"Interaction of space and matter elements during melting and cristallisation of metals and alloys" - читать интересную книгу автора (Igor V. Gavrilin)

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 Issue

1.1. 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

G_S=G_L (1)

where G_S is Gibbs’ free energy index for solid phase; G_L 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 T_M= 0.000⁰C (by definition) whereas the triple point temperature is T_TR=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.

Надпись: Fig. 1 The change of the free energies of pure metal in liquid and solid states depending on temperature 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 S_f=R ln W_l /W_s, (2)

where W_l is the number of independent ways of substance realization in liquid state, W_s - 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 20^th 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 = -ΔG_v + ΔG_s,

where ΔG_v is specific volumetric free energy; ΔG_s 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.

Надпись: Fig. 2 The change of DG depending on the radius of a crystalline nucleus 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 G_s, which equals to one third of the spare surface energy of the nucleus mentioned:

Δ G_s= 1/3 S S_is_i,

where S_iis the specific surface of the i^th zone of an equilibrium crystal at nucleation; s_iis 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 r_cis the radius of the sphere inscribed into a cube of the critical dimension.

To calculate the quantity of r_cof a spherical nucleus the following formula is used /65,67,68/:

r_c= 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 r_cthere 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 s³T₀ / 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 s³T₀lg e / L²(Δ T)²k

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 10¹⁰erg/sq.cm.

Hence

lgI/K = -32 200³1800 0.434/4 10²⁰1.38 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

ΔG_heterog/ Δ G_homog= 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 = K_lexp(-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 = K_lexp(-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/.

Надпись: Fig. 3 The diagram of the dependency of crystalline centers nucleation rate n on the overcooling of the melt DT 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.

Надпись: Fig. 4 The diagram of the dependencies of crystalline centers nucleation rate n and the linear rate of crystalline growth v on the overcooling of the melt DT for metals 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.

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.

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.

Надпись: Fig. 5 Theoretical dependency of the linear rate of crystal growth v on the overcooling DT of the melt 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.

Надпись: Fig. 6 The dependency of the linear rate of pure gallium crystals growth on overcooling 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 = K₄Δ T,

where K₄is 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.

Надпись: Fig. 7 The linear rate of deformed gallium crystals growth depending on overcooling 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.

Надпись: Fig. 8 The rate of arborescent tin crystals growth depending on overcooling 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.

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:

p_v = 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, p_v, 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.

p_v = σ_в_, (12)

p_v is vacancy gas pressure; σ_вbeing metal durability of elongation.

The point of the intersection of the curves p_v = f(T) and σ_в= 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.

2.3. Melting centers

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.

2.4. The correlation of melting centers with the centers of crystallization

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.

Chapter 3. The Mechanism of Metals and Alloys Melting Process

3.1. The elementary act of melting and the formation of the structural units of matter and space in 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.

3.2. Calculating the elementary act of melting

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:

F_v= F_η + F_σ, (13)

where F_v 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. F_v = m_cd²x/ dt²; F_η is the reactive force of viscous friction within the existing liquid; F_σ are surface forces.

In this connection

F_η= 6 p η r_c dx/dt ;

F_σ = 2p r_cσ x;

where x is the displacement of a crystalline lattice zone, which comprises one vacancy, from the equilibrium position; m_c is the mass of such a zone (after the separation of a cluster); r_c is cluster radius; η is liquid metal viscosity at T = T_melting; 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:

m_c d²x/ dt² = 6pηr_c dx/dt + 2pr_cσx (14)

Let us introduce the designations

6pηr_c = -β; 2pr_cσ = -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) (m_c 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 =v₀exp( - βt / 2 m_c ) [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 F_v 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 m_c /6pηr_c)[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.

3.3. The structure of liquid metals at the point of their formation

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.

3.4. Calculating cluster dimensions in liquid metals at the melting temperature

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 α₁is a separate atom (a molecule); α_m 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 α_n 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 ΔH_vap. 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 ΔH_meltingmust 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 ΔH_vap and ΔH_meltingenable 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 = 6n^2/3. (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 n^1/3. 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. n^2/3. 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 δH_melting 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

δH_melting = U₁ n₁/ 4 = 6n^2/3 U₁ / 4 = 3U₁ n^2/3 / 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

2δH_vap= 3 U₁ n,

δH_vap = 3/2 U₁ n. (22)

To determine the quantity of n, let us find the correlation between the specific quantities of δH_vap and δH_melting from (22) and (21). We get

δH_vap / δH_melting = (3/2 U₁ n) / (3/2 U₁ n^2/3),

therefrom

n = (δH_vap / δH_melting)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 δH_vap and δH_melting 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 δH_vap and δH_melting for the molar ones ΔH_vap and ΔH_melting, because their proportion will be identical. Thus,

n_sc = (ΔH_vap / ΔH_melting)³, (23)

where n_sc 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:

r_sc = a (n^1/3) / 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/:

n_bcc = (9/16) (ΔH_vap / ΔH_melting)³ ; (25)

r = (a/2) (3n_bcc /4)^1/3; (26)

n_fcc = (1/4) (ΔH_vap / ΔH_melting)³ ; (27)

r_fcc = (a/2) (n_fcc/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

V_c = n a³.

In its turn, the volume may be expressed through cluster radius, too /81/:

V_c = (4/3) π r³.

Equating the right sides of the two latter equations, we can find

r_sc = (3n/4π)^1/3 a. (29)

Generalizing the latter expression for any granulation types, we get

r_c = (3z n /4 π)^1/3 a, (30)

where z = k_sc/ k_c ; k_sc = 6 - is the coordinating number of a simple cubic granulation; k_c 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 ΔH_vap / 2 ΔH_melting = 3U₁ n / (4 πr² U₁)/2a²

By inserting here the value of r from (29), for a simple cubic granulation we obtain:

ΔH_vap / ΔH_melting = 3U₁ n / 3πU₁n^2/3(3/4π )^2/3≈ n^1/3/ (1/2)^1/3.

Hence, for spherical clusters with a simple cubic granulation of atoms in neighboring order inside clusters we get

n_sc = 1/2 (ΔH_vap / Δ H_melting)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:

n_c = a_c (ΔH_vap / ΔH_melting)³, (32)

where a_c 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 a_c = 1; 9/16 and 1/4 correspondingly.

Similar to (32) we derive the generalized expression for the calculation of cluster radius r_c at the melting temperature:

r_c = ( ΔH_vap / ΔH_melting) β^1/3 a, (33)

where β = 3z a_c / 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	DH_melting, C/mole /97,98/	DH_vap, C/mole /97,98/	r_c, 10^-10 m, calculations by (33)	r_c/ a	n_c, cube, calcula-tions by (32)	n_c, 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
Fe_d	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 n_c 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 n_c for silicon, magnesium and zinc amounts to 160, 340 and 480 atoms accordingly; the maximal values of n_c for gallium and lithium are 48000 and 23400 correspondingly. However, n_c 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.

3.5. Calculating the dimensions of spatial elements in liquid metals at the melting temperature

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 r_c².

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 r_c². Admitting that a cluster performs heat oscillations along the three axes, we obtain that the area of intercluster splits per cluster equals approx. 3r_c².

Then, the total area of intercluster splits per gram-atom of any liquid metal will be equivalent to

S_cl = 3N₀ r_c² / n_c .

Allowing that N₀ = 6 10²³, r_c = 10^-9 m on average, while n_c = 10³ on average, we get that S_cl ≈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:

v_s≈r_c² αa. (34)

The quantity of α may be found from the expression cited by Ya.I.Frenkel /70/:

α = σ_max / E,

where σ_max 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

σ_max = (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:

v_s ≈r_c² αa. (36)

The number of intercluster splits N_s in a mole of liquid approximates the molar quantity of clusters N_c:

N_c ≈ N₀ / n_c. (37)

If N₀ = 6 10²³, while n_c averages 10³, the number of intercluster splits N_c = N_s = 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

V_s ≈N_c v_s = (N₀ / n_c) r_c² 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 ΔV_splout 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 V_spl = 3α.

Since length for clusters is

l = 2r_c,

the relative volume of the zone of intercluster splits will be as large as

ΔV_spl = (3α / 2r_c) 100% (39)

By inserting the value of α from (35) into (39), we get

Δ V_spl = 3 (E γ / a)^1/2 / E 2 r_c.

Expression (39) is the most suitable for calculations, since the values of r_c are already known there. The values of the quantities required for calculations are listed in Table 4 below.

Calculating ΔV_spl 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)	DV_spl, %, 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
Fe_g	1835	20000	0.177	4.84
Fe_d	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.

3.6. Calculating the energy of clusters and intercluster splits motion in liquid

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

E_i = Δ H_melting/ (N₀ + N_c),

where N₀ is Avogadro Number; N_c is the number of clusters in a gram-atom of liquid metal.

Since N_c<< N₀, the latter expression can be written without any appreciable error as

E_i = Δ H_melting / N₀. (40)

On the other hand, we know that the energy of heat oscillations of one atom makes

E_a = (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:

E_c = E_a = (3/2) kT (42)

At the melting temperature the quantities of E_cand E_i must be equal, or

E_c = (3/2) kT_melting (43)

Comparing the values of E_c and E_i 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 E_c and E_i.

The results of calculations are listed in Table 5.

Table 5. The Energy of Heat Oscillations of Clusters

Metal	E_c, J, calculation by (43)	E_i, J, calculation by (40)	E_i /E_c
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 E_i and E_c 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 E_c 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

E_i = 0.707 Δ H_melting/ 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.

3.7. Calculating the point of metal melting

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 = T_melting.

Thus, we get

Δ H_melting n_c / N₀ = (3/2) kT_melting.

Hence we derive the expression for calculating the melting temperature of metals:

T_melting = Δ H_melting / 1.5 N₀k. (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 H_melting, C/mole^-1	The Melting Temperature, Т_melting, К
Metal	Δ H_melting, C/mole^-1	Т_melting, К by (98)	Т_melting, 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.

3.8. Calculating the content of activated atoms in liquid

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 C_a.

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 C_a may serve as the measure of the disordering of liquid metals contrasted with solid metals, where the quantity of C_a 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 C_a as the relation of the number of free bonds on the surface of a cluster n to the number of atoms in a cluster n_c.

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

C_a = n₁ /2n_c = S / 2V = 4π r_c² / (4/3) π r_c³ = 3/2 r_c^-1. (46)

Expressing r_c according to (33), for spherical clusters we have

C_a = (3/2) (ΔH_melting / Δ H_vap) β^-1/3. (47)

The values of C_a 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
C_a, %, 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.

3.9. The frequency of heat oscillations in clusters and the frequency of intercluster split flickering in 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:

α_n + α_n →← 2α_n

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

E_c = (3/2) kT.

The quantity of φ is to be found from the expression suggested by the oscillations theorem /103/:

φ = (1/2π A) (2E_c/ m_c)^1/2, (48)

where A is the amplitude of cluster oscillations; m_c 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 m_c = M n_c / N₀, where M is the atomic weight of matter; n_c being the number of atoms in a cluster according to Table 2, Part 3.4.

By inserting the values of E_c, A and m_c into (48), we arrive at

φ = (1/2πaα) (3kT N₀ / n_cM)^1/2. (49)

The period of heat oscillations of clusters τ will be equivalent to τ = φ^-1, or τ = 2πaα (3kT N₀ / n_cM)^-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 n_c 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.

3.10. The period of cluster existence

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 τ_cl = τ_liq, where τ_cl is the period of cluster existence, τ_cl 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.

Chapter 4. The Change of Liquid Metals Structure at Heating and Cooling

4.1. Basic theses

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 (ΔS_f / k) exp ( -E_f / kT), (50)

where Δ S_f is the entropy of vacancy forming, while E_f 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:

C_a = B exp (Δ S_f / k) exp (- E_f / 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 E_f and Δ S_f 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 S_f / k) exp (- E_f / kT) = Δ H_melting n_c / N₀

It was demonstrated above that Δ H_melting 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 C_a

C_a = Δ H_melting / E_fz (52)

Expression (52) supplies us with the absolute value of the concentration of activated atoms per mole of substance. The relative quantity of C_a may be obtained from (52) by way of division by Avogadro number. Thus

C_a = Δ H_melting / E_fz N₀ (53)

On the other hand, we determined the same relative quantity of C_a earlier by expression (47) as

C_a = γ₁Δ H_melting / Δ H_vap, (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 Н_пл / E_fz N₀ = γ₁Δ H_melting / Δ H_vap

Hence E_a = Δ H_melting / γ₁z N₀.

At present, inserting the values of T = T_melting, C_a from (54) and E_f from (55) into (51), we find:

γ₁Δ H_melting / Δ H_vap= B exp (Δ S_f / k) exp -( Δ H_vap / γ₁ z k N₀ T_melting).

Two quantities are unknown here: B and Δ S_f. 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(ΔS_f k) at the melting temperature will be equal to:

exp (Δ S_f / k) = (γ₁Δ H_melting / Δ H_vap) exp (Δ H_vap/ γ₁ z R T_melting), (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:

C_a = (γ₁Δ H_melting /Δ H_vap) exp (Δ H_vap / γ₁ z R T_melting) exp -( Δ H_vap / γ₁z R T) (57)

At T = T_meltingexpression (57) transforms automatically into expression (54).

The dependencies of C_a = f(T) and r_c = 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.

4.2. The modification of cluster dimensions with the change of temperature

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 r_c and C_a from (46), we find

r_c = 3 C_a^-1 a / 2. (58)

Inserting the value of С from (57) here, we find the dependence of cluster dimensions on temperature

r_c = (2/3ag₁) (Δ H_vap / Δ H_melting) exp-( Δ H_vap / g₁zRT_melting) exp(Δ H_vap / 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:

n_c = (4p/3z) r_c³ .

By way of inserting here the value of r_c from (58), we get

n_c = (4p/3z) (3/2)³ (C_a )^-3.

Let us designate d = 9/2z and introduce the value of С_а.

For n_c = f (T) we get:

n_c = dp [g₁ Δ H_melting / Δ H_vap) exp(Δ H_vap / g₁zRT_melting) exp-( Δ H_vap /g₁zRT)]^-3 (60)

The subset of formula (60) is represented as

n_c = dp (C_a )^-3 (61)

The dependence of r_c = 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 n_c = f (T) and r_c = 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 n_c, it is easy to determine the number of clusters per gram-atom of the given metal in liquid state. Thus

N_c = N₀ / n_c = N₀ C_a^-3 / πδ (62)

In accordance with (62), the number of clusters increases rapidly with the rise of temperature.

4.3. The modification of the volume of spatial elements in liquid metals with the change 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) ΔV_spl. The relation between ΔV_spl and C_a can be expressed under (39):

ΔV_spl = aα (3/2 r_с) 100% = aα C_a 100%.

Inserting here the value of C_a from (57), we have

Δ V_spl = aα (γ₁ Δ H_melting/ Δ H_vap) exp (Δ H_vap/ g₁zRT_melting) exp- (Δ H_vap/ 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 ΔV_spl, 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.

Chapter 5. Modifications of the Characteristics of Metals at Melting and Structure-Sensitive Properties of Liquid Metals

5.1. Of the connection between the structure and characteristics of liquid metals

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.

5.2. The mechanism of fluidity in liquid metals

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 r_c². Hence

F = p r_c². (67)

Introducing (67) into (66), we get

Te = δ / τ p r_c². (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.

5.3. Viscosity in liquid metals

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 r_c² / 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 ≈n^2/3 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 n^2/3, 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 n^1/3 = 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 m_c n^1/3, (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 / n_c M)^1/2.

Cluster mass is known, too: m_c = М n_c / N₀ .

Let us determine the number of clusters in a unit of liquid metal volume by the expression:

n = N₀ r / M n_c, (70)

where r is the density of liquid metal.

Inserting the obtained values of j, m_c and n into (69), we get:

h = (2 / 3paа) (3kT N₀ / n_c M)^1/2 (N₀ r / M n_c)^1/3 (М n_c / N₀) (71)

Expression (71) is correct for Т = Т_melting. To find the dependency of h = f(T), the value of n = f(T) should be inserted into (71):

n_c = dp [g₁ΔH_melting /ΔH_vap) exp(ΔH_vap /g₁zRT_melting) exp-(ΔH_vap /g₁zRT)]^-3

h = B exp -(ΔH_vap /g₁zRT)^-1/2 T^1/2, (72)

where B = (2 / 3paа) (3R / n_c M)^1/2 (N₀ r / M n_c)^1/3 (М n_c / N₀) (g₁ΔH_melting /ΔH_vap)exp(ΔH_vap /g₁zRT_melting) - 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) (r^4/3 / M^5/6) exp (e / RT) T^1/2 [1 - exp - (e / RT)] (73)

Here, as well as in (72), we observe the term of Т^1/2, while the quantity of e = 2Е_vap / 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,
Metal	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.

5.4. Self-diffusion in liquid metals

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 -(E_D / kT), (74)

where D is the coefficient of diffusion; E_D 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 = D_c⁰ + D_a⁰ C_a, (75)

where D is self-diffusion coefficient; D_c⁰ is self-diffusion coefficient by the cluster mechanism; D_a⁰ is self-diffusion coefficient by the mechanism of activated atoms; C_a 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:

D_c⁰ = 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 r_c + 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 r_c + a.

Or, since a << 2 r_c at Т = Т_melting, we may admit without any noticeable error that

d = 2 r_c . (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/:

E_с = ( m_с A² w² )/2, (79)

where m_c is the mass of a cluster; E_c is the energy of cluster oscillations; w is the angular frequency of oscillations; А is amplitude.

m_c = М n_c / N₀.

E_c = (3/2) kT.

w = 2p n.

А = d = 2 r_c.

n= w /2p

Having accomplished the corresponding substitutions, out of (79) we derive

w = (2Е_c /m_c)^1/2 / d; (80)

and

n = (2Е_c /m_c)^1/2 / 2pd (81)

In turn, expressing n_c through С_a under the expression (61) derived earlier, we get

n_c = dp (C_a )^-3.

Introducing the obtained values of d, E_c and m_cinto (81), for t

n = (3 k T N₀ / M dp C_a^-3 )^1/2/ 4 pr_c

Let us allow for the previously derived expression for r_c:

r_c = (3/2) а C_a^-1.

After the introduction of the concluding value we obtain:

n = (3 R T / M dp C_a^-3 )^1/2/6 p a C_a^-1

n = C_a^5/2 (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:

D_c⁰ = k d² n = (a / 4p) C_a^1/2(3 R T / M dp)^1/2 (83)

The found values of D_c⁰ by equation (83) at Т = Т_meltingare 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

D_а⁰ =(a/12pa) (3RT /dp M)^1/2C_a^3/2 (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 C_a / 4p) (3RT /dp M)^1/2[( C_a² /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	D_c⁰×10⁵ sq.cm/s, calculation by (83)	D⁰×10⁵sq.cm/s, calculation by (85)	D⁰×10⁵sq.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.

5.5. Comparing the effects of mass transfer in various states of aggregation of matter in 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

D_l⁰ = k_l l_l² n_l

for liquid state, whereas for solid state

D_s⁰ = k_s l_s² n_s.

Here l_l and l_s are the spacings of the elementary act of the prevalent diffusion mechanism for liquid and solid states correspondingly; n_l and n_s represent the frequency of the elementary acts of diffusion for liquid and solid states correspondingly; k_l and k_s 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

D_l⁰ / D_s⁰ = k_l l_l² n_l / k_s l_s² n_s

up to a constant.

It is known that l_s = а, where а is the shortest interatomic spacing in a crystal.

We recognized l_l as d_c above for liquid state, where d_c is the diameter of a cluster.

It follows from Table 1 that d_c » 10 а.

Correspondingly

D_l⁰ / D_s⁰ » (10а)²n_l / a² n_s

D_l⁰ / D_s⁰ » 10² n_l / n_s (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 D_l⁰ / D_s⁰ » 100.

That means that the correlation between the frequencies of diffusion jumps in equation (90) is n_l / n_s = 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 l_l = l_g and n_l = n_g 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.

5.6. Admixture diffusion in liquid metals and alloys

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/:

ψ_cl + ψ_dis = 1, (91)

where ψ_cl is the relative cluster contents in the structure of liquid metal; ψ_dis 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 = ψ_cl D_cl+ ψ_dis D_dis, (92)

where D_cl and D_dis 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 = D_cl + D_a C_a, (93)

where D_a is the coefficient of diffusion by the activated atoms mechanism; D_cl is diffusion coefficient by the basic cluster mechanism; C_a 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 C_a 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 S_cl as equal to the solubility of the same admixture in solid metals S_sol in the vicinity of the melting temperature, i.e.

S_cl = S_sol (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 S_aa can be found as the difference in general admixture solubility in liquid metals S_l and its solubility in solid metals near the temperature of melting. So

S_aa = S_l – S_sol (95)

S_aa = S_l – S_cl (96)

Correspondingly

S_l = S_cl + S_aa (97)

It seems rather difficult to allow for the discriminate solubility of admixtures in (93), because the quantities of D_cl and D_a in (93) can differ from partial self-diffusion coefficients of D_c⁰ and D_a⁰ in (75) due to the presence of admixtures. So let us apply here another way of calculating D by omitting the partial coefficients of D_cl and D_a.

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 r_cl and atoms r_me:

r⁰ = r_cl + r_ме C_а, (101)

where C_а 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 = r_cl (S_cl/ 0,5) + r_а C_а (S_аа/0,5), (102)

where r_а 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 S_cl= S_аа = 0,5, while r_а = r_ме, so (101) lets arrive at (102).

Inserting (101) and (102) into (100), we get:

D = D⁰ (r_cl + r_ме C_а) / [r_cl (S_cl/ 0,5) + r_а C_а (S_аа / 0,5)] (103)

All the values of the quantities constituting (103) are known, in which connection the given formula is quite applicable to calculations.

5.7. Admixture diffusion in liquid iron

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 D⁰_Fe = 1.7 10^-6 sq.cm/s /138/ and the value of D⁰_Fe=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 D⁰_Fe 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:

S_cl + S_аа = 1; S_cl = S_аа. (104)

Hence for the specified group of admixtures

S_cl = S_аа = 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 S_cl, S_аа, r_cl, C_а into the basic expression (103). We shall use the experimental value of D⁰_Fe in our calculation. Thus, at Т= 1600 ⁰С:

D_Co = 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 D⁰_Fe we get D_Co = 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 S_cl for liquid iron can be equalized to the maximum solubility of the given admixture in liquid g - iron (S_g).

For the second group of admixtures in liquid iron

S_cl = S_g;

S_аа = 1 - S_g,

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 S_aa=1 - S_gvalue 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^-6for 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

(S_cl / S_l) + (S_аа / S_l) = 1,

where S_l 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:

S_cl^H ® 0. (105)

Hence in correspondence with (104) for hydrogen and the whole third group of admixtures in liquid iron:

S_аа = 1. (106)

Introducing the values of S_cl^Н from (105) here and S_аа from (106) into (102), we obtain that the effective diffusion radius of admixtures belonging to the third group equals

r = r_а C_а / 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	S_cl, % at. /119-120/	Atomic radius, /66/	D, sq.cm /s calculation by exp. D⁰_Fe	D, sq.cm /s calculation by D⁰_Fe 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.

5.8. Of the change of coordinating numbers at melting

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 / 3V_a, (107)

where n is the number of particles in an elementary cell; V_a 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 k_c = 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 k_d, 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 = T_melting. Finally, we should single out the coordinating number for activated atoms k_a. 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

k_a = (k_c + k_d / 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 = [(k_c + k_d)/ 2 - 1] C_a + k_d(1 - C_a). (109)

The multiplier (1 - C_a) is introduced here in order not to double our consideration of activated atoms.

There are two variable quantities forming (109): C_a and k_d, 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 C_a and k_d into (109), we derive

k = 12 - C_a ; k = 10; k = 6 + 2 C_a; k = 4 + 3 C_a, (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 k_a and k_d in such metals with the rise of T. It should be pointed out that k_п 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 k_c.

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	-
Fe_g	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 – k_a, k_c and k_d - 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 k_d /17,44-46,128/.

5.9. Of the change of the electrical resistivity of metals at melting

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 f_s. Thus, f_s = 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 f_v.

f_v = ΔV_p,

where ΔV_pis 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 f_p.

In accordance with /129/, the factor of compacting equals

f_p = ΔV_c = k_p C_a,

where k_p = 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 f_c. 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 f_s that doubly reduces the number of neighbors for any given cluster to contact with (electrically) at any given moment. The f_s factor is numerically equivalent to the ratio of the coordinating numbers in metals in solid state to 12. Or

f_s = k_s / 12

Let us mark that the f_s 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

r_L / r_S = (f_s + f_v + f_p) f_s (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	f_s	f_v	f_p	f_p	r_L / r_S, calculation by (111)	r_L / r_S, 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 ρ_L / ρ_S 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.

Chapter 6. Of the Mechanism of Crystallization of Metals and Alloys

6.1. Precrystallization as the mutual extrusion of dominant and latent elements of matter and space in liquid state

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 G_l and solid state G_s. Thus, for solid state /1-2/:

G_s = H_s - T S_s (112)

and for liquid state

G_l = H_l – T S_l, (113)

where H_s and H_l represent enthalpy of solid and liquid states correspondingly; S_s and S_l being the respective entropy of solid and liquid states.

It is reckoned by right that G_s= G_l 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 G_s and G_l 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 α_n 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.

6.2. The formation of crystallization centers

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 DF_v + S s, (114)

where V is the volume of a crystal; S is its surface; DF_v 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 DF_v + 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 DF_v + 8pr s, (116)

has the extremum, while the radius r that corresponds to the bending point can be obtained under the condition that

4pr² DF_v + 8pr s = 0

Hence originates the idea of the so-called critical radius of the solid phase nucleus:

r_c = 2s/ DF_v (117)

Then, with the use of the known value of DF_v /66-68/ found under rather debatable premises, the following is derived:

r_c = 2sТ_melting / DН DТ (118)

The physical meaning of the critical radius of crystallization center is that the growth of all the crystals with the r > r_c 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 < r_c 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 r_c.

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

a_n +ds + a_n - dН_cr ® a_2n - ds + dН_cr (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 dН_cris 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 DF_v - S s, (120)

while expression (120) takes the shape of

DF = -(4/3)pr³ DF_v - 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.

6.3. The overcooling problem at crystallization

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 + α_n = α_2n + δH_cr (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 δH_cr.

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 δH_cr can be found through the following expression:

δH_cr = DH_meltingn_c / 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 δH_cr 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 δH_cr 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 δH_cr 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 δH_c_r 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 δH_cr 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

Q_m = Q_n.c. (124)

where Q_n.c_. is the heat of the elementary crystallization act according to (123), i.e.

Q_n.c. = δH_cr = DH_meltingn_c / N₀, (125)

while Q_m 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,

Q_m = (T_melting- T_i) c v ρ, (126)

where T_i is the maximum temperature of crystallization start; T_melting- T_i = 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 Q_n.c. during the time equal to one period of heat cluster oscillations.

The latter is of immense importance. The heat of Q_n.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 V_m n_c /N₀ (127)

where V_m 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

Q_m = g DТ c r V_m n_c /N₀.

Let us stipulate that V_m = M / ρ, where M is the atomic metal mass.

Finally we obtain:

Q_m = g DТ c M n_c /N₀. (128)

Now, introducing the value of Q_m from (128) and the value of Q_n.c_. from (125) into the original heat balance equation (124), we arrive at

g DТ c M n_c /N₀ = DН_melting n_c / 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 = DН_melting / 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/	DH_melting, 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 = DН_melting / 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.

6.4. Spontaneous and forced crystalline centers nucleation in liquid metals

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.

6.5. The frequency of crystalline centers nucleation

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 / n_c 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 N_c,

where N_c is the number of clusters in a mole (gram-atom) of liquid metal at the temperature of melting.

In turn, N_c = N₀ / n_c. Finally we obtain the expression for calculating the highest possible frequency of crystalline centers nucleation per gram-atom of metal:

n =( N₀ / n_c)(1/2paа) (3kT N₀ / n_c 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.

6.6. Time factor at crystallization

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 (n_c 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.

6.7. The problem of mass crystalline centers nucleation

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 N_max = N₀ / 2 n_c..The quantity of N_max 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.

6.8. The competition theory of crystallization

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 α_n is a cluster within the composition of liquid; 2α_n is the elementary crystal obtained by the accretion of two clusters; iα_nis 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> T_melting goes continuously and reciprocally providing the basis for the flickering interaction mechanism between material and spatial elements in liquid metals and alloys:

α_n + α_n→← 2α_n.

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:

1^st microcrystal

α_n + α_n→ 2α_n

2^nd microcrystal

α_n + α_n→ 2α_n

2α_n + α_n→ 3α_n

3^rd microcrystal

α_n + α_n→ 2α_n

2α_n + α_n→ 3α_n

_{…………………………..}

iα_n + α_n→ (i+1)α_n

2^nd and 3^rd microcrystal accretion:

2α_n +3 α_n→ 5α_n (134)

The accretion of the remaining microcrystals:

(i+1)α_n +5 α_n→(i+6) α_n

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 t_e, (135)

where r represents the maximal crystalline dimensions; k is the coefficient; t_e 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.

6.9. Of the change in the volume of metals at melting and crystallization

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 ΔV_spl. The value of the factor of ΔV_spl was determined earlier by expression (39) in Part 3.5.

ΔV_spl = (3α / 2r_c) 100%

Allowing for the relation between cluster radius r_c and the concentration of activated atoms on the basis of (45) as C_a = 3/2 r_c^-1, we obtain for ΔV_spl

ΔV_spl = α C_a (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 ΔV_c. The quantity of ΔV_c 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 ΔV_c 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 n_c, or

DV_c = - k_compn^’ / 2 n_c, (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; k_comp 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 С_а from (46), we derive:

DV_c = - k_comp С_а (138)

On the basis of /101/ we obtain k_comp= 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 = DV_spl + DV_c = a С_а – k_comp С_а = С_а (a - k_comp)

If we express it as a percentage, as it is accepted, we arrive at

DV = С_а (a - k_comp) 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 k_comp.

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)	DV_comp, %, calculation by (39)	DV_c, % 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
Fe_g	1835	20000	0.177	4.84	0	4.84	2.8-3.58
Fe_d	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.

6.10. The formation of shrinkage cavities and blisters in metals and alloys

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

(a_n + b) + (a_n + b) ® 2a_n + 2b;

(2a_n + 2b) + (a_n + b) ® 3a_n + 3b;

......................................................

(ia_n + ib) + (a_n + b) ® (i + 1)a_n + (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.:

r_cav = k_cav 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.

Chapter 7. Alloy Formation and the Structure of Liquid Metals

7.1. Of the mechanism of the formation of liquid alloys

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.

7.2. The point of metal dissolution and contact phenomena

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 C_A, the other one having the concentration of C_B. Let us presume that C_A exceeds C_B.

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 = C_A - C_B

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.

7.3. The formation of alloy structure in liquid state

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.

7.4. Cluster mixing stage in the formation of liquid alloys

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.

7.5. The stage of atomic diffusive mixing

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 A_mB_n. 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.

7.6. The stage of convective mixing in the formation of liquid metals

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.

7.7. Of the function of gravity in liquid metals formation

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.

7.8. The production of locally inhomogeneous alloys

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.

7.9. The formation of alloy hardening interval

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.

7.10. The structure of liquid metals with the unrestricted solubility of elements

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.

7.11. Alloys with the restricted solubility of elements in solid state and the unrestricted solubility in liquid state

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) a_a and a_b;

3. clusters, ‘covered’ or ‘separated’ by individual admixture atoms or monatomic admixture layers. For example, v. the clusters of one of the original components a_a, ‘covered’ with individual activated admixture atoms b: _b(a_a)_b.

Let us conditionally label such clusters as clad.

Let us emphasize that admixture atoms b also enter into cluster composition a_a 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 { a_a + a_b + a_{(am +bn)} + _b(a_a )_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.

7.12. The structure of liquid eutectic 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 {a_a + a_b}, (144)

where a_a and a_b 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 {a_a + a_ab + a_b}. (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.

Chapter 8. The Structure and Crystallization of Liquid Cast Iron

8.1. General data on iron-carbon alloys

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.

8.2. The formation of liquid cast iron structure

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.

8.3. The structure of liquid cast iron

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 {na_a + ma_g}, (146)

where a_a denotes austenite-like clusters; a_g 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 {na_a ®Fe₃C ¬ma_g}, (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:

a_a ®Fe₃C ¬a_g + t/2,

a_a « a_g + t,

a_a ®Fe₃C ¬a_g + 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).

8.4. The peculiarities of cast iron crystallization. The formation of gray cast iron

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.

8.5. The formation of white cast iron

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.

Chapter 9. Modifying

9.1. General data on modifying

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.

9.2. The mechanism of the influence of modifiers of the first type upon the process of crystallization

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 a_n + a_n® a_2ntwo 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 DF_v - 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 s_м.

By definition, s >s_м by the absolute quantity.

Expression (126) in case if we introduce some modifier will be presented as

DF_м = -(4/3)pr³ DF_v - 4pr²s_м, (150)

Subtracting (150) from (148), we obtain DF_м - DF = - 4pr² s_м +4pr² s or DF - DF_м = 4pr² (s-s_м) > 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.

9.3. The elements of the electron theory and the practical choice of modifiers of the first type for alloys

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 U_ef, 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 U_ef 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:U_me– U_mod

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.

U_me– U_mod> 0 – a modifier,

U_me– U_mod < 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 C_S. We shall conditionally assume that modifiers have the solubility in a hard matrix of this or that alloy that does not exceed one percent: C_S<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 = (U_me – U_mod)/ С_S, (151)

m being the coefficient of modifying activity.

Expression (151) is very simple and convenient for calculations. The quantities of C_S 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	C_S, % at. /119,120/	U_mod, /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	C_S, % at. /119,120/	U_mod, /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.

9.4. The choice of the amount of modifiers of the first type. Gas-like modifying mechanism

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.

9.5. Complex modifying

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.

9.6. Time factor at modifying

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.

9.7. Selective modifying

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.

9.8. Of the nucleation of solid phase on the surface of the modifier particles of the second type in cast alloys

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 r_p. 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 r_p 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>r_p) – or it does not (r=r_p).

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 (r_p). (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 DF_v + S DF_s, (153)

where DF_v is the change of the volumetric free energy of the system; DF_s 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 DF_v = (4/3)p(r³ – r_p³) DF_v. (154)

The change of the system’s free surface energy at the formation of solid phase on the solid inclusion surface is

S DF_v = S_p s_p – S s, (155)

where S_p is the particle surface area; S is the area of the external surface of a solid shell; s_p 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 DF_s = 4pr² s_p – 4pr² s. (156)

By inserting the values of S DF_v and S DF_s from (153) and (155) into (152), we obtain:

DF = (4/3)p(r³ – r_p³) DF_v + 4pr² s_p – 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, r_p). However, this is but the particle radius of r_p that acts as the independent argument in a physical sense, since the radius of the solid phase shell on the particle surface depends on r_p in its turn, or r = j (r_p).

If we consider that, let us find the first derivative DF = f (r, r_p) 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 = r_p = 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

4pr²DF_v (dr/dr_p) - 4pr²DF_v - 8prs (dr/dr_p) + 8pr_ps_p = 0.

Having completed the requisite cancellations, we obtain

DF_v r² (dr/dr_p) - DF_vr_p² - 2rs (dr/dr_p) + 2s_p r_p = 0. (159)

For further analysis, we are to know the mode of the function r = j (r_p). As a first approximation, the correlation between the quantities of r and r_p can be expressed by a linear dependency presented as r = A (r_p), 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 (r_p) is quite acceptable.

In this case dr/dr_p = А, and equation (159) assumes the form

А r² DF_v - 2А s r - DF_v r_p² + 2s_p r_p = 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 s_p through the independent argument of r_p.

Referring to B.Chalmers /65/, we obtain

s_p = r_p DF_vp /2 (161)

s = r_p DF_v /2 (162)

r = r_p is used in expression (161) to simplify the problem, because the shell radius may assume any values r ³ r_p 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 s_p and s from (161) and (162) into (160), we obtain:

А DF_v r² - А DF_v r r_p - DF_v r_p² + DF_vp r_p² = 0.

А DF_v r² - А DF_v r r_p + r_p² (DF_vp - DF_v) = 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 DF_vp >> DF_vtakes place.

Consequently, we can assume without any considerable error that DF_vp - DF_v @ DF_vp.

Then expression (163) will assume the form of

А DF_v r² - А DF_v r_p r + r_p² DF_vp = 0 (164)

Equation (164) as related to r is a regular quadratic equation of the type ax²+bx+c=0, where а = А DF_v; b = - А DF_v r_p; c = r_p² DF_vp .

The solution of the given expression is presented as:

r_1,2 = r_p {(1/2) + [(1/4) - (DF_vp /А DF_v)]^1/2} (165)

Now we can determine the nature of the extremum of the function of DF = f(r). Its flexion is

DF² = 2А DF_v r - А DF_v r_p . (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 DF_v r_p {(1/2) + [(1/4) - (DF_vp / А DF_v)]^1/2} - А DF_v r_p (167)

Since DF_vp < 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 DF_v r_p {(1/2) + [(1/4) - (DF_vp / А DF_v)]^1/2}

is more than the second one А DF_v r_p . 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.

9.9. Of the characteristics and choice of second-type modifiers for alloys

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 DF_vp and DF_vthan on r_p. By B.Chalmers /65/, we obtain:

DF_vp = DН_p DТ_p/ Т_Lp;

DF_v = DН DТ/ Т_Lm, (168)

where DН_p is the enthalpy of the forming of modifier substance; DТ_p = Т_Lp - Т; Т_Lp is the melting temperature of the substance of a second-type modifier.

DН, DТ and Т_Lm represent the same for the metal of the melt; Т being the current temperature.

By introducing (168) into (165), we obtain:

r_1,2 = r_p {(1/2) + [(1/4) - (DН_p DТ_p Т_Lm / А DН DТ Т_Lp )]^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 DН_p is, the more r increases.

In the third place, the greater the difference of the melting temperatures of the modifier and the given alloy (DТ_p) 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.

9.10. Second-type demodifiers

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:

Т_Lp @ Т_Lm and DН_p = D Н.

Then, on the basis of (168) we obtain:

DF_vp = DF_v and DF_vp - DF_v = 0.

In this case, equation (163) assumes the form:

А DF_v r² - А DF_v r r_p = 0 (169)

Hence we derive

А DF_v r(r - r_p) = 0 (170)

Equation (171) can be solved in two ways:

r₁ = 0; r₂ = r_p (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.

T_Lp << T_Lm .

Then DF_vp << DF_v and DF_vp - DF_v @ - DF_v

In this case, equation (162) will assume the form of

А DF_v r² - А DF_v r r_p - r_pDF_v = 0 (173)

Equation (173) has the following roots

r_1,2 = r_p {(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 < r_p.

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.

Chapter 10 Experimental Research of Liquid Alloy Structure

10.1. Of the possibility of the experimental determination of the dimensions of structural elements of matter - i.e. clusters - in liquid alloys

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.

10.2. The theory of sedimentation experiments aiming at the study of the structural units of matter in liquid alloys

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

F_v = m dV/dt

There being no auxiliary effects, the given force equates with gravity:

F_g = mg = 4pr³Dg/3.

We should also take into consideration the force of internal friction (viscosity):

F_h = 6 phrV.

Archimedean force resists gravity: F_d = 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

F_v = F_g - F_h - F_d , (177)

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^-A^t - B/A,

where С is a constant.

It follows from the initial conditions that V = 0 at t = 0, thus

V = B (e^-A^t - 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]

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 V_e. The given value can be determined if we assume that е^-А^t®0 at t®¥, so according to (180)

V_e = 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

F_g = 4pr³ (D - d)g dh/3,

and the Brownian motion force

F_b = -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.

10.3. Of the Brownian motion 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 + N_c + N_b » N₀ ,since N_c + N_b << N₀ , where N_c is the number of clusters per mole of liquid; N_b 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 (N_p = 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.

10.4. Sedimentation research procedures of the structure of liquid alloys

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)[(b_tDT/a) + (b_cDC/D)],

where h is the height of the metal layer; DT is the in-layer temperature drop; DC is the difference in concentrations; b_c 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.

10.5. Sedimentation in Pb-Sn liquid alloy

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.

10.6. Sedimentation of the elements of matter in Bi-Cd liquid alloy

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.

10.7. Sedimentation in Zn-Al liquid alloy

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.

10.8. Sedimentation in Al-Si liquid alloys /149/

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.

10.9. Sedimentation in liquid casting bronze /152/

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.

10.10. Sedimentation in liquid cast iron /144/

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.

10.11. The influence of the sample height upon the degree of inhomogeneity obtained in liquid alloys

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	C_upper, %, experiment	C_lower, %, experiment	C_upper, %, calculation by (183)	C_lower, %, 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.

10.12. Conglomeration in liquid alloys as a result of sedimentation

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.

10.13. The calculation of the dimensions of conglomerates that are formed during sedimentation process in liquid alloys

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	r_Bi = 9.05

Pb-Sn	60	200	r_Bi = 4.0-4.9	r_Sn = 5.6
		250	r_Bi = 3.6-4.2
		350	r_Bi = 2.1-3.7

Al-Si	6.0	650	r_Si = 11.2
	6.9	650	r_Si = 11.5
	8.0	650	r_Si = 11.5
	9.2	650	r_Si = 12.5
	10.8	650	r_Si = 12.1
	12.1	650	r_Si = 12.1
	13.3	650	r_Si = 11.3
	13.8	650	r_Si = 11.4

Cu-Sn	5.0	1100	r_Sn = 4.2-2.1
	5.0	1050	r_Sn = 8.1-7.0

Cu-Sn-Pb	5.0%Sn+4.9%Pb	1050	r_Sn = 3.8-5.6
		1100	r_Sn = 1.8-2.7

Fe-C	4.2	1200	r_C = 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.

10.14. Considering the experimental data of sedimentation in liquid alloys within gravity field at suppressed convection. Formation of cluster conglomerates and the hierarchy of labile structure levels in liquid alloys

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.

10.15. Of the mechanisms of metallurgical heredity

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.

Conclusion

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!

References *

1. Ubbelode A.R. Melting and Crystalline Structure. – Moscow: Mir, 1969

2. Ubbelode A.R. The Molten State of Matter. – Moscow: Metallurgy, 1982

3. Stewart G.W. //Rev. Modern Physics - 1930, 2, - p.116*

4. Stewart G.W., Benz C.A. //Phys. Rev. - 1934, 46, - p.703*

5. Kirkwood I.G. //J. Chem. Phys. - 1935, 3, - p.300*

6. Kirkwood I.G. //J. Chem. Phys. - 1936, 19, - p.275*

7. Bernal J.D. //Trans. Faraday Soc. - 1936, 33, - p.27*

8. Bernal J.D. //Nature - 1960, 185, - p.68*

9. Gingrich N.S. //Rev. Modern Phys. - 1943, 15, - p.99*

10. Danilov V.I. The Structure and Crystallization of Liquids. – Kiev: The Ukrainian Academy of Sciences Press, 1937

11. Danilov V.I., Radchenko A.V. //Physics, Technics, Fuel, - 1937, 12, - p.756

12. Wilson D.R. The Structure of Liquid Metals. – Moscow: Metallurgy, 1973

13. Brout R. Changes of Phases. – Moscow: Mir, 1967

14. March N.H. Liquid Metals. - Moscow: Metallurgy, 1972

15. Brown N., March N.H. Liquid Metals and Alloys //Met. Chem. Proc. Symp. - 1972, - p.245*

16. Vertman A.A., Samarin A.M. //Proceedings of the USSR Academy of Sciences. Technics Department of Metallurgy and Fuel. - 1960, #3, - p.17

17. Vertman A.A., Samarin A.M. Iron Melts Properties. - Moscow: Naooka, 1969

18. Born M., Green H.S. //Proc. Roy. Soc. - 1946, A188, - p.10*

19. Percus J.K., Yewick G.I. // Phys. Rev. - 1958, v.110, - p.1*

20. Byelashtchenko D.K. Transfer Phenomena in Liquid Metals and Semi-Conductors. – Moscow: Atomizdat, 1970

21. The Physics of Simple Liquids. Ed. by H.N.V. Temperley. – Moscow //Mir, 1971, - 308 p.

22. Ross R.G.// Phil. Mag. - 1970, 22, - p.573*

23. Halder N.C., Wagner C.N.// Naturforsch. - 1968, 23a, N7, - p.992*

24. Waseda J., Suzuki K. // Phys Letters - 1971, A35, N5, - p.315*

25. Byelashtchenko D.K., Spektor Ye. Z. – ‘The Theory of Metallurgical Processes’ collected works, v.3, Moscow: Metallurgy, 1973, - p.5

26. Vatolin N.A., joint authorship // Reports of the USSR Academy of Sciences. - 1974, 219, #6, - p.1394

27. Paskin A., Raman A. //Phys. Rev. Letters - 1966, 16, - p.300*

28. Bernal J.D.// Nature. - 1960, 185, - p.68*

29. Lindeman F.A.// Phys. Zs. - 1910, N11, - p.609*

30. Gavrilin I.V., Ryizhikov A.A. Certain Peculiarities of the Process of Melting and the Structure of Liquid Metals // Extracted from the work of ‘Progressive Foundry Technology’ - Gorky, 1969 - p.3

31. Arkharov V.I., Novokhatsky I.A. // Reports of the USSR Academy of Sciences. - 1969, 188, #5 - p.1069

32. Zhoukov A.A. Thermodynamic Characteristics of Melts. - Novokuznetzk. – 1969, –p.149

33. Izmailov V.A., Vertman A.A., Samarin A.M. //Foundry - 1971, #1, - p.30

34. Romanova A.V., Melnik B.A. // The Physics Magazine of Ukraine. - 1970, #5, - p.712

35. Spektor Ye. Z., Snezhko O.M., Baum B.A., Geld P.P. //Collegiate Reports. Physics. - 1971, #10, - p.68

36. Filippov Ye.S., Krestovnikov A.V. // Collegiate Reports. Ferrous Metallurgy. - 1969, #9, - p.111

37. Kumar R. An extract from ‘The Physical-Chemical Basics of Metallurgical Processes’. Moscow: Naooka, 1973, - p.288

38. Filippov Ye.S., Krestovnikov A.V. // A Physical Chemistry Magazine. - 1971, #4, - p.946

39. Yershov G.S., Poznyak L.A. The Microinhomogeneity of Metals and Alloys. – Moscow: Metallurgy. - 1985

40. Romanova A.V., Melnik B.A. // Reports of the USSR Academy of Sciences. - 1969, 159, #2, - p.294

41. Grinberg B.M., Spektor Ye. Z. //Proceedings of the USSR Academy of Sciences. Metals. - 1971, #4, - p.131

42. Spektor Ye. Z. // Reports of the USSR Academy of Sciences. - 1970, 190, #6, - p.1322

43. Arkharov V.I., Gorokh A.V., Novokhatsky I.A. // Reports of the USSR Academy of Sciences. - 1972, 206, #6, - p.1377

44. Gorokh A.V., Arkharov V.I. // Reports of the USSR Academy of Sciences. - 1970, 213, #6, - p.1353

45. Filippov Ye.S. // Proceedings of the USSR Academy of Sciences. Metals. - 1973, #1, - p.92

46. Filippov Ye.S. // Collegiate Reports. Ferrous Metallurgy. - 1975, #9, - p.126

47. Arsenyev P.P., Filippov Ye.S. // Collegiate Reports. Ferrous Metallurgy. - 1971, #1, - p.123

48. Fisher I.Z. Statistic Theory of Liquid. – Moscow: Physmatgiz. - 1961

49. Levin Ye.S., Ayushina G.D., Geld P.P. // Proceedings of the USSR Academy of Sciences. Metals. - 1970, #4, - p.49

50. Yevseyev A.M., Vertman A.A., Samarin A.M. // Proceedings of the USSR Academy of Sciences. Metals. - 1969, #6, - p.195

51. Tyagounov G.V., Baum B.A., Geld P.P. //The Works of the Ural Polytechnical Institute. - 1970, #186, - p.182

52. Spektor Ye. Z., Khokhlov S.V. //The Problems of Metal Research and Physics of Metals. – Moscow: Metallurgy, - 1968, - p.55

53. Doutchak L.I., Frenchko V.S., Voznyak O.M. // Collegiate Reports. Physics. - 1974, #4, - p.101

54. Styles G.A.// Advances in Physics. - 1967, 16, N62, - p.275*

55. Kumar R., Sivaramakrishnan C.S.//J. Mat. Sci. - 1969, 4, N11, - p.48*

56. Korsounsky V.I., Naberoukhin Yu.I. Of the Effect of Centrifuging on the Microheterogeneous Structure of Metallic Eutectic Melts. // Proceedings of the USSR Academy of Sciences. Metals. - 1973, #5, - p.1982

57. Zalkin V.M. Of the Structure of Melts in Binary Metallic Systems with a Eutectic Diagram of State. //Physkhimiya. - 1972, 46, #9

58. Liquid Alloys Structure Discussion Materials. // Proceedings of the USSR Academy of Sciences. ОТ Metallurgy and Fuel. - 1960, #6,; 1961, #3

59. Bounin K.P. To the Issue of the Structure of Metallic Eutectic Melts. Proceedings of the USSR Academy of Sciences. ОТН. - 1946, #3, - p.308

60. Taran Yu.N., Mazur V.I. The Structure of Eutectic Alloys. – Moscow: Metallurgy. - 1978, - 311p.

61. Taran Yu.N., joint authorship The Effect of Melts Centrifuging Conditions on Silicon Distribution in Silumins. //’The Problems of Forming a Metastable Structure of Alloys.’ Collected works. – Dnepropetrovsk. - 1983, - p.51

62. Zhoukov A.A., Snezhnoy R.L. ‘The Properties of Molten Metals’ collected works. – Moscow: Naooka. - 1974, - p.15

63. Hollomon J.H., Turnbull D. // Progress in Metal Physics. - 1963, - p.333*

64. Kuznetzov V.D. Crystals and Crystallization. – Moscow. - 1953

65. Chalmers B. The Hardening Theory. – Moscow: Metallurgy, - 1968

66. Cahn R. Physical Metallurgy. V. 2-3. – Moscow: Mir, - 1968 - 1972

67. Chalmers B. Physical Metallurgy. - Moscow: Metallurgy, - 1963

68. Winegard W. An Introduction into the Physics of Metal Crystallization. - Moscow: Mir - 1967

69. Frenkel Ya.I. A Kinetic Theory of Liquids. – Moscow: The USSR Academy of Sciences Press. - 1945

70. Frenkel Ya.I. An Introduction into the Theory of Metals. – Moscow: Physmatgiz, - 1973

71. Volmer M. Kinetik der Phasenbildung - Dresden - Leipzig: Steinkopff, - 1939*

72. Becker M., Doring W. // Ann. Phys. - 1955, N24, - s.719*

73. Saratovkin D.D. Arborescent Crystallization. – Moscow: Metallurgizdat, - 1957

74. Balandin G.F. The Forming of the Crystalline Structure of Castings. – Moscow: Mashinostroyeniye (Machine-Building), - 1973

75. Ryizhikov A.A. Theoretical Basics of Foundry. – Moscow: Mashgiz, - 1971

76. Tamman G. Krystallizieren und Schmelzen. - Leipzig, - 1903*

77. Flemings M. Hardening processes. - Moscow: Mir, - 1977

78. Kolmogorov A.N. To the Issue of the Statistic Theory of Crystallization of Metals // Proceedings of the USSR Academy of Sciences (mathematical series), - 1937, #3

79. J.W. Christian. The Theory of Transportation in Metals and Alloys. - Oxford-New York: Pergamon Press. – 1974*

80. Gavrilin I.V. The Basics of the Model of Microinhomogeneous Structure of Liquid Metals. // ‘The Improving of Castings and Ingots’ collected works. – Gorky: Gorky Polytechnic Institute Press, - 1979, 2^nd ed., - p.5.

81. Gavrilin I.V. The Control of the Crystallization Centers Formation Process in Castings on the Basis of Analyzing the Structure and Properties of Metallic Melts. //A Thesis for a Doctor’s Degree. – Vladimir: Vladimir Polytechnic Institute Press. - 1977

82. Gavrilin I.V. Of the Mechanism of Chemical and Structural Inhomogeneity Developing in Liquid Alloys in Relation to Their Structure. // ‘Heredity in Cast Alloys’ collected works. – Kouybyishev, - 1987, - p.18

83. Gavrilin I.V. Dynamic Microinhomogeneity in Liquid Alloys. //Foundry. - 1996, #11, - p.9

84. Vernadsky V.I. Naturalist’s Reflections. Space and Time in Living and Inorganic Matter. – Moscow: Naooka, - 1975.

85. Kurnakov N.S. Selected Works. V.1-3. - Leningrad: The USSR Academy of Sciences Press, - 1939

86. Feynman R., Layton R., Sands M. Feynman Lectures on Physics. V. 1-2. – Moscow: Mir, - 1966

87. Damask А.C., Dienes G.J. Point Defects in Metals. - Moscow: Mir, - 1966

88. Schottky W. // Zs. Phys. Chem. - 1935, 29, s.335*

89. Guegousin Ya.Ye. Microscopic Defects in Metals. - Moscow: Metallurgy, - 1962

90. Pinness B.Ya. //JTP, - 1967, 10^th ed., - p.2314

91. Gabidoullin R.I., joint authorship // Collegiate Reports. Non-Ferrous Metallurgy. - 1976, #4, - p.87

92. Kotelnikov R.B., Bashlyikov S.N., Galiakbarov Z.G., Kashtanov A.I. Extra-refractory Elements and Compounds. – Moscow: Metallurgizdat, - 1969

93. Blandin F., Laniel E., Friedel I. // Phyl. Mag. - 1959, v.4, - p.1806*

94. Hirs D., Pound G. Vaporization and Condensation. – Moscow: Metallurgy, - 1966

95. Gavrilin I.V., Yershov G.S. // Collegiate Reports. Ferrous Metallurgy. - 1973, #4, - p.149

96. Ryizhikov A.A., Gavrilin I.V. ‘Heat Processes in Castings and Molds’ collected works. – Moscow: Naooka, - 1972, p.41

97. Krestovnikov A.V., Vladimirov L.P., Goulyanitzky B.S., Fisher A.Ya. A Reference Book on Calculating Metallurgical Reactions. - Moscow: Metallurgizdat, - 1963

98. Properties of Elements. V.1. Ed. By Samsonov G.V. - Moscow: Metallurgy, - 1976

99. Kumar R. // Proc. Nucl. Phys. and Solid State Symp. - Bombay, - 1968, v.3. - P.258*

100. Kisounko V.Z., Ladianov V.I., Arkharov V.I. //Physics of Metals and Physical Metallurgy, - 1973, 36, #3, - p.52

101. Bokshtein S.Z. The Structure and Properties of Metallic Melts. - Moscow: Metallurgy, - 1971

102. Regel L.A. Outer Space Exploration. V.21. Metal Research in Outer Space. – Moscow, - 1989

103. Eyring H.J., Gleston S., Leyder K. The Absolute Rates of Reactions Theory. – Moscow: ИЛИ, - 1948

104. Trakhtenberg B.F., Krestyanov V.I. A Physical Model of Heredity By the Third-Type Modifying Scheme. //Foundry. - 1999, #1, - p.16

105. Nikitin V.I. Heredity in Cast Alloys. – Samara: Samara State Technical University Press, - 1995, - 248p.

106. Kisounko V.Z., joint authorship // Proceedings of the USSR Academy of Sciences. Metals. - 1975, #2, - p.176

107. Shvidkovsky Ye.G. Certain Problems of the Viscosity of Molten Metals. – Moscow: Gostekhteorizdat. - 1958

108. Panchenkov G.M. Liquids Viscosity Theory. – Moscow-Leningrad: Gostopizdat. - 1947

109. Born M, Green H.C. General Kinetic Theory of Liquids. - Cambridge. – 1949*

110. Andrade E.N. da C. //Phyl. Magazin. - 1934, 17, - p.497*

111. Saxton H.J. and Sherby O.D.// Amer. Soc. Metal. Trans. Quart. - 1962, 55, - p.826*

112. Cohen M.H., Turnbull D.J.// Chem. Phys. - 1959, 31, - p.1164*

113. Swalin R.A.// Acta Metallurgica. - 1961, 9. - P.379*

114. Gavrilin I.V., Yershov G.S. Admixture Diffusion in Liquid Iron. // Proceedings of the USSR Academy of Sciences. Metals. - 1974, #1, - p.23

115. Gavrilin I.V., Yershov G.S. Of the Mechanism of Self-Diffusion in Liquid Metals. // Proceedings of the USSR Academy of Sciences. Metals. - 1976, #5, p.87

116. Semenchenko V.K. Selected Chapters on Theoretical Physics. – Moscow: Prosveshtcheniye. - 1966

117. Onopriyenko G.I., Kouzmenko G.N., Kharkov Ye.I. The Physics Magazine of Ukraine. - 1967, #12, - p.39

118. Arkharov V.I., Novokhatsky I.A., Yershov G.S., Kovalenko A.M. // Reports of the USSR Academy of Sciences. - 1970, #6, - p.1329

119. Hansen M., Anderko K. The Structures of Binary Alloys. V.1-2. – Moscow: Metallurgizdat. - 1962

120. Elliot R.P. The Structures of Binary Alloys. – Moscow: Metallurgy. - 1962

121. Arkharov V.I., joint authorship // Proceedings of the USSR Academy of Sciences. Metals. - 1969, #6, - p.46

122. Elliot D.P., Glazer М., Ramakrishna R. Thermochemistry of Steel-Smelting Processes. – Moscow: Metallurgy. - 1969

123. Povolotzky D.Ya., Roshtchin R.Ye. // Proceedings of the USSR Academy of Sciences. Metals. - 1969, #1, - p.239

124. Povolotzky D.Ya., Roshtchin R.Ye., Keis А.N.// Proceedings of the USSR Academy of Sciences. Metals. - 1972, #5, - p.84

125. Povolotzky D.Ya., Roshtchin R.Ye., Keis А.N.// Proceedings of the USSR Academy of Sciences. Metals. - 1972, #5, - p.222

126. Gourevitch Yu.G. Titanium Nitrides in Stainless Steel. Thesis for a Doctor’s Degree. - Zlatoust, - 1970

127. Zhdanov G.S. Physics of Solids. – Moscow: Moscow State University Press, - 1962

128. Yershov G.S., Gavrilin I.V. // Proceedings of the USSR Academy of Sciences. Metals. - 1976, #2, - p.98

129. Gavrilin I.V., joint authorship Of the Change of the Volume of Metals at Melting. // Proceedings of the USSR Academy of Sciences. Metals. - 1974, #2, p.317

130. Gavrilin I.V. Of the Competition Theory of Crystallization. // Foundry. - 1998, #10, p.4

131. Zalkin V.M. The Nature of Eutectic Melts and Contact Melting Effect. – Moscow: Metallurgy. - 1987, - 152p.

132. Rebinder P.A., Lihtman M.S. Researches in Applied Physical Chemistry of Surface Phenomena. Moscow: Scientific and Technical Publishing House. - 1937

133. Goldshtein L.Ye. Steel and Cast Iron Microalloying. – Moscow: Mashgiz, 1959

134. Arsenyev P.P., Filippov Ye.S. // Collegiate Reports. Ferrous Metallurgy. - 197-, #3, - p.18

135. Goulyayev B.B. Foundry Processes Theory. Leningrad: Mashinostroyeniye (Machine-Building), - 1976

136. Tiller W., Takahashi T. // Acta metallurgica. - 1969, 17, N4, p.483*

137. Kreshtchanovsky P.S., Sidorenko M.F. Steel Modifying. – Moscow: Metallurgy, - 1970, - 295p.

138. Grigorovitch V.K. The Electron Structure and Thermodynamics of Iron Alloys. – Moscow: Naooka. - 1970

139. Vozdvizhensky V.M. The Forecast of Binary Diagrams of State. – Moscow: Metallurgy. - 1975

140. Gavrilin I.V. The Forming of Cast-Iron Structure at Melting and Crystallization. //Foundry. - 1998, #6, - p.4

141. Gavrilin I.V., Sharshin V.N., Tikhonov N.P. Chemical and Structural Inhomogeneity in Liquid Alloys. // Proceedings of the USSR Academy of Sciences. Metals. - 1988, #4, - p.44

142. Gavrilin I.V. Sedimentation Experiment in Liquid Alloys Study. // Proceedings of the USSR Academy of Sciences. Metals. - 1985, #2, - p.86

143. Gavrilin I.V., Frolova T.B. Of Segregation in Liquid Eutectic Alloys. // Proceedings of the USSR Academy of Sciences. Metals. - 1984, #3, - p.191

144. Gavrilin I.V. Carbon Distribution in Liquid Cast Iron. //Foundry. - 1982, #4, - p.2

145. Gavrilin I.V., Yershov G.S. The Application of the Model of the Microinhomogeneous Structure of Liquid Metals to Calculating Their Viscosity. // Proceedings of the USSR Academy of Sciences. Metals. - 1979, #3, - p.72

146. Gavrilin I.V. To the Issue of Calculating Coordinating Numbers in the Model of the Microinhomogeneous Structure of Molten Metals. // Proceedings of the USSR Academy of Sciences. Metals. - 1979, #5, - p.49

147. Gavrilin I.V., Ryizhikov A.A., Kalliopin I.K. Of the Choice of the Rational Modifiers of the First Type for Steel. // Collegiate Reports. Ferrous Metallurgy. - 1977, #4.

148. Gavrilin I.V., Kalliopin I.K., Yershov G.S. Titanium Modifying of Nitrogenated Steel. // Collegiate Reports. Ferrous Metallurgy. - 1976, #2, - p.64

149. Gavrilin I.V. Silicon Segregation in Liquid Silumins. //Foundry. - 1982, #1, - p.3

150. Gavrilin I.V. Equilibrium Chemical Inhomogeneity in Liquid Cast Alloys. //Foundry. - 1986, #1, - p.3

151. Perren J. The Brownian Motion and the Reality of Molecules. – S.-Petersburg: Obshtchestvennaya Polza (‘Public Benefit’). - 1912, - 96p.

152. Gavrilin I.V., Korogodov Yu.D., Sharshin V.N., Tikhonov N.P. Equilibrium Chemical Inhomogeneity in Liquid Cast Alloys. // Foundry. - 1984, #1, - p.3

153. Gavrilin I.V., Choudner R.V., Staroselsky M.A. //Shipbuilding Problems. - 1975, 3^rd ed., - p.3

154. Gavrilin I.V. The Equilibrium Inhomogeneous Distribution of Components in Liquid Alloys. //The Works of the All-Union Conference on Metallic Melts and Melted Slags Structure. - Sverdlovsk: The Ural Scientific Center of the USSR Academy of Sciences. - 1986, - p.242

155. Gavrilin I.V. Of the Mechanism of Chemical and Structural Inhomogeneity Developing in Liquid Alloys in Relation to Their Structure. //’Heredity in Cast Alloys’ collected works. - Kouybyishev, - 1987, - p.1

[1] Celsius

* Works issued otherwise than in Russian are marked with an asterisk.

Igor

Igor V. Gavrilin

Melting and Crystallization of Metals and Alloys

Table of Contents

Annotation. 5

Introduction. 7

Chapter 1. Melting and Crystallization. State of Issue. 9

1.1. Thermodynamics of melting and crystallization. 9

1.2. Structural theories of melting and crystallization and the structure of liquid metals. 11

1.3. Statistic and monatomic theories of liquid state of metals. 13

1.4. Models of microinhomogeneous structure of liquid metals. 15

1.5. The interrelation of monatomic and cluster approaches and the connection of various theories to experimental data. 17

1.6. The elements of the thermodynamic theory of crystalline centers nucleation. 18

1.7. The elements of the heterophase fluctuations theory as applied to the theory of crystallization. 20

1.8. Of the combination of the thermodynamic theory of nucleation with the heterophase fluctuations theory. 21

1.9. Of the growth of crystals theory. 23

1.10. Of the correlation of the structural theory of crystallization with the theory of hardening. 25

1.11. Statistic theory of crystallization. 26

1.12. Of the links between the thermodynamic, fluctuation and probability theories of crystallization. 27

Chapter 2. General Principles of the Structure of Liquid and Solid Metals as Systems of Interacting Elements of Matter and Space. 28

2.1. General definition of the states of aggregation of matter as different ways of matter-in-space distribution. 28

2.2. Premelting. 31

2.3. Melting centers. 33

2.4. The correlation of melting centers with the centers of crystallization. 34

Chapter 3. The Mechanism of Metals and Alloys Melting Process. 35

3.1. The elementary act of melting and the formation of the structural units of matter and space in liquid state. 35

3.2. Calculating the elementary act of melting. 36

3.3. The structure of liquid metals at the point of their formation. 39

3.4. Calculating cluster dimensions in liquid metals at the melting temperature. 40

3.5. Calculating the dimensions of spatial elements in liquid metals at the melting temperature. 45

3.6. Calculating the energy of clusters and intercluster splits motion in liquid. 48

3.7. Calculating the point of metal melting. 49

3.8. Calculating the content of activated atoms in liquid. 50

3.9. The frequency of heat oscillations in clusters and the frequency of intercluster split flickering in liquid metals. 50

3.10. The period of cluster existence. 52

Chapter 4. The Change of Liquid Metals Structure at Heating and Cooling. 54

4.1. Basic theses. 54

4.2. The modification of cluster dimensions with the change of temperature. 55

4.3. The modification of the volume of spatial elements in liquid metals with the change of temperature. 56

Chapter 5. Modifications of the Characteristics of Metals at Melting and Structure-Sensitive Properties of Liquid Metals. 58

5.1. Of the connection between the structure and characteristics of liquid metals. 58

5.2. The mechanism of fluidity in liquid metals. 58

5.3. Viscosity in liquid metals. 60

5.4. Self-diffusion in liquid metals. 62

5.5. Comparing the effects of mass transfer in various states of aggregation of matter in metals. 66

5.6. Admixture diffusion in liquid metals and alloys. 68

5.7. Admixture diffusion in liquid iron. 71

5.8. Of the change of coordinating numbers at melting. 74

5.9. Of the change of the electrical resistivity of metals at melting. 77

Chapter 6. Of the Mechanism of Crystallization of Metals and Alloys. 81

6.1. Precrystallization as the mutual extrusion of dominant and latent elements of matter and space in liquid state. 81

6.2. The formation of crystallization centers. 84

6.3. The overcooling problem at crystallization. 86

6.4. Spontaneous and forced crystalline centers nucleation in liquid metals. 89

Silver 90

6.5. The frequency of crystalline centers nucleation. 91

6.6. Time factor at crystallization. 92

6.7. The problem of mass crystalline centers nucleation. 93

6.8. The competition theory of crystallization. 94

6.9. Of the change in the volume of metals at melting and crystallization. 97

6.10. The formation of shrinkage cavities and blisters in metals and alloys. 101

Chapter 7. Alloy Formation and the Structure of Liquid Metals. 103

7.1. Of the mechanism of the formation of liquid alloys. 103

7.2. The point of metal dissolution and contact phenomena. 103

7.3. The formation of alloy structure in liquid state. 106

7.4. Cluster mixing stage in the formation of liquid alloys. 106

7.5. The stage of atomic diffusive mixing. 107

7.6. The stage of convective mixing in the formation of liquid metals. 108

7.7. Of the function of gravity in liquid metals formation. 108

7.8. The production of locally inhomogeneous alloys. 109

7.9. The formation of alloy hardening interval 110

7.10. The structure of liquid metals with the unrestricted solubility of elements. 110

7.11. Alloys with the restricted solubility of elements in solid state and the unrestricted solubility in liquid state. 111

7.12. The structure of liquid eutectic alloys. 111

Chapter 8. The Structure and Crystallization of Liquid Cast Iron. 114

8.1. General data on iron-carbon alloys. 114

8.2. The formation of liquid cast iron structure. 114

8.3. The structure of liquid cast iron. 115

8.4. The peculiarities of cast iron crystallization. The formation of gray cast iron. 116

8.5. The formation of white cast iron. 117

Chapter 9. Modifying. 118

9.1. General data on modifying. 118

9.2. The mechanism of the influence of modifiers of the first type upon the process of crystallization. 119

9.3. The elements of the electron theory and the practical choice of modifiers of the first type for alloys. 120

9.4. The choice of the amount of modifiers of the first type. Gas-like modifying mechanism.. 125

9.5. Complex modifying. 126

9.6. Time factor at modifying. 126

9.7. Selective modifying. 127

9.8. Of the nucleation of solid phase on the surface of the modifier particles of the second type in cast alloys. 128

9.9. Of the characteristics and choice of second-type modifiers for alloys. 132

9.10. Second-type demodifiers. 135

Chapter 10 Experimental Research of Liquid Alloy Structure. 137

10.1. Of the possibility of the experimental determination of the dimensions of structural elements of matter - i.e. clusters - in liquid alloys. 137

10.2. The theory of sedimentation experiments aiming at the study of the structural units of matter in liquid alloys. 137

10.3. Of the Brownian motion experiments. 140

10.4. Sedimentation research procedures of the structure of liquid alloys. 141

10.5. Sedimentation in Pb-Sn liquid alloy. 143

10.6. Sedimentation of the elements of matter in Bi-Cd liquid alloy. 144

10.7. Sedimentation in Zn-Al liquid alloy. 145

10.8. Sedimentation in Al-Si liquid alloys /149/ 147

10.9. Sedimentation in liquid casting bronze /152/ 148

10.10. Sedimentation in liquid cast iron /144/ 149

10.11. The influence of the sample height upon the degree of inhomogeneity obtained in liquid alloys. 149

10.12. Conglomeration in liquid alloys as a result of sedimentation. 150

10.13. The calculation of the dimensions of conglomerates that are formed during sedimentation process in liquid alloys. 151

10.15. Of the mechanisms of metallurgical heredity. 154

Conclusion. 156

References. 158

Annotation

The principles mentioned above enable the author to initiate a new quantitative theory of the melting and crystallization of metals.

Introduction

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?

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.

The book furnishes clues to all of them.

A new theory of liquid cast iron structure and crystallization was worked out.

Some additional possibilities and perspectives of the new approach are outlined in the conclusion.

Chapter 1. Melting and Crystallization. State of Issue

1.1. Thermodynamics of melting and crystallization

The most general and undoubtedly correct description of melting and crystallization processes originates from thermodynamics.

Still, it is accepted to start from thermodynamics in order to establish the generalized truth criteria touched upon before.

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/.

G_S=G_L (1)

where G_S is Gibbs’ free energy index for solid phase; G_L 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).

Until present, it has been impossible to find any convincing proofs of the real existence of such dualism.

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.

If it is so, what are those factors, unknown to science?

Questions generating...

Δ S_f=R ln W_l /W_s, (2)

where W_l is the number of independent ways of substance realization in liquid state, W_s - the same for solid state.

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’.

The last substantial experimental discovery in this sphere was made in the early 20^th century together with the development of X-rays- and further on - neutron diffraction.

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.

However, the supporters of the gaseousness of liquid metals structure did have a considerable amount of data in their favor.

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.

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.

1.3. Statistic and monatomic theories of liquid state of metals

The connection of the statistic theory of liquid metals with melting is not established directly, displaying the weakness of the theory under analysis.

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.

1.4. Models of microinhomogeneous structure of liquid metals

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/.

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.

1.5. The interrelation of monatomic and cluster approaches and the connection of various theories to experimental data

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.

1.6. The elements of the thermodynamic theory of crystalline centers nucleation

Let us consider the problem how the concept of the critical dimensions of crystalline centers arose in present theory.

ΔG = -ΔG_v + ΔG_s,

where ΔG_v is specific volumetric free energy; ΔG_s is specific surface energy.

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.

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.

Δ G_s= 1/3 S S_is_i,

where S_iis the specific surface of the i^th zone of an equilibrium crystal at nucleation; s_iis 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 r_cis the radius of the sphere inscribed into a cube of the critical dimension.

To calculate the quantity of r_cof a spherical nucleus the following formula is used /65,67,68/:

r_c= 2s T₀/ L Δ T. (4)

Graphically, the last expression corresponds to the minimum in Pict.2.

Δ G = 1/3 Ss = 32 s³T₀ / L²(Δ T)²

1.7. The elements of the heterophase fluctuations theory as applied to the theory of crystallization

(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.

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 s³T₀lg e / L²(Δ T)²k

Ya.S.Umansky considers a particular example of homogeneous crystallization of iron at the discriminate overcooling of 100, 200 and 295⁰C /68/.

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 10¹⁰erg/sq.cm.

Hence

lgI/K = -32 200³1800 0.434/4 10²⁰1.38 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

ΔG_heterog/ Δ G_homog= 0.06.

K = K_lexp(-U/ RT),

n = K_lexp(-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.

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

It is assumed that crystal growth depends on the geometry of growing crystalline planes as well as growth direction /66–67/.

v =K₂ exp (-U / RT ) exp [-E ( s )² / T (Δ T)²], (7)

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.

v = K₄Δ T,

where K₄is 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.

1.10. Of the correlation of the structural theory of crystallization with the theory of hardening

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

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/.

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.

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

The basic premises of the new approach are the following.

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/.

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.

Unfortunately, those were nothing but separate phrases in the works by N.S.Kurnakov, which never found their evolution /85/.

Such a definition of gas quite agrees with the way of description accepted in this work.

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).

2.2. Premelting

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.

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 p within the crystalline lattice can be calculated out of the known relation:

p_v = 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, p_v, 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