Growth of domains under global constraints

Physica A 188 (1992) 436-442 North-Holland

Growth of domains under global constraints L. S c h i m a n s k y - G e i e r a, Ch. Ziilicke a a n d E. Sch611 b alnstitut fiir Theoretisehe Physik, Humboldt-Universitiit zu Berlin, Invalidenstr. 42, 0-1040 Berlin, Germany blnstitut fiir Theoretische Physik, Technische Universitiit Berlin, Hardenbergstr. 36, W-IO00 Berlin 12, Germany

We develop growth rates for domains in bistable systems far from equilibrium. The governing equation is a bistable reaction-diffusion equation including a nonlocal inhibiting term. The latter one will be responsible for the existence of stable localized domains. The growth of the size of different domains shows quite surprising similarities with coalescence phenomena of droplets in van der Waals gases. Eventually we discuss the possibility of breathing of the domains induced by a time-delayed nonlocal interaction.

I. Introduction

We will be concerned with nucleation, formation and breathing of domains in one-variable bistable dynamics far from equilibrium. The main aim of our paper is to show that the formation of stable domains in bistable nonequilibrium systems behaves quite similar to processes of Ostwald ripening in thermodynamic systems. A domain will be defined as a solitary front-like (d = 1) or droplet-like (d = 2, 3) configuration with stationary interfaces. In this sense it is a localized stable inhomogeneous pattern [1-4]. For our purpose in addition to the considered local dynamics we introduce an inhibitory nonlocal interaction in the reaction-diffusion equation. In thermodynamic systems with Ostwald ripening, e.g. in van der Waals gases, this nonlocal process consists in the depletion of free monomers during the nucleation of droplets [5]. If the particle number is fixed the pressure becomes a function of the actual droplet configuration. Analogously, similar situations happen frequently in systems far from equilibrium: (i) During the dielectric breakdown in a nonlinear semiconductor the voltage drop across the semiconductor generally depends on the size of the generated current filament [6-8]. (ii) If laser light is used to excite front propagation in a bistable optical element and the absorption coefficient depends on the density of the excited particles, the intensity of the light becomes a function of the depth to which the front has propagated into the optical element [9]. (iii) In thermokinetic reactions on surfaces with a 0378-4371/92/$05.00

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large heat conductivity the t e m p e r a t u r e becomes a global p a r a m e t e r and depends on the size of the surface where the reaction takes places and where heat is produced or absorbed [10, 11]. T h e r e f o r e we consider a general nonlocal coupling due to ([12, 13] see also refs. [14, 15])

Ot - f ( n ) + n A n -

a n,

~

K [ r - r', n(r', t)]

(1.1)

.

vo As shown by the examples such nonlocal terms occur either due to specific external constraints or due to internal different length scales of different variables of the considered system. There is no difference between both p r o b l e m s in our description by phenomenological balance equations. In both situations a stable propagating front or droplet can be slowed down to come to rest at a certain value forming thereby a stable inhomogeneous domain.

2. Kinetics of domain formation

We consider a o n e - c o m p o n e n t chemical reaction system with the density n(r, t). We assume a bistable local reaction rate. Then the evolution of n(r, t) is governed by a nonlinear reaction-diffusion equation. We include a simple inhibitory nonlocal interaction and look for spherically symmetric d-dimensional droplet solutions

On d- 1 On 02n f ddr ' 0t =f(n) + --r D ~r + D --Or2 - ak j ~ O(nv0

n2) n(r', t ) .

(2.1)

H e r e @(n) is the Heaviside step function which controls the nonlocal interaction. It has the effect that if the density at any point of the space is larger than the unstable state n 2 it will diminish the growth of the density globally at every point. V0 denotes the characteristic d-dimensional volume of the range of the nonlocal interaction and k controls its time scale. The prefactor a is a constant. We will assume that the steady states are changed only slightly by the new term. Further we suppose that a stationary droplet solution has built up in the system obeying the boundary conditions n(r = O, t) = n3, n(r = w, t) = I/1, i.e, we imagine that a domain of the state n 3 is imbedded in the surrounding state n~. The position of the interface dividing both regions will be defined by the position of the unstable state n 2 and denoted by R(t),

L. Schimansky-Geier et al. / Growth of domains under global constraints

438

n(R(t), t)

=

(2.2)

n 2 .

Following Schl6gl [16, 17] we can derive a first integral by multiplying eq. (2.1) with On~Or and integrating over r from r = 0 to ~. The further analysis uses the assumption of sharp fronts. We eventually derive a dynamic equation for the domain growth [12, 13, 18]

/~=(d-1)D

(1

Rk(t )

•

(2.3)

,

where we have introduced the time-dependent critical radius

Rk(t)

=

X"-

( d - 1)Db ak(n, - n3)n3V(t)/V~, "

(2.4)

We abbreviated b = f{~-dr (On~Or) 2, V(t) = Cd R'l for the d-dimensional volume of the domain and X °= f " ) d n f(n) for the supersaturation without nonlocal coupling. The effect of the inhibitory global interaction results in a dependence of the critical radius on time via a dependence on the size V(t) of the domain. The d e n o m i n a t o r of eq. (2.4) we can call supersaturation of a bistable nonequilibrium system,

X(t) = X ° - ak(n 3

V(t) nt)n 3 V~,

(2.5)

The supersaturation will decrease for an increasing domain. Accordingly the critical radius increases as well. This is the typical behaviour taking place during the process of Ostwald ripening in a van der Wools gas [19]. We have thus obtained the same p h e n o m e n o n for the domain formation of a dissipative high-density steady state surrounded by a low-density steady state in a bistable reaction system. The investigation of the steady states of the kinetics (2.3) shows that eventually by this mechanism a stable d-dimensional domain can be formed. setting R = 0 for initial supersaturations X ° exceeding some critical value,

d-~ 1 X">

X~,

-

[I

( d Db 2 - 1

{.) '`''~'' Db

"

(2.6)

two steady solutions 0 < R{L)< R (2) occur, the smaller of which is unstable, while the larger one is stable. The dependence of the steady state radii R (~) and R (2) on the initial supersaturation X ° can be estimated for X°>> X',!r as

439

L. Schimansky-Geier et al. / Growth of domains under global constraints

R (L)--~(d-1)D

xO-

v

- Rk'

= R°~"

'

(2.7) c

ak =

77,

%

- n,) n

c..

It is seen that the smaller value corresponds to the unstable critical radius of the purely local dynamics. If now a supercritical domain is formed ( R ( 0 ) > R(1)), it will grow only until it reaches the stable steady size R (2), and hence forms a stable domain. Its size depends on the supersaturation and on the ratio of the different rate coefficients. It is proportional to the characteristic range L , = ( V , / C a ) ~/ J of the nonlocal interaction. In a former paper [13] we applied this procedure to the three applications mentioned in the introduction. To our opinion Ostwald ripening, therefore, represents a very general rule for the growth of stable domains in nonequilibrium thermodynamics as well.

3. Competition by global interaction Lifshitz and Slyozow [19] underlined that the coalescence of droplets in van der Waals gases is a selection process of clusters of different size due to the b o u n d e d total particle number. An analogous picture holds in nonequilibrium systems, where the inhibitory global interaction limits the number of domains. Starting with eq. (2.1) and using the same assumptions as in section 2 one derives for an ensemble of N domains inside the characteristic volume of the nonlocal interaction N

/~i = v - ( d - 1)

D

C ~Rj, b j=l a

Ri

(3.1)

where R~ is the size of the ith domain. For supercritical growing domains this equation represents a kind of Eigen-Fisher dynamics [20, 21]. It can be rewritten in such a way that the ensemble of domains is divided into sub- and supercritical ones. This can be done by introducing the averaged critical radius (Rk(tl)=(d-110/

( c2 :) v-

R

.

(3.2)

j=l

Thus the dynamic equation for the ensemble of domains reads [12, 13] /~i=(d-1)

D

1

(Rk(t))

1 )

Ri

"

(3.3)

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L. Schimansky-Geier et al. / Growth of domains under global constraints

Therefore the dynamics of a certain domain i depends on the actual configuration of the other domains inside the considered volume via the dependence of the mean critical radius on the volume fraction of the established domains. It will grow until the mean critical radius ~R k) reaches its size R i. After that it becomes subcritical and vanishes. The winner of this competition process always is the initially largest droplet.

4. Breathing of domains Additionally we now include a time delay in the nonlocal part of the dynamics. Several authors [1-3, 6, 22] pointed out that a time delay will be responsible for an oscillatory motion of a stable domain. We will study here the most simple situation replacing in the integrand the actual time t by t - 7. Such a delay is very general and obviously could serve as a first approximation of real physical systems only. The dynamics for the density reads On f ddr ' O~ = f ( n ) + D A n -- a k ~ O(n(r', t%

, ) - n2) n(r', t -

"r) .

(4.1)

Repeating the former analysis yields R(t)=D(d-1)

nk,_ -

(4.2)

where now the critical radius has to be taken at times delayed by the value ~-. Dynamics like eq. (4.2) are known quite well. The most fascinating point is that it allows the generation of limit cycles as a result of the overshooting of R ( t ) . Indeed, linearizing around the former stable radius R (2) = R 0 (2.7) of the domain R ( t ) = R o + ~ R ( t ) we obtain in first order of 8R a / ~ ( t ) - ( d - 1)D R~ [aR(t) - c 8 R ( t - ~-)]

(4.3)

c = d ( R , , / R ~ - 1).

(4.4)

with

Following Gurija and Lifshits [23] we will perform the stability analysis of the Radius R 0 with respect to small deviations ~R. Looking for solutions ~R*cexp(At) the complex eigenvalues A= A' +iA" obey two transcendental

L. Schimansky-Geier et al. / Growth of domains under global constraints

441

e q u a t i o n s . Let us look for e i g e n v a l u e s with v a n i s h i n g real part A ' = 0. F o r A" va 0 we find

a"(O) = - ( d - --~-1 ) D (c 2 _ 1),/2 ,

(4.5)

R0 which leads to a real s o l u t i o n for c > 1, i.e. 0 R o > Rk(d + 1)/d.

(4.6)

T h e b i f u r c a t i o n set is d e t e r m i n e d by cos[A"(0) r] = 1 / c .

(4.7)

A"(0) s t a n d s for the oscillatory m o d e t a k e n at A' = 0. Fixing c a n d c h a n g i n g z eq. (4.7) gives the critical v a l u e of ~- w h e r e a limit cycle regime is established. For

r>~-c-

R~ (d-1)D

arccos)(1/c) (c 2 - 1 ) 1/2

(4.8)

the t i m e delay r g e n e r a t e s a stable b r e a t h i n g m o d e of the d o m a i n .

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