Kinetics of reactions in random solid mixtures

Physics LettcrsA 171 (1992) 137-140 North-Holland

PHYSICS LETTERS A

Kinetics of reactions in random solid mixtures S.F. B u r l a t s k y Institute of Chemical Physics, Russian Academy of Sciences, Kosygin Street 4, l 17334, Moscow V-334, Russian Federation and A.I. C h e r n o u t s a n Department of Physics, State Academy of Oil and Gas, Leninsky Prospekt 65. 117296 Moscow, Russian Federation Received 3 September 1992; accepted for publication 15 September 1992 Communicated by V.M. Agranovich

The kinetics of the reversible contact reaction A+ Be;C in a solid random mixture of reagents is investigated. The solid mixture is modelled by a cubic lattice, each cell containing initially one of the two reagents with equal probability. The mean concentrations approach their equilibrium values by a power (not an exponential ) law. In contrast to the case of random Poisson mixtures, the reaction is diffusion controlled and fluctuation induced from the earliest stages, when only a small fraction of the particles has entered inlo the reaction.

The many-particle aspects o f the theory o f diffusion-controlled processes ( D C P ) have recently attracted considerable attention. This is due, on the one hand, to the close relation to fundamental problems o f statistical physics in which fluctuation induced behavior is essential. On the other hand, there exists a wide variety o f applications which range over an incredibly broad spectrum - the recombination o f ions and defects in solids, energy transfer in liquid and solid solutions, chemical reactions, electron dynamics in disordered systems o f repulsive scatterers and biological processes with migration, annihilation and multiplication. The D C P investigations were first stimulated by the work o f Smoluchowsky [ l ] concerning the kinetics o f irreversible coagulation o f colloid particles. His approach, based on the one-particle diffusion equation with adsorbing boundary, leads to the following mass equation for the mean density of

particles, hA(t) = h n ( t ) = --K~ernA ( t )nB( t) ,

(la)

where the overdot denotes the time derivative and

the "effective" rate constant in three-dimensional systems reads Keer=4~DR[ 1 + R / ( n D t ) 1/2 ] ,

(lb)

(for 4gDR <
(2)

which displays a slower kinetic behavior for d < 4 compared to the mean-field predictions. The point is that the small fluctuation induced initial A and B concentration difference z(r, t) is not affected by the recombination reaction. The diffusive decay o f

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z(r, t) is a slower process than the mean-field recombination decay entailed by eq. ( 1 ). For large t the reaction bath becomes separated into large domains (which characteristic size L ~ t ~/2) containing particles of only one type and the kinetic behavior is controlled by diffusive "transport" from regions enriched by species A and regions enriched by B to each other, which entails the decay (2). It was proved in ref. [ 8 ] that the fluctuation induced asymptotic law (2) is exact. A similar proof was published by Bramson and Lebowitz in a set of fundamental papers [ 9 ]. For the reversible reactions there again exists a linear combination of local concentrations which is not affected by the reaction due to the mass conservation law. As a consequence a wide variety or reversible bimolecular reactions are defined by a power-law approach to equilibrium for t ~ o o [10-15] instead of an exponential mean-field decay law. However, it should be mentioned that in the simplest systems with random homogeneous mutually independent initial particle distributions without correlations and with small mean concentrations, fluctuations influence only the long-time kinetic tails. This means that only a small amount of panicles nf reacts in the fluctuation induced regime. Here we have a straightforward analogy with the second order phase transitions in systems with small coupling, where fluctuations are important in the narrow region near the critical point. The latter is the analog of t--,ov in DCP. Consequently there is great interest in systems where the spatial fluctuations are decisive from the earliest times and govern the conversion of the bulk of the reactive species [ 16,17 ]. Examples are: systems with high reagent densities or with correlations in the reagent distributions (polymers) [18], systems with external particle random input [1215,19,20 ] or reproduction reactions [ 21 ], solids with topological defects [22] and random systems with high disorder or distant reactions, where the random motion is not diffusion [23-25 ]. In this paper we present one more example of a system where fluctuations govern the system kinetics from the early stages of the system evolution. In this system the two reagents A and B undergo the reversible bimolecular contact reaction 138

30 November 1992

K

A+B ~ C,

(3)

K-

but contrary to the systems described above, the initial state is a random mixture of finite fragments, each fragment containing only one of the reagents. More specifically, we investigate the kinetics of reaction (3) in a stereometric system which at t = 0 consists of cubic domains of size/, each domain containing with the probability ½ only A or B reagents of constant concentration no/2 (no is the initial mean concentration: n0A= noB=no). This model should describe correctly the chemical reactions in solids, where the initial state is usually prepared by grinding the components, mixing them and exposing them to high pressure and/or temperature. As the reagents are initially separated in space, the reaction is going on only due to the transport through domain borders. Hence the process should from the beginning be of diffusion controlled character. The equations for the local concentrations nA(r, t), riB(r, t) and nc(r, t) have the form hA.a (r, t) = --KnA (r, t)nB(r, t) + K_ nc(r, t)

+ DA.a /XnA.B(r, t) , hc(r, t) =KnA(r, t)nB(r, t) - K

nc(r, t)

+Dc/Xnc(r, t),

(4)

where K and K_ are the effective constants of the forward and backward reactions. In order to obtain the equations for the mean densities n ( t ) = (nA(r, t ) ) = ( n B ( r , t)) and n c ( t ) = ( n c ( r , t)) we (as in ref. [4]) introduce the density deviations aA,B,c(r, t):

nA.a(r, t) = n ( t ) +aA.B(r, t ) , nc(r, t)=nc(t)+ac(r, t) ((oA.a,c(r, t ) ) = 0 ) . From (4) we then derive the following equation for n (t), n=

-Kin2+ GAB(0, t) ] +K_nc

(n(0)=no), function,

(5)

where GAB(it, t) is the correlation

GAa(it, t) = (oA(r, t)tra(r+it, t) ) . The exact equation for

(6)

GAa(it, t) includes correlators

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of higher order, etc. As was proved in ref. [8], if we break this chain of equations by replacing the fourth order correlation function by the product of two second order correlation functions, we shall get a lower bound to the exact solution. Moreover, for some diffusion controlled problems this approach yielded the exact decay law. In our case the equation for G,aa(2, t) contains other second order correlation functions, defined in the same way as (6): (3AS = -- Kn ( GAA + GBs + 2GAs) + K_ (GAc + Gsc)

+ (DA + D e ) AGAe,

30 November 1992

hi (r, t) = ½DAnt (r, t) .

In our case the initial condition for n, is n,(r, O) = 2no. This means that the local concentrations obey the local conservation law hA(r, t ) + n B ( r , t ) + 2nc(r, t ) = 2 n o ,

and for the deviations a(r, t) we get a , ( r , t ) + aB(r, t ) + 2ac(r, t ) = 0 . Hence the correlation functions G~: may be excluded, 6Ac = - ½( 6 ~ , + GAB),

6 s c = -- ½(6Bs + 6 A . ) ,

(3AA= --Kn(2GAA+2GAB) +2K_ GAC+ 2DA AGAA,

and eqs. (7) assume the form

(3Be = -- Kn (2GBB + 2GAB) + 2K_ Gsc + 2De A GsB. (7)

(3A" = -- Kn ( G ~ + GsB + 2GAB)

The initial conditions for correlators are as follows,

(9)

+ K _ (GAA +Gss + 2GAB) + D A G A B , (3,,A = - K n ( 2 G A A + 2GAB)

G ~ ( , t , 0 ) = Gs~(,t, 0)=fo(~t),

+ K _ (GAA +GAB) + D A G A A , GAa(A, 0 ) = - f o ( 2 ) ,

Gic(L 0 ) = 0 ,

(3BS= --Kn(2GBB + 2GAaB) where fo(~) =no2(l - I ) . x l / l ) ( 1 -

+ K _ (GsB + G ~ ) + D A G B B . 12y I/1)...

(8)

(10)

Note that for a less symmetric system (for example, a nonhomogeneous one or with domains of different size) the initial function n(r, 0) would not be a simple constant and the local conservation law (9) would not be valid. In this case one should deal with the full system of six equations for six correlators. The combination of correlators

if all 12,1
with initial condition S(r, t) = 4f00.), and the equation for GAB assumes the form

hA(t) + h e ( t ) + 2nc(t) = 2 n o ,

( 3 ~ = - ( K s + ½K_ ) ( S + 4G~aB)+ D A GAe .

which in our case reduces to

These equations for S and GAB are solved by Laplace transformation. The Laplace image of GAB is equal to

n+nc~n o . The corresponding combination of local concentrations

S = G ~ + GBe - 2GAaB, obeys the pure diffusion equation k=z)zxs,

gas ( P ) = - F o ( p ) e x p ( - p Z D t ) , where

n, (r, t ) = n A ( r , t ) + ns(r, t ) + 2nc(r, t)

satisfies the pure diffusion equation 139

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p f o ( P ) = ( 2 ~ ) d/2 j f o ( X ) e x p ( - i p . i t ) dd~.

30 November 1992

the q u a s i s t a t i o n a r y state n ~ ~/GAB, a n d for r >> 1 n approaches zero according to the fluctuation law (2).

=n~(2n~pz/Sin2(½Pxl) ) References 4

• 2

where d is the space dimension. Calculating the back t r a n s f o r m for ~.=0, we get

GAB(0, t) -- - ( 2 7 t W 2 J F o ( p ) e x p ( - p 2 D t )

4 = - r i g F~/

=

p - 2 sin2(~pl) e x p ( - p 2 D t )

- n g { e f f ( r -1/2) - ( r / ~ ) 1/2[ 1 - e x p ( - r - t )

dep dp

' 1} ~ ,

where r = 4 D t / l 2. F o r r>> 1, i.e. w h e n the characteristic diffusive length is m u c h larger t h a n l,

aAB~ngr-~/2 However, in a m u c h earlier stage o f system evol u t i o n begins the specific " q u a s i s t a t i o n a r y " regime o f kinetics. T h e n h in eq. ( 4 ) (where n c is c h a n g e d to n o - n ) b e c o m e s negligibly small c o m p a r e d with Kn 2 a n d KGAB, i.e. n ( t ) is equal to

n ( t ) = ½[ ( K 2 + g l n o --4GAs)'/2--Kl ],

K~ = K _ / K .

T h i s stage b e g i n s w h e n (Dt) ~/2 is yet small c o m p a r e d to l ( r < < 1 ), i.e. the t h i c k n e s s o f a layer at a d o m a i n b o r d e r c o n t a i n i n g reagent C is m u c h s m a l l e r t h a n I. In these layers the reaction is going o n o n l y due to the t r a n s p o r t o f reagents, as the local reaction e q u i l i b r i u m is established practically i m m e d i a t e l y at each point. F o r e x a m p l e , in the case o f the irreversible reaction ( K _ = 0 ) a n d the pure diffusion limit o f the rate c o n s t a n t ( K = 4 n R D ) , this q u a s i s t a t i o n ary regime b e g i n s at v / - r ~ ( n o R 3 ) - ~ ( r / l ) 2~ (noRl 2) -i, w h i c h is small for solids or small solvent c o n c e n t r a t i o n (noRl 2 is the n u m b e r o f reagent particles in the n a r r o w layer n e a r the b o r d e r o f a cell). F o r r>> 1 n ( t ) a p p r o a c h e s the e q u i l i b r i u m value with a power, i n s t e a d o f a n e x p o n e n t i a l law,

~)n ~ no r - a/2 . In the case o f the irreversible reaction, b e g i n n i n g from

140

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