DIFFUSION POROSITY IN METALS AND ALLOYS

Ya. E. Geguzin

CONTENTS

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

II. Diffusion porosity arising during the mutual diffusion of metals forming substitutional solid solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
II.1. Mutual diffusion of metals forming substitutional solid solutions . . . . . . . . 219
II.2. Regularities in the occurrence of diffusion porosity . . . . . . . . . . . . . . . 221
II.3. Protective edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
II.4. Change in the volume of a diffusion specimen during diffusion . . . . . . . . . 225
II.5. Evaporation of the volatile component from an alloy . . . . . . . . . . . . . . 229

III. Diffusion porosity in one-component systems . . . . . . . . . . . . . . . . . . . 230

IV. Supersaturation of the crystal lattice with vacancies . . . . . . . . . . . . . . . 232
IV.1. Supersaturation with vacancies arising during mutual diffusion in substitutional solid solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
IV.2. Concentration of excess vacancies in one-component systems . . . . . . . . . 235

V. Nucleation of diffusion pores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
V.1. Critical nucleus of diffusion pores (negative crystals) . . . . . . . . . . . . . . 238
V.2. Role of “impurities” in the nucleation of diffusion pores . . . . . . . . . . . . 239

VI. Diffusion porosity and sintering of mixtures of metallic powders . . . . . . . . 241

VII. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

Cited literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

I. INTRODUCTION

It is known that in a crystal lattice, under conditions of thermodynamic equilibrium, not all sites are occupied by atoms; vacant sites (vacancies) are present, whose concentration follows from the condition of the minimum free energy of the solution of vacancies in the lattice and is determined[^1] by the relation \(\xi \simeq e^{-\frac{u}{kT}}\), where \(u\) is the energy of vacancy formation. However, there are many cases in which the true concentration of vacancies \((\xi)\) exceeds the equilibrium value. In some systems the vacancy concentration is increased because of the prior history of the specimen. Thus, it is known[^2,^4] that pure metals that have undergone rapid quenching possess, at temperatures below the quenching temperature, an elevated vacancy concentration. Subsequent anneals after quenching lead to the establishment of the equilibrium vacancy concentration; in this case the process of establishing the equilibrium concentration can be characterized by a certain relaxation time, depending on

coefficient of self-diffusion at the annealing temperature. To this same group of systems may be assigned various natural crystals in whose lattice some sites have turned out to be vacant during crystal growth. The totality of these sites constitutes excess vacancies, which at elevated temperatures must leave the lattice.

It is possible to indicate systems in which, for various reasons, excess vacancies arise in the course of diffusion annealing. In this sense, below we shall speak of systems with a “source” of vacancies. An example of such a system may be a diffusion specimen consisting of two metals \(A\) and \(B\), provided that the diffusion flux from \(A\) into \(B\) exceeds the flux from \(B\) into \(A\) \((D_{A\to B} > D_{B\to A})\). In this case metal \(A\) will become enriched with vacancies \(^{4,5}\). Another example is an alloy from which a volatile component is removed at high temperature. To this same group of systems with a “source” of vacancies should be assigned objects whose crystal lattice is strongly distorted. The process of removal of distortions at high temperatures may be accompanied by the disappearance of dislocations and, as a consequence, by the appearance of excess vacancies \(^{6,7}\). Excess vacancies in the crystal lattice may arise upon irradiation with heavy particles of high energy (neutrons, protons) \(^{8}\).

In a crystal lattice in which the concentration of vacancies exceeds the equilibrium concentration, processes must take place that reduce the number of excess vacancies. The following processes may be indicated as leading to the departure of excess vacancies from a supersaturated solution of vacancies in the crystal lattice): 1) The escape of vacancies to the external boundaries of the specimen, i.e., their removal from the specimen. This process must be accompanied by a decrease in the linear dimensions of the specimen. 2) Coagulation of excess vacancies, as a result of which macroscopic pores must appear in the specimen. In nonporous bodies the first of the named processes, in which the external boundaries of the specimen are sinks for excess vacancies, may prove significant only in the case when the characteristic linear dimension of the specimen is smaller than the linear size of the grain (thin wires, foils). Let us note that under conditions in which a vacancy concentration gradient directed toward the external boundaries is artificially maintained, the process of removing excess vacancies beyond the limits of the specimen will substantially determine the kinetics of the system’s approach to equilibrium. The second process—the coagulation of excess vacancies into macroscopic pores, each of which is a sink for them—may play a substantially greater role in reducing the vacancy concentration than the first. This is connected with the fact that, in this process, the path which a vacancy must traverse in order to obtain the possibility of leaving the supersaturated solution is of the order of fractions of the grain size. Let us note that this process is associated with the appearance of additional interfaces between pores and the lattice and, thus, leads to the establishment of a quasi-equilibrium state; true equilibrium can occur after all excess vacancies have left the specimen, i.e., after completion of the “sintering” of the macroscopic pores that have arisen. In what follows, the totality of pores arising in the crystal lattice as a result of the coagulation of excess vacancies, which move to the site of coagulation by means of a diffusion mechanism, we shall call diffusion porosity*.

*) Strictly speaking, in addition to the two processes under discussion that lead to a decrease in the concentration of excess vacancies, a third may also be indicated, consisting in the replacement of excess vacancies by dislocated atoms \(^{6,7}\). This process may play a substantial role in objects whose crystal lattice is strongly distorted (plastically deformed crystals, metals of galvanic origin, crystals having distortions of radiation origin \(^{66}\)).

The study of the processes accompanying the development of diffusion porosity in objects in which the concentration of vacant lattice sites exceeds equilibrium has proved essential both for understanding the mechanism of diffusion processes occurring in a crystal lattice and for solving practical problems in the field of kinetic phenomena in alloys. Thus, the existing ideas about the mechanism of self-diffusion and mutual diffusion in substitutional solid solutions, which consisted in the exchange of places between two atoms or groups of atoms, must be abandoned, since the fundamentally important fact of the occurrence of diffusion porosity cannot be explained with the aid of these ideas, but is naturally explained within the framework of ideas about the vacancy (hole) mechanism of atomic displacement (Frenkel\(^1\)). Thus, the work of recent years has created a firm experimental basis for the theory of diffusion phenomena in metals and substitutional solid solutions, at the foundation of which lie ideas about vacant sites in the crystal lattice. In particular, these ideas have proved essential not only for establishing the mechanism and kinetics of the process of diffusion homogenization, but also for the development of theories of such phenomena as the sintering of metal powders (Frenkel–Pines theory\(^7\)), diffusion creep of metals (Frenkel\(^9\), Pines\(^10, 13\), Herring\(^11\), Nabarro\(^12\)).

The occurrence of diffusion porosity has proved to be closely connected with such phenomena as the sintering of mixtures of powders of mutually diffusing metals\(^53\). It may be assumed that the presence of pores of diffusion origin should exert a substantial influence on the kinetics of phase transformations in alloys. An example is the process of graphitization of cast irons, the kinetics of which, as shown by the work of K. P. Bunin and his co-workers\(^14, 15\), is determined by the presence of fine porosity of diffusion origin.

In the present article, experimental studies of the process of formation of diffusion porosity and certain phenomena accompanying this process are discussed mainly.

II. DIFFUSION POROSITY ARISING DURING MUTUAL DIFFUSION OF METALS FORMING SUBSTITUTIONAL SOLID SOLUTIONS

II.1. Mutual diffusion of metals forming substitutional solid solutions

A new direction of work devoted to the study of mutual diffusion of components forming substitutional solid solutions was determined to a considerable extent by the work of Kirkendall and co-workers\(^3, 16, 17\). In these works the authors experimentally investigated the diffusion of copper atoms and zinc atoms in specimens made of brass and copper. The most essential results of these investigations are set forth in the paper by Smigelskas and Kirkendall\(^3\). In this work the authors studied the mutual diffusion of copper and zinc in a specially prepared specimen. On two opposite polished end faces of a rectangular block of \(\alpha\)-brass (30% Zn—70% Cu) six thin molybdenum wires were placed on each, after which a thick layer of copper was deposited from a galvanic bath on all sides of the block; thus, the molybdenum wires were located at the interface \(\alpha\)-brass—copper (Fig. 1). After two diffusion anneals at \(T = 785^\circ\)C of different duration (\(\tau = 6\) days and \(\tau = 56\) days), using the method of layer-by-layer grinding and X-raying, the authors established the curves of zinc-concentration distribution in the diffusion zone. It was found that zinc atoms from brass diffuse into copper faster than copper atoms into brass, i.e. that

Ya. E. GEGUZIN

\(D_{\mathrm{Zn}\to\mathrm{Cu}} > D_{\mathrm{Cu}-\mathrm{brass}}\). As a result of the inequality of the diffusion fluxes from brass into copper and from copper into brass, the tungsten markers placed at the interface between \(\alpha\)-brass and copper are displaced toward the brass. Figure 2 shows the time dependence of the displacement of one of the initial interfaces between \(\alpha\)-brass and copper. The effect observed in Kirkendall’s works

Fig. 1. Schematic representation of the specimen in Kirkendall’s experiments: \(1\)—molybdenum wires, \(2\)—\(\alpha\)-brass, \(3\)—copper.

Fig. 2. Curve of the dependence of the magnitude of displacement of the initial boundary on the annealing time in an \(\alpha\)-brass—copper specimen.

*) , as will be set forth below, is directly connected with the phenomena of the appearance of diffusion porosity and of the change in the volume of the diffusion specimen.

The results of Kirkendall’s investigations were confirmed in a large number of experimental works and, in particular, in the extensive study by Bueche and Blein\(^{23}\). These authors experimentally studied the dependence of the magnitude of displacement of the initial contact plane on the zinc content in \(\alpha\)-brass, the influence of the mutual arrangement of copper and brass on the direction of displacement of the contact plane. In addition, in order to make sure that the phenomena described by Kirkendall are not connected with the fact that one of the components of the system (zinc) has a high vapor pressure, in work\(^{23}\) experiments were carried out with specimens composed of a single-phase Cu—Al alloy and copper. Bueche and Blein showed that, regardless of the nature of the arrangement in the specimen of the copper and of the alloy in contact with it, the plane of the initial contact in the process of diffusion is displaced toward the alloy (Cu—Zn and Cu—Al), and the magnitude of the displacement is the greater, the greater the concentration of zinc (or aluminum) in the alloy specimen under investigation. They also showed that abundant porosity forms in the specimen on the alloy side during diffusion. The occurrence of diffusion porosity and the displacement of the marked plane during the mutual diffusion of metals that form solid solutions of substitution are consequences of the inequality of the partial diffusion coefficients (\(D_A\) and \(D_B\)). Let us note that these phenomena and, in particular, the motion of the marked plane can be used to estimate the partial diffusion coefficients of the components of the solution. As Darken\(^{24}\) showed, the velocity of motion of the marked plane

*) Let us note that long before the appearance of Kirkendall’s works, phenomena accompanying the so-called “Kirkendall effect” had been described in the literature. Thus, as early as 1929, Pfeil\(^{63}\) observed the occurrence of porosity in the case of reaction diffusion in the iron—oxygen system (V. I. Arkharov\(^{64}\) drew attention to this circumstance). In 1930 Grube and Lieberwirth\(^{18}\) described the appearance of diffusion porosity in the process of diffusion homogenization of a mixture of copper and iron powders. F. A. Santalov\(^{19,20,21}\) described numerous experiments testifying to the occurrence of porosity when a volatile component is removed from an alloy. In Johnson’s work\(^{22}\) an increase in the volume of the diffusion specimen was observed. What is fundamentally new in Kirkendall’s works is the convincing proof that the inequality of the partial diffusion coefficients has as its consequence the displacement of the boundary of the initial contact between the mutually diffusing components.

\(v(x;\tau)\) and the partial diffusion coefficients are related by the relations*):

\[ v=(D_A-D_B)\frac{\partial N_A}{\partial x}, \]

or

\[ v=(D_B-D_A)\frac{\partial N_B}{\partial x}. \tag{1} \]

The diffusion coefficient, experimentally determined by Matano, is, according to Darken, expressed in terms of the partial diffusion coefficients by means of the following relation:

\[ D=N_A D_B+N_B D_A. \tag{2} \]

In (1) and (2), \(N_A\) and \(N_B\) are the relative concentrations of components \(A\) and \(B\) \((N_A+N_B=1)\). Formulas (1) and (2) may be used for separate determination of \(D_A\) and \(D_B\). Using experimental data on the rate of motion of the marked plane and on the distribution of the zinc concentration in the diffusion zone\(^3\), Darken estimated the partial diffusion coefficients of copper and zinc in brass, showing that at \(T=785^\circ\mathrm{C}\)

\[ D_{\mathrm{Zn}}=5.1\cdot 10^{-9}\ \mathrm{cm}^2/\mathrm{sec}, \quad D_{\mathrm{Cu}}=2.2\cdot 10^{-9}\ \mathrm{cm}^2/\mathrm{sec}, \]

i.e. that

\[ \frac{D_{\mathrm{Zn}}}{D_{\mathrm{Cu}}}\simeq 2.3. \]

The relation found, \(D_{\mathrm{Zn}}>D_{\mathrm{Cu}}\), is in agreement with the fact that, in brass–copper specimens, diffusion porosity arises in the brass.

II.2. Regularities in the Formation of Diffusion Porosity

Since diffusion porosity arises in the process of coagulation of excess vacancies in a lattice supersaturated with vacancies as a result of the inequality of the diffusion fluxes \(G_{A\to B}\) and \(G_{B\to A}\), it is natural to consider the following questions:

a) what is the direction of the predominant diffusion flux in a diffusion couple composed of two given metals?

b) what is the distribution of excess vacancies in the diffusion zone?

The solution of the first question was given by B. Ya. Pines\(^5\). Following\(^5\), let us consider the question of the direction of the predominant diffusion flux. If two metals \(A\) and \(B\), mutually soluble without limit, are in contact, then a decrease in the free energy of the system will occur both in the diffusion of \(A\) into \(B\) and in the diffusion of \(B\) into \(A\). Since, however, the lowering of the free energy of the system may be different for two different directions of the diffusion flux, let us compare two states: 1) that obtained upon the transfer of \(n\) atoms from \(A\) into \(B\), and 2) that obtained upon the transfer of \(n\) atoms from \(B\) into \(A\). In both cases, in the metal from which \(n\) atoms have departed, \(n\) excess vacancies will appear. We shall assume\(^5\) that there is a mixture of phases I and II composed of pure components \(A\) and \(B\). Before the start of diffusion annealing, phase I contains \(N_1\) atoms of \(A\), phase II—\(N_2\) atoms of \(B\). If \(n\) atoms pass from phase I into phase II and, simultaneously, \(n\) vacancies appear in phase I, the free energy of the system will change by an amount which, in the configurational approximation (successfully used in calculations of ordering phenomena in solutions\(^ {25}\) and calculations of equilibrium diagrams of the simplest systems\(^ {26}\)), is written as follows

*) Darken’s theoretical work was carried out without taking into account the occurrence of diffusion porosity; therefore its results are very approximate, and the formulas given (1) and (2) can be used only for estimating the magnitudes of \(D_A\) and \(D_B\) (see\(^ {24,67}\)).

thus:

\[ \Delta F_1=\Delta F_1^{\mathrm I}+\Delta F_1^{\mathrm{II}}=n\left[u_0^{\mathrm{II}}\frac{N_2}{N_2+n}+\frac12\left(u_{AA}^{\mathrm{II}}-u_{AA}^{\mathrm I}\right)-\frac{u_{AA}^{\mathrm I}}{2}\frac{N_1-n}{N_1}\right]+ \]

\[ {}+kT\left[n\ln\frac{n}{N_2+n}+n\ln\frac{n}{N_1}+N_2\ln\frac{N_2}{N_2+n}+(N_1-n)\ln\frac{N_1-n}{N_1}\right]. \tag{3} \]

Similarly, for the transition of \(n\) atoms \(B\) from phase II into phase I and the formation of \(n\) vacancies in phase II, we write

\[ \Delta F_2=\Delta F_2^{\mathrm I}+\Delta F_2^{\mathrm{II}}=n\left[u_0^{\mathrm I}\frac{N_1}{N_1+n}+\frac12\left(u_{BB}^{\mathrm I}-u_{BB}^{\mathrm{II}}\right)-\frac12 u_{AB}^{\mathrm{II}}\frac{N_2-n}{N_2}\right]+ \]

\[ {}+kT\left[n\ln\frac{n}{N_1+n}+N_1\ln\frac{N_1}{N_1+n}+n\ln\frac{n}{N_2}+(N_2-n)\ln\frac{N_2-n}{N_2}\right]. \tag{4} \]

In (3) and (4), \(u_{AA}\) and \(u_{BB}\) are the potential energies of interaction of neighboring atoms of the same kind, \(u_0=u_{AB}-\dfrac{u_{AA}+u_{BB}}{2}\) is the energy of mixing (the quantities \(u\) include the factor \(z\), the coordination number). The difference of the changes in free energies in the two processes indicated, calculated per one particle from among those taking part in the diffusion flux, under the assumption that \(\dfrac{n}{N_i}\ll 1\), may be written as follows:

\[ \frac{\Delta F}{n}=\frac{\Delta F_1-\Delta F_2}{n} =\frac{u_{BB}-u_{AA}}{2} +\frac{n}{N_1}\left[kT+u_0+\frac{u_{AA}}{2}\right] -\frac{n}{N_2}\left[kT+u_0+\frac{u_{BB}}{2}\right]. \tag{5} \]

Because of the smallness of the ratios \(\dfrac{n}{N_1}\) and \(\dfrac{n}{N_2}\), the principal term in (5) is the term \(\dfrac{u_{BB}-u_{AA}}{2}\). Bearing in mind that the latent heats of vaporization of the components are related to the quantities \(u_{AA}\) and \(u_{BB}\) by the relations \(Q_A=-\dfrac{Nu_{AA}}{2}\), \(Q_B=-\dfrac{Nu_{BB}}{2}\), one may write

\[ \frac{\Delta F}{n}\simeq \frac{1}{N}(Q_A-Q_B). \tag{6} \]

If \(Q_A<Q_B\), then one-way diffusion of \(A\) into \(B\) will lead to a greater decrease in free energy*). Thus, it follows from (6) that the predominant diffusion flux must proceed from the component possessing the smaller latent heat of vaporization. The applicability of this thermodynamic criterion can be confirmed by the results of experimental investigations. Thus, work \(^{27}\) was undertaken especially in order to verify the correctness of the criterion formulated \(^{5}\). In \(^{27}\) diffusion was studied on specimens made up of two polished plates of metal \(A\), between which there was placed a thin (\(\varnothing\sim 0.1\text{--}0.4\) mm) wire of metal \(B\). The systems Co—Ni, Co—Fe, Cu—Pt, Co—Pt were studied. Table I gives data obtained in \(^{27}\) and in other works in order to illustrate

* ) Generally speaking, the direction of the predominant diffusion flux could also turn out to be different if, for \(Q_A<Q_B\), it were found that \(D_{A\to B}\ll D_{B\to A}\). However, as experiments carried out with various pairs of metals show, such circumstances do not occur.

fulfillment of the criterion determining the direction of the predominant diffusion flux.

Table 1

System \(A—B\)	\(Q_A\), kcal/mol	\(Q_B\), kcal/mol	Direction of predominant flux, expected	Direction of predominant flux, observed	Site of pore formation, expected	Site of pore formation, observed	Source
Copper—nickel	73—81	86—90	\(A \to B\)	\(A \to B\)	\(A\)	\(A\)	5 et al.
Cobalt—iron	93—98	85—94	\(B \to A\)	\(B \to A\)	\(B\)	\(B\)	27
Cobalt—nickel	93—98	86—90	\(B \to A\)	\(B \to A\)	\(B\)	\(B\)	27
Copper—platinum	73—81	112—122	\(A \to B\)	\(A \to B\)	\(A\)	\(A\)	27
Cobalt—platinum	93—98	112—122	\(A \to B\)	\(A \to B\)	\(A\)	\(A\)	27
Silver—gold	56—59	69—72	\(A \to B\)	\(A \to B\)	\(A\)	\(A\)	28
Silver—palladium	56—59	95—100	\(A \to B\)	\(A \to B\)	\(A\)	\(A\)	28

In Fig. I*) are shown the microstructures of diffusion specimens of the systems cobalt—nickel and copper—platinum, on which it is seen that, in accordance with the thermodynamic criterion under discussion, the diffusion porosity is located in the metal with the smaller latent heat of evaporation. Let us note that in some works\(^{28,29}\), solely on the basis of the observations made, the assertion was put forward that the predominant diffusion flux is directed from the component with the smaller heat of fusion. Since the heats of fusion and evaporation, in passing from element to element, vary in parallel, the observations made in\(^{28,29}\) are in agreement with the thermodynamic criterion formulated in\(^{5}\).

Let us turn to the question of the distribution of the concentration of excess vacancies arising in the diffusion zone in the process of diffusion between two mutually soluble metals. Consider\(^{30}\) the magnitudes of the diffusion flux of atoms of sort \(A\) and of sort \(B\) as a function of the quantity \(x\) (\(x\) is the distance from the contact plane between the metals \(A\) and \(B\)). Since the magnitude of the diffusion flux of both components reaches its maximum value at the initial boundary of contact between the components \((x=0)\), then, assuming that \(D_{A\to B} > D_{B\to A}\), the dependence of the flux magnitude \(M\) on \(x\) may schematically be represented as is done in Fig. 3, a. Since the fluxes of atoms \(A\) (from \(A\) to \(B\)) and atoms \(B\) (from \(B\) to \(A\)) are not equal to one another, there will always be some difference of fluxes. In a certain region of width \(\Delta x\), lying to the left (or to the right) of the plane of the initial contact, the difference of the fluxes of atoms may be written as:

\[ \Delta M = m_A - m_B \simeq \frac{\partial}{\partial x} \left\{ D_A \frac{\partial c}{\partial x} - D_B \frac{\partial c}{\partial x} \right\} \Delta x . \]

The dependence of the magnitude \(\Delta M\) on \(x\) is shown in Fig. 3, b. It is obvious that the course of the dependence of the concentration of vacant lattice sites that interests us

Fig. 3. Schematic representation of the dependence of the diffusion fluxes \((M)\) and of the concentration of vacant sites \((\xi)\) on the distance from the plane of the initial contact.\(^{30}\)

*) Figures having Roman numeration are placed on an insert (p. 232).

from the magnitude of \(x\) can be qualitatively determined by differentiating curve 3, \(b\) (see Fig. 3, \(v\)). The dependence \(\xi=\xi(x)\) shown in Fig. 3, \(v\) corresponds to some fixed instant of time. As the time of diffusion annealing increases, the concentration distribution of the mutually diffusing components will change, which will change the curve \(\xi=\xi(x)\). Namely, the maximum and the minimum on the curve \(\xi=\xi(x)\) will move away from the point \(x=0\), while the magnitude of the derivative at all points of the curve will decrease.

The curve of the distribution of the concentration of vacant sites in the diffusion zone shown in Fig. 3, \(v\) may be used to explain many observations made in the study of the diffusion zone. The presence of a maximum on the curve \(\xi=\xi(x)\), located on the side of the component from which the predominant diffusion flux is taking place, corresponds to the presence of a region of maximum supersaturation with vacancies \((\Delta \xi=\xi-\xi_0)\). In connection with this, the process of formation of diffusion porosity will be most intense in this region. It has been shown by many authors\(^{31,32,27,67}\) that pores of diffusion origin are predominantly arranged in a chain parallel to the plane of contact between the components of the specimen (Fig. 11); moreover, with time of diffusion annealing the region of most intense pore formation moves away from the contact plane \((x=0)\). The above account of the kinetics of the processes occurring during mutual diffusion and the formation of diffusion porosity is shown schematically in Fig. 4\(^{36}\).

Fig. 4. Schematic representation of the displacement of the boundary of the initial contact and of the region where porosity arises\(^{36}\).

Excess vacancies arising near the boundary of the specimen may, without taking part in the coagulation process, diffuse to this boundary, as a result of which, near the plane of the initial contact on the diffusion specimen, a depression may form, the appearance of which has been described in many experimental works\(^{28,30,31}\). On the side of component \(B\), where the minimum on the curve \(\xi=\xi(x)\) is located, the concentration of vacant sites, according to the accepted description, should be below equilibrium; in fact, this corresponds to the fact that the part of the specimen where component \(B\) is located is enriched with atoms \(A\) to a greater extent than it loses atoms \(B\). As a result of this, an elongation of the specimen occurs, or a thickening appears on it in the region where the minimum on the curve \(\xi=\xi(x)\) is located. Heumann and Kottmann\(^{30}\), discussing the dependence \(\xi=\xi(x)\), point to the principal possibility of the presence of two maxima located on different sides of the initial contact plane.

Indirect confirmation that the dependence \(\xi=\xi(x)\) is represented by a curve with a maximum may be seen in the data on the dependence of the Vickers microhardness \(H_V\) on the distance from the initial contact plane\(^{32*}\). In Fig. 5, \(a\), the experimentally found

Fig. 5. Microhardness in the diffusion-zone region of a copper—nickel specimen\(^{32}\).

*) Similar data are contained in the work of Bokle and Blin\(^{23}\).

the dependence \(H_V = \varphi(x)\) for copper—nickel specimens. Comparing the course of the curve \(H_{V1} = \varphi(x)\) with the course of the curve of the dependence of microhardness on the concentration of Ni in the Cu—Ni solution, which, taking into account the distribution of the concentration of copper and nickel in the diffusion zone in Fig. 5, б, is shown by the dashed line \(H_{V2} = \varphi(x)\), one can determine the value \(\Delta H_V = H_{V1} - H_{V2}\) (Fig. 5, в). Assuming that the function \(\xi = \xi(x)\) specifies the distribution of pores in the diffusion zone, the presence of which is the cause of the decrease in microhardness, one may suppose that the maximum on the \(\Delta H_V\) curve is a consequence of the fact that the curve \(\xi = \xi(x)\) also has a maximum in the same region of the diffusion zone where the maximum of the \(\Delta H_V\) curve is located.

II.3. Faceting of pores

Developing the analogy between the system crystal—nonequilibrium (increased) number of vacancies and a supersaturated solution, it is natural to assume that the phase crystallizing out of the solution, i.e. the pores, must have a shape determined (in accordance with the Curie—Wulff rule) by the anisotropy of the coefficient of interphase surface tension at the boundary precipitating phase—solution. Since, in the case of pore formation, the interphase surface tension coincides with the surface tension at the crystal—vacuum boundary, the shape of the pores must coincide with the equilibrium shape of the crystals in which the pores arise, i.e. so-called negative crystals must form. The question of the shape of pores arising in a supersaturated solution of vacancies in a crystal has been studied in works \(^{23,34}\). The experiments described in \(^{23,34}\) were performed on two-layer specimens composed of an aluminum—copper alloy (7% aluminum) \(^{23}\) and copper, and of a zinc—copper alloy (30% zinc) and copper \(^{34}\). On the basis of a metallographic study of sections perpendicular to the contact plane between the copper and the alloy, in \(^{34}\) the following facts are reported: a) the pores forming in \(\alpha\)-brass have faceting (Fig. III), b) in the plane of the polished section there occur pore sections having the form of triangles, rectangles, pentagons, and hexagons, c) within a given grain, in the plane of the polished section, pore sections have almost the same shape and are identically oriented.

The totality of the observations described gives grounds for assuming that the “negative crystals” forming in \(\alpha\)-brass have octahedral faceting. The character of the volume faceting of the pores in \(^{34}\) was traced by successive polishing and metallographic examination of sections of a specimen spaced 2–3 microns from one another. Thus it was established that the shape of the sections of individual pores agrees with the assumption made about their octahedral faceting. Experiments with copper—nickel specimens showed \(^{34}\) that the phenomena observed in the system \(\alpha\)-brass—copper are not a peculiarity of the case in which unipolar diffusion from a solution into a pure metal occurs, but also take place in the mutual diffusion of pure metals.

II.4. Change in the volume of a diffusion specimen in the course of diffusion

A natural consequence of the occurrence of porosity in a diffusion specimen is an increase in its volume, i.e. a decrease in density, determined pycnometrically. An increase in the volume of a diffusion specimen had been observed qualitatively earlier as well \(^{35,22}\). However, this phenomenon was systematically studied in connection with the process of formation of diffusion porosity. One of the first works \(^{28}\) devoted to the quantitative study of the phenomenon of change in the volume of a diffusion specimen composed of metals forming substitutional solid solutions contains a description of experiments performed with

layered specimens. In specimens consisting of three plates \(A—B—A\), thin tungsten wires were mounted (Fig. 6). After diffusion isothermal anneals of various durations, the distance was measured in the plane of the section between the tungsten wires located outside the diffusion zone \([1, 4]\) and on the planes of the initial contact \(A—B\) and \(B—A\) \([2, 3]\). The results of experiments carried out on specimens Cu—Ni—Cu, Ag—Au—Ag and Ag—Pd—Ag\(^*\) are summarized in Table II.

Table II

	Cu—Ni—Cu	Ag—Au—Ag	Ag—Pd—Ag
Linear dimension in cm	2.0	1.3	1.4
Annealing \(T\), °C	1030	900	815
Annealing duration in hours	144	98	192
\(\Delta(2—3)\), cm	\(+0.0105\)	\(+0.018\)	\(+0.0165\)
\(\Delta(1—4)\), cm	\(+0.007\)	\(+0.004\)	\(-0.013\)

The following results attract attention: a) marks located at the boundaries between the metals are displaced toward the component possessing the lower heat of evaporation; b) the length of the specimens changed in all three cases. In contrast to the Cu—Ni—Cu and Ag—Au—Ag specimens, where an increase in the linear dimension takes place, the Ag—Pd—Ag specimen shortened. This, however, does not contradict the fact that the diffusion process is accompanied by an increase in the volume of the specimen, since in the immediate vicinity of the initial contact on the side of metal \(A\) (for \(D_{B\to A} > D_{A\to B}\)) thickenings are formed, whose volume in the case of the Ag—Pd—Ag specimen must exceed the decrease in volume associated with the shortening of the specimen. It is evident that the total increase in the diffusion specimen must be equal to the volume of the diffusion pores that have arisen.\(^ {**}\)

Fig. 6. Diagram of the specimens used in work \(^{28}\).

The question of the change in volume of a diffusion specimen during diffusion was studied in detail in \(^{36}\); experiments were carried out on specimens of the systems \(\alpha\)-brass—copper and copper—nickel. Specimens of the \(\alpha\)-brass—copper system were prepared in the same way as the specimen in the experiment of Smigelskas and Kirkendall \(^{3}\), with the only difference that, in addition to two rows of marks at the boundaries between brass and copper, two more rows of marks were provided, located in the copper, away from the diffusion zone, similarly to how this was done in \(^{28}\). The results of dilatometric measurements performed with the described specimen are shown in Fig. 7 and are in agreement with the results of the experiments of Smigelskas and Kirkendall \(^{3}\) and of the experiments of Seitz and Kottman \(^{28}\). The dependence of \(\Delta(1—4)\) and \(\Delta(2—3)\) as a function of \(\tau^{1/2}\) is expressed by a straight line, since the change in the distances between the corresponding planes is a consequence of a process occurring by means of the diffusion mechanism.

\(^*\) Phenomena accompanying the so-called “Kirkendall effect” have also been observed in nonmetallic systems. Thus, in \(^{48}\) the appearance of diffusion porosity and a change in the volume of a diffusion specimen composed of aluminum oxide (\(\mathrm{Al_2O_3}\)) and spinel (\(\mathrm{FeO\cdot Al_2O_3}\)) are described.

\(^ {**}\) We note that in the considerations set forth, and hereafter, the possible difference in atomic volumes of metals \(A\) and \(B\) and a possible change in the volume of transformation of the solid solution \(AB\) are not taken into account; i.e., it is assumed that \(\Delta V_{AB} = V_{AB} - \dfrac{1}{2}(V_{AA} + V_{BB}) \approx 0\).

DIFFUSION POROSITY IN METALS AND ALLOYS

The study of volume change during diffusion in the copper—nickel system was carried out \(^{36}\) on specimens composed of a large number of alternating thin plates of copper and nickel. During annealing of multilayer specimens the relative changes in linear dimensions are considerable, owing to which the measurements were performed with high accuracy. Observations were made of changes in linear dimensions in the direction of the diffusion flux and perpendicular to it. Along with measurements of the linear dimensions, the density of the specimens was measured. The results of experiments carried out at \(T = 935^\circ\text{C}\), \(1052^\circ\text{C}\), and \(1060^\circ\text{C}\), are shown in Fig. 8. Let us note that the increase in the volume of the diffusion specimen, associated with a possible growth of the lattice parameter during the formation of a solid solution, is negligibly small in comparison with the observed effect. In the formation of a copper—nickel solution, in connection with the change in lattice parameter, the volume of the specimen should have decreased somewhat, since in this case the curve of the concentration dependence of the lattice parameter is convex downward relative to the straight line connecting the values of the lattice parameters of copper and nickel.

Fig. 7. Dependence of the displacement of marked planes in a diffusion specimen \(\alpha\)-brass—copper on annealing time \(^{36}\).

Fig. 8. Change in thickness of multilayer copper—nickel specimens as a function of annealing time \(^{36}\).

The experimental facts established in studying the change in volume of a diffusion specimen in the process of diffusion lead to the following conclusions: a) the change in volume of the diffusion specimen \(\Delta V\) accompanies the process of formation of diffusion porosity; b) the magnitude \(\Delta V\) is proportional to the square root of time. This proportionality may be violated at long annealing times both because overlap of different diffusion zones occurs (in the case of multilayer specimens), and because, along with the process of formation of diffusion pores, the process of their sintering begins to make itself felt; c) the rate of growth of \(\Delta V\) with time increases with increasing temperature of diffusion annealing.

From the works discussed earlier it has been established that the process of mutual diffusion of metals forming a substitutional solid solution is accompanied by a noticeable increase in the volume of the specimen. In this connection it seems natural to suppose \(^{5,37}\) that if the diffusion experiment is carried out under conditions in which the specimen is under a pressure hindering the change of its volume, the appearance of porosity will likewise be hindered and may not occur at all. At the same time, the relation between the partial diffusion coefficients should change. If diffusion in the given system is characterized by the inequality \(D_{A \to B} > D_{B \to A}\), then, when the volume change is restricted, the coefficient \(D_{A \to B}\) should decrease so that

so that the diffusion process would proceed in proportion to the equality of the fluxes \(G_{A \longrightarrow B}\) and \(G_{B \longrightarrow A}\). The considerations set forth follow from Le Chatelier’s principle, applied to mutual diffusion in solid substitutional solutions.

In \({}^{37}\) the system \(\alpha\)-brass—copper was studied, in which, as is known \({}^{3,24}\), the diffusion coefficient \(D_{\mathrm{Zn}\longrightarrow\mathrm{Cu}}\) considerably exceeds \(D_{\mathrm{Cu}\text{–brass}}\). A brass cylinder, the ends of which were polished, was placed between the polished ends of two copper cylinders. The system of three cylinders was placed in the channel of an ampoule between two punches, of which the upper one, being movable, could transmit pressure to the specimen. After diffusion annealing, a metallographic section was prepared in the plane of the axial section of the specimen, and from the measured width of the diffusion layer the diffusion coefficient was estimated as

\[ D_{\mathrm{Zn}\text{–Cu}} \simeq \frac{x_{\mathrm{Zn}\longrightarrow\mathrm{Cu}}^{2}}{\tau}. \]

The totality of the results obtained in the experiments described is collected in Table III. In Fig. IV photographs are given illustrating the results of the experiments described.

Table III

Temperature	\(\tau\,10^{-3}\), sec	\(P\), kg/cm²	\(X_{\mathrm{Zn}\longrightarrow\mathrm{Cu}}\,10^{4}\), cm	\(D\,10^{9}\), cm²/sec	Character of the porosity arising in brass
850° C	25.2	0	95—100	3.7	Abundant porosity \(\bar L \sim 9\cdot10^{-4}\)
850° C	25.2	2.5	82—86	3.0	Porosity present, \(\bar L \sim 3\cdot10^{-4}\)
850° C	25.2	12	65—70	2.0	Porosity practically absent
850° C	25.2	\(2\cdot10^{3}\) *)	60—65	2.0	Porosity practically absent

*) High pressure (substantially exceeding \(P \gg 10\) kg/cm²) has a very small influence on the change in the diffusion coefficient \(D_{\mathrm{Zn}\longrightarrow\mathrm{Cu}}\). This, apparently, indicates that after elimination of diffusion porosity the influence of pressure is due to other processes than before its elimination.

The data contained in Table III indicate that pressures \(P \gg 10\) kg/cm² applied to the diffusion pair lead to the disappearance of diffusion porosity. One may, generally speaking, suppose that the metallographically observed absence of porosity is a consequence of the “pressing shut” of microscopic pores by the applied external pressure. Such an assumption, however, contradicts the following facts: a) in the experiments described, the diffusion layer on copper decreases; b) application of pressure up to 29 kg/cm² at a temperature of 1000° C \({}^{13,61}\) to porous specimens obtained by pressing copper did not lead to the disappearance of porosity, which should have occurred if “pressing shut” of pores had taken place. Experiments with copper—nickel specimens led \({}^{37}\), qualitatively, to the same results as the experiments described with \(\alpha\)-brass—copper specimens.

In connection with the data set forth on the influence of pressure on the mutual diffusion of metals, one should dwell on the observation of Seith and Ludwig \({}^{32}\). Studying metallographically the mutual diffusion in the copper—nickel system, these authors observed that, on approaching the outer surface of the specimen, the width of the diffusion layer both on copper and on nickel increases. The width of the region on copper in which diffusion porosity is located also increases. A schematic representation of the diffusion zone near the surface of a specimen after prolonged diffusion annealing is presented in Fig. 9. The authors see the explanation of the observed phenomenon in the fact that the near-surface layer of the specimen, which had undergone mechanical treatment, is distorted, owing...

thereby facilitating diffusion processes in this layer. This explanation cannot be considered exhaustive, since the phenomenon observed by Zeitz and Leudwig also occurs in specimens that have not been subjected to mechanical treatment under severe conditions. The increase in the width of the region in which the diffusion porosity is located may be a consequence of the fact that, as the surface of the specimen is approached, the diffusion layer can change its volume more freely than the deeper portions of the diffusion zone. These portions may be regarded as being under a certain pressure.

II.5. Evaporation of a volatile component from an alloy

The formation of diffusion porosity during evaporation of a volatile component from a binary alloy was observed and described by F. A. Santalov[^19][^20][^21], who traced this phenomenon on a large number of objects. In recent years this phenomenon has been subjected to comprehensive study by a number of authors[^39][^45][^38]. It is known that removal of a volatile component from an alloy is accompanied by a decrease in the weight and volume of the specimen. The decrease in weight is a consequence of the escape from the specimen of atoms of the volatile component, while the decrease in volume is a consequence of the escape beyond the specimen of additional vacancies that arise when atoms of the volatile component are removed. In[^38] it was shown that the additional vacancies arising in the specimen at an early stage of the process of removing atoms of the volatile component practically all leave the specimen, whereas at later stages[^38][^39] all excess vacancies are divided into two approximately equal parts, one of which leaves the specimen and the other remains in the specimen. The excess vacancies remaining in the specimen either coagulate, forming macroscopic pores, or, being in the form of isolated vacancies, take part in the diffusion process.

Fig. 9. Schematic representation of the diffusion zone near the surface of a specimen[^32].

Fig. 10. Curves of the dependence of the number of pores and their average size \(L\) in \(\alpha\)-brass on the distance from the surface of the specimen[^38].

Let us consider the question of the distribution of the concentration of excess vacancies in the near-surface layer of a specimen from which atoms of the volatile component are removed. In Fig. 10 are shown curves of the dependence of the number of pores \((N)\) and their average linear size \((L)\) on the distance from the surface of the specimen \((Y)\), constructed on the basis of data from a metallographic study of a specimen of \(\alpha\)-brass annealed in vacuum for 15 hours at a temperature of \(520^\circ\mathrm{C}\)[^38]. The presence of a maximum on the curve \(L = L(Y)\) indicates that the concentration of excess vacancies with increasing distance from the surface of the specimen cannot change monotonically (in proportion to the amount of zinc that has escaped). Indeed, the linear size of a precipitate growing from a supersaturated solution (in the solid phase), other conditions being equal, is proportional to the magnitude of the supersaturation \(\bigl(L \sim (\Delta \xi)^{1/2}\bigr)\), and thus the maximum on the curve indicates

that the concentration of excess vacancies, and consequently also the magnitude of supersaturation, likewise passes through a maximum. Note that at the surface of the specimen the vacancy concentration is equal to the equilibrium concentration in the metal—the base of the alloy (in the case of brass, in copper—\(\xi_m\)), while far from the surface the vacancy concentration is the equilibrium one for the given alloy \((\xi_0 > \xi_m)\).

III. DIFFUSION POROSITY IN ONE-COMPONENT SYSTEMS

Diffusion porosity can arise not only in processes of mutual diffusion, when, as a consequence of the inequality of the partial diffusion coefficients of the components of a two-component system, excess vacancies arise, but also in one-component systems which, by one feature or another, are removed from the state of thermodynamic equilibrium.

Bearing in mind one-component systems, let us consider two different sources of excess vacancies.

In one-component systems with a strongly distorted crystal lattice, excess vacancies may appear as a consequence of the “annealing” of microdistortions of the lattice \(^{5,7}\). According to the model proposed in \(^{5}\) and \(^{7}\), microdistortions of the crystal lattice may be represented as an aggregate of “\(\perp\)” and “\(-\)” distortions (“\(\perp\)” and “\(-\)” dislocations). By a “\(\perp\)” distortion is meant a region containing \((P+1)\) atoms, which are distributed in a volume \(P\delta^3\), and by a “\(-\)” distortion—an aggregate of \((P-1)\) atoms distributed in a volume \(P\delta^3\), where \(\delta^3\) is the volume per one atom in a lattice free from distortions. The diffusional “annealing” of “\(-\)” distortions occurring at high temperature must have as its consequence the appearance of vacancies, whereas the “annealing” of “\(\perp\)” distortions is associated with absorption of vacancies. Thus, according to the model described, the regions of “\(-\)” distortions are peculiar sources of vacancies, and the regions of “\(\perp\)” distortions are sinks. Indirect evidence in favor of the idea that the process of “annealing” of microdistortions is accompanied by the appearance of excess vacancies is the circumstance that the coefficient of self-diffusion in distorted lattices \((D_{ai})\) exceeds the coefficient of self-diffusion in an equilibrium lattice \((D_{a0})\). The latter assertion is based on the fact that between the coefficient of self-diffusion of atoms \((D_a)\), the coefficient of self-diffusion of vacancies \((D_b)\), and the concentration of vacancies there is the relation

\[ D_a = \xi D_b, \tag{7} \]

which follows naturally from the concepts of the vacancy mechanism of self-diffusion, according to which the elementary act of displacement of an atom in the crystal lattice consists in its transition from the occupied site to a neighboring vacant site. Thus, one of the causes of an increase in the diffusion coefficient of atoms may be an increase in the concentration of vacant lattice sites.

If indeed the concentration of vacancies in a metal with a strongly distorted crystal lattice \((\xi_i)\) at elevated temperatures exceeds the concentration of vacancies in an equilibrium lattice \((\xi_0)\), then it may be assumed that a process of coagulation of excess vacancies will take place, as a result of which macroscopic diffusion porosity should arise. The macroscopic pores arising in the lattice, together with the regions of “\(\perp\)” distortions, are sinks for excess vacancies.

Experimentally, the occurrence of diffusion porosity in metals with a distorted metallic lattice can most simply be observed

on objects in which there is a considerable excess of the self-diffusion coefficient over its equilibrium value \((D_{ai} \gg D_{a0})\) and, consequently, a significant supersaturation of the lattice with vacancies (see the following section). This requirement is satisfied by electroplated metal, for which, as has been shown by direct measurements\(^{54}\), the self-diffusion coefficients are greatly overestimated.

The process by which diffusion porosity arises during annealing of electroplated copper and nickel was observed in\(^{47}\). In this work, carried out in order to trace the temperature and time dependence of the growth process of pores of diffusion origin, the following observations were made: 1) Diffusion pores are formed mainly at the boundaries between the grains of the specimen and have a faceting determined by the mutual orientation of the adjoining grains. Within some grains, pores having faceting are also observed. 2) With the time of isothermal annealing, the mean linear size of the pores increases. 3) Diffusion pores, being the cause of separation of grains over a considerable part of the surface between them, hinder interphase recrystallization. The results described in work\(^{47}\) are seen in Fig. V. Thus, the appearance of diffusion porosity is a stage on the way toward the establishment of true equilibrium in a lattice rich in microdistortions.

It should be pointed out that the process by which macroscopic pores arise as a result of the coagulation of excess vacancies, which appear during the annealing of “−” distortions, must take place simultaneously with the annealing process of “+” distortions, which are also “sinks” for excess vacancies.

As a result of the process of coagulation of excess vacancies, a state is established in the lattice that is far removed from the state of thermodynamic equilibrium, owing to the presence of a developed surface of macroscopic pores. Therefore, generally speaking, the process of absorption of excess vacancies by “+” distortions is thermodynamically more justified than their coagulation, since as a result of this process the remaining distortions in the lattice are annealed out and no interfaces arise with which additional energy is associated. Meanwhile, as experience shows, the process of coagulation does take place and, apparently, proceeds in parallel with the process of annealing of “+” distortions.

The coagulation of excess vacancies, as a stage on the way toward the establishment of equilibrium in a distorted lattice, can be justified kinetically, since those vacancies which on their path first encounter a pore or microcrack (see below), rather than a region of “+” distortion, will be absorbed by the indicated inhomogeneities. The annealing of the remaining “+” distortions can occur with the aid of those excess vacancies which will arise in the lattice in connection with the process of “sintering” of pores, i.e., the escape of excess vacancies to external boundaries. It is also not excluded that the number of “−” distortions \((N_-)\) exceeds the number of “+” distortions \((N_+)\). In this case \(N = N_- - N_+\) vacancies must take part in the process of coagulation (or escape to external boundaries), independently of the process of annealing of “+” distortions.

There may be excess vacancies in the lattice whose appearance is connected not with the process of removal of microdistortions, but with the process of annealing of macroscopic cracks and voids present in the specimen by virtue of its prior history. Indeed, in a crystalline lattice there may be microvoids of quenching or deformation origin, having various sizes. At elevated temperatures there will occur a process of coagulation of small voids, the molecular mechanism of which consists in the “evaporation” of vacancies from the surface of smaller voids with their subsequent deposition

(“condensation”) on the surfaces of larger cavities*). This process is analogous to the process of coagulation of finely dispersed precipitates that occurs during dispersion hardening of alloys^55. Thus, in the present case the “source” of vacancies is the smallest cavities, whose “healing” leads to enrichment of the lattice with excess vacancies deposited on the surfaces of large pores.

Let us note that the described process of the occurrence of macroscopic porosity during annealing of a metal of galvanic origin can hardly be a consequence of the coalescence of macroscopic cracks and voids present in the specimen, since such an assumption is contradicted by the experimentally observed time dependence of the linear size of the pores that are formed (see below).

In metallic objects the process of diffusion growth of some pores at the expense of neighboring small ones was studied in detail in work^56, whose authors metallographically investigated the coagulation of pores during sintering of powder compacts. In the works of K. P. Bunin and E. N. Pogrebnoi^57 and of E. N. Pogrebnoi^58, coagulation of pores arising in steel during quenching was observed.

Interesting facts concerning the process of coagulation of microscopic cavities present in specimens of optically inhomogeneous crystals of rock salt are reported in the work of R. I. Garber, V. S. Kogan, and L. M. Polyakov^59. These authors observed large faceted pores that appeared as a result of prolonged annealing of such crystals at a temperature close to the melting temperature. The appearance of large pores was accompanied by clarification of the crystal owing to the “healing” of microscopic cavities with linear dimensions of the order of the wavelength of visible light \((L \sim 10^{-5}\,\text{cm})\). In Fig. VI are shown pores formed during the coagulation of excess vacancies, which arose as a result of the “healing” of microscopic cavities.

IV. SUPERSATURATION OF THE CRYSTAL LATTICE WITH VACANCIES

IV.1. Supersaturation with vacancies arising during interdiffusion in solid substitutional solutions

In discussing the question of the causes of the occurrence of diffusion porosity, we have made use of an analogy between a crystal lattice containing excess vacancies and a supersaturated solution. In the present section the question of the degree of supersaturation of the lattice with vacancies \((\Delta \xi = \xi - \xi_0)\) is considered. To find the quantity \(\Delta \xi\), one may turn to observation of the processes of coagulation of excess vacancies and of their escape beyond the boundaries of the specimen.

The following two ways may be proposed for experimentally finding the magnitude of the supersaturation. One of them involves observation of the kinetics of growth of macroscopic pores (the kinetics of the coagulation process of excess vacancies), and the second involves observation of the kinetics of volume contraction of the specimen in connection with the escape of vacancies beyond its boundaries. Both proposed methods may be used for experimentally determining the magnitude of supersaturation both in solid substitutional solutions and in one-component systems. The kinetics of coagulation of excess vacancies in a solid solution of zinc in copper, with the aim of determining the magnitude of the supersaturation, was studied in works^40, ^41. The idea of the experiments carried out was as follows. According to Zener^43, the time dependence of the linear size of a precipitate growing from a supersaturated solution

*) The directed flux of vacancies from the surface of small cavities to the surface of larger cavities is established under the influence of the concentration gradient of excess vacancies \(\nabla C \sim 1/l\,(1/r_1 - 1/r_2)\), where \(L\) is the distance between pores, and \(r_1\) and \(r_2\) are the radii of curvature of the surfaces bounding the pore cavities.

Fig. I. Microstructures of diffusion specimens⁽²⁷⁾.
a) Co—Ni, b) Cu—Pt.

Fig. II. Copper—nickel specimen. $T = 1010^\circ\mathrm{C}$,
$t = 68$. Arrangement of pores in a chain⁽³⁰⁾.
Magnification $\times 150$.

Fig. III. Bounded pores in an $\alpha$-brass specimen⁽³⁴⁾.

Fig. IV. Structures of α-brass—copper specimens annealed at various pressures, \(T = 850^\circ\text{C}\); a) \(P = 0\), b) \(P = 2.5\ \text{kg}/\text{cm}^2\); c) \(P = 12\ \text{kg}/\text{cm}^2\). Magnification \(\times 100\).

Fig. V. Microstructures of galvanic copper specimens after annealing in vacuum at \(T = 1000^\circ\text{C}\). a) \(\tau = 5\) min; b) \(\tau = 30\) min; c) \(\tau = 90\) min; d) \(\tau = 900\) min.\(^{47}\)

Fig. VI. Pores arising during annealing of optically inhomogeneous rock salt. \(T = 780^\circ\text{C}\). Magnification \(\times 400\).\(^{59}\)

Fig. VII. Arrangement of pore chains within a single grain. α-brass. Magnification ×270^41.

Fig. VIII. Structures of copper–nickel wire models: a) initial state; b) copper in the center, nickel at the periphery. \(T = 1040^\circ\text{C}\), \(\tau = 30\) h; c) nickel in the center, copper at the periphery. \(T = 1040^\circ\text{C}\), \(\tau = 34\) h. Magnification ×150^5.

Fig. IX. Structures of wire models of a porous body at various stages of the sintering process. Copper, \(T = 1060^\circ\text{C}\). a) \(\tau = 3\) h; b) \(\tau = 18\) h; c) \(\tau = 35\) h; d) \(\tau = 43\) h; e) \(\tau = 51\) h.^51

(in the solid phase), is determined by the relation

\[ L(\tau)=k\left(\frac{n_\infty-n_r}{n_0-n_r}\right)^{1/2}\cdot (D\tau)^{1/2}; \tag{8} \]

where \(k \cong 1\), \(n_\infty\) is the concentration of the supersaturating phase far from the precipitate, \(n_r\) is the concentration of the supersaturating phase near the interface between the precipitate and the parent phase, and \(n_0\) is the concentration of the supersaturating phase in the precipitate. Bearing in mind the process of coagulation of excess vacancies, i.e., the formation and growth of negative crystals, when \(n_0=1 \gg n_r\), \(n_\infty=\xi\), \(n_r=\xi_r\), \(L(\tau)\) may be written in the form*)

\[ L(\tau)\simeq(\Delta \xi)_1^{1/2}(D_b\tau)^{1/2}, \tag{9} \]

where \(D_b\) is the diffusion coefficient of vacancies, \(\Delta \xi=\xi-\xi_r\), \(\xi_r\) is the concentration of vacant sites near a pore of radius \(r\). The quantity \((\Delta \xi)_1\) may be written as follows: \((\Delta \xi)_1=\xi-\xi_r-(\xi-\xi_0)-(\xi_r-\xi_0)=\Delta \xi-\Delta \xi_r\). As will be shown below, for \(L \simeq 10^{-4}\) cm the quantity \(\Delta \xi \gg \Delta \xi_r\), and thus \((\Delta \xi)_1 \simeq \Delta \xi\). The self-diffusion coefficient of vacancies \(D_b\) may be expressed through the atomic self-diffusion coefficient by means of the known relation \(D_a=\xi D_b\). Taking into account the above and formula (9), we find an expression determining the significant supersaturation of the lattice with vacancies:

\[ \frac{\Delta \xi}{\xi_0}\simeq \frac{1}{D_a} \left[ \frac{d}{d\tau^{1/2}}\,L(\tau) \right]. \tag{10} \]

According to (10), in order to determine \(\Delta \xi/\xi_0\) experimentally, the time dependence of the linear pore size and the value of \(D_a\) must be determined. A convenient object for finding the dependence \(L=\varphi(\tau)\) is a solid solution from which, at high temperature, a volatile component is removed. In \(^{40}\) the kinetics of the growth of diffusion pores was studied on specimens of \(\alpha\)-brass at a temperature of \(780^\circ\) C, and in \(^{41}\) at three temperatures—\(650^\circ\) C, \(750^\circ\) C, and \(820^\circ\) C. In the cited works the value \(L\), after isothermal annealing of a prescribed duration, was determined from the position of the maximum on the distribution curve of pores visible in the field of a metallographic section, according to their linear dimensions. The data obtained by the method described on the dependence \(L(\tau)\) are presented as graphs of \(L=\varphi(\tau^{1/2})\) in Fig. 11. Having determined from the graphs in Fig. 11 the value

\[ \frac{d}{d\tau^{1/2}}\,L(\tau) \]

and using the known\(^{2}\) data on the temperature dependence of the diffusion coefficient in \(\alpha\)-brass (found in experiments on the evaporation of a volatile component), one can, from (10), find the magnitude of the relative supersaturation \(\Delta \xi/\xi_0\).

Fig. 11. Dependence of the mean pore size in \(\alpha\)-brass on the time of zinc evaporation \(^{40,41}\).

Data on the temperature dependence of the quantity \(\Delta \xi/\xi_0\) are summarized in Fig. 12.

*) Strictly speaking, formulas (8) and (9) may be applied only for sufficiently large precipitate sizes, when the growth of the precipitate can be described approximately as occurring as a result of the motion of a plane boundary. Under these same conditions one may assume that \(k \cong 1\).

Ya. E. Geguzin

The observed increase of \(\dfrac{\Delta \xi}{\xi_0}\) with temperature is a consequence of the increase in the strength of the “source” of excess vacancies, since the rate of evaporation of zinc from \(\alpha\)-brass increases exponentially with temperature. As follows from the experiments described, the supersaturation arising in \(\alpha\)-brass upon evaporation of zinc in the temperature range \(650\text{--}850^\circ\) is a quantity of the order of several percent.

Fig. 12. Dependence of the relative supersaturation of the lattice of \(\alpha\)-brass with vacancies on temperature\({}^{41}\).

Using data on the kinetics of growth of pores arising during mutual diffusion in the copper—nickel system\({}^{5}\), one can also estimate the magnitude of the supersaturation of the lattice by vacancies arising in diffusion specimens of the usual type. An estimate of

\[ \frac{d}{d\tau^{1/2}}\,L(\tau), \]

made on the basis of data on the time dependence of the linear size of pores forming in copper, leads to a value \(\sim 7\cdot 10^{-6}\ \mathrm{cm}/\mathrm{sec}^{1/2}\). Taking into account that \(D_{\mathrm{Cu}-\mathrm{Ni},\,T=1040^\circ\mathrm{C}}\simeq 2.3\cdot 10^{-9}\ \mathrm{cm}^2/\mathrm{sec}\)\({}^{5}\), we find that at \(T=1040^\circ\mathrm{C}\) the ratio

\[ \frac{\Delta \xi}{\xi}\simeq 2\cdot 10^{-2}. \]

Bearing in mind the process of removal of the volatile component from an alloy, one can estimate the concentration of vacancies participating in the diffusion process by observing not the kinetics of pore growth but the decrease in the weight and volume of a specimen\({}^{38}\). Since the vacancy concentration and the diffusion coefficients of atoms are related by the known relation \(\xi=\dfrac{D_a}{D_b}\), the problem of finding the magnitude \(\xi\) reduces to finding, by independent means, the quantities \(D_a\) and \(D_b\). To determine the quantity \(D_a\), it is natural to use data on the decrease in the weight of a specimen, and to determine \(D_b\), data on the decrease in its volume. It is known\({}^{45}\) that the diffusion coefficient of atoms and the magnitude of the weight loss of a specimen \(\Delta P\), from which a volatile component is removed by diffusion (in the case where the specimen models an infinite half-space), are related by

\[ \Delta P \simeq \frac{2}{\pi^{1/2}}\,C\,(D_a\tau)^{1/2}, \tag{11} \]

where \(C\) is the initial concentration of the volatile component in the specimen in units of \(\mathrm{g}/\mathrm{cm}^3\), and \(\tau\) is time. Just as the decrease in weight is a consequence of the departure of atoms of the volatile component beyond the limits of the specimen, so the decrease in volume is a consequence of the departure of vacancies beyond the limits of the specimen. The following expression may be written, relating the quantities \(D_b\) and \(\Delta V\) (the decrease in the volume of the specimen):

\[ \Delta V \simeq \frac{2}{\pi^{1/2}}\,\xi\,(D_b\tau)^{1/2}, \tag{12} \]

where \(\xi\) is the concentration of vacant sites in the region of maximum supersaturation by vacancies, expressed in dimensionless units\({}^{*}\). From (7), (11), and (12) it follows that

\[ \xi \simeq \left(\frac{\Delta V}{\Delta P}\right)^2 C^2. \tag{13} \]

\({}^{*}\) Strictly speaking, in (12) there should stand not \(\xi\), but \(\Delta \xi=\xi-\xi_{\mathrm{eq}}\), where \(\xi_{\mathrm{eq}}\) is the concentration of vacancies in copper of the near-surface layer. Since \(\xi_{\mathrm{eq}}\simeq 10^{-2}\xi_0\) (\(\xi_0\) is the equilibrium concentration in brass), while \(\xi_{\mathrm{eq}}\ll \xi-\xi_0\), the value \(\xi_0\) in comparison with \(\xi\) in (12) may be neglected.

In [38], using formula (13), on the basis of data on the quantities \(\Delta P\) and \(\Delta V\) found in experiments with single-phase \(\alpha\)-brass, it was determined that at \(T=820^\circ\mathrm{C}\), \(\xi=3\cdot 10^{-2}\). If it is assumed that, in a solid solution, the values of the hole-formation energy \((u)\) and the activation energy of the diffusion process \(\theta\) are in the same ratio as in pure metals, then, following [2, 62; 46], one may write

\[ u \simeq \frac{\theta}{3}, \]

i.e., in the case of brass \(u \sim 1\cdot 10^4\ \text{cal/mol}\). Taking into account this value of \(u\), one can estimate the equilibrium concentration of vacancies, which turns out to be equal to

\[ \xi_{0,T=820^\circ}\simeq 10^{-2}. \]

The value of \(\xi\) was determined by means of formula (13) only very approximately. Bearing this in mind, it may be considered that comparison of the quantities \(\xi_{T=820^\circ}\) and \(\xi_{0,T=820^\circ}\) does not contradict the result, following from experiments on the kinetics of coagulation of vacancies [40, 41], according to which the relative supersaturations of the lattice by vacancies arising during interdiffusion in substitutional solid solutions are small and amount to a quantity of the order of several percent.

IV.2. Concentration of Excess Vacancies in One-Component Systems

As in the case of solid solutions, the question of supersaturation by vacancies in a one-component system can be studied experimentally by using the phenomenon of coagulation of excess vacancies and the dilatometric effect of contraction of the specimen when vacancies leave it.

The phenomenon of coagulation of excess vacancies in copper of galvanic origin was used in [47] to estimate the quantity \(\dfrac{\Delta \xi}{\xi_0}\). On the basis of data from metallographic control of specimens that had undergone isothermal annealing of various durations, curves

\[ L=\varphi(\tau^{1/2})^* \]

were constructed, from the slope of which the quantity

\[ \frac{d}{d\tau^{1/2}}\,L(\tau) \]

was determined (see formula (10)), and then, using literature data [44] on the value of the self-diffusion coefficient in copper, the relative supersaturation \(\dfrac{\Delta \xi}{\xi_0}\) was calculated.

In processing the data obtained in experiments with specimens whose lattice is strongly distorted, it is assumed that the only reason for the overestimated coefficient of self-diffusion of atoms in such a lattice \((D_{ai})\) is the excess of the vacancy concentration over the equilibrium one. Since \(D_{ai}=\xi_i D_{bi}\), the inequality \(D_{a0}>D_{ai}\) may occur in the following cases: 1) \(\xi_i=\xi_0;\ D_{bi}\gg D_{b0}\); 2) \(\xi_i>\xi_0;\ D_{bi}>D_{b0}\); 3) \(\xi_i\gg \xi_0;\ D_{bi}=D_{b0}\). The first case is excluded in connection with the presence of the effect of coagulation of excess vacancies (the formation of pores). It would be natural to suppose that, in addition to the excess of \(\xi_i\) over \(\xi_0\), the reason why the coefficient of self-diffusion in the distorted lattice is overestimated also consists in a lowering of the energy barriers for vacancy migration, i.e., that the second case occurs. It is easy, however, to show that the observed considerable excess of \(D_{ai}\) over \(D_{a0}\) (by about 3–4 orders of magnitude) can only to a small extent be a consequence of a lowering of the barriers for vacancy migration. In connection with this, in [47], when carrying out estimated calculations it was assumed that the third case occurs, i.e., that

\[ D_{bi}=D_{b0}=\frac{D_{a0}}{\xi_0}, \]

and thus the values of \(\dfrac{\Delta \xi}{\xi_0}\) found in [47] are somewhat overestimated. This assumption

* In the field of the metallographic section it was always possible to distinguish a group of approximately isometric large pores and a group of substantially finer pores. In finding the dependence \(L(\tau)\), pores of the first group were taken into account.

and leads to the fact that in (10) the quantity \(D_a\), determined in experiments with equilibrium specimens\({}^{44}\), appears. The dependences of \(\dfrac{\Delta \xi}{\xi_0}\) on \(\tau\), found in\({}^{47}\) in the form of graphs \(\log \dfrac{\Delta \xi}{\xi_0}\)—\(\log \tau\), are shown in Fig. 13. The curves presented indicate that, with increasing temperature of isothermal holding, the magnitude of the supersaturation becomes smaller. This quantity also decreases with the time of isothermal holding\(*\). Both of these results are in qualitative agreement with the ideas on the kinetics of removal of distortions and the formation of excess vacancies developed by B. Ya. Pines\({}^{5,7}\).

Fig. 13. Dependence of the relative supersaturation by vacancies of copper of galvanic origin according to data on the rate of pore growth\({}^{47}\).

\[ \bullet\!-\!\bullet\!-\!\bullet \;-\; T=500^\circ\mathrm{C}, \qquad \blacktriangle\!-\!\blacktriangle\!-\!\blacktriangle \;-\; T=750^\circ\mathrm{C}, \qquad \circ\!-\!\circ\!-\!\circ \;-\; 1000^\circ\mathrm{C}. \]

Let us discuss the question of supersaturation by vacancies of copper of galvanic origin on the basis of data on the kinetics of volume shrinkage of compacts made from active copper powders obtained by the galvanic method. The corresponding data are contained in work\({}^{47}\), in which dilatometric experiments were carried out under the regime of “isothermal” annealing.

Starting from the expression known in the diffusion theory of sintering\({}^{10}\), which determines the dependence of the linear shrinkage on time,

\[ \frac{\Delta L}{L_0}(\tau) = \frac{8}{3}\,\pi\,\frac{\sigma\delta^{3}}{kT}\cdot ND(\tau)\cdot \tau, \tag{14} \]

and bearing in mind that

\[ D(\tau)=\xi(\tau)D_b=\frac{\xi(\tau)}{\xi_0}D_a, \]

one can determine the quantity

\[ \frac{\xi(\tau)}{\xi_0}, \]

and consequently also

\[ \frac{\Delta \xi}{\xi_0}=\frac{\xi(\tau)}{\xi_0}-1. \]

For experiments carried out in the isothermal regime, we find:

\[ \left. \frac{\Delta \xi}{\xi_0} \right|_{T=\mathrm{const}} \simeq \frac{3kT}{8\pi N\sigma\delta^{3}}\, \frac{1}{D_a}\, \frac{d}{d\tau} \left( \frac{\Delta L}{L_0} \right) \left.\right|_{T=\mathrm{const}} -1. \tag{15} \]

In formulas (14), (15), and (16), \(\sigma\) is the surface tension. Figure 14 shows the time dependence of \(\dfrac{\Delta \xi}{\xi_0}\), found from the data of dilatometric experiments; we note that the degree of accuracy in estimating the quantity \(\dfrac{\Delta \xi}{\xi}\) by formula (15) decreases as the quantity \(\dfrac{\Delta \xi}{\xi_0}\) decreases (i.e., as \(\tau\) increases). When \(\dfrac{\Delta \xi}{\xi_0} \simeq 1\), this estimate becomes very crude. A comparison of Figs. 13 and 14 indicates that, qualitatively, the dependences found from data on the kinetics of pore coagulation and the kinetics of volume shrinkage during sintering have one and the same character. The observed quantitative differences can be explained by the difference in the initial distortion of the lattice of the metal of the dense electroplated coating (experiments on coagulation) and of the dispersed powder (experiments on sintering).

Thus, in metals whose crystal lattice is strongly distorted, in particular in metals of galvanic origin, there may be

\[ \text{*} \]

\(*\) We note that the data under discussion from work\({}^{47}\) do not correspond to a strictly isothermal regime, since during the time of reaching the experimental temperature the quantity \(\dfrac{\Delta \xi}{\xi_0}\) could have decreased (see\({}^{5,7}\)).

... rather substantial supersaturations by vacancies occur, the consequence of which is the appearance of diffusion porosity. Let us note that the estimates made in \(^{47}\) of

\[ \frac{\Delta \xi}{\xi_0} \]

in order of magnitude and in the character of the temperature and time dependences are in agreement with data on diffusion coefficients in metals of galvanic origin. In objects in which the “source” of excess vacancies is regions where microdistortions are localized (“—” distortions), very large supersaturations may occur, the magnitude of which decreases together with the weakening of the “source strength.” This weakening takes place during annealing of the specimen.

Let us consider the question of supersaturation of a lattice by vacancies in the case where the source of excess vacancies is microscopic cavities that “heal” at high temperatures.

The magnitude of the supersaturation can be estimated in two independent ways: from data on the size of the “healing” microcavities and from data on the time dependence of the linear dimension of growing pores. In \(^{59}\) the following data are reported on the process of coagulation of pores in crystals of optically inhomogeneous rock salt. During 100 hours in a crystal with microcavity sizes \(L \sim 10^{-5}\) cm, in the course of annealing at a temperature of \(780^\circ\)C, pores with linear size \(l \sim 3 \cdot 10^{-3}\) cm grow. For estimating the value of

\[ \frac{\Delta \xi}{\xi_0} \]

from data on the size of the “healing” cavities one may use formula (19) (see the following section), according to which

\[ \frac{\Delta \xi}{\xi_0} = \frac{2\delta^3}{kTL} \simeq 10^{-3}, \]

since \(\sigma \simeq 5 \times 10^2\) erg/cm\(^2\), \(a_0^3 \simeq 2 \cdot 10^{-23}\) cm.

From data on the kinetics of pore growth, assuming in formula (10) that \(D_a \simeq 10^{-9}\) cm\(^2\)/sec \(^{60}\), we likewise obtain

\[ \frac{\Delta \xi}{\xi_0} \simeq 10^{-3}. \]

Using data on the kinetics of “healing” of microcracks that arose during quenching of steel \(^{57}\), one can likewise estimate the value of

\[ \frac{\Delta \xi}{\xi_0}, \]

which proves to be equal to \(10^{-2}\)—\(10^{-3}\).

Fig. 14. Dependence of the relative supersaturation of copper vacancies of galvanic origin on data on the shrinkage kinetics of powder compacts \(^{47}\).

\[ \circ-\circ-\circ \;-\; T = 750^\circ \mathrm{C}; \qquad \triangle-\triangle-\triangle \;-\; T = 1000^\circ \mathrm{C}. \]

Fig. 15. Distribution of concentrations of vacant sites in a supersaturated solution near pore surfaces.
a) \(\Delta \xi_r > \Delta \xi\); b) \(\Delta \xi_r = \Delta \xi\); c) \(\Delta \xi_r < \Delta \xi\) \(^{41}\).

Let us note that the “source” under discussion cannot be the cause of significant supersaturations, since already the supersaturation

\[ \frac{\Delta \xi}{\xi_0} \simeq \frac{2\delta^3}{kTL} \simeq 1 \]

presupposes the presence of cavities whose surface curvature is \(10^{-7}\) cm, which is meaningless \(^{47}\). The foregoing is an argument in favor of the assertion that porosity arising during annealing of a strongly distorted metal of galvanic origin \(^{47}\) cannot appear as a result of “healing” of microcavities, since the magnitude of the supersaturation estimated from the kinetics of pore growth proves to be very large

\[ \left( \frac{\Delta \xi}{\xi_0} \simeq 10^2—10^3 \right). \]

V. NUCLEATION OF DIFFUSION PORES

V.1. Critical nucleus of diffusion pores

(negative crystals)

The regularities described earlier for the appearance of diffusion porosity (the presence of a boundary, data on growth kinetics) provide a basis for the idea that a lattice which, for various reasons, is enriched with vacancies may be regarded as a supersaturated solution of vacancies in the lattice, from which a phase corresponding to the vacancies must “crystallize out,” i.e., negative crystallites (diffusion pores) must appear. In this connection there arises the question of the existence of a “critical” nucleus of a negative crystal, of the relation between the dimensions of the nucleus and the magnitude of the supersaturation \((\Delta \xi = \xi - \xi_0)\), etc.

Using primitive model notions, one might suppose that the appearance of negative crystals is not connected with the occurrence of a viable critical nucleus, since the coalescence of only two vacancies*) from among the excess vacancies already leads to an energy gain, consisting in the fact that, when a pair of vacancies forms, the number of uncompensated bonds decreases. Indeed, for coordination number 12, instead of \(2 \cdot 6 = 12\) there are 11 of them, since one bond is “saturated” when a pair of vacancies forms. It can be shown, however, that in principle a critical nucleus must exist, and its dimensions can be estimated.

In comparison with the elasticity of vapor near a plane liquid boundary \((P_0)\), the elasticity of vapor near the curved surface of a droplet, whose radius of curvature is \(r\), is known to be increased by the amount

\[ \Delta P = \frac{2\sigma}{r}\cdot \frac{\delta^3}{kT}\cdot P_0, \tag{16} \]

where \(\delta^3\) is the volume of one molecule in the condensed phase. In the case of dilute solutions, the quantities \(\Delta P\) and \(P_0\) are proportional to the concentrations of molecules in the gaseous phase, and, since formula (16) contains no particle masses, it may be used to find the quantity \(\Delta \xi_r\), which determines the increase in vacancy concentration near a pore of radius \(r\) due to the presence of curvature\({}^{10}\):

\[ \Delta \xi_r = \frac{2\sigma}{r}\frac{\delta^3}{kT}\cdot \xi_0, \tag{17} \]

where \(\xi_0\) is the equilibrium concentration of vacancies in the lattice.

The distribution of the concentration of vacant sites in the region adjoining a pore which is in a supersaturated solution of vacancies, according to (17), will be different for pores of different size. For a given value of the supersaturation there may be three different types of distribution of the concentration of vacant sites\({}^{41}\) (Fig. 15). At \(r\), when \(\Delta \xi_r > \Delta \xi\) (\(\Delta \xi\) is the magnitude of the specified supersaturation of vacancies in the lattice), the distribution

*) We note that pairs of vacancies in a supersaturated solution at high temperatures arise very frequently. Indeed, assuming that the aggregate of vacancies forms an ideal gas, the mean free path of a vacancy (before formation of a “pair,” i.e., before encounter with another vacancy) \(\lambda\) can be determined with the aid of the formula:

\[ \lambda \simeq \frac{1}{n\chi}, \]

where \(n\) is the number of vacancies per unit volume, \(\chi\) is the effective cross section of the collision process. Since in our case \(\chi \sim \delta\), and \(n=\xi N\) (\(\xi\) is the vacancy concentration, \(N \sim 1/\delta^3\) is the number of lattice sites per unit volume), then \(\lambda \sim \delta/\xi\). The “waiting time” for a collision can be determined as:

\[ \tau \sim \frac{\lambda^2}{D} \sim \frac{\delta^2}{D\xi^2} \]

for \(\delta \simeq 3 \cdot 10^{-8}\,\text{cm}\), \(\xi = 10^{-2}\), \(D_{T \simeq 1000^\circ \mathrm{C}} \simeq 10^{-8} - 10^{-9}\,\text{cm}^2/\text{sec}\). We find \(\tau = 10^{-2} - 10^{-3}\,\text{sec}\).

concentration of vacancies is such (Fig. 15, a) that their flux will proceed from the surface of the pore. This corresponds to the process of “sintering” of the pore, or its reduction. For \(r\), when \(\Delta \xi_r < \Delta \xi\) (Fig. 15, c), the distribution of vacancy concentrations near the pore will ensure their influx to the pore surface. Obviously, the size of a viable nucleus must be determined by the condition

\[ \Delta \xi_r = \Delta \xi, \tag{18} \]

when the distribution of vacancy concentrations will be as shown in Fig. 15, b. Taking into account formula (17) and condition (18), one can find an expression for the radius of the critical nucleus

\[ r^* = \frac{\xi_0}{\Delta \xi}\, 2\sigma \cdot \frac{\delta^3}{kT}. \tag{19} \]

Let us note that in (19) there appears the quantity of the surface tension*) at the crystal—vacuum boundary, i.e., the proper surface tension of the substance. It is known\(^{49}\) that this quantity is approximately two orders of magnitude greater than the interphase surface tension \((\sigma_{ik})\) at the boundary between the parent phase and the nucleus, if the substance of the nucleus consists of the same atoms as the substance of the parent phase. As a result of this, for the same deviation from equilibrium, which can be quantitatively characterized by the magnitude of the relative supersaturation, the formation of a nucleus of a negative crystal presupposes a larger fluctuation than in the case when the boundary nucleus—parent phase is characterized by the surface tension \(\sigma_{ik}\). Obviously, for a given magnitude of supersaturation the work of formation of a nucleus of a negative crystal is determined by the formula

\[ \Delta \Phi_3^* = \frac{4}{3}\,\pi r^{*2}\sigma \simeq \frac{16\pi\sigma^3\delta^6}{3(kT)^2} \left(\frac{\xi_0}{\Delta \xi}\right)^2 . \tag{20} \]

V.2. The role of “impurities” in the nucleation of diffusion pores

Using formula (19) and the estimates of the supersaturation made earlier, one can approximately estimate the linear dimensions of the critical nucleus. Bearing in mind the formation of negative crystals during mutual diffusion in solid solutions, where very small supersaturations occur

\[ \left(\frac{\Delta \xi}{\xi_0} \simeq 5 \cdot 10^{-2}\right), \]

it is easy to show that the appearance of a nucleus presupposes a very large fluctuation: the resulting nucleus must consist of \(\simeq 10^9\) vacancies, which seems improbable.

By analogy with known facts from the field of crystallization of liquid solutions and melts, one may assume that the nucleation of negative crystals occurs not on nuclei of fluctuation origin (although such a possibility is not excluded in principle), but on “impurities.”

In pure polycrystalline bodies, free from foreign inclusions, two essentially different types of “impurities” may occur. The first type of “impurity” is various kinds of interfaces between

*) The quantity \(r^*\) can, as usual, be determined by minimizing the change in the free energy \(\Delta \Phi\) of the supersaturated solution that is associated with the formation of a pore of radius \(r\): \(\Delta \Phi = S\sigma - V\Delta \varphi\), where \(S = 4\pi r^2\), \(V = \frac{4}{3}\pi r^3\), \(\Delta \varphi = \frac{1}{\delta^3}(\mu'' - \mu')\), \(\Delta \varphi\) is the change in the thermodynamic potential of the solution upon precipitation from it of a pore of unit volume, \(1/\delta^3\) is the number of lattice sites per unit volume, and \(\mu''\) and \(\mu'\) are the chemical potentials of vacancies in solutions with concentrations \(\xi\) and \(\xi_0\). Since

\[ \mu_i = kT \ln \xi_i + \psi(P,T), \]

then

\[ \Delta \varphi \simeq \frac{kT}{\delta^3}\ln\frac{\xi}{\xi_0} \simeq \frac{kT}{\delta^3}\,\frac{\Delta \xi}{\xi_0}. \]

Having in view the expression found for \(\Delta \varphi\), from the condition

\[ \frac{\partial}{\partial r}\Delta \Phi = 0 \]

we find expression (19).

in direct contact with elements of the metal structure (grains, mosaic blocks). Such boundaries may be characterized by a surface tension \(\sigma_{ik}\). Let us note that the type of free boundaries under discussion is free of discontinuities, which in deformed objects most often arise precisely between mosaic blocks and grains. The second type of “impurities” should include various kinds of discontinuities (microscopic cracks). Such discontinuities may be present in the specimen in connection with its previous history (arising during crystallization, of deformation origin, etc.); they may also appear in the diffusion process as a result of stresses arising in the diffusion zone \(^{33,36,64}\). Impurities of this type may be characterized by the actual surface tension of the substance \(\sigma\). It is easy to show that “impurities” of the second type play a substantially greater role in the nucleation of diffusion pores than “impurities” of the first type. Indeed, the work of formation of a three-dimensional nucleus in the immediate vicinity of a boundary \((\Delta\Phi_3^{*1})\) is reduced by an amount \(\sim \pi r^2\sigma_{ik}\) in comparison with the work of formation of a nucleus far from the interface between structural elements \((\Delta\Phi_3^*)\). Considering the quantity

\[ k=\frac{\Delta\Phi_3^*}{\Delta\Phi_3^{*1}}=\frac{\sigma}{\sigma-\frac{3}{4}\sigma_{ik}}, \]

we find that, since for interblock boundaries \(\sigma_{ik}\sim 10^{-4}—10^{-3}\ \mathrm{erg/cm^2}\), and for intergranular boundaries \(\sigma_{ik}\sim 10^{-1}\ \mathrm{erg/cm^2}\), boundaries of this type cannot substantially affect the probability of spontaneous nucleation of a diffusion pore \(^{49}\).

In order for an “impurity” of the crack type to become the nucleus of a diffusion pore, it is necessary that the area of the crack exceed the area of a viable two-dimensional nucleus \((\sim L^{*2})\), the appearance of which must occur for the development of crack growth. Cracks of smaller dimensions must be healed by diffusion and thus cannot serve as a “sink” for excess vacancies. The magnitude of the linear size of a two-dimensional nucleus can be estimated by minimizing the change in free energy due to the occurrence of such a nucleus. The calculation leads to the expression

\[ L^*\simeq \frac{\xi_0}{\Delta\xi}\cdot 2\sigma_L\frac{\delta^2}{kT}, \]

where \(\sigma_L\) is the “linear” tension. Assuming that \(\frac{\Delta\xi}{\xi_0}\simeq 10^{-1}\), \(\sigma_L\sim \sigma\delta\), we obtain \(L^*\sim 10^{-6}\ \mathrm{cm}\). Thus, cracks with linear dimensions of the order of \(10^{-2}\ \mu\) may serve as nuclei of negative crystals. In \(^{51}\), on the basis of data on the scattering of X-rays at small angles, it is reported that in specimens of \(\alpha\)-brass there are discontinuities whose linear size is \(\sim 10^{-2}\ \mu\). Let us note here that the absence of diffusion pores in the immediate vicinity of the external boundary of a brass specimen from which zinc is removed is indirect evidence of the important role of free boundaries possessing a large surface tension in the nucleation of diffusion pores. This is also indirectly indicated by observations according to which very often a group of pores, identically oriented with respect to the plane of the section, i.e. evidently situated in one grain \(^{34}\), is arranged regularly, forming a certain chain along which, apparently, a crack extended (Fig. VII). It is very difficult to understand the reason for the arrangement of many pores along a chain within one grain without invoking the idea of a crack, since the spontaneous nucleation of all the pores of the chain is vanishingly improbable*).

*) Numerous experiments \(^{20,38,65}\) indicate that diffusion pores arising during the evaporation of zinc from brass are located in considerable numbers near intergranular boundaries. Discontinuities arise at these boundaries because zinc diffuses preferentially along grain boundaries.

The considerations set forth in this section concerning the role of free surfaces in the process of coagulation of excess vacancies make it possible to assert that excess vacancies are an “alloying impurity,” and a highly pore-forming one. In the present case, however, the emergence of the “pore-forming impurity” at the surface does not, as usual, lead to a decrease in surface tension, but leads to a decrease in the total area of free surfaces, if it is assumed that an isolated vacancy is associated with a surface \(\sim \pi\delta^{2}\). Thus, owing to the pore-forming character of vacancies, the surface energy \((E \sim \sigma S)\) decreases not in connection with a decrease in \(\sigma\), but because of a decrease in \(S\). Let us note that the assumption that an isolated vacancy is associated with a surface \(\sim \pi\delta^{2}\) and, correspondingly, a surface energy \(\sim \pi\delta^{2}\sigma\), is reasonable, since the estimate made on its basis for the magnitude of the energy of hole formation \(\left(u \sim \pi\delta^{2}\sigma \sim 10^{4}\ \frac{\text{cal}}{\text{mole}}\right)\) leads to a value of the correct order.

Let us turn to the question of pore nucleation during annealing of a metal of galvanic origin. The considerable supersaturation present in this case could make spontaneous nucleation of diffusion pores probable; however, the circumstance that in the initial state the metal of galvanic origin has a large number of microscopic discontinuities, i.e., ready-made nuclei, makes the mechanism of nucleation of diffusion pores on “impurities” predominant in this case as well. As the photographs testify, in a metal of galvanic origin the embryonic cracks have the character of intergranular discontinuities.

VI. DIFFUSION POROSITY AND SINTERING OF MIXTURES OF METALLIC POWDERS

The process of the formation of diffusion porosity and the phenomena accompanying it can substantially affect the kinetics of volumetric shrinkage during sintering of compacts obtained by pressing mixtures of powders of mutually diffusing metals.

It is known that during high-temperature annealing of a single-phase compact, in addition to the process of volumetric shrinkage itself (“removal of voids” from the compact), there also occur processes of recrystallization, removal of distortions of the crystal lattice of the powders, etc. All these processes proceed by means of a self-diffusion mechanism and are mutually interdependent. During sintering of two-phase compacts, in addition to the processes listed, there also occurs the process of mutual diffusion, leading to homogenization of the powder mixture and accompanied by the formation of diffusion porosity. This process must contribute to the magnitude of the dilatometric effect that takes place during sintering. Thus, in a real two-phase powder compact the total volumetric shrinkage is simultaneously governed by many phenomena, and therefore direct study of the influence exerted by mutual diffusion on volumetric shrinkage is very difficult.

Certain details of the interrelation between the kinetics of volumetric shrinkage governed by the process of self-diffusion and the kinetics of the process of heterodiffusion leading to equalization of concentrations within the compact were clarified in ⁵, in which the porous body was modeled by wires of different metals. A substantial difference between real powder porous bodies and the wire models studied in ⁵ consists in the fact that the crystal lattice of powders is usually strongly distorted, as a result of which the coefficients of self- and heterodiffusion are overestimated and change with time by virtue of the process of removal of distortions occurring during sintering ⁶. Thus, modeling, by depriving a very complex phenomenon of some of its features, makes it possible to trace others in a purer form.

In \({}^{5}\), specimens were studied in the form of a bundle of seven copper and nickel wires of identical diameter (\(\sim 0.13\ \mathrm{mm}\)), mounted in a copper or nickel tube (Fig. VIII, \(a\)). Among the specimens investigated were those in which six copper wires surrounded one nickel wire, and those in which six nickel wires surrounded one copper wire. In such specimens the cross sections of the wires imitate powder particles, and the gaps between them—pores. Diffusion annealing of all specimens was carried out at a temperature of \(1040 \pm 10^\circ\), both in a stream of hydrogen and in vacuum. The processes occurring during diffusion annealing were observed by metallographic methods. The totality of observations made in these experiments reduces to the following: a) homogenization is effected mainly by diffusion of copper into nickel, i.e., the diffusion is predominantly unipolar in character; b) diffusion porosity arises on the copper; nickel wires swell; c) the distance between the centers of the peripheral wires increases \(\left(L_{t=0} \simeq 2.6 \cdot 10^{-2}\ \mathrm{cm},\ L_{t=34r} \simeq 3.0 \cdot 10^{-2}\ \mathrm{cm}\right)\). In the initial state the contacting wires, after prolonged diffusion annealing, prove to be separated. Thus, at the early stage the process of homogenization is accompanied by an increase in the volume of the specimen, i.e., it counteracts volumetric shrinkage. The latter becomes especially evident if the observations made in \({}^{5}\) are compared with the results of \({}^{52}\), where, on analogous models, but consisting of seven identical copper wires, the kinetics of shrinkage of single-phase compacts was studied. In this case, when during annealing of a specimen modeling a porous body only the process of self-diffusion takes place, the centers of the peripheral wires approach one another with time, the area of contact between neighboring wires grows, and the area of the gap between them (the pore!) decreases (Fig. IX). The observations made on copper–nickel models pertain to intermediate stages on the way to an equilibrium state, in which both concentration gradients and porosity—both initial and that arising in the process of heterodiffusion—must be absent. Thus, at the stage when homogenization has not yet been completed and its course entails supersaturation of one of the phases with vacancies and, as a consequence, the appearance of pores, heterodiffusion makes a negative contribution to the magnitude of volumetric shrinkage. As follows from the observations described in \({}^{5}\), the negative contribution of heterodiffusion at this stage may exceed the positive contribution associated with self-diffusion of vacancies, as a result of which not shrinkage but swelling of the specimen will occur. It should be pointed out that over the course of the stage of the sintering process under discussion, conditions are created which should subsequently promote volumetric shrinkage. Thus, as a result of heterodiffusion the vacancy concentration increases and fine additional porosity appears; both of these effects should promote sintering at more advanced stages.

A substantial influence on the kinetics of the appearance of diffusion porosity and on the magnitude of volumetric shrinkage in real powder compacts may be exerted by the process of removal of distortions of the crystal lattice of the powders of the mixture. In order to trace the interrelation of the processes of self- and heterodiffusion during sintering of powder porous bodies, dilatometric experiments were undertaken in \({}^{53}\) with specimens obtained by pressing mixtures of copper and nickel powders. Mixtures of approximately equal-sized powders were studied. Such an arrangement of the experiments is the next step on the path from the study of wire models to the study of powder porous bodies obtained by pressing mixtures of powders of different metals differing in grain size and granulometric composition. In \({}^{53}\), dilatometric experiments performed by two methods are described. One of the methods consisted in continuous observation with a dilatometer of the linear shrinkage of a compact subjected to stepwise heating \({}^{6}\). At each temperature step, the interval

between which was equal to \(100^\circ\)C, isothermal holds of duration either \(\Delta\tau=3\) min or \(\Delta\tau=15\) min were carried out. The second procedure consisted in prolonged isothermal sintering of sets of three specimens pressed from copper and nickel powders and from a mixture of copper—nickel powders \((50\%—50\%)\), having the same geometry and initial porosity. In \(^{53}\) it was experimentally established that the linear shrinkage of porous specimens pressed from mixtures of powders of mutually diffusing metals is less than the shrinkage of compacts from the powders constituting the mixtures. In \(^{53}\) the experimental curves for the dependence of the magnitude of linear shrinkage on the concentration of the mixture were treated under the assumption that the measured value of the linear shrinkage of the compact \((\Delta L)\) is the result of summing the shrinkages at individual contacts between powder particles intersected by a straight line coinciding with the direction along which the dilatometric measurement is carried out. In this case, if \(C\) is the concentration of particles of grade \(A\), then:

\[ \Delta L=L_{AA}C^2+L_{BB}(1-C)^2+2L_{AB}C(1-C), \tag{21} \]

where \(L_{AA}\) and \(L_{BB}\) are the linear shrinkages of single-phase compacts, and \(L_{AB}\) is the linear shrinkage of a compact of a \(50\%\) mixture, in which all paired contacts are of the \(A—B\) type (“ordered mixture”). The deviation of linear shrinkage from the law of additivity may be a consequence both of purely geometrical causes, when contacts of the \(A—B\) type drop out from the number of “active” contacts (which, apparently, may occur in the case of mutually nondiffusing metals), and of processes occurring at unlike contacts, when the process of mutual diffusion causes the appearance of diffusion porosity, a change in volume, etc. Using the concept of “shrinkage per contact,” in \(^{53}\) the experimental curve of the dependence of the relative linear shrinkage of a compact on the concentration of the mixture is decomposed into “configurational” (the first two terms in (21)) and “diffusional” (the third term in (21)) components. The latter is the source of information on the interrelation of the processes of mutual diffusion and shrinkage of the powder mixture.

Fig. 16. Dependence of “shrinkage per contact” on temperature under a stepwise sintering regime. —●— \(\Delta\lambda_{AB};\ \Delta\tau=15\) min; —○— \(\Delta\lambda_{AA};\ \Delta\tau=15\) min; —△— \(\Delta\lambda_{AB};\ \Delta\tau=3\) min; —□— \(\Delta\lambda_{AA};\ \Delta\tau=3\) min. \(^{53}\)

In addition to the formation of porosity observed on wire models, in mixtures of active powders it may be expected \(^{5}\) that the kinetics of the early stage of the shrinkage process will be further complicated by the fact that heterodiffusion facilitates the removal of distortions of the crystal lattice of the “active” powders. Bearing in mind the remarks made, one may, on the basis of the experimental data, consider the question of the temperature dependence of the negative contribution made by the process of mutual diffusion to the magnitude of linear shrinkage at the early stage of the sintering process. This contribution is characterized by the value of the “shrinkage per one contact” between unlike particles \((\lambda_{AB})\), referred to one isothermal step \((\Delta\lambda_{AB})\). The corresponding curves of the dependence \(\Delta\lambda_{AB}=\varphi(T)\) are shown in Fig. 16. A feature of these curves is the presence of a maximum near \(T=900^\circ\)C. When comparing the curves corresponding to \(\Delta\tau=3\) min and \(\Delta\tau=15\) min, it is noteworthy that, up to the temperature of the maximum, the curve \(\Delta\lambda_{AB\,\Delta\tau=3\ \mathrm{min}}=\varphi(T)\) lies above the curve \(\Delta\lambda_{AB\,\Delta\tau=15\ \mathrm{min}}=\varphi(T)\), and accordingly the maximum on the first curve is more

high. These features of the curves may be interpreted as follows. Under more rapid heating (\(\Delta \tau = 3\) min), distortions of the crystal lattice in the temperature range lower than that at which the maximum is located have less time to be removed than under slow heating (\(\Delta \tau = 15\) min). In this connection, the processes of mutual diffusion under rapid heating proceed to a greater extent, and correspondingly their negative contribution to the magnitude of the linear shrinkage is larger. Under slow heating, the process of removing distortions is more fully realized, as a result of which, at subsequent temperature stages, the processes of mutual diffusion prove to be slowed down. The presence of descending branches on the curves \(\Delta \lambda_{AB}=\varphi(T)\) may be a consequence of the fact that, owing to the small size of the powders (\(\sim 4 \cdot 10^{-3}\)), the process of homogenization has time, over an interval of the order of tens of minutes, to proceed to a considerable extent. Owing to this, the volume of material transported by means of the mechanism of mutual diffusion decreases, and correspondingly the value of \(\Delta \lambda_{AB}\) must also decrease.* The curves \(\Delta \lambda_{AA}=\varphi(T)\), given in Fig. 16 for comparison and constructed on the basis of data on the shrinkage kinetics of one-component compacts made of copper powder, indicate that the presence of a descending branch on the curve \(\lambda_{AB}=\varphi(T)\) is not connected with a loss of the “activity” of the compact (because of a decrease in porosity), but expresses the features of the processes occurring at unlike contacts.

Information on the time dependence of the negative contribution to the magnitude of the linear shrinkage, connected with the appearance of diffusion porosity, follows from experiments consisting in the simultaneous prolonged sintering of compacts made up of three specimens (see above), with subsequent X-ray study of the degree of homogenization of the mixture and measurements of changes in volume. These experiments showed that the dependence of the magnitude of the diffusion contribution on annealing time is described by a nonmonotonic curve having a minimum (Fig. 17, curve \(a\)). The presence of the minimum may be a consequence of the fact that the process of mutual diffusion, first, causes “growth” of the specimen in connection with the appearance of diffusion porosity and, second, promotes acceleration of the process of self-diffusion owing to an increase in the concentration of vacant sites. The curve of the diffusion contribution may, in view of the considerations set forth, be represented as the sum of two components, as has been done in Fig. 17 (curves \(б\) and \(в\)).**

Fig. 17. Graphical decomposition of the curve of the diffusion contribution (\(a\)) to the shrinkage into components. Curve \(б\): \(\dfrac{\Delta l}{l}\sim t^{1/2}\); curve \(в\): the time dependence of the effect due to diffusion porosity. Copper—nickel mixtures (50%—50%).\(^{52}\)

The negative contribution caused by the effect of the appearance of diffusion porosity is described by curve \(в\) (Fig. 17). X-ray experiments

* The assumption concerning the rate of homogenization appears plausible, since the estimate of the coefficient of mutual diffusion made on its basis leads to a reasonable value. Indeed, for a powder-particle size of \(\sim 4—5 \cdot 10^{-3}\) cm and times of the order of 20 min, \(D \simeq 10^{-8}\ \text{cm}^2/\text{sec}\).

** In decomposing the curve into components, attention was paid to the fact that, in the region of long times, it obeys the law \(\tau^{1/2}\). One of the components (curve \(б\)) was constructed according to this law.

witness to the fact that the growth of the negative contribution, caused by the appearance of diffusion porosity, stops precisely when the diffusion transfer of substance in the homogenization process is, in the main, completed.

The interrelation of the kinetics of removal of distortions with the processes of formation of diffusion porosity and volume shrinkage of the powder mixture is qualitatively manifested in experiments in which, using a dilatometer in a step-heating regime, observations were made of the linear shrinkage of compacts obtained by pressing mixtures of copper and nickel powders of different initial “activity” (in the sense of the degree of distortion of the crystal lattice). In the region of low temperatures, “growth” (swelling) of the compact was observed; with increasing temperature this was replaced by volume shrinkage. The experiments described indicate that: 1) a decrease in the “activity” of one of the components of the mixture shifts the temperature interval in which the negative effect associated with diffusion porosity appears toward higher temperatures; 2) the magnitude of the negative shrinkage effect is greater in the case of powders of high activity. These observations can be qualitatively explained as follows.

The degree of manifestation of unilateral diffusion should be the greater, the greater the difference in the heats of evaporation of the diffusing components. If the fact of the removal of distortion during preliminary annealing is regarded as the cause of an increase in the effective heat of evaporation, then unilateral diffusion of copper into nickel in the mixture passive copper—active nickel should manifest itself in the region of higher temperatures than in the mixture active copper—active nickel. Experiments with compacts of mixtures of iron and nickel powders—two metals whose heats of evaporation differ little—lead to analogous conclusions. The magnitude of the negative contribution introduced by mutual diffusion into the value of the linear shrinkage is small in this case; it decreases with temperature, and already at \(T = 900^\circ\mathrm{C}\) the total shrinkage of the mixture exceeds the value that it should have had under the assumption of additivity of the shrinkages of the components of the mixture.

Naturally, one should expect that in inhomogeneous porous bodies the negative contribution to the value of the linear shrinkage, associated with the appearance of diffusion porosity, should depend on the initial porosity of the compact. The magnitude of this contribution should increase as the initial porosity of the compact decreases. Such a dependence may result from two causes: a) with decreasing porosity the number of dissimilar contacts near which diffusion porosity arises decreases; b) the change in volume due to the appearance of diffusion porosity will affect the change in the external dimensions of the compact to a greater extent when the “internal free volume” is small, i.e. when the initial porosity is small.

Fig. 18. Curve of the dependence of the magnitude of “growth” at one unlike contact on the initial porosity of the compact.

Figure 18 shows the curve of the dependence of the magnitude of “growth” at one unlike contact, found from dilatometric experiments with copper–nickel compacts having different initial porosity. This curve supports the considerations set forth concerning the interrelation of the initial porosity with the magnitude of the negative contribution to volume shrinkage caused by the appearance of pores of diffusion origin. We note that, by extrapolating the curve in Fig. 18 to zero porosity, one can estimate the magnitude of the “growth” referred to one unlike contact in the case when there is no internal free volume (i.e. in the case of zero porosity—\(\lambda^0_{AB}\)). This quantity can be

compare with the magnitude of the “growth” per one contact of type \(A—B\), as follows from the experiments of Barnes\({}^{36}\), who studied the change in thickness of a multilayer specimen during diffusion annealing. The specimen in Barnes’s experiments was composed of alternating plates of copper and nickel and thus is a model of a compact with an initial porosity equal to zero, in which all contacts are of type \(A—B\). From the data on the “growth” effect reported in \({}^{36}\), it follows that \(\lambda_{AB}^{0}\simeq 3—5\cdot 10^{5}\ \text{cm}\), which is close to the value following from Fig. 18.

VII. CONCLUSION

The totality of the experimental facts set forth in the article indicates that the appearance of diffusion porosity accompanies many processes occurring in crystalline (metallic and nonmetallic) systems. In all the cases considered—during diffusion homogenization, the removal of distortions, the coagulation of micropores, etc.—the process of formation of diffusion porosity, causing a decrease in the free energy of a system removed from the state of thermodynamic equilibrium, is a stage on the path toward the establishment of the state of true equilibrium in the specimen.

Diffusion porosity, with which a highly developed system of additional internal surfaces is associated, may to a considerable extent determine the kinetics of phase transformations in crystalline systems. This, in our opinion, is one of the reasons why investigations of the process of the occurrence of diffusion porosity—already a source of much information on the details of phenomena occurring in crystalline systems—will continue to be of great interest.

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