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Kinetics of first-order phase transitions in a solid solution
PHYSICS
ELSEMER
REPORTS
Physics Reports 288 (1997) 389408
Kinetics of first-order phase transitions in a solid solution Vitaly V. Slezov Kharkov Institute of Physics and Technology, National ScientiJic Center, Ul. Akademicheskaya d. I, Kharkov 108, Ukraine 310108
Abstract
A wide variety of characteristics during the stage of nucleation of a single atom phase in a solid solution is obtained under steady-state external conditions for a broad range of sizes. These include, flux in the space of sizes, number of new phase particles and their distribution function. A technique is put forward to account for the influence of the elastic field, formed in the vicinity of the nucleated phase particles and interaction of the solute components in a solid solution on the kinetics of the phase transitions. The proposed theory is extended to phase transitions when the nucleated new phase has the complex stoichiometric composition. PACS: 81.30.-t Keywords: Solid solution;
Diffusion; Decomposition
1. Introduction The kinetics of first-order phase transitions is addressed in a number of studies [l-9]. However, these publications mainly discuss one-component systems and deal with calculation of only the new phase nuclei flux in the space which corresponds to the size of the nuclei. Also, in practice, no study addressed the temporal evolution of the number of nuclei in the new phase and distribution function for the whole range of sizes. Since the investigation of phase transition kinetics is known to be extremely important, a consistent and comprehensive research of this problem, at least in the event of a solid solution, is a problem of actual importance. It would be a fairly accurate approximation to subdivide the kinetics of phase transition of the first order in a solid solution into three stages. Let us note that the stationary solution which would cover all nuclei sizes at any stage of nucleation does not exist. The formation of the new phase nuclei proceeds during the first stage. Due to the strong dependence of the nucleation rate upon the supersaturation (degree of metastability of a solid solution) the formation of nuclei is completed after the time interval when the degree of metastability shows only a small variation. During the second stage, the growth of new phase particles takes place while their number remains constant. During 0370-1573/97/$32.00
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1997 Elsevier
PII so370-1573(97)00034-3
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V. V. Slezov lf’hysics Reports 288 (1997) 389-408
this stage the degree of metastability decreases substantially. During the third and final stage, the “recondensation” of atoms which constitute new phase particles takes place along with dissolution of small particles and growth of large ones. This process is accompanied by a fall in particle number and growth of their average size while the number of excess atoms remains unchanged. This third (asymptotic in time) stage is essentially a nonlinear process and was consistently addressed in a number of papers by Lifshitz and Slezov [IO-121 and generalized in publication [13]. The current paper continues this line. The results presented in the current paper are of both review and original interest and advance the theory of phase transitions in a solid solution. If the direct coalescence of particles of precipitating new phase can be neglected, then the approach elaborated in the current study can be extended to the investigation of phase transition kinetics in liquid and gaseous media.
2. Basic relations Since we consider the new phase formation in a solid solution with substantially high thermal conductivity, the heat emission or absorption, related with transition of atoms from one phase to another, almost does not change the temperature of the solution. Consequently, the process of nucleation in these materials can with sufficient accuracy be considered as isothermal. The absorption or emission of atoms by new phase particles proceeds by individual atoms. Actually, if the probability to absorb or emit II atoms for the time At is proportional to ( WAty , where W denotes the probability per unit time to emit or absorb one atom, then for At + 0 we can restrict to the first-order processes of n = 1. Hence, the kinetic equation has the well-known form af(rt,t)/at
= rq_],J(l2
L-1 = K-,,nf(n
- 1,t) - &&j-(~,t>
- l,t) - K,n-lf(%t> ,
+ K+l,nf(n + l,t> - K,n+lf(%t> = J-1 -In
P
In = @i,,+,f(n,t> - K+l,nf(n + l,t> >
where f(n, t) satisfies the distribution function of new phase particle over the number of atoms y1 at the time moment t per single lattice node, F&+l, Wn--l,n are the probabilities for the new phase particles of n and n - 1 size to absorb one atom per unit time and to come from state iz - 1 --+II, Iz +)2 + 1. K,n--l, K+l,n are the probabilities for the new phase particle of n and n + 1 size to emit one atom per unit time and to come from state n --) II - 1, n + 1 + ~1. The probability W depends on the degree of the system’s metastability (excess number of atoms in a solid solution with respect to that in thermodynamically equilibrium solid solution). Consequently, to describe the kinetics of the first-order phase transitions by the complete set of equations, Eq. ( 1) has to be supplemented by the conservation law for the total number of atoms in a solid solution and new phase particles CO
=
C(t)
+
C
nif(ni,
t)
.
Here Co denotes the initial concentration the definite time moment.
(1) and C is the number of solute atoms per lattice node at
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3. Ratio of probabilities of atoms absorption and emission by the new phase particles New phase particles are known [lo] to subdivide into two classes: the particles with number of atoms IZ n, for which the solution is supersaturated. In the former case, the new phase particle dissolves and in the latter it grows. The particles of size y1= IZ, (n, is the critical size) are in equilibrium with the solid solution. Let us now consider particles of size IZ< ~1,. Since the solution is undersaturated for these particles, they are subject to conditions which make heterophase fluctuations possible. This means that their coefficients of emission and absorption are the same as those for the heterophase fluctuations and, consequently, the flux in the space which corresponds to the particle sizes 1, vanishes when substituting the distribution function of the heterophase fluctuations f(n) = exp[-A@(n)/T] in the expression for the flux. Here, A@(n) is the formation work of such fluctuation which expresses the difference of potentials of IZ atoms in a new phase and a solid solution with respect to the surface energy A@(n) = n(,$ - $)
+ 47~a~n~‘~o.
(2)
Hereinafter the following notations are used: yi is the chemical potential of atom in the new phase, pT is the chemical potential of atom in a solid solution, 4rt a2 n2j3CTis the surface energy on the boundary of new phase particle of size n with a solid solution, G is the surface tension, a = (3c0/2rc)‘/~ or o = 47ca3/3 is the volume per atom in the matrix. By substituting the distribution of the heterophase fluctuations in the expression for the flux of nuclei in the space of sizes the latter vanishes In(h) = 0. Consequently, we obtain K,n+llK+l,n
= exp[pT -
P~(~I/T~
p’(n) = pi + (87c/3)a2m-“3
n < n, ,
(3)
where T denotes the temperature in energy units. For the particles of size II > n, the situation is more complex because the heterophase fluctuations do not exist in such conditions. Hence, let us envisage the virtual (auxiliary) solution with chemical potential per atom being the same as that for the new phase particle of size YEwith respect to the surface energy. Therefore, for each new phase particle of size IZ there is a specific virtual solution being in the equilibrium with this particle, i.e. K+l,n = E++,, where Fn,n+l is the probability per unit time for new phase particle of size II to absorb atom from the virtual solution. For the dilute solution where the interaction of solute components is neglected, the virtual solution is equal to the solution with equilibrium concentration of solute components for the new phase particle of definite size. In this case, solute atoms in both true and virtual solution interact only with the matrix and, consequently, the energetical barriers and environments of solute atoms are identical in both solutions. For the concentrated solution one has to take into account the interaction of solute atoms. In this case, the virtual solution for the new phase particle of size IZ appears to be the true solution with part of atoms immobilized (frozen) and the rest of mobile atoms being in equilibrium with the given particle of new phase. Therefore, the number of jumps per unit time of mobile atoms to the new phase particle equals to the number of atoms, emitted by a particle to the solution. On the other hand, the mobile atom from the virtual solution has the same probability per unit time to jump to the new phase particle as that for the atom to jump from the true solution by virtue of their potential barrier and environments being equal because in both events they jump from the solution to the particle. The environment does not change during the jump. Consequently, the
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probability of jump from the true and virtual solution (which by definition is equal to the probability of atom emission by a new phase particle) have the same kinetic multiplier and differ only by the configuration, favorable for the jumps. Since the frequency of atom jumps from one phase to another is small compared to the frequency of atomic oscillations, with sound accuracy one can consider atoms to be in thermodynamical~quilibrium with the solution of a given composition. Consequently, by using the condition K+l,n = IKl;n+lwe obtain
(4) where p and b signifies the probability of the favorable configurations for the atoms to jump to the new phase particle of size n from the true and virtual solutions with average (thermodynamical) energy of these subsystems. Since p and j coincide with the most probable states of subsystems with the definite composition, the following relation is known to be valid for p and F under such conditions: pnr=1,
@Af=l,
where Ar and Ap denote the statistical weights of the states with the given thermodynamical energy for the configuration which corresponds to this energy and has one atom in the conditions, favorable for the jump. Thus, we obtain p/j? =
Al’jAr
= exp(AS - AS),
(5)
where AS and AS signify the entropies per atom in the virtual and true solutions. The interaction in the neutral condensed media is known to be short range, so the interaction of the new phase particle and solid solution proceeds only through the interface solution. Since true and virtual solutions have the same temperature, they can be considered as a closed system being in thermal equilibrium with the given composition. This implies AS-AS=AS,,
(6)
where ST denotes the change of the total entropy of the closed system virtual-true solution with definite composition caused by the transition of one atom from true solution to the virtual one. Since AS and AS can be always added by So the entropy of the rest of the system virtual-true solution, which does not change in course of transition of one atom from the true solution to the virtual one, as we have already mentioned, the chemical potential per atom in the virtual solution for the new phase particle of size it is equal to the chemical potential in the new phase particle of size IZ with respect to the surface energy b = $(n)= pi + 8rca20n -‘13/3. Consequently, for n > n&r > ,2(n)) change of the total entropy ASr > 0 and at the same time it is known that R = -T max = -IRmaxI = ~IA@1 = pT - ~‘(4 (7) T T ’ T where R,,, is the maximal work which the closed system can make in course of reversible transition to the state of more equilibrium. In this event at the constant temperature /A@1 = pT - $(n), IZ> n,. Thereby, the ratios of probabilities of absorption and emission in event of 12> n, can be written in the form AS, AS, = --&,,,
WI&n+1 AF PT - P%) = exp AS,, = exp wn+l,n = ar T
.
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The ratio of probabilities is known from Eqs. (3) and (8) to have universal form for any II, i.e. the relations Eqs. (3) and (8) have the equal form and cover all size range of new phase particles. Such a universal definition of probabilities ratio yields a universal, simple method to obtain the coefficients of the kinetic equation ( 1). This method can be easily extended to various cases, e.g. when the new phase particles have the stochiometric composition [ 14, 151, when the interaction of solute components with elastic field which arises in the vicinity of new phase particle needs to be taken into account [ 141, interaction of the solute components [ 161, nucleation of gas-filled voids [ 171, nucleation in overcooled melts [18], influence of surface effects [14], etc. The steady-state solution (1) apparently describes the heterophase fluctuations in the event of the equilibrium solution. Thus, by writing Eq. (1) in the form
af tg
=
In-1
-
Ll = K-l,n.f (n>
-K+l,n.f(n + 1) and after substituting
[
Wn,n_,
1
“f-(n)
1
[~f(;‘;‘l, - 1] =o,
(9)
in Eq. (8) with the following:
PT- PW _-----=T
A@(n + 1) - A@(n) T
we obtain f(n)=Aexp
K-l,n .f(n - 1) _
[
-F
>
/L-$(nT
1) = - A@(n) - A@(n - 1) ,
c1oj
T
1.
(11)
To provide a consistent macroscopic description of first-order phase transitions, one should treat the transition of atom to the new phase as requiring formation work which depends on the particle size IZ only through the surface energy. Naturally, when n m 1, such an extrapolation describes new phase particles only qualitatively because the surface energy for n N 1 becomes a poorly determined parameter. But, provided that the volume of the new phase particles is fairly large and their average size fi B 1, the number of small particles is low and, consequently, their contribution to the various characteristics of solid state is small as well. Thus, the accepted approximation is fairly accurate in this case. The same approximation permits us to treat the solute components as the new phase nuclei with zero formation work, which implies f WI n_O = A = c .
4. Transition from the difference to the differential form of the kinetic equation To describe the kinetics of all stages of the diffusive decomposition derivative of distribution function d2f/an2 with respect to n has to be is this derivative which makes possible for new phase particles of size size n >n,. Higher derivatives in the case of yt, % 1 are substantially can be neglected. Consequently, the condition which permits to apply y2, % 1.
of a solid solution the second taken into account because it n < IZ, to achieve supercritical less than the second one and the derived equation reads as
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394
The size spectrum of new phase particles of size range n manner. This is related to the fact that although the difference spectrum, the character of its solution forces the differential result as well. Thus, by using the relations (8) and (10) we rewrite the form af,,_ at
_I n ’
=-_
1 is also described in a straightforward equation must be applied to this size equation to give qualiatively correct difference
equations
in the differential
=-S n
an
% n,n+l e -A~(n)lT[eA~(n)lrf(,, an
t> _ eA@W)/~f(,
_ 1, t>l
(12) Evidently, for the equilibrium case when aflat = 0 and A@(n) > 0 for any n the heterophase fluctuations become the solution of Eq. ( 12). Thereby, the conservation law Eq. ( 1) takes the straightforward form co = C(t) +
s0
c*3f(n, t)n dn .
(13)
The relations ( 12) and ( 13) form the complete set of equations which describes phase transition kinetics for a solid solution in any stage. The first term of the expression for I,, is seen from Eq. (12) to denote the “hydrodynamical” flux in space of sizes and the coefficient by this term signifies conventional growth rate for the new phase particle of size n: dn/dt = - K,n+l l/T 6A@/6n .
(14)
The second term in the expression for 1, (12) is proportional to the first derivative of the distribution function and represents diffusion term in the expression for I,, in the space of sizes and coefficient by this term is the diffusion factor in the space of sizes, (15)
Q, = %,n+i >O . It can be seen from Eq. (14) that for &A@/% > 0
dn/dt < 0 ,
the relation which determines
6A@/6n < 0
dnldt > 0,
(16)
n, reads
6A@/6n = 0.
(17)
In case of a dilute solid solution we have
A@(n)
p=-nlnC+-n T
c 00
47ma2 2,3 _ pn213 -y-“B($-A) T
P-0.
(18)
I/ V. Slezov I Physics Reports 288 (1997)
where C,, denotes the equilibrium solution. 1 6AQi = _~ -ln$ T 6n aAl3=
ln
_-1
of the solute in a solid
$
-1
-1
C
s2A@
at the definite temperature
cx)
CX
B[
concentration
395
389408
P
’
-d/3
Here the following notations are used: CIis the number ( CIN 1) [ 14,151, D is the diffusion factor of solute in the lattice; a is the lattice parameter (distance between the nodes of a lattice).
5. Steady-state distribution function and flux in the space of sizes on the stage of nucleation in the domain 1 0 holds everywhere on the stage of nucleation in the event of diffusive decomposition of a solid solution. It means that the behavior of the distribution function at the definite point is not affected by the value of distribution function to the right from this point, hence, the value of distribution function at this point can be treated as the boundary value for the states to the right. So, the state of distribution function to the left of certain point can be found independently on the states to the right of this point. The following relation can be obtained from the equation for In: I, = I = - Wn,+, (df Jdn + f dA@/dn) f In-4 + c 3 This expression
f I”=9=
f(s)>
yields 9 exp(A@(n’)/T)
I=
(C - &)
f(n)=Cexp
Wn’,n’+l
(-q)
(20)
n
bev(A@(n’)lT) dn, J(a,b) = > s 1’
dn’ -’
u
Wn’,n’+l
‘$$ +f(g)exp (-Am(nTi ““‘“‘)$$$,
(21)
.4 I 11 --I
$(g)=f(g)exp
[y]
1
f(g)=1
At the point g = n, + Fn the term with derivative
$
can be neglected
in the expression
for 1.
396
V. V.
Since the expression
J
’ ewWn’)lT
0
Reports 288 (1997) 389408
for A@(n) has the sharp peak at y1= n, we obtain dn,
Wn',n'+l
where 6n, is expressed
SlezovlPhysics
=
O”ex@W’)lT
dn,
=
Wn',n'+l
J -cxY
hfi
-exp WnC,&+l
A@(%>
[
~
T
(22)
1’
by the relation
By using Eq. (22) we rearrange the expression
for the flux I as
expWW/T) ] -’ =
Wdd,
g”Clerp
(-F)
,
(23)
where the definition of J(a,b) is given by Eq. (21). Since the expression $g < C holds at 6n > &zc, which can be easily verified, the second term in Eq. (23) can be neglected because it is small. Consequently, the expression f(n)=Cexp
(-F)
=Cexp(-F)
kerfc (2) k [l -erf
(&$)I
(24)
holds. Here erf (x) = - erf (-x) signifies the error tinction. Naturally, in such approximation Eq. (24) is not valid in the immediate vicinity of n = g, but the function f(g) is known. In the immediate vicinity of n N g the relation (2 1) is to be used.
6. Estimate of the relaxation time for the establishment of the system’s steady state in the domain l
(25) A&=
A@-lnW+const.
at n>l,
V. K Slezov! Physics Reports 288 (1997)
--1 &A$ =_--1 6A@ T 6n T Fn
389408
391
l “W=B(&&)-;? w 6n
=&kc&) 1 h2A& -------_ T 6n2
g(l-&)=-$r&.
Firstly, we consider the solution within the interval 1
= a af/ih
+ b 8f/i3n2 - df ,
(26)
where a = -dn/dt
= WFA@/Gn > 0,
b = W(n),
Let us rearrange f”=fit + $, where $ is the steady-state which has to vanish reads
d = - Wh2A@/6n2 > 0. solution of Eq. (1). So the perturbation
$(n, 0) = - ~tO(n, - 6n, - n)l?(n - 1). Hereby we take into account that fj+, = f(n) < 0, n > 1, where f(n) is the heterophase fluctuations which remain in the solution after the transition to the metastable state. The exact solution of Eq. (26) with the initial condition $(n, z)Izzo = $(n, o) can be found rewriting f” in the form f”= exp[-dT+cp(z+n/a)]p(n, z), where q is chosen to retain only the second derivative of p with respect to n on the right-hand side of Eq. (26). ( -d r - -i;r)
$(n,t)=exp
&[~$(n’)exp(-a(n~n’)
s
=
exp(-W
=
exp(-dz) exp(-(“lc)‘) 2&z
O”
- (nTbz)z)
dn’
dn’
~~$(n’)dn’.
(27)
In the above relation we accounted for the fast descent of $(n’) starting from n’ = 1. Decaying factor in the exponent is the Green function of Eq. (26). Let us now obtain the maximal relaxation time z,,,. It follows from Eq. (27) that 1 ”
=
d + a2/4b ’
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398
(28)
b max- Wn,n+, In+ = 3aCnf/3 ) Therefore,
afnin
=
4bmx
we obtain (29)
In the dimensional
variables Eq. (29) reads
7, = f(a2/DcnC)[ln C/C,lW2 = i(a2/DrxC)nY3/fi.
(30)
Provided that a = 0, Eq. (26) within the interval -6n, 5 n - n, 5 6n, produces $(n, r) = &
exp(-dr)
11
i&n’) dn’ (31)
fs‘t lIn-n,1&, $(n) = { 0 beyond this domain, Therefore,
the relaxation
time can be written in the form
7, = $(a2/cdX)n~J3/~.
(32)
7. Equation for the flux I@, z) and the value of dcldz at the moment zF of cessation of the new phase particles formation and transition to the process of coalescence By differentiating I(n, r) with respect to time and using the kinetic equation we shall obtain important relations for the flux in the space of sizes on the stage of nucleation [14] dI/dr = H$,++,{a21/an2 + (( l/T)6A@/6n)iX/an}
,
(33)
where 0
( >= -1
(C/C)Z ;
qq/T, + 1 .
The steady flux will be established during the time of system relaxation to the steady-state of the distribution function (32) in the domain 1
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As it is seen from Eq. (28), the less the absolute value of the coefficient by the first derivative the longer is the relaxation time. Thus, to obtain upper boundary of relaxation time we shall set the small parameter in Eq. (3) (l/T)SA@/Sn
E
0
equal to zero within the considered from Eq. (33) that 1, = ISt 1 - erf [ (
n - n, + 6n, 2Jwz
interval
I
-6n, 5 n - n, < 6n,. Consequently,
n 2 n, - ih, ,
it can be derived
(34)
’
provided that the boundary conditions Il,,=,,-s,,, =Ist and Wn,n+l= K,,n,+l = const. are obeyed. By using Eq. (34) we can derive the time of steady-state establishment z1 at the point n = n, + 6n, or, otherwise, the time of translation of the value of I at the point n, - 6n, to the point n, + 6n,. so that the time r1 reads 2P zl N 6n~/W = --&ln C/C,)-2.
(35)
Therefore, the times of relaxation to the steady state of distribution function and flux within the interval n < n, + ih, are of the same order and differ only by the numerical coefficients, which are of the order of 1. Let us now obtain one more important expression, related with fall of concentration in course of new phase particles growth and rise of the critical size n,. Growth of n,(z) with time permits us to treat the flux as the steady state during the finite time when n
(36)
It follows from Eqs. (35) and (36) that at r1 ~~~ the flux adjusts in the space of sizes and the steady-state flux n < nmax exists. On this stage the “diffusive” term of the kinetic equation plays an important role. Otherwise, if z1 >z2 the flux does not adjust and decreases to the left from critical size n = n, down to negative values at the last stage of a solid solution decomposition and, respectively, there is no formation of new phase particles. At this stage, the “diffusive” term apparently does not play an essential role in the whole range of sizes, for even in the domain of critical size, where the “hydrodynamical” term is virtually equal to zero, the particle stays only for extremely small time and, in view of the critical point motion, the particle virtually does not change its state. Equality of these two characteristic times determine li, at the moment when the nucleation ceases, z1 = &f/W
= z2 = &2,/f& ,
ri, = W/&n, = aC
3jl/2 .
J--
(37)
As it is seen, Eq. (37) is accurate to the factor which is close to unity, but the relation between the determining physical parameters appears correct tic = (dn,/dC)
C ,
400
V. V Slezovl Physics Reports 288 (1997) 389d.M
For the case of the dilute solution, Eq. (37) gives the following time r=rr
relation between
C and C at the
(38) In the stage of nucleation Eq. (38) by CO=C~z=,,.
the concentration
shows insignificant
variations
8. The flux in the space of sizes and distribution function for On, is established
and can be substituted
in
+ 6n after the steady state
Within this interval the relation $
j’~nE,lc+ATl -ggr’=&
(g;
holds at 6n > 6n,(&z, Q n,). describes the flux reads
Consequently,
aI/& = W,,,+1(6A@,lT6n)i31/&
within the interval
n > n, + &I the expression
which
= -(an/aqi2/an, (39)
$=g=n,+~n, =I(n,)
,
nc = nc(C(~>>.
Hereby we are setting equality 6n = 6n, because the quasi-steady state has already been established in the domain 6n - 6n,. The solution of Eq. (39) can be derived in the form
g=nc + h,
I(n,z>=l(s,C(zo(z,n)>),
where zo(r,n)
is the characteristics
!f!!! = 3acn2/3 ds
~o(%Y)=z >
To(~, nmax >=
[+-$j=3cKn2i3(ln~-&).
n’:3 - g’/3 = EC ln
=
1
(40)
of Eq. (39). (41)
Concentration C experiences small variation on the stage of nucleation in high susceptibility of the nucleation rate I -exp[-( l/T@@(n)] barrier (l/T)A@(n,) = 1/2/?ni’3 $1. Consequently, the characteristic found by taking into account the infinitesimal&y of the j/n ‘j3 term.
.-l/3
0,
(z - Q)[l - nj’3n_]
’ dz’
z - zo .Ira P
1
1
dz’
T
In 1 + [
of p 9 1, which results height of the potential n, + 6n, can be easily of this we can write
)
- r - r. s TOg1i3 + aC In C/C,(r’
a(z - zo)C In C/Cm
because to the for n 2 In view
&(r
-To) 9
Ii3
lnC
c ,I
- ~0) ln(n/g)‘13
= SI(Z -
ro)C In C/C,
.
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401
Finally, we obtain the relation for ro, zo(z, n) = z -
nli3 - g113+ nAi3ln(n/g))“3
(42)
cxc
Let us now deduce nmax(T) from Eq. (42). Here nmax(r) is the value, which the steady state has extended to at the moment z if started from the point g(z 5 TN). The relation zo(z, nmax) = 0 is obeyed for n,,,(r), which in turn yields the relation niiX - g1’3 + nbi3 ln(nmax/g)‘i3 = az , In event of nmax$g
a = aCo ln(Co/C,)
.
(43)
one can deduce to a fairly high accuracy that
n maxN (g’i3 + UT)3 . The time r has the origin at the moment of steady-state establishment z, for 0 5 n 5 n, + 6n, = g. Thus, in the course of time, nonzero flux and respective distribution function in the space of sizes appear in the domain 05 n <_nmax(z). To derive them explicitly, one has to employ the law of substance conservation (13). It can be conveniently rewritten in the differential form, as follows: dC dz=-0
Oirafndn= at
s
O3-n arn dn = -Ig -s o an
I,, dn .
(44)
Since the expression I(n, z) =
I(& ~o(S,n,7)) >
9
5 n 5 %ax(~),
0,
n >&lax(r) ;
(45)
holds, the integral term in Eq. (44) can be rewritten
in the form
-zo) dz,, =-JI(ro)gdzo = JI(ro)$(t T
T
.
0
The following
dz =
0
(46)
relations have been used while deducing the above formula:
ro(g, 9, r) = r 3 dn
0
dn
---_=
dro
~o(S, %nax,7) = 0 > dn
-d$r - ro) = $(g1’J
+ a(r - 2lJ))3 = 3a(g”3 + a(z - ro))2 ;
(47)
which follow from the relation for zo(n, r) within the domain 0 2 r < rN, where rN designates the time interval of the new phase particles formation. Hence, from Eq. (44) we have 7 dC - = -Z(r)g I(ro) $(r - ro)dro, (48) dz J’0 if provided that the conditions CjrZO= Co and z 5 z N are obeyed. The relation Eq. (48) is fairly straightforward. Variation of the concentration (quantity of substance in the solution) with time is
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determined by the first term on the right-hand side, which stands for the rate of the new phase particles growth at the point g. The second term on the right-hand side designates the rate of the substance build up on the existing particles within the interval 122 g. The expression dn/dz(T - zo) denotes the rate of the particle growth at the moment z if the particle has emerged at the point g at the time moment zo, while I(Q) stands for the number of the particles which have emerged at the point g during the time interval dzo. By introducing cp= 1 - C/Co and noting that cpB 1 at the stage of nucleation z < TN, we obtain
(49)
In fact, the explicit expression for I(cp) has been used in the above relation. exponent for I at the small 4p we get
I=
10expbwl
,
By expanding
the
(50)
where the equalities
(51) hold. Relation (49) apparently represents the complete equation for q within the interval z
ci,= cp(0) ( 1 +
E g1,3] - ~(0)~,~e~“c~~‘a~~(~o)
4=0=0,
&=T, =
(I + ‘(~~~)]
dzo,
$.
(52)
Eq. (52) can be solved by a method of sequential approximations, if substituting known function under the integral instead of q(2). The account of this integral is seen to give a correction - Q(O)’ 4 1. Hence, retaining the first term on right-hand side in the first-order approximation over N @(O) we obtain cp = cj(o)(g”3/4a){(
1 + az/g”3)4 - 1)
Consequently, under this approximation Eq. (52). Evidently, the relation Eq. (53) by substituting f(n, z) at n > g, but the flux in the space of sizes in the domain
,
T
&=7,
=
t
.
we set exp[-n,cp] 21 1 to an accuracy which follows from can be obtained directly from the conservation law Eq. (13) accuracy of such an approach is harder to estimate. The n 2 g is determined by the “hydrodynamical” term, which
V V. Slezovl Physics Reports 288 (1997)
is substantially
f(n37) = 9
larger than the “diffusive”
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403
one in this domain. Hence, we obtain
Z(g, fo(n, 7)) = Zoe-n~rp(To(n,z)) _ dn,d7
- &(ln
3an2i3
5 n I nmax(7),
0<7<7N,
GICJ’
2
(54)
0
ndp41.
At the tip of the interval n E n,,, (z), where the distribution declines to zero, predominant role is played by the “diffusion term”. As it can be seen from the kinetic equation this decline proceeds within the interval
Relation (54) apparently satisfies the respective boundary condition at n = g. Therefore, the distribution function and flux show smooth variations within the domain n g can be obtained as follows:
hllax N=JTI(g,z') J n
dz’=
o
f(n’,z)dn’=loz,
(55)
The time moment TN, when the growth of new phase particles ceases, can be derived from Eq. (53). Hence, we have rN 2i (fi”4/&O)(ln
CO/C,)-‘((4%C$n,)
ln CO/Cm)“4 .
(56)
The peak size of new phase particle on the stage of nucleation ni2X(rN) ? RrNCOln co/C, The distribution f(%r)=.@(g
= &“4((4aCi/n,)
can be deduced from Eq. (43) as
ln CO/C,)‘i4 .
function formed during the stage of nucleation - n) + f(& r)&n - g)@nmaX(r) - n) ,
(57) has the form
75
7N .
(58)
Here we do not take into account the detailed behavior of the distribution function at the edge of the interval for it is not important on the majority of occasions. When the nucleation stage is finished and formation of viable particles of new phase ceases, the transition stage begins. During this stage the number of particles virtually does not change iv F! IV($) = N,,, and the particles simply grow. Initial distribution function for this stage is the distribution of particles over the sizes, which has been formed on the stage of nucleation. Since the particles at this stage are fairly large (n % 1 ), the “diffusion clouds” of solute around the new phase particles need to be taken into account when determining their growth rate. These “clouds” in view of the concentration being small, Co < 1, are quasi-steady state and adjust to the size of particle
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[ 121. Which means that the flux of solute to the particle can be deduced by solving the respective diffusion problem with certain concentration C near the surface of the particle, which is determined by the joining of this flux with that in the vicinity of the particle [14, 151. So, the relation dnJdz = 3aC,, ln(C,,/C,)n2’3
= 3n’i3(Co - e,,)
(59)
holds. The second term of Eq. (59) designates the flux from the immediate vicinity of the particle, similar to the one addressed by Eq. (48), and the third term denotes the flux to the immediate vicinity of particle from the bulk of a solution. If the relation an’i3 + CO/Cw > 1 is satisfied, then by using Eq. (59), we obtain C, = C,
•t G/an
113
,
dn/dz = 3n’j3 Co .
(60)
Therefore, if CI= a(T), which denotes the height of the barrier for the solute atom to jump to the particle from the immediate vicinity is not anomalously small, then Eq. (60) can be used for the description of the transition stage. The reversed inequality an1/3
nf
p
CO/&,,
,
dn/dr = 3n1’3Co( 1 - N,,,n/Co)
(61)
hold, which results in Dt/a2 = T&f r” n;13/Co = l/N,$ CiJ3 , Let us note that if the particles reach the sizes of an:& % Co/C, during the stage of nucleation, then Eq. (60) should be applied to the growing nuclei. The results obtained are naturally different, but the technique is still the same. In this case, we derive n2/3
zo(n, 7) = 7 -
2
_
2co
9
213
’
n,,,(T)
= (g2j3 + 2Cor)3’2 ,
= 3cg(g2’3 + 2Co(z - ro))“2 ,
(62)
,
N = Zoz,
N,,,,, = Io~N =,/j&
. 0
The transition stage is followed by the final and asymptotic [l&13, 19-211.
in time stage of diffusive decomposition
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The developed approach to the investigation of the first-order phase transitions permits us to take into account various factors which influence this process. If the process of incorporation of the solute atom by the particle is not instantaneous and has the absorption frequency v per node and the solute atom “exists” for the finite time z in the adsorbed state on the surface of the particle, then the coefficient a can be derived by taking into account that the barrier for the jumps in the interface layer is equal to that in the bulk of the solution. Therefore, we can infer CI, as follows [ 14,221; z=vr&l
+ vzs).
(63)
On account of the elastic fields being formed in the vicinity of the new phase particle its growth rate will naturally change [ 14, 151, but this change is not essential. It mainly leads to the change of equilibrium concentration C,, as it can be seen from the expression below: C, --+ C, exp((eSwS - E~CO - u)/T) .
(64)
Here eSc.? denotes the elastic energy per atom in the new phase particle, E,,W is the same energy in the bulk of a solid solution, cs and cp are the densities of elastic energy in the interface layer, u = u + co&/&, u is the energy of a solute interaction with the elastic field, as/% is the variation of elastic energy density in the interface layer, related with variation of elasticity modulus upon the concentration. If the solute interaction is strong enough, then it needs to be taken into account, which, as shown in the study [ 161, leads to the replacement of C by cp= C exp[pC/r]. Here p designates the coefficients which account for the interaction of solute atoms (p + p + PC).
9. Kinetics of decomposition
of a solid solution with the complex stoichiometric
composition
In the case of a complex stoichiometric composition of the precipitating phase, the size of a new phase particle is determined by a number of structural elements ~1,and its volume equals to V = YEWS, where co, denotes the volume of a structural element. If the change of volume per atom in the new phase is not essential, then o, = L: viai, where co, denotes the volume of a single solute atom of ith kind and vi are the stochiometric coefficients. Distribution function in the space of sizes is the function of number of structural elements in the new phase particles and time. The growth rate in this event can be deduced [ 191 by using the conditions of stoichiometricity ji/Vi
=jk/vk
=
.. . ,
as follows:
(65) where o, = 4naz/3 and o, = 47cai/3 is the volume per atom of lattice. The term before last on the right-hand side of Eq. (65) denotes the flux to the particle from the interface layer where the concentration equals to Ci, (Gin stands for the equilibrium concentration
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of ith component of solute in the interface layer by the particle of size n). The last term on the right-hand side of Eq. (65) designates the flux from the bulk of a solid solution to the interface layer near the new phase particle. Quantity Ci denotes concentration of ith kind solute atoms in a solid solution. To derive this flux, one has to solve the respective diffusive problem [23]. In view of these fluxes continuity they are equal, which yields the equation for Z;n ln(Z;,/Ci,)
= l/C!i?Z1’3(Cj- 2;,),/2;, .
The above expression
(66)
yields:
‘j3 ln(Ci/Ci,) < 1 or ln(n CT/K,) 4 n-1’3 C(vi/cli), where K,, by definition is K,, = (a) Z;., r” C, for s(iy1 Q Gz or K, = Km exp[-j3n-1/3]. (b) C, P Ci* for ain’i3 9 CiCi’ or K”~ gK,(n C’~)-’ C up’. It follows from the above relations that for the stage of nucleation expression dn/dz in the form
at Cli< 1 it is convenient
to use
(67) Eq. (67) should be used on the stage of nucleation if the values of ai are not anomalously small. At the same time, the growth rate at the stage of nucleation, when nz& % 1, and at the transition stage can be conveniently calculated by the following expression:
(68) The quantity C’inin Eq. (68) needs to be determined from the condition of the fluxes stoichiometricity for the different kinds of atoms and condition of chemical equilibrium ni Cz = K,,. To determine the coefficients in the kinetic equation for f(n, t) one has to compare Eq. (67) with general expression dnldz = - Wn,+, 6A@/6n , where the relation 6A@/Fn = - In (n
CT/Kn)
holds [ 151. For the transition stage and late stage, when the “diffusive” term in the kinetic equation can be neglected, the knowledge of only dn/dr will be sufficient. The law of solute atoms conservation yields the expressions for the relative supersaturation of a solid solution d relatively to the new phase precipitate A=-q=lnF,
Ci,o=Cj+vj/m/dn, cxz cc
#$=-vi .io
a.f-
-dn= ar
0 T
-v,
Ig (I
Gro& 0
- To>dzo
, >
(69)
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Here I is the steady-state flux at the point g or yt,. This flux is obtained by making the following substitutions [ 151 in the expression for single-atom phase I:
with the replacement I + ID/at preceding the above substitutions. After that, the solution and outcomes are equivalent to those of the kinetics of single atom phase with respect to above substitutions. In order to employ Eq. (68) for the derivation of integral term in Eq. (69) the quantity A = const. = Di(Cf - Cl&)/vi needs to be found from the condition of chemical equilibrium for C;,, as follows: (C; - (vi/Di)A)“’ = K, exp[ -fi/n”3]
n
.
i
To estimate the time of the transition stage in the event of the average size n’/3 9 /?, one can let A N (DiCi/Vi),i,, which means that this time estimate can be obtained by substituting DC0 + system. (DiCilvi bn in the expression for a single-component
10. Discussion In conclusion, let us note that the picture of the first-order phase transition in a solid solution for the homogeneous nucleation and under the steady-state conditions is fairly clear. During the stage of nucleation, the fast establishment of a quasi-steady state of undercritical nuclei (n n,). This growth virtually ceases when the metastability experiences a small decline (C N C,). The subsequent growth of the nuclei number contributes little to the total number of already formed nuclei. After that the “transition” stage begins. During this stage the number of new phase particles shows a small variation, but the metastability of the system (supersaturation of a solid solution) almost vanishes. Only the estimate of time lag for this stage is presented in the current study. A comprehensive analysis of this stage, formation of the distribution function from the one which was formed on the stage of nucleation is given by the papers [ 13,201, where the distribution function is shown to transform to the universal one, “forgetting” its past states during this transformation. Then, the final and asymptotic-in-time late stage begins (C, N C,). A detailed discussion of this stage is given in the publications [ 10, 12, 19,2 11. Finally, we would like to mention that the developed approach can be extended to the nonsteady state external conditions and presence of new phase sources as well as to the case of heterogeneous nucleation of new phase, when the formation work of the nuclei decreases due to the various reasons, which requires only the improvement of homogeneous nucleation theory. If the buildup of new particles proceeds on the already existing nucleation centers in course of the heterogeneous nucleation in a solid solution, then the “transition” stage begins at once. Such a case requires special consideration.
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Acknowledgements This work was performed as a part of a project financed by the Bundesministerium fiir Bildung, Wissenschaft, Forschung und Technologie (BMBF), Germany. The author also acknowledges the financial support given by the SOROS foundation (Grant No. ISSEP-SPU042062).
References [I] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] [12] [ 131 [14] [ 151 [ 161 [ 171 [ 181 [19] [20] [21] [22] [23]
M. Volmer, Kinetik der Phasenbilding, Th. Steinkopf, Dresden, 1933. R. Becker, W. Doring, Ann. Phys. 24 (1935) 719. R. Kaischew, I. Stranski, J. Phys. Chem. A 170 (1934) 295. K. Binder, D. Stauffer, Adv. Phys. 25 (1976) 343. H. Trinkaus, H. Yoo, Phil. Mag. A 55 (1987) 269. H. Wiedersich, J. Katz, Adv. Colloid Interface Sci. 10 (1979) 33. F. Kuni, A. Grinin, Sov. Colloid J. 46 (1984) 460. F. Kuni, A. Grinin, Sov. ZhTMF 80 (1989) 418. J. Katz, M. Donahue, Adv. Chem. Phys. 40 (1979) 137. 1. Lifshitz, V. Slezov, Sov. ZhETF 35 (1958) 475. 1. Lifshitz, V. Slezov, Sov. Phys. Solid State 1 (1959) 1401. 1. Lifshitz, V. Slezov, J. Phys. Chem. Solids 19 (1961) 35. V. Slezov, Theory of Diffusive Decomposition of Solid Solutions, Sov. Sci. Rev. A 17 (1995). V. Slezov, J. Schmelzer, Sov. Phys. Solid State 36 (1994) 353. V. Slezov, J. Schmelzer, J. Phys. Chem. Solids 55 (1994) 243. V. Slezov, J. Schmelzer, Y. Tkach, Sov. Phys. Solid State 37 (1995) 3212. V. Slezov, Sov. Phys. Solid State 37 (1995) 2879. V. Slezov, Sov. Phys. Solid State 38 (1996) 433. V. Slezov, V. Sagalovich, Sov. Phys. Solid State 17 (1975) 1497. V. Slezov, J. Phys. Chem. Solids 39 (1978) 367. V. Slezov, V. Sagalovich, J. Phys. Chem. Solids 38 (1977) 943. L. Tanatarov, V. Slezov, Zh. Metallofiziki 10 (1988) 199. V. Slezov, V. Sagalovich, Phys. Uspekhi (UFN) 151 (1987) 67.
ELSEMER
REPORTS
Physics Reports 288 (1997) 389408
Kinetics of first-order phase transitions in a solid solution Vitaly V. Slezov Kharkov Institute of Physics and Technology, National ScientiJic Center, Ul. Akademicheskaya d. I, Kharkov 108, Ukraine 310108
Abstract
A wide variety of characteristics during the stage of nucleation of a single atom phase in a solid solution is obtained under steady-state external conditions for a broad range of sizes. These include, flux in the space of sizes, number of new phase particles and their distribution function. A technique is put forward to account for the influence of the elastic field, formed in the vicinity of the nucleated phase particles and interaction of the solute components in a solid solution on the kinetics of the phase transitions. The proposed theory is extended to phase transitions when the nucleated new phase has the complex stoichiometric composition. PACS: 81.30.-t Keywords: Solid solution;
Diffusion; Decomposition
1. Introduction The kinetics of first-order phase transitions is addressed in a number of studies [l-9]. However, these publications mainly discuss one-component systems and deal with calculation of only the new phase nuclei flux in the space which corresponds to the size of the nuclei. Also, in practice, no study addressed the temporal evolution of the number of nuclei in the new phase and distribution function for the whole range of sizes. Since the investigation of phase transition kinetics is known to be extremely important, a consistent and comprehensive research of this problem, at least in the event of a solid solution, is a problem of actual importance. It would be a fairly accurate approximation to subdivide the kinetics of phase transition of the first order in a solid solution into three stages. Let us note that the stationary solution which would cover all nuclei sizes at any stage of nucleation does not exist. The formation of the new phase nuclei proceeds during the first stage. Due to the strong dependence of the nucleation rate upon the supersaturation (degree of metastability of a solid solution) the formation of nuclei is completed after the time interval when the degree of metastability shows only a small variation. During the second stage, the growth of new phase particles takes place while their number remains constant. During 0370-1573/97/$32.00
0
1997 Elsevier
PII so370-1573(97)00034-3
Science B.V. All rights reserved
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this stage the degree of metastability decreases substantially. During the third and final stage, the “recondensation” of atoms which constitute new phase particles takes place along with dissolution of small particles and growth of large ones. This process is accompanied by a fall in particle number and growth of their average size while the number of excess atoms remains unchanged. This third (asymptotic in time) stage is essentially a nonlinear process and was consistently addressed in a number of papers by Lifshitz and Slezov [IO-121 and generalized in publication [13]. The current paper continues this line. The results presented in the current paper are of both review and original interest and advance the theory of phase transitions in a solid solution. If the direct coalescence of particles of precipitating new phase can be neglected, then the approach elaborated in the current study can be extended to the investigation of phase transition kinetics in liquid and gaseous media.
2. Basic relations Since we consider the new phase formation in a solid solution with substantially high thermal conductivity, the heat emission or absorption, related with transition of atoms from one phase to another, almost does not change the temperature of the solution. Consequently, the process of nucleation in these materials can with sufficient accuracy be considered as isothermal. The absorption or emission of atoms by new phase particles proceeds by individual atoms. Actually, if the probability to absorb or emit II atoms for the time At is proportional to ( WAty , where W denotes the probability per unit time to emit or absorb one atom, then for At + 0 we can restrict to the first-order processes of n = 1. Hence, the kinetic equation has the well-known form af(rt,t)/at
= rq_],J(l2
L-1 = K-,,nf(n
- 1,t) - &&j-(~,t>
- l,t) - K,n-lf(%t> ,
+ K+l,nf(n + l,t> - K,n+lf(%t> = J-1 -In
P
In = @i,,+,f(n,t> - K+l,nf(n + l,t> >
where f(n, t) satisfies the distribution function of new phase particle over the number of atoms y1 at the time moment t per single lattice node, F&+l, Wn--l,n are the probabilities for the new phase particles of n and n - 1 size to absorb one atom per unit time and to come from state iz - 1 --+II, Iz +)2 + 1. K,n--l, K+l,n are the probabilities for the new phase particle of n and n + 1 size to emit one atom per unit time and to come from state n --) II - 1, n + 1 + ~1. The probability W depends on the degree of the system’s metastability (excess number of atoms in a solid solution with respect to that in thermodynamically equilibrium solid solution). Consequently, to describe the kinetics of the first-order phase transitions by the complete set of equations, Eq. ( 1) has to be supplemented by the conservation law for the total number of atoms in a solid solution and new phase particles CO
=
C(t)
+
C
nif(ni,
t)
.
Here Co denotes the initial concentration the definite time moment.
(1) and C is the number of solute atoms per lattice node at
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3. Ratio of probabilities of atoms absorption and emission by the new phase particles New phase particles are known [lo] to subdivide into two classes: the particles with number of atoms IZ
+ 47~a~n~‘~o.
(2)
Hereinafter the following notations are used: yi is the chemical potential of atom in the new phase, pT is the chemical potential of atom in a solid solution, 4rt a2 n2j3CTis the surface energy on the boundary of new phase particle of size n with a solid solution, G is the surface tension, a = (3c0/2rc)‘/~ or o = 47ca3/3 is the volume per atom in the matrix. By substituting the distribution of the heterophase fluctuations in the expression for the flux of nuclei in the space of sizes the latter vanishes In(h) = 0. Consequently, we obtain K,n+llK+l,n
= exp[pT -
P~(~I/T~
p’(n) = pi + (87c/3)a2m-“3
n < n, ,
(3)
where T denotes the temperature in energy units. For the particles of size II > n, the situation is more complex because the heterophase fluctuations do not exist in such conditions. Hence, let us envisage the virtual (auxiliary) solution with chemical potential per atom being the same as that for the new phase particle of size YEwith respect to the surface energy. Therefore, for each new phase particle of size IZ there is a specific virtual solution being in the equilibrium with this particle, i.e. K+l,n = E++,, where Fn,n+l is the probability per unit time for new phase particle of size II to absorb atom from the virtual solution. For the dilute solution where the interaction of solute components is neglected, the virtual solution is equal to the solution with equilibrium concentration of solute components for the new phase particle of definite size. In this case, solute atoms in both true and virtual solution interact only with the matrix and, consequently, the energetical barriers and environments of solute atoms are identical in both solutions. For the concentrated solution one has to take into account the interaction of solute atoms. In this case, the virtual solution for the new phase particle of size IZ appears to be the true solution with part of atoms immobilized (frozen) and the rest of mobile atoms being in equilibrium with the given particle of new phase. Therefore, the number of jumps per unit time of mobile atoms to the new phase particle equals to the number of atoms, emitted by a particle to the solution. On the other hand, the mobile atom from the virtual solution has the same probability per unit time to jump to the new phase particle as that for the atom to jump from the true solution by virtue of their potential barrier and environments being equal because in both events they jump from the solution to the particle. The environment does not change during the jump. Consequently, the
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probability of jump from the true and virtual solution (which by definition is equal to the probability of atom emission by a new phase particle) have the same kinetic multiplier and differ only by the configuration, favorable for the jumps. Since the frequency of atom jumps from one phase to another is small compared to the frequency of atomic oscillations, with sound accuracy one can consider atoms to be in thermodynamical~quilibrium with the solution of a given composition. Consequently, by using the condition K+l,n = IKl;n+lwe obtain
(4) where p and b signifies the probability of the favorable configurations for the atoms to jump to the new phase particle of size n from the true and virtual solutions with average (thermodynamical) energy of these subsystems. Since p and j coincide with the most probable states of subsystems with the definite composition, the following relation is known to be valid for p and F under such conditions: pnr=1,
@Af=l,
where Ar and Ap denote the statistical weights of the states with the given thermodynamical energy for the configuration which corresponds to this energy and has one atom in the conditions, favorable for the jump. Thus, we obtain p/j? =
Al’jAr
= exp(AS - AS),
(5)
where AS and AS signify the entropies per atom in the virtual and true solutions. The interaction in the neutral condensed media is known to be short range, so the interaction of the new phase particle and solid solution proceeds only through the interface solution. Since true and virtual solutions have the same temperature, they can be considered as a closed system being in thermal equilibrium with the given composition. This implies AS-AS=AS,,
(6)
where ST denotes the change of the total entropy of the closed system virtual-true solution with definite composition caused by the transition of one atom from true solution to the virtual one. Since AS and AS can be always added by So the entropy of the rest of the system virtual-true solution, which does not change in course of transition of one atom from the true solution to the virtual one, as we have already mentioned, the chemical potential per atom in the virtual solution for the new phase particle of size it is equal to the chemical potential in the new phase particle of size IZ with respect to the surface energy b = $(n)= pi + 8rca20n -‘13/3. Consequently, for n > n&r > ,2(n)) change of the total entropy ASr > 0 and at the same time it is known that R = -T max = -IRmaxI = ~IA@1 = pT - ~‘(4 (7) T T ’ T where R,,, is the maximal work which the closed system can make in course of reversible transition to the state of more equilibrium. In this event at the constant temperature /A@1 = pT - $(n), IZ> n,. Thereby, the ratios of probabilities of absorption and emission in event of 12> n, can be written in the form AS, AS, = --&,,,
WI&n+1 AF PT - P%) = exp AS,, = exp wn+l,n = ar T
.
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The ratio of probabilities is known from Eqs. (3) and (8) to have universal form for any II, i.e. the relations Eqs. (3) and (8) have the equal form and cover all size range of new phase particles. Such a universal definition of probabilities ratio yields a universal, simple method to obtain the coefficients of the kinetic equation ( 1). This method can be easily extended to various cases, e.g. when the new phase particles have the stochiometric composition [ 14, 151, when the interaction of solute components with elastic field which arises in the vicinity of new phase particle needs to be taken into account [ 141, interaction of the solute components [ 161, nucleation of gas-filled voids [ 171, nucleation in overcooled melts [18], influence of surface effects [14], etc. The steady-state solution (1) apparently describes the heterophase fluctuations in the event of the equilibrium solution. Thus, by writing Eq. (1) in the form
af tg
=
In-1
-
Ll = K-l,n.f (n>
-K+l,n.f(n + 1) and after substituting
[
Wn,n_,
1
“f-(n)
1
[~f(;‘;‘l, - 1] =o,
(9)
in Eq. (8) with the following:
PT- PW _-----=T
A@(n + 1) - A@(n) T
we obtain f(n)=Aexp
K-l,n .f(n - 1) _
[
-F
>
/L-$(nT
1) = - A@(n) - A@(n - 1) ,
c1oj
T
1.
(11)
To provide a consistent macroscopic description of first-order phase transitions, one should treat the transition of atom to the new phase as requiring formation work which depends on the particle size IZ only through the surface energy. Naturally, when n m 1, such an extrapolation describes new phase particles only qualitatively because the surface energy for n N 1 becomes a poorly determined parameter. But, provided that the volume of the new phase particles is fairly large and their average size fi B 1, the number of small particles is low and, consequently, their contribution to the various characteristics of solid state is small as well. Thus, the accepted approximation is fairly accurate in this case. The same approximation permits us to treat the solute components as the new phase nuclei with zero formation work, which implies f WI n_O = A = c .
4. Transition from the difference to the differential form of the kinetic equation To describe the kinetics of all stages of the diffusive decomposition derivative of distribution function d2f/an2 with respect to n has to be is this derivative which makes possible for new phase particles of size size n >n,. Higher derivatives in the case of yt, % 1 are substantially can be neglected. Consequently, the condition which permits to apply y2, % 1.
of a solid solution the second taken into account because it n < IZ, to achieve supercritical less than the second one and the derived equation reads as
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394
The size spectrum of new phase particles of size range n manner. This is related to the fact that although the difference spectrum, the character of its solution forces the differential result as well. Thus, by using the relations (8) and (10) we rewrite the form af,,_ at
_I n ’
=-_
1 is also described in a straightforward equation must be applied to this size equation to give qualiatively correct difference
equations
in the differential
=-S n
an
% n,n+l e -A~(n)lT[eA~(n)lrf(,, an
t> _ eA@W)/~f(,
_ 1, t>l
(12) Evidently, for the equilibrium case when aflat = 0 and A@(n) > 0 for any n the heterophase fluctuations become the solution of Eq. ( 12). Thereby, the conservation law Eq. ( 1) takes the straightforward form co = C(t) +
s0
c*3f(n, t)n dn .
(13)
The relations ( 12) and ( 13) form the complete set of equations which describes phase transition kinetics for a solid solution in any stage. The first term of the expression for I,, is seen from Eq. (12) to denote the “hydrodynamical” flux in space of sizes and the coefficient by this term signifies conventional growth rate for the new phase particle of size n: dn/dt = - K,n+l l/T 6A@/6n .
(14)
The second term in the expression for 1, (12) is proportional to the first derivative of the distribution function and represents diffusion term in the expression for I,, in the space of sizes and coefficient by this term is the diffusion factor in the space of sizes, (15)
Q, = %,n+i >O . It can be seen from Eq. (14) that for &A@/% > 0
dn/dt < 0 ,
the relation which determines
6A@/6n < 0
dnldt > 0,
(16)
n, reads
6A@/6n = 0.
(17)
In case of a dilute solid solution we have
A@(n)
p=-nlnC+-n T
c 00
47ma2 2,3 _ pn213 -y-“B($-A) T
P-0.
(18)
I/ V. Slezov I Physics Reports 288 (1997)
where C,, denotes the equilibrium solution. 1 6AQi = _~ -ln$ T 6n aAl3=
ln
_-1
of the solute in a solid
$
-1
-1
C
s2A@
at the definite temperature
cx)
CX
B[
concentration
395
389408
P
’
-d/3
Here the following notations are used: CIis the number ( CIN 1) [ 14,151, D is the diffusion factor of solute in the lattice; a is the lattice parameter (distance between the nodes of a lattice).
5. Steady-state distribution function and flux in the space of sizes on the stage of nucleation in the domain 1
f I”=9=
f(s)>
yields 9 exp(A@(n’)/T)
I=
(C - &)
f(n)=Cexp
Wn’,n’+l
(-q)
(20)
n
bev(A@(n’)lT) dn, J(a,b) = > s 1’
dn’ -’
u
Wn’,n’+l
‘$$ +f(g)exp (-Am(nTi ““‘“‘)$$$,
(21)
.4 I 11 --I
$(g)=f(g)exp
[y]
1
f(g)=1
At the point g = n, + Fn the term with derivative
$
can be neglected
in the expression
for 1.
396
V. V.
Since the expression
J
’ ewWn’)lT
0
Reports 288 (1997) 389408
for A@(n) has the sharp peak at y1= n, we obtain dn,
Wn',n'+l
where 6n, is expressed
SlezovlPhysics
=
O”ex@W’)lT
dn,
=
Wn',n'+l
J -cxY
hfi
-exp WnC,&+l
A@(%>
[
~
T
(22)
1’
by the relation
By using Eq. (22) we rearrange the expression
for the flux I as
expWW/T) ] -’ =
Wdd,
g”Clerp
(-F)
,
(23)
where the definition of J(a,b) is given by Eq. (21). Since the expression $g < C holds at 6n > &zc, which can be easily verified, the second term in Eq. (23) can be neglected because it is small. Consequently, the expression f(n)=Cexp
(-F)
=Cexp(-F)
kerfc (2) k [l -erf
(&$)I
(24)
holds. Here erf (x) = - erf (-x) signifies the error tinction. Naturally, in such approximation Eq. (24) is not valid in the immediate vicinity of n = g, but the function f(g) is known. In the immediate vicinity of n N g the relation (2 1) is to be used.
6. Estimate of the relaxation time for the establishment of the system’s steady state in the domain l
(25) A&=
A@-lnW+const.
at n>l,
V. K Slezov! Physics Reports 288 (1997)
--1 &A$ =_--1 6A@ T 6n T Fn
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391
l “W=B(&&)-;? w 6n
=&kc&) 1 h2A& -------_ T 6n2
g(l-&)=-$r&.
Firstly, we consider the solution within the interval 1
= a af/ih
+ b 8f/i3n2 - df ,
(26)
where a = -dn/dt
= WFA@/Gn > 0,
b = W(n),
Let us rearrange f”=fit + $, where $ is the steady-state which has to vanish reads
d = - Wh2A@/6n2 > 0. solution of Eq. (1). So the perturbation
$(n, 0) = - ~tO(n, - 6n, - n)l?(n - 1). Hereby we take into account that fj+, = f(n) < 0, n > 1, where f(n) is the heterophase fluctuations which remain in the solution after the transition to the metastable state. The exact solution of Eq. (26) with the initial condition $(n, z)Izzo = $(n, o) can be found rewriting f” in the form f”= exp[-dT+cp(z+n/a)]p(n, z), where q is chosen to retain only the second derivative of p with respect to n on the right-hand side of Eq. (26). ( -d r - -i;r)
$(n,t)=exp
&[~$(n’)exp(-a(n~n’)
s
=
exp(-W
=
exp(-dz) exp(-(“lc)‘) 2&z
O”
- (nTbz)z)
dn’
dn’
~~$(n’)dn’.
(27)
In the above relation we accounted for the fast descent of $(n’) starting from n’ = 1. Decaying factor in the exponent is the Green function of Eq. (26). Let us now obtain the maximal relaxation time z,,,. It follows from Eq. (27) that 1 ”
=
d + a2/4b ’
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398
(28)
b max- Wn,n+, In+ = 3aCnf/3 ) Therefore,
afnin
=
4bmx
we obtain (29)
In the dimensional
variables Eq. (29) reads
7, = f(a2/DcnC)[ln C/C,lW2 = i(a2/DrxC)nY3/fi.
(30)
Provided that a = 0, Eq. (26) within the interval -6n, 5 n - n, 5 6n, produces $(n, r) = &
exp(-dr)
11
i&n’) dn’ (31)
fs‘t lIn-n,1&, $(n) = { 0 beyond this domain, Therefore,
the relaxation
time can be written in the form
7, = $(a2/cdX)n~J3/~.
(32)
7. Equation for the flux I@, z) and the value of dcldz at the moment zF of cessation of the new phase particles formation and transition to the process of coalescence By differentiating I(n, r) with respect to time and using the kinetic equation we shall obtain important relations for the flux in the space of sizes on the stage of nucleation [14] dI/dr = H$,++,{a21/an2 + (( l/T)6A@/6n)iX/an}
,
(33)
where 0
( >= -1
(C/C)Z ;
qq/T, + 1 .
The steady flux will be established during the time of system relaxation to the steady-state of the distribution function (32) in the domain 1
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399
As it is seen from Eq. (28), the less the absolute value of the coefficient by the first derivative the longer is the relaxation time. Thus, to obtain upper boundary of relaxation time we shall set the small parameter in Eq. (3) (l/T)SA@/Sn
E
0
equal to zero within the considered from Eq. (33) that 1, = ISt 1 - erf [ (
n - n, + 6n, 2Jwz
interval
I
-6n, 5 n - n, < 6n,. Consequently,
n 2 n, - ih, ,
it can be derived
(34)
’
provided that the boundary conditions Il,,=,,-s,,, =Ist and Wn,n+l= K,,n,+l = const. are obeyed. By using Eq. (34) we can derive the time of steady-state establishment z1 at the point n = n, + 6n, or, otherwise, the time of translation of the value of I at the point n, - 6n, to the point n, + 6n,. so that the time r1 reads 2P zl N 6n~/W = --&ln C/C,)-2.
(35)
Therefore, the times of relaxation to the steady state of distribution function and flux within the interval n < n, + ih, are of the same order and differ only by the numerical coefficients, which are of the order of 1. Let us now obtain one more important expression, related with fall of concentration in course of new phase particles growth and rise of the critical size n,. Growth of n,(z) with time permits us to treat the flux as the steady state during the finite time when n
(36)
It follows from Eqs. (35) and (36) that at r1 ~~~ the flux adjusts in the space of sizes and the steady-state flux n < nmax exists. On this stage the “diffusive” term of the kinetic equation plays an important role. Otherwise, if z1 >z2 the flux does not adjust and decreases to the left from critical size n = n, down to negative values at the last stage of a solid solution decomposition and, respectively, there is no formation of new phase particles. At this stage, the “diffusive” term apparently does not play an essential role in the whole range of sizes, for even in the domain of critical size, where the “hydrodynamical” term is virtually equal to zero, the particle stays only for extremely small time and, in view of the critical point motion, the particle virtually does not change its state. Equality of these two characteristic times determine li, at the moment when the nucleation ceases, z1 = &f/W
= z2 = &2,/f& ,
ri, = W/&n, = aC
3jl/2 .
J--
(37)
As it is seen, Eq. (37) is accurate to the factor which is close to unity, but the relation between the determining physical parameters appears correct tic = (dn,/dC)
C ,
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V. V Slezovl Physics Reports 288 (1997) 389d.M
For the case of the dilute solution, Eq. (37) gives the following time r=rr
relation between
C and C at the
(38) In the stage of nucleation Eq. (38) by CO=C~z=,,.
the concentration
shows insignificant
variations
8. The flux in the space of sizes and distribution function for On, is established
and can be substituted
in
+ 6n after the steady state
Within this interval the relation $
j’~nE,lc+ATl -ggr’=&
(g;
holds at 6n > 6n,(&z, Q n,). describes the flux reads
Consequently,
aI/& = W,,,+1(6A@,lT6n)i31/&
within the interval
n > n, + &I the expression
which
= -(an/aqi2/an, (39)
$=g=n,+~n, =I(n,)
,
nc = nc(C(~>>.
Hereby we are setting equality 6n = 6n, because the quasi-steady state has already been established in the domain 6n - 6n,. The solution of Eq. (39) can be derived in the form
g=nc + h,
I(n,z>=l(s,C(zo(z,n)>),
where zo(r,n)
is the characteristics
!f!!! = 3acn2/3 ds
~o(%Y)=z >
To(~, nmax >=
[+-$j=3cKn2i3(ln~-&).
n’:3 - g’/3 = EC ln
=
1
(40)
of Eq. (39). (41)
Concentration C experiences small variation on the stage of nucleation in high susceptibility of the nucleation rate I -exp[-( l/T@@(n)] barrier (l/T)A@(n,) = 1/2/?ni’3 $1. Consequently, the characteristic found by taking into account the infinitesimal&y of the j/n ‘j3 term.
.-l/3
0,
(z - Q)[l - nj’3n_]
’ dz’
z - zo .Ira P
1
1
dz’
T
In 1 + [
of p 9 1, which results height of the potential n, + 6n, can be easily of this we can write
)
- r - r. s TOg1i3 + aC In C/C,(r’
a(z - zo)C In C/Cm
because to the for n 2 In view
&(r
-To) 9
Ii3
lnC
c ,I
- ~0) ln(n/g)‘13
= SI(Z -
ro)C In C/C,
.
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401
Finally, we obtain the relation for ro, zo(z, n) = z -
nli3 - g113+ nAi3ln(n/g))“3
(42)
cxc
Let us now deduce nmax(T) from Eq. (42). Here nmax(r) is the value, which the steady state has extended to at the moment z if started from the point g(z 5 TN). The relation zo(z, nmax) = 0 is obeyed for n,,,(r), which in turn yields the relation niiX - g1’3 + nbi3 ln(nmax/g)‘i3 = az , In event of nmax$g
a = aCo ln(Co/C,)
.
(43)
one can deduce to a fairly high accuracy that
n maxN (g’i3 + UT)3 . The time r has the origin at the moment of steady-state establishment z, for 0 5 n 5 n, + 6n, = g. Thus, in the course of time, nonzero flux and respective distribution function in the space of sizes appear in the domain 05 n <_nmax(z). To derive them explicitly, one has to employ the law of substance conservation (13). It can be conveniently rewritten in the differential form, as follows: dC dz=-0
Oirafndn= at
s
O3-n arn dn = -Ig -s o an
I,, dn .
(44)
Since the expression I(n, z) =
I(& ~o(S,n,7)) >
9
5 n 5 %ax(~),
0,
n >&lax(r) ;
(45)
holds, the integral term in Eq. (44) can be rewritten
in the form
-zo) dz,, =-JI(ro)gdzo = JI(ro)$(t T
T
.
0
The following
dz =
0
(46)
relations have been used while deducing the above formula:
ro(g, 9, r) = r 3 dn
0
dn
---_=
dro
~o(S, %nax,7) = 0 > dn
-d$r - ro) = $(g1’J
+ a(r - 2lJ))3 = 3a(g”3 + a(z - ro))2 ;
(47)
which follow from the relation for zo(n, r) within the domain 0 2 r < rN, where rN designates the time interval of the new phase particles formation. Hence, from Eq. (44) we have 7 dC - = -Z(r)g I(ro) $(r - ro)dro, (48) dz J’0 if provided that the conditions CjrZO= Co and z 5 z N are obeyed. The relation Eq. (48) is fairly straightforward. Variation of the concentration (quantity of substance in the solution) with time is
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determined by the first term on the right-hand side, which stands for the rate of the new phase particles growth at the point g. The second term on the right-hand side designates the rate of the substance build up on the existing particles within the interval 122 g. The expression dn/dz(T - zo) denotes the rate of the particle growth at the moment z if the particle has emerged at the point g at the time moment zo, while I(Q) stands for the number of the particles which have emerged at the point g during the time interval dzo. By introducing cp= 1 - C/Co and noting that cpB 1 at the stage of nucleation z < TN, we obtain
(49)
In fact, the explicit expression for I(cp) has been used in the above relation. exponent for I at the small 4p we get
I=
10expbwl
,
By expanding
the
(50)
where the equalities
(51) hold. Relation (49) apparently represents the complete equation for q within the interval z
ci,= cp(0) ( 1 +
E g1,3] - ~(0)~,~e~“c~~‘a~~(~o)
4=0=0,
&=T, =
(I + ‘(~~~)]
dzo,
$.
(52)
Eq. (52) can be solved by a method of sequential approximations, if substituting known function under the integral instead of q(2). The account of this integral is seen to give a correction - Q(O)’ 4 1. Hence, retaining the first term on right-hand side in the first-order approximation over N @(O) we obtain cp = cj(o)(g”3/4a){(
1 + az/g”3)4 - 1)
Consequently, under this approximation Eq. (52). Evidently, the relation Eq. (53) by substituting f(n, z) at n > g, but the flux in the space of sizes in the domain
,
T
&=7,
=
t
.
we set exp[-n,cp] 21 1 to an accuracy which follows from can be obtained directly from the conservation law Eq. (13) accuracy of such an approach is harder to estimate. The n 2 g is determined by the “hydrodynamical” term, which
V V. Slezovl Physics Reports 288 (1997)
is substantially
f(n37) = 9
larger than the “diffusive”
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403
one in this domain. Hence, we obtain
Z(g, fo(n, 7)) = Zoe-n~rp(To(n,z)) _ dn,d7
- &(ln
3an2i3
5 n I nmax(7),
0<7<7N,
GICJ’
2
(54)
0
ndp41.
At the tip of the interval n E n,,, (z), where the distribution declines to zero, predominant role is played by the “diffusion term”. As it can be seen from the kinetic equation this decline proceeds within the interval
Relation (54) apparently satisfies the respective boundary condition at n = g. Therefore, the distribution function and flux show smooth variations within the domain n
hllax N=JTI(g,z') J n
dz’=
o
f(n’,z)dn’=loz,
(55)
The time moment TN, when the growth of new phase particles ceases, can be derived from Eq. (53). Hence, we have rN 2i (fi”4/&O)(ln
CO/C,)-‘((4%C$n,)
ln CO/Cm)“4 .
(56)
The peak size of new phase particle on the stage of nucleation ni2X(rN) ? RrNCOln co/C, The distribution f(%r)=.@(g
= &“4((4aCi/n,)
can be deduced from Eq. (43) as
ln CO/C,)‘i4 .
function formed during the stage of nucleation - n) + f(& r)&n - g)@nmaX(r) - n) ,
(57) has the form
75
7N .
(58)
Here we do not take into account the detailed behavior of the distribution function at the edge of the interval for it is not important on the majority of occasions. When the nucleation stage is finished and formation of viable particles of new phase ceases, the transition stage begins. During this stage the number of particles virtually does not change iv F! IV($) = N,,, and the particles simply grow. Initial distribution function for this stage is the distribution of particles over the sizes, which has been formed on the stage of nucleation. Since the particles at this stage are fairly large (n % 1 ), the “diffusion clouds” of solute around the new phase particles need to be taken into account when determining their growth rate. These “clouds” in view of the concentration being small, Co < 1, are quasi-steady state and adjust to the size of particle
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K V. Slezovl Physics Reports 288 (1997) 389-408
[ 121. Which means that the flux of solute to the particle can be deduced by solving the respective diffusion problem with certain concentration C near the surface of the particle, which is determined by the joining of this flux with that in the vicinity of the particle [14, 151. So, the relation dnJdz = 3aC,, ln(C,,/C,)n2’3
= 3n’i3(Co - e,,)
(59)
holds. The second term of Eq. (59) designates the flux from the immediate vicinity of the particle, similar to the one addressed by Eq. (48), and the third term denotes the flux to the immediate vicinity of particle from the bulk of a solution. If the relation an’i3 + CO/Cw > 1 is satisfied, then by using Eq. (59), we obtain C, = C,
•t G/an
113
,
dn/dz = 3n’j3 Co .
(60)
Therefore, if CI= a(T), which denotes the height of the barrier for the solute atom to jump to the particle from the immediate vicinity is not anomalously small, then Eq. (60) can be used for the description of the transition stage. The reversed inequality an1/3
nf
p
CO/&,,
,
dn/dr = 3n1’3Co( 1 - N,,,n/Co)
(61)
hold, which results in Dt/a2 = T&f r” n;13/Co = l/N,$ CiJ3 , Let us note that if the particles reach the sizes of an:& % Co/C, during the stage of nucleation, then Eq. (60) should be applied to the growing nuclei. The results obtained are naturally different, but the technique is still the same. In this case, we derive n2/3
zo(n, 7) = 7 -
2
_
2co
9
213
’
n,,,(T)
= (g2j3 + 2Cor)3’2 ,
= 3cg(g2’3 + 2Co(z - ro))“2 ,
(62)
,
N = Zoz,
N,,,,, = Io~N =,/j&
. 0
The transition stage is followed by the final and asymptotic [l&13, 19-211.
in time stage of diffusive decomposition
V. V. Slezovl Physics Reports 288 (1997)
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The developed approach to the investigation of the first-order phase transitions permits us to take into account various factors which influence this process. If the process of incorporation of the solute atom by the particle is not instantaneous and has the absorption frequency v per node and the solute atom “exists” for the finite time z in the adsorbed state on the surface of the particle, then the coefficient a can be derived by taking into account that the barrier for the jumps in the interface layer is equal to that in the bulk of the solution. Therefore, we can infer CI, as follows [ 14,221; z=vr&l
+ vzs).
(63)
On account of the elastic fields being formed in the vicinity of the new phase particle its growth rate will naturally change [ 14, 151, but this change is not essential. It mainly leads to the change of equilibrium concentration C,, as it can be seen from the expression below: C, --+ C, exp((eSwS - E~CO - u)/T) .
(64)
Here eSc.? denotes the elastic energy per atom in the new phase particle, E,,W is the same energy in the bulk of a solid solution, cs and cp are the densities of elastic energy in the interface layer, u = u + co&/&, u is the energy of a solute interaction with the elastic field, as/% is the variation of elastic energy density in the interface layer, related with variation of elasticity modulus upon the concentration. If the solute interaction is strong enough, then it needs to be taken into account, which, as shown in the study [ 161, leads to the replacement of C by cp= C exp[pC/r]. Here p designates the coefficients which account for the interaction of solute atoms (p + p + PC).
9. Kinetics of decomposition
of a solid solution with the complex stoichiometric
composition
In the case of a complex stoichiometric composition of the precipitating phase, the size of a new phase particle is determined by a number of structural elements ~1,and its volume equals to V = YEWS, where co, denotes the volume of a structural element. If the change of volume per atom in the new phase is not essential, then o, = L: viai, where co, denotes the volume of a single solute atom of ith kind and vi are the stochiometric coefficients. Distribution function in the space of sizes is the function of number of structural elements in the new phase particles and time. The growth rate in this event can be deduced [ 191 by using the conditions of stoichiometricity ji/Vi
=jk/vk
=
.. . ,
as follows:
(65) where o, = 4naz/3 and o, = 47cai/3 is the volume per atom of lattice. The term before last on the right-hand side of Eq. (65) denotes the flux to the particle from the interface layer where the concentration equals to Ci, (Gin stands for the equilibrium concentration
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V. V. Slezov I Physics Reports 288 (1997) 389-408
of ith component of solute in the interface layer by the particle of size n). The last term on the right-hand side of Eq. (65) designates the flux from the bulk of a solid solution to the interface layer near the new phase particle. Quantity Ci denotes concentration of ith kind solute atoms in a solid solution. To derive this flux, one has to solve the respective diffusive problem [23]. In view of these fluxes continuity they are equal, which yields the equation for Z;n ln(Z;,/Ci,)
= l/C!i?Z1’3(Cj- 2;,),/2;, .
The above expression
(66)
yields:
‘j3 ln(Ci/Ci,) < 1 or ln(n CT/K,) 4 n-1’3 C(vi/cli), where K,, by definition is K,, = (a) Z;., r” C, for s(iy1 Q Gz or K, = Km exp[-j3n-1/3]. (b) C, P Ci* for ain’i3 9 CiCi’ or K”~ gK,(n C’~)-’ C up’. It follows from the above relations that for the stage of nucleation expression dn/dz in the form
at Cli< 1 it is convenient
to use
(67) Eq. (67) should be used on the stage of nucleation if the values of ai are not anomalously small. At the same time, the growth rate at the stage of nucleation, when nz& % 1, and at the transition stage can be conveniently calculated by the following expression:
(68) The quantity C’inin Eq. (68) needs to be determined from the condition of the fluxes stoichiometricity for the different kinds of atoms and condition of chemical equilibrium ni Cz = K,,. To determine the coefficients in the kinetic equation for f(n, t) one has to compare Eq. (67) with general expression dnldz = - Wn,+, 6A@/6n , where the relation 6A@/Fn = - In (n
CT/Kn)
holds [ 151. For the transition stage and late stage, when the “diffusive” term in the kinetic equation can be neglected, the knowledge of only dn/dr will be sufficient. The law of solute atoms conservation yields the expressions for the relative supersaturation of a solid solution d relatively to the new phase precipitate A=-q=lnF,
Ci,o=Cj+vj/m/dn, cxz cc
#$=-vi .io
a.f-
-dn= ar
0 T
-v,
Ig (I
Gro& 0
- To>dzo
, >
(69)
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Here I is the steady-state flux at the point g or yt,. This flux is obtained by making the following substitutions [ 151 in the expression for single-atom phase I:
with the replacement I + ID/at preceding the above substitutions. After that, the solution and outcomes are equivalent to those of the kinetics of single atom phase with respect to above substitutions. In order to employ Eq. (68) for the derivation of integral term in Eq. (69) the quantity A = const. = Di(Cf - Cl&)/vi needs to be found from the condition of chemical equilibrium for C;,, as follows: (C; - (vi/Di)A)“’ = K, exp[ -fi/n”3]
n
.
i
To estimate the time of the transition stage in the event of the average size n’/3 9 /?, one can let A N (DiCi/Vi),i,, which means that this time estimate can be obtained by substituting DC0 + system. (DiCilvi bn in the expression for a single-component
10. Discussion In conclusion, let us note that the picture of the first-order phase transition in a solid solution for the homogeneous nucleation and under the steady-state conditions is fairly clear. During the stage of nucleation, the fast establishment of a quasi-steady state of undercritical nuclei (n
V. V. Slezov I Physics Reports 288 (1997) 389408
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Acknowledgements This work was performed as a part of a project financed by the Bundesministerium fiir Bildung, Wissenschaft, Forschung und Technologie (BMBF), Germany. The author also acknowledges the financial support given by the SOROS foundation (Grant No. ISSEP-SPU042062).
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