SINTERING, CREEP, RECOVERY, RECRYSTALLIZATION, AND OTHER PHENOMENA CAUSED BY SELF-DIFFUSION IN CRYSTALLINE SOLIDS

B. Ya. Pines

CONTENTS

1. The phenomenon of sintering . . . 502
2. Sintering as viscous flow. The theory of Ya. I. Frenkel . . . 504
3. The diffusion theory of sintering and its connection with the theory of viscous flow . . . 507
4. Complication of the phenomenon of sintering under practical conditions . . . 514
5. Some experimental data on the rate of sintering . . . 517
6. Sintering as a form of diffusion creep under the action of surface-tension forces. Laws of diffusion creep. Sintering under pressure. Creep of solid bodies . . . 527
7. The phenomenon of recovery and its influence on the kinetics of diffusion creep (sintering) . . . 536
8. Recrystallization . . . 545
9. Sintering and creep of amorphous bodies . . . 554
10. On the connection of heterodiffusion phenomena with self-diffusion phenomena in crystalline bodies . . . 556
11. Conclusion . . . 558

Until comparatively recently it was believed¹ that manifestations of self-diffusion in solid bodies were very limited; as one of the few observed consequences of self-diffusion, the displacement (change in distribution) of radioactive isotopes was considered—for example, isotopes previously deposited on the surface and then diffusing into the interior of the body. Even at the present time, to measure the coefficient of self-diffusion, use is made almost exclusively of the “radioactive-isotope method.”² However, it is now known that a broad group of processes, previously not directly associated with self-diffusion, is in fact caused by this phenomenon. These include the processes of so-called sintering, diffusion creep, recovery, recrystallization, surface

creep of atoms, and certain others. Self-diffusion manifests itself especially distinctly in the indicated processes in simple bodies composed of atoms of one kind (for example, in metals). But even in the case of more complex bodies (in particular, alloys), the number of facts indicating the role of self-diffusion in phenomena in which it had previously not been taken into account is steadily increasing. Thus, it is becoming clear³ that processes of heterodiffusion are closely connected with self-diffusion (cf. below).

1. THE PHENOMENON OF SINTERING

Let us first turn to the phenomenon of sintering, the role of self-diffusion in which has already been clarified rather fully. As is known, this phenomenon is widely used in technology, in particular in the production of ceramic and refractory materials and in metal ceramics.

Outwardly, sintering manifests itself in the fact that a porous body (for example, one obtained by pressing powders), after being heated to a suitable temperature, exhibits “shrinkage” (i.e., a reduction of its “external” linear dimensions) and becomes proportionally denser (its porosity decreases); at the same time the mechanical strength of the body also increases.

Sometimes, in addition to sintering, one also distinguishes the phenomenon of mutual “baking together” of several bodies between which contact has been established; after heating, a single body is formed from such a group of bodies, and the initial contact, which may have existed only at a few points, proves to be continuous, up to the complete “disappearance” of all traces of the surfaces separating the individual bodies.

Both variants of the phenomenon essentially correspond to the “spontaneous” filling with substance of free volume inside bodies or between bodies*).

In practical conditions, the phenomenon of sintering is often used not in a “pure” form. For example, refractory articles are prepared from powders of refractory oxides with the addition of a so-called “binder,” whose purpose may vary. Sometimes it serves to bring about a reaction in the solid phase and the formation of new compounds cementing the grains of the starting material. In other cases the “binder” is added in order to obtain, upon

*) Let us note that the definition of sintering sometimes encountered in the technical literature⁴ as a phenomenon involving change of the contact surface inside bodies or between bodies should be recognized as insufficient and imprecise. The surface is a two-dimensional boundary of a three-dimensional body, and any change in it (including that which occurs during sintering) can take place only as a result of a volume redistribution of substance.

during heating, at the grain boundaries of the main substance, a small amount of a relatively fusible eutectic is formed, facilitating rapid crystallization and the formation of a unified “crystalline intergrowth” in the body.

The introduction of a “binder” is also practiced solely to preserve the “strength” of the pressed powder until adhesion between the grains is achieved as a result of “sintering.” The need for such a “binder” arises when a powder of a brittle substance is subjected to pressing. Usually, in the case of refractory oxides, an organic binder is introduced for this purpose (for example, dextrin), which burns out when the body is heated in an oxidizing atmosphere and does not affect the state of the oxides.

The amount of binder is taken so small that there is also direct contact between the grains of the body, permitting “coalescence” of the grains to take place (upon heating).

Only the last of the cases indicated corresponds to sintering in its pure form. When the introduction of a binder leads to a reaction in the solid phase or to the appearance of a liquid phase, the processes that develop are more complex, and we shall not consider them here.

It should be pointed out that, when powders of sufficiently plastic bodies (metals, plastic salts) are pressed, it is not necessary to introduce a “binder” to preserve the strength of the compacts. Plastic deformation of the grains leads to such considerable mutual “interlocking” of them that, during heating as well, the strength of the body and good contacts between the grains are preserved.

In practice, sintering is used both for one-component and for multicomponent bodies (made up of powders of different substances). In the latter case, besides sintering, upon heating of the body there may also occur processes of alloy formation (i.e. solid solutions, new crystalline phases, etc.). We shall not touch upon such processes at first, i.e. we shall restrict ourselves to considering the sintering of one-component bodies, uncomplicated by the formation of an alloy or the appearance of new phases.

However, even in the case of one-component porous bodies, heating will not always involve only the phenomenon of sintering. If a body is formed by pressing a powder, then it is fine-grained, and the crystal lattice in the individual grains is distorted. Upon heating to temperatures at which self-diffusion becomes appreciable, the phenomena of recovery and recrystallization will occur simultaneously with sintering. Only in a porous single-crystal body with an undistorted crystal lattice could sintering be observed in its pure form. At first we shall set aside these complicating circumstances and assume that the phenomenon of sintering is being considered in a “pure” form, without accompanying processes of recovery and recrystallization in a one-component body.

2. SINTERING AS VISCOUS FLOW. THE THEORY OF Ya. I. FRENKEL

The pioneer in the development of the physical theory of sintering (as well as of a number of other fruitful theories, among which we note the so-called “hole” theory of self-diffusion, related to the question under consideration\(^5\)) was the outstanding Soviet scientist Prof. Ya. I. Frenkel.

Frenkel put forward the idea of viscous flow of solids\(^6\) by means of a diffusion mechanism, regarding this flow as analogous to that observed in liquids and describing it as the directed displacement of a small number of “holes.” In liquids, by “holes” Frenkel means very small cavities (of linear dimensions of the order of several angstroms), opening and closing owing to thermal fluctuations.\(^7\) In crystals the “holes” are simply “vacancies,” i.e., lattice sites not occupied by atoms. The number of these “vacancies” increases when a crystal is heated, since they are formed by atoms leaving their sites as a result of thermal motion. The equilibrium number of vacancies \(N'\) at each temperature \(T\) is given by the formula

\[ \frac{N'}{N}=e^{-\frac{U}{kT}}, \tag{1} \]

where \(N\) is the total number of sites, \(k\) is Boltzmann’s constant, and \(U\) is the “energy of hole formation,” of the same order of magnitude as the latent heat of evaporation.

In his theory of self-diffusion Frenkel proceeds from the fact that the displacement of atoms in the lattice consists in their successive replacement by “vacancies.” It is therefore possible to describe self-diffusion also as the displacement of “vacancies” and to introduce the concept of a vacancy diffusion coefficient \(D'\), related to the self-diffusion coefficient of the crystal \(D\) by the formula

\[ D=\frac{N'}{N}D'=cD', \tag{2} \]

where \(c=\frac{N'}{N}\) is the concentration of vacancies.

As for the magnitude \(D'\), according to Frenkel it is equal\(^5,7\) to

\[ D'=\frac{\delta^2}{6\tau}=\frac{\delta^2}{6\tau_0}e^{-\frac{\Delta U}{kT}}, \tag{3} \]

where \(\delta\) is the distance between neighboring sites, \(\tau\) is the mean residence time of a “vacancy” at one and the same lattice site, \(\tau_0\) is the period of oscillation of an atom about its equilibrium position, and \(\Delta U\) is the activation energy necessary for the transition of an atom from the initial site to a neighboring one (initially vacant).

When describing the viscous flow of solids Frenkel used the formula for the viscosity coefficient \(\eta\)

\[ \frac{1}{\eta}=\frac{D}{kT}\delta, \tag{4} \]

which he had previously established for the case of liquids (amorphous bodies). Formula (4) is obtained by comparing Stokes’ formula for the resistance of a sphere (of radius \(\delta\)) moving in a viscous liquid, and Einstein’s relation between the diffusion coefficient and mobility.

Next, viscous flow is described purely phenomenologically, as in an isotropic amorphous body. We shall present Frenkel’s solution of two problems relating to the phenomenon of sintering. These are: a) the problem of the viscous filling-in of a spherical pore in a continuous body, and b) the problem of the “coalescence” of two drops. In both cases the process is considered as viscous flow under the action of surface-tension forces.

a) A spherical cavity in a viscous body must continuously decrease under the action of capillary forces. Owing to spherical symmetry, the displacements at all points of the body will have a radial direction, and their velocity for an incompressible body must be expressed by the formula

\[ v=\frac{\beta}{r^2}, \]

where \(r\) is the distance of the corresponding point from the center of the cavity. The coefficient \(\beta\) can be determined through the rate of change of the cavity radius \(a\):

\[ \beta=a^2\frac{da}{dt}. \]

Equating the work of the internal-friction forces in the entire volume of the body \(V_0\), expressed as

\[ \int_{V_0} 2\eta \sum_i \sum_k v_{ik}^2\,dV, \]

where

\[ v_{ik}=\frac{1}{2}\left[\frac{dv_i}{dx_k}+\frac{dv_k}{dx_i}\right] \]

is the strain-rate tensor (in the present problem reducing to the radial component \(v_{rr}=\dfrac{dv}{dr}=-\dfrac{2\beta}{r^3}\)), to the work of the surface-tension forces (i.e., to the decrease of the free energy of the cavity surface)

\[ 2\eta\int_a^\infty v_{rr}^2\,4\pi r^2dr = \frac{32\pi}{3}\eta\frac{\beta^2}{a^3} = -\frac{d}{dt}(4\pi\sigma a^2) = -8\pi\sigma a\frac{da}{dt}, \]

where \(\sigma\) is the coefficient of surface tension, Frenkel obtains:

\[ \frac{da}{dt}=-\frac{3}{4}\frac{\sigma}{\eta}. \tag{5} \]

This equation shows that the decrease in the radius of the cavity must occur at a constant rate. The time for complete closing of the pore is equal to

\[ t_0=\frac{4}{3}\frac{\eta a_0}{\sigma}, \tag{5a} \]

where \(a_0\) is the initial value of its radius (at the moment \(t=0\)).

b) Two “viscous” drops of spherical shape, touching at the initial moment of time at a single point, after coalescence will be in contact along a circle of radius \(y(t)\) (Fig. 1).

Fig. 1.

Since what is involved is an estimate of the rate of the process (for particles that may also have different shapes), it is sufficient, for simplicity of calculation, to assume that after coalescence the remaining part of the drops also retains the shape of a sphere of radius \(a(t)\). Under this condition \(y=a\sin\theta\), where \(\theta\) is the central angle determining the “common” area of coalescence. The total volume of the drops is conserved; consequently, the condition must be satisfied

\[ \frac{\pi a^3}{3}\left(2+3\cos\theta-\cos^3\theta\right)=\frac{4\pi}{3}a_0^3, \]

where \(a_0\) is the initial radius of a drop.

The decrease in the free surface of both drops is

\[ S_0-S=8\pi a_0^2-4\pi a^2(1+\cos\theta), \]

which, for \(\theta\ll 1\), to terms of second order in smallness, is equal to:

\[ S_0-S=2\pi a_0^2\theta^2. \]

The calculation below is carried out for the initial stage of coalescence \((\theta\ll 1)\). The work of the surface-tension forces (per unit time) is equal to

\[ -\sigma\frac{dS}{dt}=-4\pi a_0^2\sigma\frac{d}{dt}\left(\frac{\theta^2}{2}\right). \]

The work of the internal-friction forces may be estimated from the mean gradient of the deformation velocity \(\gamma\sim \frac{dv}{dx}\). The deformation is characterized by the quantity \(a(1-\cos\theta)\simeq a\frac{\theta^2}{2}\), representing

a decrease in the distance between the center of one of the drops and the surface of its contact with the other. Consequently, in order of magnitude, \(\dot{\gamma}\sim\dfrac{d}{dt}\left(\dfrac{\theta^2}{2}\right)\), and the work of the frictional forces (per unit time)

\[ \sim \frac{8\pi}{3}a^3\eta\dot{\gamma}^{\,2}. \]

Equating it to the work of the surface-tension forces, one can obtain an equation for \(\dot{\gamma}\): \(\dot{\gamma}=\dfrac{3}{4}\dfrac{\sigma}{a\eta}\), so that

\[ \theta^2=\frac{3}{2}\frac{\sigma}{a\eta}\,t \]

or, since \(\pi y^2=\pi a^2\sin^2\theta\sim \pi a^2\theta^2\),

\[ \pi y^2=\frac{3\pi}{2}\frac{a\sigma}{\eta}\,t. \tag{6} \]

The time of “complete coalescence” of the drops according to formula (6) should be of the order \(\dfrac{\eta a}{\sigma}\), i.e. of the same order as the time for closing a cavity with initial radius \(a\).

Frenkel considers the coalescence of drops as the first stage of sintering, during which the gaps between particles are filled in to such an extent that the remaining pores become isolated. The second stage of sintering is reduced to the closure (i.e. likewise to viscous filling) of the residual pores that do not communicate with one another.

He also pointed out a possible retardation of the second stage due to the release of absorbed gases, which, accumulating in the pores, should hinder their filling. Assuming that the amount of gas in a spherical cavity remains constant and, consequently, that its pressure increases with decreasing radius according to the formula \(p=p_0\left(\dfrac{a_0^3}{a^3}\right)\), one may conclude that, at \(p=\dfrac{2\sigma}{a}\), the capillary pressure will be completely compensated and further sintering will cease. Hence one obtains the formula for the limiting (minimum) radius of the cavity:

\[ a_{\min}=\sqrt{\frac{p_0a_0^3}{2\sigma}}. \tag{7} \]

3. DIFFUSION THEORY OF SINTERING AND ITS CONNECTION WITH THE THEORY OF VISCOUS FLOW

Independently of Ya. I. Frenkel (and almost simultaneously with him), the author of these lines developed a description of sintering directly as a process of diffusion displacement of atoms\(^8\); in this approach, the concept of viscous flow was not introduced, and sintering was regarded as a consequence of the redistribution of matter by self-diffusion.

In principle the two points of view are very close, all the more so since in the “diffusion” calculation the theory of self-

diffusion. However, the results of the diffusion calculation did not agree with those given above, for a reason that at first was unclear; this gave grounds for considering the two existing points of view on the mechanism of sintering as different,* which, as will be seen from what follows, must now be regarded as having fallen away (with a certain qualification, to be discussed below). We shall, however, adhere to a chronological presentation of the question and therefore first consider the diffusion theory of sintering as an independent one.

In describing sintering as a manifestation of self-diffusion it is necessary to indicate the cause of the directed displacement of atoms leading to the healing of pores, the coalescence of droplets, etc. Within the framework of the diffusion problem this cause can only be a difference in concentrations (or, more generally, a difference in chemical potentials) at different points of the volume.

As was already mentioned above, using Frenkel’s theory of self-diffusion one may with equal success speak of self-diffusion of atoms or diffusion of “vacancies,” provided only that relation (2) between the corresponding diffusion coefficients is taken into account.

It is therefore possible, in the language of diffusion theory, to describe sintering also as the diffusion of “vacancies.” Directed diffusion will take place only if at different points of the body’s volume there is a different concentration of “vacancies.” It is easy to show that in a porous body, or in a body having internal surfaces, or, finally, in a continuous homogeneous body of irregular shape whose surface does not correspond to a minimum of free energy, there must necessarily be differences in the “equilibrium concentration of vacancies” at different points of the body.

Consider, for example, a body of large dimensions (radius \(R\)) possessing at its center a spherical cavity (pore) of small radius \(a \ll R\). Near the outer surface of the body the equilibrium concentration of vacancies \(c_0\) will practically not differ from that determined by formula (1).

At the boundary with the surface of the internal cavity, however, the concentration must be increased. Indeed, the problem of “vacancies” arising in a crystal may be regarded as analogous to the problem of evaporation. Instead of the internal cavity of the body we have, as it were, a droplet of substance; instead of “vacancies,” vapor molecules. In thermodynamics it is shown that near a droplet whose surface has radius of curvature \(a\), the equilibrium vapor pressure \(p_0\)

* In addition to volume self-diffusion, work \(^{8}\) also considered the occurrence of sintering by means of “surface self-diffusion,” as well as evaporation of atoms from convex portions of the surface and condensation on concave portions. It was noted in the work that the indicated sintering processes must have a subordinate character.

increased by the amount

\[ \Delta p=\frac{2\sigma}{a}\,\frac{v_0}{kT}\,p_0 . \tag{8} \]

Here \(v_0\) is the proper volume of one vapor molecule in the condensed phase. The remaining notation has been indicated above. The formula presented, which is obtained by a very general thermodynamic method and is valid not only for vapor elasticity, but also for concentrations of dilute solutions, contains no mass of the particles. It therefore can also be applied to a dilute solution of “vacancies” in a crystal.

In the case of a dilute solution, the values \(p\) and \(\Delta p\) in formula (8) are proportional to the concentrations. Thus, the increase in the equilibrium concentration of “vacancies” near a pore of radius \(a\) can be determined from the formula

\[ \Delta c=\frac{2\sigma}{a}\,\frac{v_0}{kT}\,c_0 \quad \text{or} \quad c_a=c_0+\Delta c=c_0\left(1+\frac{2\sigma}{a}\,\frac{v_0}{kT}\right). \tag{9} \]

Here \(c_0\) is the equilibrium concentration near a plane surface, and for \(v_0\) we have: \(v_0\sim\delta^3\), if \(\delta\) is the constant of the crystalline lattice of the body.

Since, as already mentioned, near the outer surface of a body having radius of curvature \(R\gg a\), the equilibrium concentration \(c_R\simeq c_0\), inside the body there must arise a gradient of the equilibrium concentration, i.e., a diffusion flux of vacancies from the pore to the outer surface must be established; through the latter the vacancies will leave the body. It is clear that the process described means nothing other than the diffusional “healing” of the pore by atoms.

The rate of the process can be determined by using the solution of the diffusion equation

\[ D\nabla^2 c=\frac{\partial c}{\partial t} \]

for vacancies. Neglecting at first the change in the concentration distribution caused by the displacement of the pore boundaries, one may, for the spherically symmetric case, take a solution of the form \(c=\dfrac{A}{r}+B\) (\(r\) is the distance of the point under consideration from the center of the cavity); the constants \(A\) and \(B\) are determined from the boundary conditions at \(r=a\) \((c=c_a)\) and \(r=R\) \((c=c_0)\). The flux of vacancies “leaving” the body through its entire outer surface \(\Omega\), per unit time, is

\[ S=D\int \nabla c\,d\Omega . \]

Substituting the values of the constants \(A\) and \(B\) and calculating the gradient, we find:

\[ S=4\pi D'a(c_a-c_0)=8\pi D'c_0\frac{\mathfrak{v}^3}{kT}=8\pi D\frac{\mathfrak{v}^3}{kT}. \]

If \(D\) is expressed in \(\text{cm}^2/\text{sec}\), then \(S\) represents the flux of the volume of “vacancies” leaving the body in \(1\) sec., i.e. the rate of decrease of the pore volume

\[ S=-\frac{d}{dt}\left(\frac{4\pi}{3}a^3\right). \]

Equating the two values of \(S\), we obtain (see also \({}^{9}\)):

\[ \frac{da}{dt}=-\frac{2\sigma}{a^2}\frac{\mathfrak{v}^3}{kT}D, \tag{10} \]

which differs essentially from Frenkel’s formula (5) for the equivalent problem of “viscous filling” of a cavity of radius \(a\), if relation (4) between the coefficients of viscosity and self-diffusion is adopted.

Namely, according to (10) the quantity \(\dfrac{da}{dt}\) turns out to be smaller by \(\left(\dfrac{\delta}{a}\right)^2\) times than according to formula (5), if \(\eta\) is expressed by means of formula (4); for pores of radius \(a\sim10^{-3}—10^{-4}\ \text{cm}\) this means an increase in the duration of sintering by \(10^6—10^8\) times. The resulting difference is so great that it is not difficult to choose which of the formulas agrees with experiment, although the calculation in the form presented is still schematic and requires further refinement (see below).

According to (10), the time of complete sintering is

\[ t_0=\frac{a_0^3}{\mathfrak{v}^3}\frac{kT}{6D\sigma}. \]

It is known from experiment that, for example, for a pressed powder of such a metal as Cu, with pores of diameter \(\sim10^{-3}\ \text{cm}\), the time of complete sintering at \(900—1000^\circ\text{C}\) is several hours, i.e. \(\sim10^4\ \text{sec}\). Taking \(\sigma\sim10^3\ \text{dyn}/\text{cm}\), we obtain from (10), at \(T\sim10^3\), \(\mathfrak{v}^3\sim10^{-23}\ \text{cm}^3\), \(D\sim10^{-8}\ \text{cm}^2/\text{sec}\). With the same figures, by formula (5a), when formula (4) is used for \(\eta\), the value \(D\sim10^{-17}\ \text{cm}^2/\text{sec}\) is obtained.

The latter figure is underestimated by \(6—7\) orders of magnitude in comparison with known values of the self-diffusion coefficient at temperatures of intense sintering (\(10^{-10}—10^{-11}\ \text{cm}^2/\text{sec}\)). The value following from (10), however, is \(2—3\) orders of magnitude higher than the value for a solid body, which may be explained by the nonequilibrium character of such a system as a pressed powder (the presence of distortions of the crystal lattice; cf. below). Thus, the experimental data confirm formula (10), and not formula (5), taken together with formula (4).

This still does not mean that, in describing sintering (of a crystalline body), the “diffusion theory” is preferable to the theory of “viscous flow,” since in the language of the theory of viscous flow the result will also be correct if one abandons formula (4) and introduces another relation between the coefficients of viscosity and self-diffusion, namely the relation following from comparison of (10) and (5):

\[ \frac{1}{\eta}=\frac{D\delta^3}{kTa^2} \tag{11} \]

(we here omit the inessential numerical coefficient \(2/3\)). With relation (11), both concepts will be equivalent and will lead to the same result.

The validity of the latter conclusion was shown by us\({}^{10}\) not only as applied to the problem of sintering considered here (“overgrowth” or “filling in” of a spherical cavity), but also more generally—for viscous flow of crystalline bodies effected by self-diffusion. Indeed, let us compare the equations of viscous flow with the equation of self-diffusion of “vacancies.”

The latter equation may be written, in the case of a porous body, regarding it as possessing vacancy “sources” (for a quasistationary problem), in the form*)

\[ D'\nabla^2 c+M=0, \tag{12} \]

where \(M\) is the strength of the “sources” (which may be distributed over the whole volume).

Introducing the vacancy flux

\[ \mathbf{q}=D'\operatorname{grad} c, \tag{12a} \]

we write (12) in the form

\[ \operatorname{div}\mathbf{q}+M=0. \tag{12б} \]

From dimensional considerations, \(M\sim \dfrac{D'\Delta c}{L^2}\), where \(L\) is some linear dimension of the problem under consideration; as for the magnitude \(\Delta c\), for a body in which pressure differences \(\Delta p\) exist,

\[ \Delta c=\Delta p\,\frac{v_0}{kT}\,c_0. \tag{13} \]

(This formula is a generalization of formula (9); \(2\sigma/a\) has been replaced by \(\Delta p\).) Thus,

\[ M\sim \frac{D\delta^3}{kTL^2}\,\Delta p =\frac{D\delta^3}{kTL^2}(p_0-p). \tag{13a} \]

*) The conclusions below, based on dimensional considerations, also apply completely to the diffusion equation in the form

\[ D'\nabla^2 c=\frac{\partial c}{\partial t}. \]

If the diffusion of vacancies (i.e., atoms) leads to the flow of the body, then the displacement velocities \(\dfrac{d\mathbf u}{dt}=\dot{\mathbf u}\) are determined by the particle flux

\[ \mathbf v=\dot{\mathbf u}=\mathbf q . \tag{14} \]

Applying the operation \(\operatorname{grad}\) to (12b), we obtain, taking into account (12a), (13), and (14):

\[ \nabla^2 \mathbf q+\operatorname{grad} M=\nabla^2 \mathbf v-\frac{D\delta^3}{kTL^2}\operatorname{grad} p=0, \tag{15} \]

which coincides with the particular form of the equations of viscous flow (the Navier–Stokes equations)

\[ \eta\nabla^2\mathbf v-\operatorname{grad}p=0, \tag{15a} \]

for

\[ \dot{\mathbf v}=\frac{d\mathbf v}{dt}=0 \]

and when the velocities are small, if the relation

\[ \frac{1}{\eta}=\frac{D\delta^3}{kTL^2} \tag{16} \]

is satisfied.

It follows from what has been said that Frenkel’s idea of the viscous flow of crystalline bodies, effected by self-diffusion, is fully justified. However, it turns out that relation (4) between the coefficients of viscosity and self-diffusion in the case of crystalline bodies must be replaced by relation (16), of which (11) is a special case.

It should be noted that the “model” considerations underlying the derivation of formula (4) (the displacement of an atom during self-diffusion as the Stokes motion of a small sphere in a viscous liquid) are clearly inapplicable to the case of self-diffusion in crystals and can be justified only for liquids (or amorphous bodies).

On the other hand, formulas (16) and (11) go back to relation (9), which assumes the presence within the body of “vacancies” of constant size, which does not correspond to reality in the case of a liquid. It should therefore be assumed that relation (16) will not be valid for liquid (amorphous) bodies and can apply only to crystalline bodies.

We also point out that the difference between formulas (4) and (11) (or (16)) is not reduced merely to a change in the order of magnitude of the viscosity coefficient. The linear dimensions entering into (11) (and also (16)) (\(L\) or \(a\)) may change in the course of viscous flow, which will lead to a change in the kinetics of the phenomenon. Thus, in the problem of sintering a spherical pore, formulas (5) and (4) give a linear dependence of the pore radius \(a\) on time (if \(\eta=\mathrm{const}\)); whereas according to formula (10) the volume of the pore decreases linearly with time.

Taking these differences into account, one can with equal success describe sintering effects as corresponding to “viscous flow” or directly to self-diffusion.

Let us turn, in particular, to the process of “coalescence of droplets” (i.e. sintering of spherical particles), the calculation of which was given by Frenkel in the language of the theory of viscous flow, and describe this phenomenon as a direct manifestation of self-diffusion. In doing so, let us note that, by the conditions of symmetry, a variant of the same problem is the sintering of one spherical particle to a plane polished section. This variant, for which the required graphical construction is simpler, is the one we shall consider (Fig. 2).

The center of the droplet, initially at a distance from the plane equal to its radius \(a\), after sintering (over a time interval \(t\)) will approach the plane by a segment \(h\). Let us denote by \(y\) the radius of the circle of contact formed by the particle and the plane. As is clear from Fig. 2, sintering means a redistribution of the substance of the particle, namely, its transfer from the spherical segment \(I\) of height \(h\) into the region of the ring \(II\) with outer diameter \(y\). Let us denote the radius of curvature of the annular layer \(II\) in the plane of the drawing by \(\rho\). The excess concentration of vacancies near the outer surface of layer \(II\) is equal to

Fig. 2.

\[ \Delta c=\sigma\left(\frac{1}{\rho}+\frac{1}{y}\right)c_0\frac{\delta^3}{kT}\sim \frac{\sigma}{\rho}c_0\frac{\delta^3}{kT}\quad(\text{for }y\gg \rho). \]

Since the influx of atoms into layer \(II\) takes place from layer \(I\), the concentration gradient will, in order of magnitude, be equal to \(\dfrac{\Delta c}{y}\), and the rate of the diffusion process is determined by the equation
\(D'\dfrac{\Delta c}{y}S=\dfrac{dV}{dy}\), where \(dV=S\,dy\) is an element of the volume of layer \(II\), and \(S\) is its outer surface. Thus:

\[ \frac{dy}{dt}=\frac{D\delta^3}{kT}\frac{\sigma}{\rho y}. \tag{17} \]

The quantity \(\rho\) is easy to find using the relation \(\rho+h=\dfrac{y^2}{2a}\) (see Fig. 2) and the condition of conservation of volume during sintering, i.e. the equality of the volumes of layers \(I\) and \(II\). Approximately one obtains: \(\rho=\dfrac{1}{4}\dfrac{y^2}{a}\).

Substituting this value into (17) and integrating, we find:

\[ \frac{y^4}{a}=16\sigma \frac{D\delta^3}{kT}\cdot t . \tag{18} \]

For the case of adhesion of two particles all the parameters of the problem are preserved, with the exception of the quantity \(\rho\), which is doubled. Therefore in the latter case the formula

\[ \frac{y^4}{a}=8\sigma \frac{D\delta^3}{kT}\cdot t \tag{18a} \]

is valid.

Up to the numerical factor, this coincides with Frenkel’s formula (6), if one substitutes:

\[ \frac{1}{\eta}=\frac{D\delta}{kT}\cdot \frac{\delta^2}{y^2}. \]

In the present case the “characteristic” linear dimension is \(L=y\).

4. COMPLICATION OF THE SINTERING PHENOMENON UNDER PRACTICAL CONDITIONS

The simplest problems considered correspond to idealized conditions both with respect to the geometry of the system and with respect to the physical phenomena occurring in it.

For real cases of “adhesion” of two bodies of arbitrary shape or of sintering of a pressed powder, the kinetics of the process may differ from that described.

Let us point, for example, to the retardation of the process of closure of closed pores, already noted by Frenkel, owing to the accumulation of gases in them. Suppose there is a specimen pressed from powder. If pressing was not carried out in vacuum, then there is gas in the pores under a pressure equal to atmospheric \((p_0=10\ \mathrm{dyn}/\mathrm{cm}^2)\) or higher. Since sintering is usually carried out at a temperature \(T_1\) exceeding the pressing temperature \(T_0\), at the sintering temperature the gas pressure in the closed pores (if diffusion of the gas through the body is disregarded) is further increased by a factor \(n=\dfrac{T_1}{T_0}\). As the pore radius decreases from \(a_0\) to \(a\), this pressure increases from \(np_0\) to

\[ p=np_0\frac{a_0^3}{a^3}, \]

and when the excess pressure over atmospheric,

\[ p_0\left(n\frac{a_0^3}{a^3}-1\right), \]

reaches the capillary “negative” pressure acting on the inner surface of the pore, i.e., when

\[ p_0\left(n\frac{a_0^3}{a^3}-1\right)=\frac{2\sigma}{a}, \tag{19} \]

the decrease in the pore radius will cease and will no longer take place.

At each given moment the rate of sintering is determined by the excess of the capillary pressure over the gas pressure, i.e., it will be determined by the equation

\[ \frac{da}{dt}=-\frac{2\sigma}{a}\frac{\delta^3}{kT}D\left[1-\frac{p_0 n}{2\sigma}\frac{(a_0^3-a^3)}{a^2}\right], \tag{20} \]

which is a generalization of equation (10).

Condition (19) shows that, in the case of pores with a sufficiently large initial radius, not sintering but the reverse process of increasing pore volume may occur, if the capillary pressure is less than the gas pressure inside the pore. Taking, for example, \(\sigma\sim 10^3\) dyn/cm, \(p_0=10^6\) dyn/cm\(^2\) and \(n=\frac{3}{2}\), we find from (19):

\[ 1>\left(\frac{a}{a_0}\right)^2=\frac{np_0a_0}{2\sigma}\left(1-\frac{a^3}{na_0^3}\right) =\frac{3}{4}10^3a_0\left(1-\frac{2a^3}{3a_0^3}\right), \]

whence it is seen that pores with a diameter greater than \(4\cdot 10^{-2}\) mm (when they contain gas at the indicated pressure) will expand, causing a decrease in the density of the body until an equilibrium diameter satisfying condition (19) is reached. For \(2a_0=2\cdot 10^{-1}\) mm the equilibrium diameter will be \(\sim 2.8\cdot 10^{-1}\) mm. For pores with \(2a_0<4\cdot 10^{-2}\) mm, sintering will occur, but it will stop upon reaching the equilibrium radius; for example, for \(2a_0=2\cdot 10^{-2}\) mm, \(2a_p\sim 1.4\cdot 10^{-2}\) mm.

In the calculations made, diffusion of gas through the body has been neglected. Taking diffusion into account, the braking effect will prove to be reduced*).

We shall further point out the effect of nonuniform overgrowth of pores of different sizes. In the idealized case, when the pores are spherical, have the same radius \(a\), and are located at large distances \(R_i\gg a\) from one another, as already indicated, the volume of each of them must decrease at a constant rate, so that the relative change in the volume of a body containing \(N\) pores in \(1\ \mathrm{cm}^3\) will be determined by the formula (cf. (10))

\[ \frac{\Delta V}{V_0}=8\pi\sigma\frac{N\delta^3}{kT}Dt, \tag{21} \]

where the number of pores \(NV_0\) will remain unchanged. However, if the body contains pores of different size, then a process of increasing the volume of large pores is inevitable, accompanied by complete overgrowth of the small pores adjacent to them. Indeed,

*) If the coefficient of “heterodiffusion” of the gas through the body substantially exceeds the coefficient of self-diffusion of the substance, the presence of the gas will practically not hinder the phenomenon of sintering.

near small pores, the increase in the concentration of “vacancies” will be greater than in the vicinity of large ones (cf. formula (9)). Consequently, within the body there will arise, over comparatively small distances (between pores), rather considerable differences in the concentrations of “vacancies,” which will lead to diffusion fluxes of atoms from large pores to small ones. This means that the small pores will heal up, while the large ones will increase. The process described may lead to a decrease in the number of pores per unit volume \(N\), i.e., to a slowing of the sintering of the body as a whole (a decrease in the rate of volumetric “shrinkage” \(\dfrac{\Delta V}{V_0}\)).

The phenomenon noted may be further complicated by the presence of gas in the pores. Since, during the sintering of small pores, the gas pressure in them will increase faster than in large ones, diffusion of gas will also arise from small pores into large ones; in this case the process of coarsening of the pores and decrease in their total number will develop, but possibly more slowly than in the absence of gas.

The sintering of a porous body containing pores of different size is, in general, a more complex phenomenon, not yet described theoretically with sufficient completeness. The question of what distribution of vacancy concentration is obtained inside the body and how it changes with time (along with the change in the distribution of pores by size, as well as the localization of pores) has not yet been solved.

Even in the case when all pores have the same diameter, the phenomenon may proceed more complexly than was described above. If the dimensions of the body are large, nonuniform sintering may occur—more considerable in the part of the body adjoining the external surface (the “vacancies” have time to diffuse out of it earlier). In very large bodies, sintering should proceed in such a way that first, near the surface, a completely sintered (dense) crust is formed, which will then thicken further\(^8\).

In real systems, a retardation of sintering may occur, caused by the presence of layers that impede self-diffusion, for example oxide layers in pressed metallic powders. However, if sintering is carried out in a reducing atmosphere (and reduction of the oxides to the metallic state takes place), the process proceeds even more rapidly than in a system containing no oxides\(^ {30}\), since the crystal lattice of the metal formed during reduction is nonequilibrium and the phenomena of self-diffusion proceed in it more rapidly (cf. below).

We have not yet dwelt on the question of the dependence of the sintering rate on the simultaneously occurring processes of relaxation and recrystallization (in connection with the presence of distortions of the crystal lattice). We shall return to this below. Let us point out only that

details of the geometric conditions during sintering, if they do affect the kinetics of the phenomenon, do so only insignificantly. One may, for example, compare the rate of filling of the volume of so-called “open” pores having different shapes. If these pores are hemispherical, the kinetics of the decrease in their volume (at least in the first stage) will be the same as for closed spherical pores, i.e., it is described by formula (10), according to which the rate of change of volume is constant in time (for \(D=\mathrm{const}\)). Open pores, or “surface depressions,” formed by the contact of spherical particles differ radically in shape from “hemispherical” pores; for them the kinetics of growth is given by equation (18a), which outwardly does not resemble formula (10). But if, using (18a), one calculates the rate of filling of the free volume between spherical particles, it turns out that this rate: a) is constant in time and b) is, to within a numerical coefficient, the same as in the case of “hemispherical” pores.

5. SOME EXPERIMENTAL DATA ON THE RATE OF SINTERING

In view of the possible complications of sintering kinetics by various side phenomena mentioned above, when comparing theoretical conclusions with experiment, attention is drawn above all to experimental investigations concerned with the phenomenon of “bonding,” and, moreover, to those in which determinations of the change in the contour of the contacting bodies were carried out directly.

Among the investigations indicated is the work of G. Kuczynski\(^{11}\), published in 1949. The author observed the bonding of spherical Cu and Ag granules to flat polished sections of the corresponding metals. The Cu granules had diameters from 4 to 100 \(\mu\). The experiment with them was carried out in such a way that, after separation into three fractions: a) \(>35\,\mu\), b) \(15\text{--}35\,\mu\), and c) \(<15\,\mu\), particles of one fraction were placed on the surface of a copper polished section and subjected to heating for different durations at various temperatures in an atmosphere of hydrogen.

After heating, the polished section with the granules bonded to it was coated with bakelite and ground from the end face so as to obtain a section of the granules along the diameter at the place of contact with the polished section. The section was examined under a microscope, and, with the aid of an ocular micrometer, the following were measured: 1) the diameter of the granules, 2) the length of the “chord” along which bonding of the granule to the polished section had occurred. To obtain greater accuracy, the grinding was performed gradually, and at intermediate stages measurements of the particle diameter were also made; the length of the “chord” was read at the maximum measured value of the diameter of the given particle.

An important feature of the measurements carried out was the use of grains that had been preliminarily annealed (in a hydrogen atmosphere) at \(900^\circ\) for 18–24 hours (to avoid mutual contact, the particles were immersed in carbon powder during annealing). The author of the work writes that annealing was carried out in order to reduce oxides that might have been present. But in addition to this result, annealing undoubtedly also led to the removal of lattice distortions, i.e. to the restoration of the equilibrium coefficient of self-diffusion (cf. below).

Reliable measurements could be made only by using grains of the first two fractions mentioned above. Grains with a diameter \(<10\mu\) proved to have a less perfect spherical shape and gave a large scatter of the measured values.

In the paper the author gives only the data of the processing of mean values, indicating that in each experiment (at a given temperature and duration of heating) measurements were made on at least ten grains.

Analogous measurements were performed on silver grains that had adhered to silver emery. In this case grains of large size (\(\sim 350\mu\)) were used; heating was carried out in air.

The author compared the results of measurements of the chord length \((y)\), along which adhesion took place, with calculated values based on various concepts.

He was familiar with Frenkel’s work\(^6\) and with work\(^8\), in which the diffusion theory of sintering is set forth. In addition, as possible mechanisms of adhesion he also considers the filling of the near-contact region: a) by evaporation of atoms from the convex parts of the grains and their condensation on the concave surface (where the equilibrium vapor pressure is lower), and b) by “surface” diffusion. Both of the latter mechanisms had already been noted in work\(^8\), where it was pointed out that they should be of subordinate importance.

Unfortunately, in presenting the theoretical formulas, G. Kuchinskii made a number of errors which will have to be corrected below. For the time dependence of \(y\) according to the theory of viscous flow, he gives Frenkel’s formula\(^6\):

\[ \frac{y^2}{a}=\frac{3}{2}\frac{\sigma t}{\eta}. \tag{22a} \]

He tried to derive the corresponding formula for \(y\) according to the diffusion theory himself, using the calculations given in \(8^*)\).

\[ \underline{\phantom{xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx}} \]

\(^*)\) Let us note that only in his first article\(^ {11}\) does Kuchinskii cite (although incompletely and incorrectly) work\(^8\), from which he borrowed the description of the diffusion mechanism of sintering (an increase in the concentration of vacancies near the concave surface) and the basic formulas for calculating the partial

obtained the result:

\[ \frac{y^5}{a^2}=\frac{40\delta^3}{kT}Dt, \tag{226} \]

which is incorrect, as is seen from the preceding exposition (cf. formula (18a), § 3).

Kuchinskii’s error in this question is due to the fact that, in the calculation, he assumed the concentration gradient to be of the order of magnitude \(\frac{\Delta c}{\rho}\), whereas in the present problem \(|\operatorname{grad} c|\sim \frac{\Delta c}{y}\). The correct formula for this case (see (18a)) has the form:

\[ \frac{y^4}{a}=\frac{8\delta^3}{kT}Dt. \tag{22в} \]

Kuchinskii also gives formulas for the dependence \(y=y(t)\) for the case of “evaporation and condensation”:

\[ y^3=Kt \tag{22г} \]

(\(K\) is a constant), and also for the case of “surface diffusion”:

\[ \frac{y^7}{a^3}=\frac{5\delta\delta^4}{kT}D_n t. \tag{22д} \]

In the last formula, \(D_n\) denotes the coefficient of surface diffusion. In deriving formulas (22г) and (22д), Kuchinskii also made errors, as a result of which both formulas are incorrect and must be replaced by the following*):

a) case of “surface diffusion”:

\[ \frac{y^6}{a^2}\left(\ln \frac{y}{2a}+\frac{1}{6}\right)=\frac{4\delta^4D_n}{kT}t, \tag{22е} \]

b) case of evaporation and condensation:

\[ y^7\sim Kt. \tag{22ж} \]

problem of the sintering of a sphere to a plane. Although the formulation of this particular problem is clearly taken from Frenkel,\(^6\) in Kuchinskii’s subsequent articles (see, for example,\(^{12}\)) the diffusion theory of sintering is mentioned as “put forward and developed by I. Kuchinskii.” It is not clear how this, decisively at variance with reality, assertion, repeated in articles not only by foreign\(^{13,14,16}\) but also by individual Soviet authors,\(^{15}\) arose.

*) Derivation of the kinetic formulas for the sintering of a sphere to a plane.

a) Case of surface diffusion. The Laplace equation for the established concentration distribution \(\nabla^2 c=0\) in the two-dimensional case—on the surface of a sphere—reduces to

\[ \frac{\partial^2 c}{\partial \theta^2}+\operatorname{ctg}\theta\,\frac{\partial c}{\partial \theta}=0. \]

The solution may be written in the form \(c=A\ln\operatorname{tg}\frac{\theta}{2}+B\). Taking the boundary condi-

For greater clarity we give in the table a summary of the various formulas for the time dependence \(y\) for different mechanisms of “sintering” (according to Kuchinskii and in the corrected form).

Table 1

Mechanism of “sintering”	Dependence \(y = y(t)\): corrected calculation	Dependence \(y = y(t)\): according to Kuchinskii	Note
1. Viscous flow according to Frenkel (at a constant coefficient of viscosity)	\(y^2 \sim t\)	\(y^2 \sim t\)	Borrowed by Kuchinskii from \(^{5}\)
2. Volume diffusion	\(y^4 \sim t\)	\(y^5 \sim t\)	Borrowed by Kuchinskii from \(^{5}\)
3. Surface diffusion	\(y^6 \ln \dfrac{y}{2a} \sim t\)	\(y^7 \sim t\)	Borrowed by Kuchinskii from \(^{5}\)
4. Evaporation and condensation	\(y^7 \sim t\)	\(y^3 \sim t\)	Borrowed by Kuchinskii from \(^{5}\)

…taking \(c = c_1\) at \(\theta = \theta_1\) and \(c = c_0\) at \(\theta = \dfrac{\pi}{2}\), we determine:

\[ \operatorname{grad} c = \frac{1}{a}\cdot \frac{\partial c}{\partial \theta} = \frac{1}{y}\, \frac{c_1-c_0}{\ln \dfrac{y}{2a}} . \]

(On the surface of the sphere; the notation is the same as in § 3, see Fig. 2; the relation \(y=a\sin\theta\) is used.) Equating the surface flux \(2\pi y D'_n \operatorname{grad}c\cdot \delta\) to the rate of change of the volume of the contact region

\[ \frac{dV}{dt}\cong \frac{\pi}{a}\,y^3\,\frac{dy}{dt}, \]

we obtain (taking into account the relation \(c_1-c_0=\dfrac{\sigma\delta^3}{\rho kT}c_0\)):

\[ \frac{4\sigma\delta^4 D_n}{kT} = \frac{y^5}{a^2}\ln\frac{y}{2a}\,\frac{dy}{dt}, \]

whence, after integration, (22e) follows.

b) The case of evaporation and condensation. The decrease in vapor elasticity near the concave surface of the contact region (radius \(\rho\)) is

\[ \Delta p \sim \frac{\sigma\delta^3}{kT}\,\frac{p}{\rho} \]

(\(p\) is the elasticity of the vapor above a plane surface). Evaporation proceeds from the surface of the sphere \(\cong 2\pi a^2\); vapor condenses on the surface \(\sim 2\pi^2 y\rho\); the vapor flux is

\[ q \sim \Delta p\,\frac{a^2}{y\rho} = k'\frac{a^4}{y^5}. \]

The rate of change of the volume of the contact region is

\[ \frac{dV}{dt}\sim \frac{\pi}{a}\,y^3\,\frac{dy}{dt} = 2\pi^2 y\rho q. \]

Substituting the value of \(q\), we obtain formula (22ж) of the text.

Comparison of the experimental data with the calculation was carried out by Kuchinskii as follows. The averaged experimental values \(y\), corresponding to different isothermal holds, were plotted on a logarithmic scale as a function of \(\ln t\). The slope of the resulting straight lines made it possible to determine the exponent in the relation \(y^n \sim t\), and in this way to indicate the mechanism of sintering.

Experimentally, Kuchinskii found values of \(n\) close to 5 (4.5–5.0 in the case of Cu at a temperature of \(700\text{–}900^\circ\mathrm{C}\); 5.4–4.9 in the case of Ag at temperatures of \(500\text{–}800^\circ\mathrm{C}\)), which agreed with the exponent corresponding to volume diffusion, according to his incorrect calculation. In light of the corrections introduced above, it should be considered that in reality, alongside volume diffusion, surface diffusion also took place in Kuchinskii’s experiments, and also, possibly, evaporation with subsequent condensation*).

This circumstance also became apparent in Kuchinskii’s treatment of experimental data relating to different temperatures (used to determine the coefficient of self-diffusion). On the graphs of \(\lg D\) as a function of

\[ \frac{1}{T} \]

it was not possible to obtain satisfactory straight lines; the points are widely scattered and (for Cu specimens) reveal systematic deviations toward an increase of \(D\) at low temperatures (large \(\frac{1}{T}\)), especially for grains of small diameter. Kuchinskii himself interprets these deviations as corresponding to the effect of surface diffusion.

It should be noted, however, that the scatter of points on all of Kuchinskii’s graphs is extremely large, so that the quantitative results he gives (for example, the coincident values of the activation heats of volume and surface self-diffusion of Cu, \(56\,000\ \mathrm{cal}/\mathrm{mol}\)) do not possess the necessary persuasiveness.

For a reliable verification of the quantitative relations between the principal effects taking place—volume and surface self-diffusion—the measurements must be carried out more carefully and treated according to the corrected formulas.

Thus, on the basis of the available literature data on the experimental study of “sintering,” one can as yet draw only the qualitative conclusion that this phenomenon proceeds by volume and surface self-diffusion, the latter (and also, possibly, evaporation with subsequent condensation) manifesting itself in the early stages of the process and at small \(a\).

*) According to formulas (22e) and (22zh), the effects of surface diffusion, as well as of evaporation and condensation, must be relatively more noticeable at small \(t\) (in the early stage of the process) and for small values of \(a\) (at a small radius of curvature).

There is also no complete certainty in the results, published up to recent times, of investigations of the sintering effect in pressed powders.

The most systematic measurements have been made of the kinetics of isothermal shrinkage in pressed powders of pure metals.[^17][^18][^19] The general result of all the investigations is the establishment of the nonconstancy of the shrinkage rate. Experience shows that volume shrinkage gradually slows with time if the temperature at which sintering is carried out is kept constant. By raising the temperature, one can increase the shrinkage rate, but subsequently (at constant temperature) the process again slows. On the question of the law governing the dependence of the shrinkage rate on time, the data of different authors diverge. An exponential equation was proposed[^17] for the shrinkage rate:

\[ \frac{1}{V_0}\frac{d(\Delta V)}{dt} \simeq \dot{M}\cdot e^{-\beta t}, \tag{23} \]

where \(M\) and \(\beta\) are constants.

Ivensen[^18], on the basis of his experimental data, put forward another equation, describing the change in porosity \(v\) (i.e., in effect, also shrinkage) with time according to the law:

\[ v = v_0(qmt+1)^{-\frac{1}{m}}. \tag{24} \]

The empirical values of the two constants \(m\) and \(q\) entering into (24) proved to be nonintegral and to vary from specimen to specimen. The temperature dependence of the quantities \(m\) and \(q\), and their connection with the treatment of the specimens, were not established.

In work[^19], the volume isothermal shrinkage of specimens pressed from copper powders of electrolytic origin was described by the equation

\[ \frac{\Delta V}{V_0} = A(1-e^{-at}) + Bt, \tag{25} \]

and an attempt was made to interpret the constants \(A\) and \(B\) in accordance with the diffusion theory of sintering.

Later, however, it became clear that the experimental data of which equation (25) was a generalization do not correspond to strictly isothermal conditions; the shrinkage effect before attaining the prescribed isothermal holding is comparable with that occurring during the holding, if the specimen is immediately heated to a temperature exceeding \(500\text{–}600^\circ\mathrm{C}\). This circumstance is readily explained if one takes into account what was said above about the slowing of shrinkage with time; the shrinkage rate is greatest at the initial moment and decreases considerably over a time of the same order as the heating time of the specimens.

The noted shortcoming also applies to the experimental data obtained in works \(^{17}\) and \(^{18}\), since the isothermal holds were also carried out there by heating immediately to the prescribed high temperatures.

The true law governing the kinetics of isothermal shrinkage of compacted powders has been determined only quite recently, in experiments using stepwise heating with a small temperature step \((50—100^\circ \mathrm{C})\). These experiments will be discussed below.

Let us also note here that, in addition to studies of shrinkage, reports have recently appeared in the literature \(^{20,21}\) on measurements of gas permeability (under conditions of Knudsen flow, i.e., in a rarefied gas) in loose and compacted metal powders subjected to heating (sintering).

According to these measurements, with the aid of the formula proposed by Deryagin \(^{22}\),

\[ S = k \frac{v^2}{Q}\frac{dp}{dx} \]

(\(v\) is the volume porosity of the body, \(\frac{dp}{dx}\) is the pressure gradient, \(Q\) is the gas flow, \(k\) is a constant), the surface of the so-called open pores, \(S\), was calculated. The author regards Deryagin’s formula as absolutely exact and estimates the error in the value of \(S\) as due only to errors in the measurements of \(v\), \(Q\), and \(\frac{dp}{dx}\) (\(\sim 3\%!\)); whereas in reality the formula determines the mean value of the ratio \(d^4 l^2\) to \(d^3 l\) (\(d\) is the pore diameter, \(l\) is the length), i.e., in fact only the order of magnitude of \(S\) is estimated. It turned out that the change in the quantity \(S\), determined by the indicated method, proceeds parallel to shrinkage, which should correspond to the “healing” of the so-called open pores (the state of “closed” pores is not detected in measurements of gas permeability). In work \(^{15}\), detailed data were also obtained on the change in the quantity \(S\) during isothermal sintering of specimens compacted from powders of certain metals (Cu, Fe, Ni) that were in different initial states. However, the isothermal holds were carried out, as in works \(^{17,18,19}\), by sudden heating immediately from room temperature to the prescribed temperature; therefore here too the true kinetics of the isothermal process remained undisclosed*). Nevertheless, the data contained in the work—

*) In articles \(^{15,21}\), a “new” explanation of the sintering phenomenon is also put forward, according to which shrinkage consists in the “convergence” and denser “packing” of powder particles after the elimination of “roughness” on their surfaces (by surface diffusion). In essence, when properly formulated, this proposition would not differ from the conclusions of the theory of vis—

in experiments 15, 20, 21, the experimental results of measurements of the surface of “open” pores are of definite interest. They show that the kinetics of the decrease in “open” porosity coincides with the kinetics of the change in “total” porosity (including closed pores), i.e., that the complicating circumstances which might occur in the case of closed pores (for example, the influence of gas; cf. above) have practically no substantial effect on the course of the process.

Let us now turn to a brief account of the results of a study of the kinetics of isothermal sintering of pressed powders under conditions of stepwise heating^36. The samples studied were pressed from copper powders obtained electrolytically, and from iron powders reduced from scale. The kinetics of shrinkage was observed by means of a sensitive dilatometer distinguished by very low thermal inertia.

Heating was first carried out directly to \(400^\circ\mathrm{C}\) (below this temperature practically no measurable shrinkage was observed), and then stepwise at intervals of every \(50^\circ\) (in some experiments every \(100^\circ\)). In a given experiment, at each step the same isothermal hold \(t_0\) was used; in different experiments this hold was different (from 2.5 min to 20 min), but it always substantially exceeded the time required for heating by one step (\(\sim 30\) sec). Shrinkage readings were taken every 30 sec. Samples having an initial porosity of \(\sim 30\%\) were brought during sintering to a porosity of \(\sim 3\text{--}4\%\).

The principal results of the experiments proved to be as follows:

a) If one plots on a graph the total shrinkage attained at a given temperature (i.e., the sum of the shrinkage values at all

of Frenkel’s viscous flow, for if attention is fixed on individual particles, then, according to Frenkel, shrinkage consists in the mutual approach of particles, accompanied by a change in their shape (as a result of “viscous” flow under the action of surface-tension forces). At the same time, of course, a change in shape is implied which does not reduce only to the elimination of “roughness” of the surface, but concerns the particles as a whole; it takes place wherever this leads to a decrease in free surface. Consequently, the “new” explanation differs only in arbitrarily neglecting the change in particle shape, i.e., an effect without which dense filling of a volume by particles is impossible. The question of the possible role of surface diffusion in the sintering process has already been discussed above (see also § 9, note).

Let us also note that errors have apparently crept into the calculation given in article^15, since the final formula for the kinetics of shrinkage (corresponding to the mutual approach of particles after removal of the “roughness” of their surface by surface diffusion) is given in the form:

\[ \frac{\Delta \tau}{\tau + \Delta \tau} = D_y t \]

(\(\tau\) is porosity, \(\Delta \tau\) is the change in porosity over time \(t\), \(D_y\) is the coefficient of shrinkage diffusion). Here the dimensions of the right- and left-hand sides do not coincide.

preceding temperatures and the given one) as a function of \(\sqrt{t_0}\), a linear dependence is obtained with good accuracy for each value of the temperature. Denoting the shrinkage over the time \(t_0\) at temperature \(T\) by

\[ \left(\frac{\Delta V}{V_0}\right)_{T,t_0}=y_{T,t_0}, \]

this result may be written in the form

\[ R=\sum_T y_{T,t_0}=m(T)+\sqrt{t_0}\,n(T). \tag{26} \]

The curves \(R=R(\sqrt{t_0})\), as well as \(m(T)\) and \(n(T)\) for Cu specimens, are shown in Figs. 3 and 4.

b) The magnitude of the shrinkage at a given temperature, \(y_{T,t}\), depends on the time \(t\) at large \(t\) practically according to the law \(y_{T,t}\sim \sqrt{t}\). At small \(t\) (beginning from \(t=0\)) the indicated dependence changes into a linear one, \(y_{T,t}\sim t\). On the graph \(y_{T,t}=f(\sqrt{t})\), at small \(\sqrt{t}\) a parabolic dependence is observed, which at larger \(\sqrt{t}\) changes into a linear one. Two such graphs (\(t_0=10\) min and \(t_0=20\) min) are given in Figs. 5a and 5b.

It is thus found that proportionality to \(\sqrt{t}\) is the principal type of time dependence of shrinkage.

Applying, to describe the shrinkage of a pressed powder, the formula of diffusion theory (21) (p. 515), according to which \(y_{T,t}\sim Dt\), we see that agreement of the experimental data with the theoretical calculation can be obtained only if it is assumed that the coefficient of self-diffusion is not constant in time and changes (at large \(t\)) according to the law \(D\sim 1/\sqrt{t}\). The nonconstancy of the coefficient of self-diffusion can mean only a change in state.

Fig. 3. Curves of total shrinkage under stepwise heating as a function of \(\sqrt{t_0}\) (\(t_0\) is the duration of the steps; specimens pressed from electrolytic copper powder; initial porosity of the specimens \(\sim 30\%\)).

Fig. 4.

substance, i.e., the occurrence, besides sintering, of other processes as well. Measurement of the heat capacity of pressed metal powders shows\(^{24}\) that, when such objects are heated, a certain excess energy is released, which corresponds to a nonequilibrium initial state, i.e., to the presence of distortions of the crystal lattice*). The removal of distortions of the crystal lattice upon heating is the so-called recovery process. It follows from this that the time-varying value of the coefficient of self-diffusion, corresponding to the observed kinetics of sintering, must be due to the recovery process proceeding simultaneously with sintering. Thus, the phenomenon of sintering in a real nonequilibrium system proves to be inseparable from the recovery process. A correct description of the kinetics of sintering can be obtained only by taking into account the recovery-induced change in the coefficient of self-diffusion. We shall return to this below.

Fig. 5a. Course of shrinkage during each heating stage (the abscissa is \(\sqrt{t}\); \(t = 10\) min).

Fig. 5b. Same as in Fig. 5a, for \(t_0 = 20\) min.

It is expedient, however, first to discuss the question of the agreement with experiment of the remaining quantities entering into the calculation formulas describing sintering, and above all of the capillary pressure determined by surface tension.

* The amount of heat liberated upon heating pressed powders is tens of times greater than that corresponding to the effect of surface contraction in the formation of a compact body from a powder; therefore the indicated heat is practically only a measure of the distortions distributed in the volume of the body.

6. SINTERING AS A TYPE OF DIFFUSION CREEP UNDER THE ACTION OF SURFACE-TENSION FORCES. LAWS OF DIFFUSION CREEP. SINTERING UNDER PRESSURE. CREEP OF CONTINUOUS BODIES.

It has already been pointed out above that there is a relation between two descriptions of sintering: a) as a “viscous flow” (under the action of surface-tension forces), and b) as a directed diffusive displacement of atoms under the influence of differences in vacancy concentrations caused by capillary pressure. Under condition (16) both descriptions are identical for crystalline bodies, since the differential equations of self-diffusion and of viscous flow then prove to be the same.

But the equivalence of the differential equations means not only the coincidence of the solution of one or another particular problem (characterized by the shape of the body, its surface, the applied external forces, etc.), but corresponds to a general connection between the phenomena of “viscous flow” and directed self-diffusion. It is easy to understand that, by virtue of equations (12) and (15), and under the action of other, non-capillary bodies, phenomena of “viscous flow” must arise in crystalline bodies, consisting in directed self-diffusion, and between the coefficients of viscosity and self-diffusion the relation (16) will hold. In other words, in crystalline bodies under the action of external forces there must occur a special kind of deformation proceeding by means of the diffusive displacement of atoms and phenomenologically similar to viscous flow. We shall call this deformation diffusion creep. Sintering is a particular case of diffusion creep (or flow) occurring under the action of surface-tension forces.

It should be emphasized that diffusion creep has nothing in common with ordinary plastic deformation in crystals, which proceeds by means of so-called slip (or twinning), i.e. processes involving the simultaneous (joint) displacement of considerable groups of atoms. Deformation by slip leads, as is well known, to structural changes (the appearance of distortions of the crystal lattice) and to changes in physical properties, in particular mechanical ones (so-called hardening), which in turn affects the further course of deformation. Effects of this kind are absent in diffusion creep.

Only the phenomenon of recovery accompanying plastic deformation, arising upon sufficient heating of plastically deformed crystalline bodies (and reducible to the removal of lattice distortions, as well as of changes in physical properties caused by the preceding deformation), is effected with the aid of processes of the diffusion type, close to those taking place

in creep. Let us also note that deformation by slip or twinning begins only when the magnitude of the stresses applied to the body exceeds a certain yield limit; such a limit is absent for diffusion creep.

The distinction between diffusion creep in crystals and ordinary plastic deformation was noted quite definitely and precisely already by Ya. I. Frenkel’[^5].

The regularities of diffusion creep are contained in the equations of directed self-diffusion (12) or of “viscous flow” (15) given above, provided relation (16) is satisfied. It should be noted, however, that the latter relation is only an estimate and may include dimensionless coefficients that differ for individual special problems. It is therefore preferable, when solving concrete problems, to proceed from the equation of self-diffusion; the characteristic linear dimensions and numerical coefficients will then be determined more reliably (cf., for example, the solutions given above for the problem of the adhesion of a sphere to a plane).

Diffusion creep, of course, must occur both in continuous and in porous bodies, and be caused not only by the action of capillary forces but also by other forces, for example forces applied from outside. When the acting forces exceed the yield limit, in addition to creep there will also be ordinary plastic deformation, which (especially at low temperatures) may considerably exceed the magnitude of the creep. Experimentally the creep effect can be isolated if the applied stresses are reduced to values below the yield limit. It should also be noted that the coefficient of self-diffusion \(D\) contained in equations (12), and also (15) (implicitly, through \(\eta\)), even at constant temperature is not always a constant; in the presence of distortions of the crystal lattice, \(D\) will change substantially with time (if the body is heated to a temperature at which self-diffusion phenomena proceed at a noticeable rate, i.e. creep is appreciable).

Thus, the kinetics of the creep phenomenon may prove to be rather complicated.

It should be pointed out that there is one type of stressed state in which creep should not take place—namely the case of homogeneous all-round compression (in this case there are no pressure differences at points of the body that could give rise to differences in the equilibrium concentrations of vacancies). Since all-round compression is characterized by zero shear stresses, it is not difficult to conclude from this that diffusion creep occurs only under the action of shear stresses; this, incidentally, as is well known,[^25] follows from the equations of viscous flow (15), which describe creep.

To carry out a homogeneous all-sided compression in a porous body is practically possible only in the case when the body contains only the so-called open pores (connected by “channels” with the external surface); the indicated stressed state will be attained in such a body by placing it in a gaseous medium under pressure. If, however, the body also contains closed pores, inside which the pressure differs from the pressure in the gaseous medium, creep will, of course, arise, which will lead (depending on the sign of the difference of the total pressures, including the capillary pressure) to the filling in or expansion of the closed pores.

The study of the phenomenon of creep under the joint action of external and capillary forces may be of interest for testing the correctness of the propositions developed above. We shall therefore dwell somewhat more fully on the consideration of the question of creep in a porous body to which external forces are applied. Let us take the simplest case. Suppose that over the entire external surface of a body containing both open and closed pores there is applied (for example, mechanically) a homogeneous pressure \(p\). Of course, the elements of the body will by no means be in a state corresponding to true all-sided compression. To determine the stressed state, let us mentally add, on the internal surface of the body, two systems of mutually balancing (at each point) forces: a) the aggregate of forces producing on the internal surface a homogeneous pressure of the same magnitude as that existing on the external surface, and b) the aggregate of forces equal at each point to the forces of system a), but opposite to them in sign. The equilibrium and the stressed state of the body will not change from the simultaneous introduction of the systems of forces a) and b).

The forces of system a), together with the homogeneous pressure on the external surface, will bring about at all points of the body a true all-sided pressure that does not cause creep.

The remaining forces of system b) are equivalent to an “negative” pressure of magnitude \(p\) applied to the internal surface of the body.

Under the action of the sum of this pressure and the negative pressure \(2\sigma \frac{1}{a}\), caused by the capillary forces (\(a\) is the radius of curvature of the internal pores), creep must occur (to an appreciable extent at appropriate temperatures), the result of which in the present case will be sintering.

The rate of the sintering process will be increased in comparison with that obtained in the same body in the absence of external pressure. A quantitative estimate of the increase is not difficult to perform for the idealized case in which the body contains pores (open or closed) of one and the same radius of curvature \(a\). In the absence of external pressure, the change in pore radius is given by equation (10).

If, however, an all-around pressure \(p\) is applied (to the outer surface of the body), the kinetics of the process will be described by the relation

\[ \frac{da}{dt}=-\frac{D\delta^3}{kT}\,\frac{1}{a}\left(\frac{2\sigma}{a}+p\right) =-\frac{2D\delta^3}{kT}\,\frac{\sigma}{a^2}\left(1+\frac{ap}{2\sigma}\right), \tag{27} \]

which is a generalization of equation (10) for the present problem. In contrast to (10), according to (27) the pore volume changes nonlinearly with time (the addition \(\frac{pa}{2\sigma}\) in the brackets on the right-hand side decreases together with \(a\)).

Integrating equation (27), we find:

\[ \frac{2D\delta^3}{kT}\,\sigma t = \frac{1}{2K}(a_0^2-a^2) - \frac{a_0-a}{K^2} + \frac{1}{K^3}\ln\frac{1+Ka_0}{1+Ka}, \tag{27a} \]

where \(K=\frac{p}{2\sigma}\). As \(K\to 0\), the right-hand side of (27a) tends to the value corresponding to equation (10), namely to the quantity \(\frac{a_0^3-a^3}{3}\). If, however, \(Ka_0\) is large, i.e. \(p\gg \frac{2\sigma}{a_0}\), then the first term on the right-hand side of (27a) is practically the largest, and approximately

\[ \frac{2D\delta^3}{kT}\,\sigma t \simeq \frac{1}{2K}(a_0^2-a^2) \]

or

\[ \frac{2D\delta^3}{kT}\,pt \simeq (a_0^2-a^2), \]

which also corresponds to a nonlinear change of the pore volume with time \((t\sim v_0^{2/3}-v^{2/3})\).

Let \(\frac{N}{V_0}\) be the number of pores per unit volume, so that the porosity of the specimen is

\[ x=\frac{N}{V_0}a^3 \]

(the initial porosity is

\[ x_0=\frac{N}{V_0}a_0^3 \]

). Denote the change in porosity relative to its initial value \(x_0-x\) by \(\Delta x\), and eliminate the quantity \(a\) from the right-hand side of (27a), expressing it through \(x_0\) and \(\Delta x\). We obtain:

\[ \frac{2D\delta^3}{kT}\,\sigma t = \frac{a_0^3}{2\mu} \left\{ \left[1-\left(1-\frac{\Delta x}{x_0}\right)^{2/3}\right] -\frac{2}{\mu} \left[1-\left(1-\frac{\Delta x}{x_0}\right)^{1/3}\right] \right. \]

\[ \left. +\frac{2}{\mu^2}\ln \frac{1+\mu}{1+\mu\left(1-\frac{\Delta x}{x_0}\right)^{1/3}} \right\}. \tag{27б} \]

Here \(\mu=Ka_0=\frac{pa_0}{2\sigma}\). For small values of \(\frac{\Delta x}{x_0}\), the right-hand side of (27б) reduces to

\[ \frac{a_0^3}{3}\,\frac{\Delta x}{x_0}\,\frac{1}{1+\mu} = \frac{a_0^3}{3}\,\frac{\Delta x}{x_0}\,\frac{1}{1+\frac{pa_0}{2\sigma}}. \tag{27в} \]

If an identical specimen is subjected to sintering (under identical conditions) without external pressure (i.e., at \(p = 0\)), then at the same time \(t\) the change in porosity would be

\[ \Delta x'=\frac{2DD_0^3}{kT}\sigma t\cdot \frac{3x_0}{a_0}; \]

Thus, in the initial stage of sintering under external hydrostatic pressure \(p\), the magnitude of shrinkage (equal to the change in porosity) \(\Delta x\) is changed, in comparison with that corresponding to sintering of the same specimen without pressure, \(\Delta x'\), in the ratio

\[ \frac{\Delta x}{\Delta x'}=1+\frac{pa_0}{2\sigma}. \tag{28} \]

According to (28), the ratio \(\dfrac{\Delta x}{\Delta x'}\) does not depend on the value of the self-diffusion coefficient \(D\) (or on the change of \(D\) with time), and therefore determination of \(\dfrac{\Delta x}{\Delta x'}\) can be used to verify the correctness of the formulas of the diffusion theory of sintering and creep as applied to the remaining quantities (except \(D\)).

This was done in work\({}^{10}\), in which comparative determinations were made of the magnitude of shrinkage (for specimens of Cu and Fe, pressed from powders and having \(x_0 \simeq 40\%\)) during sintering under pressure and without pressure. Specimens sintered under pressure were subjected to uniaxial compression. In this case, owing to friction between the end faces of the specimen and the jaws of the press, hydrostatic compression was in fact also obtained, differing in magnitude in different elements of the body (decreasing with distance from the end faces). The applied uniaxial pressure \(p\) was equivalent to a certain effective hydrostatic (external) pressure \(p_{\mathrm{vn}}^{*}\); the magnitude of the latter was determined by comparing data on changes: 1) of volume:

\[ \delta\!\left(\frac{\Delta V}{V}\right) = \left.\frac{\Delta V}{V}\right|_{p} - \left.\frac{\Delta V}{V}\right|_{p=0} \]

and 2) of specimen length:

\[ \delta\!\left(\frac{\Delta L}{L}\right) = \left.\frac{\Delta L}{L}\right|_{p} - \left.\frac{\Delta L}{L}\right|_{p=0}. \]

The relation between \(\delta\!\left(\dfrac{\Delta V}{V}\right)\) and \(\delta\!\left(\dfrac{\Delta L}{L}\right)\) proved to be linear, and the points for specimens pressed from powders of different metals (copper, iron) and sintered at various temperatures, as well as for specimens (of the same dimensions and the same porosity) compressed in a press at room temperature, lay on one and the same straight line; this showed that the quantitative relation between \(\delta\!\left(\dfrac{\Delta V}{V}\right)\) and \(\delta\!\left(\dfrac{\Delta L}{L}\right)\), corresponding to the ratio \(\dfrac{p_{\mathrm{vn}}^{*}}{p}\), is determined by the “geometry” of the specimens, and not by the physical phenomena during densification. Experimentally it was found that

\[ \delta\!\left(\frac{\Delta V}{V}\right)=\varepsilon\,\delta\frac{\Delta L}{L}, \]

where \(\varepsilon \simeq 0.6\).

It follows from this that the effective hydrostatic (external-pressure) component was \(p_{\text{ext}}^{*}=0.2p\) [equality \(p^{*}=p\) would correspond to

\[ \delta\left(\frac{\Delta V}{V}\right)=3\delta\left(\frac{\Delta L}{L}\right) \]

].

The experimentally found values of \(\delta\left(\frac{\Delta V}{V}\right)\), which, in the notation adopted above (cf. (28)), are equal to \(\Delta x-\Delta x'\), proved to depend linearly on \(p\), which is in agreement with the calculation.

The value of \(\frac{2\sigma}{a_{0}}\) determined with the aid of (28) from the magnitude of \(\Delta x'\) also proved to be in quite good agreement with the value of the pore radius that could be estimated from the initial dimensions of the powder grains. The computed value of the viscosity coefficient \(\eta\) turned out to be independent of pressure.

Thus, relation (28) was confirmed in work \(^{10}\), which may be regarded as corroborating the conclusion that sintering is a particular case of diffusion creep occurring under the action of surface-tension forces (and, in the presence of an external pressure \(p\), under the combined action of external and capillary forces).

Let us note that the linear relation \(\Delta x-\Delta x'\) with the pressure \(p\) corresponding to equation (28), as already indicated, should be valid only for small \(\Delta x-\Delta x'\); at a large value of shrinkage under pressure, according to (27a) or (27b), linearity is violated. An additional cause of the violation of linearity in a body containing pores of different sizes may be a decrease in the number of pores \(N\) at large \(\Delta x\) (pores of small size will become completely overgrown and the number of active pores will decrease).

From the point of view of a general consideration of the question of diffusion creep, it is also of interest to elucidate the laws of this phenomenon in solid (nonporous) bodies.

Here too let us consider the simplest problem, namely the phenomenon of creep in thin filaments under the action of tensile or compressive forces. The latter include, in particular, surface-tension forces compressing a thin filament along its axis with a force \(2\pi R\sigma\) (\(R\) is the radius of the filament), to which there corresponds a compressive stress

\[ \frac{2\pi R\sigma}{\pi R^{2}}=\frac{2\sigma}{R}, \]

whereas the capillary pressure directed along the radius of the filament is equal to \(\frac{\sigma}{R}\). Thus, the action of capillary forces on the filament reduces: a) to hydrostatic compression by the pressure \(\frac{\sigma}{R}\), and b) to compression along the axis by the pressure \(\frac{\sigma}{R}\) (assuming the filament to be sufficiently long, one may neglect additional local stresses near the ends of the filament). Hydrostatic compression, as already indicated, does not cause creep. Consequently,

in relation to the creep of an unloaded thin filament, it may be regarded as being under an axial compression caused by the pressure \(\frac{\sigma}{R}\).

We shall present the solution of the problem of the creep of a filament in two variants, corresponding to the description of creep a) as “viscous flow” and b) as directed self-diffusion.

a) Variant of “viscous flow.”\(^6\) The components of the displacement velocity (if the origin of the coordinate system is chosen at the center of the filament, and the \(x\)-axis is placed along the axis of the filament) for the given problem may be written (assuming the deformation to be homogeneous) in the form

\[ v_x=\alpha x;\quad v_y=-\frac{\alpha}{2}y;\quad v_z=-\frac{\alpha}{2}z, \tag{29} \]

where

\[ \alpha=\frac{1}{L}\frac{dL}{dt}=\frac{1}{x}\frac{dx}{dt}=\text{const} \tag{29a} \]

(\(L\) is the length of the filament).

It is not difficult to verify that the values of the components \(\mathbf{v}\) according to (29) satisfy the equation of viscous flow (15a).

Equating the work of the internal-friction forces in the entire volume of the body to the sum of the work of the external forces and the decrease of the surface energy, we obtain:

\[ \int 2\eta\sum_i\sum_k v_{ik}^{2}\,dV = F\frac{dL}{dt} - 2\pi\sigma\left( L\frac{dR}{dt}+R\frac{dL}{dt} \right). \tag{30} \]

Here \(F\) is the applied force; \(F=\pi R^2p\), where \(p\) is the tensile stress.

According to (29), only the following values of \(v_{ik}\) are different from zero: \(v_{11}=\alpha\), \(v_{22}=-\frac{\alpha}{2}\), and \(v_{33}=-\frac{\alpha}{2}\). Further, owing to conservation of the volume of the filament, \(\pi R^2L=V_0=\text{const}\), and

\[ R=\frac{1}{\sqrt{L}}\sqrt{\frac{V_0}{\pi}},\qquad \frac{dR}{dt} = -\frac{1}{2}\sqrt{\frac{V_0}{\pi}}\,L^{-3/2}\frac{dL}{dt}. \]

Therefore equation (30) reduces to the form:

\[ \frac{1}{L^2}\frac{dL}{dt} = \frac{1}{3V_0\eta}(F-\pi\sigma R) = \frac{\pi R^2}{3V_0\eta}\left(p-\frac{\sigma}{R}\right) = \frac{1}{3L\eta}\left(p-\frac{\sigma}{R}\right) = \frac{1}{3L\eta}\left( p-\sigma\sqrt{L}\sqrt{\frac{\pi}{V_0}} \right). \tag{31} \]

The creep rate, as is evident, must be nonconstant, even if \(\eta=\text{const}\). In reality, as follows from the preceding, for a crystalline body \(\eta\) must depend on li-

linear dimensions of the filament. We determine the existing dependence by considering creep as directed self-diffusion.

b) The case of directed self-diffusion. The components of the concentration gradient must satisfy equations analogous to equations (29) (cf. (14) and (12)), i.e., it must be:

\[ D'\operatorname{grad}_{x} c = D'\frac{\partial c}{\partial x} = -\beta x;\quad D'\frac{\partial c}{\partial y}=-\frac{\beta}{2}y;\quad D'\frac{\partial c}{\partial z}=-\frac{\beta}{2}z. \tag{32} \]

Relations (32) are easy to understand if one takes into account that under homogeneous deformation, for each segment of filament of length \(dx\) (Fig. 6), the condition must be satisfied that the flux of atoms entering\(^*\) through the cylindrical surface,

\[ -\,D'\frac{\partial c}{\partial r}\bigg|_{r=R} 2\pi R\,dx, \]

and the difference of the fluxes through the end areas separated by \(dx\),

\[ +\,D'\frac{\partial^2 c}{\partial x^2}\bigg|_{x}\pi R^2 dx, \]

i.e., the equality

\[ -2\frac{\partial c}{\partial r}\bigg|_{r=R} = \frac{\partial^2 c}{\partial x^2}R \]

must hold, or

\[ \frac{\partial q_x}{\partial x} = -\frac{2q_R}{R} \quad (q_i=D'\operatorname{grad}_i c). \]

Fig. 6.

Hence it follows:

\[ q_x=-\frac{2q_R}{R}x=\beta x, \]

where

\[ \beta=-\frac{2q_R}{R}=\mathrm{const}. \]

The magnitude of the concentration gradient determining the vacancy flux \(q_R\) is, in order of magnitude, equal to \(\Delta c/R\), where \(\Delta c\) is given by the pressure difference on the lateral and end surfaces of the filament element \(dx\), i.e.,

\[ q_R \sim D'\frac{\Delta c}{R} = D'c_0\frac{\delta}{kT}\frac{1}{R} \left(p-\frac{\sigma}{R}\right) = \frac{D\delta}{kT}\frac{1}{R} \left(p-\frac{\sigma}{R}\right). \tag{33} \]

The flux of vacancies through the ends of the filament, characterizing the rate of elongation of the filament, is equal to

\[ -\,q_x\bigg|_{x=\frac{L}{2}} = \frac{2q_R}{R}\cdot\frac{L}{2} = \frac{D\delta}{kT}\frac{L}{R^2} \left(p-\frac{\sigma}{R}\right) = \frac{dL}{dt}. \tag{34} \]

Thus, for the kinetics of filament elongation by creep we obtain the equation

\[ \frac{1}{L}\frac{dL}{dt} = \frac{D\delta}{kT}\frac{1}{R^2} \left(p-\frac{\sigma}{R}\right), \tag{35} \]

\(^*\) Or, correspondingly, of vacancies leaving through the same surface, etc.

coinciding with (31) for

\[ \frac{1}{\eta}=\frac{3D\delta^{3}}{kT}\cdot \frac{1}{R^{2}}, \tag{36} \]

which corresponds to the general relation (16); the characteristic linear dimension here is the radius of the thread \(R\)*).

The solution of equation (35) (as also of equation (31)), of course, depends on the time dependence of \(p\), or \(F=\pi pR^{2}\).

If, as the simplest condition for experimental realization, one assumes that \(F=\mathrm{const}\), one obtains solutions of equations (31) and (35), which are given below in Table II.

Table II

Solutions of the equations for the kinetics of creep of threads for \(F=\mathrm{const}\)

Case	\(A\) (for \(\eta=\mathrm{const}\))	\(B\) (for \(\dfrac{1}{\eta}=\dfrac{3D\delta^{3}}{kT}\cdot\dfrac{1}{R^{2}}\))
General formula	\(\dfrac{Ft}{6V_{0}\eta}=\dfrac{1}{\gamma^{2}}\left|\ln\left(1-\dfrac{\gamma}{\lambda}\right)+\dfrac{\gamma}{\lambda}\right|_{\lambda_{0}}^{\lambda}\)	\(\dfrac{\pi D\delta^{3}}{kT}\dfrac{Ft}{V_{0}^{2}}=\dfrac{1}{\gamma^{4}}\left|\ln\left(1-\dfrac{\gamma}{\lambda}\right)+\dfrac{\gamma}{\lambda}+\dfrac{\gamma^{2}}{2\lambda^{2}}+\dfrac{\gamma^{3}}{3\lambda^{3}}\right|_{\lambda_{0}}^{\lambda}\)
\(\gamma\ll\lambda_{0}\) \(F\gg\pi\sigma R_{0}\)	\(\dfrac{L}{L_{0}}=\dfrac{1}{1-\dfrac{Ft}{3\pi\eta R_{0}^{2}}}\)	\(\dfrac{L}{L_{0}}=\dfrac{1}{\sqrt{\,1-\dfrac{2D\delta^{3}}{kT}\cdot\dfrac{Ft}{\pi R_{0}^{4}}\,}}\)
\(\gamma\gg\lambda_{0}\) \(F\ll\pi\sigma R_{0}\)	\(\dfrac{L}{L_{0}}=\dfrac{1}{\left(1+\dfrac{\sigma t}{6\eta R_{0}}\right)^{2}}\)	\(\dfrac{L}{L_{0}}=\dfrac{1}{\left(1+\dfrac{3\pi\sqrt{\pi}}{2}\dfrac{D\delta^{3}}{kT}\cdot\dfrac{\sigma t}{R_{0}^{3}}\right)^{2/3}}\)

In addition to the general solution of the creep problem, under the simultaneous action of an external force \(F\) together with the surface tension \(\sigma\), the table also gives formulas for the limiting cases of large and small values of \(F\) (which correspond to the action only of the external force or only of surface tension). To simplify the writing of the general solution, the following notation has been introduced:

\[ \lambda=\sqrt{L} \quad\text{and}\quad \gamma=\frac{\sigma}{F}\sqrt{\pi V_{0}}. \]

* It should be noted that above the case of a homogeneous, i.e., single-crystal, thread was tacitly meant; if the thread is polycrystalline and the linear size of the grains \(\rho\) proves to be \(<R\), then in relation (36) (or (10)) the characteristic linear dimension will be \(\rho\).

Although several experimental works on the study of the creep of thin filaments have been published in the literature \(^{26,27}\), the kinetics of the process has in fact not been studied. In work \(^{26}\) the surface tension \(\sigma\) of copper (in the crystalline state!) was determined from the magnitude of the force \(F\) stopping creep. As was to be expected (cf. \(^{29}\)), it was found that the surface tension in the solid state practically does not differ from that possessed by the liquid metal.

In other works \(^{27}\) only the viscosity coefficient \(\eta\) was determined and compared with the self-diffusion coefficient. In one work it was shown that for single-crystal filaments

\[ \frac{1}{\eta} \sim \frac{1}{R^n}, \]

where in the case of Sn it was found that \(n = 1.95\), and in the case of Pb \(n = 1.75\) (the theoretical value is \(n = 2\)).

It should be noted that, when single-crystal thin filaments are obtained directly from the melt or by recrystallization at premelting temperatures, one should expect the absence of distortions of the crystal lattice, so that experimental data on the creep kinetics of filaments obtained in this way should not be complicated by annealing effects. Thus, the creep of thin filaments is a phenomenon through the study of which one can successfully carry out an experimental verification of the theory of directed self-diffusion. Of course, the experiment is rather difficult, since the magnitude of the applied stresses must not exceed the yield point (in order to avoid complications from the effect of plastic deformation), and the temperature must be maintained constant with high accuracy along a filament of considerable length; moreover, the value of the temperature (in order to obtain a large elongation) must be close to the melting temperature.

7. THE PHENOMENON OF ANNEALING AND ITS INFLUENCE ON THE KINETICS OF DIFFUSIONAL CREEP (SINTERING)

The considerable influence of distortions of the crystal lattice on the kinetics of sintering is now well known. The rate of sintering may increase by several orders of magnitude if the powder used for making a porous body is obtained in a state with a nonequilibrium (distorted) lattice, for example by means of decomposition, reduction reactions, etc.

Conversely, by removing lattice distortions (as a result of high-temperature annealing, carrying out a phase transition under equilibrium conditions, etc.), one can, all other conditions being equal, reduce the rate of sintering by tens and hundreds of times.

An interesting example is the sintering of magnesium oxide. In specimens pressed from powder obtained by decomposition of the carbonate salt \(\mathrm{MgCO_3}\), sintering is already observed at \(600^\circ\mathrm{C}\).

such intense sintering that the porosity falls practically to zero[^31]. At the same time, for appreciable sintering of magnesium oxide, which has no distortions of the crystal lattice, heating to temperatures of \(\sim 1400\text{—}1500^\circ\mathrm{C}\) is required.

In the practical use of sintering, objects possessing considerable distortions of the crystal lattice are always employed. In particular, in the case of metals, so-called active powders are used for sintering, obtained by reduction of oxides, electrolytically, or by grinding in vortex mills, etc.

Thus, elucidating the kinetics of sintering occurring under practical conditions is closely connected with determining the influence of distortions of the crystal lattice on this phenomenon.

As follows from what has been said above, the presence of distortions of the crystal lattice should influence the kinetics not only of sintering, but also of diffusional creep in general.

The solution of the problem of the influence of distortions of the crystal lattice on the phenomenon of creep encounters great difficulties, since up to the present time there is no complete clarity regarding the microscopic picture of the arrangement of atoms in a region with a distorted lattice, and the question of the distribution, as well as the possible interaction, of individual distortions (or regions of the crystal containing distortions) has not been studied.

Although it is known that, in the presence of distortions, the coefficient of self-diffusion \(D\) must be changed (increased), no quantitative relation has been established between the magnitude \(D\) and any characteristic of the distortions.

The problem is further complicated by the fact that lattice distortions present in the initial state of a crystalline body do not remain unchanged; at temperatures at which self-diffusion proceeds at an appreciable rate (in the distorted lattice!), they are spontaneously removed, which constitutes the so-called phenomenon of recovery1.

In the process of recovery many physical properties of a body change, including the coefficient of self-diffusion. Consequently, the kinetics of recovery must to a considerable extent determine also the kinetics of diffusional creep.

It is known that certain physical properties, for example mechanical and electrical, are restored in metals

with distortions of the crystal lattice after annealing at comparatively low temperatures (for Cu and Fe at \(200—400^\circ\) C), much lower than the sintering temperatures customarily used (\(800—1000^\circ\) C). From this, however, it by no means follows—contrary to certain existing opinions—that at sintering temperatures the crystal lattice of the body is allegedly already completely restored and that the annealing process can have no effect on the kinetics of sintering.

The fact of the sharp influence of the presence of initial lattice distortions on the rate of sintering at high temperatures is firmly established. It inevitably indicates that the restoration of the mechanical and electrical properties of metals, which occurs during low-temperature annealing, does not exhaust the process of returning the crystal lattice to the state of equilibrium, so that annealing is a more complex phenomenon than is sometimes supposed.

As applied to the processes of diffusion creep, what is essential above all is the changes in the coefficient of self-diffusion that occur during annealing. In view of the difficulties of a detailed microdescription, one might attempt to determine the kinetics of the change in \(D\) during annealing in a purely phenomenological way. Since the lattice distortions that cause the increased value of \(D\) are removed by the same diffusion path, at first glance it seems that, for example, in an isothermal process the rate of decrease of the increased self-diffusion coefficient should be proportional to the existing value of \(D\), i.e.

\[ \frac{dD}{dt}=-aD, \]

where \(a\) is a proportionality coefficient. From the written relation there would follow an exponential time course of the quantity \(D\), and with it of the shrinkage during sintering of a porous body, which is proportional (cf. (21)) to the quantity

\[ \int_0^t D\,dt. \]

As was already indicated in § 5, this conclusion does not agree with the experimental data obtained under conditions actually close to isothermal ones. The conclusion just given also proves untenable theoretically, if one turns to the modern theory of the phenomenon of self-diffusion, developed mainly by Ya. I. Frenkel.

In § 2 it was pointed out that the coefficient of self-diffusion \(D=cD'\), where \(c\) is the concentration of vacancies in the crystal lattice, and \(D'\) is the diffusion coefficient of vacancies. The increased, relative to the equilibrium value (in a system with distortions), value of \(D\) should in practice be due only to an increase in the number of vacancies, equal, say, not to \(c\), but to \(\xi \gg c\).

As for the quantity \(D'\), then, as calorimetric measurements\(^{24}\) show, at temperatures of appreciable sintering ...

the energy state of the lattice practically does not differ from the equilibrium one; this means that the barriers for the displacement of vacancies are preserved and, hence, \(D'\) is also unchanged.

In view of what has been said, the removal of the “excess” value of \(D\) during annealing is the decrease of the quantity \(\xi\). But the disappearance of excess vacancies can occur, in practice, only when vacancies meet atoms located in interstices. Such atoms will below, for brevity, be called “inclusions.” The number of inclusions must be equal to the number of vacancies, since excess vacancies must also appear as a result of the formation of inclusions. The probability of a vacancy meeting an inclusion is proportional to the quantity \(\xi^{2}\); the number of disappearing vacancies is proportional to the same quantity and, consequently, so is the change in the self-diffusion coefficient.

It is already clear from this that \(\dfrac{dD}{dt}\) is not proportional to \(D\) to the first power, so that the above-mentioned exponential law should not occur.

The true kinetics of the change in \(D\) should be determined quantitatively by developing the above considerations, based on Frenkel’s theory of self-diffusion, as applied to an object for which the change in the arrangement of particles is known. But since the structural changes corresponding to distortions of the lattice have not yet been established, it is possible to give only qualitative estimates based on the general characterization of the distortions.

First of all it is necessary to answer the question of the origin of the excess vacancies \((\xi-c)\) that appear on heating in a distorted crystalline lattice. Without connecting the essence of the further arguments with any definite model of distortions, one may, quite generally, assume that in the presence of distortions in a crystalline lattice there appear elements of volume with an excess and a deficient number of atoms. In the linear case these would be so-called “dislocations” or “interlockings,” which may be characterized as follows. If in some row of atoms in the lattice, over a length of \(g\) periods, \(g+1\) or \(g-1\) atoms were arranged, this would mean the appearance of a “\(+\)” or, respectively, a “\(-\)” dislocation.

For simplicity of exposition, below we shall speak of linear dislocations, although in essence the arguments should also be valid for other kinds of violation of the periodicity of the lattice*).

* ) Two-dimensional and three-dimensional violations of periodicity are possible through the formation of regions with several excess and deficient atoms; linear dislocations with several excess atoms, etc., are also conceivable.

“+” and “−” dislocations must exist in the crystal in equal numbers. In this case a “+” dislocation, in the arrangement of atoms, is close to an inclusion in the sense indicated above (an atom situated in an interstice), while a “−” dislocation is close to a vacancy. It is easy to see that it is necessary to overcome a comparatively small activation energy \( \nu \) in order to convert a “−” dislocation into a vacancy, since for this it is sufficient to pass from a uniform distribution of \(g-1\) atoms over a segment of \(g\) periods to their arrangement at the lattice sites, as a result of which one empty site, i.e. a vacancy, will appear.

The situation is somewhat different with “+” dislocations. By means of a thermal fluctuation (exceeding some activation energy \( \nu' \)) the excess atom will likewise be removed from the \(g+1\) row (transferred to an interstice), but the equilibrium state of the crystal lattice will not be completely restored on the segment of the “+” dislocation, since the excess inclusion remains, and with it a certain distortion of the lattice, though one embracing a smaller number of atoms.

Let the total number of dislocations of one sign per unit volume be \(\xi_0\). On heating the body to temperature \(T\), as a result of thermal fluctuations there must be liberated:

\[ \xi_T=\xi_0 e^{-\nu/kT} \tag{37a} \]

vacancies; in addition,

\[ \xi'_T=\xi_0 e^{-\nu'/kT} \tag{37b} \]

“+” dislocations must be converted into inclusions.

In writing relations (37a) and (37b), we have assumed the activation energies \(\nu\) and \(\nu'\) (for converting “−” and “+” dislocations respectively into vacancies and inclusions) to be the same for all dislocations of each kind, i.e. as though all dislocations of one type were identical (for example, having the same order \(g\)). This is, of course, a simplification, admissible only in a schematic description of the phenomenon.

One may suppose that at a temperature \(T\) exceeding the larger of the quantities \(\dfrac{\nu'}{k}\) and \(\dfrac{\nu}{k}\), many physical properties of the crystal that had been altered by the presence of dislocations (distortions) will practically prove to be restored, even if the number of remaining inclusions and vacancies amounts, for example, to 1% of the total number of atoms.

Such a conclusion, however, is clearly inapplicable to the self-diffusion coefficient \(D\), since the number of excess vacancies mentioned may exceed the equilibrium value of the number of vacancies \(c_0\) several times even at premelting temperatures, and at lower temperatures by hundreds and thousands of times; this must be accompanied by a proportional increase of \(D\).

Thus, it should be accepted that the increased value of the self-diffusion coefficient observed when bodies with a distorted crystal lattice are heated is due to the formation of a large number of excess vacancies during the healing of the lattice in regions with an insufficient number of atoms.

The excess vacancies must migrate through the crystal lattice and, upon encountering inclusions (or “⊥” dislocations), will in a relative number of cases \(b_1=e^{-\gamma/kT}\) be annihilated (i.e., be replaced by excess atoms); here \(\gamma\) is the activation energy for the replacement of a vacancy by inclusions. The number of encounters is proportional to the length of the path of self-diffusion of the vacancies \(S\simeq \sqrt{D't}\) and to the probability that a vacancy and an inclusion appear in neighboring positions. Therefore the number of vacancies disappearing during the time \(dt\) will be

\[ d\xi=-Ab_1\xi^2\,dS, \tag{38a} \]

where \(A\) is a constant.

Integrating (38a), we obtain for an isothermal process (at some temperature \(T\)):

\[ \xi=\frac{1}{\frac{1}{\xi_{\mathrm{n}}}+Ab_1\sqrt{D't}}; \tag{38b} \]

\(\xi_{\mathrm{n}}\) is the initial concentration of excess vacancies at the moment \(t=0\).

It should be noted that while the liberated vacancies are mobile and migrate through the crystal, the inclusions should practically not be displaced from the sites they occupy (the activation energy of the transition is very large). Further, in the initial state (as is known, for example, from experience with the formation of distortions under plastic deformation), “⊥” and “←” dislocations are not distributed uniformly throughout the crystal, but are concentrated in groups in various “centers.” Therefore, when vacancies are liberated, what should remain are not inclusions uniformly distributed throughout the crystal, but centers in which the density of inclusions is greatly increased. It is quite probable that, when the initial number of liberated vacancies \(\xi_{\mathrm{n}}\) changes, the number of centers \(j\) is preserved and only the average number of inclusions in each center changes: \(\rho=\frac{\xi_{\mathrm{n}}}{j}\). This circumstance can be taken into account by assuming the constant \(A\) in formulas (33a) and (38b) to be inversely proportional to the average number of inclusions falling in each center, \(\rho=\frac{\xi_{\mathrm{n}}}{j}\). After this, formula (38b) takes the form:

\[ \xi=\frac{\xi_{\mathrm{n}}}{1+b\sqrt{D't}}, \tag{38v} \]

where \(b\) is a new constant, proportional to \(b_1\) and to the quantity \(j\).

The obtained relation, which determines the change of \(\xi\) with time in an isothermal process, should also characterize the law of isothermal change of \(D=\xi D'\), since (at high temperatures) \(D'\) is practically constant.

Using formula (38b) one can determine the kinetics of diffusional creep (sintering) of a body possessing distortions of the crystal lattice. In particular, for a porous body with an unchanged number of pores per unit volume \(N\), for which, at \(D=\mathrm{const}\), the kinetics of shrinkage would be described by equation (21), we shall have, if \(\xi\) changes with time according to (38b),

\[ y_{T,t}=\left(\frac{\Delta L}{L}\right)_{T,t}\simeq \frac{1}{3}\left(\frac{\Delta V}{V}\right)_{T,t} = \frac{8\pi}{3}\frac{\sigma\delta^{3}}{kT}ND'\int_{0}^{t}\xi\,dt \]

\[ = \frac{16\pi}{3}\frac{\sigma\delta^{2}}{kT}\frac{N\xi_{\mathrm{н}}}{b^{2}} \left[ b\sqrt{D't}-\ln\left(1+b\sqrt{D't}\right) \right]. \tag{39} \]

By \(y_{T,t}\) here is denoted the magnitude of the linear shrinkage at time \(t\) in an isothermal process \((T=\mathrm{const})\).

For small \(t\), from (39) we obtain (by expanding the logarithm in a series):

\[ y_{T,t}\simeq \frac{8\pi}{3}\frac{\sigma\delta^{3}}{kT}N\xi_{\mathrm{н}}D't, \tag{39a} \]

which corresponds to a linear dependence on \(t\), which is indeed observed experimentally at small \(t\).

For large \(t\), in formula (39) the logarithmic term may be neglected, and one obtains

\[ y_{T,t}=\left(\frac{\Delta L}{L}\right)_{T,t} \sim \frac{16\pi}{3}\frac{\sigma\delta^{3}}{kT}\frac{N}{b}\xi_{\mathrm{н}}\sqrt{D't}, \tag{39b} \]

i.e. a linear dependence on \(\sqrt{t}\), which (see § 5) agrees with the results of experiments that are maximally close to isothermal ones.

Thus, the schematic ideas set forth above concerning the liberation and subsequent “annihilation” of vacancies (in the process of recovery in bodies with distortions of the crystal lattice) lead to a qualitatively correct description of the kinetics of isothermal sintering.

It is more difficult to determine by calculation the temperature dependence of the sintering rate, since for this it is necessary to establish the initial number of excess vacancies \(\xi_{\mathrm{н}}\) at each temperature. We shall give some estimates relating to stepwise heating (see § 5). According to (37a), the number of vacancies liberated when the temperature is raised by one step \(\Delta T\) is

\[ \Delta\xi_{T}\simeq \frac{\partial \xi_{T}}{\partial T}\Delta T = \xi_{0}\frac{\gamma\Delta T}{kT^{2}}e^{-\gamma/kT} = \xi_{T}\frac{\gamma}{kT}\cdot\frac{\Delta T}{T}. \tag{40} \]

The initial number of excess vacancies at each isothermal stage is not, however, reduced to only one value \(\Delta \xi_T\). It is also necessary to take into account the excess vacancies remaining from the preceding heating stages. For sufficiently long isothermal holds at each stage \(t_0\bigl(b\sqrt{D'_{t-\Delta T}t_0}\gg 1\bigr)\), one may restrict oneself to taking into account the vacancies retained from one preceding stage, i.e., write:

\[ \xi_{\mathrm{n}}=\Delta \xi_T+\frac{\Delta \xi_{T-\Delta T}}{b\sqrt{D'_{T-\Delta T}t_0}}. \tag{41} \]

The addition to \(\Delta \xi_T\) corresponding to vacancies remaining from the preceding heating stage increases as \(t_0\) decreases, as

\[ \frac{1}{\sqrt{t_0}}. \]

With the aid of relation (40) one can determine the magnitude of the shrinkage under stepwise heating, i.e. \(R=\sum y_T t_0\), which, according to the experimental data (cf. § 5), is expressed by formula (26)

\[ R=m(T)+\sqrt{t_0}\,n(T). \]

Let the total shrinkage at the temperature \(T_1=T-\Delta T\) be

\[ \sum_{T_1} y_{T,t_0}=A_1(T_1,t_0). \]

By the time \(t\) at the temperature \(T=T_1+\Delta T\), according to (39b), we shall have:

\[ y_{T,t}+\sum_{T_1} y_{T,t_0}=A_1(T_1,t_0)+\sqrt{t}\,B(T,t_0). \]

By the end of this temperature stage, the total shrinkage will be:

\[ y_{T,t_0}+\sum_{T_1} y_{T,t_0}=A_1(T_1,t_0)+\sqrt{t_0}\,B(T,t_0). \]

It follows from this that, in turn, \(A_1(T_1,t_0)\) can be represented in the form

\[ A_1(T_1,t_0)=A_2(T_2,t_0)+\sqrt{t_0}\,B_1(T_1,t_0), \]

where \(A_2(T_2,t_0)\) is the total shrinkage by the time \(t_0\) at the temperature

\[ T_2=T_1-\Delta T. \]

Continuing this reasoning and noting that, according to (41), at each temperature

\[ B_i=P_i(T)+\frac{1}{\sqrt{t_0}}\,Q_{i+1}(T), \]

it is easy to estab-

see that the calculated value of the quantity

\[ R=\sum_{\tau} y_{\tau,t_k} \]

will be represented by the formula

\[ R(T)=m(T)+\sqrt{t_0}\,n(T), \]

in which

\[ m=\sum_{i+j} Q_i \quad \text{and} \quad n=\sum_i P_i . \]

Thus, the course of complete shrinkage is also qualitatively consistent with what is expected on the basis of the schematic description of the kinetics of relaxation developed above.

A verification of the quantitative agreement, in particular as applied to objects with distortions of the crystal lattice of different magnitudes, has not yet been carried out. It is of great interest and should contribute to a refinement and more detailed description of the microprocesses during relaxation.

Let us note here that the evaluation, practiced in a number of works \(^{10,15}\), of the temperature dependence of the sintering rate from the value of the experimentally found activation energy of the process is, in light of the foregoing, not adequate to the essence of the phenomena taking place. Indeed, the determination of the indicated activation energy is performed by plotting the logarithm of the initial shrinkage rate as a function of \(\frac{1}{T}\) and finding the angular coefficient of the resulting straight line. In doing so, it is implicitly assumed that: 1) the initial shrinkage rate is proportional to the coefficient of self-diffusion \(D\), and 2) the quantity

\[ D=Ae^{-\frac{u}{kT}}, \]

where \(u\) is the activation energy.

In reality, for a body with distortions of the crystal lattice the dependence of \(\ln D\) on \(\frac{1}{T}\) does not reflect the true value of the activation energy, since the formulas

\[ D=cD',\quad c=e^{-\frac{u_0}{kT}},\quad D'=Ae^{-\frac{\Delta u}{kT}} \]

and \(u=u_0+\Delta u\), corresponding to the equilibrium state of the crystal lattice, become invalid. As indicated above,

\[ D=\xi D' \]

(at the initial moment \(D=\xi_{\mathrm{n}}D'\)), where \(\xi\) is by no means a single-valued (and, moreover, exponential) function of \(\frac{1}{T}\).

The empirically determined value \(\tilde{u}\) can in fact correspond only to \(\Delta u\) (entering into \(D'=Ae^{-\frac{\Delta u}{kT}}\)), with some-

addend, depending on the kinetics of the change in \(\xi_n\); since the latter addend is not very large, the nonexponential character of the dependence of \(\xi_n\) on \(\frac{1}{T}\) remains graphically uncaptured, leading only to some scatter of the points and to a change in the slope of the straight line. As experience shows, for example, for pressed Cu powders there is empirically determined a value of the activation energy lying within \(\sim 1.5—2.0\ \frac{\mathrm{cal}}{\mathrm{g\text{-}mol}}\) for specimens of very different origin, exhibiting different kinetics of the process. This value is substantially smaller than the true value \(u_{\mathrm{eq}}\simeq 5.7\cdot 10^3\ \frac{\mathrm{cal}}{\mathrm{g\text{-}mol}}\), obtained for Cu specimens in the equilibrium state. But differences in the empirical values of \(u\) practically do not characterize the degree of distortion of the lattice and give no information about the magnitudes \(\xi\) that determine the kinetics of the process.

8. RECRYSTALLIZATION

Above, in considering the processes of directed self-diffusion and recovery, we disregarded the possible polycrystalline structure of the body and conducted the discussion as applied to an object having no internal boundaries, i.e., a single crystal. In reality, in a considerable majority of cases (especially applied ones), one has to deal with polycrystalline bodies composed of a large number of crystallites, at whose boundaries there is an “interphase” surface tension. It is known that in polycrystals (and, in the presence of distortions of the crystal lattice, also in single crystals), at high temperatures there occurs the so-called phenomenon of recrystallization, which amounts to a change in the sizes and number of crystallites. Since this phenomenon also proceeds by a diffusion path, it is necessary to determine what its molecular mechanism is and to what extent it affects other effects of self-diffusion in crystalline bodies.

First of all, let us note that in substances crystallizing in the cubic system, the coefficient of self-diffusion does not depend on crystallographic orientation (the coefficient of self-diffusion, like the coefficients of heterodiffusion, is a tensor of the second rank and in cubic crystals does not depend on direction). Thus, the different orientation of crystallites in a polycrystalline aggregate should not affect the magnitude of the self-diffusion coefficient (in bodies with a lattice of the cubic system). A certain change (increase) in the self-diffusion coefficient may be due to the presence of internal separation surfaces between crystallites, along which surface diffusion takes place, but this effect is substantial only at high dispersity

crystallites and low temperatures. In what follows we shall neglect it. In connection with what has been said, in a first approximation (for crystals of the cubic system, especially at high temperatures), in describing the effects of self-diffusion one may disregard the fine-crystalline structure of the body, as was done above.

Let us now turn to the phenomenon of recrystallization. Two types of this phenomenon are distinguished: 1) coalescence recrystallization, which reduces to the enlargement of crystallites and is observed in bodies with a distorted as well as an undistorted lattice; coalescence recrystallization, of course, does not occur in single crystals. 2) Recrystallization proper (or true recrystallization), manifested only in bodies that have undergone considerable plastic deformation*). Recrystallization of this kind may also lead to a decrease in the sizes of the crystals (an increase in their number). It is assumed \(^{32}\) that true recrystallization, like, for example, crystallization from the liquid phase, proceeds by the formation of “nuclei” and their subsequent growth. It is sometimes thought \(^{32}\) that the very phenomenon of true recrystallization is fundamentally due to the same causes as crystallization from another phase, i.e., to the appearance of crystals with a lower free energy (for example, free from distortions). Below we shall see that such an assumption cannot be justified.

It is simplest to indicate the molecular mechanism of coalescence recrystallization. The presence within a body of surfaces of separation between crystallites means that there exist volume elements differing in the magnitude of their chemical potential. This must inevitably lead to the appearance of differences in the equilibrium concentrations of vacancies, together with which there will arise fluxes of directed self-diffusion that reduce the concentration differences, i.e., also the initial differences in chemical potential. These fluxes will disappear only after the cause producing the difference in chemical potential, i.e., the presence of surfaces of separation inside the body, has been eliminated.

Let us consider, as an example, a simple case in which inside one large crystal there is situated another, differently oriented and having the form of a convex polyhedron (Fig. 7). At the boundary of the two crystals there must then arise a certain interphase surface tension \(\sigma_{ik}\), different on different faces and dependent on the mutual orientation of the two crystals. Mechanically, the surface tension \(\sigma_{ik}\) is equivalent to forces \(p_{ik}=l_j\sigma_{ik}\), applied parallel to the faces to the edges (\(l_j\) are the lengths of the corresponding edges). The result of compounding the forces \(p_{ik}\), acting on different faces, will be the appearance of components,

*) It is also called “recrystallization by treatment.”

perpendicular to the faces, and consequently also the pressure \(p \sim k_1 \dfrac{\sigma_{ik}}{l}\), acting on the faces. Here \(l\) is a certain linear dimension of the inner crystal, \(k_1\) is a numerical coefficient of order unity, depending on the shape of the crystal, the angle between faces, etc. In what follows we shall take \(k_1 = 1\).

Thus, the inner crystal is compressed by the pressure \(p\), while to the outer one (from its inner surface) a negative pressure \(p\) is applied.

Fig. 7.

The inner crystal, undergoing uniform compression, will have an equilibrium concentration of vacancies decreased by

\[ \Delta c = c_0 \frac{\delta^3}{kT} p. \]

In the outer crystal near the common boundary there will appear a vacancy concentration increased by \(\Delta c\). Owing to this, a flux of vacancies must be established from the outer crystal to the inner one, or a flux of atoms from the inner crystal to the outer one, equal to

\[ q \simeq 2 \frac{D' \Delta c}{\delta}. \]

In the case of an isometric inner crystal (linear dimension \(l\)), the change of its volume will be determined by the equation

\[ \frac{d(l^3)}{dt} \simeq - \frac{4D\delta^2}{kT}\,\sigma_{ik} l, \tag{42} \]

and the change of the linear dimensions by the formula

\[ l^2 = l_0^2 - \frac{4D\delta^2}{kT}\,\sigma_{ik} t. \tag{42a} \]

The inner crystal will decrease (which corresponds to a decrease of the surface energy of the system) and as it were “yield” to the outer one.

Let us note that the quantity \(\sigma_{ik}\) appearing in formula (42) is the mean value of the interphase surface tension over all boundaries. The velocity of displacement of different faces will in general be different, in accordance with the value of \(\sigma_{ik}\) present on each of them.

The example given refers to the case when only two crystals border one another. In practice, usually, many crystals border on a considerably larger number of neighbors (of order 10). On each face very diverse values of \(\sigma_{ik}\) may be obtained; moreover, edges may be common to three or more crystals.

Suppose, in particular (Fig. 8), that there is a crystal \(I\) with a number of neighbors equal to the number of faces, and that the boundaries between neighboring crystals pass through edges. Let us denote the surface tension at the mutual boundaries of neighboring crystals by \(\sigma'_{ik}\), and at the boundary of crystal \(I\) with its neighbors by \(\sigma_{ik}\).

Fig. 8.

The displacement of the boundaries of crystal \(I\) will be determined by the same equation (42), in which, however, instead of \(\sigma_{ik}\) there must appear the quantity \(\sigma'_{ik} - \sigma_{ik}\) (more precisely, \(\sigma'_{ik}\cos\alpha' - \sigma_{ik}\cos\alpha\)). For \(\sigma'_{ik} > \sigma_{ik}\) the volume of crystal \(I\) will increase; for \(\sigma'_{ik} < \sigma_{ik}\) it will decrease. Of course, there may also occur such a case when some faces of crystal \(I\) approach its center, while others move away from the center.

In light of the above, it is easy to indicate a simple rule for determining the direction of displacement of boundaries between crystals: the boundary will move from the crystal stretched by the forces of interphase surface tension toward the neighboring crystal compressed by the same forces. As is easy to verify, this will always lead to a decrease in the internal surface energy of the system. The rate of the process at each face (or near each edge) will be determined by the value of the resulting local pressure.

Although formulas (42) and (42a) apply to a very particular case, they may be used for a rough estimate of the “mean” kinetics of the recrystallization process in a polycrystalline body (determination of the mean path of displacement of boundaries and of the mean recrystallized volume).

In doing so, one must use the mean value \(\sigma_{ik}\) for different mutual orientations of the grains. As is known\(^{33}\), this quantity is approximately two orders of magnitude lower than the value of \(\sigma\)—the surface tension at the boundary with vacuum.

Comparing formulas (42) and (42a) with formula (10), we are convinced that, despite the smallness of \(\bar{\sigma}_{ik}\), the process of recrystallization (for linear dimensions of grains of the same order as the pore dimensions) must proceed much faster than the sintering process, for (42) differs from (10) by a factor of order

\[ \frac{l}{\delta}, \]

which, for \(l = 10^{-3} - 10^{-4}\ \mathrm{cm}\), exceeds \(10^{4}\). Even taking into account the fact that

\[ \bar{\sigma}_{ik} \simeq 10^{-2}\sigma \]

the recrystallization rate proves to be \(10^{2}\) times greater than the sintering rate (i.e., under conditions in which pores of diameter \(10^{-3}\ \mathrm{cm}\), for example, are “healed” by sintering, recrystallization must take place in volumes with linear dimensions \(\sim 10^{-1}\ \mathrm{cm}\)).

This conclusion immediately draws attention to the question of the kinetics of recrystallization in polycrystalline bodies with a distorted lattice, for it means that recrystallization practically precedes sintering, i.e. that sintering is carried out in a recrystallized body. It is necessary to establish what change in the rate of sintering (and of other processes of directed self-diffusion) may be caused by the occurrence of recrystallization.

As applied to processes of the diffusion type, as was already indicated above (§ 7), the main feature of a body with distortions of the crystal lattice consists in the appearance of excess vacancies, entailing an increased value of the coefficient of self-diffusion (temperatures are meant at which the removal of distortions has already occurred, as a result of which excess vacancies and interstitials have been formed; see § 7). Consequently, it is necessary to determine whether recrystallization is accompanied by a change in the number of excess vacancies.

When a process of coalescence recrystallization takes place (in a body with a distorted lattice), there is no basis for expecting a change in the number of vacancies (or interstitials), for the diffusional displacement of a boundary, as described above, cannot destroy local density inhomogeneities* consisting in the presence of excess vacancies (or excess atoms). Only when vacancies meet interstitials can their mutual annihilation occur (the annealing process). Therefore, in bodies with a crystal lattice of the cubic system (where the coefficient of self-diffusion does not depend on direction), coalescence recrystallization, affecting neither the number of vacancies present nor the kinetics of the change in their number (i.e. the annealing process), must leave the coefficient of self-diffusion and its change with time practically unaffected (we disregard changes associated with the anisotropy of the coefficient of self-diffusion in bodies of a noncubic system).

* Any neighboring atom may replace an excess vacancy; the latter, however, does not thereby disappear at all, but is merely displaced to the site previously occupied by the atom.

On the other hand, the kinetics of the recrystallization phenomenon itself will depend substantially on the magnitude of the self-diffusion coefficient [cf. (42)]. An increased value of \(D\) will lead to an acceleration of recrystallization. A fall of \(D\) in the process of recovery must, correspondingly, reduce the rate of recrystallization. Thus, it is not recrystallization (collective) that affects the recovery process and the magnitude of the self-diffusion coefficient, but, on the contrary, the latter determine the course of recrystallization.

It is more difficult to clarify the question of the relations for the process of true recrystallization. It is known from experiment that after considerable plastic deformation even relatively slight heating leads to the appearance of an increased number of crystallites of smaller size than the original ones. As has already been mentioned, this phenomenon is sometimes attributed to the fluctuation formation and growth of new crystallization centers of the “undistorted” phase. But such an explanation encounters contradictions. Namely:

1) According to the well-known regularities of the formation of crystallization centers in the solid phase (the so-called principle of dimensional and orientational correspondence of Konobeevskii–Dankov\(^{28}\)), nuclei of the undistorted phase would have to be oriented so as to create a minimum surface tension at the boundary with the original phase, i.e., practically, to be arranged parallel to the original phase. Consequently, several nuclei arising within one grain cannot form differently oriented new crystals, contrary to experiment.

2) The growth of nuclei must occur at such temperatures when the first phase of recovery, leading to the removal of distortions and the formation of excess vacancies and inclusions, has already been completed to a considerable extent (to remove one distortion requires a smaller thermal fluctuation than to form a nucleus). The residual disequilibrium of the lattice therefore consists only in the presence of the indicated inclusions and vacancies. If one assumes that the growing nucleus does not incorporate them into itself, then the rate of its growth must be limited by the process of mutual annihilation of vacancies and inclusions, ending only at very high temperatures (cf. § 7). This would practically mean that growth should not take place until recovery is completed. At the same time, the growing nucleus cannot “push” the excess vacancies and inclusions into the original phase (with a corresponding increase there in their concentration), since such a process would correspond to an increase in the free energy of the system*) (in contrast to for-

*) The change in the free energy of a body whose concentration of vacancies (or inclusions) is \(\xi\), with a good approximation, may be expressed by the formula
\[ F(\xi)=N\{\xi u_0+kT[\xi\ln\xi+(1-\xi)\ln(1-\xi)]\}, \]
where \(u_0\) is the energy of formation of one vacancy (inclusion), \(N\) is the number of atoms. As

mally similar phenomenon of decomposition of a supersaturated solid solution, where a decrease in the free energy is achieved at the expense of a decrease in the potential energy of the system).

3) The contradiction indicated in item 2) cannot be removed by the assumption that excess vacancies (and inclusions) enter the growing nucleus, for in that case the newly formed and the original phases would again not differ in lattice nonequilibrium; the growth of the nucleus of the new phase would not lead to a decrease in the volume term of the free energy and would be associated only with an increase in surface energy, which, as thermodynamics teaches, is impossible.

The remarks given show that the process of true recrystallization cannot be represented as corresponding to the growth of nuclei of an undistorted phase at the expense of an unstable and nonequilibrium initial phase; moreover, the excess vacancies and inclusions remaining after the completion of the first stage of recovery cannot in general give rise to the recrystallization effect.

The cause of recrystallization can only be other changes that arise in the crystal during plastic deformation. As is known, the appearance of lattice distortions does not exhaust the effect of structural changes in plastically deformed crystalline bodies. There also occurs a dispersion of crystalline grains, i.e., their division into finely dispersed “blocks.”

Deformed single crystals reveal bent and curved cleavage planes, along which changes in lattice orientation macroscopically appear to be continuous, which is possibly due to a regular distribution of distortions. However, X-rayographically (by the effect of the so-called “asterism” of Laue patterns), discrete differences are sometimes also found in the orientations of the blocks into which a single crystal is divided during deformation. If during deformation the blocks do not

it is known that minimization of this expression leads to the formula for the equilibrium concentration of vacancies

\[ c=e^{-\frac{u_0}{kT}}. \]

If a body with an average vacancy concentration \(\xi\) is split into two phases—one with the equilibrium vacancy concentration \(c\) (the relative amount of this phase being \(x\)) and another [its content being \((1-x)\)] with an increased concentration

\[ \xi'=\frac{\xi-cx}{1-x}, \]

then the change in free energy will be

\[ \varphi=F(\xi)-xF(c)-(1-x)F(\xi'). \]

It is easy to see that \(\varphi>0\), for, assuming the contrary, we find

\[ x>\frac{F(\xi')-F(\xi)}{F(\xi')-F(c)}. \]

Putting \(\xi=c+\varepsilon_1\), \(\xi'=c+\varepsilon_2\) and expanding \(F(\xi')\) and \(F(\xi)\) in a series in powers of \(\varepsilon_2\) and \(\varepsilon_1\) \(\left[F'(\xi)\right]_{\xi=c}=0\), which is legitimate in view of the smallness of all concentrations, we obtain (after simple transformations) \(x>1\), i.e., an absurdity. The physical meaning of the result is that nonequilibrium redistribution of vacancies lowers the entropy of the system without changing the potential energy.

are not yet completely differentiated, then after the first stage of recovery (removal of distortions) such differentiation of them must occur. Consequently, a crystal (and still more a polycrystalline aggregate), after considerable plastic deformation and low-temperature recovery, must consist of very small blocks. Neighboring blocks will be close in orientation, but the angle of rotation of the axes of the lattice can, over the extent of a single crystalline grain, reach a considerable magnitude.

Such a system, at relatively low temperatures, must exhibit a noticeable effect of collective recrystallization, since, although the interphase tension at the boundaries of the blocks is small, the sizes of the blocks are also small, and the rate of change of the linear size of a block—$l$—is given by the relation

\[ \frac{dl}{dt}\sim \frac{\sigma_{ik}}{l} \]

[cf. (42)]. As the blocks grow, the effect of their difference in orientation must increase, so that the value of $\frac{dl}{dt}$ can in practice be preserved despite the growth of $l$. When the collective recrystallization of the blocks leads to the formation of grains of substantially increased (microscopically noticeable) dimensions, the differentiation of the new grains will become quite distinct, since the differences in their orientation will prove considerable.

Thus, according to what has been set forth above, the effect of true recrystallization consists in the collective recrystallization of the blocks into which crystalline grains are divided during plastic deformation, and is not connected with the presence of lattice distortions. The latter agrees well with the known fact that true recrystallization appears only in bodies subjected to plastic deformation. Bodies containing distortions of the crystal lattice of another origin (for example, after electrodeposition or reactions of decomposition, oxidation, reduction, etc.) never exhibit recrystallization accompanied by a decrease in grain size.

The experimental explanation given above for the origin of true recrystallization has not yet been tested. Compared with the hypothesis of the formation and growth of centers of an undistorted phase, it has the advantage of internal consistency (cf. above).

As for the kinetics of the process, if for the linear size of the blocks $l$ one adopts the value $10^{-6}\ \mathrm{cm}$, which is in agreement with certain electron-microscopic data\(^{34}\), and for $\sigma_{ik}$ at the boundary of the blocks the value $\sim 10^{-1}—10^{-2}\ \mathrm{dyn}/\mathrm{cm}$, then by formula (42) we obtain a growth rate $\frac{dl}{dt}$ exceeding the rate of ordinary collective recrystallization (one should also take into account the strongly increased value of $D$ under considerable plastic deformation).

As is known, a change in the magnitude of plastic deformation affects the course of true recrystallization in the following way. With a large degree of deformation, recrystallization proceeds at low temperatures and a considerable number of comparatively small crystals is formed. With small deformation, crystallization proceeds at very high temperatures and leads to the appearance of a small number of large crystals (in the limit—to a single crystal).

According to the hypothesis of the formation and growth of centers of an undistorted phase, this is explained as the result of an increased probability of the appearance of nuclei when the lattice is substantially distorted. From the point of view of the ideas set forth above, it must be assumed that an increase in the degree of plastic deformation should contribute: a) to a higher dispersion of the blocks, i.e., to a smaller value of the linear dimension \(l\), and b) to an increase of the “mosaicity angle” \(\varepsilon\) between individual blocks.

The latter effect, apparently, is of greater importance, since the magnitude of the interphase surface tension at the boundary of blocks \(\sigma_{ik}\), according to the simplest estimate \(^{33}\), can be (in the case of cubic crystals) expressed by the formula

\[ \sigma_{ik} \simeq \sigma \left[\sum_{j=1}^{3}\cos \alpha_{ji} - \sum_{j=1}^{3}\cos \alpha_{jk}\right]^2 . \]

Here \(\sigma\) is the surface tension at the boundary with vacuum (on the crystal face most densely populated with atoms), \(\alpha_{ji}\) and \(\alpha_{jk}\) are the angles of the normal to the boundary plane between the blocks with the directions of the lattice axes in each of the blocks. For similarly oriented blocks one may take, for example, \(\alpha_{1i}=\alpha_{2i}=\frac{\pi}{2}\);

\[ \alpha_{1k}=\frac{\pi}{2}; \quad \alpha_{2k}=\frac{\pi}{2}-\varepsilon; \quad \alpha_{3i}=0, \quad \alpha_{3k}=\varepsilon, \]

which gives for \(\sigma_{ik}\) the expression \(\sigma_{ik}\sim \sigma_0 \varepsilon^2\) *); for \(\varepsilon\sim 3\cdot 10^{-3}—10^{-2}\) and \(\sigma \simeq 10^{-3}\ \text{dyn}/\text{cm}\) we obtain for \(\sigma_{ik}\) at the block boundary the above-cited value \(10^{-1}—10^{-2}\ \text{dyn}/\text{cm}\). Since at small \(\varepsilon\), \(\sigma_{ik}\) depends on \(\varepsilon\) quadratically, the influence of a change in angle must be more significant than the effect of a change in \(l\).

When the phenomenon of recrystallization is used for growing single crystals, the body is first subjected to a small plastic deformation (several percent). It is possible that collective recrystallization of the blocks does not take place in this case, and only the usual collective recrystallization of whole grains occurs. The favorable influence of preliminary plastic

*) In article \(^{33}\) an analogous formula was erroneously given in the form \(\sigma_{ik}\sim \sigma_0 \varepsilon^4\); this is an obvious misprint.

deformation is due only to the effect of an increase in the coefficient of self-diffusion.

Let us note one interesting consequence to which the ideas set forth above concerning the mechanism of coalescence recrystallization lead. If external forces causing elastic deformations are applied to a polycrystalline body in which a recrystallization process is taking place, the rate of the process should change. Indeed, owing to the different orientation of the grains and the anisotropy of the elastic constants, besides the average elastic stresses, which are the same in neighboring grains, there will also appear local stresses (which corresponds to a different elastic deformation of the free grains under the action of identical average stresses). This means that, at the boundaries between grains, in addition to the interphase capillary pressure \(\frac{\sigma_{ik}}{l}\), there will appear an additional pressure (proportional to the mean applied stress \(p\), and the greater the more significant the difference in the orientation of the grains and the effect of elastic anisotropy). For \(p\) sufficiently exceeding \(\frac{\sigma_{ik}}{l}\), the recrystallization process may be substantially accelerated (or slowed down). An obstacle to increasing \(p\) may be the crossing of the elastic limit and the onset of plastic deformation*).

9. SINTERING AND CREEP OF AMORPHOUS BODIES

As is clear from the preceding, the kinetics of the phenomena of viscous flow (creep and, in particular, sintering) in crystalline bodies is complicated by: a) the nonconstancy of the coefficient of viscosity \(\eta\) (its dependence on certain characteristic dimensions of the body) and b) in bodies with distortions of the crystal lattice, by the phenomenon of recovery, which causes changes in the coefficient of self-diffusion.

The indicated conditions do not apply in the case of amorphous bodies. In § 3 it was already mentioned that, in the case of amorphous bodies, Frenkel’s formula must be fully valid

\[ \frac{1}{\eta}=\frac{D\delta}{kT}. \]

*) Plastic deformation will also contribute to accelerating the process (increase of the value of \(D!\)), but it may lead to recrystallization of blocks, i.e. to an increase in the number of crystalline grains (cf. above in the text). If the body is under all-round compression, the pressure may be brought to very large values without plastic deformation arising; but in this case there will be no acceleration of recrystallization, since the deformation of all grains under all-round compression, regardless of their orientation, will be the same, and no additional pressure differences at the grain boundaries will arise.

In this case, in view of the absence here of any nonequilibrium states that could be caused by mechanical actions, the quantity \(D\) is a constant at a given temperature, at least for simple bodies (in alloys, nonequilibrium states corresponding to nonuniform concentration are still possible). The kinetics of the phenomena of diffusional creep (viscous flow) in amorphous bodies must therefore obey the simple formulas established by Frenkel.

Experimental verification of the formulas of Frenkel’s theory of viscous flow was carried out as applied to amorphous objects for two cases: 1) the sintering of spherical particles to a plane\({}^{35}\) and 2) the filling-in of cylindrical holes in tubes\({}^{23}\). In both cases glass was used as the material.

Experiments on the sintering of particles to a plane\({}^{35}\) fully confirmed the following formula from Frenkel’s theory:

\[ y^2=\frac{\sigma a}{\eta}\,t \]

(\(y\) is the diameter of the circle along which sintering took place, \(a\) is the radius of the sphere, \(\sigma\) is the surface tension). The experimentally found values of \(\eta\) were plotted on a logarithmic scale as a function of \(\frac{1}{T}\).

This yielded a straight line (with some deviation toward increased values of \(\ln \eta\) at large \(T\), i.e. small \(\frac{1}{T}\)). The values of \(\eta\) found have the correct order of magnitude.

Studies of the kinetics of the filling-in of cylindrical holes were carried out on glass capillaries. During heating the capillaries were arranged horizontally and, in order to avoid bending under the action of gravity, were subjected to continuous rotation about the axis. The following formula from Frenkel’s theory for the rate of change (decrease) of the capillary radius was quite well confirmed:

\[ \frac{dr}{dt}=-\frac{2}{3}\frac{\sigma}{\eta}. \]

For \(\eta\) an exponential dependence on \(\frac{1}{T}\) was likewise found.

Thus, it may be considered that, as applied to the case of amorphous bodies, the theory of viscous flow in the form developed by Ya. I. Frenkel agrees quite satisfactorily with experiment and is exhaustive*).

*) Incidentally, let us note that the good agreement of the theory of bulk viscous flow in amorphous bodies with experiment (where the effects are not complicated by distortions of the lattice) removes the “basis” for the allegedly decisive role of surface diffusion in sintering phenomena (cf. \({}^{15}\)).

10. ON THE CONNECTION BETWEEN HETERODIFFUSION AND SELF-DIFFUSION PHENOMENA IN CRYSTALLINE SOLIDS

The close connection between the effects of hetero- and self-diffusion is due to the fact that both phenomena are carried out by means of one and the same mechanism: the successive replacement of vacancies by atoms in the crystal lattice.

Without discussing here in detail the relations between the two phenomena, we shall point only to some recently discovered facts concerning heterodiffusion in inhomogeneous porous bodies.

Fig. 9a. Specimen made of six Cu wires and one Ni wire (in the center) after sintering at \(1040^\circ\mathrm{C}\) for 20.5 hours in an \(\mathrm{H}_2\) atmosphere.\(^3\)

If a porous body is made, for example, of densely packed wires of different metals (forming solid solutions with one another) and heated to high temperatures, then it is possible to follow the processes occurring in the body of two types: a) sintering (removal of pores from the body by self-diffusion) and b) equalization of concentration (heterodiffusion processes).

In particular, in the case of a body composed of Cu and Ni wires, microscopic observations established\(^3\) that the diffusion processes develop here in the following way. First, heterodiffusion processes begin, proceeding practically unipolarly—copper diffuses into nickel, whereas nickel does not diffuse into copper. The diameter of the nickel wires increases; the copper wires retain their diameter, but become porous (Figs. 9a, 9b). Only after a considerable increase in the diameter of the nickel wires and the parallel development of porosity in the (formerly solid) copper-

... in wires, sintering of the body as a whole (i.e., its densification) is observed. At the very initial stage in Cu wires, a phenomenon occurs that is, as it were, opposite to sintering (the appearance of new pores, rather than the overgrowth of existing ones).

The effects described are explained simply if the diffusion phenomena taking place here are regarded as proceeding by the replacement of vacancies in the crystal lattice by atoms.

Fig. 96. Specimen made of six Ni wires and one Cu wire (in the center) after sintering at 1040° C for 20 hours in an H₂ atmosphere.^3

Indeed: 1) practically only diffusion of Cu into Ni is observed. To clarify the reason why the diffusion has a unipolar character, let us compare mentally two processes: a) the transfer of \(n\) Cu atoms into Ni and b) the transfer of \(n\) Ni atoms into Cu. In both cases, the displacement of atoms must be accompanied by the formation of \(n\) vacancies in the phase from which the atoms have departed. The corresponding change in the entropy of the system will be the same for processes a) and b)^*. But the change (increase) in potential energy will be smaller in case a), since the work of removing a Cu atom from the lattice of this metal is less than the corresponding quantity for a Ni atom (the latent heat of evaporation of Cu is less than the latent heat of evaporation of Ni). Consequently, thermal fluctuations leading to process a) must occur much more often. 2) The departure of atoms

^*) Likewise the same is the change in potential energy associated with the formation of a solution of Ni in Cu or Cu in Ni.

The introduction of Cu into the Ni lattice is accompanied by the appearance of excess (above the equilibrium number) vacancies in Cu. When such vacancies meet, they will coalesce, since this leads to a decrease in the free energy of the system. When the number of excess vacancies in Cu becomes large (it increases because the self-diffusion coefficient of Cu is smaller than the coefficient of heterodiffusion of Cu in Ni, so that the excess vacancies do not have time to leave Cu), a significant fraction of the vacancies coalesces, forming pores.

Thus, the appearance of porosity in Cu wires is simply the precipitation of excess vacancies from their supersaturated solution in Cu.

Let us emphasize that no other conceptions of the mechanism of diffusion (the exchange-of-places hypothesis, etc.) are capable of explaining the fact of the appearance of porosity in the phase from which atoms depart during unipolar diffusion. In this connection, the experiments noted above may be regarded as direct experimental proof of Ya. I. Frenkel’s theory of the vacancy mechanism of diffusion processes.

11. CONCLUSION

The review given above is far from exhaustive. Nevertheless, it testifies to the significant development, in recent times, of ideas concerning the processes of directed self-diffusion in crystalline bodies and the connection of these processes with a variety of phenomena. In part, there is already experimental verification of the indicated ideas, although much still remains to be done to elucidate certain dependences.

It must be noted that the successes achieved in this field are due in large measure to the fruitful ideas of Ya. I. Frenkel, in particular his theory of self-diffusion, which is finding ever new, previously unforeseen applications and confirmations.

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1. In the literature the term “return” is sometimes used to denote the phenomenon of recovery; this is highly inappropriate, since “return” is customarily used for the restoration of mechanical properties upon slight heating after ageing in alloys. The phenomenon of return is caused by the reverse transition into solid solution of precipitated colloidal-size inclusions, i.e., it is connected with processes that have nothing in common with recovery. ↩