The monomer-monomer model revisited

N

surface science Surface Science 325 (1995) L392-L396

ELSEVIER

Surface Science Letters

The m o n o m e r - m o n o m e r m o d e l revisited F. Moiny, Martine Dumont *'1, j. Lion, P. Dufour, R. Dagonnier Facult~ des Sciences, Universit£ de Mons-Hainaut, B-7000 Mons, Belgium

Received 11 July 1994; accepted for publication 15 November 1994

Abstract

The monomer-monomer model for surface reactions has been investigated by means of Monte Carlo simulations performed for different kinetic regimes. The poisoning time and the width of the "chaos window" has been specified in terms of the control rate constant and the lattice size. Indicative lateral interactions acting on the adsorption process have been shown to be opposed to the chaotic behaviour of the model.

Surface reactions are reactions between gaseous species occurring on catalytic surfaces. W h e n operating as open systems, surface reactions behave as nonlinear dissipative systems. These systems can show kinetic (or non equilibrium) phase transitions, bistability and associated hysteresis, complex selfsustained oscillations, etc. [1]. A m o n g these nonlinear kinetic effects are the noise-induced phase transitions which are due to the interaction of an external noise with spatial degrees of freedom of the system [2]. A n example of these transitions consists of a transition of bistability, in a system that is monostable in the absence of noise. The m o n o m e r - m o n o m e r ( M - M ) model is the basis of a Monte Carlo surface reaction which, as demonstrated by Fichthorn et al. [3,4], exhibits such a noise-induced bistability.

* Corresponding author. E-mail: [email protected]. 1 Research Associate, National Fund for Scientific Research, Belgium.

The M - M model is a particular case of the A B model. W h e n applied to the surface reaction A g B g catalytic surface > A B g , the A B model refers to the following kinetic scheme: kA

Ag + S.

(1)

" B,

(2)

, A B g -]- S S .

(3)

kB

Bg + S , k-13

AN

kr

The subscript g refers to molecules in the gas phase. A, B and S refer to one surface site occupied by the species A, B and vacant, respectively. A B . . . . . SS represent A B , . . . , SS pairs of nearest-neighbour sites. Moreover, k i, k_i (i = A, B), and k r are the rate constants for adsorption, desorption, and surface reaction, respectively. Within the M - M model, the reactants A and B

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" A, k-A

F. Moiny et al. / Surface Science 325 (1995)L392-L396

are assumed to be identical, yet distinguishable. The rate constants for adsorption and desorption are equal: k A = k B = k,,

(4)

k _ A = k _ B = k d.

(5)

In terms of classical kinetics the exact rate equations of the M - M model reaction read

the M - M model has stimulated further mv~u~at,o.~ about the origin and the characteristics of this noiseinduced bistability transition [5-9]. It should be emphasized that this simple model has a pertinent methodological role. Indeed it is well known that other surface reactions like

Amg q- Bng

dXA k a ( 1 --XA - - X B ) -- 2 k r S A B -- kdXA'

(6)

dX B dt = k~(1 - X A - X B ) - 2krXAB - k d X B.

(7)

dt=

These equations tell us that, if k d v~ 0, there is only one stable steady state (denoted by overbars): 0 ~<£A = £ B < 1 / 2 .

(8)

If k d = 0, the two additional non-reactive steady states (XA = 1, -'YU= 0) and (-~A = 0, XB = 1) are unstable and the system remains monostable. In this context let us refer to regular Monte Carlo surface reactions when these computer experiments verify exact kinetic reactions like Eqs. (6) and (7). By means of Monte Carlo simulations (MCS) Fichthorn et al. [3,4] have shown that in the limit of vanishing desorption (i.e. k d --+ 0) the M - M model reaction exhibits a drastic departure from a regular behaviour: (i) finite size lattices eventually " p o i s o n " to fully saturated (in A or B, with equal probability) non reactive states, indicating a transition to bistability; (ii) the dynamics of this saturation is characterized by self-sustained chaotic oscillations of the fractional coverages and the rate of reaction. These authors demonstrate the emergence of this bistability (and the related chaotic behaviour) as a result of fluctuations which arise from the statistical nature of reactant adsorption. Moreover they show that, when the rate of desorption increases, the bistability - effect of the reactant adsorption fluctuations " w e a k e n s " until the system behaves regularly, i.e., becomes monostable. Therefore, there exists a critical value k d of the desorption rate constant which defines a "bistability window" (and a corresponding "chaos window") outside which this M - M reaction is regular. This intriguing kinetic behaviour of the MCS of -

catalytic . surface

) AB

g

( m = 1, 2; n = 2)

involve the canonical reaction mechanism, Eq. (3). Therefore, as raised by Fichthorn et al. [4] and investigated on a preliminary level by Mukesh [10], the following question remains open; could any surface reaction of this type exhibit chaotic self-oscillations (borne to coverage fluctuations arising from the statistical nature of the adsorption) such as those observed in MCS of the M - M model reaction? The objective of our work was to complete and specify the previous results reported in other studies [3-9]. By using the various rate constants as adjustable parameters of our MCS of the M - M model, we have been able to determine the poisoning time and the "bistability window" for different kinetic regimes. Full details about our MCS procedures may be found in Ref. [11]. Our MCS of the M - M model have been performed: (i) on plane square lattices with periodic boundary conditions and a total number N of sites in the range (10 × 10) ~
(9)

(ii) starting either with empty lattices or with the entire lattice randomly populated with 50% A and 50% B. Notice that the results we present hereafter do not significantly depend on the choice between these two initial conditions. We have investigated many kinetic regimes starting from reaction controlled ones (i.e., k a > k r = k c) up to adsorption controlled cases (i.e., kr > k a -- kc), i.e., specifically with a control rate constant k c (s -1 per site) in the range 10 - 1 ~< k c

~< 101°.

(10)

Now our results and comments (A to D below). A: For the irreversible M - M model (i.e. when k d = 0) we have found that the poisoning time t_ (s) versus the lattice size N (in the range, Eq. (9~) is

F. Moiny et aL / Surface Science 325 (1995) L392-L396

lU

i

•

,

I

,

0

•

I

%"8

~

i

:

,

,

"

=

,

I 4

MONOST

• -I

r-

0.1

~

o

t~ -2 "5

. .I

~

0-0.01

-5

0.001

. . . . . . . . O0

i 1000

. . . . . . . .

r 10000

,

BISTABLE

/

£

.

-4

n u m b e r of sites --5

Fig. 1. Illustrative plot of the poisoning time tp v e r s u s the lattice size N for kinetic regimes controlled by the reaction (O), by the adsorption (O) and intermediate (I-1), i.e., k c = k r = 104, ka = 107; kc = ka = 104, k r = 107; k~ = k a = k r = 104, respectively. The linear regression analysis performed either with a power law function yields Eq. (11) (solid line) or with the analytical form proposed by Krapivsky [9] gives Eq. (12) (dashed line).

g i v e n b y the almost equivalent expressions (see Fig. 1) 1 tp = - - N A, kc 1 tp = 6 k e N

A=1.08+0.05,

In N ,

(11)

(12)

in all kinetic regimes. This result specifies the estim a t i o n s given in Refs. [4,7,9]. A l t h o u g h it has b e e n recognized [ 3 - 9 ] that the M - M m o d e l has no direct correspondence with physical surface reactions let us m e n t i o n that b y extrapolating the formula, Eq. (12), to practical catalysis experiments we should refer to the indicative value (per c m 2 of catalyst surface) tp ~ 10 n s,

n > 8.

(13)

Indeed for a surface reaction b e t w e e n molecules reacting, at S T P conditions, on a typical catalyst surface (i.e., for N = 5 × 1014 sites c m - 2 ) textbooks (e.g. Ref. [12]) tell us that if we assume a sticking coefficient near unity we get a rate constant for a d s o r p t i o n k a -~ 10 7 s - 1 per site. Therefore for our fictitious M - M reaction we should consider - at best reaction controlled regimes such that -

107 > k a > k c = k r .

(14)

, -6

14

,

12

-

,

I 0

-

r

I 2

L o g ( k o / k r)

Fig. 2. Plot of the threshold value k~ (s -2) v e r s u s k a / k r with kr = 104 s -1 per site. For kd < kd* the MCS dynamics is chaotic (the system is bistable) while for kd > kd* the MCS reaction is regular.

Despite the " s l o w n e s s " of the p o i s o n i n g we should question the possibility of observing the selfsustained chaotic oscillations correlative to this poisoning. B: W e have estimated the " c h a o s w i n d o w " of the M - M m o d e l by letting k d increase u p to a critical v a l u e kd* where the M - M m o d e l behaves regularly. This threshold value k~ depends on the kinetic regime considered. This is illustrated in Fig. 2 where k r is fixed and k a (i.e., essentially the pressure) is the adjustable parameter.

100

*~I~ .3(10- ' ~ , 1

o.1

iMONOSTABL E

BISTABLE

,i 1000

,

,

,

,

,

,

, ,i

,

,

,

10000

n u m b e r of sites

Fig. 3. Illustrative plot of the critical desorption rate constant k~ versus the lattice size N for a reaction controlled regime (k a = 107 s -1 per site, k r = 104 S-1 per site). The linear regression analysis of the log-log plot gives the power law function Eq. (15).

F. Moiny et al. / Surface Science 325 (1995) L392-L396

Moreover we have observed that, for all kinetic regimes, k2 (s -1) depends on N as follows (see Fig.

(18)

(a) In the experiments performed subject to condition Eq. (17), we assume a repulsive interaction between adsorbing and adsorbed particles of the same species. In this way we assume that each adsorption has an activation energy which increases with the specific coverage as generally admitted in chemisorption (e.g. see textbook Ref. [12]). In this case for /z >/x* = 10 - 3 the M - M model reaction becomes regular, while f o r / z / * * = 10 - 3 the M - M model reaction becomes regular. (c) With Eq. (19) (a combination of the effects considered in (a) and (b)) the adsorption promotes the formation of AB pairs to the detriment of AA and BB pairs more efficiently than in MCS performed under Eqs. (17) or (18). In this case, for tz > p,* --~ 3 >< 10 - 4 the M - M model reaction becomes regular. In short, these last computer experiments show that infinitesimal lateral interactions (of the order of one J mo1-1 at T = 3 0 0 K!) would kill the chaos in a M - M type reaction. Conclusions of this work are the following: (i) The bistability observed in irreversible M - M model surface reactions is a finite size effect which we have quantitatively specified. (ii) The conditions for an observation of self-sustained chaotic oscillations correlative to this noise-induced bistability have also been specified. (iii) The bistability and the correlative chaotic dynamic are readily destroyed by infinitesimal lateral interactions.

(19)

References

3): k2 = A k c N -n,

A=4___2,

n=1.1_0.2.

(15)

This result specifies the N dependence of k~* previously shown by Fichthorn et al. (e.g. see Fig. 5 in Ref. [4]). Eq. (15) tells us that, except for N << 1014, the condition k a < kd* seems difficult to satisfy for simple molecular desorptions (e.g. see Ref. [12] for indicative values of k d of non-associative desorptions). C: Knowing kd* we have performed MCS of quasi M - M models, i.e., M - M models slightly perturbed towards the regular A - B model (see Eqs. (1-3)) as follows. We have considered reactants A and B with different adsorption (and desorption) rate constants in order to depart, as far as possible, from the prescriptions, Eqs. (4) and (5), but still observe the chaotic behaviour characteristic of the M - M model. MCS performed on (100 >< 100) lattices in all kinetic regimes with k_ i < kd* (i = A, B) show that for adsorption rate constants k a and k B, such that

I kA - - kB I / k A --< 5

>< 10 -5,

(16)

the A - B model exhibits self-sustained chaotic oscillations. Notice that, in practice, the adsorption condition, Eq. (16), could not be satisfied as current accuracy of flowmeters cannot be better than 10 - 3 . D: Finally we have investigated how lateral nearest-neighbour (nn) interactions could affect the chaotic kinetics of the irreversible M - M model (i.e., when k d = 0). We have performed MCS of the M - M model with adsorption rate constants depending on the nn local coverages Xi l°c in three ways (preserving the invariance of the model under the exchange x A ~ XB): (a)

k i=k a e

- ~ X !oc

' ,

X loe

(17)

i:A,B,

(b)

ki=k a e ~ J ,

(c)

k i = k a e ~(x]°c-x]°c),

i=a,

B (i-~j), i=A,

B (i4=j).

The indicative results given here have been obtained for MCS performed on (100 >< 100) lattices, in the reaction controlled regime (i.e. with k a = 107, k r = 104 s -1 per site; see Eq. (14)).

[1] G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems (Wiley, New York, 1977). [2] W. Horsthemke and R. Lefever, Noise-Induced Transitions (Springer, New York, 1984).

F. Moiny et al. / Surface Science 325 (1995) L392-L396 [3/ K. FlChthorn, E. Gulari and R. Ziff, Phys. Rev. Lett. 63 (1989) 1527. [4] K. Fichthorn, E. Gulari and R. Ziff, Chem. Eng. Sci. 44 (1989) 1403. [5] D. Considine, S. Redner and H. Takayasu, Phys. Rev. Lett. 63 (1989) 2857. [6] D. Ben-Avraham, S. Redner, D.B. Considine and P. Meakin, J. Phys. A: Math. Gen. 23 (1990) L613. [7] D. Ben-Avraham, D. Considine, P. Meakin, S. Redner and H. Takayasu, J. Phys. A: Math. Gen. 23 (1990) 4297.

[8] E. Clement, P. Leroux-Hugon and L.M. Sander, J. Stat. Phys. 65 (1991) 925. [9] P.L. Krapivsky, Phys. Rev. A 45 (1992) 1067. [10] D. Mukesh, J. Catal. 133 (1992) 153. [11] P. Dufour, M. Dumont, V. Chabart and J. Lion, Comput. Chem. 13 (1989) 25. [12] J.M. Thomas and W.J. Thomas, Introduction to the Principles of Heterogeneous Catalysis (Academic Press, New York, 1967).