Finite volume fraction effects on coarsening—II. Interface-limited growth

Acta metall, mater. Vol. 39, No. 8, pp. 2013-2016, 1991 Printed in Great Britain.All rights reserved

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FINITE VOLUME FRACTION EFFECTS ON COARSENING--II. INTERFACE-LIMITED GROWTH Y. ENOMOTO Department of Physics, Faculty of Science, Nagoya University, Nagoya 464-01, Japan (Received 22 November 1989; in revised form 20 August 1990) Abstract--We study the asymptotic behavior of the interface-limited coarsening on the basis of the statistical theory recently developed by us. This theory takes into account the cooperative effects arising from statistical correlations among precipitates. Using the theory, we discuss the finite precipitate volume fraction effects on the form of the precipitate size distribution function and the coarsening rate. Then we find that such effects play important roles on these properties, even though the volume fraction is small. Rrsumr--Nous &udions le comportement asymptotique du grossissement des grains limit6 par les interfaces, sur la base d'une throrie statistique que nous avons rrcemment drveloppre. Cette throrie tient compte des effets cooprratifs provenant des corrrlations statistiques entre prbciptrs. A l'aide de cette throrie, nous discutons les effets de la fraction volumique finie des prrcipitrs sur la forme de la fonction de distribution des tailles de prrcipitrs et sur la vitesse de grossissement. Ces effets jouent un rfle important sur ces propri&rs, mrme dans le cas d'une fraction volumique peu 61evre. Zmammenfassung---Wir untersuchen das asymptotische Verhalten der grenzfl/ichengehinderten Vergr6berung mit der yon uns kiirzlich entwickelten statistischen Theorie. Diese Theorie beriicksichtigt den yon den statistischen Korrelationen zwischen den Ausscheidungen herriihrenden kooperativen Effekte. Mit dieser Theorie diskutieren wir die Einfliisseder endgiiltigenVolumanteileder Ausscheidungen auf die Form der Verteilungsfunktion der Ausscheidungsgrrl3en und auf die Vergr6berungsrate. Daraus finden wir, dab solche Einfliisse eine wichtige Rolle spielen, auch wenn die Volumanteile klein sind.

1. INTRODUCTION Dynamics of coarsening has been widely studied as a prototype of pattern formations [1]. The coarsening process occurs by the growth of larger precipitates at the expense of smaller ones with the total precipitate volume being conserved. The driving force for coarsening is the surface energy of precipitate which manifests itself in the size dependence of the local equilibrium concentration (the Gibbs-Thompson relation). For the diffusion-controlled coarsening, Lifshitz, Slyozov [2] and Wagner [3] (LSW) have proposed a mean-field theory which is valid in the dilute limit Q ~ 0 , Q being the total precipitate volume fraction. Since then, extensions of the classical LSW theory for the case with finite volume fraction have been considered by several authors [4]. Recently we have proposed a statistical mechanical method to study the finite Q-effects systematically. This method is a systematic expansion method in powers of Q1/2 [5]. Applying this method to the diffusion-controlled coarsening, we have discussed the finite-Q effects on the asymptotic behavior of this phenomenon. Then, we have pointed out that up to order Q 1/2, such effects are remarkable, even though Q is small [6].

Very recently, we have applied the same systematic technique to the interface-limited coarsening [7], which has been studied by Wagner [3] (the Wagner theory) on a level with the LSW theory. Then, we have obtained the Q-dependent kinetic equation for the precipitate size distribution function, up to order Q I/2 [7]. In the present paper, on the basis of the previous results, we discuss the finite Q-effects on the asymptotic behavior of the interface-limited coarsening. In the next section we briefly summarized our previous results for the size distribution function. In Section 3 we discuss the Q-dependence of the asymptotic solution of the distribution function and temporal power law of the average precipitate size. Section 4 concludes the paper.

2. KINETIC EQUATION FOR THE PRECIPITATE SIZE DISTRIBUTION FUNCTION We consider an ensemble of the spherical and immobile precipitates whose growth is controlled by the interface kinetics. Let f ( R , t) denote the precipitate size distribution function per unit volume with radius R at time t. Recently, using the systematic expansion method in powers of QJ/2, we have

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ENOMOTO: FINITE VOLUME FRACTION EFFECTS ON COARSENING--II

obtained the following kinetic equation forf(R, t) in the late stage, up to order Q ~/2 [7]

~f(R, t)+f_[V(R, t)f(R, Ot~

t)] = 0

V(R, t) = d--ffTR=(2G/R){A(p)+B(p)} (It

where p ( p ) is a scaled size distribution function and is time-independent, satisfying normalization conditions

(1) (2)

;p(p)dp= fpp(p)dp= l.

In terms of p(p), the kth moment is redefined by

mk = fpkp(p) dp

with

G = ~2C~qv/ka T

(3)

A(p)=p/m2(t)-I B(p)=~p{p-m3(t)/m~(t)}/m2(t) p = R / ~ (t)

(4) (5) (6)

(12)

(13)

and thus becomes time-independent, using equation (10), we then find

n(t) = (3Q/47zm3)R(t) -3 f(R, t ) = (3Q/4nm3)~(t)-4p(p).

(14) (15)

Inserting equation (15) into equation (1), we have where /~ denotes the probability constant assumed to be independent of R [3], y the surface energy, C~q the thermal equilibrium concentration, v the atomic volume of the solute, ks T the thermal energy. Here the number density of precipitates n(t), the average radius -~(t), and the ktb moment mk(t) are defined, respectively, by

n(t)= f f(R, t) dR

(7)

R(t) = fRf(R, t) dR/n (t)

(8)

mk(t)

= fpkf(R, t) dR/n(t)

(9)

d } d~(t)2 {4 + p-~p

{A(p)+B(p)}p(p)].

=4Gdp

The r.h.s, of equation (16) does not explicitly depend on time, but the 1.h.s. does, through the term d/dt R(t) 2. Therefore, we may put

dR(t)2= K(Q)

Q --(4~/3) fRaf(R, t)dR

=(4n/3)l](t)3n(t) ma(t).

(10)

Here we note that the volume fraction Q is timeindependent, because of equations (1)-(6). The term A(p) in equation (2) is the same mean field term as that of the Wagner theory, except that in the Wagner theory the radius R is scaled as x - R/R~(t) with the critical radius R~(t) = mE(t)R (t) [3]. On the other hand, the term B(p) in equation (2) denotes the renormalized cooperative effects among precipitates, which has been studied by none of the previous authors. 3. ASYMPTOTIC SOLUTION FOR f(R, t) Now we discuss the asymptotic solution of Qdependent kinetic equation for f(R, t). Here the asymptotic form of f(R, t) is assumed to be given by [6]

f(R, t) = [n(t)/~(t)]p(p)

(11)

(17)

where K(Q) is the time-independent coarsening rate, to be determined later. This leads to /~(t) 2 - R(0) 2 =

where the integrals over R always run from 0 to oo. Moreover, the precipitate volume fraction Q is given by

(16)

K(Q)t

(18)

where R(0) is the initial average precipitate size at a starting time of the asymptotic region, to be determined experimentally. As is seen from equations (14) and (18), the average size grows as t 1/2 and the number density of precipitates decays as t-3/2. These temporal power laws are identical to those of the Wagner theory. Thus the finite volume fraction effects are found not to alter the qualitative behavior of the temporal power laws. Finally we discuss the analytic form o f p (p). From equations (16) and (17), we obtain the first-order differential equation for p(p)

p(p)+~---~{X(p, Q)p(p)/3p}=O

(19)

Q)=p2-{4G/K(Q)} {A(p)+B(p)}. (20) X(p,Q)=const. for small p, equation (19) an asymptotic solution p(p),,,p for small p,

X(p,

Since yields Therefore, it is convenient to introduce a new function G(p) by

p(p) = 3pF(p).

(21)

Then, equation (19) is rewritten by

3pF(p) + d [X(p, Q) F(p)] = 0 . dp

(22)

ENOMOTO:

FINITE VOLUME FRACTION EFFECTS ON COARSENING--II

7.5

1.75

1.25

1.50

1.20

2015

2.5

5oi 2o

K(O)

rn3

m2 1.25

1.06

I .I 5

[ 0.05

0

1.10 0.10

0

1.0 0.10

0.05

Q

o

Fig. 1. Moments rn2 and m 3 as a function of the volume fraction Q. Note that m: = 9/8 and m 3 = 1.3621 for Q = 0.

2. Relative coarsening rate K(Q)/K(O) with K(O)=(64/81)G and cut-off P0 as a function of the volume

Fig.

fraction Q. Integrating equation (22) over p and using equation (12), we find

K(Q ) = (4G/rn 2)F (0)

(23)

On the other hand, for large p, we find from equation (22) that F ( p ) ~ p - 5 , leading to a logarithmic divergence o f the third moment. Therefore, F(p) must have a finite cut-off at p = p 0 , beyond which F(p) = 0. This situation is the same as that of the diffusion-controlled coarsening. Then, the similar stability analysis for equation (10) to that of the LSW theory, concludes that the X(p, Q) must have a double root at p = P0 as

X(p, Q)=(3,z)(p - p0) 2

(24)

with z = 3{ 1 - (G/m 2K(Q)) 3 Q ~ 3

} -]

(25)

po = (2zG/3mz K(Q )) { 1 - (m3/m2)~/3 Q/rn 3} (26)

K(Q )=( G/m2) { ( I - ( m 3 / m 2 ) ~ ) +4m 2

~

2

}.

(27)

As a result, we obtain the analytic form o f p ( p ) as p(p)=

{:p(po-p)-2-=exp[-zpo/(po-p)],

for p < P0

cover the Wagner theory. Numerical results for m2, m3, P0, K(Q) and p(p) are shown as a function of Q in Figs 1-3, respectively. 4. CONCLUSION We have studied the finite Q-effects on the asymptotic behavior of the interface-limited coarsening, up to order Q ]/2. With increasing the precipitate volume fraction Q, the scaled size distribution function p(p) broadens and the coarsening rate K(Q) increases, while the temporal power laws remain. Moreover, the Q-dependence of p(p) and K(Q) is found to be remarkable, even though Q is small. These results are similar to those for the diffusion-controlled coarsening [6]. We finally note that we have examined the higher order corrections in the Q 1/2 expansion and the results above do not change essentially. At present the experimental situation is not settled. Although there are numerous experimental studies, the effects of the finite volume fraction have not been

1.51 1.o -

/

A / O L...-o

=0 : o.os

for p//> P0 (28)

=

'~

0.5

with

N = zp~ exp(z).

(29)

Here we note that there is no adjustable parameter in the above results, although we must determine the moments m2 and m3 self-consistently for each value of Q. In the dilute limit Q --* 0, we have m 2 = 9/8, po=9/4 and K(O)=(64/81)G, and thus we can reAM 39/~-T

P(P)

0

I 1

2

q 3

p

Fig. 3. Scaled precipitate size distribution function p(p) vs p = R/R(t) for Q = 0 (the Wagner theory), 0.05 and O.1.

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ENOMOTO: HNITE VOLUME FRACTION EFFECTS ON COARSENING---II

systematically investigated. Systematic experimental investigations of the finite Q-effects on interfacelimited coarsening are thus highly desirable. REFERENCES

1. J. D. Gunton, M. S. Miguel and P. S. Sahni, in Phase Transition and Critical Phenomena (edited by C. Domb and J. L. Lebowitz), Vol. 8, p. 267. Academic Press, New York (1983).

2. I. M. Lifshitz and V. V. Slyozov, J. Phys. Chem. Solids 19, 35 (1961). 3. C. Wagner, Z. Elektrochem. 65, 58 (1961). 4. For a review, see P. W. Voorhees, J. Stat. Phys. 38, 231 (1985). 5. M. Tokuyama and K. Kawasaki, Physica A123, 386 (1984): K. Kawasaki, Y. Enomoto and M, Tokuyama, Physica A135, 426 (1986). 6. Y. Enomoto, M. Tokuyama and K. Kawasaki, Acta metall. 34, 2119 (1986). 7. R. Kato and Y. Enomoto. In preparation.