42, b).
The appearance of cut crystals was observed along with the formation
of round conglomerates on the expiry of 72 hours of holding (Fig.42, c).
Having been held for 96 hours, crystals enlarge (Fig.42, c). The
composition of crystals corresponds to the intermetallic compound of Zn-Al type
that is lacking in the diagram of state of the given alloy.
Thus, the possibility of the transition of the microinhomogeneous
structure of liquid alloys into the macroinhomogeneous structure of solid
alloys was experimentally proved.
Sedimentation experiments let us
calculate the dimensions of Brownian particles in liquid alloys by equation
(185). The Brownian motion theory applies to any particles whose heat and
gravity energies are comparable. The mentioned equation is inactive toward the
dimensions of sedimentator particles and therefore quite applicable to our
calculations, - to the initial stages of the process of sedimentation, at
least.
Let us remark that when deriving (185) we make an assumption
concerning the difference between the average melt density and the density of
sedimentator particles (clusters and their conglomerates).
Assuming the density of particles equal to that of the pure
component of the eutectic, we cause certain indeterminancy, which is
unavoidable at the current level of knowledge. By prior estimation, the
indeterminancy of the value Δ imparts a relative error of 3-10% into the
calculation of r by (184).
By assuming that cluster conglomerates enriched b+
By one of the components do exist in melts, which results from our
experimental data, we get the following dimensions of cluster conglomerates in
the examined alloys on the basis of (184) (v. Table 23).
Table 23. The Dimensions of Conglomerates with the
Predominance of One of the Components in Liquid Alloys after Holding in Liquid
State within Gravity Field at Suppressed Convection for 3 Hours
Alloy |
Second
element content, % |
Temperature,
C |
Conglomerate
radius, calculation, nm |
Conglomerate radius, calculation, nm |
Bi-Cd |
50 |
180 |
rBi
= 9.05 |
|
|
|
|
|
|
Pb-Sn |
60 |
200 |
rBi
= 4.0-4.9 |
rSn
= 5.6 |
|
|
250 |
rBi
= 3.6-4.2 |
|
|
|
350 |
rBi
= 2.1-3.7 |
|
|
|
|
|
|
Al-Si |
6.0 |
650 |
rSi
= 11.2 |
|
|
6.9 |
650 |
rSi
= 11.5 |
|
|
8.0 |
650 |
rSi
= 11.5 |
|
|
9.2 |
650 |
rSi
= 12.5 |
|
|
10.8 |
650 |
rSi
= 12.1 |
|
|
12.1 |
650 |
rSi
= 12.1 |
|
|
13.3 |
650 |
rSi
= 11.3 |
|
|
13.8 |
650 |
rSi
= 11.4 |
|
|
|
|
|
|
Cu-Sn |
5.0 |
1100 |
rSn
= 4.2-2.1 |
|
|
5.0 |
1050 |
rSn
= 8.1-7.0 |
|
|
|
|
|
|
Cu-Sn-Pb |
5.0%Sn+4.9%Pb |
1050 |
rSn
= 3.8-5.6 |
|
|
|
1100 |
rSn
= 1.8-2.7 |
|
|
|
|
|
|
Fe-C |
4.2 |
1200 |
rC
= 2.7-4.9 |
|
If we take into account the relatively large effective dimensions of
Brownian particles, according to Table 23, that exceed cluster dimensions by
order of magnitude approximately, we may conclude that the dimensions of
Brownian particles in alloys considerably exceed those of separate clusters as
early as on the expiry of three hours of holding.
In case of the monatomic structure of liquid alloys no noticeable
inhomogeneity can arise in liquid samples 10-100mm tall, which we demonstrated
earlier /142, 150, 153/.
The presence of microgroups with the radius of 1-10nm, on the
contrary, must cause a certain slight chemical inhomogeneity in samples 100mm
tall at the level of 0.01-0.1%, by calculation. However, our experiments
indicated a much higher degree of inhomogeneity present. Therefore, the results
of our experiments overpassed the limits of the simplified theoretical
alternative: clusters or separate atoms.
Concentration inhomogeneity obtained in the samples of various
alloys in capillaries 50-100mm in height under the conditions of holding in
liquid state within gravity field at suppressed convection reaches tens of
percents. It also leads to the forming of structural inhomogeneity represented
by drop-shaped conglomerates. Moreover, even after holding in liquid state up
to 96 hours the equilibrium distribution is not achieved, so the process of alloys
segregating into the original components continues.
The achieved results are unprecedented by the obtained inhomogeneity
in liquid state. Such a considerable inhomogeneity was not achieved even while
conducting centrifuging experiments.
There are no precedents to the discovered tendency of the
continuation of liquid alloys segregation into the original components. Both
sedimentation theory and the theory of centrifuging prognosticate quite a rapid
establishment of equilibrium. In reality, equilibrium was not struck. This fact
cannot be interpreted either.
What processes can lead to such a substantial segregation of alloys
in liquid state?
The influence of oxidation and admixtures was eliminated by way of
testing. The influence of capillary material was also eliminated through the
varying of the materials.
We excluded the possibility of thermal diffusion and barometric
diffusion effect, too, by carrying out special experiments. The effect of
shallow diffusion was easy to eliminate, too. A series of other less probable
processes, such as Marangony surface convection, was discussed and eliminated
/102/.
As a result of such elimination, we can name two basic causes of
alloys segregation into the original components:
1.
The lack of convection in samples.
2.
Structural inhomogeneity of melts in liquid
state.
The conclusion concerning the influence of the lack of convection
points out that the given factor is primary in the formation of liquid alloys
under earth conditions. To all appearances, if it were not for convection,
there would not be such diversity in alloys, diagrams of state and structures.
Some of the diagrams of state of binary alloys would become unrecognizable.
This is convection, and not diffusion, that sustains many of the existent
alloys in liquid state as a macroscopic homogeneous mix.
As for structural inhomogeneity, the presence of the elements of
matter - clusters - and the elements of space in liquid metals provides the
incensive for the process of segregation to set on. Yet if there were nothing
but clusters in liquid alloys, segregation process would have established
equilibrium at a very low inhomogeneity degree with the difference in
concentrations in a 100mm tall sample that does not exceed 0.1% and stopped,
because the process of diffusion, according to the theory, must balance the
subsequent process of the redistribution of single clusters.
Still, unexpectedly for us, the process of segregation in
experiments developed very extensively, so the process continued even on the
expiry of 96 hours of holding in liquid state, although the difference in
concentrations 10-20 times as large (1000 – 2000%) was reached in samples and
considerable structural changes were observed.
The existence of clusters only cannot explain such a result.
Besides, the given result does not keep within the limits of the monatomic
alloy structure theory. Only the formations that are larger than clusters can
ensure such a speed and depth of the processes of spontaneous segregation of
liquid alloys.
We suppose that here we handle a synergetic process going
simultaneously at several levels and developing in time. This process causes
the formation of a whole hierarchy of liquid state structures. The term of the
hierarchy of structures of liquid state was introduced by the author in works
/143, 152, 154, 155, 141/ on the basis of sedimentation researches of our own
in liquid alloys.
A vertical displacement of like clusters under the influence of
gravity and Archimedean force triggers the process. I.e. the process starts
according to the Brownian motion theory.
However, the Brownian motion theory does not allow for the
interaction between Brownian particles. Judjing from experimental results, such
an interaction occurs and determines a further development of segregation
process in liquid alloys.
A slow displacement of like clusters in one direction cannot but
result in a phenomenon similar to orthokinetic coagulation. The frequency of
encounters of like clusters and the time of their side-by-side stay increase in
the process of such a motion.
Under such conditions, even an inconsiderable prevalence of
interaction forces between like particles (A-A and B-B forces symbolically, the
composition of a binary alloy being AB) over the forces of interaction between
unlike particles AB will inevitably lead to the forming of agglomerations of
like clusters, or cluster conglomerates, inside liquid. The above-mentioned
hierarchy of structures of liuqid state arises in this way.
Such processes of conglomeration of like clusters go at any moment
of time within any liquid alloy but unstable cluster conglomerates get easily
decomposed through convection. This is convection that sustains liquid alloys
in an as-homogeneous mixes of heterogeneous clusters condition.
The given data is corroborated by the already ample data of orbital
experiments /102/, where sedimentation processes in melts turned out to be
unexpectedly significant under the conditions of the lack of natural
convection.
Diffusion, which used to be considered as the principal motive force
of alloy forming, participates in the mentioned process to a certain extent,
yet it can resist neither gravity nor Archimedean force, to say nothing about
the forces of interaction between like clusters.
The dimensions of the elements of matter in liquid alloys – i.e.
clusters – that numerous authors, the author of the present work included,
calculated by a variety of methods, fluctuate within the limits of 1-10nm,
whereas the dimensions of drop-shaped conglomerates formed as a result of
sedimentation amounts from 0.1 to 1.0mm. Consequently, there can exist three or
four hierarchical dimensional levels of the agglomerations of like clusters
between clusters and visible conglomerates in liquid alloys within the limits
of 10nm-0.01mm. These levels correspond to the dimensions of colloidal
particles.
Thus, there may arise the following hierarchical levels of cluster
conglomerates in the process of sedimentation in liquid alloys.
Separate clusters – 1-10nm dimensional level
Cluster conglomerates - 10-100nm dimensional level
Colloidal cluster conglomerates - 100-1000nm dimensional level
Colloidal microdrops - 1000-10000nm (1-10mkm) dimensional level
Drop-shaped macroconglomerates - 10-1000mkm dimensional level.
Colloidal microdrops and drop-shaped conglomerates are the only two
largest dimensional levels of conglomerates registered in the course of the
experiment. Smaller cluster aggregations act as a constituent part of solid
alloys in the process of crystallization and remain undisclosed by existent
methods. We should underline that cluster conglomerates are unstable, labile
formations at any dimensional levels so they can exist and be developing only
if convection is lacking. However, under certain conditions these formations
may become stabilized and even form new phases. And, of course, conglomerates
can directly affect the structure of cast metal, which was demonstrated
earlier. Still, the basic result of the forming of conglomerates at all
dimensional levels is the acceleration of the process of segregating this or
that liquid alloy into its original components or solutions.
The growth of cluster conglomerates, as well as the competitive
growth of crystals, is possible at different levels: conglomerates can grow
both by way of separate clusters adjunction and the coalescence of neighboring
conglomerates.
Therefore, cluster conglomerates are able to dimensionally evolve
very far from the original clusters while remaining nothing but a constituent
of liquid.
In the sequel, conglomerates can evolve into new phases.
In the process of sedimentation conglomerates behave as indivisible
formations of far larger dimensions than clusters. Correspondingly, the larger
conglomerates are, the faster they emerge, since the speed of the floating of
particles, according to Stokes’ equation, is proportionate to the square of
their radius. So sedimentation process, developing synergetically, i.e. going
simultaneously at different dimensional levels, accelerates and develops
towards the complete segregation of the melt into its original components.
Such complete melt segregation was not achieved in our experiments,
yet we noted a definite tendency to the development of the process toward the
segregation of melts. The obtained twentyfold segregation is quite a quantity,
though, that seems to be comparable to the degree of segregation reached after
approx. ten passings when purifying metals of admixtures by the zone refining
method /68/.
Consequently, the phenomenon that we disclosed – that of liquid
alloys segregation developing within gravity field – can be applied both
scientifically and practically as the real alternative to the method of zone
refining of metals and alloys, as well as a way of segregating certain alloys
into their original components.
A wide spectrum of the phenomena of metallurgical heredity
corroborates the inference concerning the complex hierarchical structure of
liquid alloys. The above-cited data of Ch.10 on sedimentation experiments
lasting many hours confirm both the stated conclusion and the inference that we
made earlier: the basic structural elements of liquid alloys are extremely
stable in time.
It was shown above that the period of cluster existence equals the
period of the existence of liquid state. Sedimentation experiments lasting
24-96 hours corroborate the latter conclusion within the limits of the
experiment– up to 96 hours, at any rate.
Metallurgical heredity is a highly complicated phenomenon. Most
often heredity is meant when we touch upon the inheritance of a crystalline
structure. However, the phenomena of metallurgical heredity have a wider
implication than a structural factor as such.
Mechanical properties, as well as the tendency to cracking,
shrinkage, and other technologically significant properties, are inherited
under certain conditions, too.
Therefore, it would be more correct to speak about the spectre of
metallurgical heredity phenomena. The common feature of the given
manifestations, widely different one from another, concerning the connection
between the structure and properties of liquid and solid metals is that these
or those properties or alloy parameters are passed from charge materials to
final castings through liquid state.
Consequently, when broaching the phenomena of metallurgical
heredity, we deal with a certain mechanism, or mechanisms, of data transfer
from the original solid charge through melting, liquid state and
crystallization to a final casting.
The feature that integrates the phenomena of metallurgical heredity
is their wide prevalence and practical importance, on the one hand, and their
being unstable, labile, on the other hand. In particular, it is well known that
some manifestations of metallurgical heredity can be eliminated by the
overheating of liquid metal, for instance, and its thorough mixing. It refers
to structural heredity in the first place.
The problem of heredity carriers in liquid metals, as well as the
hereditary information transfer mechanism, is of practical importance. The
problem mentioned has been given insufficient scrutiny so far.
Traditionally, solid insoluble particles of various inclusions that
are strictly indeterminable are reckoned as information carriers in liquid
metals /10/.
V.I.Nikitin systematized the ideas on the carriers of hereditary
information adding clusters to admixture particles, as well as cluster
conglomerations and other elements of the structure of melts /105/.
Unfortunately, the mechanism of passing hereditary information remains unknown.
The zone of the ‘responsibility’ of this or that hereditary information carrier
for this or that type of hereditary phenomena has not been determined either.
V.I.Nikitin also introduced the concept of gene engineering for
melts and castings that is based upon the control of the structure and
properties of castings with the application of metallurgical heredity phenomena
and modifying; yet first it is necessary to study the mechanisms of
transferring variant hereditary features from liquid to solid state in order to
practically apply gene engineering to melts.
Our research shows that the particles of modifiers of the second
type can preserve the shell of solid phase on their surface at temperatures
higher than the melting temperature of the alloy. So a considerable overcooling
is required to liquidate these particles, as well as time to deactivate the
surface of such inclusions. Thus, the operation mechanism of second-type
modifiers that was disclosed here is one of the mechanisms of structural
heredity.
Undoubtedly, clusters and intercluster splits fuction as the
carriers of hereditary information, too. This follows from the melting and
crystallization theories initiated by the author that reveal the connection
between liquid and solid state structures. However, clusters carry but the
principal information of the crystalline lattice type and the way it is built.
Clusters cannot carry information concerning the number of crystals, their
dimensions, etc. It is another hierarchical level of metal structure – and
another informational level.
More specific information may be carried by different cluster
conglomerates and conglomerates of intercluster splits that are often to be
found in liquid alloys, which was proved by our sedimentation experiments in
melts. Possible variants of such conglomerates are actually unlimited both by
their composition and their dimensions, structure and other parameters.
Therefore, the possible variants of their hereditary influence are also
extremely diverse and unpredictable as yet. Unlike clusters proper, their conglomerates
are far less stable. The overheating of the melt, as well as natural convection
or artificial intermixing of the melt, etc. may cause their decomposition.
Thus, many types of metallurgical heredity are highly sensitive to
the overheating and intermixing of melts.
Metallurgical heredity researches closely relate to the studies of
the structure of melts and appear to be quite challenging as far as the forming
of castings with controllable high nonequilibrium properties is concerned.
Proceeding from
the most general considerations of the structure of real bodies that include
both the elements of matter and the elements of space, we succeeded in
developing a new unified non-contradictory theory of the melting and
crystallization of metals and alloys.
The given theory differs fundamentally from the existent theory and
turns out to be incompatible with it.
A surprising persistence of science, accumulated for more than a
hundred-year period of the existence of crystal nucleation theory, with respect
to this field keeps the new theory from gaining ground. Prevailing ideas will
obviously offer a strong and long-term resistance.
However, the drawbacks of existent theory seem to be so substantial
that only the lack of a more or less reasonable alternative can account for the
existence of the former for such a long time period. Moreover, the lack of
ideas on a far more complex, or even a fundamentally different, structure of
aggregation states and real bodies in general aggravates the situation.
Still, on the expiry of a certain time, with the accumulation of new
data, the former theory will inevitably be relegated to the past.
We ought not to blame anyone for the founding of the wrong crystal
nucleation theory, since the structure of the states of aggregation of matter
used to be presumed monatomic until recently, while the concept of the inner
elements of space was never introduced. The ideas of the flickering nature of
the interplay of material and spatial elements inside real systems were
non-existent all the more. Those are the mentioned ideas that prove to be basic
ones for the new theory of melting, crystallization and the liquid state of
metals.
The given concepts will take the longest time to engraft, for the
store of knowledge in this sphere is too insufficient being confined to the
material of the present book. Still, the road is clear, and anyone can take it.
The methods of experimental research of the elements of space,
vacancies excepting, are to be established yet.
We may hypothetically propose to study subtle oscillations of
electrical resistivity in capillary samples of liquid metals and alloys. The
flickering nature of the interaction between the elements of matter and space
in liquid metals at the level of clusters and intercluster splits must result
in the flickering character of electrons passing through the melt, too.
Certainly, the samples should be very small; otherwise subtle oscillations of
electrical resistivity with the order period of a billionth fraction of a
second in large samples will level because of mutual multiple superposition of
such oscillations.
The discrete character of the elementary acts of melting and
crystallization can also become a subiect of experimental study, as well as the
discrete mechanism of liquid metals fluidity. The measurement of the elementary
amount of the latent heat of crystallization or melting could generate a lot of
new actual information on the specified processes.
These are precision experiments requiring precision instrumentation
that the author was unable to get on account of science subsistence conditions
at this time in this country. Somebody may be luckier, though.
Certainly, the importance of the above-stated methods of precision
sedimentation experiments oriented on the study of liquid alloys structure,
cannot be denied.
In fact, this is the first direct method of measuring the dimensions
of sedimentator units of matter in liquid alloys that was specially dedicated
to serve the given purpose and proved highly sensitive to the dimensions of
sedimentator particles in liquid alloys.
The tendency displayed by many liquid alloys to segregate into their
original components – we discovered it in the course of our experiments –
within gravity field is an essentially new experimental result that can also be
applied to industrially purify metals of admixtures instead of the method of
zone melting of metals.
The results that we obtained in this field have been repeatedly
published including the publications in magazines issued by the USSR Academy of
Sciences and Russian Academy of Sciences. The comment was but favorable.
In conclusion, we would like to make a hypothesis concerning the
applicability of the equation of state to condensed aggregation states – liquid
and solid.
The attempts at adapting the equation of state to liquids and solids
were, and are, continuously being made. On the one hand, there are no
theoretical bars to it – which brings no practical results either, on the other
hand.
When calculating vacancy gas pressure in solids earlier, it was
stated and corroborated by quantitative data that the equation under analysis
is applicable to vacancy gas pressure, at least, at the point of melting, with
high accuracy.
In this connection, there arises a hypothesis that the equation of
state PV =RT should be applied if we allow for the existence of
different types of the elements of matter and space at many levels.
For gases, the measured pressure and volume turned out to
incidentally coincide with the pressure and volume of material and spatial elements
that are characteristic of the given state. Namely, the intrinsic pressure of
the elements of matter and space in gases coincided with that measured by
customary equipment.
Our hypothesis of the applicability of the equation of state
consists in the following: this is the intrinsic – and not ambient – pressure
of the elements of matter and space in the state specified that must be taken
into account while applying this equation to condensed states.
The concept of the intrinsic pressure of the elements of space in
solids and liquids, and at other levels of the structure of real bodies, is a
new one.
To know how it can be measured is a new experimental trend in the
research of matter-space real systems, too. To be more exact, a series of new
trends should propagate, since specific elements of matter and space are
peculiar to every level of matter-space systems, as well as their specific
internal interaction and its parameters, intrinsic pressure including.
Certainly, various forms of the elements of space characteristic of
every form of the elements of matter are yet to be disclosed to build up a
system similar to the periodic law of the elements of space at corpuscular
level and at other levels, too. There is a demand for the periodic law of
matter-space element complexes.
An extra hypothesis refers to the flickering nature of the
interaction between the elements of matter and space. Evidently, flickers in
the diversity of their forms, oscillatory, rotary and other flickers including,
are typical of many levels of real systems’ structure. Interatomic interaction
in solids and liquids and molecules is very likely to have a flickering nature,
which determines all practically important properties of solids - for instance,
durability, plasticity, electric conductivity, density, etc. etc. The
parameters of spatial elements, including the characteristics of their
flickers, are specified in the present work for the liquid state of aggregation
of matter only. Solid state has parameters of its own. They are to be determined
as yet. We are to focus on the contribution of the latter to the real
properties of solids, which will surely turn out to be as significant as the
contribution of material elements – atoms - within solid crystals.
Concluding the book by a series of hypotheses, the author suggests
that all interested people – both the supporters and adversaries of the
proposed new concepts - should volunteer and test the new potentialities. New
ways always yield new results.
Good luck!
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