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Theory of diffusive decomposition of supersaturated solid solution under the condition of simultaneous operation of several mass-transfer mechanisms
THEORY OF DIFFUSIVE DECOMPOSITION OF SUPERSATURATED SOLID SOLUTION UNDER THE CONDITION OF SIMULTANEOUS OPERATION OF SEVERAL MASS-TRANSFER MECHANISMSt V. V. SLEZOV, V. V. SAGALOVICHand L. V. TANATAROV Khar’kov
Physicotechnical
Institute of the Academy of Sciences of the Ukrainian,
(Received 18 July 1977: Accepted
29 October
U.$!jK.
1977)
kinetics of diffusive decomposition of a supersaturated solid solution under the condition of simultaneous operation of several mass-transfer mechanisms are developed. It is shown that the expressions for the growth rate of the precipitate and for the size distribution function have one peak. The location of these peaks, which slowly changes with time, can be considered a slowly changing parameter.
~ktrrt-_The
Sometimes the key factor determining the particle growth is the formation rate of the chemical bond between the particle and the atom that was brought to it by one of the mass-transfer mechanisms. Since the particle tends to assume the shape corresponding to the lowest surface energy, any accumulation of atoms produces a strong, balancing surface diffusion flow that tends to minimize the surface energy. We may assume that the atoms are uniformly distributed along the surface and that their concentration c, which is different from the equilibrium concentration CR. is the same for any mass-transfer mechanism. As a result, we must replace CR by c’ in all the expressions for J, in Refs. [ 1,4,5]. Thus we obtain the following expression for the flow density determined by the volume diffusion:
tWRGOUCIlON
The kinetics of diffusive decomposition of supersaturated solutions involving only one mass-transfer mechanism were developed in Refs. [l-S]. In a real situation, however, several such mechanisms can be involved simultaneously. Thus, for example, a transfer of mass always occurs along the gram boundaries and dislocation lines concurrently with the volume diffusion. It should be noted that all these mechanisms operate concurrently in all the new-phase particles, because only the particles with the fastest growth rate survive after undergoing diffusive interaction at a later stage of decomposition (coalescence). In this paper we investigate the kinetics of decomposition of a solution when several mass-transfer mechanisms are operating concurrently. l.TEECEowmRATEoFAPARllcLe IN THE PrlEmrATE
I,=+‘).
Generally, the growth rate of a spherical particle is related ‘to the density of the diffusion flow by dR dr=
7
(2)
where E is the concentration at some distance from the inclusion. If the diffusion initially starts in the bulk of the material, spreads to the boundary, and then propagates along the boundary crossing the inclusion, then the flow is given by [4]
Jr = I,
where J, is the density of the flux of atoms to the particle, which is determined by the ith mass-transfer mechanism. The equations for the flux, which are given in Refs. [I, 4.51. were obtained by solving the quasi-stationary diffusion equation under quasi-equilibrium conditions at the particle’s surface:
Jz=
jfz ND, 2
In
(2C,R)
(E - c’),
(3)
where CI is the lattice constant, D, is the coefficient of the diffusion along the surface, N is the number of surfaces crossing the inclusions, and
cl,-Jr = CA,
2 _
c s
where CR is the equilibrium concentration at the boundary. Because of the nonstationary conditions and the motion of the boundary, the corrections obtained are negligible and quadratic relative to supersaturation which is presumed to be small.
DJ’a
20 ’
where I’ is the average diameter of the object. The mass transfer, a two-stage diffusion, can also occur along the dislocations. The flow density corresponding to this diffusion is[S]
s3=;~(t-c’).
tgreprint No. 390(1977). 705
M-l
v. v.
706
SLEZOV
where I is the average length of the dislocation line (one section of the dislocation network). If the diffusion occurs predominantly along the dislocation lines directly connecting the neighboring inclusions, then a direct mass transfer will occur along these lines. In this case the flow density is given by ,, _
NnsoD, 3Qo“3cc _
c,)
4nR=R ( 4r >
where NH is the number of lines entering into the inclusion, So is the cross section of the dislocation tube through which the mass transfer takes place, R is the average inclusion radius, and Go is the relative excess quantity of the material per unit volume. Summing the expressions for the flow densities, we obtain
,JWO R
-
(c - c’),
where
W/(4) 2DIn (2C/R)+d(8Dln(//o) (5) The right-hand side of D(R) can have other terms proportional to (l/R)” (n = 2.3,. . .), which are determined by other mass-transfer mechanisms. Of importance is the fact that when R+aD(R)+D and all the flows are proportional to E- c’. ’ Under quasi-stationary conditions the total flow density I must be equal to the number of bonds formed per unit time. In the linear approximation from the deviation from the equilibrium concentration the last value can be written as k(c’- cn), where k is the rate of formation of the bond. Thus we can write D(R)
R
(c - c’) = k(c’ - CR).
By taking c’ from eqn (6) and substituting it in eqn (4), we can write eqn (I) as follows:
where
D(R) Den = I t (D(R)/kR)’
When k+mD.dR)+D(R). When k-+0, the D(R) coe.fficientdrops out from the equation for the growth rate and we have the so-called diffusionless growth process in which the kinetics are determined by the formation rate of the chemical bonds. Notice that the expressions for the growth rates for different mass transfer mechanisms can be distinguished by their coefficient and the exponent of l/R in front of the factor A - (a/R). This makes it possible to write the general expression for the growth rate determined by one mass-transfer mechanism as follows: (n > 2).
(A-$.)
For a single mass-transfer mechanism eqn (6) corresponds to the boundary condition of the third kind.
(8)
It can be expressed in terms of the equations given above when the parameters n and D,a”-’ are assigned specific values. We obtain eqn (2) when n = 3 and 4 = D and eqn (3) when n = 4 and D.a = (ND,a/Z In (26/R&). etc. So far we examined the new-phase inclusions having a spherical shape, because the physical properties of the inclusion and the matrix are isotropic. Generally, the shape of the inclusions is not spherical in the presence of anisotropy. However, if all the dimensions of the inclusion change with time in such a manner that their ratio remains constant (the shape is preserved), then we can introduce the characteristic size
The growth rate uR denotes the time derivative of the parameter R and is a function of c - cR. R, and the time 1: OR =
dR x
=
f(C
-
CR,
R, 1).
Here CR is the equilibrium concentration for the given size R at which uR vanishes, i.e. f(0, R, t) = 0. If t S to= R2/D and the supersaturation A = c - CR-+ 1 (these conditions characterize the later stage of the decomposition process known as coalescence when almost all the material from which the grains of the new phase are produced is located in the inclusions), then the time dependence in the expression uR can be ignored because the characteristic time of the variation in size R is tn - t,,/A B to. In fact, if we have volume diffusion and spherical grains
(7)
As is well known[l], cR = c-t (u/R). where cm is the equilibrium concentration at the plane boundary, Q = (Zp/kT)c,, w = a’, and y is the surface-tension coefficient. By introducing the supersaturation A = E - c,, we can rewrite the expression for the growth rate dR/dr as follows:
!+v
er al.
$=:A,
i.e.
R2-DAtR,
where R= tR-a=x.
to
If the diffusion occurs on the surface, according to eqn (3) ‘p-a$A
and
R-‘--aAD,tR
Theory of diffusive decomposition of supersaturated
Since A = c - CRis small, we can write f(C
-
CR, R)
=
A(R) *(C
-
CR).
A(R) is a composite function whose asymptotic behavior at R
A(R)=+(o+;+.
j.
The constants (I, b, . . . are associated with the tensor components of the diffusion coefficients, of the surfacetension coetlicients, and the stress tensor produced as a result of the growth of a new-phase grain. Equation (5) is used to determine a and b for the isotropic case. The value of R. is determined from the exact solution of the diffusion equation for the anisotropic case. As can be seen from eqn (7). for Den in the isotropic case
solid solution
707
considered stationary (effects such as the motion of pores induced by temperature gradients are not taken into account), and only the dissolved atoms can be in motion. We assume that the excess quantity of the new-phase material Qc,is small, i.e. Q,,* I. Therefore the collision integral in the kinetic eqn (9), which is of the order of Qo2, can be ignored[3]. This approximation, which takes into account only the self-consistent diffusion field, is usually called “hydrodynamic approximation.” Note that eqn (10) whose right-hand side has the function A(R) instead of the factor D/R distinguishes the set of eqns (9-11) from that for the one mass-transfer mechanism. In Refs. [I, 4, S] a dimensionless time suitable for the mass transfer mechanism under consideration was used in solving eqs (9-11). This cannot be done now because all the mechanisms are “working” simultaneously. Since at sufficiently long times the mass transfer is diffusion attributable primarily to the volume mechanism, we chose dimensionless canonical variables T and u corresponding to the canonical form of this mechanism:
RI=;,
u=$,
R,,=;.
k
Rko=$. (12)
k = A(r)],-o This equation can also be used in the anisotropic case. 2 mNDAMENTAL
EQu.4TloNs AND THm
soLtJTtoN
The equations describing the diffusive disintegration of the super-saturated solution are the continuity equation for the size distribution function of particles at a given time f(R, t), the conservation law for the excess material which produces the new phase in the solution and grains, and the equation for the growth rate of the grain [ I]:
dR uR = z = A(R)(E - CR)
Let us assume that the mass transfer is accomplished through the vo!ume diffusion and through the diffusion along the surface of the object and along the dislocation lines. In this case,
A(R)=-
R ’
where
1
OD: ND, Mx4DD.v) D(R) = D [ ’ + % ’ D = 2 In (2C/Ro) + d/(8 In (l/a))’ Using the variables[l2] follows:
eqns (9-l I) can be written as
Qo=A+~$~-R3f(R.r)dk. Here o, is the volume per atom of pure solvent, o is the volume of a single atom in the precipitate, Q0 is the relative excess quantity of the substance in the solution [ I] and A is the supersaturation of the solution at the time r. The right-hand side of eqn (9) should be equated to the collision integral which describes the direct diffusion interaction of two inclusions separated a short distance from one another 131. Moreover, we are considering a spatially homogeneous problem. In the inhomogeneous case, however, the left-hand side of eqn (9) should have a dependence on the space coordinates and a corresponding derivative[8]. In this case the balance eqn (I I) is replaced by the diffusion equation with sources and sinks [8]. Since in this paper we examine only the solid solutions, the macroscopic new-phase precipitates can be
l-ke-r13=&peT
Qo
_ fo(u)u’(u,r)du
I UocT)
where f,,(o) is the initial size distribution function of the new-phase grains u = R/Rko, u(u, T) is the solution of eqn (13) with the initial condition u(u, T)]~-~ = u, A,, is the initial supersaturation &= (4n/3)(R:dQ,), u,,(v) is the root of the equation u(u, T) = 0: i.e. udr) is the initial size of the grain dissolved at the time 7, and D3a
B=DR~’
~=3aD~.
dr
(14)
A detailed analysis in Refs. [I, 21 shows that to satisfy condition (11) the Y(T) and r3(~) functions for large T’S
708
V. V. SLuov
should vary in such a way that at each instant of time the g&r) plot, just as the T and Y functions, would tie below the u axis and in zeroth approximation would touch it at one point, which we shall henceforth call the blocking point [ 11. The tangency condition reduces to the following equations:
et ul.
dependence of Rk on t. In fact, u. is a function of fi and 6 is related to Rk through tqn (14). Substituting (18) for uo in eqn t 16) and expressing #I through R* according to (141,we obtain a differential equation connecting R and 1. The solution of this equation, which has separable variables. yields a rather compkx dependence of & on t:
B
t _ 27 &93(o) 16 aD
I
dx (I/( 1+ (1419)xf x’) - x - (7(9)][t/( I + (1419)x+ x2) - x + 1J3 7 [~/(tt~1419)x+x2~-x-(1/3)1'
BtOMRdR~b
When t is large, we have
i gtu.TLug=O lwu, 7)
au
i Denoting
I“_“o=o*
Rk +at, consistent with the volume di!Mon mechanism. If B(O)* 1, but the times are such that p(ONRkdR)b 1, we can easily obtain the dependence
V(U,T)“Y(T) 1+! I ( > we obtain from the first equation:
&‘- R:o = (+kat. y&,7)
= uo’ uo- 1
(15)
and from the definition r(7): ~=3~(,+~)~.
0
( 16)
The second equation, together with eqn (15). reduces to ud3 - 2uo) = (3uo - 4)@.
characteristic of the mass transfer along the surface. The dependence of Rd, on t is cumbersome because of the difficulty of expressing u. in terms of fl and using the exact eqn ( 18). We introduce in eqn (17) a new unknown variable x = 3 - ~0, which can be expressed as follows:
(17)
Solving this equation for uo, we obtain This equation can be replaced by a more simpfe equation uo=~(l-B)iJ(~(I-8).+28)
(18)
We can represent #I as /3(r) = @(OMR~RL), but according to (12) R&R, = eVd3,i.e. /S = B(O)tmd3 and 3 uo= ;5f t - /?fO)eed3~(~(1-~(O}e-“‘)2+28(0~e-~3).
B x=6(8+1) which can be used for small and large values of /3. For u. we obtain MO+;&]
(1%
It can be seen from eqn (18) that when T--, m (B --, 0) 3 uo-‘-. 2 When fl -+VJwe obtain from the same equation u,, = 4/3. #I = =corresponds to the case D = 0, i.e. purely surfsee diffusion. By rewriting eqn (17) in the form B
_ un(3 - 2Uol 3uo-4
we can see tbat uo is in the range&Z) bcuuse fl is positive. It can be shown that for all the values of it0 lying in the range (j, $ the derivative duo>() dr ’ i.e. the u&r) function, which is an increasi~ function, varies from its initial value u&). which is also in this range, to the value 312 when T*=, Relation (16) makes it possible to determine the
When~=O~o=3/2andwhen~~~uo=4/3. We substitute u. into eqn (16) and replace in it & by fi according to eqn (14). Thus we obtain a differential equation that can be solved as follows: B(O) dx (x t 9/S)’ I p (Xt l)‘(x +(3/2)Xx’ + (7/3)x + (3/2) 0 =+LM(-$* When /3 * 1, expressing @ in terms of R* we obtpin R+IlL characteristic of the mass transfer through volume diffusion. ~n~~l,weobt~n Rk4- R:o= ($ht.
Theory of diffusive decompositionof supersaturatedsolid solution characteristic of the mass transfer through surface diffusion. To determine the kinetics of motion of the blocking point for an arbitrary number of mass-transfer mechanisms. we must determine its location when the growth rate is described by eqn (8). In the variables
we obtain du” -&=y(?Hu-I)-u” RI” 7=In--;;-, R 10
y(r) = D&u”-‘n
-&.
(20)
Ir
UC@=-
.!,
w40)=~~
(II > uo)
g(u, uo) is determined by the growth rate of the particle. If the surface mass-transfer mechanism and the volume mass-transfer mechanism are in operation, ~(u, uo) can be determined from eqn (13) and udr) from (IS) or (19). By determining the change in ud7) more precisely, we can obtain an additional term in the expression for +J(u.7). which is close to unity because of the small derivative dudr)/dT. We now determine the number of peaks in the distribution function over the variable u for a given 7 by solving the following equation with respect to u api(u, 7) _ o
au
For the blocking point we obtain
709
.
If there is only one mass-transfer mechanism, we have:
@a21
ag au= I.
and for y(r): n”
y(r)=(n_l)“-I=nuo
n--l
.
If z = u/u0 is sued instead of II. then eqn (20) becomes universal dZ”
~=-(Z-l)‘(~~(n-/~Zf~‘)
(nb2).
(22)
It follows from eqn (21) that when many mass-transfer mechanisms are involved simultaneously the blocking point can be only in the interval 1-z u. < 2. Since the average radius R increases with time, the relative contribution of the volume mechanism increases and becomes dominant at large times and the blocking point approaches 3/2. The growth rate of the inclusion is proportional to the sum of the flow densities for the different mechanisms and each one of them in the dimensionless coordinates is analogous to the right-hand side of eqn (22); i.e. it is essentially a negative value. Therefore, u. approaches monotonically the point 3/2; i.e. in fact, II,, lies in a much narrower range depending on the initial conditions: either I c u. < 312 or 312< u. < 2. Since the variation of u. is very small. its time derivative can be ignored in zeroth approximation if the characteristics of the continuity equation are determined. The solution is the same as that for the volume diffusion because T was chosen for this mechanism: A _,-1++*(u.
Pi@, 7) =
L
a1
s(a uo) 0
(u
<
uo)
’ (u>uo)
Because the left-hand side of the last equation decreases monotonically, it has only one time-independent root corresponding to one peak of the cp function. This has a simple physical explanation: the distribution peak is attributable to the presence of a single blocking point. For the same reason there is only one peak when several mass-transfer mechanisms are in operation, because there is only one blocking point, although in this case it is not stationary with respect to time. Several peaks may be observed in the distribution function as a result of experimental preparation of the samples. This is attributable to spatial inhomogeneity of the solutions investigated, which results in a mass Gansfer in different sections via different mechanisms. A composite curve with several peaks can be obtained by superimposing the distribution curves. Apparently this was done in Ref. 17) where two peaks were observed. I. Lifshitz I. M. and Slerov V. V.. I.E.T.P. X,2 (1958). 2. Lifshitz 1. hi. and Skzov V. V., 1. Phys. Chem. So/ids 19.35 (l%l). 3. Lifshitz I. M. and Skzov V. V.. Fiz. Tucrd. Te/a (Soviet. 1. Sol. Slate Phys. 1, 1402 (1959). 4. Skzov V. V.. Fir. Tuenf. Trio (So&i 1. Sol. State Phys.) 9,4 (190. 5. Slezov V. V. and Levi0 D. M., Fir. Tuerd. Te/a Stute Phys.) 12,6 (1970).
(SovietJ. So/.
6. Kosevich A. M.. Skzov V. V. and Saralidze2. K., J.ET.P. 52, 1079(lW7). 7. Swift W. hf., M&lug. Trans.4. I53 (1973). 8. Skzov and Ryabukhin V. I., Atomnaya Energiya (Soviet 1. Afom. Energy)II, I (1976).
Physicotechnical
Institute of the Academy of Sciences of the Ukrainian,
(Received 18 July 1977: Accepted
29 October
U.$!jK.
1977)
kinetics of diffusive decomposition of a supersaturated solid solution under the condition of simultaneous operation of several mass-transfer mechanisms are developed. It is shown that the expressions for the growth rate of the precipitate and for the size distribution function have one peak. The location of these peaks, which slowly changes with time, can be considered a slowly changing parameter.
~ktrrt-_The
Sometimes the key factor determining the particle growth is the formation rate of the chemical bond between the particle and the atom that was brought to it by one of the mass-transfer mechanisms. Since the particle tends to assume the shape corresponding to the lowest surface energy, any accumulation of atoms produces a strong, balancing surface diffusion flow that tends to minimize the surface energy. We may assume that the atoms are uniformly distributed along the surface and that their concentration c, which is different from the equilibrium concentration CR. is the same for any mass-transfer mechanism. As a result, we must replace CR by c’ in all the expressions for J, in Refs. [ 1,4,5]. Thus we obtain the following expression for the flow density determined by the volume diffusion:
tWRGOUCIlON
The kinetics of diffusive decomposition of supersaturated solutions involving only one mass-transfer mechanism were developed in Refs. [l-S]. In a real situation, however, several such mechanisms can be involved simultaneously. Thus, for example, a transfer of mass always occurs along the gram boundaries and dislocation lines concurrently with the volume diffusion. It should be noted that all these mechanisms operate concurrently in all the new-phase particles, because only the particles with the fastest growth rate survive after undergoing diffusive interaction at a later stage of decomposition (coalescence). In this paper we investigate the kinetics of decomposition of a solution when several mass-transfer mechanisms are operating concurrently. l.TEECEowmRATEoFAPARllcLe IN THE PrlEmrATE
I,=+‘).
Generally, the growth rate of a spherical particle is related ‘to the density of the diffusion flow by dR dr=
7
(2)
where E is the concentration at some distance from the inclusion. If the diffusion initially starts in the bulk of the material, spreads to the boundary, and then propagates along the boundary crossing the inclusion, then the flow is given by [4]
Jr = I,
where J, is the density of the flux of atoms to the particle, which is determined by the ith mass-transfer mechanism. The equations for the flux, which are given in Refs. [I, 4.51. were obtained by solving the quasi-stationary diffusion equation under quasi-equilibrium conditions at the particle’s surface:
Jz=
jfz ND, 2
In
(2C,R)
(E - c’),
(3)
where CI is the lattice constant, D, is the coefficient of the diffusion along the surface, N is the number of surfaces crossing the inclusions, and
cl,-Jr = CA,
2 _
c s
where CR is the equilibrium concentration at the boundary. Because of the nonstationary conditions and the motion of the boundary, the corrections obtained are negligible and quadratic relative to supersaturation which is presumed to be small.
DJ’a
20 ’
where I’ is the average diameter of the object. The mass transfer, a two-stage diffusion, can also occur along the dislocations. The flow density corresponding to this diffusion is[S]
s3=;~(t-c’).
tgreprint No. 390(1977). 705
M-l
v. v.
706
SLEZOV
where I is the average length of the dislocation line (one section of the dislocation network). If the diffusion occurs predominantly along the dislocation lines directly connecting the neighboring inclusions, then a direct mass transfer will occur along these lines. In this case the flow density is given by ,, _
NnsoD, 3Qo“3cc _
c,)
4nR=R ( 4r >
where NH is the number of lines entering into the inclusion, So is the cross section of the dislocation tube through which the mass transfer takes place, R is the average inclusion radius, and Go is the relative excess quantity of the material per unit volume. Summing the expressions for the flow densities, we obtain
,JWO R
-
(c - c’),
where
W/(4) 2DIn (2C/R)+d(8Dln(//o) (5) The right-hand side of D(R) can have other terms proportional to (l/R)” (n = 2.3,. . .), which are determined by other mass-transfer mechanisms. Of importance is the fact that when R+aD(R)+D and all the flows are proportional to E- c’. ’ Under quasi-stationary conditions the total flow density I must be equal to the number of bonds formed per unit time. In the linear approximation from the deviation from the equilibrium concentration the last value can be written as k(c’- cn), where k is the rate of formation of the bond. Thus we can write D(R)
R
(c - c’) = k(c’ - CR).
By taking c’ from eqn (6) and substituting it in eqn (4), we can write eqn (I) as follows:
where
D(R) Den = I t (D(R)/kR)’
When k+mD.dR)+D(R). When k-+0, the D(R) coe.fficientdrops out from the equation for the growth rate and we have the so-called diffusionless growth process in which the kinetics are determined by the formation rate of the chemical bonds. Notice that the expressions for the growth rates for different mass transfer mechanisms can be distinguished by their coefficient and the exponent of l/R in front of the factor A - (a/R). This makes it possible to write the general expression for the growth rate determined by one mass-transfer mechanism as follows: (n > 2).
(A-$.)
For a single mass-transfer mechanism eqn (6) corresponds to the boundary condition of the third kind.
(8)
It can be expressed in terms of the equations given above when the parameters n and D,a”-’ are assigned specific values. We obtain eqn (2) when n = 3 and 4 = D and eqn (3) when n = 4 and D.a = (ND,a/Z In (26/R&). etc. So far we examined the new-phase inclusions having a spherical shape, because the physical properties of the inclusion and the matrix are isotropic. Generally, the shape of the inclusions is not spherical in the presence of anisotropy. However, if all the dimensions of the inclusion change with time in such a manner that their ratio remains constant (the shape is preserved), then we can introduce the characteristic size
The growth rate uR denotes the time derivative of the parameter R and is a function of c - cR. R, and the time 1: OR =
dR x
=
f(C
-
CR,
R, 1).
Here CR is the equilibrium concentration for the given size R at which uR vanishes, i.e. f(0, R, t) = 0. If t S to= R2/D and the supersaturation A = c - CR-+ 1 (these conditions characterize the later stage of the decomposition process known as coalescence when almost all the material from which the grains of the new phase are produced is located in the inclusions), then the time dependence in the expression uR can be ignored because the characteristic time of the variation in size R is tn - t,,/A B to. In fact, if we have volume diffusion and spherical grains
(7)
As is well known[l], cR = c-t (u/R). where cm is the equilibrium concentration at the plane boundary, Q = (Zp/kT)c,, w = a’, and y is the surface-tension coefficient. By introducing the supersaturation A = E - c,, we can rewrite the expression for the growth rate dR/dr as follows:
!+v
er al.
$=:A,
i.e.
R2-DAtR,
where R= tR-a=x.
to
If the diffusion occurs on the surface, according to eqn (3) ‘p-a$A
and
R-‘--aAD,tR
Theory of diffusive decomposition of supersaturated
Since A = c - CRis small, we can write f(C
-
CR, R)
=
A(R) *(C
-
CR).
A(R) is a composite function whose asymptotic behavior at R
A(R)=+(o+;+.
j.
The constants (I, b, . . . are associated with the tensor components of the diffusion coefficients, of the surfacetension coetlicients, and the stress tensor produced as a result of the growth of a new-phase grain. Equation (5) is used to determine a and b for the isotropic case. The value of R. is determined from the exact solution of the diffusion equation for the anisotropic case. As can be seen from eqn (7). for Den in the isotropic case
solid solution
707
considered stationary (effects such as the motion of pores induced by temperature gradients are not taken into account), and only the dissolved atoms can be in motion. We assume that the excess quantity of the new-phase material Qc,is small, i.e. Q,,* I. Therefore the collision integral in the kinetic eqn (9), which is of the order of Qo2, can be ignored[3]. This approximation, which takes into account only the self-consistent diffusion field, is usually called “hydrodynamic approximation.” Note that eqn (10) whose right-hand side has the function A(R) instead of the factor D/R distinguishes the set of eqns (9-11) from that for the one mass-transfer mechanism. In Refs. [I, 4, S] a dimensionless time suitable for the mass transfer mechanism under consideration was used in solving eqs (9-11). This cannot be done now because all the mechanisms are “working” simultaneously. Since at sufficiently long times the mass transfer is diffusion attributable primarily to the volume mechanism, we chose dimensionless canonical variables T and u corresponding to the canonical form of this mechanism:
RI=;,
u=$,
R,,=;.
k
Rko=$. (12)
k = A(r)],-o This equation can also be used in the anisotropic case. 2 mNDAMENTAL
EQu.4TloNs AND THm
soLtJTtoN
The equations describing the diffusive disintegration of the super-saturated solution are the continuity equation for the size distribution function of particles at a given time f(R, t), the conservation law for the excess material which produces the new phase in the solution and grains, and the equation for the growth rate of the grain [ I]:
dR uR = z = A(R)(E - CR)
Let us assume that the mass transfer is accomplished through the vo!ume diffusion and through the diffusion along the surface of the object and along the dislocation lines. In this case,
A(R)=-
R ’
where
1
OD: ND, Mx4DD.v) D(R) = D [ ’ + % ’ D = 2 In (2C/Ro) + d/(8 In (l/a))’ Using the variables[l2] follows:
eqns (9-l I) can be written as
Qo=A+~$~-R3f(R.r)dk. Here o, is the volume per atom of pure solvent, o is the volume of a single atom in the precipitate, Q0 is the relative excess quantity of the substance in the solution [ I] and A is the supersaturation of the solution at the time r. The right-hand side of eqn (9) should be equated to the collision integral which describes the direct diffusion interaction of two inclusions separated a short distance from one another 131. Moreover, we are considering a spatially homogeneous problem. In the inhomogeneous case, however, the left-hand side of eqn (9) should have a dependence on the space coordinates and a corresponding derivative[8]. In this case the balance eqn (I I) is replaced by the diffusion equation with sources and sinks [8]. Since in this paper we examine only the solid solutions, the macroscopic new-phase precipitates can be
l-ke-r13=&peT
Qo
_ fo(u)u’(u,r)du
I UocT)
where f,,(o) is the initial size distribution function of the new-phase grains u = R/Rko, u(u, T) is the solution of eqn (13) with the initial condition u(u, T)]~-~ = u, A,, is the initial supersaturation &= (4n/3)(R:dQ,), u,,(v) is the root of the equation u(u, T) = 0: i.e. udr) is the initial size of the grain dissolved at the time 7, and D3a
B=DR~’
~=3aD~.
dr
(14)
A detailed analysis in Refs. [I, 21 shows that to satisfy condition (11) the Y(T) and r3(~) functions for large T’S
708
V. V. SLuov
should vary in such a way that at each instant of time the g&r) plot, just as the T and Y functions, would tie below the u axis and in zeroth approximation would touch it at one point, which we shall henceforth call the blocking point [ 11. The tangency condition reduces to the following equations:
et ul.
dependence of Rk on t. In fact, u. is a function of fi and 6 is related to Rk through tqn (14). Substituting (18) for uo in eqn t 16) and expressing #I through R* according to (141,we obtain a differential equation connecting R and 1. The solution of this equation, which has separable variables. yields a rather compkx dependence of & on t:
B
t _ 27 &93(o) 16 aD
I
dx (I/( 1+ (1419)xf x’) - x - (7(9)][t/( I + (1419)x+ x2) - x + 1J3 7 [~/(tt~1419)x+x2~-x-(1/3)1'
BtOMRdR~b
When t is large, we have
i gtu.TLug=O lwu, 7)
au
i Denoting
I“_“o=o*
Rk +at, consistent with the volume di!Mon mechanism. If B(O)* 1, but the times are such that p(ONRkdR)b 1, we can easily obtain the dependence
V(U,T)“Y(T) 1+! I ( > we obtain from the first equation:
&‘- R:o = (+kat. y&,7)
= uo’ uo- 1
(15)
and from the definition r(7): ~=3~(,+~)~.
0
( 16)
The second equation, together with eqn (15). reduces to ud3 - 2uo) = (3uo - 4)@.
characteristic of the mass transfer along the surface. The dependence of Rd, on t is cumbersome because of the difficulty of expressing u. in terms of fl and using the exact eqn ( 18). We introduce in eqn (17) a new unknown variable x = 3 - ~0, which can be expressed as follows:
(17)
Solving this equation for uo, we obtain This equation can be replaced by a more simpfe equation uo=~(l-B)iJ(~(I-8).+28)
(18)
We can represent #I as /3(r) = @(OMR~RL), but according to (12) R&R, = eVd3,i.e. /S = B(O)tmd3 and 3 uo= ;5f t - /?fO)eed3~(~(1-~(O}e-“‘)2+28(0~e-~3).
B x=6(8+1) which can be used for small and large values of /3. For u. we obtain MO+;&]
(1%
It can be seen from eqn (18) that when T--, m (B --, 0) 3 uo-‘-. 2 When fl -+VJwe obtain from the same equation u,, = 4/3. #I = =corresponds to the case D = 0, i.e. purely surfsee diffusion. By rewriting eqn (17) in the form B
_ un(3 - 2Uol 3uo-4
we can see tbat uo is in the range&Z) bcuuse fl is positive. It can be shown that for all the values of it0 lying in the range (j, $ the derivative duo>() dr ’ i.e. the u&r) function, which is an increasi~ function, varies from its initial value u&). which is also in this range, to the value 312 when T*=, Relation (16) makes it possible to determine the
When~=O~o=3/2andwhen~~~uo=4/3. We substitute u. into eqn (16) and replace in it & by fi according to eqn (14). Thus we obtain a differential equation that can be solved as follows: B(O) dx (x t 9/S)’ I p (Xt l)‘(x +(3/2)Xx’ + (7/3)x + (3/2) 0 =+LM(-$* When /3 * 1, expressing @ in terms of R* we obtpin R+IlL characteristic of the mass transfer through volume diffusion. ~n~~l,weobt~n Rk4- R:o= ($ht.
Theory of diffusive decompositionof supersaturatedsolid solution characteristic of the mass transfer through surface diffusion. To determine the kinetics of motion of the blocking point for an arbitrary number of mass-transfer mechanisms. we must determine its location when the growth rate is described by eqn (8). In the variables
we obtain du” -&=y(?Hu-I)-u” RI” 7=In--;;-, R 10
y(r) = D&u”-‘n
-&.
(20)
Ir
UC@=-
.!,
w40)=~~
(II > uo)
g(u, uo) is determined by the growth rate of the particle. If the surface mass-transfer mechanism and the volume mass-transfer mechanism are in operation, ~(u, uo) can be determined from eqn (13) and udr) from (IS) or (19). By determining the change in ud7) more precisely, we can obtain an additional term in the expression for +J(u.7). which is close to unity because of the small derivative dudr)/dT. We now determine the number of peaks in the distribution function over the variable u for a given 7 by solving the following equation with respect to u api(u, 7) _ o
au
For the blocking point we obtain
709
.
If there is only one mass-transfer mechanism, we have:
@a21
ag au= I.
and for y(r): n”
y(r)=(n_l)“-I=nuo
n--l
.
If z = u/u0 is sued instead of II. then eqn (20) becomes universal dZ”
~=-(Z-l)‘(~~(n-/~Zf~‘)
(nb2).
(22)
It follows from eqn (21) that when many mass-transfer mechanisms are involved simultaneously the blocking point can be only in the interval 1-z u. < 2. Since the average radius R increases with time, the relative contribution of the volume mechanism increases and becomes dominant at large times and the blocking point approaches 3/2. The growth rate of the inclusion is proportional to the sum of the flow densities for the different mechanisms and each one of them in the dimensionless coordinates is analogous to the right-hand side of eqn (22); i.e. it is essentially a negative value. Therefore, u. approaches monotonically the point 3/2; i.e. in fact, II,, lies in a much narrower range depending on the initial conditions: either I c u. < 312 or 312< u. < 2. Since the variation of u. is very small. its time derivative can be ignored in zeroth approximation if the characteristics of the continuity equation are determined. The solution is the same as that for the volume diffusion because T was chosen for this mechanism: A _,-1++*(u.
Pi@, 7) =
L
a1
s(a uo) 0
(u
<
uo)
’ (u>uo)
Because the left-hand side of the last equation decreases monotonically, it has only one time-independent root corresponding to one peak of the cp function. This has a simple physical explanation: the distribution peak is attributable to the presence of a single blocking point. For the same reason there is only one peak when several mass-transfer mechanisms are in operation, because there is only one blocking point, although in this case it is not stationary with respect to time. Several peaks may be observed in the distribution function as a result of experimental preparation of the samples. This is attributable to spatial inhomogeneity of the solutions investigated, which results in a mass Gansfer in different sections via different mechanisms. A composite curve with several peaks can be obtained by superimposing the distribution curves. Apparently this was done in Ref. 17) where two peaks were observed. I. Lifshitz I. M. and Slerov V. V.. I.E.T.P. X,2 (1958). 2. Lifshitz 1. hi. and Skzov V. V., 1. Phys. Chem. So/ids 19.35 (l%l). 3. Lifshitz I. M. and Skzov V. V.. Fiz. Tucrd. Te/a (Soviet. 1. Sol. Slate Phys. 1, 1402 (1959). 4. Skzov V. V.. Fir. Tuenf. Trio (So&i 1. Sol. State Phys.) 9,4 (190. 5. Slezov V. V. and Levi0 D. M., Fir. Tuerd. Te/a Stute Phys.) 12,6 (1970).
(SovietJ. So/.
6. Kosevich A. M.. Skzov V. V. and Saralidze2. K., J.ET.P. 52, 1079(lW7). 7. Swift W. hf., M&lug. Trans.4. I53 (1973). 8. Skzov and Ryabukhin V. I., Atomnaya Energiya (Soviet 1. Afom. Energy)II, I (1976).