Elementary derivation of kinetic equations for Ostwald ripening

Physica 135A (1986) 426-445 North-Holland, Amsterdam

ELEMENTARY DERIVATION OF KINETIC EQUATIONS FOR OSTWALD RIPENING Kyozi

KAWASAKI,

Yoshihisa

Department of Physics, Faculty of Science,

ENOMOTO

Kyushu University 33, Fukuoka

812, Japan

and Michio General

Education,

TOKUYAMA

Tohwa University, Fukuoka

Received

23 September

815, Japan

1985

A direct and elementary derivation is presented for the kinetic equation of the single droplet distribution function and the variance equation that describe Ostwald ripening. The derivation is basically algebraical and the screening effects of diffusional interactions among droplets is taken into account automatically by a projection technique. The results correct up to the square root of the volume fraction confirm the kinetic equations derived previously by Tokuyama and Kawasaki except for a correction to the variance equation.

1. Introduction In the previous paper by two of us’), to be referred to as TK hereafter, have derived a kinetic equation for the single droplet distribution function and the variance equation which describe finite volume fraction corrections to the Ostwald ripening. The derivation makes use of the scaling expansion in the volume fraction. The scaling expansion method, however, is not so familiar among workers in this field. Thus it is desirable to have a more direct and elementary derivation in view of the anticipated usefulness of these kinetic and variance equations. Here we present such a derivation. Our derivation is algebraical in nature and we develop an expansion scheme in diffusional interactions among droplets in which screening effects are automatically taken into account by a projection technique. In section 2 we explain our basic ideas and introduce necessary notations. In section 3 we derive a kinetic equation for the single droplet distribution function correct up to +l” where 4 is the droplet volume fraction. This is followed in section 4 by a derivation of the variance equation correct up to +l” where the result of TK is corrected. The last section concludes the paper. 0378-4371/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

KINETIC EQUATIONS FOR OSTWALD RIPENING

427

2. Basic ideas and notations Our starting point is eqs. (1.2) and (1.3) of TK I), which describe the change of the radius Ri(t) of the ith droplet2’3), as

$

; rRi(t)3 = -4?TDMi(t)

M,(t)=

hi +

i

g,M,(t)

)

(2.1)

,

(2.2)

j=l

where hi = a(1 - Ri(t) lR,) )

eij= e( IXi - Xjl -

Ri

(2.3)

- Rj)

(2.5)

)

with (Y the capillarity length, R, the initial critical radius, D the diffusion coefficient and Xi the position vector of the center of the ith droplet. The 0(x) is the usual step function equal to 0 for x < 0 and to 1 for x > 0. The function g, reduces to zero for i = j. It is convenient to introduce a matrix notation as M for the N-dimensional vector (Ml,M2,...,MN) and g for the N x N matrix whose (i, j)-component is gij. Then we write eq. (2.2) formally as M=A+g-M

or M=(l

-g)-‘.A=A.(l

-ST)-‘,

(2.6)

where the symbol ‘ * ’ denotes the contraction of index, 1 the N x N unit matrix and gT the transposed matrix of g. Since time variation occurs only through Ri(t), we have

(2.7) where we have used eq. (2.1) and Ri = dR,ldt

and

K. KAWASAKI

428

a

D

L,+=

et al

(2.8)

R; JR, is an adjoint

operator

of

L,=$.

(2.9) ’

Therefore,

I

inserting

eq. (2.6) into eq. (2.7) we have the following

time variation

operator: ;

-gT)-'.L+.

=A.(1

An advantage account. First, fluctuation

Al)

A(1)

(2.10)

of the method is that the screening effects are readily taken into we define the distribution function operator f( 1) and the

A(1)

by

= 2 %I, ;=I -f(l)

(2.11)

i) ,

-f(l)

(2.12)

,

where

6(i, j) = 6(R, - R,)S(X;

-

X,)

(2.13)

and

f(l) = is the

oil))

distribution

(2.14) function

of a single

droplet,

the

angular

bracket

(. . .)

denoting the average over the initial ensemble for which TK used the bar, and n = (R,,X,).We also need multidroplets distributions functions. Thus we first note

mow=L(L 2) + @l, 2vm

(2.15)

with L(L

2) = c 6(1, i,)6(2,
i*) .

(2.16)

429

KINETIC EQUATIONS FOR OSTWALD RIPENING

(i, . . . i,)

Here the angular bracket Then f,( 1,2) defined by

denotes the sets of m different

droplets.

f-2(1> 2) = (LO72))

(2.17)

is the 2-droplet distribution f(l)f(2).

function. This can be further generalized

**f(m) = 5 i

6(1,2, . . . , l)fm-l+l(l,

l+

as (2.18)

1, . . . , m) ,

I=1

where 6(1,2, . . . ) I) = 6(1,2)S(2,3). UL

(2.19)

. . S(Z - 1, I) )

(2.20)

2, . . . 3m) = C C * . . C 6( 1, i,)S(2, i2). . . s(m, i,) , (i,...i,)

and the symbol i denotes the summation over all the combinations arguments. Eqs. (2.14) and (2.17) are also generalized as

(fm(l,2,.

. . , m)) =f,(l,

among

(2.21)

2, . . . , m) ,

where the f,(l, 2, . . . , m) is the distribution function of m droplet defined by TK (2.1). We now write the matrix product for any two matrices Y and 2 in terms of the distribution function operator as

(Y *Z),, =

=

IX YmjZ,ns 1d(l) i

i

1d(l) Ym,f(l)Zln+ /d(l)L,Af(wl,

= (Y - PZ),,

Here we have introduced

+ (Y - QZ),,

Qz>,, = /d(l)

.

the projection

(Y- ~%, = j- 41) Y,n,f(l)Z,, (Y.

Ym,C ‘(1, i)Z,,

L,AfUM,,

(2.22) operators

P and Q = 1 - P as (2.23a)

7

2

(2.23b)

430

K. KAWASAKI

which will simplify the subsequent = (1 - PgT)-‘[l

(1 -ST)-’

et al

developments.

- Qg’(1

Using the operator

- PST)-‘I-’

,

identity (2.24)

we write eq. (2.6) as M = A-(1 -

PST)-‘[l

- Qg’(1 - PgT)-‘I-’

We then define the renormalized

iz

A-(1 - PgT)-’

gLg’.

(2.25)

n and g’ through (2.26a)

)

(1 - PgT)-’

.

.

(2.26b)

We now find that the operator (1 - PgT) takes into account the screening effects. Indeed from eq. (2.26) we have $j'(l- PgT) = gT

or, more explicitly,

g; =

s;+i d(l) dgT,f(l)

kF!j =

gij

and

+

I d(1)

SiltTlj.R1)

(2.27)

Eq. (2.27) is identical to TK (3.11). Similarly eq. (2.26b) can be written as i=h+h.PgT(l-PST)-‘=A+A.Pij’

or

&=4

+

I Wk,,A,f(l)

(2.28)

which is identical to TK (3.12). It is important to note that the &,‘s are in fact not zero in contrast to the gij’s. Thus & given by TK (4.5) is not correct for i = j, although, fortunately, the error introduced there is higher order in the volume fraction 4 than fi. The difference, however, is important in the present formalism.

KINETIC

EQUATIONS

FOR OSTWALD

Now we turn to eq. (2.25) and rewrite quantities (2.26a) and (2.26b) as A4 = i-(1

- Qtj’)-’

RIPENING

431

it in terms of the renormalized

.

(2.29)

The ring-diagram expansion expressed only in terms of screened quantities is thus obtained as M=~+~.QQ’+~.Q(j’Q~‘+....

(2.30)

In the following section, we shall rederive the results of TK by using the ring-diagram expansion method. Therefore, it is convenient to use the same diagrammatic representations as those of TK. The correspondences between the diagrammatic elements and the algebraic expressions are given in table I. For example, the renormalized g,, is shown diagramatically in fig. 1. In table II, we TABLEI The diagram elements used in the present paper. Diagram elements

Algebraic expression

Ofl

vertex

ln

d(n)

Diagram elements m n 0

Algebraic expression

“6”

W, n)f(n)

Wc n)

f i An

h”

I

An

i n

k

1 d(n)i, 2

0

ati,i, k)

4 6(1,2,3,4) 3

G,(m,

m

n

..’ I

l ‘\

b--Sk

\

n)

Cd4 i, k)

-0

g11 =

Fig. 1. The diagrammatic

representation

of &,

K. KAWASAKI

432

TABLE

The orders

et al.

II

of the diagram

Diagram elements

Order

n f(n) G,(l,. 0

elements.

s” So’ s-’ s

,m)

2

d,rn

L,

s-’

s?

0

summarize orders of these elements in powers of S -’ = R/I - C#I Ii2 where R and 1 are the average droplet radius and the screening length, respectively.

3. The kinetic equation for f(1) In the present section, we derive the kinetic equation for the distribution function f(1). Using eqs. (2.7), (2.14) and (2.29), we obtain

y$f(l)

= bZ(1)

(3.1)

1

where we have used the relation L:6(1,

i) = L,6(1, i)

(3.2)

and Z(l)=(CM[fi(l.i))=(i.(l

-Qg’))‘.S(l))

(3.3)

and have introduced the N-component vector S( 1) whose ith component is S(1, i). Then we consider the ring-diagram expansion of eq. (3.3) corresponding to eq. (2.30) as %

Z(1) = c Z’“‘(1) ) ??I=0

(3.4)

where Z(O)(l) = (A. S(1)) ) Z(“)(l)=

(i.(QgT)‘V(l))

(3.5) (~~21).

(3.6)

KINETIC EQUATIONS

FOR OSTWALD

RIPENING

Eq. (3.5) reproduces the starting equation of Lifshitz-Slyozov replaced by the average radius Z?:

433

in which R, is

Z”‘(l) = (C[Ai + /d(2) &,A,f(2)]‘(1, i)) I

=

‘1 f(l) +

=

I d(2)tii,A,f(2).f(l)

&f(l).

Moreover,

(3.7)

in general we write eq. (3.6) as follows:

Z’“‘(1) = (&(QCJT)“.S(l)) =j-d(2).../

d(m + 1) g&3.

x (f(l)Af(2)...A(m+

X [f(1)9*(2,3,.

* . g,,m+&n+~

1))

. . , m + 1) + $m+l(l,

2,. . . , m + l)] ,

(3.8)

where we have used eqs. (2.12) and (2.23b), and have introduced the moments of fluctuations as .Fm(1,2,.

. . , m)=

(A(l)***A(m))

.

(3.9)

Before we go on to the next step, we investigate the properties of the new functions .Fm(l, . . . , m). We first note that 5, is related to the droplet correlation functions G,( 1,2, . . . , m) introduced in TK through the following relations: Wl,

2) = G,(l, 2) + S(l, 2)f(l),

W,

2,3) = G,(l, 2,3) + E 6(1,2)~,(2,3)

s‘,(l, 2,3,4)

= G,(l, 2,3,4)

(3.10a)

+ 8(1,2,3)f(l)

+ 5 G,(l, 2)G,(3,4)

,

(3.10b)

+ 5 8(1,2)G,(2,3,4)

+ i

6(1,2)f(2)G,(3,4)

+ 5 %l, 2)G,(3,4)f(l)f(2)

+ i

S(l, 2,3)G,U, 4) + 6(1,2,3,4)f(l) ,

(3.1Oc)

434

K. KAWASAKI

where g is the summation example, we have

et al.

over all possible CYcombinations

of indices. For

3

2 S(L 2)G,(2,3)

= 8(1,2)G,(2,3)

+ 8(2,3)G,(3,1)

+ %3,1)G,(L

2) . (3.10d)

Eqs. (3.10) follow since in G,( 1,2, . . . , m) all the m indices must be distinct. We now employ the cumulant expansion [4] for 9, as . . ,rn)=C...C

Sm(1,2,.

11 with the constraint 9;(1,2,.

.

(. . -), denoting

9Fl...Sf

1,

(3.11)

P

I, + 1, + +* ++ Z, = m, where , m) = (Af(1).

. . A(m)).

(3.12)

,

the cumulant average. For example, we obtain

(a)

(b)

(4

(4

Fig. 2. The diagrammatic representations direct product of the diagrams.

of eqs. (3.10).

(3.11) and (3.13).

The symbol

C9 expresses

KINETIC EQUATIONS FOR OSWALD

RIPENING

435

Fig. 3. The general diagram representation of Icm’(l)

(4

(b)

Fig. 4. The diagram expressions of 1”‘(l) and 1”‘(l).

ST(l) = 0 )

SS(1,2)

= B,(l, 2) )

~;(1,2,3,4)=%(1,2,3,4)-i

%(l,

293) = .%(l, 23)

,

%(1,2)&(3,4) (3.13)

= G,(l,2,3,4)+f: +i

where 5 has the representation of Now we turn to (3.8) and (3.10),

6(1,2)G,(2,3,4)

6(1,2,3)G,(3,4)+S(1,2,3,4),

same meaning as in (3.10). Fig. 2 shows the diagrammatic eqs. (3.10), (3.11) and (3.13). eq. (3.8)) which is given diagrammatically by fig. 3. Using eqs. we have algebraic expressions of fig. 4 as

Z”‘(l) = j- 42) &&[f(W1(2) + 95;2(1,2)1 (3.14a)

K. KAWASAKI

436

1’2’(1)

=

+

j-

I421

il

+

d(3)

42)

S,,

43)

et al.

+

&,ti2,&%(1.2,3)

Wk52~2,~,G2(2,1>

+ I d(2)

3)f(I)

~,,&G2(2~

+

j

42)

d(2)

I

&,&,~,fW(l)

&1&2h2G2(2,1)

&2~22~2G2(23

1)

I

Using table II, it turns out that all the terms below the second line in eq. (3.14b) are of higher orders in S - ’ than the terms in eq. (3.14a) and those in the first line of eq. (3.14b). Similarly we also find that I’“‘( 1) with m 2 3 are at most of order Y2. Thus eq. (3.4) is diagrammatically shown in fig. 5 to order Y’, where we have indicated the order of each term in S-‘. The corresponding algebraic expression is given by Z(1) = &f(l)

+ C’[V,(f)

+ J,(f)]

(3.15a)

+ Q(P),

with the drift term

V,(f)

=

~J1.m

+

and the soft-collision

J

42)

&2~2*&.fWf(~)

(3.15b)

*

term (3.15c)

Substitution of (3.15) into (3.1) yields the kinetic equation for f(1) a(-‘) = O(C#/‘~). Th’IS reduces to the kinetic equation (3.30) of TK.

1(l) =A + S-’

J +

O(S

Fig. 5. The relevant

-‘)

diagrams

in 1(l)

up to order

S-l.

up to

KINETIC EQUATIONS FOR OSWALD

RIPENING

437

4. The kinetic equation for x(1,2)

In the present section we derive the kinetic equation for the variances x( 1,2) defined by (4.1)

x(1,2) = %l, W(l) + G,(l, 2) .

First, we consider the equation for f2( 1,2), which is the distribution function of two droplets. Using eqs. (2.6), (2.16), (2.17) and (3.2), we thus find -$ f*(l,2)

= WL+L(l,

= cl+

P,,Q

= cl+

Wl(

= cl+

~12Mfw,

I

2)) = (7

M/L:

Jk~,~(L

W2~

A)

M,S(l? WCL

i,>

;) I

1)

Zj S(l, i)6(2,i)) I

+f(w(1)1,

(4.2)

where H2 is defined as Hz(291) s (C Mia(l, (0)

i)6(2, j)) -f(2)~(1)

9

(4.3)

and Pij is the exchange operator between i and j. Eq. (4.3) can be rewritten as

=(CE Mi’(l,

H2(2,1)

1

= (W6(1)Af(2))

Moreover

i)6(2, j,> - [6(1,2)

+f(2)](C MiS(l, i)) I

i

-S(1,2)(M.6(1)).

we can expand eq. (4.4) in ring-diagrams

H,(2,1)

(4.4)

summations

= 2 H;““(2,1), m=O

Him’(2, l)=

(~.(Q~‘)“.S(l)Af(2)) _ _ =giigiZ”

as (4.5)

-S(1,2)(~.(Qt~~)“.S(l))

.g;;;Ti,,A,[(~(1)A(2)A(i)...A~(m))

-S(1,2)(~(l)A(i)...Af’(m))] -

_

=giigiZ*’

-g~,,A,~,(i2iZ-m),

(4.6)

K. KAWASAKI

438

where the integrations function @, as

et al

over the barred indices are implied and we define the

@,(121’2’. . . m’) = sm+2(121’ . .*rn’)+f(l)%~+,(21’...~‘) - 6(1, 2)Sm+,(11’.

. . m’) - 6(1,2)f(l)%Jl’.

* ’ m’)

(4.7) Eqs. (4.6) as well as CD* are shown diagrammatically in fig. 6. For first few terms of expression (4.5) it is easier to use (4.5) rather than (4.6). For example we have for Hr’ and Hi” the following: HY’(2,l)

= (2 h;S(l - i)6(2 (il) = h,G2(l,

&“(2,1)

j))- f(2)P’(l) (4.8a)

2) ,

[i Qo'];s
= (2

+ (G,,

+ &&V&(2>

1) + J d(3) g,,&~,(l,2,3)

(4.8b)

The second line of eq. (4.8b) represent parts of X,, and Y,, given by TK and are of O(S-*). The diagrammatic expressions of Hr’(2,l) + Hi”(2, 1) are shown in fig. 7. Next we consider Hy’(2, 1) which is expressed as HF’(2,l)

= g ,jg&@*(

- 1,2,3,4)

= &&&[%~(1235)

+f(1)s3(234)

- 6(1,2)9#34)

- 6(1, 2)f(1)sz(3,

4)] .

(4.9)

Using table II, the third term in the square bracket of eq. (4.9) turns out to be of higher order than other terms. The Hy’(2,l) are represented in fig. 8, up to S-l, which are parts of X,, and Y,, . Since the diagrams for X,, and Y2, given in TK are not yet exhausted so far, we also need to consider Him’(2, 1) with m 2 3. Using table II, we find that parts of Hy’(2,l) are of order S-‘, and the rest of Hy’(2,l) and ail the terms of Him’(2, 1) with m 2 4 are of higher order in S-l. The Hy’(2,l) are represented diagrammatically in fig. 9, up to S-‘, which complement the parts of X,, and Y,, contained in eqs. (4.8b) and (4.9). In

KINETIC EQUATIONS

FOR OSWALD

439

RIPENING

(4

Hirn’(2, 1) =

a

+

$zJ

1

1

d

2 -

9 m+,

. . l

-

%_._

0

(b)

1

2

3

4

2

:

.

@*(1,2,3,4)=

3

=c

w--o o--o

+

Fig. 6. (a) The general diagram representation Qz( 1,2,3,4) up to order S-l.

OfR ‘I+

fo

&La

of Hi”‘(2,l).

A

f

(b) The diagrammatic expression of

440

K. KAWASAKI

H:0’(2,1)

+ @‘(2,1)

et al.

=

f

f

&---Q

+s-’

+

L

Cy-dl---Q

+

f

-

C-a

(cc9---o

+

-

+

Q--+ ‘-L-e’

07

(XI

9 I

VI

Fig. 7. The relevant diagrams of HF’(2, 1) + Hi”(2, 1) up to order S-l. X and Y under indicate that these diagrams are parts of X,, and Yz, of TK, respectively.

H:2’(2,1)

= I /d(3)

d(4) &,&,&@~(l,

2,374)

gT+E+E+G

=s-’

[ +

(Y)

gJ+ Go

+

+-j

w

(Y)

W)

(Y)

p_+ E+ 07

(Xl

+ I

&,/

+ y-J

W)

(Y)

o(s-*)

-+.

(X)

(X)

Fig. 8. The relevant diagrams of Hy’(2, 1) up to S-‘. X and Y under diagrams are parts of X,, and Yz, of TK, respectively.

summary,

(X)

(Y)

L+qf+T VI

+

diagrams

we have for I&(2,1)

up to S-’

diagrams

indicate

that these

the following:

f&(2,1) = m,,G,(L 2) + &2&f(l)f(2)+ S-‘[X,,+ Ynl

+ OF*)

3

(4.10)

where (4.11) Eqs. (4.10) is identical to TK (3.16). Using (3.1), (4.1) and (4.2), we have

KINETIC EQUATIONS

H:9’(2,1) = s-’

FOR OSTWALD RIPENING

441

[c%+k.+++-# (X)

(X)

(X)

L+p&+E

+

07

07

07

jc_+~o+$

+

(Y)

w

(X)

+F+k+pq (X)

w

(Y)

+ B(P). Fig. 9. The relevant diagrams of Hy’(2,l) up to S-l. X and Y under diagrams indicate that these diagrams are parts of X,, and Y2, of TK, respectively.

-$ x(1,2)

= 6(1,2)L,Z(l)

+ &&(2,1)

= (1+ MWW,

+ L&(1,2)

2)1(l) + H*(l, 2)] .

(4.12)

Since we want to compare (4.12) with TK (3.36), we now consider SV,(f)lSf(l) and SJ,(f)/Sf(l). For this purpose, we need S&,/Sf(l) and 6);,/6f(l). From (2.27) we obtain (4.13) where the last equality is justified diagrammatically (2.28) and (4.13) we have

s/i*

sfo

in fig. 10. Similarly from

-

(4.14)

= &t4 .

Now we have

SW)

-

d(3) &f(3)

x(3,1)

and

d(3) 6fo 6J(2) x(391) *

K. KAWASAKI

442

with 2

3=-

1

Fig. 10. The diagram

In fact, the algebraic expression W2) I d(3) 6f(3)

%,

hf(l) representation

+ x,2 + /I

d(3) 6fo ‘“(‘)

of eq. (4.13).

of fig. 11 is given by 6V) + 1 d(3) -cq(3) G,(3,1)

x(391) = Zf(l)

= a(19 2)Vl)

et al.

d(3) Wk,,&,&,&x(4~

l)f(3)m

x(37 1)

+b +b f

f

f f

f f

--0

f

Fig. 11. The diagram

representation

of eq. (4.15).

KINETIC EQUATIONS FOR OSWALD

This result is a correction to TK (3.19). representation of fig. 12 as

JW I 43) 68fc3j

= Ylz - g,,m,,G,(l, -

I

Similarly we have the algebraic

2) - j d(3) &,&,&G,(3,l)f(2)

d(3) &m&,(1,2,3)

+ S(l, 2) 1 j- d(3) d(4) &A~G(3~ +

443

W2) f(l) + j-d(3) -aft31 ‘X3,1)

$f$

x(331) =

RIPENING

II

d(3)d(4)&,m,,

aG,(4,3) 6f(3)

4)f(l)

x(3,1) .

(4.16)

Eq. (4.16) is identical to TK (3.20). Comparing eqs. (3.15a), (4.10) and (4.11) with eqs. (4.15) and (4.16), we can thus transform (4.12) into

6V(2) -[ 6f(3)

+/d(3)

1

6 JW ~d.0 + &f(3) x(3,1) (4.17a)

where D,,(f)

= &,m,,G,(L

2) + I 43)

&J&W,

2) + &&f(2)P,,]G,(2,3)

,

(4.17b)

I

-6JW x(3,1)= d(3) 6f(3)

Y*, +

+

W(4,2)

II d(3) d(4) & ~ bf(3)

Fig. 12. The diagram representation of eq. (4.16)

x(31 l)

K. KAWASAKI

444

Z,?(f)

= 1 d(3) &P&%(

F,,(t) = i

1,233)

et al.

- 1 d(4)

x(471) ] 3

‘;;;;‘)

(4.17c)

d(3)g,,g,,g?,A,(3)f(l)

The term F,,(f)

is a new term which is not considered

to SV(2)/6f(3)

of TK.

5. Concluding

remarks

Except

for F,,(f)

by TK and is a correction

we recover

TK (3.36).

In the present paper we have rederived the kinetic equations that include the variance equation for Ostwald ripening and have corrected some errors of the TK theory by using the new method we had developed. This method is found to be more direct and elementary than the TK method where one can automatically take into account screening effects. The results of the corrections to the variance equation of TK are given correction to the kinetic corrections the scaling

by eqs. (4.15), (4.17d) and the appendix, equation for f(1) was found. The roles

while no of these

will be discussed in a subsequent pape?), where we shall investigate function of the scattering structure function which is obtained from

x(132).

Acknowledgements This work was supported in part by the Scientific Education, Science and Culture.

Research

Fund of Ministry

of

Appendix In the present appendix we correct the error in the appendix of TK. We consider eq. (4.17~) in the long time limit. Inserting (4.1) into (4.17~) we have

-&(f)

= j d(3) 22,m,,[

G,(l,

2,3)

- 1 d(4) s;$)2)

G(4,

I)]

(A.1)

KINETIC

EQUATIONS

FOR

OSTWALD

RIPENING

445

In order to calculate the last term of (A.l) we use the following relation:

C(l) = j- d(n)&m,,G;h

1)

64.2)

9

where C(1) and GZJ(1,2) are the late stage form of the soft-collision term and the correlation function, respectively, and Gy(1,2) is defined as

G;(l, 2) = %fWV)

(A-3)

7

l)U*,A,

(A.4)

.

Thus we obtain in the late stage

- i d(3) &,&,m,,G;(37

-

I 43) “‘sf(l)am,, -

2)f(l)

G;(3,2)f(l)

.

(A.51

Moreover, calculating the first two terms on the r.h.s. of (A.l), taking into account the late stage properties, we obtain the last two terms on the r.h.s of (A.5) with opposite signs. Therefore, we have

in the last stage. This result corrects the appendix of TK.

References 1) 2) 3) 4) 5)

M. J.J. K. R. M.

Tokuyama and K. Kawasaki, Physica 123A (1984) 386, to be referred Weins and J.W. Cahn, Materials Research 6 (1973) 151. Kawasaki and T. Ohta, Physica 118A (1983) 175. Kubo, J. Phys. Sot. Jpn. 17 (1962) 1100. Tokuyama, Y. Enomoto and K. Kawasaki, to appear in Physica A.

to as TK.