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The time dependent behavior of the ostwald ripening for the finite volume fraction
~cto metall. Vol. 35, No. 4, pp. 915-922, 1987 Printed in Great Britain. All rights reserved
Copyright
0
OOOI-6160/87 $3.00 + 0.00 1987 Pergamon Journals Ltd
THE TIME DEPENDENT BEHAVIOR OF THE OSTWALD RIPENING FOR THE FINITE VOLUME FRACTION Y. ENOMOTO, K. KAWASAKI and M. TOKUYAMAt Department of Physics, Faculty of Science, Kyushu University 33, Fukuoka 812, Japan (Received
30 June 1986; in revised ,firm 6 August 1986)
Abstract-On the basis of the theory of Tokuyama and Kawasaki on the Ostwald ripening, we investigate the time dependent behavior from the late stage into the scaling region. We calculate numerically the transient behavior of the droplet size distribution function and the average droplet radius for various initial conditions. Moreover we compare the obtained results with the experiments in which the transient behavior of the Al-Li, Ni-Al and Ni-Si alloy systems were investigated for various ageing temperatures. The present theoretical results are in good agreement with these experimental results. R&mm&-En nous basant sur la theorie du murissement d’ostwald prop0G.e par Tokuyama et Kawasaki, nous etudions le comportement en fonction du temps. Nous calculons numeriquement le comportement transitoire de la fonction de repartition de taille des gouttes et le rayon moyen des gouttes pour differentes conditions initiales. En outre, nous comparons les resultats obtenus avec des experiences dans lesquelles le comportement transitoire des systemes d’alliages Al-Li, Ni-Al et Ni-Si Ctait etudib pour differentes temperatures de vieillissement. Les rtsultats theoriques que nous presentons sont en bon accord avec ces resultats experimentaux. Zusammenfassung-Mit der Theorie der Ostwald-Vergriiberung von Tokuyama und Kawasaki wird das Zeitverhalten zwischen dem letzten Stadium und dem Skalierungsbereich untersucht. Das Ubergangsverhalten der Verteilungsfunktion der TriipfchengrijBen und der mittlere Tropfchenradius werden fur verschiedene Anfangsbedingungen numerisch berechnet. Die erhaltenen Ergebnisse werden mit Experimenten verglichen, bei denen das Ubergangsverhalten der Al-Li-, Ni-Al- und Ni-Si-Legierungssysteme bei verschiedenen Auslagerungstemperaturen gemessen worden ist. Die theoretischen Ergebnisse stimmen mit den experimentellen gut iiberein.
1. INTRODUCTION The later stage of the ordering
process of decomposition in binary alloy systems quenched from the on-phase region into the two-phase region, the so called Ostwald ripening [l], has been studied for a long time. The first serious theoretical investigation was accomplished by Lifshitz, Slyozov and Wagner [2] (henceforth referred to as the LSW theory). For systems in the dilute limit where the volume fraction q defined by equation (2.4) below, of precipitates is very close to zero, the LSW theory predicted that in the long time limit (what is called the scaling regioc: the stationary scaled droplet size distribution function does exist, where the average radius of droplets grows in time t as t’13and the droplet number density decays as t-‘. However, experimentally observed distribution functions were more symmetric and broader than that of the LSW theory, although the predicted temporal power laws seem to be correct. Therefore, for
tGenera1 Education, Tohwa University, Fukuoka 815, Japan.
the last 20 years or so, various theoretical attempts (31 have been made to include the cooperative effects of droplets which were neglected in the LSW theory. Recently, Marqusee and Ross (MR) [4] as well as Tokuyama and Kawasaki (TK) [5] have used a systematic statistical method and investigated the effects of finite droplet volume fraction q, up to order q ‘P. The TK theory differs from the theories in the followings. (1) The TK theory argued that there are two characteristic stages of coarsening; the intermediate stage where the volume fraction is still changing and the late stage where it is timeand obtained the time evolution independent, equation of the distribution function for each stage. (2) The TK theory pointed out that not only the collisionless drift processes for competitive growth studied by the previous theories but also the softcollision processes play important roles in coarsening. The soft-collision processes, which have been investigated by none of the previous authors, originate from the interaction between droplets which are immobile but are correlated. Such correlations are generated by long time cumulative effects of droplet interactions through the diffusion of the concentra915
916
ENOMOTO
et al.:
OSTWALD
RIPENING
tion field. Very recently Marder [6] investigated such correlation effects. Hereafter we define the late stage to be the stage where the droplet volume fraction is time-independent but the scaled droplet distribution function is varying, while the scaling region means the one which follows the late stage and where the scaled droplet size distribution function is time-independent. All these theories mentioned, however, discussed the distribution function only in the scaling region. Starting from the LSW theory, Venzl[7] investigated the time dependent behavior of the droplet distribution functions and then concluded that the distribution function of the LSW theory in the scaling region is an attractor for various initial conditions. Venzl also discussed the effects of the finite volume fraction [8], but did not obtain the concrete forms of the distribution function. In the present paper, we use the TIC theory to study the transient behavior of the distribution function for the finite volume fraction from the late stage into the scaling region, and compare the results with experimental data. Until now many experiments [9] have been performed for various binary alloy systems. However, only a few experiments are suitable for comparison with the theory as will be discussed below. In some experiments the droplet volume fraction changed in time throughout periods of observations, in some the precipitating droplets were not spherical, and in some the necesssary quantities such as the diffusion coefficient and the capillarity length were not investigated. Therefore, we pick up the following three experiments for comparison with our computational results; (1) the Ni-6.5 wt% Si system studied by Rastogi and Ardell [lo], (2) the Ni-6.05 wt% Al system studied by Hirata and Kirkwood [l I], and (3) the Al-l.87 wt% Li system studied by Eguchi et al. [12]. In particular, the experiments by Eguchi et al. are the first systematic investigations to focus on the transient behavior from the beginning of the late stage into the scaling region. In Section 2 we briefly summarize the kinetic equation of the distribution function of the TK theory and reduce it to a more tractable form. In Section 3 we calculate the resulting’equation for various initial conditions and investigate the transient behavior. In Section 4 we compare the obtained results with those of the experiments. Section 5 concludes the paper. In the Appendix we discuss the corrections due to the time dependence of the droplet volume fraction.
2. DERIVATION
OF THE KINETIC
EQUATIONS
In the present section we briefly summarize the previous results about the kinetic equations of the late stage and reduce these equations to more tractable forms. Let f(R, t) denote the single droplet size distribution function per unit volume with. radius R at time
FOR FINITE
VOLUME
FRACTION
t. We have obtained the time evolution equation of in the late stage up to order q”’ [5] as
f(R,t)
(Rt).
(2.1)
Here the drift terms contain 1(p) and u(p) defined as I(P)=l-P
(2.2)
a(P)=P+J,(r)-_p)
(2.3)
where p is the relative droplet radius given by p = R/R(t), R(t) the average radius, ecthe capillary length and D the diffusion coefficient of the solute in the matrix. Here the droplet volume fraction q(t), the average radius ir(t) and nth moment m,(t) are defined, respectively, by R3f(R, t) dR
q(t) = (4x/3)
(2.4)
R(t)=lnl(R,t)dRIS/(R,I)dR (2.5)
m.(t)=lpli(R.i)dRlSI(R,l)dR (2.6) where the integrals over R always run from 0 to co. The term with S(R,t) in equation (2.1) describes the effect of the soft-collision and is defined as t(R, >tlf(R, >t) = hlJ3qlm,o) X
dR, s
dX,g,zK,~d-VG,
tlf@,>t)
(2.7)
I
where M, and Fi, are the operators defined as Mij = J(P,) - P,f(Rj,t) x
Cj
s
d&
s
dX,gtil(p&Pnj
(2.8)
d.9exp[s(l + Pji).Y,Mi,]gij
=
s ’ C1 + ‘jilsiPin(Pj)
(2.9)
with gi. = (R(t)/lX, - Xj])exp[ - IX, - Xjj/Z(t)], l(t) = l/dand
Here n(t) denotes the droplet number density defined by n(t) = /f(R, t)dR, Xi the position vector of the
ENOMOTO et al.:
OSTWALD RIPENING
center of the ith droplet with radius Ri, pi = RJR(t), and Pij the exchange operator between i and j. The droplet volume fraction q(t) is conserved in the late stage, since we have n(P)f(Rt)
dR =
a(P)f(R,t)
FOR FINITE VOLUME FRACTION
917
with Yi= (2 -
Pi)/P
1
1
dR (2.18)
=
t(R,t)f(R,t)
dR = 0.
(2.10)
I Thus we can set in the late stage q(r) =
Q
(2.11)
where Q is the initial total supersaturation. In the previous work [13], we investigated the scaling behavior of the soft-collision term t(R,t)f(R,t) using the assumption that in the scaling region we have j-&t)
r = {aD/R(t)3}s,
t(R,t)f(R,t)=
+ &t)-$(R,R)
(R,t)
(2.19)
with c(R,B) = NCR)-
{P/n(f)} (2.20)
D(R,R)= Pi{&% x
- 1) - R(r)
-&RR)- -W,&/P
(2.21)
Lj = {Ti(t)3/CtD}Zi. E(R,R)
In terms of these variables, we can rewrite equation (2.7) as
W,,tlf(R,,t)
c(R,ii) [
(2.12)
= {n(t)l&t))P,(P)
where p,,(p) is a normalized scaled time-independent distribution function. Thus the moment m, = JP”Po(P)dP 1sa 1so t’rme-independent. In Ref. [14] we obtained numerically the stationary solutions P,,(p) for various Q. In the present paper, since we are dealing with the transient behavior from the late stage region into the scaling region, the ansatz (2.12) cannot be accepted. However, the present method of analysis is similar to the previous one [13]. First, we introduce the dimensionless variables xi = X,//(t),
Since the second term of equation (2.17) gives rise to higher derivative terms, we neglect it, as was done in Ref. [ 131.Thus employing the same procedure as that described in Appendix of Ref. [13], we can finally transform the term c^(R,t)f(R, t) into a simpler form as
= [(B+A+B_/A_){l
xJCZ+(A+
= f(R,)- bM~>~
x (J_ (P,,t)
(2.13)
where the operator f(R,) is given by
A_)
-A_) - l)]/(A+ + A_)
(2.22)
where e(x) is the step function defined by 0(x) = 1 and 0 for x > 0 and x < 0, respectively. Here A, and B, are given by A, =
&R,)f(R,,t)
- 0(1-
dRl(p){h(R,K)
+ h(R,E)}-’
s
= {47~,lN)1
dR, s
dx,G,,l(Pz)f,zGz,(l
+ P,,)
s
x L,p,~(p,)f(R,,t)f(R,,t)
x
(2.14)
& {f(RWR)
1
&t)2/W)
(2.23)
with fi, =
dz exp[r(l + Pji)LiMii]
(2.15)
B, =
dRl(p)‘{h(R,R)
+ h(R,R)j-’
xf(Rt)ln(t) Gij=exp[--(xi-xjl]/(xi-x,1.
(2.16)
Secondly, we reduce the operator 1, to a more tractable form. We divide the term L,Mi, in equation (2.15) into two parts as L,M, = -yi + {R(t)31(pi)/R;}
(2.17)
(2.24)
with h(R,K) = (2 - R/R(t)}/{R/R(t)}3.
These results are the same as those of Ref. [13], except that the scaling assumption (2.12) has not yet been made.
ENOMOTO
918
et al.:
OSTWALD
RIPENING FOR FINITE VOLUME
In conclusion, we obtain a Fokker-Planck kinetic equation for f(R, t) as
type
&W,
,/‘m
on a lattice as
0 c a, < a2 < a3 C 1. .
~I(R,t)=.D~~[-B(R,R)+K(t) x
discrete approximation
FRACTION
9lW. t) (2.25)
70 <
7,
<
72
<
7,
<
with the equal spacing Aa (i.e. a, = nAa) and the discrete dimensionless time with the variable step size A7i
i.e. 7, = 70+ i
with the average source/sink strength B(R,R) given
j=l
by B(R,B) = -1(p) - ,/‘~I~(P)
+ c(RR)J.
(2.26)
3. METHODS OF THE NUMERICAL ANALYSIS AND THE RESULTS Now we discuss our methods of the numerical analysis which in a manner are similar to those to Venzl [7]. First, introducing the dimensionless variables as
a = R/cc,
7 = t/t,,
F(a,z)da
= t~‘f(R, t) dR
(3.1)
with t, = ir(to)3/cxD and t, the time at which the late stage process begins, we can rewrite equation (2.25) and (2.26) as F(a, 7) = lW(a,r)
(3.2)
with
(3.4)
R(z)/a
m,(7) = Sp”F(a,r)
da/lF(a,r)
.
>
(1) the initial distribution is a smooth function of a, (2) it decays rapidly for large a, (3) it has one maximum at some value of a/Z(O) = O(l),
a
=
A71
Since the computational errors generally introduce variations in the volume fraction, sizes of the time steps Azj are automatically adjusted in such a way to satisfy the mass conservation law (2.11) within a 98% accuracy at each step. Hereafter we set Aa = 10-2Z(7), t0 = 0 and AZ, = 10e2. The partial differential equation (3.2) is now solved using the standard implicit formula. The initial distribution function F(a, 0) is the result of the nucleation process and the early growth of droplets. Thus within the present discussion we cannot determine the exact form of F(a,O). However, F(a,O) is supposed to satisfy the following requirements:
(4) - [F(a,0)/a2] aa (3.3)
* . .
(5) ii(
is finite for a-0,
- ii(O)3 is positive.
Assumptions (I)--(3) are justified by experimental observations, (4) is required to avoid divergences in such quantities as am3F(a,0) for a +O, and (5) is to avoid the negative coarsening rate. Let us consider a modified Gaussian as a special class of functions which fulfil the above conditions as
(3.5)
Y(a) = a2 exp[ - (a - aJ2/(2a2,)]
p = a/ii(r)
(3.6)
with the parameters a, and a,,, roughly corresponding to the position of the maximum and the width of Y(a), respectively. The we choose
70 = t&.
(3.7)
da
Experimentally the time to is not determined precisely. However, we find that the operator fi does not depend on the time variable explicitly, but depends on it only implicitly through the moments and the average radius that contain F(a, 7). Therefore, as long as t,, is chosen to be a time in the late stage where equation (3.2) is valid, different choices of to only lead to shifts of the origin of the time and do not affect our analyses in any substantial way. In terms of these variables equation (3.2) is defined on the two dimensional domain (a,7)e[O, co) x [7,,, co) with 7. = to/t,. We solve these equations numerically by using
F(a,O) = const x Y(a)
(3.8)
(3.9)
where const is given by const = 3Q/bnja3Y(a)
da}
(3.10)
according to the mass conservation (2.11). The computations are performed for the parameter values (a,,a,)=(0.5,0.2), (O.S,O.S), (1.5,0.2) and (1.5,0.5) with the initial distributions shown in Fig. 1. From now on we set Q = 0.1 but can obtain the similar results for the different volume fractions. In Fig. 2 we show the average radii Z(7) for various initial conditions. It can be seen that the behavior of
ENOMOTO et al.:
:\ \
I
:
/-7.
\ \ \
919
o5
XLu=Ol,rJ ..L . . .
,<-
_.
i
:\ h :
-_(15,051 -'~-l15,021
: -.-(05,051 : : ---(05,02~
i
i
\
y;
oc,
:
i \ \ ,\j
IOL
:
i j
I
ti_.A
:
:
. ..’
(%I,%)
‘,
:
I\
‘...,\
..
:
I I
FOR FINITE VOLUME FRACTION l.O-.\,
,... :
I I .ok
OSTWALD RIPENING
".,,
10
'...,_ .. 20
30
0
OOl 16'
IO0
IO'
a = R/a
Fig. 1. The initial distribution function Y(a) against for the parameter values (~,,,,a,) = (O&0.2), (O.S,O.S), (1.50.2) and (l.S,OS).
Fig. 3. The time variation of the coarsening rates K(Q,r)/ K(0) for the same parameter values as in Fig. 1 for Q = 0.1.
depends on the initial conditions almost up to with T, = 1, while for 7 > T, all the curves coincide, grow as 7 '/' and approach a common asymptotic value. In Fig. 3 we plot the relative coarsening rate K(Q,z)/K(O), in which K(Q,z) is defined by
Thus we also show the graph of pO(p) by dots in Fig. 4 for comparison. These figures indicate p(p,7) = pO(p) for 7 g T2 = 10’. That is, the system almost falls into the scaling region for 7 > T2, and moreover p,(p) is indeed an attractor for various initial conditions. To confirm these conclusions, in Figs 5 and 6
Z(7) 7 _
T,
K(Q,7)Z(O)’
-$(7)'=
and K(0) = 4/9. Both Fig. 2 and Fig. 3 indicate that beyond 7 = T, the values of K(Q,z) do not depend on the initial conditions and approach the same value. Of course, the asymptotic value of K(Q, 7) is the same as that of Ref. [ 141at Q = 0.1. We should remark that for the systems which have larger average initial radii(i.e.(a,,a,)=(1.5,0.2),(1.5,0.5))K(Q,7)grows towards a common value, while for the systems which have smaller initial radii [i.e. (0.5,0.2), (0.5,0.5)] K(Q,7) decreases towards a common value. The time evolutions of the normalized scaled distribution function p(p, 7) = Z(7)@, r)/jF(a, 7)da are shown in Fig. 4 (a),(b) for (a,,~,) = (0.5,0.2) and (1.5,0.5), respectively. One of our interests is whether these time-dependent distribution functions p(p,r) reach the time-independent scaling solution pO(p) given by Ref. (141, as the time increases to infinity.
0
1.o 2.0
20
(b)
(r?-6 (07 0
lOI3
Fig. 2. The time variations of the average radius for the same parameter as in Fig. 1 for Q = 0.1.
Fig. 4. The time evolution of the normalized droplet size distribution functionsp(p,r) with Q = 0.1 for various times; (a) @,,,,a,) = (0.5,0.2), (b) (1.5,0.5). The dots indicate the result of the time-independent distribution function p,,(p) given by Ref. [14].
ENOMOTO et al.:
920
r
0.5
04
OS'= 0.2
OSTWALD RIPENING
FOR FINITE VOLUME FRACTION 448K
aging
0
.... _ ',_
1 0.1 t
0.01
I
I
I1111111
Id
I
I IIIIII IO3
IO2 T
Fig. 5. The time variation of the standard deviations of the p(p,r) with Q =O.l for the same parameter values as in Fig. 1.
r
10
KS
0=0.1
aoi&
Fig. 7. The time evolution of the normalized droplet size distribution function for an AI-Li alloy with Q = 0.1 and T. = 448 K at various ageing times; (a) 72 ks, (b) 180 ks and (c) 720 ks.
-1.0
IO'
102
IO3
r
Fig. 6. The time variation of the skewness of p(p,r) with Q = 0.1 for the same parameter values as in Fig. 1.
we show the behavior of the standard deviation a and the skewness k, of the distribution function p(p,r), defined by a = dm and k, = ((p - l)‘)/a’. Here (...) denotes the average over p(p,r). These quantities also approach the common values, a = 0.267 and k, = -0.557 at Q = 0.1, which are obtained by Ref. [14]. In conclusion we have investigated the transient behavior from the late stage into the scaling region on the basis of the TK theory. The numerical results show that for 7 > Tl the dependence on the initial condition disappears and, in addition, for T > T, the system falls into the scaling region. The results obtained for r > T2 are in agreement with those of Ref. [14]. 4. COMPARISON OF THE NUMERICAL RESULTS WITH EXPERIMENTS
In the present section we compare the numerical Table
1. The
experimental data for the ageing temperature T., and 1,. See the text for these symbols T. (K)
AI-Li AI-L1 AI-Li Ni-AI Ni-Si
results of the preceding section with experimental data. First, we determine the time scale t, = a(t,J3/uD using experimental data. As was discussed in Section 3, the precise choice of to is not important in our analyses as long as it is in the late stage. Therefore we choose t, to be a time after which the observed droplet volume fraction is almost constant. Next we choose initial distribution functions which closely reproduce experimentally observed distribution functions at t = to. In Table 1 we summarize the time scales and other necessary experimental data for each experiment. In Fig. 7 (a)-(c) we show the time evolution of the normalized distribution function for the G’(Al,Li) precipitates of the Al-l 87 wt% Li alloy with Q = 0.1 and aged at T, = 448 K. The histograms indicate the experimental results. Beyond t = 720 (ks) both the experimentally observed scaled distribution function and the computed one are almost steady or the system seems to be in the scaling region. Thus in Fig. 7 (c) the results of the LSW theory and the Ardell theory are also shown for comparison. In Fig. 8 temporal variations of the average radius,
448 473 498 853 1048
Q 0.1 0.1 0.1 0.0122 0.062
a @ml 0.262 0.248 0.236 0.026 63
to04 12 18 3.6 1.2 18
as well as Q, c(, c, ii
&J (nm) 8.4 11.4 IO.5 4.2 124.7
1,W 25.1 66.1 54.6 1.3 20.6
ENOMOTO et al.:
OSTWALD RIPENING
2.0 30 -g
-
25-
0
20
c IE 2
ng,ng
921
FOR FINITE VOLUME FRACTION
(a)
tampsrofure r
448K
-
15-
0 2 6
10 -
s 6 035 -
Ardell b
030-
i
0.25
-
m &
0.20
-
u+
LSW
z 0 ;j 0.15 -
I
I
I
0.0
P
I -1.0
LSW -
I
10'
lo4
103
Time
(3)
Fig. 8. The time evolution of (a) the average radii, (b) the standard deviations, and (c) the skewness for various ageing temperatures of the AI-Li alloy with Q = 0.1. the standard deviations o and the skewness k, are shown for the Al-Li alloy at various ageing temperatures. The solid lines in the figures indicate the results
of the present calculation. 2.0
Fig. 9. The time evolution of the normalized droplet size distribution function for a Ni-Al alloy with Q = 0.0122 and T, = 853 K at various ageing times; (a) 7.2 ks and (b) 14.4 ks.
In Fig. 9 (a),(b) and Fig. 10 (a),(b) the time evolution of the normalized distribution functions are shown for the 7 '(N&Al) precipitates of the Ni6.05 wt% Al alloy with Q = 0.0122 and T, = 853 K, and for the y’(Ni,Si) precipitates of the Ni-6.5 wt% Si alloy with Q = 0.062 and Ta = 1048 K, respectively. From Table 1 we find that Figs 9(b) and 10(b) 2.0
(a)
10
1.0
0.0
0.0 0.0
(b)
I
1.o
20
P
Fig. 10. The time evolution of the normalized droplet size distribution function for a Ni-Si alloy with Q = 0.062 and T, = 1048 K at various ageing times; (a) 18 ks and (b) 28.8 ks.
ENOMOTO et al.: OSTWALD RIPENING FOR FINITE VOLUME FRACTION
922
REFERENCES 1.0
0.51 16’
I
I
l11111l
I 100
I
I
IllilL Id
Fig. 11.The time evolution of the ratio R,/K for @,,a,) = (0.5,0.2) and (1.5,0.5).
correspond to 7 z 1 and 7 z 0.5, respectively, and thus these systems have not reached the scaling region in the concerned stage. From the above figures we conclude that (1) the present calculation based on the TK theory can satisfactorily reproduce the tendency of the transient behavior of the experimental data, and (2) in the scaling region the present results are in good agreement with those of the experiments.
1. W. Ostwald, Analyrische Chemie, 3rd edn, p. 23. Engleman, Leipzig (1901). 2. I. M. Lifshitz and V. V. Slyozov, J. Phys. Chem. Solid 19, 35 (1961). 3. A. J. Ardell, Acta metall. 20,61 (1972); A. D. Brailsford and P. Wynblatt, Acta metall. 27, 489 (1979). 4. J. A. Marqusee and J. Ross, J. Chem. Phys. 80, 536 (1984). 5. M. Tokuyama and K. Kawasaki, Physicu A123, 386 (1984); K. Kawasaki, Y. Enomoto and M. Tokuyama, Physica A135, 426 (1986). 6. M. Marder, Phys. Rev. Lett. 55, 2953 (1985). I. G. Venzl, Ber. Bunsenges. Phys. Chem. 87, 318 (1983). 8. G. Venzl, Phys. Rev. A31, 3431 (1985). 9. A. J. Ardell and R. B. Nicholson, J. Phys. Chem. Soli& 27, 1793 (1966); D. J. Chellman and A. J. Ardell, Acta metall. 22, 577 (1974); V. Munjall and A. J. Ardell, Acta meraN. 24, 827 (1976). 10. P. K. Rastogi and A. J. Ardell, Acta metall. 19, 321 (1971). 11. T. Hirata and D. H. Kirkwood, Acta metall. 25, 1425 (1977). 12. T. Eguchi, Y. Tomokiyo and S. Matsumura, Phase Transitions. To be published. 13. M. Tokuyama, K. Kawasaki and Y. Enomoto, Physica A134, 323 (1986). 14. Y. Enomoto, M. Tokuyama and K. Kawasaki, Acta metall. 35, 907 (1987).
APPENDIX
5. SUMMARY Starting from the TK theory we first reduced the kinetic equation for the droplet size distribution function in the late stage. Then we numerically solved the resulting equation and investigated the transient behavior of the distribution function as well as its standard deviation and skewness, and the average droplet radius and the coarsening rates. From the above calculations we found that the scaled distribution functions and the coarsening rates gradually approach those given by Ref. [14] regardless of the initial conditions. That is, we concluded that the time-independent scaling solution given by Ref. [14] is an attractor for various initial conditions. Moreover, we compared the present theoretical results with the experiments where the transient behavior of the binary Al-Li, Ni-Al and Ni-Si alloy systems were investigated. The present results well reproduce the tendency of the transient behavior of the above experiments. Acknowledgements-The authors would like to thank Professor T. Eguchi, Dr Y. Tomokiyo and Dr S. Matsumura for many valuable comments on this work and for discussing their experimental results prior to publication. They are also grateful to Professor T. Mori for a number of useful comments on the earlier version of the manuscript. This work was partly financed by the Scientific Research Fund of the Ministry of Education, Science and Culture.
In this Appendix we investigate effects of time dependence of the droplet volume fraction on the results obtained. The TK theory indicates that such corrections mainly come from the difference between the average radius R(1) and the critical radius R,(r) defined by &E(t)= a/A(r) where A(r) is the supersaturation at time 1. In the late stage of the TK theory k(f) reduces to K(r). Therefore we simply replace F?(I)of equation (2.25) and (2.26) by K(t) and study the time variation of R,(I). For the initial conditions we choose, instead of equation (3.10), the following c = 0.8 x 3,/(4$da~(a)r$)
(Al)
so that q(t,) = 0.8 x Q with (a,,a,)=(O.5,0.2), and we determine equation
(1.5,0.5)
the supersaturation
A(r) +
A(r) from the
daF(a,T)’ = Q. (A2) s Other conditions and the computational method are the same as those in Section 3. The ratio &(7)/R(r) = l/[A(r)ii(r)] are shown in Fig. 11. From this figure we find that R, rapidly reduce to R after 7 - 0.7. Since the dependence on the initial condition remains till r z T, = 1 as was discussed in Section 2, the correction of the ttme dependence of the volume fraction is not important since it can be absorbed into the choice of the initial conditions and thus does not alter the results obtained in Section 2.
Copyright
0
OOOI-6160/87 $3.00 + 0.00 1987 Pergamon Journals Ltd
THE TIME DEPENDENT BEHAVIOR OF THE OSTWALD RIPENING FOR THE FINITE VOLUME FRACTION Y. ENOMOTO, K. KAWASAKI and M. TOKUYAMAt Department of Physics, Faculty of Science, Kyushu University 33, Fukuoka 812, Japan (Received
30 June 1986; in revised ,firm 6 August 1986)
Abstract-On the basis of the theory of Tokuyama and Kawasaki on the Ostwald ripening, we investigate the time dependent behavior from the late stage into the scaling region. We calculate numerically the transient behavior of the droplet size distribution function and the average droplet radius for various initial conditions. Moreover we compare the obtained results with the experiments in which the transient behavior of the Al-Li, Ni-Al and Ni-Si alloy systems were investigated for various ageing temperatures. The present theoretical results are in good agreement with these experimental results. R&mm&-En nous basant sur la theorie du murissement d’ostwald prop0G.e par Tokuyama et Kawasaki, nous etudions le comportement en fonction du temps. Nous calculons numeriquement le comportement transitoire de la fonction de repartition de taille des gouttes et le rayon moyen des gouttes pour differentes conditions initiales. En outre, nous comparons les resultats obtenus avec des experiences dans lesquelles le comportement transitoire des systemes d’alliages Al-Li, Ni-Al et Ni-Si Ctait etudib pour differentes temperatures de vieillissement. Les rtsultats theoriques que nous presentons sont en bon accord avec ces resultats experimentaux. Zusammenfassung-Mit der Theorie der Ostwald-Vergriiberung von Tokuyama und Kawasaki wird das Zeitverhalten zwischen dem letzten Stadium und dem Skalierungsbereich untersucht. Das Ubergangsverhalten der Verteilungsfunktion der TriipfchengrijBen und der mittlere Tropfchenradius werden fur verschiedene Anfangsbedingungen numerisch berechnet. Die erhaltenen Ergebnisse werden mit Experimenten verglichen, bei denen das Ubergangsverhalten der Al-Li-, Ni-Al- und Ni-Si-Legierungssysteme bei verschiedenen Auslagerungstemperaturen gemessen worden ist. Die theoretischen Ergebnisse stimmen mit den experimentellen gut iiberein.
1. INTRODUCTION The later stage of the ordering
process of decomposition in binary alloy systems quenched from the on-phase region into the two-phase region, the so called Ostwald ripening [l], has been studied for a long time. The first serious theoretical investigation was accomplished by Lifshitz, Slyozov and Wagner [2] (henceforth referred to as the LSW theory). For systems in the dilute limit where the volume fraction q defined by equation (2.4) below, of precipitates is very close to zero, the LSW theory predicted that in the long time limit (what is called the scaling regioc: the stationary scaled droplet size distribution function does exist, where the average radius of droplets grows in time t as t’13and the droplet number density decays as t-‘. However, experimentally observed distribution functions were more symmetric and broader than that of the LSW theory, although the predicted temporal power laws seem to be correct. Therefore, for
tGenera1 Education, Tohwa University, Fukuoka 815, Japan.
the last 20 years or so, various theoretical attempts (31 have been made to include the cooperative effects of droplets which were neglected in the LSW theory. Recently, Marqusee and Ross (MR) [4] as well as Tokuyama and Kawasaki (TK) [5] have used a systematic statistical method and investigated the effects of finite droplet volume fraction q, up to order q ‘P. The TK theory differs from the theories in the followings. (1) The TK theory argued that there are two characteristic stages of coarsening; the intermediate stage where the volume fraction is still changing and the late stage where it is timeand obtained the time evolution independent, equation of the distribution function for each stage. (2) The TK theory pointed out that not only the collisionless drift processes for competitive growth studied by the previous theories but also the softcollision processes play important roles in coarsening. The soft-collision processes, which have been investigated by none of the previous authors, originate from the interaction between droplets which are immobile but are correlated. Such correlations are generated by long time cumulative effects of droplet interactions through the diffusion of the concentra915
916
ENOMOTO
et al.:
OSTWALD
RIPENING
tion field. Very recently Marder [6] investigated such correlation effects. Hereafter we define the late stage to be the stage where the droplet volume fraction is time-independent but the scaled droplet distribution function is varying, while the scaling region means the one which follows the late stage and where the scaled droplet size distribution function is time-independent. All these theories mentioned, however, discussed the distribution function only in the scaling region. Starting from the LSW theory, Venzl[7] investigated the time dependent behavior of the droplet distribution functions and then concluded that the distribution function of the LSW theory in the scaling region is an attractor for various initial conditions. Venzl also discussed the effects of the finite volume fraction [8], but did not obtain the concrete forms of the distribution function. In the present paper, we use the TIC theory to study the transient behavior of the distribution function for the finite volume fraction from the late stage into the scaling region, and compare the results with experimental data. Until now many experiments [9] have been performed for various binary alloy systems. However, only a few experiments are suitable for comparison with the theory as will be discussed below. In some experiments the droplet volume fraction changed in time throughout periods of observations, in some the precipitating droplets were not spherical, and in some the necesssary quantities such as the diffusion coefficient and the capillarity length were not investigated. Therefore, we pick up the following three experiments for comparison with our computational results; (1) the Ni-6.5 wt% Si system studied by Rastogi and Ardell [lo], (2) the Ni-6.05 wt% Al system studied by Hirata and Kirkwood [l I], and (3) the Al-l.87 wt% Li system studied by Eguchi et al. [12]. In particular, the experiments by Eguchi et al. are the first systematic investigations to focus on the transient behavior from the beginning of the late stage into the scaling region. In Section 2 we briefly summarize the kinetic equation of the distribution function of the TK theory and reduce it to a more tractable form. In Section 3 we calculate the resulting’equation for various initial conditions and investigate the transient behavior. In Section 4 we compare the obtained results with those of the experiments. Section 5 concludes the paper. In the Appendix we discuss the corrections due to the time dependence of the droplet volume fraction.
2. DERIVATION
OF THE KINETIC
EQUATIONS
In the present section we briefly summarize the previous results about the kinetic equations of the late stage and reduce these equations to more tractable forms. Let f(R, t) denote the single droplet size distribution function per unit volume with. radius R at time
FOR FINITE
VOLUME
FRACTION
t. We have obtained the time evolution equation of in the late stage up to order q”’ [5] as
f(R,t)
(Rt).
(2.1)
Here the drift terms contain 1(p) and u(p) defined as I(P)=l-P
(2.2)
a(P)=P+J,(r)-_p)
(2.3)
where p is the relative droplet radius given by p = R/R(t), R(t) the average radius, ecthe capillary length and D the diffusion coefficient of the solute in the matrix. Here the droplet volume fraction q(t), the average radius ir(t) and nth moment m,(t) are defined, respectively, by R3f(R, t) dR
q(t) = (4x/3)
(2.4)
R(t)=lnl(R,t)dRIS/(R,I)dR (2.5)
m.(t)=lpli(R.i)dRlSI(R,l)dR (2.6) where the integrals over R always run from 0 to co. The term with S(R,t) in equation (2.1) describes the effect of the soft-collision and is defined as t(R, >tlf(R, >t) = hlJ3qlm,o) X
dR, s
dX,g,zK,~d-VG,
tlf@,>t)
(2.7)
I
where M, and Fi, are the operators defined as Mij = J(P,) - P,f(Rj,t) x
Cj
s
d&
s
dX,gtil(p&Pnj
(2.8)
d.9exp[s(l + Pji).Y,Mi,]gij
=
s ’ C1 + ‘jilsiPin(Pj)
(2.9)
with gi. = (R(t)/lX, - Xj])exp[ - IX, - Xjj/Z(t)], l(t) = l/dand
Here n(t) denotes the droplet number density defined by n(t) = /f(R, t)dR, Xi the position vector of the
ENOMOTO et al.:
OSTWALD RIPENING
center of the ith droplet with radius Ri, pi = RJR(t), and Pij the exchange operator between i and j. The droplet volume fraction q(t) is conserved in the late stage, since we have n(P)f(Rt)
dR =
a(P)f(R,t)
FOR FINITE VOLUME FRACTION
917
with Yi= (2 -
Pi)/P
1
1
dR (2.18)
=
t(R,t)f(R,t)
dR = 0.
(2.10)
I Thus we can set in the late stage q(r) =
Q
(2.11)
where Q is the initial total supersaturation. In the previous work [13], we investigated the scaling behavior of the soft-collision term t(R,t)f(R,t) using the assumption that in the scaling region we have j-&t)
r = {aD/R(t)3}s,
t(R,t)f(R,t)=
+ &t)-$(R,R)
(R,t)
(2.19)
with c(R,B) = NCR)-
{P/n(f)} (2.20)
D(R,R)= Pi{&% x
- 1) - R(r)
-&RR)- -W,&/P
(2.21)
Lj = {Ti(t)3/CtD}Zi. E(R,R)
In terms of these variables, we can rewrite equation (2.7) as
W,,tlf(R,,t)
c(R,ii) [
(2.12)
= {n(t)l&t))P,(P)
where p,,(p) is a normalized scaled time-independent distribution function. Thus the moment m, = JP”Po(P)dP 1sa 1so t’rme-independent. In Ref. [14] we obtained numerically the stationary solutions P,,(p) for various Q. In the present paper, since we are dealing with the transient behavior from the late stage region into the scaling region, the ansatz (2.12) cannot be accepted. However, the present method of analysis is similar to the previous one [13]. First, we introduce the dimensionless variables xi = X,//(t),
Since the second term of equation (2.17) gives rise to higher derivative terms, we neglect it, as was done in Ref. [ 131.Thus employing the same procedure as that described in Appendix of Ref. [13], we can finally transform the term c^(R,t)f(R, t) into a simpler form as
= [(B+A+B_/A_){l
xJCZ+(A+
= f(R,)- bM~>~
x (J_ (P,,t)
(2.13)
where the operator f(R,) is given by
A_)
-A_) - l)]/(A+ + A_)
(2.22)
where e(x) is the step function defined by 0(x) = 1 and 0 for x > 0 and x < 0, respectively. Here A, and B, are given by A, =
&R,)f(R,,t)
- 0(1-
dRl(p){h(R,K)
+ h(R,E)}-’
s
= {47~,lN)1
dR, s
dx,G,,l(Pz)f,zGz,(l
+ P,,)
s
x L,p,~(p,)f(R,,t)f(R,,t)
x
(2.14)
& {f(RWR)
1
&t)2/W)
(2.23)
with fi, =
dz exp[r(l + Pji)LiMii]
(2.15)
B, =
dRl(p)‘{h(R,R)
+ h(R,R)j-’
xf(Rt)ln(t) Gij=exp[--(xi-xjl]/(xi-x,1.
(2.16)
Secondly, we reduce the operator 1, to a more tractable form. We divide the term L,Mi, in equation (2.15) into two parts as L,M, = -yi + {R(t)31(pi)/R;}
(2.17)
(2.24)
with h(R,K) = (2 - R/R(t)}/{R/R(t)}3.
These results are the same as those of Ref. [13], except that the scaling assumption (2.12) has not yet been made.
ENOMOTO
918
et al.:
OSTWALD
RIPENING FOR FINITE VOLUME
In conclusion, we obtain a Fokker-Planck kinetic equation for f(R, t) as
type
&W,
,/‘m
on a lattice as
0 c a, < a2 < a3 C 1. .
~I(R,t)=.D~~[-B(R,R)+K(t) x
discrete approximation
FRACTION
9lW. t) (2.25)
70 <
7,
<
72
<
7,
<
with the equal spacing Aa (i.e. a, = nAa) and the discrete dimensionless time with the variable step size A7i
i.e. 7, = 70+ i
with the average source/sink strength B(R,R) given
j=l
by B(R,B) = -1(p) - ,/‘~I~(P)
+ c(RR)J.
(2.26)
3. METHODS OF THE NUMERICAL ANALYSIS AND THE RESULTS Now we discuss our methods of the numerical analysis which in a manner are similar to those to Venzl [7]. First, introducing the dimensionless variables as
a = R/cc,
7 = t/t,,
F(a,z)da
= t~‘f(R, t) dR
(3.1)
with t, = ir(to)3/cxD and t, the time at which the late stage process begins, we can rewrite equation (2.25) and (2.26) as F(a, 7) = lW(a,r)
(3.2)
with
(3.4)
R(z)/a
m,(7) = Sp”F(a,r)
da/lF(a,r)
.
>
(1) the initial distribution is a smooth function of a, (2) it decays rapidly for large a, (3) it has one maximum at some value of a/Z(O) = O(l),
a
=
A71
Since the computational errors generally introduce variations in the volume fraction, sizes of the time steps Azj are automatically adjusted in such a way to satisfy the mass conservation law (2.11) within a 98% accuracy at each step. Hereafter we set Aa = 10-2Z(7), t0 = 0 and AZ, = 10e2. The partial differential equation (3.2) is now solved using the standard implicit formula. The initial distribution function F(a, 0) is the result of the nucleation process and the early growth of droplets. Thus within the present discussion we cannot determine the exact form of F(a,O). However, F(a,O) is supposed to satisfy the following requirements:
(4) - [F(a,0)/a2] aa (3.3)
* . .
(5) ii(
is finite for a-0,
- ii(O)3 is positive.
Assumptions (I)--(3) are justified by experimental observations, (4) is required to avoid divergences in such quantities as am3F(a,0) for a +O, and (5) is to avoid the negative coarsening rate. Let us consider a modified Gaussian as a special class of functions which fulfil the above conditions as
(3.5)
Y(a) = a2 exp[ - (a - aJ2/(2a2,)]
p = a/ii(r)
(3.6)
with the parameters a, and a,,, roughly corresponding to the position of the maximum and the width of Y(a), respectively. The we choose
70 = t&.
(3.7)
da
Experimentally the time to is not determined precisely. However, we find that the operator fi does not depend on the time variable explicitly, but depends on it only implicitly through the moments and the average radius that contain F(a, 7). Therefore, as long as t,, is chosen to be a time in the late stage where equation (3.2) is valid, different choices of to only lead to shifts of the origin of the time and do not affect our analyses in any substantial way. In terms of these variables equation (3.2) is defined on the two dimensional domain (a,7)e[O, co) x [7,,, co) with 7. = to/t,. We solve these equations numerically by using
F(a,O) = const x Y(a)
(3.8)
(3.9)
where const is given by const = 3Q/bnja3Y(a)
da}
(3.10)
according to the mass conservation (2.11). The computations are performed for the parameter values (a,,a,)=(0.5,0.2), (O.S,O.S), (1.5,0.2) and (1.5,0.5) with the initial distributions shown in Fig. 1. From now on we set Q = 0.1 but can obtain the similar results for the different volume fractions. In Fig. 2 we show the average radii Z(7) for various initial conditions. It can be seen that the behavior of
ENOMOTO et al.:
:\ \
I
:
/-7.
\ \ \
919
o5
XLu=Ol,rJ ..L . . .
,<-
_.
i
:\ h :
-_(15,051 -'~-l15,021
: -.-(05,051 : : ---(05,02~
i
i
\
y;
oc,
:
i \ \ ,\j
IOL
:
i j
I
ti_.A
:
:
. ..’
(%I,%)
‘,
:
I\
‘...,\
..
:
I I
FOR FINITE VOLUME FRACTION l.O-.\,
,... :
I I .ok
OSTWALD RIPENING
".,,
10
'...,_ .. 20
30
0
OOl 16'
IO0
IO'
a = R/a
Fig. 1. The initial distribution function Y(a) against for the parameter values (~,,,,a,) = (O&0.2), (O.S,O.S), (1.50.2) and (l.S,OS).
Fig. 3. The time variation of the coarsening rates K(Q,r)/ K(0) for the same parameter values as in Fig. 1 for Q = 0.1.
depends on the initial conditions almost up to with T, = 1, while for 7 > T, all the curves coincide, grow as 7 '/' and approach a common asymptotic value. In Fig. 3 we plot the relative coarsening rate K(Q,z)/K(O), in which K(Q,z) is defined by
Thus we also show the graph of pO(p) by dots in Fig. 4 for comparison. These figures indicate p(p,7) = pO(p) for 7 g T2 = 10’. That is, the system almost falls into the scaling region for 7 > T2, and moreover p,(p) is indeed an attractor for various initial conditions. To confirm these conclusions, in Figs 5 and 6
Z(7) 7 _
T,
K(Q,7)Z(O)’
-$(7)'=
and K(0) = 4/9. Both Fig. 2 and Fig. 3 indicate that beyond 7 = T, the values of K(Q,z) do not depend on the initial conditions and approach the same value. Of course, the asymptotic value of K(Q, 7) is the same as that of Ref. [ 141at Q = 0.1. We should remark that for the systems which have larger average initial radii(i.e.(a,,a,)=(1.5,0.2),(1.5,0.5))K(Q,7)grows towards a common value, while for the systems which have smaller initial radii [i.e. (0.5,0.2), (0.5,0.5)] K(Q,7) decreases towards a common value. The time evolutions of the normalized scaled distribution function p(p, 7) = Z(7)@, r)/jF(a, 7)da are shown in Fig. 4 (a),(b) for (a,,~,) = (0.5,0.2) and (1.5,0.5), respectively. One of our interests is whether these time-dependent distribution functions p(p,r) reach the time-independent scaling solution pO(p) given by Ref. (141, as the time increases to infinity.
0
1.o 2.0
20
(b)
(r?-6 (07 0
lOI3
Fig. 2. The time variations of the average radius for the same parameter as in Fig. 1 for Q = 0.1.
Fig. 4. The time evolution of the normalized droplet size distribution functionsp(p,r) with Q = 0.1 for various times; (a) @,,,,a,) = (0.5,0.2), (b) (1.5,0.5). The dots indicate the result of the time-independent distribution function p,,(p) given by Ref. [14].
ENOMOTO et al.:
920
r
0.5
04
OS'= 0.2
OSTWALD RIPENING
FOR FINITE VOLUME FRACTION 448K
aging
0
.... _ ',_
1 0.1 t
0.01
I
I
I1111111
Id
I
I IIIIII IO3
IO2 T
Fig. 5. The time variation of the standard deviations of the p(p,r) with Q =O.l for the same parameter values as in Fig. 1.
r
10
KS
0=0.1
aoi&
Fig. 7. The time evolution of the normalized droplet size distribution function for an AI-Li alloy with Q = 0.1 and T. = 448 K at various ageing times; (a) 72 ks, (b) 180 ks and (c) 720 ks.
-1.0
IO'
102
IO3
r
Fig. 6. The time variation of the skewness of p(p,r) with Q = 0.1 for the same parameter values as in Fig. 1.
we show the behavior of the standard deviation a and the skewness k, of the distribution function p(p,r), defined by a = dm and k, = ((p - l)‘)/a’. Here (...) denotes the average over p(p,r). These quantities also approach the common values, a = 0.267 and k, = -0.557 at Q = 0.1, which are obtained by Ref. [14]. In conclusion we have investigated the transient behavior from the late stage into the scaling region on the basis of the TK theory. The numerical results show that for 7 > Tl the dependence on the initial condition disappears and, in addition, for T > T, the system falls into the scaling region. The results obtained for r > T2 are in agreement with those of Ref. [14]. 4. COMPARISON OF THE NUMERICAL RESULTS WITH EXPERIMENTS
In the present section we compare the numerical Table
1. The
experimental data for the ageing temperature T., and 1,. See the text for these symbols T. (K)
AI-Li AI-L1 AI-Li Ni-AI Ni-Si
results of the preceding section with experimental data. First, we determine the time scale t, = a(t,J3/uD using experimental data. As was discussed in Section 3, the precise choice of to is not important in our analyses as long as it is in the late stage. Therefore we choose t, to be a time after which the observed droplet volume fraction is almost constant. Next we choose initial distribution functions which closely reproduce experimentally observed distribution functions at t = to. In Table 1 we summarize the time scales and other necessary experimental data for each experiment. In Fig. 7 (a)-(c) we show the time evolution of the normalized distribution function for the G’(Al,Li) precipitates of the Al-l 87 wt% Li alloy with Q = 0.1 and aged at T, = 448 K. The histograms indicate the experimental results. Beyond t = 720 (ks) both the experimentally observed scaled distribution function and the computed one are almost steady or the system seems to be in the scaling region. Thus in Fig. 7 (c) the results of the LSW theory and the Ardell theory are also shown for comparison. In Fig. 8 temporal variations of the average radius,
448 473 498 853 1048
Q 0.1 0.1 0.1 0.0122 0.062
a @ml 0.262 0.248 0.236 0.026 63
to04 12 18 3.6 1.2 18
as well as Q, c(, c, ii
&J (nm) 8.4 11.4 IO.5 4.2 124.7
1,W 25.1 66.1 54.6 1.3 20.6
ENOMOTO et al.:
OSTWALD RIPENING
2.0 30 -g
-
25-
0
20
c IE 2
ng,ng
921
FOR FINITE VOLUME FRACTION
(a)
tampsrofure r
448K
-
15-
0 2 6
10 -
s 6 035 -
Ardell b
030-
i
0.25
-
m &
0.20
-
u+
LSW
z 0 ;j 0.15 -
I
I
I
0.0
P
I -1.0
LSW -
I
10'
lo4
103
Time
(3)
Fig. 8. The time evolution of (a) the average radii, (b) the standard deviations, and (c) the skewness for various ageing temperatures of the AI-Li alloy with Q = 0.1. the standard deviations o and the skewness k, are shown for the Al-Li alloy at various ageing temperatures. The solid lines in the figures indicate the results
of the present calculation. 2.0
Fig. 9. The time evolution of the normalized droplet size distribution function for a Ni-Al alloy with Q = 0.0122 and T, = 853 K at various ageing times; (a) 7.2 ks and (b) 14.4 ks.
In Fig. 9 (a),(b) and Fig. 10 (a),(b) the time evolution of the normalized distribution functions are shown for the 7 '(N&Al) precipitates of the Ni6.05 wt% Al alloy with Q = 0.0122 and T, = 853 K, and for the y’(Ni,Si) precipitates of the Ni-6.5 wt% Si alloy with Q = 0.062 and Ta = 1048 K, respectively. From Table 1 we find that Figs 9(b) and 10(b) 2.0
(a)
10
1.0
0.0
0.0 0.0
(b)
I
1.o
20
P
Fig. 10. The time evolution of the normalized droplet size distribution function for a Ni-Si alloy with Q = 0.062 and T, = 1048 K at various ageing times; (a) 18 ks and (b) 28.8 ks.
ENOMOTO et al.: OSTWALD RIPENING FOR FINITE VOLUME FRACTION
922
REFERENCES 1.0
0.51 16’
I
I
l11111l
I 100
I
I
IllilL Id
Fig. 11.The time evolution of the ratio R,/K for @,,a,) = (0.5,0.2) and (1.5,0.5).
correspond to 7 z 1 and 7 z 0.5, respectively, and thus these systems have not reached the scaling region in the concerned stage. From the above figures we conclude that (1) the present calculation based on the TK theory can satisfactorily reproduce the tendency of the transient behavior of the experimental data, and (2) in the scaling region the present results are in good agreement with those of the experiments.
1. W. Ostwald, Analyrische Chemie, 3rd edn, p. 23. Engleman, Leipzig (1901). 2. I. M. Lifshitz and V. V. Slyozov, J. Phys. Chem. Solid 19, 35 (1961). 3. A. J. Ardell, Acta metall. 20,61 (1972); A. D. Brailsford and P. Wynblatt, Acta metall. 27, 489 (1979). 4. J. A. Marqusee and J. Ross, J. Chem. Phys. 80, 536 (1984). 5. M. Tokuyama and K. Kawasaki, Physicu A123, 386 (1984); K. Kawasaki, Y. Enomoto and M. Tokuyama, Physica A135, 426 (1986). 6. M. Marder, Phys. Rev. Lett. 55, 2953 (1985). I. G. Venzl, Ber. Bunsenges. Phys. Chem. 87, 318 (1983). 8. G. Venzl, Phys. Rev. A31, 3431 (1985). 9. A. J. Ardell and R. B. Nicholson, J. Phys. Chem. Soli& 27, 1793 (1966); D. J. Chellman and A. J. Ardell, Acta metall. 22, 577 (1974); V. Munjall and A. J. Ardell, Acta meraN. 24, 827 (1976). 10. P. K. Rastogi and A. J. Ardell, Acta metall. 19, 321 (1971). 11. T. Hirata and D. H. Kirkwood, Acta metall. 25, 1425 (1977). 12. T. Eguchi, Y. Tomokiyo and S. Matsumura, Phase Transitions. To be published. 13. M. Tokuyama, K. Kawasaki and Y. Enomoto, Physica A134, 323 (1986). 14. Y. Enomoto, M. Tokuyama and K. Kawasaki, Acta metall. 35, 907 (1987).
APPENDIX
5. SUMMARY Starting from the TK theory we first reduced the kinetic equation for the droplet size distribution function in the late stage. Then we numerically solved the resulting equation and investigated the transient behavior of the distribution function as well as its standard deviation and skewness, and the average droplet radius and the coarsening rates. From the above calculations we found that the scaled distribution functions and the coarsening rates gradually approach those given by Ref. [14] regardless of the initial conditions. That is, we concluded that the time-independent scaling solution given by Ref. [14] is an attractor for various initial conditions. Moreover, we compared the present theoretical results with the experiments where the transient behavior of the binary Al-Li, Ni-Al and Ni-Si alloy systems were investigated. The present results well reproduce the tendency of the transient behavior of the above experiments. Acknowledgements-The authors would like to thank Professor T. Eguchi, Dr Y. Tomokiyo and Dr S. Matsumura for many valuable comments on this work and for discussing their experimental results prior to publication. They are also grateful to Professor T. Mori for a number of useful comments on the earlier version of the manuscript. This work was partly financed by the Scientific Research Fund of the Ministry of Education, Science and Culture.
In this Appendix we investigate effects of time dependence of the droplet volume fraction on the results obtained. The TK theory indicates that such corrections mainly come from the difference between the average radius R(1) and the critical radius R,(r) defined by &E(t)= a/A(r) where A(r) is the supersaturation at time 1. In the late stage of the TK theory k(f) reduces to K(r). Therefore we simply replace F?(I)of equation (2.25) and (2.26) by K(t) and study the time variation of R,(I). For the initial conditions we choose, instead of equation (3.10), the following c = 0.8 x 3,/(4$da~(a)r$)
(Al)
so that q(t,) = 0.8 x Q with (a,,a,)=(O.5,0.2), and we determine equation
(1.5,0.5)
the supersaturation
A(r) +
A(r) from the
daF(a,T)’ = Q. (A2) s Other conditions and the computational method are the same as those in Section 3. The ratio &(7)/R(r) = l/[A(r)ii(r)] are shown in Fig. 11. From this figure we find that R, rapidly reduce to R after 7 - 0.7. Since the dependence on the initial condition remains till r z T, = 1 as was discussed in Section 2, the correction of the ttme dependence of the volume fraction is not important since it can be absorbed into the choice of the initial conditions and thus does not alter the results obtained in Section 2.