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Bistable kinetics of simple reactions on solid surfaces: lateral interactions, chemical waves, and the equistability criterion
Physica D 70 (1994) 383-395 North-Holland
SSDI: 0167-2789(93)E0283-H
Bistable kinetics of simple reactions on solid surfaces: lateral interactions, chemical waves, and the equistability criterion V.P. Z h d a n o v a,b a n d B. K a s e m o b s Department of Applied Physics, Chalmers University of Technology, S-412 96 Gt~teborg, Sweden b Institute of Catalysis, Novosibirsk 630090, Russian Federation Received 24 May 1993 Revised mannseript received 25 August 1993 Accepted 3 September 1993 Communicated by Y. Kuramoto
In heterogeneous reactions, the rate constants of desorption, diffusion and elementary reactioh steps are usually strongly dependent on reactant covetagas due to adsorbate-adsorbate lateral interactions. We analyze the effect of this factor on the bistable regime of the reaction kinetics. As an example, we consider CO oxidation on Pt(l 11 ). The equistability lines in the bistable region for this reaction are calculated by analyzing propagation of chemical waves and taking into account the coverage dependence of the CO diffusion coefficient. The results of simulations are compared with the available experimental data. We show that it is possible to obtain the relationship between various kinetic parameters, for example, between CO and oxygen sticking probabilities, by studying special features of the bistable kinetics.
1. Introduction
Critical phenomena, such as kinetic phase transitions, chemical waves and oscillations, may take place in strongly nonequilibrium systems [ 1,2 ]. Physically, this behaviour is driven by the Gibbs-free-energy decrease of an overall chemical reaction. Mathematically, critical phenomena are connected with nonlinear features of kinetic equations describing physicochemieal rate processes. The interplay of chemical kinetics and diffusion may result in formation of chemical waves, inducing transitions from a metastable kinetic "phase" to a stable state, or, more generally, may cause the homogeneous state of the system to become unstable, leading to spontaneous formation of time-dependent or stationary spatial patterns. During the past decade, general ideas [ 1,2] formulated in this
field of chemical kinetics have been actively employed and developed in experimental studies and theoretical simulations of reactions at solid surfaces [3-5]. Particular attention has been paid to CO oxidation on Pt because this reaction exhibits bistable kinetics on the (111 ) face and bistable kinetics with oscillations on the (001) and (110) faces [3]. The bistable regime of CO oxidation on Pt(111 ) has been explored in detail by Ehsasi et al. [6] under ultrahigh vacuum (UHV) conditions (Po2 - 2.66× 10-9 bar, Pco <10 -9 bar and T = 340-680 K). To interprete their experimental data, they have employed the wellknown ZGB model [7]. This model is, however, not quite representative of the actual physical situation since it assumes that the reaction between nearest-neighbour reactants occurs instantly and that CO adsorption is strongly in-
016%2789/94/$07.00 © 1994- Elsevier Science B.V. All rights reserved
384
V.P. Zhdanov, B. Kasemo / Bistable kinetics of simple reactions
hibited by the presence of preadsorbed oxygen. These assumptions are not fulfilled for the system under consideration. The ordinary kinetic equations predicting bistable kinetics for CO oxidation on Pt were analysed in refs. [8-11 ]. In particular, B~ir et al. [10] have recently proposed a realistic kinetic model for CO oxidation on P t ( l l l ) (and on Pt (110) as well [ 11 ] ), taking into account the measured coverage dependences of the CO and 02 sticking coefficients and using the values for the rate constants of CO desorption and CO + O reaction, obtained at low coverages. The coverage dependence of these rate constants was ignored. The kinetic behaviour of the model has been studied in detail at Po2 = 2.66x 10 -7 bar, PCO ~ 5× 10 - s bar and T = 400-600 K but has not been compared with the experiment. The models employed in refs. [6] and [10] reproduce a sudden, steplike decrease in the reaction rate with increasing CO pressure but considerably overestimate the magnitude of this effect. The latter seems to be connected with neglecting lateral adsorbate-adsorbate interactions (see the results of our calculations presented below). Attempts to study the effect of lateral interactions on the bistable kinetics of CO oxidation on Pt have been made by Kaukonen and Nicminen [ 12 ], Luque et al. [ 13 ] and Satulovsky and Albano [14]. All three groups have employed Monte Carlo simulations. The treatment of lateral interactions used by Kaukonen and Nieminen [ 12 ] appears to be inconsistent from the physical point of view. The reason is the following. If one intends to incorporate lateral interactions into the model, it is necessary to distinguish between lateral interactions ofnonactivated adsorbed particles and lateral interactions in the activated state [ 15 ] (the terms "activated" and "nonactivated" correspond here to the language of the transition state theory). In the case of the adsorption/desorption process, for example, the former interactions contribute to the desorption rate but not to the ad-
sorption rate. The latter interactions affect both adsorption and desorption rates because the activated complex for adsorption and desorption is the same. Describing CO desorption, Kaukonen and Nieminen do not distinguish explicitly between these interactions. From the context of their paper it follows however that they are considering the interactions of nonactivatcd particles, because these interactions are used to calculate the distribution of nonactivatcd particles on the surface. Thus, these interactions should not affect the CO adsorption process, while Kaukohen and Nieminen incorporate just the same interactions into the CO adsorption rate constant. Under such circumstances, the physical meaning of the results obtained is not quite clear. The approach employed by Luque et al. [ 13 ] is principally more correct because generally they use different lateral interactions for CO adsorption and desorption. In their real calculations, they assume that the lateral interactions for adsorption have the same value but the opposite sign compared to that for desorption. This assumption does not reflect characteristics of the CO/Pt system. Indeed, the sticking coefficient for CO adsorption on Pt is well known to be close to unity and weakly dependent on coverage. The latter means that the lateral interaction in the activated state is very weak. Thus, to describe the CO adsorption/desorption process, we should incorporate lateral interactions only into the desorption rate constant. In addition, Laque et al. do not take into account CO diffusion while this process is very rapid compared to other elementary steps in CO oxidation. The same comments are applicable to the simulations by Satulovsky and Albano [ 14], where CO diffusion is ignored and lateral interactions are taken into account only for the activated state. In the present paper, we explore theoretically the bistable kinetics of CO oxidation on Pt ( 111 ) under UHV conditions, taking into account the coverage dependence of the rate constants for CO desorption, CO + O reaction, and CO diffusion. The mean-field model employed
V..P. Zhdanov, B. Kasemo / Bistable kinetics o f simple reactions
385
Table 1 Kinetic parameters [I0] for CO oxidation on P t ( l l l ) : /C are the rate constants for the reactant fluxes, s the sticking coefficients, ~, the preexponential factors, and E ( 0 ) the activation energies at low coverages. CO adsorption: 0 2 adsorption: CO desorption: C O + O reaction: CO diffusion:
kl = sco/Cco k4 = so2K:o2 k2 k3 D
K~co = 1.919 x I0 s s - l m b a r -1 /(:02 = 3.589 x 105 s - l m b a r - I t/co = 1.25 x 1015 s -1 ~'r = 1.645 x 1014 s -1 D O = 10 - a m2s -1
is described in section 2. The effect of various parameters on the position of the bistable region in the stability diagram is demonstrated in section 3. Chemical waves are discussed in section 4. The latter contains a derivation of the equistability criterion, taking into account the coverage-dependent coefficient for CO diffusion. In section 5, our calculated results are compared with experimental data measured by Ehsasi et al. [6]. Extrapolation of the results obtained under UHV conditions to high (atmospheric) pressures is briefly discussed in section 6.
2. Model
The well-established three-step LangmuirHinshelwood mechanism of CO oxidation on Pt (111 ) at high-vacuum pressure conditions contains reversible monomolecular CO adsorption, irreversible oxygen adsorption, and CO + O reaction between adsorbed species to form product CO2 molecules which desorb rapidly [3]. The corresponding kinetic equations are as follows [10]: d0co dt = klPco [ 1 - (0co/0~o) 2 ] - k20co
-k30coOo,
( 1)
d0o _ k4Po2 [1 - 0co/0~o - 0o/0~] 2 dt
-k30coOo,
(2)
where 0~o = 0.5 and 0 h =0.25 are the saturation coverages. Eq. (1) takes into account that the
sco=0.84 so2=0.06 E c o ( 0 ) = 3 4 . 9 kcal/mol E r ( 0 ) = 2 4 . 1 kcal/rnol E a i f ( 0 ) = 7 keal/mol
coverage dependence of the CO sticking coefficient is weak up to saturation (which is suggested to be due to a precursor mechanism of adsorption). The model also takes into account that preadsorbed CO inhibits dissociative adsorption of oxygen and that no such site-blocking effect is exerted by adsorbed oxygen on incoming CO molecules. Biir et al. [ 10 ] have employed eqs. ( 1 ) and (2) assuming that the rate constants for CO desorption, k2, and C O + O reaction, k3, are independent of coverage. In their simulations, they have used the measured rate constants corresponding to the low-coverage limit (table 1 ). Thus, the model [ 10] does not contain any adjustable parameters. The reliability of the measured rate constants is however open for discussions (as is usual in surface science). The preexponential factor and activation energy for CO desorption seem to be well established. The situation with the Arrhenius parameters for CO + O reaction is more complex because different studies report different values of these parameters (cf. refs. [ 10 ] and [ 16 ] and see the review [ 17 ] ). The results of simulations are however not too sensitive to the details of describing the CO + O reaction as long as this step is rapid compared to CO desorption. In our study, we will employ essentially the same model with the same kinetic parameters as in ref. [ 10] (some minor changes in the value of the oxygen sticking coefficient will be made and motivated in section 5, where we will discuss the experimental results). However, one additional important element is introduced, namely the coverage dependence of the above mentioned
386
V.P. Zhdanov, B. Kasemo / Bistable kinetics of simple reactions
rate constants. This is a primary new feature of the present work compared to the study of Bfir et al. [ 10 ]. Our preliminary inspection showed that at steady-state conditions the oxygen coverage is always low inside the bistable region in the stability diagram (because the reaction is rather rapid). Thus we do not need to introduce into the model the dependence of the rate constants k2 and k3 on oxygen coverage. On the other hand, the CO coverage is often large and accordingly we should focus our attention on the dependence of the CO desorption and CO + O reaction rate constants on 0co. On the Pt ( 111 ) surface, CO is adsorbed in ontop sites at all coverages up to saturation. In addition, bridge-site adsorption takes place above 0co = ½ [ 17-19 ]. The binding energy difference between the on-top and bridge sites is low (~_2 kcal/mol), and the CO-CO lateral interactions are repulsive [ 18,19 ]. In principle, all these facts can be described in the framework of the lattice-gas model as in ref. [ 19 ]. If, however, one takes into account not only CO adsorption but also oxygen adsorption occuring in threefold sites and CO + O reaction, this approach becomes too cumbersome. For this reason, we will in our simulations employ the mean-field approximation. In particular, the effect of lateral interactions and of bridge-site adsorption on the CO resorption and C O + O reaction rate constants will be described by a linear dependence of the activation energies on CO coverage, Eco (0co) = Eco (0) - AOco/O~o,
(3)
E r ( 0 C O ) = E r ( 0 ) - BOco/O~o.
(4)
The inspection of available TPD data for CO desorption (see refs. in [19]) shows that A __4-5 kcal/mol. We will use A = 4 kcal/mol. This value of A corresponds in fact to rather weak repulsive next-nearest-neighbour CO-CO lateral interactions. The strong nearest-neighbour COCO interactions can be taken into account under UHV conditions implicitly by introducing the
constraint 0co < 0~o as it has been done in eqs. (1) and (2) (for oxygen, by analogy, we have
0o _< 0~). The activation energy for the CO + O reaction on the Pt-group metals is known to decrease with increasing coverage [ 17 ]. The TPD data [ 16 ] indicate that B < 10 kcal/mol. Our analysis of the data [20] on titration of oxygen by CO has however shown that B < 5 kcal/mol. Below, the results are presented for B = 0 and 4 kcal/mol, respectively. It is also necessary to express the coverage dependence of the activation energy for CO diffusion, Edif (0CO) = Edif ( 0 ) -- COco~ 0~o,
(5)
because this dependence is significant for describing chemical waves and calculating the equistability lines (section 4). Experimental data allowing a reliable estimate of the parameter C are missing. We will use C = 0 or 4-3 kcal/mol. The repulsive CO-CO interaction excludes first-order phase transitions in the CO overlayer [18], and accordingly the CO islands are not formed during CO adsorption. The available TPD data for oxygen adsorption on Pt(111 ) (e.g., ref. [21]) indicate that the O - O lateral interactions are repulsive as well. Thus, if the surface is covered by oxygen, formation of oxygen islands is also negligible. The situation when both the CO and oxygen coverages are not low is not so clear, and in the latter case one cannot certainly exclude formation of islands. Such situations are however rare at the steady-state conditions. All these facts justify application of the mean-field kinetic equations. Note also that the positive parameters A = 4 kcal/mol and B = 4 kcal/mol introduced above correspond to rather weak repulsive adsorbate-adsorbate interactions (< 0.5-1 kcal/mol). In this case, the correlations in the adsorbate-adsorbate arrangenent are not too strong. This is an additional argument in favour of the mean-field approximation.
387
EP. Zhdanov, B. Kasemo / Bistable kinetics o f simple reactions
(a)
C o n c l u d i n g this section, we n o t e t h a t i f the rate c o n s t a n t k3 is d e p e n d e n t o n l y o n C O c o v e r age, eq. ( 2 ) c a n be easily s o l v e d at s t e a d y - s t a t e conditions. We then have
0o/0~- 1 -x
+ gx
,
r
°
<
-x
+ g x ) 2 - (1 - x ) 2 ]
1/2,
.
.
.
.
.
,
~,,
*%
>
-[(1
,
*
(6) 0.2
where x = 0co/0~o and g = k 3 0 ~ o O ~ / 2 k 4 P 0 2 . Thus, at steady-state conditions, we need in fact only to analyze eq. (1) in order to obtain CO coverage because oxygen coverage in this equation is dependent implicitly (via eq. (6)) on CO coverage.
%÷
0
ttttl
b)
6
. . . . ~ o ÷ - ~ . ~
ttttlli
i,
i
,J....
i
i
t
i
i
,
tO
t,
2O
2.'s
30
3,
40
,,,,,,,
_
,
-a,,
4,
~
.~0
,
Po2=2.66x 10-7 bar 7
S::
3. Bistable region and hysteresis At steady-state conditions, eq. (1) for CO coverage has one or three solutions. In the latter case, the intermediate solution (with respect to absolute coverage) is unstable and the other two solutions are stable. In particular, figs. 1-4 show CO coverage, reaction rate and stability diagrams calculated for various values of the parameters ,4 and B at Poe -- 2.66 x 10-7 bar (this pressure is the same as in ref. [10]). The coverage dependence of the activation energy for CO desorption is seen to reduce the extension in CO pressure of the bistable region (cf. figs. lc and 2c). The effect of the coverage dependence of the activation energy for CO + O reaction on the bistable region is o p p o s i t e (fig. 3b). Explanation of these results from the mathematical point of view is rather tedious because eqs. (1) and (2) with expressions (3) and (4) are strongly nonlinear. Physically, however, the effects obtained appear to be quite clear. In particular, a decrease of the activation energy for CO desorption with increasing coverage results in an increase of the CO desorption rate. This means that the reaction system tends in this case to an adsorption/desorption equilibrium. At equilibrium, however, the bistable region
: *+
\
o = 540 K
0
.<
2 520 K
•o
F
!
~_
•
÷**.. ,¢,f_~*°~50 K ........................ : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :..... : i
i
i
i
i
i
i
L
l0
15
20
25
30
3'
40
4'
'0
CO PRESSURE (10 -9 bar)
]
~o
PO2=2.66xi0 -T bar
J
8, I 0
400
420
440
460
48O
,00
520
540
,6O
TEMPERATURE (K)
Fig. I. (a) CO coverage and (b) reaction rate as a function of CO pressure, and (c) stability diagram (the bistable region is shadowed) at P02 = 2.66 x 10-7 bar. The results are calculated for A = B = 0, i.e. no coverage dependence of the rate constants for CO desorption and CO+O reaction.
KP. Zhdanov, B. Kasemo / Bistable kinetics of simple reactions
388
[a)
.
.
.
.
.
, ....
,
' .
:
......
:
.....
l
"-~
0.8
os
%%
~.
.++"
%¢¢
06
o.o
+e T = 5 2 0 K
~
~. r~ ¢2
T=52(
""
/~"
5
4
0
K
~
.
÷
% %
r~
+
04
%
*
0.2
... :
m
.
~
e
o,2
\
% .....%
o o
L ~
i
o
5
i
a
15
Io
i
i
i
20
25
30
h ........
35
J
~
40
45
..........
5O
C O P R E S S U R E ( 1 0 - " b*~r)
(b)
PO2=2.66x10-T b a r
35
b)
,'
-
,--
P o ~ : 2 . 6 6 x 10 -7
7
~,) 10
q'a°'iF L 5
0 5
I%
15
20
25
Lc)
~
]
30 35 40 45 50 C O P R E S S U R E (10 -9 b a r ) ,.
.
.
.
.
.
P o 2 = 2 . 6 6 x 10 -7 b a r '
~
~°
aa 1:~
15
040~t
420
440
460
480
500
520
5~0
TEMPERATURE
56O (K)
Fig. 2. As fig. 1 for A---4 kcal/mol and B = 0 , i.e. decreasing CO desorption energy and constant activation energy for C O + O reaction with increasing CO coverage.
is known to be absent at all. Thus, the coverage dependence of the activation energy (with A > 0) for CO desorption should suppress the bistability. An increase of the activation energy
oo
42o
440
460
48o
5oo
520
54~]
TEMPERATURE
5(~3 (K)
Fig. 3. (a) CO coverage as a function of CO pressure and (b) stability diagram at A--0 and B = 4 kcal/mol, i.e. constant activation energy for CO desorption and decreasing activation energy for CO + O reaction with increasing CO coverage.
for CO oxidation with increasing coverage leads the system far from an adsorption/desorption equilibrium, and in this case the bistable region of course increases. It is of interest that the cusp position in the stability diagram is almost independent of the coverage dependence of the activation energies for CO desorption and CO + O reaction (cf. figs. lc, 2c, 3b and 4c). This is because near the cusp point the reaction rate is maximum and both adsorbate coverages are consequently low. The latter allows us to obtain simple analytical expressions for the cusp position. In particular, lin-
V.P. Zhdanov, B. Kasemo / Bistable kinetics of simple reactions
(a)
k4Po2(1 - 2 0 c o / 0 ~ o - 2 0 0 ) - k 3 0 c o O o = O.
1
it
" - ~ o.8
.....
..........
o.6
r~
>
(8)
i
t
o
~
389
0.4
T=520 K
"..
©
......"
.
°
e02
~K ~
. i 5
i I0
i 15
l 20
215
i 30
~"~'~
-
f
~-~-
m
~,*~"'" 560 K i 35
I 40
415
Taking into account that the reaction rate conslant is large, we can conclude that one of the reactants should dominate on the surface. If oxygen dominates (i.e., 0o >> 0co), CO desorption is negligible compared to C O + O reaction, and one can rewrite eq. (7) and (8) as (9)
k l P c o - k30coOo = O, 50
k4Po2(1 - 200/00) - k30coOo = O,
(b) Po~=2.66 x 10.7 bar
(10)
or 00/05
g Z
kleco)/2k4e02.
(11 )
The latter equation shows that the oxygen domination is possible at k 4 P o 2 > kl Pco. If CO dominates on the surface, eq. (8) can be rewritten as
*
©
~--- ( k 4 P o 2 -
<
f***,,***" T=450 K I
k4Po2 (1 - 20co/0~o) - k30coO 0 = O. 5
15
10
20
25
2~0 ('()
35
35
40
PRESSURE
45
From (7) and (12), we obtain
( 1 ( ) -~' h a k )
c) Po2=2.66 x 10.7 bar
~
(12)
50
k4Po2(1 -20co/0~o) - k l P c o
3o
+ k 2 0 c o = O,
(13) or 0co = (k4P02 - klPco)/(2k4P02/O~o - k2).
~ lO
(14) 5
0 400
I 420
I 440
I 460
I 4g0
I 5~
i 520
i 540
560
TEMPERATURE (K) Fig. 4. A.~ fig. l for A = B = 4 k c a l / m o l , i.e. decreasing activation energies for C O d e s o r p t i o n a n d C O + O reaction with increasing C O coverage.
earizing the adsorption terms in eqs. ( 1 ) and (2) at smal 0, we have near the cusp point klPco - k20co - k30coOo - O,
(7)
The latter equation shows that at k4P02 > k]Pco we can construct an additional (unstable) solution to eqs. (1) and (2) provided that 2kAP02/O~. 0 > k2. The third (stable) solution is obtained only if one does not linearize the adsorption terms. The analysis presented shows that at a given oxygen pressure the cusp position takes place at P~o ~- k4Po2/kl
(15)
390
EP. Zhdanov, B. Kasemo / Bistable kinetics of simple reactions (a)
and 1
Po2=2.66x
2k4Po2/O~o = k2,
(16)
T=520
10 -7 bar
K
L
"-~ 0.3
...z
The latter equation can be rewritten as (we set o.6
kB=l)
I
<
T* =
Eco (0) In (vcoS~o/2k4Po2)"
( 17 )
Eqs. ( 15 ) and (17) indicate that the cusp position in fact does not depend on the reaction rate constant, as long as this constant is large compared to the adsorption rate constants. Concluding this section, it is of interest to note that in the bistable region one may in principle observe a hysteresis in the reaction rate and reactant coverages (fig. 5) as the CO/O2 pressure ratio is varied up and down.
p;
,-...
0.4
> ©
I I
I
© r.D
%
...., r ....,
r 1 I
0.2
I
i
i
5
10
15
| 20
25
i
I
I
I
30
35
dO
45
50
(b) [
Po2=2.66x10 T=520
-7 bar
K
P~
/ /n
/ ooO
4. Chemical waves and equistability criterion
I
g
1 f 1 I I I t
L-" , * °
ooo
If the temperature is not too high, the model presented by eqs. ( 1 ) and (2) predicts a bistable reaction regime and hysteresis at P~o < Pco < Pc°o (Pc°o and P~o are the critical CO pressures at a given temperature and oxygen pressure). In this regime, one stable solution corresponds to the surface covered predominantly by oxygen and the other to the surface covered predominantly by CO. These solutions are stable with respect to fluctuations of the gas-phase concentrations if we employ the mean-field approximation and do not take into account fluctuations of adsorbate coverages due to surface diffusion. Including the latter process results in the possibility of formation of chemical waves [ 2,5,10 ]. When considering the effect of surface diffusion, a new important parameter Pdo (P~o < P~o < P° o) should be introduced, corresponding to equistability of different "kinetic phases". In the bistable region, the oxygen-domination regime is absolutely stable at Pco < Pdo and metastable at P~o < Pco < p O . The CO-domination regime is metastable at P~x) < Pco < P~o and absolutely
1 I I I I
,oO
k T :
, 5
i I0
i 15
n 20
II
1
. . . . . . . . ,,, , , .....
1111
i
~
i
o
i
25
30
35
40
45
CO
PRESSURE
(10
50
~j b a r )
Fig. 5. (a) CO coverage and (b) reaction rate as a function of CO pressure for A = B = 0. Pointers show a hysteresis. The stepwise transition corresponding to the equistability criterion (solid lines) is calculated assuming that the CO diffusion coefficient is independent of coverage (i.e., C = 0).
stable at Pco __ P~o. ("Absolutely stable" here means stable not only with respect to fluctuations of gas-phase concentrations but also with respect to the influence of surface diffusion. ) The stability analysis of the mean-field solutions in the bistable region is known to be directly connected with calculating the velocity of chemical waves [2]. If we assume, for example, that at a given value of Pco (P~o < Pco < P° o) one part of the surface, initially covered predominantly by CO, and the other part, initially covered by oxygen, are separated by a linear interface, then with increasing time the interface will move from the CO-covered region
V.P. Zhdanov, B. Kasemo / Bistable kinetics of simple reactions
to the oxygen-covered region or in the opposite direction due to diffusion and reaction (this is a chemical wave). In the former case, the velocity of the chemical wave is positive, v > 0, the CO-domination regime is stable, and the oxygendomination regime is metastable. If v < 0, the situation is the reverse. The parameter P~o introduced above corresponds to v = 0. To calculate the velocity of a chemical wave, one must include surface diffusion into the reaction scheme. In our case, taking into account that oxygen diffusion is negligible compared to CO diffusion [ 3,10 ], we should solve the following equation:
-O0 ~
=
-O ~ D ( O ) -O0 ~
(18)
where 0 is the CO coverage (we omit here the subscript CO), D (0) the coverage-dependent CO diffusion coefficient, and W (0) the righthand part of eq. ( 1 ). Eq. (18) assumes that the oxygen coverage is given by the steady-state solution of eq. (6). This assumption is justified because the reaction is rapid compared to the wave propagation (see also ref. [10] ). A chemical wave, realizing the transition from one state to another and moving at a velocity v, is given by a special solution
0 = 0(~),
~ = x-vt,
(19)
that satisfies the relevant boundary conditions at = q:oo. Substituting expression (19) into eq. (18), multiplying the result by D (0) d0/d~, and then integrating the equation obtained from - c ~ to +c~ and taking into account that d 0 / d ~ = 0 in both limits, we derive
v =
Equistability occurs for v = 0. Thus, the equistability criterion can be written as 03
D(O) W(O) dO -- 0.
f~3D(O)W(O)dO f~_+°°D(O)(dO/cl~)2d~'
(21)
Ol
This equation defines implicitly the value of Pdo (through the dependence of W (0) on Pco) and indicates that P~o may be strongly affected by the coverage dependence of the diffusion coefficient. If D ( 0 ) = const., eq. (21) reduces to the following well-known result [ 2 ]: 03
f w(o) dO= + W(O),
391
0.
(22)
Ol
Eq. (21) (or eq. (22)) is an analog to the Maxwell rule employed in the theory of firstorder phase transitions [ 15,22 ]. The position of the equistability lines in the Pco-temperature plane for CO oxidation is shown in fig. 6. The full curves indicate the equistability lines in the case when the diffusion coefficient is independent on coverage (C = 0). The dashed lines are obtained at C = =1=3 kcal/mol. An increase of the diffusion coefficient with increasing coverage (C > 0) shifts the equistability lines to lower CO pressures. If the diffusion coefficient decreases with increasing coverage (C < 0), the effect is opposite. Finally, we note that if the creation and propagation of chemical waves are rapid (this is expected to be the case for CO oxidation on Pt( 111 )), one does not expect a hysteresis in the reaction rate. Instead, we should have a stepwise change in the reaction rate and coverage at the CO pressure Pdo, corresponding to the equistability criterion (fig. 5).
(20) 5. Comparison with experimental data
where 01 and 03 are the steady-state solutions to eq. ( 1 ) corresponding to the surface covered predominantly by oxygen or by CO, respectively.
As pointed out in the introduction, the bistable regime for CO oxidation on Pt (111 ) has been
V.P. Zhdanov, B. Kasemo / Bistable kinetics of simple reactions
392 a)
¢-
35
-t-
Po2=2.66× i0-~ bax
~-' ........ ' Experimental Data for
.'*?
'
CO+O Reaction on Pt ( 1 1 1 ) f at Pm:2.66x10 -0 bar / X \
¢
...
/-
.../'
'
~ \
"\,
667 K | ................. I "~3K] "'-q
*
Z ©
.7 /
(3
/ a \i
....T...=49! ........K......
5
o ,~O0
42O
46O
440
480
50~
520
540
560
CO PRESSURE (10-9
bar)
~ (b)
Fig. 7. Measured rate of CO oxidation on Pt(l I I ) as a function of the CO pressure (Ehsasi et al. [6]).
Po~=2.66x 10-7 bar
....:'i;F 2o
©
%
.
420
44O
(c)
46~
,
480
5O0
,
......
Po2=2.66x 10-7 bar
,
.
• •
52O
..'"
540
................
56O
,
~
..z /"
a
~o
oo
....*" o~
4O0
420
~4q
460
480
5O0
52O
5~0
TEMPERATURE (I()
Fig. 6. Stability diagram at (a) A = B = 0, (b) A = 4 k c a l / m o l a n d B = 0, a n d (c) A = B = 4 kcal/mol.
Dots confine the bistable region. The cquistability lines am calculated assuming that the activation energy for CO diffusion is described by eq. (5) with C = 0 (the central full curve), 3 kcal/mol (the lower dashed curve), and - 3 kcal/mol (the higher dashed curve).
expedmentaly explored in detail by Ehsasi et al. [6]. The data obtained at P o 2 = 2 . 6 6 x 10 -9 bar are shown in fig. 7. The corresponding results of our simulations, with the parameters presented in table 1, are displayed in fig. 8. Comparing figs. 7 and 8, two features of the bistable kinetics arc of particular intererst, namely, the stcpwise change in the reaction rate and the cusp position, respectively. If the coverage dependences of the CO desorption and C O + O reaction rate constants arc ignored, the theory overestimates the stepwise change in the reaction rate (cf. figs. 7 and 8a). If we take into account the coverage dependence of the CO desorption rate constant, the agreement between the simulation and the experiment becomes considerably better (cf. figs. 7 and 8b). If both the desorption and reaction rate constants are allowed to depend on coverage, the agreement is also good (cf. figs. 7 and 8c). The measured cusp position is at Pco ~- 8 x 10 -1° bar and T ~-555 K. The calculated values are Pco ~- 3.5 x 10 - l ° bar and T "=-495 K. The theory thus considerably underestimates the cusp pressure. This pressure is expected to depend only on the sticking coefficients of CO and 02 (cq. (15)). Taking into account that the value of the CO sticking coefficient is well established, we may conclude that the 02 stick-
V.P. Zhdanov, B. Kasemo / Bistable kinetics of simple reactions
393
0.14
o.o6(a) Po2=2.66xlO -° bar ~
0.12
0.05 s
o.1 0,04
i!
__
0.03
T=475 K ~ o.o6
N
~ 0.02 <
/i
O.Ol
420 K
~ 0.04
..................
......................../...........i2....2............................................................. ....
i
0.02 0
o
i
i
0.05
0.1
i
i
0.15
0.2
i
i
0.25
0.3
i
i
0,35
i
0.4
0.45
0.5
~ '
o.o6 (b) Po2=2.66x 10-0 bar 0.05
i
0.2
i
0.4
i
O.6
r
i
0,8 I CO PRESSURE (10 -9 bar)
Fig. 9. As fig. 7 for So2 = 0.14, A = 4 kcal/mol, B -- 0, and C = 3 kcal/mol.
7
0.04
ing coefficient (So2 = 0.06) presented in table 1 ....................does .......not . correspond to the experimental data
j
0.03 0.02 < 0.01
/
..................................
0 i
i
i
i
:
i
i
a
i
0.05
0.1
0,15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
o.= (c)
0.05
P°2=2"66x 1
0
-
~
"7 o,o4 ~
0.03
¢~ o.ol
o " 0.05 ,
...................
,
0.1
0.15
" - i ......... : ........ 0.2 0.25 0.3 0.35 0.4 0.45 0.5 CO PRESSURE (10 -9 bar)
Fig. 8. Reaction rate as a function o f CO pressure. The resuits have been calculated employing the kinetic parameters presented in table 1 (in particular, So2(0 ) -- 0.06). The coverage dependence o f the activation energies for CO desorption and oxidation is described respectively by the following parameters: (a) A = B ffi 0; (b) A ffi 4 kcal/mol, B ffi 0; (c) A = B -- 4 kcal/mol.
reported by Ehsasi et al. [6]. Under these circumstances, it is necessary to note that there is considerable disagreement between the literature data on the value and functional dependence upon coverage of the O2 sticking coefficient [23 ]. In particular, the initial sticking coefficient ranges from 0.02 to 0.2 in different studies. To reproduce the cusp position, we should use in our simulations So2 ( 0 ) = 0 . 1 4 (cf. figs. 7 and 9). In this case, however, the theory still underestimates a little the cusp temperature. This temperature is dependent first of all on the Arrhenius parameters for CO desorption at low coverages (eq. (17) ). From our point of view, the Arrhenius parameters presented in table 1 are reliable and it is not quite clear why the cusp temperature is underestimated. Perhaps this is connected with some minor contribution of deflect sites (e.g., steps) to the reaction kinetics. The influence of defects may also explain why the experimental changes in the reaction rate with increasing CO pressure are not exactly of the stepwise shape. The latter may in principle be also connected with nonequilibrium kinetic effects. Such effects however should result in a hysteresis in the reaction rate, but Ehsasi et al. [6] do not report a hysteresis.
394
EP. Zhdanov, B. Kasemo / Bistable kinetics of simple reactions
6. Comments on the "pressure-gap" problem The bistable kinetics of CO oxidation on Pt were observed not only under UHV conditions but also at high (atmospheric) pressures (see, e.g., the review in ref. [9] ). It is of interest to extrapolate to high pressures the results obtained under UHV conditions in the framework of the standard three-step Langmuir-Hinshelwood mechanism. Solving this "pressure-gap" problem (the details will be published elsewhere [24] ), we have arrived at the following conclusions. (i) If one employs in simulations the kinetic parameters obtained under UHV conditions but ignores the coverage dependence of the CO desorption and C O + O reaction rate constants (as in ref. [10]), the Langmuir-Hinshelwood scheme fails to describe the apparent activation energy and reaction order with respect to CO at high pressures. In addition, the predicted values of the ratio Pco/Po2 corresponding to the ignition locus on the stability diagram are in poor agreement with the experiment at high pressuires. (ii) The mean-field corrections (section 2), taking into account a weak coverage dependence of the rate constants for CO desorption and C O + O reaction under UHV conditions, also do not allow to describe the reaction kinetics at high pressures. (iii) If however one explicitly introduces into the model a strong repulsive nearest-neighbour CO-CO lateral interaction, the results predicted are reasonable both at low and high pressures. In particular, the calculated apparent activation energy and reaction orders are in the same range as the measured ones. The ignition locus on the stability diagram is also predicted at reasonable ratios Pco/Po2. In summary, the standard Langmuir-Hinshelwood scheme with proper choice of the coverage dependence of the activation energy for CO desorption allows in principle to describe the kinetics of CO oxidation
on Pt both at low and high pressures.
7. Conclusion Employing the CO oxidation on P t ( l 11 ) as a model system, we have studied in detail the effect of the coverage dependence of the rate constants of reactant desorption and of elementary reaction steps on the bistable kinetics of simple heterogeneous reactions. Our simulations show that the coverage dependence of the activation energy for reactant desorption reduces the bistable region in the stability diagram. The effect of the coverage dependence o f the activation energy for the reaction steps on the bistable region is opposite. The equistability lines in the stability diagram can be constructed by analyzing the propagation of chemical waves. We have derived the equistability criterion taking into account the coverage dependence of the reactant diffusion coefficient. The kinetic model employed in our study reproduces the main special features of the bistable kinetics of CO oxidation on Pt ( 111 ) measured by Ehsasi et al. [6 ] under UHV conditions provided that (i) the coverage dependence of the activation energies for CO desorption and CO oxidation is described using the realistic parameters .4 _ B -~ 4 kcal/mol, and (ii) the 02 sticking coefficient is equal to 0.14 at low coverages. This example shows that, studying special features o f the bistable kinetics, it is possible to obtain the relationship between various kinetic parameters, e.g. between reactant sticking probabilities.
Acknowledgements Financial support from the Swedish Research Council for Engineering Sciences (Contract 92951 ) is gratefully acknowleged. One of us (V.P. Zh. ) is also thankful to the Swedish Natural Science Research Council for supporting his stay at
V.P. Zhdanov, B. Kaaemo / Bistable kinetics of simple reactions
Chalmers University of Technology (Contract F-GF-3734-306). References [I ] P. Gray and S.IC Scott, Chemical Oscillations and Instabilities (Clarendon, Oxford, 1990); S.K. Scott, Chemical Chaos (Clarendon, Oxford, 1991); A. Okninski, Catastrophe Theory, Comprehensive Chemival Kinetics, Vol. 33, C.H. Bamford, C.F.H. Tipper and R.G. Compton, eds. (Elsevier, Amsterdam, 1992). [2]A.S. Mikhailov, Foundations of Synersetics I (Springer, Berlin, 1990). [3] G. Ertl, Adv. Catal. 37 (1990) 231; Science 254 (1991) 1750; F. Schiith, B.E. Henry and L.D. Schmidt, Adv. Catalysis 39 (1993) 51. [4] V.I. Bykov, V.I. Elokhin, A.N. Gorban and G.S. Yahlonskii, Kinetic Models of Catalytic Reactions, Comprehensive Chemical Kinetics, Vol. 32, C.H. Bamford, C.F.H. Tipper and R.G. Compton, eds. (Elsevier, Amsterdam, 1991 ). [5] J.W. Evans, Langmuir 7 (1991) 2514. [6] M. Ehsasi, M. Matloch, O. Frank, J.H. Block, IC Christmann, F.S. Rys and W. Hirschwald, J. Chem. Phys. 91 (1989) 4949. [7] R.M. Ziff, E. Gulari and Y. Barshad, Phys. Rev. Lctt. 56 (1986) 2553. [8] R.K. Herz and S.P. Matin, J. Catal. 65 (1980) 281; J.R. Creighton, F.-H. Tseng, J.M. White and J.S. Turner, J. Phys. Chem. 85 (1981) 703;
395
D.J. Kaul, R. Sant and E.E. Wolf, Chem. Eng. Sci. 42 (1987) 1399; M. Dumont, P. Duffour, B. Scnte and R. Dagonier, J. Catalysis 122 (1990) 95. [9] M.P. Harold and M.E. Garske, J. Catalysis 127 (1991) 524; 127 (1991) 553; Chem. Eng. Sci. 47 (1992) 623. [ 10] M. Bit, Ch. Ziilicke, M. Eiswirth and G. Ertl, J. Chem. Phys. 96 (1992) 8595. [ 11 ] K. Krischer, M. Eiswirth and G. Ertl, J. Chem. Phys. 96 (1992) 9161. [ 12] H.-P. Kaukonen and R.M. Nieminen, J. Chem. Phys. 91 (1989) 4380. [ 13 ] J.-J. Luque, F. Jinenez-Morales and M.C. Lemos, J. Chem. Phys. 96 (1992) 8535. [14] J. Satulovsky and E.V. Albano, J. Chem. Phys. 97 (1992) 9440. [ 15 ] V.P. Zhdanov, Elementary Physicochemical Processes on Solid Surfaces (Plenum, New York, 1991 ). [16] J.L. Gland and E.B. Kollin, J. Chem. Phys. 78 (1983) 963. [17] V.P. Zhdanov, Surf. Sci. Rep. 12 (1991) 183. [18] B.N.J. Persson, Phys. Rev. B. 40 (1989) 7115; Surf. Sci. Rep. 15 (1992) I. [19] L.V. Lutsevich, O.A. Tkachenko and V.P. Zhdanov, Langmuir 8 (1992) 1757. [20] C.T. Campbell, G. Ertl and J. Scgner, J. Chem. Phys. 73 (1980) 5862. [21] C.T. Campbell, G. Ertl, H. Kuipers and J. Scgner, Surf. Sci. 107 (1981) 220. [22] R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics (Wiley, New York, 1975) p. 304. [23] A.C. Luntz, M.D. Williams and D.S. Buthune, J. Chem. Phys. 89 (1988) 4381. [24] V.P. Zhdanov and B. Casemo, Appl. Surf. Sci., in press.
SSDI: 0167-2789(93)E0283-H
Bistable kinetics of simple reactions on solid surfaces: lateral interactions, chemical waves, and the equistability criterion V.P. Z h d a n o v a,b a n d B. K a s e m o b s Department of Applied Physics, Chalmers University of Technology, S-412 96 Gt~teborg, Sweden b Institute of Catalysis, Novosibirsk 630090, Russian Federation Received 24 May 1993 Revised mannseript received 25 August 1993 Accepted 3 September 1993 Communicated by Y. Kuramoto
In heterogeneous reactions, the rate constants of desorption, diffusion and elementary reactioh steps are usually strongly dependent on reactant covetagas due to adsorbate-adsorbate lateral interactions. We analyze the effect of this factor on the bistable regime of the reaction kinetics. As an example, we consider CO oxidation on Pt(l 11 ). The equistability lines in the bistable region for this reaction are calculated by analyzing propagation of chemical waves and taking into account the coverage dependence of the CO diffusion coefficient. The results of simulations are compared with the available experimental data. We show that it is possible to obtain the relationship between various kinetic parameters, for example, between CO and oxygen sticking probabilities, by studying special features of the bistable kinetics.
1. Introduction
Critical phenomena, such as kinetic phase transitions, chemical waves and oscillations, may take place in strongly nonequilibrium systems [ 1,2 ]. Physically, this behaviour is driven by the Gibbs-free-energy decrease of an overall chemical reaction. Mathematically, critical phenomena are connected with nonlinear features of kinetic equations describing physicochemieal rate processes. The interplay of chemical kinetics and diffusion may result in formation of chemical waves, inducing transitions from a metastable kinetic "phase" to a stable state, or, more generally, may cause the homogeneous state of the system to become unstable, leading to spontaneous formation of time-dependent or stationary spatial patterns. During the past decade, general ideas [ 1,2] formulated in this
field of chemical kinetics have been actively employed and developed in experimental studies and theoretical simulations of reactions at solid surfaces [3-5]. Particular attention has been paid to CO oxidation on Pt because this reaction exhibits bistable kinetics on the (111 ) face and bistable kinetics with oscillations on the (001) and (110) faces [3]. The bistable regime of CO oxidation on Pt(111 ) has been explored in detail by Ehsasi et al. [6] under ultrahigh vacuum (UHV) conditions (Po2 - 2.66× 10-9 bar, Pco <10 -9 bar and T = 340-680 K). To interprete their experimental data, they have employed the wellknown ZGB model [7]. This model is, however, not quite representative of the actual physical situation since it assumes that the reaction between nearest-neighbour reactants occurs instantly and that CO adsorption is strongly in-
016%2789/94/$07.00 © 1994- Elsevier Science B.V. All rights reserved
384
V.P. Zhdanov, B. Kasemo / Bistable kinetics of simple reactions
hibited by the presence of preadsorbed oxygen. These assumptions are not fulfilled for the system under consideration. The ordinary kinetic equations predicting bistable kinetics for CO oxidation on Pt were analysed in refs. [8-11 ]. In particular, B~ir et al. [10] have recently proposed a realistic kinetic model for CO oxidation on P t ( l l l ) (and on Pt (110) as well [ 11 ] ), taking into account the measured coverage dependences of the CO and 02 sticking coefficients and using the values for the rate constants of CO desorption and CO + O reaction, obtained at low coverages. The coverage dependence of these rate constants was ignored. The kinetic behaviour of the model has been studied in detail at Po2 = 2.66x 10 -7 bar, PCO ~ 5× 10 - s bar and T = 400-600 K but has not been compared with the experiment. The models employed in refs. [6] and [10] reproduce a sudden, steplike decrease in the reaction rate with increasing CO pressure but considerably overestimate the magnitude of this effect. The latter seems to be connected with neglecting lateral adsorbate-adsorbate interactions (see the results of our calculations presented below). Attempts to study the effect of lateral interactions on the bistable kinetics of CO oxidation on Pt have been made by Kaukonen and Nicminen [ 12 ], Luque et al. [ 13 ] and Satulovsky and Albano [14]. All three groups have employed Monte Carlo simulations. The treatment of lateral interactions used by Kaukonen and Nieminen [ 12 ] appears to be inconsistent from the physical point of view. The reason is the following. If one intends to incorporate lateral interactions into the model, it is necessary to distinguish between lateral interactions ofnonactivated adsorbed particles and lateral interactions in the activated state [ 15 ] (the terms "activated" and "nonactivated" correspond here to the language of the transition state theory). In the case of the adsorption/desorption process, for example, the former interactions contribute to the desorption rate but not to the ad-
sorption rate. The latter interactions affect both adsorption and desorption rates because the activated complex for adsorption and desorption is the same. Describing CO desorption, Kaukonen and Nieminen do not distinguish explicitly between these interactions. From the context of their paper it follows however that they are considering the interactions of nonactivatcd particles, because these interactions are used to calculate the distribution of nonactivatcd particles on the surface. Thus, these interactions should not affect the CO adsorption process, while Kaukohen and Nieminen incorporate just the same interactions into the CO adsorption rate constant. Under such circumstances, the physical meaning of the results obtained is not quite clear. The approach employed by Luque et al. [ 13 ] is principally more correct because generally they use different lateral interactions for CO adsorption and desorption. In their real calculations, they assume that the lateral interactions for adsorption have the same value but the opposite sign compared to that for desorption. This assumption does not reflect characteristics of the CO/Pt system. Indeed, the sticking coefficient for CO adsorption on Pt is well known to be close to unity and weakly dependent on coverage. The latter means that the lateral interaction in the activated state is very weak. Thus, to describe the CO adsorption/desorption process, we should incorporate lateral interactions only into the desorption rate constant. In addition, Laque et al. do not take into account CO diffusion while this process is very rapid compared to other elementary steps in CO oxidation. The same comments are applicable to the simulations by Satulovsky and Albano [ 14], where CO diffusion is ignored and lateral interactions are taken into account only for the activated state. In the present paper, we explore theoretically the bistable kinetics of CO oxidation on Pt ( 111 ) under UHV conditions, taking into account the coverage dependence of the rate constants for CO desorption, CO + O reaction, and CO diffusion. The mean-field model employed
V..P. Zhdanov, B. Kasemo / Bistable kinetics o f simple reactions
385
Table 1 Kinetic parameters [I0] for CO oxidation on P t ( l l l ) : /C are the rate constants for the reactant fluxes, s the sticking coefficients, ~, the preexponential factors, and E ( 0 ) the activation energies at low coverages. CO adsorption: 0 2 adsorption: CO desorption: C O + O reaction: CO diffusion:
kl = sco/Cco k4 = so2K:o2 k2 k3 D
K~co = 1.919 x I0 s s - l m b a r -1 /(:02 = 3.589 x 105 s - l m b a r - I t/co = 1.25 x 1015 s -1 ~'r = 1.645 x 1014 s -1 D O = 10 - a m2s -1
is described in section 2. The effect of various parameters on the position of the bistable region in the stability diagram is demonstrated in section 3. Chemical waves are discussed in section 4. The latter contains a derivation of the equistability criterion, taking into account the coverage-dependent coefficient for CO diffusion. In section 5, our calculated results are compared with experimental data measured by Ehsasi et al. [6]. Extrapolation of the results obtained under UHV conditions to high (atmospheric) pressures is briefly discussed in section 6.
2. Model
The well-established three-step LangmuirHinshelwood mechanism of CO oxidation on Pt (111 ) at high-vacuum pressure conditions contains reversible monomolecular CO adsorption, irreversible oxygen adsorption, and CO + O reaction between adsorbed species to form product CO2 molecules which desorb rapidly [3]. The corresponding kinetic equations are as follows [10]: d0co dt = klPco [ 1 - (0co/0~o) 2 ] - k20co
-k30coOo,
( 1)
d0o _ k4Po2 [1 - 0co/0~o - 0o/0~] 2 dt
-k30coOo,
(2)
where 0~o = 0.5 and 0 h =0.25 are the saturation coverages. Eq. (1) takes into account that the
sco=0.84 so2=0.06 E c o ( 0 ) = 3 4 . 9 kcal/mol E r ( 0 ) = 2 4 . 1 kcal/rnol E a i f ( 0 ) = 7 keal/mol
coverage dependence of the CO sticking coefficient is weak up to saturation (which is suggested to be due to a precursor mechanism of adsorption). The model also takes into account that preadsorbed CO inhibits dissociative adsorption of oxygen and that no such site-blocking effect is exerted by adsorbed oxygen on incoming CO molecules. Biir et al. [ 10 ] have employed eqs. ( 1 ) and (2) assuming that the rate constants for CO desorption, k2, and C O + O reaction, k3, are independent of coverage. In their simulations, they have used the measured rate constants corresponding to the low-coverage limit (table 1 ). Thus, the model [ 10] does not contain any adjustable parameters. The reliability of the measured rate constants is however open for discussions (as is usual in surface science). The preexponential factor and activation energy for CO desorption seem to be well established. The situation with the Arrhenius parameters for CO + O reaction is more complex because different studies report different values of these parameters (cf. refs. [ 10 ] and [ 16 ] and see the review [ 17 ] ). The results of simulations are however not too sensitive to the details of describing the CO + O reaction as long as this step is rapid compared to CO desorption. In our study, we will employ essentially the same model with the same kinetic parameters as in ref. [ 10] (some minor changes in the value of the oxygen sticking coefficient will be made and motivated in section 5, where we will discuss the experimental results). However, one additional important element is introduced, namely the coverage dependence of the above mentioned
386
V.P. Zhdanov, B. Kasemo / Bistable kinetics of simple reactions
rate constants. This is a primary new feature of the present work compared to the study of Bfir et al. [ 10 ]. Our preliminary inspection showed that at steady-state conditions the oxygen coverage is always low inside the bistable region in the stability diagram (because the reaction is rather rapid). Thus we do not need to introduce into the model the dependence of the rate constants k2 and k3 on oxygen coverage. On the other hand, the CO coverage is often large and accordingly we should focus our attention on the dependence of the CO desorption and CO + O reaction rate constants on 0co. On the Pt ( 111 ) surface, CO is adsorbed in ontop sites at all coverages up to saturation. In addition, bridge-site adsorption takes place above 0co = ½ [ 17-19 ]. The binding energy difference between the on-top and bridge sites is low (~_2 kcal/mol), and the CO-CO lateral interactions are repulsive [ 18,19 ]. In principle, all these facts can be described in the framework of the lattice-gas model as in ref. [ 19 ]. If, however, one takes into account not only CO adsorption but also oxygen adsorption occuring in threefold sites and CO + O reaction, this approach becomes too cumbersome. For this reason, we will in our simulations employ the mean-field approximation. In particular, the effect of lateral interactions and of bridge-site adsorption on the CO resorption and C O + O reaction rate constants will be described by a linear dependence of the activation energies on CO coverage, Eco (0co) = Eco (0) - AOco/O~o,
(3)
E r ( 0 C O ) = E r ( 0 ) - BOco/O~o.
(4)
The inspection of available TPD data for CO desorption (see refs. in [19]) shows that A __4-5 kcal/mol. We will use A = 4 kcal/mol. This value of A corresponds in fact to rather weak repulsive next-nearest-neighbour CO-CO lateral interactions. The strong nearest-neighbour COCO interactions can be taken into account under UHV conditions implicitly by introducing the
constraint 0co < 0~o as it has been done in eqs. (1) and (2) (for oxygen, by analogy, we have
0o _< 0~). The activation energy for the CO + O reaction on the Pt-group metals is known to decrease with increasing coverage [ 17 ]. The TPD data [ 16 ] indicate that B < 10 kcal/mol. Our analysis of the data [20] on titration of oxygen by CO has however shown that B < 5 kcal/mol. Below, the results are presented for B = 0 and 4 kcal/mol, respectively. It is also necessary to express the coverage dependence of the activation energy for CO diffusion, Edif (0CO) = Edif ( 0 ) -- COco~ 0~o,
(5)
because this dependence is significant for describing chemical waves and calculating the equistability lines (section 4). Experimental data allowing a reliable estimate of the parameter C are missing. We will use C = 0 or 4-3 kcal/mol. The repulsive CO-CO interaction excludes first-order phase transitions in the CO overlayer [18], and accordingly the CO islands are not formed during CO adsorption. The available TPD data for oxygen adsorption on Pt(111 ) (e.g., ref. [21]) indicate that the O - O lateral interactions are repulsive as well. Thus, if the surface is covered by oxygen, formation of oxygen islands is also negligible. The situation when both the CO and oxygen coverages are not low is not so clear, and in the latter case one cannot certainly exclude formation of islands. Such situations are however rare at the steady-state conditions. All these facts justify application of the mean-field kinetic equations. Note also that the positive parameters A = 4 kcal/mol and B = 4 kcal/mol introduced above correspond to rather weak repulsive adsorbate-adsorbate interactions (< 0.5-1 kcal/mol). In this case, the correlations in the adsorbate-adsorbate arrangenent are not too strong. This is an additional argument in favour of the mean-field approximation.
387
EP. Zhdanov, B. Kasemo / Bistable kinetics o f simple reactions
(a)
C o n c l u d i n g this section, we n o t e t h a t i f the rate c o n s t a n t k3 is d e p e n d e n t o n l y o n C O c o v e r age, eq. ( 2 ) c a n be easily s o l v e d at s t e a d y - s t a t e conditions. We then have
0o/0~- 1 -x
+ gx
,
r
°
<
-x
+ g x ) 2 - (1 - x ) 2 ]
1/2,
.
.
.
.
.
,
~,,
*%
>
-[(1
,
*
(6) 0.2
where x = 0co/0~o and g = k 3 0 ~ o O ~ / 2 k 4 P 0 2 . Thus, at steady-state conditions, we need in fact only to analyze eq. (1) in order to obtain CO coverage because oxygen coverage in this equation is dependent implicitly (via eq. (6)) on CO coverage.
%÷
0
ttttl
b)
6
. . . . ~ o ÷ - ~ . ~
ttttlli
i,
i
,J....
i
i
t
i
i
,
tO
t,
2O
2.'s
30
3,
40
,,,,,,,
_
,
-a,,
4,
~
.~0
,
Po2=2.66x 10-7 bar 7
S::
3. Bistable region and hysteresis At steady-state conditions, eq. (1) for CO coverage has one or three solutions. In the latter case, the intermediate solution (with respect to absolute coverage) is unstable and the other two solutions are stable. In particular, figs. 1-4 show CO coverage, reaction rate and stability diagrams calculated for various values of the parameters ,4 and B at Poe -- 2.66 x 10-7 bar (this pressure is the same as in ref. [10]). The coverage dependence of the activation energy for CO desorption is seen to reduce the extension in CO pressure of the bistable region (cf. figs. lc and 2c). The effect of the coverage dependence of the activation energy for CO + O reaction on the bistable region is o p p o s i t e (fig. 3b). Explanation of these results from the mathematical point of view is rather tedious because eqs. (1) and (2) with expressions (3) and (4) are strongly nonlinear. Physically, however, the effects obtained appear to be quite clear. In particular, a decrease of the activation energy for CO desorption with increasing coverage results in an increase of the CO desorption rate. This means that the reaction system tends in this case to an adsorption/desorption equilibrium. At equilibrium, however, the bistable region
: *+
\
o = 540 K
0
.<
2 520 K
•o
F
!
~_
•
÷**.. ,¢,f_~*°~50 K ........................ : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :..... : i
i
i
i
i
i
i
L
l0
15
20
25
30
3'
40
4'
'0
CO PRESSURE (10 -9 bar)
]
~o
PO2=2.66xi0 -T bar
J
8, I 0
400
420
440
460
48O
,00
520
540
,6O
TEMPERATURE (K)
Fig. I. (a) CO coverage and (b) reaction rate as a function of CO pressure, and (c) stability diagram (the bistable region is shadowed) at P02 = 2.66 x 10-7 bar. The results are calculated for A = B = 0, i.e. no coverage dependence of the rate constants for CO desorption and CO+O reaction.
KP. Zhdanov, B. Kasemo / Bistable kinetics of simple reactions
388
[a)
.
.
.
.
.
, ....
,
' .
:
......
:
.....
l
"-~
0.8
os
%%
~.
.++"
%¢¢
06
o.o
+e T = 5 2 0 K
~
~. r~ ¢2
T=52(
""
/~"
5
4
0
K
~
.
÷
% %
r~
+
04
%
*
0.2
... :
m
.
~
e
o,2
\
% .....%
o o
L ~
i
o
5
i
a
15
Io
i
i
i
20
25
30
h ........
35
J
~
40
45
..........
5O
C O P R E S S U R E ( 1 0 - " b*~r)
(b)
PO2=2.66x10-T b a r
35
b)
,'
-
,--
P o ~ : 2 . 6 6 x 10 -7
7
~,) 10
q'a°'iF L 5
0 5
I%
15
20
25
Lc)
~
]
30 35 40 45 50 C O P R E S S U R E (10 -9 b a r ) ,.
.
.
.
.
.
P o 2 = 2 . 6 6 x 10 -7 b a r '
~
~°
aa 1:~
15
040~t
420
440
460
480
500
520
5~0
TEMPERATURE
56O (K)
Fig. 2. As fig. 1 for A---4 kcal/mol and B = 0 , i.e. decreasing CO desorption energy and constant activation energy for C O + O reaction with increasing CO coverage.
is known to be absent at all. Thus, the coverage dependence of the activation energy (with A > 0) for CO desorption should suppress the bistability. An increase of the activation energy
oo
42o
440
460
48o
5oo
520
54~]
TEMPERATURE
5(~3 (K)
Fig. 3. (a) CO coverage as a function of CO pressure and (b) stability diagram at A--0 and B = 4 kcal/mol, i.e. constant activation energy for CO desorption and decreasing activation energy for CO + O reaction with increasing CO coverage.
for CO oxidation with increasing coverage leads the system far from an adsorption/desorption equilibrium, and in this case the bistable region of course increases. It is of interest that the cusp position in the stability diagram is almost independent of the coverage dependence of the activation energies for CO desorption and CO + O reaction (cf. figs. lc, 2c, 3b and 4c). This is because near the cusp point the reaction rate is maximum and both adsorbate coverages are consequently low. The latter allows us to obtain simple analytical expressions for the cusp position. In particular, lin-
V.P. Zhdanov, B. Kasemo / Bistable kinetics of simple reactions
(a)
k4Po2(1 - 2 0 c o / 0 ~ o - 2 0 0 ) - k 3 0 c o O o = O.
1
it
" - ~ o.8
.....
..........
o.6
r~
>
(8)
i
t
o
~
389
0.4
T=520 K
"..
©
......"
.
°
e02
~K ~
. i 5
i I0
i 15
l 20
215
i 30
~"~'~
-
f
~-~-
m
~,*~"'" 560 K i 35
I 40
415
Taking into account that the reaction rate conslant is large, we can conclude that one of the reactants should dominate on the surface. If oxygen dominates (i.e., 0o >> 0co), CO desorption is negligible compared to C O + O reaction, and one can rewrite eq. (7) and (8) as (9)
k l P c o - k30coOo = O, 50
k4Po2(1 - 200/00) - k30coOo = O,
(b) Po~=2.66 x 10.7 bar
(10)
or 00/05
g Z
kleco)/2k4e02.
(11 )
The latter equation shows that the oxygen domination is possible at k 4 P o 2 > kl Pco. If CO dominates on the surface, eq. (8) can be rewritten as
*
©
~--- ( k 4 P o 2 -
<
f***,,***" T=450 K I
k4Po2 (1 - 20co/0~o) - k30coO 0 = O. 5
15
10
20
25
2~0 ('()
35
35
40
PRESSURE
45
From (7) and (12), we obtain
( 1 ( ) -~' h a k )
c) Po2=2.66 x 10.7 bar
~
(12)
50
k4Po2(1 -20co/0~o) - k l P c o
3o
+ k 2 0 c o = O,
(13) or 0co = (k4P02 - klPco)/(2k4P02/O~o - k2).
~ lO
(14) 5
0 400
I 420
I 440
I 460
I 4g0
I 5~
i 520
i 540
560
TEMPERATURE (K) Fig. 4. A.~ fig. l for A = B = 4 k c a l / m o l , i.e. decreasing activation energies for C O d e s o r p t i o n a n d C O + O reaction with increasing C O coverage.
earizing the adsorption terms in eqs. ( 1 ) and (2) at smal 0, we have near the cusp point klPco - k20co - k30coOo - O,
(7)
The latter equation shows that at k4P02 > k]Pco we can construct an additional (unstable) solution to eqs. (1) and (2) provided that 2kAP02/O~. 0 > k2. The third (stable) solution is obtained only if one does not linearize the adsorption terms. The analysis presented shows that at a given oxygen pressure the cusp position takes place at P~o ~- k4Po2/kl
(15)
390
EP. Zhdanov, B. Kasemo / Bistable kinetics of simple reactions (a)
and 1
Po2=2.66x
2k4Po2/O~o = k2,
(16)
T=520
10 -7 bar
K
L
"-~ 0.3
...z
The latter equation can be rewritten as (we set o.6
kB=l)
I
<
T* =
Eco (0) In (vcoS~o/2k4Po2)"
( 17 )
Eqs. ( 15 ) and (17) indicate that the cusp position in fact does not depend on the reaction rate constant, as long as this constant is large compared to the adsorption rate constants. Concluding this section, it is of interest to note that in the bistable region one may in principle observe a hysteresis in the reaction rate and reactant coverages (fig. 5) as the CO/O2 pressure ratio is varied up and down.
p;
,-...
0.4
> ©
I I
I
© r.D
%
...., r ....,
r 1 I
0.2
I
i
i
5
10
15
| 20
25
i
I
I
I
30
35
dO
45
50
(b) [
Po2=2.66x10 T=520
-7 bar
K
P~
/ /n
/ ooO
4. Chemical waves and equistability criterion
I
g
1 f 1 I I I t
L-" , * °
ooo
If the temperature is not too high, the model presented by eqs. ( 1 ) and (2) predicts a bistable reaction regime and hysteresis at P~o < Pco < Pc°o (Pc°o and P~o are the critical CO pressures at a given temperature and oxygen pressure). In this regime, one stable solution corresponds to the surface covered predominantly by oxygen and the other to the surface covered predominantly by CO. These solutions are stable with respect to fluctuations of the gas-phase concentrations if we employ the mean-field approximation and do not take into account fluctuations of adsorbate coverages due to surface diffusion. Including the latter process results in the possibility of formation of chemical waves [ 2,5,10 ]. When considering the effect of surface diffusion, a new important parameter Pdo (P~o < P~o < P° o) should be introduced, corresponding to equistability of different "kinetic phases". In the bistable region, the oxygen-domination regime is absolutely stable at Pco < Pdo and metastable at P~o < Pco < p O . The CO-domination regime is metastable at P~x) < Pco < P~o and absolutely
1 I I I I
,oO
k T :
, 5
i I0
i 15
n 20
II
1
. . . . . . . . ,,, , , .....
1111
i
~
i
o
i
25
30
35
40
45
CO
PRESSURE
(10
50
~j b a r )
Fig. 5. (a) CO coverage and (b) reaction rate as a function of CO pressure for A = B = 0. Pointers show a hysteresis. The stepwise transition corresponding to the equistability criterion (solid lines) is calculated assuming that the CO diffusion coefficient is independent of coverage (i.e., C = 0).
stable at Pco __ P~o. ("Absolutely stable" here means stable not only with respect to fluctuations of gas-phase concentrations but also with respect to the influence of surface diffusion. ) The stability analysis of the mean-field solutions in the bistable region is known to be directly connected with calculating the velocity of chemical waves [2]. If we assume, for example, that at a given value of Pco (P~o < Pco < P° o) one part of the surface, initially covered predominantly by CO, and the other part, initially covered by oxygen, are separated by a linear interface, then with increasing time the interface will move from the CO-covered region
V.P. Zhdanov, B. Kasemo / Bistable kinetics of simple reactions
to the oxygen-covered region or in the opposite direction due to diffusion and reaction (this is a chemical wave). In the former case, the velocity of the chemical wave is positive, v > 0, the CO-domination regime is stable, and the oxygendomination regime is metastable. If v < 0, the situation is the reverse. The parameter P~o introduced above corresponds to v = 0. To calculate the velocity of a chemical wave, one must include surface diffusion into the reaction scheme. In our case, taking into account that oxygen diffusion is negligible compared to CO diffusion [ 3,10 ], we should solve the following equation:
-O0 ~
=
-O ~ D ( O ) -O0 ~
(18)
where 0 is the CO coverage (we omit here the subscript CO), D (0) the coverage-dependent CO diffusion coefficient, and W (0) the righthand part of eq. ( 1 ). Eq. (18) assumes that the oxygen coverage is given by the steady-state solution of eq. (6). This assumption is justified because the reaction is rapid compared to the wave propagation (see also ref. [10] ). A chemical wave, realizing the transition from one state to another and moving at a velocity v, is given by a special solution
0 = 0(~),
~ = x-vt,
(19)
that satisfies the relevant boundary conditions at = q:oo. Substituting expression (19) into eq. (18), multiplying the result by D (0) d0/d~, and then integrating the equation obtained from - c ~ to +c~ and taking into account that d 0 / d ~ = 0 in both limits, we derive
v =
Equistability occurs for v = 0. Thus, the equistability criterion can be written as 03
D(O) W(O) dO -- 0.
f~3D(O)W(O)dO f~_+°°D(O)(dO/cl~)2d~'
(21)
Ol
This equation defines implicitly the value of Pdo (through the dependence of W (0) on Pco) and indicates that P~o may be strongly affected by the coverage dependence of the diffusion coefficient. If D ( 0 ) = const., eq. (21) reduces to the following well-known result [ 2 ]: 03
f w(o) dO= + W(O),
391
0.
(22)
Ol
Eq. (21) (or eq. (22)) is an analog to the Maxwell rule employed in the theory of firstorder phase transitions [ 15,22 ]. The position of the equistability lines in the Pco-temperature plane for CO oxidation is shown in fig. 6. The full curves indicate the equistability lines in the case when the diffusion coefficient is independent on coverage (C = 0). The dashed lines are obtained at C = =1=3 kcal/mol. An increase of the diffusion coefficient with increasing coverage (C > 0) shifts the equistability lines to lower CO pressures. If the diffusion coefficient decreases with increasing coverage (C < 0), the effect is opposite. Finally, we note that if the creation and propagation of chemical waves are rapid (this is expected to be the case for CO oxidation on Pt( 111 )), one does not expect a hysteresis in the reaction rate. Instead, we should have a stepwise change in the reaction rate and coverage at the CO pressure Pdo, corresponding to the equistability criterion (fig. 5).
(20) 5. Comparison with experimental data
where 01 and 03 are the steady-state solutions to eq. ( 1 ) corresponding to the surface covered predominantly by oxygen or by CO, respectively.
As pointed out in the introduction, the bistable regime for CO oxidation on Pt (111 ) has been
V.P. Zhdanov, B. Kasemo / Bistable kinetics of simple reactions
392 a)
¢-
35
-t-
Po2=2.66× i0-~ bax
~-' ........ ' Experimental Data for
.'*?
'
CO+O Reaction on Pt ( 1 1 1 ) f at Pm:2.66x10 -0 bar / X \
¢
...
/-
.../'
'
~ \
"\,
667 K | ................. I "~3K] "'-q
*
Z ©
.7 /
(3
/ a \i
....T...=49! ........K......
5
o ,~O0
42O
46O
440
480
50~
520
540
560
CO PRESSURE (10-9
bar)
~ (b)
Fig. 7. Measured rate of CO oxidation on Pt(l I I ) as a function of the CO pressure (Ehsasi et al. [6]).
Po~=2.66x 10-7 bar
....:'i;F 2o
©
%
.
420
44O
(c)
46~
,
480
5O0
,
......
Po2=2.66x 10-7 bar
,
.
• •
52O
..'"
540
................
56O
,
~
..z /"
a
~o
oo
....*" o~
4O0
420
~4q
460
480
5O0
52O
5~0
TEMPERATURE (I()
Fig. 6. Stability diagram at (a) A = B = 0, (b) A = 4 k c a l / m o l a n d B = 0, a n d (c) A = B = 4 kcal/mol.
Dots confine the bistable region. The cquistability lines am calculated assuming that the activation energy for CO diffusion is described by eq. (5) with C = 0 (the central full curve), 3 kcal/mol (the lower dashed curve), and - 3 kcal/mol (the higher dashed curve).
expedmentaly explored in detail by Ehsasi et al. [6]. The data obtained at P o 2 = 2 . 6 6 x 10 -9 bar are shown in fig. 7. The corresponding results of our simulations, with the parameters presented in table 1, are displayed in fig. 8. Comparing figs. 7 and 8, two features of the bistable kinetics arc of particular intererst, namely, the stcpwise change in the reaction rate and the cusp position, respectively. If the coverage dependences of the CO desorption and C O + O reaction rate constants arc ignored, the theory overestimates the stepwise change in the reaction rate (cf. figs. 7 and 8a). If we take into account the coverage dependence of the CO desorption rate constant, the agreement between the simulation and the experiment becomes considerably better (cf. figs. 7 and 8b). If both the desorption and reaction rate constants are allowed to depend on coverage, the agreement is also good (cf. figs. 7 and 8c). The measured cusp position is at Pco ~- 8 x 10 -1° bar and T ~-555 K. The calculated values are Pco ~- 3.5 x 10 - l ° bar and T "=-495 K. The theory thus considerably underestimates the cusp pressure. This pressure is expected to depend only on the sticking coefficients of CO and 02 (cq. (15)). Taking into account that the value of the CO sticking coefficient is well established, we may conclude that the 02 stick-
V.P. Zhdanov, B. Kasemo / Bistable kinetics of simple reactions
393
0.14
o.o6(a) Po2=2.66xlO -° bar ~
0.12
0.05 s
o.1 0,04
i!
__
0.03
T=475 K ~ o.o6
N
~ 0.02 <
/i
O.Ol
420 K
~ 0.04
..................
......................../...........i2....2............................................................. ....
i
0.02 0
o
i
i
0.05
0.1
i
i
0.15
0.2
i
i
0.25
0.3
i
i
0,35
i
0.4
0.45
0.5
~ '
o.o6 (b) Po2=2.66x 10-0 bar 0.05
i
0.2
i
0.4
i
O.6
r
i
0,8 I CO PRESSURE (10 -9 bar)
Fig. 9. As fig. 7 for So2 = 0.14, A = 4 kcal/mol, B -- 0, and C = 3 kcal/mol.
7
0.04
ing coefficient (So2 = 0.06) presented in table 1 ....................does .......not . correspond to the experimental data
j
0.03 0.02 < 0.01
/
..................................
0 i
i
i
i
:
i
i
a
i
0.05
0.1
0,15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
o.= (c)
0.05
P°2=2"66x 1
0
-
~
"7 o,o4 ~
0.03
¢~ o.ol
o " 0.05 ,
...................
,
0.1
0.15
" - i ......... : ........ 0.2 0.25 0.3 0.35 0.4 0.45 0.5 CO PRESSURE (10 -9 bar)
Fig. 8. Reaction rate as a function o f CO pressure. The resuits have been calculated employing the kinetic parameters presented in table 1 (in particular, So2(0 ) -- 0.06). The coverage dependence o f the activation energies for CO desorption and oxidation is described respectively by the following parameters: (a) A = B ffi 0; (b) A ffi 4 kcal/mol, B ffi 0; (c) A = B -- 4 kcal/mol.
reported by Ehsasi et al. [6]. Under these circumstances, it is necessary to note that there is considerable disagreement between the literature data on the value and functional dependence upon coverage of the O2 sticking coefficient [23 ]. In particular, the initial sticking coefficient ranges from 0.02 to 0.2 in different studies. To reproduce the cusp position, we should use in our simulations So2 ( 0 ) = 0 . 1 4 (cf. figs. 7 and 9). In this case, however, the theory still underestimates a little the cusp temperature. This temperature is dependent first of all on the Arrhenius parameters for CO desorption at low coverages (eq. (17) ). From our point of view, the Arrhenius parameters presented in table 1 are reliable and it is not quite clear why the cusp temperature is underestimated. Perhaps this is connected with some minor contribution of deflect sites (e.g., steps) to the reaction kinetics. The influence of defects may also explain why the experimental changes in the reaction rate with increasing CO pressure are not exactly of the stepwise shape. The latter may in principle be also connected with nonequilibrium kinetic effects. Such effects however should result in a hysteresis in the reaction rate, but Ehsasi et al. [6] do not report a hysteresis.
394
EP. Zhdanov, B. Kasemo / Bistable kinetics of simple reactions
6. Comments on the "pressure-gap" problem The bistable kinetics of CO oxidation on Pt were observed not only under UHV conditions but also at high (atmospheric) pressures (see, e.g., the review in ref. [9] ). It is of interest to extrapolate to high pressures the results obtained under UHV conditions in the framework of the standard three-step Langmuir-Hinshelwood mechanism. Solving this "pressure-gap" problem (the details will be published elsewhere [24] ), we have arrived at the following conclusions. (i) If one employs in simulations the kinetic parameters obtained under UHV conditions but ignores the coverage dependence of the CO desorption and C O + O reaction rate constants (as in ref. [10]), the Langmuir-Hinshelwood scheme fails to describe the apparent activation energy and reaction order with respect to CO at high pressures. In addition, the predicted values of the ratio Pco/Po2 corresponding to the ignition locus on the stability diagram are in poor agreement with the experiment at high pressuires. (ii) The mean-field corrections (section 2), taking into account a weak coverage dependence of the rate constants for CO desorption and C O + O reaction under UHV conditions, also do not allow to describe the reaction kinetics at high pressures. (iii) If however one explicitly introduces into the model a strong repulsive nearest-neighbour CO-CO lateral interaction, the results predicted are reasonable both at low and high pressures. In particular, the calculated apparent activation energy and reaction orders are in the same range as the measured ones. The ignition locus on the stability diagram is also predicted at reasonable ratios Pco/Po2. In summary, the standard Langmuir-Hinshelwood scheme with proper choice of the coverage dependence of the activation energy for CO desorption allows in principle to describe the kinetics of CO oxidation
on Pt both at low and high pressures.
7. Conclusion Employing the CO oxidation on P t ( l 11 ) as a model system, we have studied in detail the effect of the coverage dependence of the rate constants of reactant desorption and of elementary reaction steps on the bistable kinetics of simple heterogeneous reactions. Our simulations show that the coverage dependence of the activation energy for reactant desorption reduces the bistable region in the stability diagram. The effect of the coverage dependence o f the activation energy for the reaction steps on the bistable region is opposite. The equistability lines in the stability diagram can be constructed by analyzing the propagation of chemical waves. We have derived the equistability criterion taking into account the coverage dependence of the reactant diffusion coefficient. The kinetic model employed in our study reproduces the main special features of the bistable kinetics of CO oxidation on Pt ( 111 ) measured by Ehsasi et al. [6 ] under UHV conditions provided that (i) the coverage dependence of the activation energies for CO desorption and CO oxidation is described using the realistic parameters .4 _ B -~ 4 kcal/mol, and (ii) the 02 sticking coefficient is equal to 0.14 at low coverages. This example shows that, studying special features o f the bistable kinetics, it is possible to obtain the relationship between various kinetic parameters, e.g. between reactant sticking probabilities.
Acknowledgements Financial support from the Swedish Research Council for Engineering Sciences (Contract 92951 ) is gratefully acknowleged. One of us (V.P. Zh. ) is also thankful to the Swedish Natural Science Research Council for supporting his stay at
V.P. Zhdanov, B. Kaaemo / Bistable kinetics of simple reactions
Chalmers University of Technology (Contract F-GF-3734-306). References [I ] P. Gray and S.IC Scott, Chemical Oscillations and Instabilities (Clarendon, Oxford, 1990); S.K. Scott, Chemical Chaos (Clarendon, Oxford, 1991); A. Okninski, Catastrophe Theory, Comprehensive Chemival Kinetics, Vol. 33, C.H. Bamford, C.F.H. Tipper and R.G. Compton, eds. (Elsevier, Amsterdam, 1992). [2]A.S. Mikhailov, Foundations of Synersetics I (Springer, Berlin, 1990). [3] G. Ertl, Adv. Catal. 37 (1990) 231; Science 254 (1991) 1750; F. Schiith, B.E. Henry and L.D. Schmidt, Adv. Catalysis 39 (1993) 51. [4] V.I. Bykov, V.I. Elokhin, A.N. Gorban and G.S. Yahlonskii, Kinetic Models of Catalytic Reactions, Comprehensive Chemical Kinetics, Vol. 32, C.H. Bamford, C.F.H. Tipper and R.G. Compton, eds. (Elsevier, Amsterdam, 1991 ). [5] J.W. Evans, Langmuir 7 (1991) 2514. [6] M. Ehsasi, M. Matloch, O. Frank, J.H. Block, IC Christmann, F.S. Rys and W. Hirschwald, J. Chem. Phys. 91 (1989) 4949. [7] R.M. Ziff, E. Gulari and Y. Barshad, Phys. Rev. Lctt. 56 (1986) 2553. [8] R.K. Herz and S.P. Matin, J. Catal. 65 (1980) 281; J.R. Creighton, F.-H. Tseng, J.M. White and J.S. Turner, J. Phys. Chem. 85 (1981) 703;
395
D.J. Kaul, R. Sant and E.E. Wolf, Chem. Eng. Sci. 42 (1987) 1399; M. Dumont, P. Duffour, B. Scnte and R. Dagonier, J. Catalysis 122 (1990) 95. [9] M.P. Harold and M.E. Garske, J. Catalysis 127 (1991) 524; 127 (1991) 553; Chem. Eng. Sci. 47 (1992) 623. [ 10] M. Bit, Ch. Ziilicke, M. Eiswirth and G. Ertl, J. Chem. Phys. 96 (1992) 8595. [ 11 ] K. Krischer, M. Eiswirth and G. Ertl, J. Chem. Phys. 96 (1992) 9161. [ 12] H.-P. Kaukonen and R.M. Nieminen, J. Chem. Phys. 91 (1989) 4380. [ 13 ] J.-J. Luque, F. Jinenez-Morales and M.C. Lemos, J. Chem. Phys. 96 (1992) 8535. [14] J. Satulovsky and E.V. Albano, J. Chem. Phys. 97 (1992) 9440. [ 15 ] V.P. Zhdanov, Elementary Physicochemical Processes on Solid Surfaces (Plenum, New York, 1991 ). [16] J.L. Gland and E.B. Kollin, J. Chem. Phys. 78 (1983) 963. [17] V.P. Zhdanov, Surf. Sci. Rep. 12 (1991) 183. [18] B.N.J. Persson, Phys. Rev. B. 40 (1989) 7115; Surf. Sci. Rep. 15 (1992) I. [19] L.V. Lutsevich, O.A. Tkachenko and V.P. Zhdanov, Langmuir 8 (1992) 1757. [20] C.T. Campbell, G. Ertl and J. Scgner, J. Chem. Phys. 73 (1980) 5862. [21] C.T. Campbell, G. Ertl, H. Kuipers and J. Scgner, Surf. Sci. 107 (1981) 220. [22] R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics (Wiley, New York, 1975) p. 304. [23] A.C. Luntz, M.D. Williams and D.S. Buthune, J. Chem. Phys. 89 (1988) 4381. [24] V.P. Zhdanov and B. Casemo, Appl. Surf. Sci., in press.