ostwald ripening in open systems - nihon universitynakahara/text/nakahara...ostwald ripening in open...

Ostwald ripening in open systems Akio Nakahara, Toshihiro Kawakatsu, and Kyozi Kawasaki Department of Physics, Faculty of Science, Kyushu University 33, Fukuoka 812, Japan

(Received 28 May 1991; accepted 13 June 1991)

We consider coarsening processes of precipitates in open systems where homogeneous cells are coupled together by the diffusion of the solute. Using dynamical scaling assumptions and performing numerical simulations on a two-cell model, we have obtained the following results: First, the supersaturation in each cell asymptotically obeys the same Lifshitz-Slyozov-Wagner (LSW) scaling law for a closed system. Second, other physical quantities, such as the size distribution function of precipitates, obey new scaling laws, when the total amount of both the solute and the precipitates in the cell flows out into the other cell. The asymptotic long time behavior of such physical quantities are highly dependent on the initial conditions.

1. INTRODUCTION

Recently, the dynamics of first-order phase transitions has been fascinating physicists, chemists, and scientists in many other fields, due to its rich nonlinear phenomena, in- cluding the formation of spatiotemporal patterns.’ For example, the spatial patterns of precipitations are known to originate from the spatial inhomogeneity in two main processes of the pattern formation, i.e., the nucleation process of droplets of precipitates and the coarsening process of these droplets.* The former produces a spatial inhomogeneity in the number of droplets in the earlier stage, and the latter induces the mass transport into the region where large droplets exist. In this paper, we mainly study effects of the spatial inhomogeneity in the coarsening process in order to under- stand the pattern formation in such systems.

The late stage dynamics of the coarsening process, where the supersaturation of the solution is very small, is characterized by the Ostwald ripening. When the total amount of the minority phase, i.e., the solute and the precipitates, is vanishingly small, each droplet evolves under the influence of the supersaturation in a mean field sense. The growth of droplets in this stage is driven in such a way that the surface energy of droplets is minimized, and therefore only large droplets grow at the expense of smaller ones.

Lifshitz, Slyozov3 and Wagner4 (LSW) studied the Ostwald ripening in a closed system, where the total amount of both the solute and the precipitates is conserved. Assum- ing spatial homogeneity inside the system, they found the following results. First, the supersaturation (T (t) obeys a scaling law given by

t - “3, (1.1)

where t is the time. [In this paper, variables, such as t and a(t), are reduced ones, whose definitions are described in Ref. 5. See also Appendix A.] Next, they found that the size distribution function of droplets, i.e., the probability of find- ing a droplet with radius rat time t, denoted as f( r, t), obeys a universal scaling law given by

f(r,t) = t -4/3FLSW(rt -l/3), (1.2)

where q is the total amount of both the solute and the precipitates per unit volume and

s

m MLSW=

n - dzz”FLSW(z). (1.3) 0

Here, F Lsw (z) is the LSW scaling function given by

PO2

FLsw(z) G (22, + zpyzo - z)‘1’3 exp[ -*I

KKz<zo 1,

lo x=0 > (1.4)

where z, = (3) “3 and the constant p. is determined by the normalization condition MkSW = 1. They also found that the total number of droplets per unit volume n(t), and the mean radius of droplets 7(t) are given by

(1.5)

T(t) EM ywt “3,

respectively. (1.6)

Since the LSW theory is based on the assumption that the total amount ofboth the solute and the precipitates in the system is conserved, it is not obvious whether the LSW theory is applicable to the coarsening process in an open system, where the total amount changes through the diffusion of the solute with their environments.

Venzl pioneered in investigating the Ostwald ripening in open systems.6 The system he studied is a set of homogeneous cells (subsystems) coupled with each other by the diffusion of the solute. He reduced the degrees of freedom of the system and derived a set of ordinary differential equations for the density of droplets, the mean radius of droplets, the width of the size distribution function of droplets (the second order cumulant of the size distribution function), and the supersaturation. He insisted that the mass transport is induced by the differences in both the supersaturation and the quantities characterizing the size distribution function of droplets between the adjacent cells. Since he neglected the third and higher order cumulants of the size distribution function, his approximate equations could not correctly describe the asymmetric functional form of the size distribution function, such as the LSW scaling function.6 Therefore it seems that his theory is not enough to characterize the

J. Chem. Phys. 95 (6), 15 September 1991 0021-9606191 I1 84407-l 1$03.00 0 1991 American Institute of Physics 4407 Downloaded 26 Jun 2004 to 133.43.121.32. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

asymptotic long time behavior of physical quantities of these systems.

Beenakker and Ross’ studied the Ostwald ripening in an open system, which is immersed in a reservoir of a constant solute concentration. By taking the screening effects into consideration, they insisted that, when the system is infinitely large, the screening effects make the evolution of physical quantities of the system independent of the value of the supersaturation in the reservoir, and therefore the evolution of physical quantities of such an infinite open system is equiva- lent to that of an infinite closed system. They also insisted that their theory is applicable to the case when the system size is larger than the screening length. As they discussed, their theory describes only the earlier time behavior of physical quantities of the system, when the diffusion flux from the outside of the system has not yet changed the total amount of both the solute and the precipitates considerably. On the other hand, as will be shown in the following sections, the change of the total amount is expected to be dominant in the limiting long time behavior of physical quantities of the system.

This paper is organized as follows. In Sec. II, we investi- gate the Ostwald ripening of a system which consists of two homogeneous cells coupled together by the diffusion of the solute. On the basis of a certain set of assumptions, we will find the following results: First, the supersaturation in each cell actually obeys the LSW scaling law. On the other hand, the size distribution function of droplets may or may not obey a new scaling law, depending on the initial conditions. Therefore the LSW theory cannot always be used when we discuss the size distribution function in open systems. We will also find that we should not truncate the information on the size distribution function of large droplets, because the asymptotic long time behavior of physical quantities of each cell, such as the size distribution function, is determined by the initial size distribution functions of large droplets. The validity of our discussions will be examined by numerical simulations in Sec. III. Finally, in Sec. IV, we summarize our results and discuss the applicability of our arguments to real situations.

II. ANALYTIC TREATMENTS In this section, we study the Ostwald ripening of a sys-

tem which consists of two homogeneous cells of the same volume, labeled a and b, which are open subsystems, coupled with each other by the diffusion of the solute (see Fig. 1). First, in Sec. II A, we present basic formulas which describe the late stage dynamics of the coarsening process for such a two-cell model. Using dynamical scaling assumptions, we develop our arguments in Sec. II B and Sec. II C. Unlike the LSW scaling laws which hold in closed homogeneous systems, the asymptotic long time behavior of physical quantities of the open subsystem is not unique, and therefore is expected to depend on the initial conditions. We study its dependence on the initial conditions in Sec. II D.

A. Basic formulas In this subsection, we present basic formulas which de-

scribe the late stage dynamics of the coarsening process for

4408 Nakahara, Kawakatsu, and Kawasaki: Ostwald ripening in open systems

cell a) , ! ceN 6) I

FIG. 1. A schematic illustration of the two-cell model with the coupling constant d.

the two-cell model. 5,6 We label each cell by the suffix i (i= a,b).

First, we comment on the difference between a closed system and an open subsystem. In the closed system, the mass conservation law holds as

-$9(t) = 0, (2.1)

where q(t) is the total mass per unit volume in the closed system. In the open subsystem (cell), however, the total mass per unit volume qi (t) is no longer conserved due to the diffusional coupling with the other cell.

We consider droplets of precipitates in a solution con- tained in a cell. Each droplet is assumed to be at rest. We also assume spatial homogeneity inside the cell. Thus we can de- fine a mean concentration field in each cell and each droplet is characterized by its radius r and by the cell suffix in which the droplet is located.

We denote the size distribution function of droplets of radius r in cell i at time t as A (r,t). For positive r, it obeys a continuity equation

$-&(r,t) +-$[~i(TrtM;(rJ)] =O, (2.2)

where ui (r,t> is a growth rate of the radius of the droplets in cell i. From the Gibbs-Thomson condition, the growth rate is given by

ui(r,t) = f[“i(f) -$]9 (2.3)

where oi (t) is the supersaturation of the solution in cell i [see Appendix A for the derivation of Eq. (2.3) 1. Introduc- ing a critical radius in cell i defined by

l-$(+1, ai(t>

(2.4)

Eq. (2.3) is rewritten as

vi(rJ> =+-[-$-g--+1. (2.5)

Thus the droplet in cell i grows when r > c(t), and shrinks when r<c(t).

The total number and the total volume of droplets per unit volume in cell i, denoted as ni (t) and lci (t), are given by

m ni(t) = dr.fi PA, (2.6)

u,(t) =$ s m dr ?L (r,t),

0 (2.7)

J. Chem. Phys., Vol. 95, No. 6, 15 September 1991

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respectively. Then, the total amount qi (t) is expressed as

qi tr) = uj tt) + ui tt>* (2.8) These formulas derived above also hold in the closed system by omitting the suffix i.

Now, we present the evolution of qi (t) for the open subsystem. For simplicity, we set the volume of each cell to be unity. Since the mass transport takes place due to the difference in the supersaturation between the two adjacent cells, the evolution of qi (t) is given by

$jtt) =d [“j(f) -ui(t)], (2.9)

wherejf i and d is a coupling constant between the two cells. Equation (2.9) immediately leads to the mass conservation in the whole system

4, (t) -I q*(t) 2

= QUJ,

where q,,, is a total amount of both the solute and the precipitates in the whole system divided by the total volume and is a constant of time.

Nakahara, Kawakatsu, and Kawasaki: Ostwald ripening in open systems 4409

U,(f)“4$J -ai, (2.16) &(r,t)d -BiciFi(rr -“‘). (2.17)

Such a regime is usually called a scaling regime. Substituting Eqs. (2.13) and (2.14) into Eq. (2.7)) the

total volume of droplets ui (t) is given by

u,(t) ==t -% + [ I

m dzi zFi (zi) + r;(z,,t)

0 1 %.t -“t, (2.18) where

xisfli - 4ai (2.19) and I’Y(z,,t) is a correction term which becomes asymptotically negligible in the limit t +oo.FromEq. (2.18),wefind that ui (t) goes to infinity for negative xi, which is obviously unphysical and should be excluded.

(2.10) From Eqs. (2.8), (2.11), and (2.18), the total amount in cell i, qi (t), is given by

qi(t)-max(t -n’,t -+) (2.20)

in the scaling regime. Substituting Eq. (2.20) into Eq. (2. lo), which describes the mass conservation in the whole system, we obtain

min(x,,x,) = 0, (2.21) owing to the positivity of ai. Without loss of generality, we can set

x, = 0, (2.22)

where the total amount qa ( t) tends asymptotically to a finite value in the limit t-t 00. In cell b, the relation

x,>o (2.23) holds. Therefore for vanishing xb, the total amount in cell 6, qb ( t), also tends asymptotically to a finite value in the limit t-r 00. For positive xb , however, qb (t) flows out into cell a and it eventually goes to zero for large t.

Substituting Eq. (2.20) into Eq. (2.9), the following relation holds in the scaling regime:

la,(t) -~(Tg(t)~-max(t-‘l~a”,t-“f”i’)-~(t -‘>. (2.24)

Equations (2.2), (2.3), and (2.7)-(2.9) are the basic equations for the late stage dynamics of the coarsening process for the two-cell model.

In the following subsections, we show that the time dependence of qi (t) gives a new feature of the coarsening process.

6. Dynamical scaling in open subsystems

In this subsection, we study the asymptotic long time behavior of physical quantities of the open subsystem on the basis of dynamical scaling assumptions.

Following Marqusee and ROSS,’ we assume that the supersaturation in cell i, ui ( t), is expressed as

u,(t) = t -“‘[gi + rg(t)]9 (2.11) where aj and gj are positive constants and r;(t) is a correction which is assumed to become asymptotically negligible in the limit t-rco. From Eqs. (2.4) and (2.11), the critical radius in cell i grows as

rf(r) ate’. (2.12) We also assume that the size distribution function of droplets in cell i, A (r,t) , is expressed as

J;trYt) =r -8’[CiFj(zi) + r{(zi,t>]j (2.13)

where pi and ci are positive constants, Fi (zi ) is a scaling function in cell i, and zi is a scaled radius in cell i given by

zi=rt --=I. (2.14) Here we require that the scaling function Fi (zi ) satisfies the normalization condition

I

m dz, Fi ( zi ) = 1.

0 (2.15)

The correction term I’{(zi,t) in Eq. (2.13) is also assumed to become asymptotically negligible in the limit t-r CX). Thus the feature of the late stage dynamics of the coarsening process is characterized by the slowest time scale, i.e.,

From Eq. (2.24)) we may assume that, in the scaling regime, the difference in the supersaturation between the two adjacent cells becomes asymptotically negligible as compared with the value of the supersaturation itself, i.e.,

l”aCt) -ab(t)141ai(t)l* (2.25) Then, the whole system behaves asymptotically as a closed system and the supersaturation in each cell in the scaling regime obeys the LSW scaling law ( 1.1);

ai( $ 0 20 t - l/3, (2.26)

i.e.,

(2.27)

ai =-. 3

J. Chem. Phys., Vol. 95, No. 6,15 September 1991 Downloaded 26 Jun 2004 to 133.43.121.32. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

Equations (2.24) and (2.26) satisfy the relation (2.25). From Eqs. (2.14) and (2.28), the scaled radius zi is given by

z = z, = z, = rt - ‘I3 , (2.29) where we have introduced z for simplicity of the notation. From Eq. (2.28),the exponent ai, which characterizes the asymptotic long time behavior of both the supersaturation CT~ (t) and the scaled radius zi, takes the same value in both cells. On the other hand, the exponent xi, which characterizes the asymptotic long time behavior of other physical quantities, such as the total volume of droplets u i ( t) , can be different from one cell to another, depending on the initial conditions.

Next, we study the asymptotic long time behavior of other physical quantities, such as the size distribution function of droplets. From Eqs. (2.13), (2.19), (2.28), and

I

15;: (z) = Fxi(z)

with


(2.29), the size distribution function of droplets in cell i is expressed as

x(r,t) = t -(4’3+x1)[ciFi(z) + l?{(z,t)]. (2.30)

Substituting Eqs. (2.11) and (2.27)-( 2.30) into Eqs. (2.2) and (2.3), and neglecting the correction terms which become asymptotically negligible in the limit t+ 03, the continuity equation in the scaling regime is written as

Fi (z) + +z-$F~ (z)

(2.31)

Therefore the scaling function Fi (z) is a solution of the ordinary differential equation (2.3 1) and is given by

(2.32)

PX2 F”(z)s (22, +z)l+4/3(1+~)(zo -z)2+5/3(1+x) exp

-(l+x)L z, -2 1 (O&Z<Zo 1 , (2.33)

where z, = (3/2) 1’3. The constant px is determined by the normalization condition M,” = 1 where

M;E Co s

dzz”F”(z). (2.34) 0

The scaling function F”(z) has a cutoff at z = z, and is characterized by the parameter x. As x increases, the peak position of F;“(z) shifts to the smaller z side and the functional

FIG. 2. The LSW scaling function FLSW [Eq. (1.4) 1, denoted by the solid line, and the scaling functions F”(r) [Eq. (2.33) ] in the open subsystem, denoted by dotted lines, are shown as functions of the scaled size z. The parameter x in F”(z) is, from the larger z side to the smaller zside, x = 2,4, 6,8, and 10, respectively. Asx increases, the peak position ofF”(z) shifts to the smaller z side and the functional form ofF”(z) becomes more symmetric.

(z>zo )

form of F”(z) becomes more symmetric (see Fig. 2). We note that, for vanishing x, Eq. (2.33) coincides with the LSW scaling function ( 1.4);

F’(z) = FLsw(z). (2.35) From Eqs. (2.30) and (2.32), the size distribution function of droplets in cell i in the scaling regime is written as

A(r,t>-t -~4’3+x’)ciFxi(z). (2.36) FromEqs. (2.29), (2.34),and (2.36),themeanradiusFi((t) in cell i in the scaling regime is given by

s’,(t) -M:‘t “3. (2.37) Substituting Eqs. (2.26), (2.34), and (2.36) into Eqs. (2.6)-(2.8), the total number of droplets n,(t), the total volume of droplets ui (t), and the total amount of both the solute and the precipitates qi (t), in the scaling regime are expressed as

u (t)?c.M”1 -T , - 313 ’ and

j&,,fXzt --*< 3’3

qi(t)- F&f:“+ w+ 2’3 *-l/3 01

3 0

2/3

z t - l/3

(2.38)

(2.39)

C”Gxi <j)

(Xi = f), (2.40)

(“i >f)

respectively. From Eq. (2.40), ci for vanishingx, is given by

J. Chem. Phys., Vol. 95, No. 6,15 September 1991


3qi(co) Ci(Xi =O) =-. 47TM’:

When the system is closed, i.e., qi ( 00 ) = q, Eq. (2.41) agrees with the LSW prediction (1.2).

If we compare Eqs. (2.37)-( 2.40) with the results as for the closed system, i.e., Eqs. (1.5) and (1.6) and q=u(t)-O(l), we find that n,(t), ui(t), and qi(t) obey new scaling laws for positive xi. Only Fi (t) obeys the same power law in time as

Ti (I) a t “3, (2.42)

(2.41) Thus in this case, xb is related toj( t), i.e., the new scaling laws of cell b, such as the scaling function (2.33)) are related to the diffusion flux flowing out of the cell. For x,, > 1, we obtain

Ij(t)l at -4’3. (2.45)

In this case, we cannot derive such a relation between xb and j(t). For vanishing x,, we need the information on the correction r: (t) in Eq. (2.18) to obtain the asymptotic long time behavior ofj( t).

with the modified coefficient MF. It now becomes clear that xi is an important parameter

which characterizes the asymptotic long time behavior of physical quantities of the cell, such as the size distribution function of droplets. For vanishing xi, the total amount in the cell qi (t) tends asymptotically to a finite value in the limit t-r 03, andJ; (r,t) and ni (t) in the scaling regime obey the same LSW scaling laws. The coarsening process in the closed system is the familiar example of this case. On the other hand, for positive xi, we see, from Eq. (2.40), that the total amount in the cell flows out into the neighboring cells and it eventually goes to zero for large t. In this case, more droplets dissolve than in the case of a closed system where the LSW scaling law is realized [see Eqs. ( 1.5) and (2.38) 1, andJ; ( r,t ), ni (t), ui ( t) , and qi ( t) in the scaling regime obey new scaling laws which depend strongly on xi.

From Eqs. (2.22) and (2.23), we find that the LSW scaling laws hold in cell a, where more droplets exist as compared to the other neighboring cell, i.e., cell 6. On the other hand, physical quantities of cell b obey new scaling laws characterized by x,, which are different from the LSW predictions.

In the next subsection, we derive the relation between xb and the diffusion flux. We expect that xb depends strongly on the number of large droplets in the initial state, because only the large droplets at the initial time to survive at a later time t. We will see how xb is determined by the initial size distribution functions of large droplets of the both cells in Sec. II D.

Nakahara, Kawakatsu. and Kawasaki: Ostwald ripening in open systems 4411

D. Dependence on the initial conditions

In this subsection, we study how xb depends on the initial size distribution functions of droplets in both cells. Let Zi (c) be the initial size of droplets in cell i which are dissolved at time c (see Fig. 3). Then, n, (t) can be expressed as

s m ?$(l) = dr& (r,t, 1, (2.46)

I,(r) where to is the initial time. From Eqs. (2.38) and (2.46), we obtain

s m drA(r,to)Nqt --1+xr). (2.47)

l,(l) We assume that, in the limit t -+ 00, the behavior of phys-

ical quantities of cell b with the finite coupling constant d asymptotically approaches that of cell b with the infinite d, owing to Eq. (2.25 ) . This assumption will be confirmed by the numerical simulations in Sec. III. Thus we concentrate here on the infinite d case.

When d is infinite, the supersaturation in each cell takes the same value at any time;

o,(t) =ab(t). (2.48)

As the critical radius is determined only by the supersaturation oi (t) owing to Eq. (2.4)) the droplets in each cell with the same radius evolve in the same manner. Therefore we have

C. Relation with the diffusion flux In this subsection, we derive the relation between the

parameter xb and the diffusion flux j(t) . For positive xb, fromEqs. (2.22), (2.23),and (24O),thediffusionfluxj(t), which is given by the right-hand side of Eq. (2.9), is expressed as

x &fXbt -(1+x,) 3

bb 3 (O<x, <f>

(j(t)l, +[$,kf;‘3 + ($)2’3]t -4’3 (xb =$)

1 3 -- 0

2’3* -‘$,3

3 2 cxb >f)

(2.43)

in the scaling regime. For 0 < xb <f, we obtain

Ij(l)(at --(‘+Q). (2.44)

0 r

FIG. 3. A schematic illustration representing Z,(t) in Fq (2.46). Z,(t) denotes the initial size of droplets in cell i which are dissolved at time t.


l(t)=,,(t) = l,(t), (2.49) where we have introduced I(t) for simplicity of the notation. FromEqs. (2.22), (2.23), (2.47),and (2.49),weobtainthe value of xb .

Now, we apply our arguments to a special case and actually find the value of xb. As a simple example, we consider a situation, where the initial size distribution function of large droplets takes the following asymptotic form:

Atr,*o)-+xp[ - (;)m i], r+ 00, (2.50)

where m, and si are positive constants characterizing the value of the initial size distribution function of large droplets of cell i. It includes the Gaussian distribution for the special case m i = 2.

First, we study the case when m i takes the same value m in both cells;

m,=m,=m.

For the case (2.51)

So >sb, (2.52) there are more larger droplets in cell a as compared to cell b at the initial time to. In this case, more droplets survive at time t in cell a than in cell 6, and therefore the total amount in cell 6, qb (t), flows out into cell a. Thus as we mentioned above, the physical quantities of cell (I in the scaling regime obey the same LSW scaling laws as for the closed system;

x, = 0. (2.53) From Eqs. (2.47), (2.50), (2.51), and (2.53), we obtain

Jyt$exp[ - (c)m’] --C -‘. (2.54)

Using the asymptotic form of the incomplete l? function, Z, (t) is given by

I,(t)--s,(ln t)“m ”. (2.55) Similarly, we obtain

We use a scaled size in cell i defined by yi- [rui(t>]3.

Defining hi (vi,t) by

Jbyt,drexp[ - ($)mb] --t -‘l+xb),

and therefore

(2.56)

qi (t) is expressed as Z,(t)--s,[(l +xb)Int]““b. (2.57)

FromEqs. (2.49), (2.51), (2.55),and (2.57),x, isgivenby qi(*) = ui (*I + s m&j yihi (.Yi,*)s (3.3) 0

XbC sb 0

--m -110. (2.58)

s.2 and, then, the continuity equation (2.2) with the growth rate (2.3) is rewritten as

Note that xb increases as m increases, and also as the ratio 1 sb/s, decreases. From Eq. (2.58), we find that the total ‘hi (YiPt) [ui(*)]3 at amount in cell b, qb (t), flows out into cell a, and it eventually goes to Zero in the lim it f+ CO. Then, fb (r,f), nb ( f), ub (f), and qb (t) in the scaling regime obey new scaling laws characterized by xb which are different from the LSW predictions.

For the case

a +3ayI K

1 Joi (*I Yi1’3- l+ [ui(t)l4 at Yi hi(Yi9*) ) 1

+ 3 Is:t, l4 85 (*I

at hi(Yi*t) = O.

so =s,, (2.59) these are as many large droplets in cell b as in cell a at the initial time to. In this case, the total amount in each cell qi (t)

From Eq. (3.4), we find that the second term in the left- hand side behaves singularly as yi- 2’3 for yi + 0. In order to eliminate such a singularity, we introduce a new variable pi (vi,t) defined by6s8


tends asymptotically to a finite value in the lim it t-+ CO, and x (r,t) and ni (t) in each cell in the scaling regime obey the same LSW scaling laws;

x, = Xb = 0. (2.60) Next, we study the case when m i differs from one cell to

another, say, ma <mb- (2.61)

In this case, we obtain x, = 0, (2.62) xb - (In * )(mb- m ”)‘maa (2.63)

Thus xb goes to infinity in the lim it t-+ 03. Therefore the scaling assumption breaks down in this case.

From these results, we find that the parameter xb strongly depends on the ratio of the initial size distribution function of large droplets of cell b to that of cell a. As the ratio decreases, the number of droplets in cell b at time t, nb (t), decreases, and therefore the parameter xb characterizing the new scaling laws increases.

In the next section, we perform numerical simulations on the two-cell model and demonstrate the validity of our arguments in this section.

III. NUMERICAL SIMULATIONS

In this section, we perform numerical simulations on the two-cell model. First, we describe the procedure of numerical simulations in Sec. III A. Next, we present our numerical results in Sec. III B.

A. Procedure of numerical simulations Here we briefly describe the procedure of the numerical

simulations which is based on the method developed by Venz1.6,8

(3.1)

(3.2)

(3.4)

J. Chem. Phys., Vol. 95, No. 6.15 September 1991



hi(+Yjpt) = exp pi (YiPt) +Yj’3 + + * [ yf’3 1

Then, the continuity equation (3.4) becomes

(3.5)

lci;t) ]3 $icyJ) + 3 Yi”3- l+ Ioi:f)14

1

aai (t) at Yi $i(YiJ)

1

+ [Oj:t)]4 ?+j + yf/3 +yf”> + 1 = 0. (3.6)

Equation (3.5) also guarantees the positivity of the new size distribution function hi (yi,t>.

Next, we introduce a scaled time defined by d7qa,(t)]3dt, (3.7)

where the mean supersaturation ow (t) of the whole system is defined by

o;(t) = ca(t) +a,ta 2 .

(3.8)

Note that the scaled time r is asymptotically proportional to In t in the scaling regime, owing to Eq. (2.26). From Eq. (3.7), the continuity equation (3.4) is rewritten as

dd 3 a L I a -T$?i(Yi,T)

+ 3(yi’3 - 1 + [$$13& ~Yj]$pi(yj9T)

m.$336+y;"+y:/') + 1 =O. (3.9)

Substituting Eq. (3.7) into Eq. (2.9)) the evolution of qi (7) is expressed as

-$qitTl = [a fTl13 [aj(T> -ai( (3.10) w

wherejf: i. From Eqs. (3.3) and (3.5)) qi (7) is expressed as

qj(T) = a,(r)

s -dYj Yj exp Q”j (Yi77) + Jf3 + + - 1

JW +

0 I

(3.11) Equations (3.7)-( 3.11) are the basic equations for our

numerical simulations. We rewrite these equations into a set of finite difference equations, which are correct up to the second order in time mesh Ar and in size space mesh Ayi. At the boundaries in size space yi, we use the forward and back- ward finite differences, which are also correct up to the second order in Ayi. We solve these equations numerically using the usual implicit iterative method.’ We employ the variable time mesh AT and use the constant size space mesh Ay, . In Appendix B, we describe how we determine Ar in our numerical simulations.

B. Results of numerical simulations

Here we present the results of our numerical simulations. We set the initial time as to = 1 and also set

qa(fo) =z,(f,) = 1 and o,(t,) =o,(t,) =O.l. To study the situations where the initial size distribution function of droplets takes the asymptotic form (2.50), we use as the initial size distribution function of droplets the following function:

(Vi -Pi)*

5: +yfj3 +:I], (3.12)

where vi is a normalization constant, ,LL~ and ci are positive constants, and H”(yi ) is the LSW scaling function ( 1.4) expressed by the scaled variable yi. As is shown in Appendix C, Eq. (3.12) coincides with the asymptotic form (2.50) withm, =mb =6andsi = ~OXC,!‘~.

We fix,ui = 1 iii Eq. (3.12) for both two cells. We study the two cases having two different values ofgi in Eq. (3.12), i.e., case (i) with 5, = 1 and gb = 3”*/2, and case (ii) with & = 1 and &, = tt) 6 I”. Thus the value of si is given by s, = 10 and sb = 10X (3)“” in case (i), and s, = 10 and Sb = 10x ,q,“” in case (ii). From the predictions (2.53) and (2.58), the value of xi are expected to be x, = 0, xb = 4 in case (i), and x, = 0, xb = 4 in case (ii), respectively. To verify our assumption that the asymptotic long time behavior of physical quantities of the system is independent of the coupling constant d, we study, in case (iii), the system with the infinite d using the same initial conditions as that in case (i), and compare these results. We summarize the parameters used in our numerical simulations and the predictions of the values of xi in Table I.

Now, we present our numerical results. Figures 4,5 (a), and5(b) showtheevolutionofqi(t),ai(t),andt1’3ai(t), respectively, for case (i). From Fig. 4, we see that the total amount of both the solute and the precipitates in cell b q6 (t) flows out into cell a. From Fig. 5 (a), we ensure that ai (t) obeys a power law in time as (T! (t) a t - 1’3. Figure 5 (b) shows that t “30i (t) approaches the finite value (s)*“, which is expected from the LSW scaling law ( 1.1) . Note that the difference between o, (t) and a,, (t) is too small to be distinguished in these figures. From Figs. 5 (a) and 5 (b), we can verify our prediction that the supersaturation oi (t) in each cell obeys the same LSW scaling law 0. (t) 2 ($) 2’3t - “3. We obtain similar results in the two oth- e; cases, namely in case (ii) and case (iii).

TABLE I. Parameters used in our numerical simulations and the theoreti- cal predictions of the values of x, for these parameters obtained from Eqs. (2.53) and (2.58).


2.0 ."'I'~"I'~~'I"~'I~'~'1~"'1"~'I~"*1**~' ,/-- - 0.0

/ / / / 1.5 -

/ /

/ / -

h -1.0 / 1cr I---------

/ L h _ /' t - 1.0 --\ V c '. .LL e t

t -2.0 I 0.5

1 L -‘\ -1 E ..t*...r....l....I....I....I.~.~I**..I~T.* -A- - 0.0 . .

0.0 3.0 6.0 9.0

Zog,,( t 1 .

-3.0 * 0.0 390 6.0 9.0

FIG. 4. The total amount of both the solute and the precipitates in each cell qi (r) is shown as a function of t ime t for case (i) with d = 10. The broken line denotes qa (t) in cell a and the dash-dot line denotes qb ( t) in cell b.

Next, we present the evolution of ni (t) and the size dis- V .- tribution function of droplets. Here we use the normalized b

size distribution function Gi (z,t> in the scaled size space z 3 defined by

dz Gi (ZJ) = dr.6 (r,t)

So” drf;: (r,t> ’ (3.13)

From the argument in Sec. II, G,(z,t) is expected to ap- proach the scaling function FXi( z) .

First, we present the results of case (i) . Figure 6 shows the evolution of ni (t). We see that n, (t) obeys the same LSW scaling law n, (t) a t - ‘, while nb (t) obeys the new power law in time as nb (t) a t - (I +Xb) with xb = 1, which is expected from E$. (2.58). Figures 7(a) and 7(b) show the evolution of G. (z,t) and Gb (z,t), respectively. It is clear that G, (ZJ) approaches the LSW scaling function F’(z), while Gb (z,t) approaches the scaling function F”“(z) withx, = f. Figure 8 shows the evolution of a mean value of z of cell i, Zi, which is averaged over Gi (zJ). We see that Z, approaches the mean value of the LSW scaling function F’(z), i.e., My, while Zb approaches the mean value of the scaling function FXb(z), i.e., My, with xb = f.

We obtain similar results for ni (t), G, (z,t), and Zi in case (iii) with the infinite d. As an example, we present, in Fig. 9, the evolution of Gb(z,t) for the finite d, which is almost the same as G,, (z,t) for the finite d in Fig. 7(b).

Finally, we present the results of case (ii) with the different initial condition. Figure 10 shows the evolution of G,, (z,t). We see that Gb (z,t) obeys the new scaling law characterized by xb = d, which is expected from Eq. (2.58).

Now, we compare our results with those obtained by Venzl.‘j As was mentioned in Sec. I, Venzl insisted that the mass transport is induced by the differences in both the supersaturation and the quantities characterizing the size distribution function of droplets between two cells. It was con-


m ,J t 1

h 4

i

hlO( t 1

FIG. 5. (a) The supersaturation in each cell o, (t) is shown as a function of t ime t for case (i) with d = 10. The solid guide line represents the LSW scaling law ff( t) oc f - I” Note that o, (t) and o,(t) are almost overlapped. . (b) The quantity t %, (t) in each cell is shown as a function of t ime t for case (i) with d = 10. The solid guide line represents the finite value (j)“‘, which is expected from the LSW scaling law ( 1.1). Note that t “‘a. (t) and t “2ub (t) are almost overlapped.

f irmed by the present work that the mass transport is actually induced by the difference in the size distribution function of droplets between the two cells. On the other hand, we have found, from the analytical treatments, that the difference in the supersaturation between two cells at the initial time to has no influence on the asymptotic properties of other quantities, such as the size distribution function of droplets, in the scaling regime. We have also found, from both the analytical treatments and the numerical simulations, that the asymptotic long time behavior of physical quantities are determined by the initial size distribution functions of large droplets in both cells. Therefore, to obtain these asymptotic long time behavior of physical quantities, we should not truncate the information on the size distribu-

J. Chem. Phys., Vol. 95, No. 6,15 September 1991



-2.0

-4.0

-6.0

-8.0

-10.0 8 0.0 3.0 6.0 9.0

k,,( t >

FIG. 6. The total number of droplets per unit volume in each cell ni (t) is shown as a function of time t for case (i) with d = 10. The broken line denotes n, ( t) in cell a and the dash-dot line denotes trb (t) in cell b. The solid guide line represents the LSW scaling law n(t) (r f - ‘. The dotted guide line represents the different power law n(t) Q f - (’ + ‘) with x = f.

tion function of large droplets, which Venzl did not take into account explicitly.

IV. CONCLUDING REMARKS

We have studied the coarsening process of a system which consists of two homogeneous cells coupled with each other by the diffusion of the solute. We obtained the following results.

(i) The supersaturation (TV (t) in each cell in the scaling regime obeys the same LSW scaling law as for closed systems.

3.0 L I I ‘~‘J’/“I’~~‘I~‘~(

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 z 2

0-Q ,,. , ,.,,..‘,,,,,‘\,.“.,“,.,.‘,‘,,...i”’.

I:- 0.8

---------------------------------4------------.

o,7 ~,,,

0.0 3.0 6.0 9.0

FIG. 8. The mean value ofz in each cell Si is shown as a function of time t for case (i) with d = 10. The broken line denotes Z?., in cell (2 and the dash-dot line denotes Z:b in cell b. The solid guide lines represents the mean value of the LSW scaling function F’(z), i.e., My. The dotted guide lines represent the mean values of the scaling function F”, i.e., M;, where the parameter x is, from top to bottom, x = 4, 4, and 1, respectively.

(ii) The number of droplets n, (t) and the total volume of droplets ui (t) per unit volume and the size distribution function of droplets A (r,t) in cell i depend strongly on their initial conditions. When there are relatively many large droplets in the cell as compared to the other cell at the initial time, the total amount of both the solute and the precipitates in the cell qi (t) asymptotically approaches a finite value in the limit t -+ 00, and ni (t) andJ; (r,t) in the scaling regime obey the same LSW scaling laws. Otherwise, the total amount qi (t) in the cell flows out into the other cell, and it

3.0

FIG. 7(a) The evolution of the normalized size distribution function of droplets in cell a G. (z,t) is shown as a function of the scaled size z for case (i) with d = 10. Broken lines denote G, (zJ), where the time is, from bottom to top, t = lo’,‘, 106.‘, lo’.‘, and lo*,‘, respectively. The solid line denotes the LSW scaling function FLSW (z) . Dotted line denotes the scaling function F”(z) withx = f. Note that G, (z,r) approaches F Lsw(z) forlargei. (b) SameasFig. 7(a) but for cell b, with dash-dot lines denoting Gb (z,~), where the time is, from bottom to top, r = lo’,‘, 106,0, lo’,‘, and lo’.‘, respectively. Note that Gb (z,l) approaches F”(z) with x = f for large t.


0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

2

FIG. 9. Same as Fig. 7(b) but for case (iii) with the infinite d. The time is, from bottom to top, r = 10 5o 106q lo’,‘, and lo*.‘, respectively. Note that , the evolution of Gb (ZJ) for the infinite d is almost the same as those for the finite d in Fig. 7(b).

eventually goes to zero for large t. Then& (r,t), ni ( t), uj (t), and qi (t) in the scaling regime obey new scaling laws characterized by the non-negative parameter xi as:

f;:(r,t)4it - (4/3 + x’)I;%(rt - l/3), (4.1) ni(t)dit --1+xi), (4.2)

ui (t) + c,M:‘t -+,

and qi(t)-maX[ui(t),ai(t)], (4.4)

respectively. Here ci is a positive constant, F”‘(z) is the scaling function (2.33) of the open subsystem and

M;: = s

*dzz”F”(z). 0

(4.5)

3.OC."'i' " " '1 I"","","*'

0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

z

FIG. 10. Same as Fig. 7(b) but for case (ii) with the different initial condition, with the dotted line denoting F"(z) with x = A. The time is, from the bottom to top, r= 105’, IO”‘, lo’.“, and lo*‘, respectively. Note that Gb (z,r) approaches F'(z) with x = &, which is expected from E$. (2.58).


The parameter xi is determined by the initial size distribution functions of droplets.

(iii) The mean radius Ti (t) in each cell grows as Fi (t) du?t “3. (4.6)

Thus for positive xi, only the coefficient M;” differs from the LSW prediction, i.e., MkSW in Eq. (1.6).

(iv) For 0 < xb <f, xb is related with the diffusion flux j(t) as

lj(t)lat -(‘+xb). (4.7) Although our model seems to be too simple, it may be

possible to apply our results to real situations. In the formation of the precipitation pattern, the space is often separated into two regions, i.e., the region L where larger precipitates can be found and the region S where only smaller precipitates are observed. In this case, larger precipitates in the region L let smaller precipitates in the region S dissolve by the Ostwald ripening. Thus, the growth of precipitates in the region S may be regarded as the coarsening process in the open subsystem, where the total amount of both the solute and the precipitates flows out into neighboring regions. Ap- plying our arguments to this situation, we may expect that the scaling function of the size distribution of precipitates in the region S becomes more symmetric than the LSW scaling function, as was reported in Ref. 2. We may also expect that the parameter xg, which characterizes the scaling function, is related with the diffusion flux from the region S to the region L, when the inequality 0 < X~ <; is satisfied.

ACKNOWLEDGMENTS

One of the authors (A.N.) thanks Dr. K. Sekimoto, Dr. H. Hayakawa, and N. Suematsu for valuable discussions. This work was supported by the Scientific Research Fund of the Ministry of Education, Science and Culture of Japan.

APPENDIX A In this Appendix, we derive the growth rate (2.3) from

the Gibbs-Thomson condition. From the Gibbs-Thomson condition, the equilibrium

concentration at the surface of a droplet with radius R is given by

C,(R) =C,(m, +-$ (Al)

where C, ( CO ) is the equilibrium concentration at the surface of an infinitely large droplet and a is a capillary length. Since we are interested in the diffusion limited growth of droplets, the growth rate is given by

R&T) = +7 - C,(R)], (A21

where T is the time, D is a diffusion coefficient of the solute, and C( 7”) is the mean concentration of the solution. Substi- tuting Eq. (A2) into Eq. (Al), we obtain

V(R,T) =; [an -;],

J. Chem. Phys., Vol. 95, No. 6, 15 September 1991



where Z ( 7’) = C( 7) - Cbg ( CO ) is the supersaturation of the solution.

Now, we introduce a reduced time t and a reduced radius r as follows:5

t~w&J)13T I a'

(A4)

(AS) a

We also rewrite the supersaturation in reduced form as B(T) u(t) = -.

C,(cQ) (A6)

Using these reduced variables, the growth rate is expressed as

v(r,t) = +t) - $1. (A7)

APPENDIX B

In this Appendix, we describe how we determine the variable time mesh AT. We also describe some technicalities used in our numerical simulations.

We determine Ar in such a way that Ar satisfies the following relation:

I I Ap (E PO ’

hr<6,

(Bl) W)

where E and 6 are positive constants,p( T) denotes a physical quantity at time T, and Ap is the change ofp(T) per small time interval AT. At the initial time to in each numerical simulation, we set E = 5.0X 10d4 and S = 1.0X 10e4. In order to avoid increase in the number of iterations in the implicit iterative method, we automatically decrease the values of criteria E and 8.

We set space size mesh as Ayi = 5.0 X 10 - 3. In order to avoid increase in the number of iterations, we also change the boundary at the largest end ofy, by eliminating the mesh point at the end when the value of hi (yj,r) at the point becomes smaller than 1 .O x 10 - 60.

APPENDIX C

In this Appendix, we verify that the initial size distribution function of droplets (3.12) used in our numerical simulation coincides with the asymptotic form (2.50) with m, =mb =6andsi = 10~fi’~.

From Eqs. (1.4), (2.29), (3.1), and (3.12), the initial size distribution function of large droplets in the size spacey is expressed by a Gaussian distribution with the variance < :;

hi(yi,to)-exp Yf [ 1 -- f .Yi+CO* s: (Cl)

UsingEqs. (3.1)and(3.2),werewriteEq. (Cl) inthesize space r as

B(r,t,)-exp( - [+]j, r-00. (C2)

If we take a, (to ) = 0.1, Eq. (C2) coincides with Eq. (2.50) withm,=m,=6andsi=10X<;‘3.

‘J. D. Gunton, M. San Miguel, and Paramdeep S. Sahni, in Phase Transi- tions and Critical Phenomena, edited by C. Domb and J. L. Lebowitz (Academic, New York, 1983), Vol. 8, and the references therein.

%. Kai and S. C. Mtiller, Sci. Form 1,l ( 1985), and the references therein. 31. M. Lifshitz and V. V. Slyozov, J. Phys. Chem. Solids 19, 35 (1961). %. Wagner, 2. Elektrochem. 65,58 1 ( 196 1) . ‘J. A. Marqusee and John Ross, J. Chem. Phys. 79,373 (1983). 6G. Venzl, Phys. Rev. A 31,343l (1985); J. Chem. Phys. 85, 1996,2006

(1986). ‘C. W. J. Beenakker and John Ross, J. Chem. Phys. 83,471O (1985). *G. Venzl, Ber. Bunsenges. Phys. Chem. 87,318 (1983). 9W. H. Press, B. P. Flannery, S. A. Teokolsky, and W. T. Vetterling, Nu- merical Recipes in C (Cambridge, Cambridge, 1988).


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