PHYSICAL REVIEW A

VOLUME 45, NUMBER 2

Kinetics of monomer-monomer

15 JANUARY 1992

surface catalytic reactions

P. L. Krapivsky Central Aerohydrodynamic

Institute, Academy

U. S.S.R., 140160 Zhukovsky 3, Moscow Region, U. S.S.R. (Received 29 April 1991)

of Sciences of the

We study the kinetics of an irreversible monomer-monomer model of heterogeneous catalysis. In this model, two reactive species, A and B, adsorb and stick to single sites of a catalytic substrate. Surface reactions are assumed to occur only between dissimilar species that are nearest neighbors on the substrate. The kinetics of the process are studied in the reaction-controlled limit. We map the monomer-monomer model of heterogeneous catalysis onto a kinetic Ising model and find that the dynamics of the process is spin-exchange dynamics. a superposition of zero-temperature spin-flip dynamics and infinite-temperature We solve the kinetics analytically and determine the rate at which the catalyst becomes "saturated, i.e., completely covered by only one species. We show that the saturation time is proportional to N ln(Nj, where N is the number of catalyst sites. We also discuss the monomer-monomer process with desorp-

"

tion.

PACS number(s): 05.70.Ln, 82.65.Jv, 82.20.Mj

I. INTRODUCTION Heterogeneous catalysis is a fundamental kinetic process of considerable interest in many apparently unrelated fields of science and technology. In this process, the rate of a chemical reaction is enhanced by the presence of a suitable catalyst material [1]. An example with practical applications is the platinum surface used to catalyze the reaction: 2CO+Oz~2COz (the motivation for accelerating the oxidation of carbon monoxide is clear in the context of automobile emissions). Consequently, it is of great practical and theoretical interest to possess mathematical predictions concerning the kinetics of catalytic reactions. Typically, such processes are described by Langmuir-Hinshelwood kinetics, where the molecules are assumed to be randomly distributed on the surface [1]. These are assumptions of a mean-field character, as microscopic details, such as spatial Auctuations in concentration and excluded volume interactions, are neglected. However, recent studies suggest that Auctuations are a crucial element in driving the kinetics [2]. Therefore the investigations of microscopic models have begun to identify the general principles underlying the dynamical behavior of heterogeneous catalysis [3 —10]. Depending on the relative deposition rates, there may be the phenomenon of "poisoning" or saturation, where the catalytic substrate eventually becomes covered by one of the species only, thereby terminating the catalysis process, or there may be an apparent reactive steady state. Kinetic phase transitions which demarcate these possibilities have recently been elucidated by many authors [3—5, 7 —12]. In the present study, we investigate the kinetics of heterogeneous catalysis for the monomer-monomer process. In this model, two reactive species, A and B, adsorb and stick to single sites of a catalytic substrate. Surface reactions are assumed to occur only between dissimilar species which are nearest neighbors on the substrate. For unequal adsorption probabilities, the substrate quickly 45

saturates with the preferred species. For equal adsorption probabilities, a finite substrate still saturates, but at a much slower rate with equal probability of saturation by either of the two species. There are two basic limiting cases of catalysis. When the reaction on the substrate occurs quickly, the process is limited by the adsorption rate. This is the adsorptioncontrolled limit. On the other hand, in the reactioncontrolled limit the adsorption occurs readily so that the overall process is limited by the conversion of unlike monomers to AB pairs. Much of the previous work [3 —5, 7, 8, 10 —12] has considered the adsorption-controlled limit. However, many catalytic processes occur in the opposite limit [1]. In the present paper, we focus on the reactioncontrolled limit for the monomer-monomer process. It is clear that the model is a gross oversimplification of the actual catalytic process. A more realistic treatment should consider at least the effects of surface diffusion, desorption, finite adsorption and reaction rates, and nearest-neighbor interactions. Very recently, some of these effects have been studied numerically [6, 13,14]. However, the present model contrasts with more realistic ones because it can be solved analytically. Therefore the model appears to be an extremely rare problem of nonequilibrium statistical mechanics which may be solved exactly in arbitrary dimensions. In the present work we describe the kinetics of the monomer-monomer catalytic process in the most interesting case of a two-dimensional (2D) substrate (the kinetics of the process on a 1D substrate have been reported on in my previous work [15]). It is important that for the monomer-monomer process differences between the reactiononly quantitative controlled and adsorption-controlled limits have been observed [9,16]. Thus the relatively simple reactioncontrolled limit gives an insight into the general case of arbitrary adsorption and reaction rates. The rest of this paper is organized as follows. In Sec. II we map the monomer-monomer catalysis model in the 1067

1992 The American Physical Society

P. L. KRAPIVSKY

1068

there is a net bias of either A s or saturates exponentially in time with the preferred species. Therefore we shall discuss the most interesting case of equal adsorption probabilities, where diffusive fluctuations drive the system to saturation. We shall consider the kinetics of catalysis in a reaction-limited regime, introduced in a pioneer work [17], on the infinite square lattice. Other types of lattices can a1so be dealt with. It is useful to map the model onto a kinetic Ising model. Identifying A's with +1 and B's with —1, a full substrate may be described in terms of Ising variables S = [S;, ]. It is not difficult to derive the following master equation for the probability distribution P (S, t) of the Ising state S at time t:

reaction-controlled limit onto a kinetic Ising model with competing dynamics. In Sec. III we calculate one- and two-spin correlation functions. From this solution, basic information about the rate of saturation is obtained. In Sec. IV we discuss the catalysis model with desorption. The conclusions are summarized in Sec. V.

sorption probabilities,

B's, and the substrate

II. DYNAMICS In the recent-controlled limit, the substrate quickly becomes full. Thus we start our study with the substrate that is randomly filled by equal amounts of A and B units. If two neighboring sites are occupied by opposite species, a reaction occurs in which the reactants desorb, each unoccupied site then being immediately refilled by either A or B with equal probability. For unequal ad-

P(s, t)=— g [U~(FJS)P(F; S, t) t&

U~(S)P—(s,

t)]++ [V~(F~F;+,Js, t)P(F, F, +, S,"t)

J

l&

+g [WJ(FJFi+,S, t)P(F; FJ+,S, t) 17

(S, t)] . W; (S)P—

sions

[4 — S~(s;+ij+S;+i +S; +i+S;

1

8vi

i)], (2a)

Vij(S) =

= WJ(S)=

1

8rz

(1 — SJS;+ij ),

(2b)

(1 — SJS~+i) .

(2c)

1

8&2

Notice that the process (2a) defines the single-flip dynamics at temperature T =0 while the processes (2b) and (2c) define the Kawasaki spin-exchange dynamics at temperature T = ~. A simple analysis shows that the dynamics of the surface catalytic reaction coincides with the dynamics (2) under the following constraint between the time scales of spin-flip and -exchange processes: 1

V()(S)P—(s, t)]

J

J

In this equation, UJ(S), V; (S), and Wt(S) denote the rate for the system to jump from the state S to the state FJS, F; F;+& S, and F; F; +,S, respectively, and F;. for all denotes the spin-flip operator, F; S = [ S — (p, q)A(i, j); S;J]. Flip rates are given by the expres-

UJ(S)=

45

2

Notice that a general case of arbitrary spin-flip and -exchange time scales can also be dealt with. When both spin-lip and -exchange processes are model since the syspresent, we have a nonequilibrium tem is in contact with two heat baths which are at zero As we shall show below, this and infinite temperature. model is solvable in the sense that the time evolution of the one- and two-spin correlation functions can be calculated exactly. In closing we note that our model is an example of Ising-type systems with competing dynamics

which have been investigated in a series of recent studies (see, e.g. [18,19] and references therein).

III.

ANALYTICAL RESULTS FOR CORRELATIONS

The advantage of the spin formulation of the problem is that one can easily derive a closed set of differential equations for the spin-correlation functions &S; . .

.S

„&=gS;

S „P(S,t) .

(4)

S

First, we discuss the one-spin correlation functions. Multiplying both sides of Eq. (1) by S,i and summing over all configurations S we find that the average values of the magnetization at different sites &S;t & are related by the following set of differential equations:

4r„&S, , &=a,, &S,, . dt

(5)

&

Here we introduce a renormalized 1

1

1

T}

+

time scale

1 T2

and a difference Laplace operator the relation

6,.

which is defined by

s,, &s,, = —4&s„.&+ &s,. „.&+ s,. „. +&s„„&+&s. . . &

&

.

&

&

A remarkable feature of Eq. (5) is that it coincides with equation of the single-flip zerothe corresponding temperature kinetic model with the time scale ~. Henceforth, we take this time scale to be unity. Proceeding with a solution of Eq. (5) we introduce the generating function

KINETICS OF MONOMER-MONOMER

45

G(X, Y, r)=

y g=

oo

y J=

X'YJ&S,, & .

oo

—,

') —1]G .

&S„.&(r

R q(t

— . I,

o „I, t)g— m, n

where I„denotes the modified Bessel functions. ing (11) we have used the identity

In deriv-

XJI (t)= exp[ —,'(X+X ')t] .

(12}

j

.

Eq.

(13b)

we

introduced

the

)

„s,

&

(13b) shorthand

We now turn to the transient kinetics of the system on the basis of the simplifying assumption that the initial Then invariant. distribution P(s, t) is translationally (SJS depends on (m i) and (n——j) at all times. Introducing R~ = (S,"S;+ +v ) we can rewrite (13a) and (13b) as follows:

„)

2

d

Rpq =&pqRpq

(Ipl+

d R io=R20+R ii dt

+R ]

Iql &

i

1),

(14a)

(3+Z)R io+Z

(14b)

Instead of presenting the solution to the discrete equation (14), it is simpler and intuitively more revealing to employ an approximate continuum description which becomes exact asymptotically at p +q 1. Replacing the difference Laplace operator by the continuum one we find that the correlation function R (p, q, t} obeys the difFusion

))

equation

F+

leads to the

(19)

g=r/~t dF dg

1

where C =dC/dt.

=finite .

2tCF C

(20)

(21)

A self-consistency condition requires

tC/C =n =const

&

Z =~~(7 i+Kg) '. One can investigate the general case of arbitrary Z, 0 & Z & 1, although the constraint (3} gives the appointed value Z = —,'.

d dt

(18)

By inserting (19) into (15) we get

—ml

„, „.&+&s,, „s,„,

+&s. .s, —2(3+Z)(SJS;+ij ) . In

t~~, r~~,

equali

(sJs;+i ) =2z+(s; s;+i gi)+(s; s;+2

„,

(17)

R (p, q, r) = C (t)F (rJ)

I

+&s,, s,

origin

in the scaling region

These equations are identical to the corresponding tions of the single-flip model. Equations for + n =1,— however, are slightly different, e.g. ,

, &+&s,.

on R (p, q, t) at the

condition

where b is a scale factor, this rather naturally ansatz

(13a)

,

4

(16)

I

4— &sJs.„)=(b, J+5.„)&SJS „& . dt

j

.

r ~~br, t ~bt

oo

I

for all (p, q)%(0, 0)

Proceeding with a solution of the diffusion equation (15) subject to the conditions (16) and (17), we exploit the rotational symmetry of the problem and search the solution depending on the radial coordinate r = (p + q ) ' ~ and time t. Further, taking into account that the problem is invariant under the transformation

Second, we consider the two-spin correlation functions. n &— After some algebra one finds, for li —m I+ 2,

I

=0)=0

R (0, 0, r}=1 .

(10}

yields

J=

In the present problem, the initial state has the following form:

The boundary p =q =0 reads

=0) =~,J

(S;, ) = exp(

(15)

2

(9)

Solving (9) at arbitrary initial conditions

1069

R

(8)

Using (5} and (8} we derive a simple equation for the generating function,

=[ '(X+X '+ Y+ Y

SURFACE CATALYTIC. . .

(22)

and gives C(t)-t" The discr. ete boundary condition (17) in a framework of continuum is actually valid at Therefore one finds C(t)F(t '~~)~1 at description. i.e., the asymptotic behavior of F(rJ) long times t takes the form

r-1

»1,

F(rJ)-rJ" If

at

g~O .

(23)

(22) and (23) into (21), we find that This makes the right side of the equation vanish. Integrating this equation yields n

we substitute

=0.

F(g)=

p2

f

exp

(24)

2

behavior of (24} is F(rJ)~ — ln(g) at the boundary condition C(t)F(t '~ )~1 at defines the function C(t),

The asymptotic

g~O. So,

t~ao C(t)~2/ln(t)

at t

+~

.

— (25)

behavior of the sum the summation by integration, and using (24} and (25), we arrive at the following asymptotic relation:

Let us consider an asymptotic

g~ ~R~. Replacing +2m t 0 rJ — dg,

f

27rt

+ gR&& — lnt

at t ~DO

(26)

P. L. KRAPIVSKY

1070

On the basis of Eq. (26) one can conclude that any finite substrate becomes saturated, i.e., completely covered by only one species. Furthermore, a requirement 2m T/ln(T)-N gives an estimate of the saturation time

T,

T-N

ln(N)

(27)

where N is a number of catalyst sites. Notice that a linear dependence of the saturation time with possible corrections was first predicted by benlogarithmic Avraham et al. [9] on the basis of numerical simulations.

IV. CATALYSIS MODEL WITH DESORPTION

It is clear that the present model of monomermonomer surface catalytic reaction is a gross oversimplification of the actual catalytic process. A more realistic treatment should consider at least the effects of surface diffusion, desorption, finite adsorption and reaction rates, and nearest-neighbor interactions. Very recently, some of these effects have been studied numericalmodel of catalysis [6, 13]. We ly in the monomer-dimer have not succeeded in solving such realistic models on a two-dimensional substrate. However, if one takes into account only the desorption process, the model proves to be solvable by our methods. cataActually, let us consider the monomer-monomer lytic process in the reaction-limited regime with desorption. This model was introduced and investigated numerically by Fichthorn, Gulari, and Ziff [14] and then theoretically in the mean-field approximation by Considine, Redner, and Takayasu [14]. The model can be mapped onto a kinetic Ising model again. When a particle desorbs, the unoccupied site is then immediately refilled with an A or a 8 with equal probability, i.e., we get the additional noise spin lips. Therefore the dynamics of the process consists of nearest-neighbor spin exchanges with the previous rate (2b) and spin fiips with the modified rate

[4 —S; (S;+,,

1

UJ(S)=

7i

+

+S;, +S; +, +S;,

1

, )] (28)

273

45

— „(S)=ye„.(S„)—4(1 —y)(S, )

(29)

J

and solve this system with general initial conditions (10):

(S;J ) = exp( —t)g

o

„I,

yt

I,

yt

(30)

m, n

Here we introduced a renormalized 1

1

7

7i

+

1

72

+

1

73

and a spin-flip parameter

time scale

(31)

1

(32)

72

~

2

~

~

~

d — —4(1 —y)R R =yb, WW

(33)

W

Here we take the time scale 7 to be unity again. Let us first consider a system with a finite desorption probability, i.e., we assume that 7&-73. Then a simple analysis shows that a system relaxes to the steady state with equal amounts of A's and 8's. The relaxation time T= 1/(2 —2y)= ,'+r3—/r, is finite. We now turn to a more interesting case when the number of catalyst sites N tends to infinity and, simultaneously, the desorption probability tends to zero. Furthermore, we assume that the latter quantity scales with the " with some exponent a &0. former as 7, /73 N Taking advantage of the continuum description, we recast Eq. (33) in the steady state into the following equation:

'+" R=4('

y)R

(34)

Bg

Bp

Combining Eqs. (32), (31), and (3) we express the numerical factor in the right-hand side of Eq. (34)

4(1 — ) y

(~Na)

—1

(35)

73

y-

where 1. We look for solutions to Eq. (34) subject to the periodicity boundary condition

R(p+&N, q)=R(p, q+&N )=R(p, q) . When a&1, one can neglect the influence of

(36) the

periodicity condition and recast Eq. (34) as follows:

It is not difficult to find a set of differential equations

4r d

+

We now turn to the two-spin correlation functions. On this level the description of the system in terms of a single-flip model breaks down, but corrections are rather trivial and displayed only near the origin. For simplicity, we write the equations only for translationally invariant initial conditions and at p + q & 1:

d2

for magnetizations

1

7

dp2

R=(yN +— T df

)

'R .

(37)

We use WKB procedure to find the asymptotic behavior and obtain

R

-r

So, if a

(

'~

exp[

r/(yN

)'~ ] at r &&1

.—

(38)

g-

N and a 1, a correlation length g scales as system relaxes to the steady state (38) with approximately equal amounts of A's and 8's after the relaxation time

T= 1/(2 —2y)-N When a) 1, a system R = 1 and, consequently,

relaxes to the trivial steady state the saturation occurs. The saturation time has been estimated in the preceding section, T-N lnN. Finally, we discuss the borderline case a=1. A simple

KINETICS OF MONOMER-MONOMER

45

analysis gives the steady-state tions

R= cosh(Ax} cosh(A, )

(39)

cosh(A, }

where

A,

(40)

=(8y)

..

1071

substrate in the reaction-controlled limit. For equal adsorption probabilities, we have found that diffusive fluctuations eventually drive a finite-sized system to saturation in a time proportional to N ln(N) where N is the number of catalyst sites. Thus our exact results confirm the assumption of ben-Avraham et al. [9] about possible logarithmic correction in the saturation time. Comparison of the present results with the mean-field predictions [9] confirms the conclusion of ben-Avraham et al. [9] that the upper critical dimension of the substrate for the monomer-monomer process is d, = 2. We have also confirmed the occurrence of a noise-induced transition from monostability to bistability in a model of heterogeneous catalysis with desorption. The methodology described in the present study in the most interesting case of a 2D substrate is immediately Since the monomerapplicable to other dimensions. monomer model on a 1D substrate has been investigated previously [15], we outline such a generalization on a d dimensional hypercubic lattice at d &2. Following the lines of Sec. III, it is not difficult to find the general solution for magnetizations,

two-spin correlation func-

cosh(Ay)

SURFACE CATALYTIC.

A system relaxes onto the nontrivial steady state (39) after the relaxation time T-N. Thus our exact approach reproduces the occurrence of a noise-induced transition [14] from monostability to bicatalysis with stability in a model of heterogeneous desorption. V. CONCLUSIONS AND DISCUSSIONS

We have solved the kinetics of the monomer-monomer model of heterogeneous catalysis on a two-dimensional I

(S;; )=exp(

t)

g. . . —

—

I;

cr

m&,

For the two-spin correlation functions we again derive the diffusion equation (15) in continuum approximation. The properties of diffusion processes with a steady source [see the boundary condition (17)] suggest that above the upper critical dimension, d d, =2, a system relaxes onto ' '. After a source is the Laplacian steady state, R =0, see Eqs. (16) and (17), the leading edge turned on at t . . ,pd, t) of the profile of the correlation function R will advance diffusively, i.e., as &t. So,

) -r

(p„.

Pd

R(p„. . . ,pd, t)- f0

dr

r"

'R

-t

.

(42)

On the basis of this equation we conclude that any finite substrate becomes completely covered by only one species after the saturation time proportional to the number of catalyst sites in full agreement with mean-field predictions [9]. Our methods can be usefully applied to the monomer-

Kinetics of and G. Djega-Mariadassou, Catalytic Reactions (Princeton University Press, Princeton, NJ, 1984); I. A. Campbell, Catalysis at Surfaces (Chapman and Hall, New York, 1988). [2] T. Engl and G. Ertl, Aduances in Catalysis (Academic, New York, 1979), Vol. 28, p. 1. [3] R. M. Ziff, E. Gulari, and Y. Barshad, Phys. Rev. Lett. 56, 2553 (1986); R. M. Ziff and K. Fichthorn, Phys. Rev. B 34,

[1] M. Boudart

Heterogeneous

—

2038 (1986) [4] R. Dickman, Phys. Rev. A 34, 4246 (1986). ~

(41)

I;

, md

monomer process with desorption at arbitrary dimension. A system again relaxes to a steady state, like (38} or (39), which is determined by the competition between desorption, which tends to empty the substrate, and the catalytic reaction, which drives a system towards saturation. This competition underlies the transition from monostability to bistability. It is clear that idealized models discussed above cannot represent complex physical and chemical processes occurring even in the most simple catalyst systems. Nevertheless, I hope that an analytical description of the monomer-monomer model of heterogeneous catalysis can help in understanding the behaviors of real catalysis. ACKNOWLEDGMENT

I am greatly indebted to S. Redner for sending a copy of Ref. [9] prior to publication.

[5] P. Meakin and D. Scalapino, J. Chem. Phys. 87, 731 (1987); M. Kolb and Y. Boudeville, ibid. 92, 3935 (1990). [6] H. P. Kaukonen and R. M. Nieminen, J. Chem. Phys. 91, 4380 (1989); D. G. Vlachos, L. D. Schmidt, and R. Aris, ibid. 93, 8306 (1990). [7] D. A. Brown and P. Kleban, Phys. Rev. A 40, 161& (1989); T. Aukrust, D. A. Brown, and I. Webman, Europhys. Lett. 10, 249 (1989). [8] D. ben-Avraham, S. Redner, D. B. Considine, and P. Meakin, J. Phys. A 23, L613 (1990).

1072

P. L. KRAPIVSKY

[9] D. ben-Avraham, D. B. Considine, P. Meakin, S. Redner, and H. Takayasu, J. Phys. A 23, 4297 (1990); D. B. Considine, S. Redner, and H. Takayasu, ibid. 23, L1111 (1990). [10] E. R. Grannan and G. Swindle, J. Stat. Phys. 61, 1085 (1990). [11]G. Grinstein, Z. W. Lai, and D. A. Brown, Phys. Rev. A 40, 4820 (1989). [12] I. Jensen, H. C. Fogedby, and R. Dickman, Phys. Rev. A 41, 3411 (1990). [13] P. Araya, W. Porod, R. Sant, and E. E. Wolf, Surf. Sci. 208, L80 (1989). [14] K. Fichthorn, E. Gulari, and R. M. Ziff, Phys. Rev. Lett. 63, 1527 (1989); D. Con sidine, S. Red ner, and H. Takayasu, ibid. 63, 2857 (1989).

45

[15] P. L. Krapivsky (unpublished). [16] K. A. Fichthorn, E. Gulari, and R. M. Ziff, in Catalysis 1987, edited by J. W. Ward (Elsevier, Amsterdam, 1988). [17] E. Wicke, P. Kumman, W. Keil, and J. Scheifler, Ber. Bunsenges. Phys. Chem. 4, 315 (1980). [18] A. de Masi, P. A. Ferrari, and J. L. Lebowitz, Phys. Rev. Lett. 55, 1947 (1985); P. L. Garrido, A. Labarta, and J. Marro, J. Stat. Phys. 49, 551 (1987); P. L. Garrido and J. Marro, Phys. Rev. Lett. 62, 1929 (1989); P. L. Garrido, J. Marro, and J. M. Gonzales-Miranda, Phys. Rev. A 40, 5802 (1989). [19] M. Droz, Z. Rasz, and J. Schmidt, Phys. Rev. A 39, 2141 (1989); M. Droz, Z. Rasz, and P. Tartaglia, ibid. 41, 6621 (1990).

VOLUME 45, NUMBER 2

Kinetics of monomer-monomer

15 JANUARY 1992

surface catalytic reactions

P. L. Krapivsky Central Aerohydrodynamic

Institute, Academy

U. S.S.R., 140160 Zhukovsky 3, Moscow Region, U. S.S.R. (Received 29 April 1991)

of Sciences of the

We study the kinetics of an irreversible monomer-monomer model of heterogeneous catalysis. In this model, two reactive species, A and B, adsorb and stick to single sites of a catalytic substrate. Surface reactions are assumed to occur only between dissimilar species that are nearest neighbors on the substrate. The kinetics of the process are studied in the reaction-controlled limit. We map the monomer-monomer model of heterogeneous catalysis onto a kinetic Ising model and find that the dynamics of the process is spin-exchange dynamics. a superposition of zero-temperature spin-flip dynamics and infinite-temperature We solve the kinetics analytically and determine the rate at which the catalyst becomes "saturated, i.e., completely covered by only one species. We show that the saturation time is proportional to N ln(Nj, where N is the number of catalyst sites. We also discuss the monomer-monomer process with desorp-

"

tion.

PACS number(s): 05.70.Ln, 82.65.Jv, 82.20.Mj

I. INTRODUCTION Heterogeneous catalysis is a fundamental kinetic process of considerable interest in many apparently unrelated fields of science and technology. In this process, the rate of a chemical reaction is enhanced by the presence of a suitable catalyst material [1]. An example with practical applications is the platinum surface used to catalyze the reaction: 2CO+Oz~2COz (the motivation for accelerating the oxidation of carbon monoxide is clear in the context of automobile emissions). Consequently, it is of great practical and theoretical interest to possess mathematical predictions concerning the kinetics of catalytic reactions. Typically, such processes are described by Langmuir-Hinshelwood kinetics, where the molecules are assumed to be randomly distributed on the surface [1]. These are assumptions of a mean-field character, as microscopic details, such as spatial Auctuations in concentration and excluded volume interactions, are neglected. However, recent studies suggest that Auctuations are a crucial element in driving the kinetics [2]. Therefore the investigations of microscopic models have begun to identify the general principles underlying the dynamical behavior of heterogeneous catalysis [3 —10]. Depending on the relative deposition rates, there may be the phenomenon of "poisoning" or saturation, where the catalytic substrate eventually becomes covered by one of the species only, thereby terminating the catalysis process, or there may be an apparent reactive steady state. Kinetic phase transitions which demarcate these possibilities have recently been elucidated by many authors [3—5, 7 —12]. In the present study, we investigate the kinetics of heterogeneous catalysis for the monomer-monomer process. In this model, two reactive species, A and B, adsorb and stick to single sites of a catalytic substrate. Surface reactions are assumed to occur only between dissimilar species which are nearest neighbors on the substrate. For unequal adsorption probabilities, the substrate quickly 45

saturates with the preferred species. For equal adsorption probabilities, a finite substrate still saturates, but at a much slower rate with equal probability of saturation by either of the two species. There are two basic limiting cases of catalysis. When the reaction on the substrate occurs quickly, the process is limited by the adsorption rate. This is the adsorptioncontrolled limit. On the other hand, in the reactioncontrolled limit the adsorption occurs readily so that the overall process is limited by the conversion of unlike monomers to AB pairs. Much of the previous work [3 —5, 7, 8, 10 —12] has considered the adsorption-controlled limit. However, many catalytic processes occur in the opposite limit [1]. In the present paper, we focus on the reactioncontrolled limit for the monomer-monomer process. It is clear that the model is a gross oversimplification of the actual catalytic process. A more realistic treatment should consider at least the effects of surface diffusion, desorption, finite adsorption and reaction rates, and nearest-neighbor interactions. Very recently, some of these effects have been studied numerically [6, 13,14]. However, the present model contrasts with more realistic ones because it can be solved analytically. Therefore the model appears to be an extremely rare problem of nonequilibrium statistical mechanics which may be solved exactly in arbitrary dimensions. In the present work we describe the kinetics of the monomer-monomer catalytic process in the most interesting case of a two-dimensional (2D) substrate (the kinetics of the process on a 1D substrate have been reported on in my previous work [15]). It is important that for the monomer-monomer process differences between the reactiononly quantitative controlled and adsorption-controlled limits have been observed [9,16]. Thus the relatively simple reactioncontrolled limit gives an insight into the general case of arbitrary adsorption and reaction rates. The rest of this paper is organized as follows. In Sec. II we map the monomer-monomer catalysis model in the 1067

1992 The American Physical Society

P. L. KRAPIVSKY

1068

there is a net bias of either A s or saturates exponentially in time with the preferred species. Therefore we shall discuss the most interesting case of equal adsorption probabilities, where diffusive fluctuations drive the system to saturation. We shall consider the kinetics of catalysis in a reaction-limited regime, introduced in a pioneer work [17], on the infinite square lattice. Other types of lattices can a1so be dealt with. It is useful to map the model onto a kinetic Ising model. Identifying A's with +1 and B's with —1, a full substrate may be described in terms of Ising variables S = [S;, ]. It is not difficult to derive the following master equation for the probability distribution P (S, t) of the Ising state S at time t:

reaction-controlled limit onto a kinetic Ising model with competing dynamics. In Sec. III we calculate one- and two-spin correlation functions. From this solution, basic information about the rate of saturation is obtained. In Sec. IV we discuss the catalysis model with desorption. The conclusions are summarized in Sec. V.

sorption probabilities,

B's, and the substrate

II. DYNAMICS In the recent-controlled limit, the substrate quickly becomes full. Thus we start our study with the substrate that is randomly filled by equal amounts of A and B units. If two neighboring sites are occupied by opposite species, a reaction occurs in which the reactants desorb, each unoccupied site then being immediately refilled by either A or B with equal probability. For unequal ad-

P(s, t)=— g [U~(FJS)P(F; S, t) t&

U~(S)P—(s,

t)]++ [V~(F~F;+,Js, t)P(F, F, +, S,"t)

J

l&

+g [WJ(FJFi+,S, t)P(F; FJ+,S, t) 17

(S, t)] . W; (S)P—

sions

[4 — S~(s;+ij+S;+i +S; +i+S;

1

8vi

i)], (2a)

Vij(S) =

= WJ(S)=

1

8rz

(1 — SJS;+ij ),

(2b)

(1 — SJS~+i) .

(2c)

1

8&2

Notice that the process (2a) defines the single-flip dynamics at temperature T =0 while the processes (2b) and (2c) define the Kawasaki spin-exchange dynamics at temperature T = ~. A simple analysis shows that the dynamics of the surface catalytic reaction coincides with the dynamics (2) under the following constraint between the time scales of spin-flip and -exchange processes: 1

V()(S)P—(s, t)]

J

J

In this equation, UJ(S), V; (S), and Wt(S) denote the rate for the system to jump from the state S to the state FJS, F; F;+& S, and F; F; +,S, respectively, and F;. for all denotes the spin-flip operator, F; S = [ S — (p, q)A(i, j); S;J]. Flip rates are given by the expres-

UJ(S)=

45

2

Notice that a general case of arbitrary spin-flip and -exchange time scales can also be dealt with. When both spin-lip and -exchange processes are model since the syspresent, we have a nonequilibrium tem is in contact with two heat baths which are at zero As we shall show below, this and infinite temperature. model is solvable in the sense that the time evolution of the one- and two-spin correlation functions can be calculated exactly. In closing we note that our model is an example of Ising-type systems with competing dynamics

which have been investigated in a series of recent studies (see, e.g. [18,19] and references therein).

III.

ANALYTICAL RESULTS FOR CORRELATIONS

The advantage of the spin formulation of the problem is that one can easily derive a closed set of differential equations for the spin-correlation functions &S; . .

.S

„&=gS;

S „P(S,t) .

(4)

S

First, we discuss the one-spin correlation functions. Multiplying both sides of Eq. (1) by S,i and summing over all configurations S we find that the average values of the magnetization at different sites &S;t & are related by the following set of differential equations:

4r„&S, , &=a,, &S,, . dt

(5)

&

Here we introduce a renormalized 1

1

1

T}

+

time scale

1 T2

and a difference Laplace operator the relation

6,.

which is defined by

s,, &s,, = —4&s„.&+ &s,. „.&+ s,. „. +&s„„&+&s. . . &

&

.

&

&

A remarkable feature of Eq. (5) is that it coincides with equation of the single-flip zerothe corresponding temperature kinetic model with the time scale ~. Henceforth, we take this time scale to be unity. Proceeding with a solution of Eq. (5) we introduce the generating function

KINETICS OF MONOMER-MONOMER

45

G(X, Y, r)=

y g=

oo

y J=

X'YJ&S,, & .

oo

—,

') —1]G .

&S„.&(r

R q(t

— . I,

o „I, t)g— m, n

where I„denotes the modified Bessel functions. ing (11) we have used the identity

In deriv-

XJI (t)= exp[ —,'(X+X ')t] .

(12}

j

.

Eq.

(13b)

we

introduced

the

)

„s,

&

(13b) shorthand

We now turn to the transient kinetics of the system on the basis of the simplifying assumption that the initial Then invariant. distribution P(s, t) is translationally (SJS depends on (m i) and (n——j) at all times. Introducing R~ = (S,"S;+ +v ) we can rewrite (13a) and (13b) as follows:

„)

2

d

Rpq =&pqRpq

(Ipl+

d R io=R20+R ii dt

+R ]

Iql &

i

1),

(14a)

(3+Z)R io+Z

(14b)

Instead of presenting the solution to the discrete equation (14), it is simpler and intuitively more revealing to employ an approximate continuum description which becomes exact asymptotically at p +q 1. Replacing the difference Laplace operator by the continuum one we find that the correlation function R (p, q, t} obeys the difFusion

))

equation

F+

leads to the

(19)

g=r/~t dF dg

1

where C =dC/dt.

=finite .

2tCF C

(20)

(21)

A self-consistency condition requires

tC/C =n =const

&

Z =~~(7 i+Kg) '. One can investigate the general case of arbitrary Z, 0 & Z & 1, although the constraint (3} gives the appointed value Z = —,'.

d dt

(18)

By inserting (19) into (15) we get

—ml

„, „.&+&s,, „s,„,

+&s. .s, —2(3+Z)(SJS;+ij ) . In

t~~, r~~,

equali

(sJs;+i ) =2z+(s; s;+i gi)+(s; s;+2

„,

(17)

R (p, q, r) = C (t)F (rJ)

I

+&s,, s,

origin

in the scaling region

These equations are identical to the corresponding tions of the single-flip model. Equations for + n =1,— however, are slightly different, e.g. ,

, &+&s,.

on R (p, q, t) at the

condition

where b is a scale factor, this rather naturally ansatz

(13a)

,

4

(16)

I

4— &sJs.„)=(b, J+5.„)&SJS „& . dt

j

.

r ~~br, t ~bt

oo

I

for all (p, q)%(0, 0)

Proceeding with a solution of the diffusion equation (15) subject to the conditions (16) and (17), we exploit the rotational symmetry of the problem and search the solution depending on the radial coordinate r = (p + q ) ' ~ and time t. Further, taking into account that the problem is invariant under the transformation

Second, we consider the two-spin correlation functions. n &— After some algebra one finds, for li —m I+ 2,

I

=0)=0

R (0, 0, r}=1 .

(10}

yields

J=

In the present problem, the initial state has the following form:

The boundary p =q =0 reads

=0) =~,J

(S;, ) = exp(

(15)

2

(9)

Solving (9) at arbitrary initial conditions

1069

R

(8)

Using (5} and (8} we derive a simple equation for the generating function,

=[ '(X+X '+ Y+ Y

SURFACE CATALYTIC. . .

(22)

and gives C(t)-t" The discr. ete boundary condition (17) in a framework of continuum is actually valid at Therefore one finds C(t)F(t '~~)~1 at description. i.e., the asymptotic behavior of F(rJ) long times t takes the form

r-1

»1,

F(rJ)-rJ" If

at

g~O .

(23)

(22) and (23) into (21), we find that This makes the right side of the equation vanish. Integrating this equation yields n

we substitute

=0.

F(g)=

p2

f

exp

(24)

2

behavior of (24} is F(rJ)~ — ln(g) at the boundary condition C(t)F(t '~ )~1 at defines the function C(t),

The asymptotic

g~O. So,

t~ao C(t)~2/ln(t)

at t

+~

.

— (25)

behavior of the sum the summation by integration, and using (24} and (25), we arrive at the following asymptotic relation:

Let us consider an asymptotic

g~ ~R~. Replacing +2m t 0 rJ — dg,

f

27rt

+ gR&& — lnt

at t ~DO

(26)

P. L. KRAPIVSKY

1070

On the basis of Eq. (26) one can conclude that any finite substrate becomes saturated, i.e., completely covered by only one species. Furthermore, a requirement 2m T/ln(T)-N gives an estimate of the saturation time

T,

T-N

ln(N)

(27)

where N is a number of catalyst sites. Notice that a linear dependence of the saturation time with possible corrections was first predicted by benlogarithmic Avraham et al. [9] on the basis of numerical simulations.

IV. CATALYSIS MODEL WITH DESORPTION

It is clear that the present model of monomermonomer surface catalytic reaction is a gross oversimplification of the actual catalytic process. A more realistic treatment should consider at least the effects of surface diffusion, desorption, finite adsorption and reaction rates, and nearest-neighbor interactions. Very recently, some of these effects have been studied numericalmodel of catalysis [6, 13]. We ly in the monomer-dimer have not succeeded in solving such realistic models on a two-dimensional substrate. However, if one takes into account only the desorption process, the model proves to be solvable by our methods. cataActually, let us consider the monomer-monomer lytic process in the reaction-limited regime with desorption. This model was introduced and investigated numerically by Fichthorn, Gulari, and Ziff [14] and then theoretically in the mean-field approximation by Considine, Redner, and Takayasu [14]. The model can be mapped onto a kinetic Ising model again. When a particle desorbs, the unoccupied site is then immediately refilled with an A or a 8 with equal probability, i.e., we get the additional noise spin lips. Therefore the dynamics of the process consists of nearest-neighbor spin exchanges with the previous rate (2b) and spin fiips with the modified rate

[4 —S; (S;+,,

1

UJ(S)=

7i

+

+S;, +S; +, +S;,

1

, )] (28)

273

45

— „(S)=ye„.(S„)—4(1 —y)(S, )

(29)

J

and solve this system with general initial conditions (10):

(S;J ) = exp( —t)g

o

„I,

yt

I,

yt

(30)

m, n

Here we introduced a renormalized 1

1

7

7i

+

1

72

+

1

73

and a spin-flip parameter

time scale

(31)

1

(32)

72

~

2

~

~

~

d — —4(1 —y)R R =yb, WW

(33)

W

Here we take the time scale 7 to be unity again. Let us first consider a system with a finite desorption probability, i.e., we assume that 7&-73. Then a simple analysis shows that a system relaxes to the steady state with equal amounts of A's and 8's. The relaxation time T= 1/(2 —2y)= ,'+r3—/r, is finite. We now turn to a more interesting case when the number of catalyst sites N tends to infinity and, simultaneously, the desorption probability tends to zero. Furthermore, we assume that the latter quantity scales with the " with some exponent a &0. former as 7, /73 N Taking advantage of the continuum description, we recast Eq. (33) in the steady state into the following equation:

'+" R=4('

y)R

(34)

Bg

Bp

Combining Eqs. (32), (31), and (3) we express the numerical factor in the right-hand side of Eq. (34)

4(1 — ) y

(~Na)

—1

(35)

73

y-

where 1. We look for solutions to Eq. (34) subject to the periodicity boundary condition

R(p+&N, q)=R(p, q+&N )=R(p, q) . When a&1, one can neglect the influence of

(36) the

periodicity condition and recast Eq. (34) as follows:

It is not difficult to find a set of differential equations

4r d

+

We now turn to the two-spin correlation functions. On this level the description of the system in terms of a single-flip model breaks down, but corrections are rather trivial and displayed only near the origin. For simplicity, we write the equations only for translationally invariant initial conditions and at p + q & 1:

d2

for magnetizations

1

7

dp2

R=(yN +— T df

)

'R .

(37)

We use WKB procedure to find the asymptotic behavior and obtain

R

-r

So, if a

(

'~

exp[

r/(yN

)'~ ] at r &&1

.—

(38)

g-

N and a 1, a correlation length g scales as system relaxes to the steady state (38) with approximately equal amounts of A's and 8's after the relaxation time

T= 1/(2 —2y)-N When a) 1, a system R = 1 and, consequently,

relaxes to the trivial steady state the saturation occurs. The saturation time has been estimated in the preceding section, T-N lnN. Finally, we discuss the borderline case a=1. A simple

KINETICS OF MONOMER-MONOMER

45

analysis gives the steady-state tions

R= cosh(Ax} cosh(A, )

(39)

cosh(A, }

where

A,

(40)

=(8y)

..

1071

substrate in the reaction-controlled limit. For equal adsorption probabilities, we have found that diffusive fluctuations eventually drive a finite-sized system to saturation in a time proportional to N ln(N) where N is the number of catalyst sites. Thus our exact results confirm the assumption of ben-Avraham et al. [9] about possible logarithmic correction in the saturation time. Comparison of the present results with the mean-field predictions [9] confirms the conclusion of ben-Avraham et al. [9] that the upper critical dimension of the substrate for the monomer-monomer process is d, = 2. We have also confirmed the occurrence of a noise-induced transition from monostability to bistability in a model of heterogeneous catalysis with desorption. The methodology described in the present study in the most interesting case of a 2D substrate is immediately Since the monomerapplicable to other dimensions. monomer model on a 1D substrate has been investigated previously [15], we outline such a generalization on a d dimensional hypercubic lattice at d &2. Following the lines of Sec. III, it is not difficult to find the general solution for magnetizations,

two-spin correlation func-

cosh(Ay)

SURFACE CATALYTIC.

A system relaxes onto the nontrivial steady state (39) after the relaxation time T-N. Thus our exact approach reproduces the occurrence of a noise-induced transition [14] from monostability to bicatalysis with stability in a model of heterogeneous desorption. V. CONCLUSIONS AND DISCUSSIONS

We have solved the kinetics of the monomer-monomer model of heterogeneous catalysis on a two-dimensional I

(S;; )=exp(

t)

g. . . —

—

I;

cr

m&,

For the two-spin correlation functions we again derive the diffusion equation (15) in continuum approximation. The properties of diffusion processes with a steady source [see the boundary condition (17)] suggest that above the upper critical dimension, d d, =2, a system relaxes onto ' '. After a source is the Laplacian steady state, R =0, see Eqs. (16) and (17), the leading edge turned on at t . . ,pd, t) of the profile of the correlation function R will advance diffusively, i.e., as &t. So,

) -r

(p„.

Pd

R(p„. . . ,pd, t)- f0

dr

r"

'R

-t

.

(42)

On the basis of this equation we conclude that any finite substrate becomes completely covered by only one species after the saturation time proportional to the number of catalyst sites in full agreement with mean-field predictions [9]. Our methods can be usefully applied to the monomer-

Kinetics of and G. Djega-Mariadassou, Catalytic Reactions (Princeton University Press, Princeton, NJ, 1984); I. A. Campbell, Catalysis at Surfaces (Chapman and Hall, New York, 1988). [2] T. Engl and G. Ertl, Aduances in Catalysis (Academic, New York, 1979), Vol. 28, p. 1. [3] R. M. Ziff, E. Gulari, and Y. Barshad, Phys. Rev. Lett. 56, 2553 (1986); R. M. Ziff and K. Fichthorn, Phys. Rev. B 34,

[1] M. Boudart

Heterogeneous

—

2038 (1986) [4] R. Dickman, Phys. Rev. A 34, 4246 (1986). ~

(41)

I;

, md

monomer process with desorption at arbitrary dimension. A system again relaxes to a steady state, like (38} or (39), which is determined by the competition between desorption, which tends to empty the substrate, and the catalytic reaction, which drives a system towards saturation. This competition underlies the transition from monostability to bistability. It is clear that idealized models discussed above cannot represent complex physical and chemical processes occurring even in the most simple catalyst systems. Nevertheless, I hope that an analytical description of the monomer-monomer model of heterogeneous catalysis can help in understanding the behaviors of real catalysis. ACKNOWLEDGMENT

I am greatly indebted to S. Redner for sending a copy of Ref. [9] prior to publication.

[5] P. Meakin and D. Scalapino, J. Chem. Phys. 87, 731 (1987); M. Kolb and Y. Boudeville, ibid. 92, 3935 (1990). [6] H. P. Kaukonen and R. M. Nieminen, J. Chem. Phys. 91, 4380 (1989); D. G. Vlachos, L. D. Schmidt, and R. Aris, ibid. 93, 8306 (1990). [7] D. A. Brown and P. Kleban, Phys. Rev. A 40, 161& (1989); T. Aukrust, D. A. Brown, and I. Webman, Europhys. Lett. 10, 249 (1989). [8] D. ben-Avraham, S. Redner, D. B. Considine, and P. Meakin, J. Phys. A 23, L613 (1990).

1072

P. L. KRAPIVSKY

[9] D. ben-Avraham, D. B. Considine, P. Meakin, S. Redner, and H. Takayasu, J. Phys. A 23, 4297 (1990); D. B. Considine, S. Redner, and H. Takayasu, ibid. 23, L1111 (1990). [10] E. R. Grannan and G. Swindle, J. Stat. Phys. 61, 1085 (1990). [11]G. Grinstein, Z. W. Lai, and D. A. Brown, Phys. Rev. A 40, 4820 (1989). [12] I. Jensen, H. C. Fogedby, and R. Dickman, Phys. Rev. A 41, 3411 (1990). [13] P. Araya, W. Porod, R. Sant, and E. E. Wolf, Surf. Sci. 208, L80 (1989). [14] K. Fichthorn, E. Gulari, and R. M. Ziff, Phys. Rev. Lett. 63, 1527 (1989); D. Con sidine, S. Red ner, and H. Takayasu, ibid. 63, 2857 (1989).

45

[15] P. L. Krapivsky (unpublished). [16] K. A. Fichthorn, E. Gulari, and R. M. Ziff, in Catalysis 1987, edited by J. W. Ward (Elsevier, Amsterdam, 1988). [17] E. Wicke, P. Kumman, W. Keil, and J. Scheifler, Ber. Bunsenges. Phys. Chem. 4, 315 (1980). [18] A. de Masi, P. A. Ferrari, and J. L. Lebowitz, Phys. Rev. Lett. 55, 1947 (1985); P. L. Garrido, A. Labarta, and J. Marro, J. Stat. Phys. 49, 551 (1987); P. L. Garrido and J. Marro, Phys. Rev. Lett. 62, 1929 (1989); P. L. Garrido, J. Marro, and J. M. Gonzales-Miranda, Phys. Rev. A 40, 5802 (1989). [19] M. Droz, Z. Rasz, and J. Schmidt, Phys. Rev. A 39, 2141 (1989); M. Droz, Z. Rasz, and P. Tartaglia, ibid. 41, 6621 (1990).