Comments on “isothermal sustained oscillations in a very simple surface reaction” by C.G. Takoudis, L.D. Schmidt and R. Aris

Llll

Surface Science 11.5 (1982) Llll-L116 North-Holland Publishing Company

SURFACE

SCIENCE

LETTERS

COMMENTS ON “ISOTHER~L SUSTAINED OSCILLA~ONS A VERY SIMPLE SURFACE REACTION” BY C.G. TAKOUDIS, L.D. SCHMIDT AND R. ARIS * M. BOUILLON,

R. DAGONNIER,

Fuirc~ui~+ des Sciewes. Received

4 August

P. DUFOUR

UnioersctL; de I’Etot. B-7000 Mom. 1981; accepted

for publication

and Martine

IN

DUMONT

**

Belgium

3 December

198 I

The oscillating CO oxidation on Pt is examined as a possible candidate for the Takoudis. Schmidt and Aris (TSA) model. Moreover the results of our Monte Carlo simulation of this model reaction are tentatively compared to TSA numerical results.

In a recent paper Takoudis, Schmidt and Aris (TSA) [I] propose a kinetic model for bimolecular surface reactions that exhibit isothermal sustained oscil1ation.s. The most challenging feature of this model consists in the inclusion, in the kinetics, of a “Langmuir-Hinshelwood” (LH) mechanism requiring two adjacent vacant sites. This note has two objectives: (i) a detailed examination of the CO oxidation on Pt as a possible candidate for the TSA model as suggested by these authors; (ii) a comparison between TSA numericai results and those we have obtained by a Monte Carlo simulation of a surface reaction involving a four-site LH mechanism and for the parameter values specific of the oscillations displayed in ref. [I]. Let us first set the model equations along TSA lines, These authors consider the surface reaction A + ; B,, - products, that involves

n = I or 2,

the following

(1)

steps

(k,p,. E,) AS-S

tk_>-,)

B,, f nS

(2)

AS,

(X,F,/,r. E,) 1X-25-2f

nBS,

n=

1 or2,

(A,‘ E3f

AS+BS+2S

-

4s + products,

* Work partly supported by the IRIS Program sponsored by the Belgian Ministry P0liLZy. ** Charge de Recherches au Pond National Beige de la Recherche Scientifique.

0039-6028/82/0000-0000/$02.75

0 1982 North-Holland

for Science

where S denotes an active site and k,, pi and E, are respectively the coverage independent rate constants, the partial pressures and the activation energies. Note that the subscripts 1 and 2, referring to A and B respectively, will be assigned to CO and oxygen respectively for the CO oxidation. According to TSA, the kinetic scheme (2)-(4) can be considered as the lumped sequence of site exchange and reaction steps. These authors describe the kinetics of reaction (1) in terms of the following dimensionless balance equations for the fractional coverages 8,, 3, and t?, = 1 - 8, - 6,: d@,/dr

=f,

d&/dr=f2

= +?, - y,& - tY,&&, =a$:,

-yz&J

with the positive

parameters

ai --k,p,/k,,

yI =:-,,‘k,.

a2

=kk,p,/nk,,

Before (a) (5) to results

(5a)

-0,8,0,“,

n= 1 or2,

(5b)

and scaled time

y2 =k_,/k,,

r=kk3t.

(61

the comments a few remarks are necessary. In ref. [ 11, TSA restrict their stability analysis of the autonomous system the case n = 1. The sole oscillation displayed in their paper ([ 11, fig. 3) from the numerical resolution of eqs. (5), case n = 1, with

LY,=0.016,

y, =O.OOl,

az =0.0278,

yz =0.002.

(7)

(b) For the CO oxidation on Pt, it is generally admitted that the adsorption of oxygen is dissociative and that the desorption of atomic oxygen (60 5 E_, 5 70 kcal) may be disregarded in the temperature range where oscillations are observed. Therefore, in our examination of the CO oxidation within the model case n = 2 we set yz = 0. Notice that TSA observed that limit cycles may exist in the case n = 2 and that one vanishing parameter (e.g. yz = 0) does not hinder the oscillations. (c) According to TSA physical explanations of the four-site LH mechanism, eq. (4), we can tentatively describe the CO oxidation within the case n = 1. Namely, we consider the oxygen initial adsorption as molecular, i.e., 0, +

s $ o,s, -2

and interpret the reaction step (4) as a combination of multiple steps involving the interme~ate species OS, the concentration of which remains always small enough to satisfy the pseudo-steady state assumption. (d) For the system (S), TrJ=(Zlf,/M, + af,/M,) is negative at the boundaries of the phase space unit triangle (( 8,) 0,) 2 0, (8, + 4,) G 1). Therefore let us determine which parameter values satisfy the necessary condition for oscillations: TrJ > 0 inside the unit triangle. In the case n = I, TrJ has a saddle point located at 8, = 8, = 0.1057 and a maximum located at 8, = 8, = 0.3943. At the saddle point TrJ is negative whatever the parameter values. The

M. Bouillon et al. / Isothermul

maximum

reaches

a positive

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sustained oscillutions

value if

a, + y, + a2 + yz < 0.0962.

(8)

Notice also that, for n = 1, the existence of mixed steady states (J; = 0, i = 1,2) in the limit yi + 0, requires aj G 0.037 (i = 1,2). In the case n = 2 with yz = 0 (see remark (b) above) TrJ still exhibits an always negative saddle point and a maximum located at 8, = r3, = [0.25 + 0.1443(1 + 4t~~)‘/~] (inside the unit triangle provided a2 5 0.5). This maximum reaches a positive value if a, + y, + a2 G 0.0962( 1 + 4a2)3’2.

(9)

In the limit a2 + 0, condition (9) reads (a, + y,) C 0.0962, in the upper limit a, -+ 0.25 (prescribed by f2 = 0, for n = 2), eq. (9) requires (a, + y,) G 0.0221. We conclude this remark by noticing that the maximum values of the parameters are of the same order of magnitude (for n = 1 or 2), i.e., a few percent. As we could plausibly multiply all the values quoted in eq. (7) by a factor 2 (at most), let us consider these values as indicative in the forthcoming comments. (1) To examine the CO oxidation on Pt as a possible candidate to TSA model we refer to table 1 and fix p, /p2 =p, /pto, = 1% throughout this analysis. The adsorption rate coefficients ki (i = 1,2) have the form (e.g., see ref. [4]) kjp, = 5.3 X lO”( A4jT)-1’2a,p,

exp( -E,/RT),

(10)

where Mj, uj and pi are the molecular weight (in g), the sticking coefficient and the partial pressure (in atm), respectively. Thus, in both cases n = 1 and n = 2, one should satisfy al/a2

~:(ei~,/e~p~)

(11)

exp(E,/T),

as E, - 0 (e.g., see refs. [5,6]). If we introduce in eq. (11) the TSA ratio a, /a2 together with the values u, = Q.54 an d a, = 0.38 [7], we obtain with T in the range of table 1: 3sEE,

55

kcal mole-‘.

(12)

This range of values for E, is fairly acceptable

Table 1 Parameter ranges corresponding to the observed oscillations denote the period and the temperature, respectively Expt. No. I 2

T

1-15 2-10

for the dissociative

in the CO oxidation

adsorption

on Pt;

tp and T

Ref.

WI

Pco/Po, @I

Pt0t (atm)

400-550 500-650

OS-I.2 0.8-S.

1

[21

0.016

131

of 0, (case n = 2), but seems unlikely for the molecular adsorption of 0, (case n = 1). In the latter case, if we set Ez = 0 and cyr /ty, 5 2, we find u, 2 50az. Thus in the case n = 1 we should rather propose the ratio (~~/a[’ * 50 (with (Ye5 0.037!) which is an order of magnitude larger than the ratio used by TSA. (2) To calculate the ratio y, /ar, in the CO-oxidation case, we use eq. (10) together with the definition of the desorption rate coefficients (e.g., see ref. [4]): k_,

=viexp(-E_,/RT).

For i=

(12)

1, eq. (10) gives (E, =O, a, ~0.54.p,/p,,,

= 1%):

k,?J, = 5 X 107T -‘/‘pPtot.

(13)

25==E _ , ~530 kcal mole-’ Eq. (12)(i= 1, Y, = 10’3s-’ for experiment 1 {see table 1; at ??* 450 K 2x

10-s

5y,/cw,

i.e., y, is practically we find 4 x 1o-4 sy,/cu,

54x

1op6,

negligible 54

161) and eq. (13) yield

(14) with respect to (Y,. For experiment

2 at T* 550 K.

x 10-2.

(15)

Notice that for the TSA example of oscillations ~,/a, = 6.25 X 10-I. In the case n = 2, yz = 0 (see remark (b) above). while in the case n = 1 no values for the activation energies E,, E_, (“molecular” adsorption/desorption for 0,) are available for the evaluation of y2. (3) If we compare the period of the oscillations displayed in fig. 3 of ref. [l] to the range of the measured periods (see table 1) we observe that 5x10-“~k,~8x10-2s~‘. As the total pressure at which the model oscillations (13)) satisfies the relation P tot $+:2 X 10-8T”2k,(u,,

(16) could be observed

(see eq.

(17)

we conclude that’the CO oxidation on Pt meets the TSA model for oscillations with a period t, 5 1 s at a total pressure ptot 5 10 -’ atm. To our knowledge. oscillations in the CO oxidation at such low pressure have not been reported up to now. Perhaps the reduction of NO by CO could be a better candidate for the TSA model. Recently, Adlhoch et al. [8] reported oscillations for this reaction at pressures less than 5 X 10 -’ Torr (400 < T < 520 K, 1
considered in two ways: (i) case 2/3, the two vacant sites are assigned to be nearest neighbours of the adsorbed moleculeB; (ii) case 2/6, the two vacant sites are assigned to be nearest neighbours of the adsorbed pair AB. Figs. 1 and 2 show the time evolution of the simulated reaction rate r,,, and of the rate rcdc = rg E S,t9,6’,‘, calculated with the values of 8, and t9, obtained at each step of the simulation. We denote by A” the mean value of any quantity A(t) over the time range (2-3.8) X 104k,t (i.e. far from the transient regime) and by A,A its mean fluctuation. In the case 2/3 the reaction seems to vanish in the long time limit; the coverages converge to the values 6y = 0.4445 and ezrn= 0.5543 (Are, = 3 X 10 -3, i = 1,2). It appears that the production is essentially governed by the desorption and therefore practically non-detectable. The case 2/6 seems more favorable to the TSA model as, for times T 2 7 X 103k3 t, the simulated production rate remains finite and the reaction is “stable”. The asymptotic coverages 8: = 0.1959 and 8,m = 0.7832 (A,&, = 2 X lo-‘, A& = 4 X 10 -3) are closer to the mean coverages of TSA (i.e., 8, = 0.1945 and 8, = 0.7025) than those obtained in the case 2/3. However, the simulation leads to a reaction rate (r& = P$$ = 10 -4) one order of magnitude lower than the mean reaction rate obtained by TSA (i.e. (r;Z) = 14.6 X 10 -4). Moreover, the coverage function 8,13,8: describing the reaction step (4) overestimates rCalCwith respect to r,,, up to times r= 1.5 X 104k,t (after that time the fluctuating robs{t) and r,& t) get tangled). More refined coverage functions for the four-site reaction step should be tested, with as a probable consequence

0 0.8

1

1.5

K3hlt?

Fig. 1. Mean evolution of the simulated reaction rate robs (solid line) and of the calculated reaction rate rCdC=r&=0,t$@S2 (broken line) in the case 2/3. The asymptotic rates are r$, =Z10 -’ and rCzC =3.55X IO-’ (see text),

Lll6

M.

Bouillon

et cd. /

Isothermul

sustained

oscillations

01 0.5

1

Fig. 2. Mean evolution of robs (solid line) and r$,=2.05X lop4 and r=T,, =0.78X 10-4.

I5 of T,,,~ (broken

Kgt do3

line) in the case

2/6.

Here

changes in the steady state geometry and stability analysis. Due to our simulation limitations (i.e. essentially the number of lattice sites), we have not been able to detect oscillations in the fluctuating noise characteristic of the long time “stable” regime of the reaction (case 2/6).

References [l] C.G. Takoudis, L.D. Schmidt and R. Aris, Surface Sci. 105 (1981) 325. [2] J.P. Dauchot and J. Van Cakenberghe, Nature Phys. Sci. 246 (1973) 61; Japan. J. Appl. Phys. Suppl. 2 (2) (1973) 533; and private communications. [3] D. Barkowski, R. Haul and U. Kretschmer, Surface Sci. 107 (1981) L329. [4] D.O. Hayward and B.M.W. Trapnell, Chemisorption (Buttetworths, London, 1964). [5] A.E. Morgan and G.A. Somojai, J. Chem. Phys. 51 (1969) 3309; Surface Sci. 12 (1968) 405. [6] H.P. Bonzel and J.J. Burton, Surface Sci. 52 (1975) 223. [7] B. Kasemo and E. Torqvist, Phys. Rev. Letters 44 (1980) 1555. [S] W. Adlhoch, H.G. Lintz and T. Weisker, Surface Sci. 103 (1981) 576. [9] The details on our Monte Carlo simulations of surface reactions will appear elsewhere.