NonArrhenius behavior in the initial rate of a catalyticsurface reaction: Theory andMonte Carlo simulationKristen A. Fichthorn and Prakash G. Balan Citation: The Journal of Chemical Physics 101, 10028 (1994); doi: 10.1063/1.467991 View online: http://dx.doi.org/10.1063/1.467991 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/101/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Communication: On the origin of the non-Arrhenius behavior in water reorientation dynamics J. Chem. Phys. 137, 031101 (2012); 10.1063/1.4737390 On the non-Arrhenius behavior of negative-bias temperature instability Appl. Phys. Lett. 88, 093506 (2006); 10.1063/1.2180438 NonArrhenius rate constants involving diffusion and reaction J. Chem. Phys. 85, 7318 (1986); 10.1063/1.451371 NonArrhenius rate constants in complex reaction systems J. Chem. Phys. 83, 2280 (1985); 10.1063/1.449832 Viscoelastic Relaxation and NonArrhenius Behavior in Diols J. Chem. Phys. 44, 965 (1966); 10.1063/1.1726850

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Non-Arrhenius behavior in the initial rate of a catalytic-surface reaction: Theory and Monte Carlo simulation

Kristen A. Fichthorn Department of Chemical Engineering and Physics, The Pennsylvania State University, University Park, Pennsylvania 16802

Prakash G. Balan Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802

(Received 23 June 1994; accepted 25 August 1994)

We have identified the factors contributing to the compensation effect in the initial rate of a bimolecular surface reaction, the kinetics of which are influenced by adsorbate lateral interactions. A simple theory, based on the quasichemical approximation, can predict the temperature ranges over which compensation is the most pronounced in Monte Carlo simulations of the initial rate. Both the simulations and the theory reveal an interesting phenomenon-apparent negative activation energies, which occur when the activation energy for reaction increases with increasing temperature faster than k BT. This phenomenon could contribute to experimentally observed decreases seen in the rate of the CO oxidation reaction on several single-crystal metal surfaces. © 1994 American Institute of Physics.

I. INTRODUCTION

The influence of adsorbate lateral interactions on kinetic processes at surfaces has been of continuing interest in sur­face science. In many experiments, the influence of adsorbate interactions on kinematic elementary steps, such as thermal desorption, surface diffusion, and chemical reaction, is un­derstood through. the framework of transition-state theory. From transition-state theory, the rate coefficient k for an el­ementary step is given by the form

(1)

in which {3=(kBT)-I, Vo is the pre-exponential factor, and E is the activation energy given by the difference between the minimum (initial state) and the saddle-point (transition-state) energies along the minimum-energy path between the initial and final states. Adsorbate-adsorbate interactions alter the adsorbate-substrate potential-energy surface and, hence, are. expected to alter the activation energy and pre-exponential tenns of the rate coefficient. Generally, adsorbate-adsorbate interactions are limited in range so that the binding energy of an adspecies is influenced only by its surrounding local en­vironment (i.e., nearest, next-nearest, etc., neighbors). In an adlayer, the configuration of adsorbed species is not gener­ally uniform and many different local environments are pos­sible. The net rate coefficient K of an elementary step under these circumstances can be viewed as a distribution of indi­vidual rate coefficients arising from each possible local en­vironment, i.e., as

(2)

where fi is the mole fraction of adspecies with a local envi­ronment of type i and a rate coefficient ki , given by

(3)

Hence, one possible influence of adsorbate-lateral interac­tions upon the kinetics of surface phenomena is that, at fixed temperature, the macroscopic Arrhenius parameters (E and vo) do not reflect the unique values indicated by Eq. (1), but, instead, they reflect the distribution given by Eq. (2).

A significant consequence of adsorbate lateral interac­tions for surface-reaction kinetics has been demonstrated in recent Monte Carlo simulation studies by Kang and co-workers. I Using a lattice-gas model with pairwise­additive adsorbate interactions, they simulated the initial rate of a surface reaction between a species B and a preadsorbed overlayer of species A. They found that changes in the dis­tribution of rate coefficients with temperature led to nonlin­ear Arrhenius plots. The apparent (macroscopic) activation energies obtained from the plots did not accurately reflect the microscopic values built into the model. Although their model assumed that the pre-exponential factor was constant and independent of the local configuration of adsorbed reac­tant, the apparent pre-exponential factor varied by over 11 orders of magnitude. They found that changes in the appar­ent activation energy and pre-exponential factor were com­mensurate, exhibiting a compensation effect. A similar effect was seen by Fichthom and Weinberg2 in a Monte Carlo simulation study of the steady-state reaction rate in a model of the CO oxidation reaction.

Over the years, there have been many attempts to ex­plain the compensation effect?-ll There have been recent reports of this phenomenon in experimental studies of surface-reaction kinetics/2

-14 as well as thermal

desorption. Is-The findings of Kang et al. l could present a plausible explanation for some of the experimental results, since this study can explain very large variations in the pre­exponential factor. An interesting aspect of their study was that the compensation etIect could be seen only for certain sets of adsorbate lateral-interaction parameters. Considering the strong implications of these findings for the interpretation of surface rate data, it is clearly desirable to have a means of

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K. A. Fichthorn and P. G. Balan: Rate of a catalytic-surface reaction 10029

discerning the conditions under which such a phenomenon could occur in an experimental system. In this paper, we investigate the factors contributing to non-Arrhenius behav­ior in the rate of a bimolecular (heteronuclear) surface reac­tion. We derive a relationship which provides a simple indi­cator of the compensation effect in the model reaction and compare the predictions of the indicator to Monte Carlo simulations.

II. MODEL

The experimental situation which we consider is one in which initial rates are measured in a bimolecular surface re­action, analogous to the situation probed by Kang et al. 1 A reactant gas of species Ary, with a pressure P Ao' is introduced into the system and allo~ed to equilioiate with the catalyst metal surfaceh A2 adsorbs dissociatively and is present on the surface as atomic A. When A2 has established equilibrium with the surface, gas-phase A2 is removed from the system and species B, with a pressure PB, is introduced. Species B adsorbs on the surface and reacts with species A to form product AB. If the rates of B adsorption, desorption, and surface diffusion are large compared to the rate of A - B sur­face reaction, and if the rate of surface reaction is large com­pared to the rates of A surface diffusion and A2 recombina­tive desorption, then species B will rapidly equilibrate with respect to an essentially "frozen" A adlayer and surface re­action will be the rate-limiting step. Such a scenario may be possible in the CO oxidation reaction on platinum surfaces. 16

In the simulation study of Kang and co-workers,1 species A was modeled as a square-lattice gas with nearest and next­nearest neighbor repulsive interactions. The configuration of species A in the preadsorbed layer was established at a fixed temperature and fractional surface coverage e A with the Me­tropolis Monte Carlo algorithmY Subsequent to equilibra­tion of species A, the adlayer configuration was frozen and the adsorption of species B was simulated. At a given vacant site i, the probability of B adsorption PB,I was obtained by a balance between the rates of adsorption and desorption, given by

aeEd•if3

PB,t E j3 1 +ae d,i • (4)

In this expression, a is the ratio of the flux of B to the surface to the pre-exponential factor for desorption and the activation energy for desorption is given by

(5)

Here, N A is the number of nearest-neighbor A:s to site i, Ed,o is the activation energy for the desorption of an isolated B, and CAB is the nearest-neighbor A-B interactio,J+ strength. Once the overlayer configuration was established, the overall reaction rate was calculated from an expression of the form

(6)

where (JAB is the fractional surface coverage of reactant (A-B) pairs and ii is the fractiOn of reactant pairs whose local elwu'onm.ent of adsorbed species dictates (through ad-

sorbate lateral interactions) an activation energy for reaction, Er,i' Accounting for A-A and nearest-neighbor A-B inter­actions, the activation energy for a given local environment is given by

Here, E r,O is the activation energy for the reaction of an isolated A-B pair, N AA is the number of nearest-neighbor A:s to A, N AA.n is the number of next-nearest-neighbor A's to A, NAB and NBA are the number of nearest-neighbor B's to A and A:s to B, respectively, and EAA, EAA n' and EAB are the associated lateral interaction energies. . ...

At a fixed coverage of A, changes in the configuration of A, the fractional surface coverage of B, and the configuration of B cause the distribution of Eq. (6) to change with tem­perature. This, in tum, can lead to nonlinear Arrhenius plots and compensation. Although compensation seen in the appar­ent (Arrhenius) activation energy should rigorously be iden­tified with changes in the distribution of Eq. (6) with tem­perature, Kang and co-workers 1 were able to interpret the observed compensation in their results in terms of changes in the mean activation energy. In our initial etIort, we derive a simple indicator of compensatory trends in bimolecular .surface-reaction kinetics, which is based upon the tempera­ture dependence of the mean (number-average) activation energy for reaction.

Assuming nearest-neighbor A-A, B-B, and A-B adsor­bate lateral interactions, a mean activation energy· for reac­tion (E r) is given by

(Er) = Er,o"'" (N AA)EAA -(NBB)EBB

- (N AB)+(NBA»EAB' (8)

Here, (Nij), i,j=A or B, reflects the average number of nearest neighbors, with subscripts as given by Eq. (7). An indicator of a compensatory trend in experimental data is a change in (E rl with temperature. The dimensionless deriva­tive is given by

a(Er) a(NAJ' a(NBB) a(1/ f3) = ~ EAA a(1/ f3) EBB a(1/ f3) ..

( a{ NAB) a( N BA) )

- € AB a( 1/ (3) + a( 1/ f3) . (9)

The quantities (N AA), (N BB)' (N AB)' and (N BA) and their de­rivatives can be calculated via various approximations. In this work, we use the quasichemical approximation. 18 The quasichemical approximation and several variants have been extensively used by Zhdanov l9- 24 and others25- 28 in theoreti­cal studies of surface-reaction kinetics. However, to our knowledge, our use of the quasichemical approximation to derive an indicator of the compensation effect for surface­reactfon kinetics is novel. In the quasichemical approxima­tion, pairs of sites on the surface are assumed to be indepen­dent of one another. Considering only A's and vacant sites comprising the preadsorbed A layer, pairs can consist of two

ft:.s, which can be found with a probability of P AA, an A and

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10030 K. A. Fichthorn and P. G. Balan: Rate of a catalytic-surface reaction

a vacant site, with a probability P AO' and two vacant sites, which have a probability of P 00' The relationships between pairs of sites are given by

P AA + P AO+ Poo= 1,

2P AA + P Ao=2()A,

(10)

(11)

and the quasichemical "equilibrium" approximation, given by

PAAPOO p2. =0.25exp(-EAAf3).

AO (12)

From Eqs. (10)-(12), we obtain an expression for each of the probabilities.

Considering the various components comprising the mean activation energy, we have

(13)

and

3P AO

(N BA) = 2 ( 1 - () A) . (14)

Equation (14) reflects the fact that the configuration of A's surrounding the B. of a reactant pair is established in the preadsorbed A layer and that this configuration cannot relax with respect to the configuration of B. To obtain (NAB)' we note that (N AB) = 3 ()B and we use a pair approximation for Bg, in which

(3) ( P AO ) i ( P 00 ) 3 - i

i 2(l-()A) (l-()A) PB,i'

Here, PB,i is given by Eq. (4) with

Ed,i=Ed,o-(i+ l)EAB'

(15)

(16)

Equations (15) and (16) give the average probability to place a gas-phase B on a vacant site adjacent to the A of an A-B reactant pair. The average encompasses each possible envi­ronment (e.g., 4 Ns, 3 Ns and one vacant site, etc.) that could surround the vacant site. Since we are considering A-AandA-B interactions only, analogous to the situation in the study of Kang,1 938=0 and (NBB) does not contribute to Eqs. (8) and (9).

Figure 1 shows a plot of (N AA)' (NAB)' and (NBA) for both attractive (EAB<O) and repulsive (EAB>O) A-B interac­tions at a low [Fig. lea)] and a high [Fig. l(b)] coverage of preadsorbed A. To attempt a comparison of our predicted compensation to the simulation results of Kang et al., I we used their values, a= 1 X 10 -26 and E d,O = 36.0 kcaVmol, in our calculations. We also used a nearest-neighbor repulsive A-A interaction of EAA =2 kcal/mol. Figure I shows that the variation in (N AA) and (N BA) is very small over the tempera­tures probed. Variations in (Er) with temperature are clearly dominated by the temperature dependence of (NAB)' At low temperatures, (N AB)-d-there is a B adsorbed on every available site next to an A. As the temperature increases, B's desorb from the surface and their coverage approaches zero

2.5 -t-NM -NBA •••••• NAB' EAB = -2 kcall mol --'- NAB' EAB = -1 kcall mol --- NAB' EAB = 1 kcall mol •• -._. NAB' EAB = 2 kcall mol

~ 2.0 .0 J=. .12> (I) z '0 Q; .0 E ::J z

(a)

!!! .8 J=. C> 'a; z '0 Q; .0 E :::> z

1.5

1.0-

0.5-

0.0 200 250

5

4

3

300 350 400 450 T. K

-t-NM -NBA •••••• NAB' EAB = ·2 kcall mol I­._ .... NAB' EAB = -1 kcal Tmol --- NAB' EAB = 1 kcall mol ...... NAB' EAB = 2 kcall mol

I-

1--...,,,\ 'I.,

\ \ "'\ ....•..... \ 2

1 -

\ \ \ \ ~ \ \. \ ~

\. \ -_-\-\ ---'\\ :----\ \ ~\ ~ \

O~ __ ~···~-.... ~.~,~.-=-=---= .. -~-=-=~·~ .. -==~~ 200 250 300 400 450

(b) 350

T. K

FIG. 1. The various terms contributing to Eq. (8). as a function of tempera­ture for both attractive (EAB= 1, -2) and repulsive (EAB= 1,2) interactions at (a) 8A=0.1 and (b) 8A =0.5.

in the high-temperature limit. Therefore, compensation in the initial rate of reaction is due to changes in the surface cov­erage of B with temperature that influence the activation en­ergy for reaction through the A - B interaction.

Differentiating (N AA), (NAB), and (N BA) with respect to temperature and substituting into Eq. (9) yields the compen­sation indicator. Nonzero values of the compensation indica­tor reflect changes in (Er) with temperature and indicate the conditions under which compensation can be seen. Figure 2 shows a plot of Eq. (9). for attractive (EAB<O), repulsive (EAB>O), and zero A-B interactions. at low [Fig. 2(a)] and high [Fig. 2(b)l coverages of pre adsorbed A. We see, from Fig. 2. that the temperature range over which compensation can be expected [nonzero a(Er )/a(lI/1)] is dependent on the magnitude and sign of the adsorbate lateral interactions. Compensation is predicted to occur at the highest tempera­tures with strong, attractive interactions (EAB=-2 kcallmoI). The temperatures. at which compensation is expected, de-

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K. A. Fichthorn and P. G. Balan: Rate of a catalytic-surface reaction 10031

ft) ,

(a)

en.

~

ft)

w~

"" ft)

(b)

100

50

o ------.. r..::-:.:.: .......

-50

-100

200

100

50

0

-50

-100

200

" ,

250

250

\ , \ , \ , " \

\~ ... \ \

300

T, K

300

T, K

- EAB = ·2 kcall mo ""-"" EAB =·1 kcall mol ••••• EAB = 0 ._- "AB = 1 kcall mol .... - EAB B 2 kcall mol

350 400

EAB = ·2 kcall mo -EAB=·1 kcal/mol •••••• eAB = 0 ._-. eAB = 1 kcall mol .•.••• EAB = 2 kcall mol

350 400

FIG. 2. The compensation indicator, given by Eq. (9), as a function of temperature for (a) 8A =0.1 and (b) 8A =0.5 with attractive and repulsive interactions.

crease with increasing tAB, with the exception of tAB =0,

which shows no compensation. These features can be ex­plained by the influence of adsorbate lateral interactions upon the activation energy for the desorption of B (cf. Fig. 1). Compensation occurs over temperature ranges where (NAB) is changing the most rapidly. With larger Ed' that arise from attractive adsorbate interactions, the temperature re­gime, over which (N AB) is rapidly changing, is shifted to higher temperatures. In the case of tAB=O, compensation is due to changes in (N AA) only. Since this change is negligible (cf. Fig. 1), no compensation is predicted.

Another interesting feature in Fig. 2 is that the indicator develops an asymmetric shape, with peaks and shoulders, as the A-B interaction becomes either strongly attractive or

strongly repulsive. This effect is the most pronounced for tAB =2 kcal/mol, where peaks and/or shoulders can be ob­served at four different temperatures. We assign, to these peak temperatures, the four different environments that a B, neighboring the A of a reactant pair, can have. At low A coverages, a B adjacent to an A is likely to have no or one additional A neighbor. As the A coverage increases, the B is likely to have an increasing number of A neighbors. For re­pulsive A-B interactions. B will fill sites with the most A neighbors at the lowest temperatures. The shifting of the peak and shoulder positions to lower temperatures with in­creasing ()A in Fig. 2 is consistent with the filling of sites with an increasing number of A neighbors. For smaller mag­nitudes of the A-B interaction, there is less of a difference between the energies of B's at sites of different A coordina­tion. In this case, the peaks assume a more symmetric shape, which is consistent with only one type of binding site.

Considering the effect of surface coverage, a comparison between Figs. 2(a) and 2(b) shows that the temperatures, over which compensation is predicted, become higher for attractive interactions and lower for repulsive interactions as the coverage of A increases. This effect can be attributed to the effect of A neighbors on the activation energy for the desorption of B. At higher coverages of A, Ed is higher for attractive interactions and lower for repulsive interactions, causing B to desorb from the surface at higher and lower temperatures, respectively.

The accuracy of the model predictions can be assessed by comparison to initial reaction rates measured from Monte Carlo simulations, which are exact. We compare our results to those found in the study of Kang et al. 1 A difference be­tween our theory and their simulations is that the simulations used both nearest- and next-nearest-neighbor A-A interac­tions, while we consider only nearest-neighbor A-A interac­tions here. Since we find that the A-A interaction has a neg­ligible effect on compensation and since their study was conducted under conditions for which the A overlayer was a disordered lattice gas, we conclude that differences between the type of A-A interaction will not significantly influence the comparison. In the simulation study, compensation was found and analyzed most extensively for tAB = -2. No sig­nificant compensation was seen for repulsive adsorbate lat­eral interactions (tAB>O) or for tAB=O. Since their study focused on the temperature range between 300 and 400 K, our predictions are consistent with their results. Inspection of Fig. 2 shows that the model does not predict significant com­pensation between 300 and 400 K for tAB>O. However. compensation is predicted in the temperature range of their study for attractive adsorbate interactions. Figure 2 shows only negligible compensation (due to the A-A interaction) for tAB=O, consistent with their findings. Hence. the model can predict all of the trends seen in the study of Kang and co-workers. 1

It is of interest to determine the extent to which the predictions of our model are quantitatively consistent with an exact, Monte Carlo simulation. For this purpose, we used a modified version of the algorithm of Kang et ai., 1 described above, to determine the initial rate of a bimolecular surface reaction. As described in Eqs. (4)-(7), the parameters of the

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10032 K. A. Fichthorn and P. G. Balan: Rate of a catalytic-surface reaction

model are the activation energy and pre-exponential factor for reaction of an isolated A-B pair, which we took as E r ,0=22 kcaVmol (Ref. 1) andyo= 1. We chose a constant pre-exponential factor so that we could focus only on changes on the rate due to changes in the distribution of Eq. (6). The exact value of Vo is of no consequence to our results. For the desorption of B [cf., Eqs. (4) and (5)], we used a=8.49XlO- 18 and Ed ,o=25.0 kcaVmol. These are plausible values for the desorption of CO from Pt(100).16 The value of a is consistent with PB=2.3XlO-8 Torr, an accessible labo­ratory value. Simulations were run on square, 128X 128 lat­tices with periodic boundary conditions. Beginning with equilibrated lattices containing A at OA=O.l, 0.2, 0.3, and 0.4, simulations were run at 14 different temperatures be­tween 200 and 460 K. At a fixed temperature and coverage of A, we determined the initial rate of reaction for tAB = - 2, 0, and 2 kcaVmol. We obtained the ,rate coefficient k from the relation

k=r/OAB, (17)

where r is given by Eq. (6). Overall rate coefficients, for a fixed temperature and surface coverage of A, were calculated as an average over 200-1500 configurations of A.

Our simulations differ from those of Kang et al. 1 in two ways. First, we used only nearest-neighbor repulsive A-A interactions with tAA =2 kcaVmol. The impetus for using only nearest-neighbor interactions was to facilitate compari­sons to theory. Another difference between the two simula­tions is that Kang et al. used a mean-field approximation to calculate NAB in Eq. (7) from the probability [given by Eq. (4)] of adsorbing a B on sites neighboring the A. This pro­cedure can produce a noninteger NAB' which could not occur physically. Here, we prefer to use the exact procedure, in which B's are actually adsorbed onto the simulation surface and NAB is counted for each reactant pair. In our method, the rate coefficient, for a given, fixed configuration of A, is cal­culated as an average over 1000 Monte Carlo steps, in which B is adsorbed on or desorbed from sites on the surface ac­cording to a probability given by Eq. (4).

Arrhenius plots of In(k) vs liT are shown in· Fig. 3 for tAB = -2.0 kcaVmol [Fig. 3(a)] and tAB=2 kcal/mol [Fig. 3(b)]. Nonlinearity of the Arrhenius plots can be seen for both cases. For repulsive interactions, the slopes are indica­tive of negative activation energies, even though none of the individual activation energies comprising the distribution [ef. Eq. (6)] is negative. We obtained the slopes and intercepts of the plots, considering successive pairs of points, at each (jA'

The results are shown in Figs. 4 and 5. From these figures, it can be seen that the activation energies and pre-exponential factors vary in a commensurable way, exhibiting the com­pensation effect. Although we have implicitly assumed that the pre-exponential factor is constant in our definition of the rate [cf., Eq. (6)], the apparent pre-exponential factors, ob­tained from Arrhenius plots, vary by over 60 orders of mag­nitude.

The source of compensation in this system is a change in the distribution of activation energies comprising the rate. Much of the observed behavior can be explained in terms of

10.10

10.12

10.14

.>t! 10-16

10.18 -0- SA ~ 0.1 ..... SA=O.2 -0- SA ~O.3 ...... SA ~ 0.4

10.20

2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.8x10·3

(a) 1fT

10.10

10.11

10.12

.x:

10.13

-0- ijA = 0.1 ..... SA~0.2

10.14 ...... SA = 0.3 -0- SA = 0.4

10.15

2.5 3.0 3.5 4.0 4.5 5.0x10-3 (b) 1fT

FIG. 3. Arrhenius plots of the rate coefficients, obtained from the Monte Carlo simulations at the four surface coverages of A probed in this study, vs, liT for (a) attractive A-8 interactions and (b) repulsive A-8 interactions.

changes in the mean (number average) activation energy, given by

LfE· (E )=' r,t r 2: Iz ' (18)

where fl is the fraction of A.,-B pairs with an activation en­ergy for reaction Er i' We calculated (Er> in the simulations and a plot is shown'in Fig. 6. It can be seen that (E r) varies significantly with temperature for both attractive [Fig. 6(a)] and repulsive [Fig. 6(b)] interactions. Taking deviations of the pre-exponential factors, shown in Fig. 5, from zero to be representative of compensation, we see that compensation occurs when there is a change in (E r) .with temperature. It is also. evident that the activation energies, obtained from the

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K. A. Fichthorn and P. G. Balan: Rate of a catalytic-surface reaction 10033

50x103

40

(5 30 E

nl ~

uJ 20

10

0

(a)

30

20

10 (5 E

nl 0 ~ -w-

-10

-20

-30x103

(b)

300

250

350 T, K

300 T, K

-o-IJA =0.1 .... eA=0.2 -0- SA= 0.3 -+- 9A = 0.4

400 450

-0- SA = 0.1 -II- SA = 0.2 -0- eA = 0.3 -+- SA = 0.4

350

EG. 4. Activation energies, obtained from the slopes of the Arrhenius plots of Fig. 3 at the four surface coverages of A probed in 'this study, for Ca) attractive A-B interactions and (bY repulsive A-B interactions.

Arrhenius plots shown in Fig. 4, exhibit the largest devia­tions from (Er) (shown in Fig. 6) over the regions where (Er) is changing the most rapidly.

A unique situation exists for repulsive €A13' in which negative activation energies are implied by the Arrhenius plot [Figs. 3(b) and 4(b)], but not by the mean activation energy [Fig. 6(b)]. For repUlsive €AB, there is an increase in the mean activation energy with increasing temperature. At a fixed activation energy, the transition-state-theory rate coef­ficient, given by Eq. (1), increases with increasing tempera­ture. In the case of repulsive interactions, the increase in (E r) with increasing temperature tends to decrease the value of k. With two opposing factors (increase in (E r ), decrease in 13), the rate coefficient could increase with increasing tempera­ture, if the decrease in 13 outweighs the increase in (Er ), or decrease with increasing temperature, if (E r) increases more

40

30

20

0 10 > --> E 0

-10

-20

-30

250 (a)

20

0

-20 >0

>

.s -40

-60

·80

200 (b)

300

250

350

T,K

300 T, K

'-o-IJA = 0.1 .... SA,,0.2 -o-,eA = 0.3 -+- SA" 0.4

400 450

-0- IJA = 0.1 .... eA z O.2 -0- SA = 0.3 -+- 9A = 0.4

350 400

FIG. 5. Pre-exponential factors, obtained from the y intercepts of the Arrhenius plots of Fig. 3 at the four surface coverages of A probed in this study. for Ca) attractive A-B interactions and (b) repulsive A-B interactions.

strongly than 13 decreases. A plot of the product (Er)f3 vs temperature is shown for repulsive A-B interactions' in Fig. 7. At high and low temperatures, when the coverage' of B does not vary significantly with temperature, the decrease in 13 with increasing' temperature governs the rate coefficient and the Arrhenius plot indicates positive activation energies [cf., Fig. 3(b)]. However, for' the intermediate temperature range, over which B is desorbing from the .surface, (E r) is increasing faster than 13 is decreasing, leading to a decrease in k with increasing temperature and negative, apparent ac­tivation energies.

A decrease in the steady-state reaction rate with increas­ing temperature has been observed in the CO oxidation' re­action on several metal surfaces. 16 Of course, the kinetics of tIte reaction at steady state are different from those that we have investigated in this study. For example, in a steady-state

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10034 K. A. Fichthorn and P. G. Balan: Rate of a catalytic-surface reaction

24x103

22

20 "0 E

~ 18

-'"

'" 16

w~ 0

• 14 ~

12

10 200

(b)

• 0

• • 0

C

250

0

• 0

300

T, K

o I:iA =0.1 • SA = 0.2 o 8A =O.3 • SA=O.4

350 400

FIG. 6. The number-averaged activation energy, given by Eq. (18) and obtained from the simulations at the four surface coverages of A probed in this study, as a function of temperature for Ca) attractive A-B interactions and (b) repulsive A-B interactions.

reaction, the kinetics of both reactant adsorption and surface reaction can influence the overall rate of reaction and surface reaction may not be the rate-determining step. In the experi­mental studies,16 the decrease in the reaction rate with in­creasing temperature was attributed to the desorption of CO from the surface, which decreases the fraction of reactant pairs and lowers the rate [cf., Eq. (6)]. This phenomenon also occurs in our simulations, since B desorbs from the surface as the. temperature increases. Our results indicate that the decrease in the reaction rate with increasing temperature could be due to both a decrease in CO coverage (which de­creases r) and an increase in the activation energy for reac­tion (whi~h decreases k and, hence, r).

Utilizing parameters appropriate for our Monte Carlo model, we calculated the compensation indicator, given by Eq; (9). With a slight adjustment, the indicator can be made

250 300

T, K

-0- eA = 0.1 ___ SA = 0.2

-0- SA= 0.3 --- SA = 0.4

350 400

FIG. 7. The product (Er)f3, where (Er> is given by Eq. (18) and obtained from the simulations, vs temperature, at the four surface coverages of A probed in this study, for repulsive A - B interactions.

to reflect the deviations of the pre-exponential factor from vo, so that more direct comparisons are possible to the pre­exponential factors shown in Fig. 5. Assuming, as we have so far, that the distribution of Eq. (6) can be represented by an effective rate coefficient containing the mean activation energy, Le.,

(19)

the pre-exponential factor, obtained from an Arrhenius plot, is given by

inC v)cx In(k) - f3a InCk)! a{3, (20)

which gives

In( vi vo)cx f3 2aEr(f3)1 a{3== - aEr(f3)1 a(11 13). (21)

Hence, the negative of the indicator in Eq. (9) should be directly proportional to In(vlvo). A plot of the modified indi­cator, for both attractive and repulsive interactions, is shown in Fig. 8. Several of the trends in Fig. 8 match those of the pre-exponential factors from the simulations, shown in Fig. 5. It is clear, from a comparison of the theory and simulation, that the theory can predict the temperature range at which compensation is the strongest. As discussed above, compen­sation is expected to occur at higher temperatures for attrac­tive interactions than for repulsive interactions, true to the trends in Fig. 5. Figure 5 also shows that the compensation effect is the most pronounced at low coverages for repulsive interactions, in agreement with Fig. 8. Hence, the indicator can predict much of the observed behavior.

There are, however, important discrepancies between the predictions _of the indicator and the simulations. The most notable of these is that compensation occurs in the simula­tions at temperatures over which the indicator predicts that compensation should be negligible. These deviations occur at low temperatures for repulsive interactions and, to a lesser extent, at high temperatures for attractive interactions. In ad-

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K. A. Fichthorn and P.G. Balan: Rate of a catalytic-surface reaction 10035

80

60 - liA E O.1 Attractive - eA~O.2

40 ...... eA =O.3

--- °A-O.4

""'- 20

(t) 0

ut (t) -20

"40

-60

-80 200 250 300 350 400 450

T, K

FlG. 8. The compensation indicator, given by Eq. (9), at the four surface coverages of A probed in this study, for both attractive and repulsive A-B interactions.

dition, the indicator incorrectly predicts the sign of the pre­exponential factor for attractive interactions at high cover­ages of A. These discrepancies result from approximations underlying the indicator, which we examine next.

The two key approximations in the indicator are the qua-

sichemical approximation and our use of a mean, number­weighted activation energy to represent the distribution of Eq. (6). While our estimates of the terms of Eqs. (8) and (9) might be improved by a more rigorous approximation than the quasichemical approximation, we believe that the key source of discrepancy between the theory and simulations is our use of a mean activation energy. Comparing Figs. 5 and 6, it becomes apparent that compensation cannot always be linked to changes in the mean activation energy obtained from the simulations. For example, for €AB=2 with BA=O.I, the apparent pre-exponential factor at 210 K is about 12 or­ders of magnitude smaller than vo, even though (E r) [Fig. 6(b)] is constant at temperatures below 250 K. Further evi­dence that the number-averaged energy does not provide the whole picture can be seen in the pre-exponential factor for attractive interactions [Fig. 5(a)], which can be either much greater than (BA=O.l) or much less than (BA =O.4) unity. If changes in the mean activation energy, which increases with decreasing temperature [cf., Fig. 6(a)J, were solely respon­sible for the trend in the pre-exponential factor, then the pre­exponential factor should always be greater than unity. Hence, although an indicator based on the mean activation energy can predict the temperatures over which compensa­tion is the strongest, there are several features of the results which suggest that "distribution effects" can be important.

An attempt can be made to improve the agreement be­tween theory and simulations by developing an estimate of the distribution of Eq. (6). Within the quasichernical approxi­mation, the rate coefficient for reaction· can be expressed as

k~ "0 (~ ~ (n (3 7i) (Po:' r(OB)1 ( ~~r'-I OXP[(i<AA + j<AB)Pl) (~ ~ (~) (37 i)

X ( 2(; ~~tJ )' (O~Y ( (1 ~~ AT'-I eXP(i<AB{l»)

where OB is given by Eqs. (15) and (16) and

3 I P oo ~

OB = ("1 ..... B)" k.J A i=O

( 3) ( P AO ) i ( P 00 ) 3.,- I

i .2(I-BA) (l-OA) PB,I'

(23)

PB,i in Eq. (23) is given by Eq. (4) with

(24)

Equation (22) consists of two major factors, involving sum­mations. The first of these gives the product of the probabil­ity, for each possible way in which the three sites, surround­ing the A of an A-B reactant pair, can be populated with Ns, B's, and vacant sites, and the influence of each environment on the activation energy for reaction. Similarly, the second factor accounts for the possible influence of A, B, and vacant sites, surrounding the B of the A-B pair, on the rate of re-

exp( - E r ,o{3) , (22)

action. Using Eq. (20), we obtained apparent pre-exponential factors from rate coefficient given by Eq. (22) and the results are shown in Fig. 9.

Examining Fig. 9, it is evident, in the case of repulsive interactions, that a compensation indicator, based on an esti­mate of the distribution of rates, more accurately predicts the behavior seen in the simulations that an estimator that is based on the mean activation energy for reaction. Comparing Figs. 9(b) and Fig. 8 to the simulation results in Fig. 5(b), we see that the relation of Eq. (22) predicts compensation at the lowest temperatures in the simulations, unlike the relation of Eg. (9). IIi the case of attractive interactions, the distribution­based indicator predicts that the apparent pre-exponential factor should, in all cases, be less than vo, providing a better agreement with the simulation results at high BA . However, a discrepancy remains between the theory and the simulation

at low () A, where the simulations predict that the apparent

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10036 K. A. Fichthorn and P. G. Balan: Rate of a catalytic-surface reaction

0 ;> -;>

E

(a)

0 ;> -;>

E

(b)

10

5

0

-5

-10

-15

-20 250 300 350

T,K

-6A'=0.1 -- aA =0.2 •.•••. SA = 0.3 .-_. SA" 0.4

400 . 450

20~-------L------~------~--------~

°t==-------:?~~~==r

-20

-40

-60

·l:IA- 0.1 -6A =0.2 .....• SA = 0.3 --- 6A = 0.4

-80~------_.-------.~----_.--------r 200 250 300

T, K

350 400

FIG. 9. Apparent pre-exponential factors, obtained from Eq. (22) at the four surface coverages of A probed in this study, for (a) attractive A-B interac­tions and (b) repulsive A-B interactions.

pre-exponential factor should be greater than vo. This feature suggests that the quasi chemical approximation represents the distribution of A on the surface as more uniform than is actually the case.

For the case of 8.-\==0.1 with attractive interactions~ the simulations show a dip in the apparent pre-exponential factor at high temperatures [cf., Fig. 5(a)], which is followed by an increase in v by over 30 orders of magnitude as the tempera­ture decreases. We attribute the high-temperature dip in v to the filling of a minority of high-activation-energy sites. Such sites could originate from small clusters of A's in close prox­imity to one another. The high-energy sites receive the low­est weighting in Eq. (6). As the temperature decreases, sites

with increasingly lower desorption and reaction energies be­gin to fill and, eventually, they saturate. These sites have a greater weighting in Eq. (6). They shift the rate-derived ac­tivation energy and pre-exponential factor to lower values, hence, the decrease in Fig. Sea). At the lower temperatures, the high-energy sites are "passivated" and the environment for B adsorption becomes virtually homogeneous. Under these circumstances, the distribution of rates can be accu­rately represented by a delta function at the mean activation energy. It is clear, comparing Figs. Sea), 8, and 9(a), that, for attractive interactions at low 8A , the mean activation energy provides a better description of the temperature dependence of the pre-exponential factor than the distribution. Instead of providing the bimodal character, evidenced by the simula­tions, Eq. (22) provides a more even representation of the various possible activation energies. The reSUlting behavior is the same as that for attractive interactions at higher (}A:

configurations associated with low activation energies domi­nate the rate. Since these configurations become increasingly probable with decreasing temperature, the result is a decrease in the apparent activation energy with decreasing tempera­ture and apparent pre-exponential factors less than Vo.

III. CONCLUSIONS

In summary, the primary focus of this work was to elu­cidate the factors contributing to the compensation eiIect in the initial rate of a bimolecular surface reaction. In the initial rate "experiment" that we chose to model, the compensation effect is caused by changes in the surface coverage of reac­tant that alter the rate of reaction through adsorbate lateral interactions. A simple theory, based on changes in the mean . (number average) activation energy with temperature, pre­dicts the temperature ranges over which compensation is the most pronounced in Monte Carlo simulations of the initial rate. Deviations of the simulations from the theory can be attributed mostly to changes in the distribution of rates that cannot be accurately described by a numbeFaverage activa­tion energy. A modified indicator, based on a quasi chemical estimate of the distribution of rates, shows improved agree­ment with the simulations concerning the temperature range over which compensation can be seen. Under most condi­tions, the modified indicator can also, predict whether the pre-exponential factor should increase or decrease with in­creasing temperature. However, the quasichemical estimate of the distribution of rates appears to be more uniform than the actual distribution and this can lead to discrepancies be­tween theory and simulation for some conditions. Both the theories and the simulations revealed an interesting phenomenon-apparent negative activation energies, which occur when the activation energy for reaction increases with increasing temperature faster than kBT. This phenomenon could contribute to experimentally observed decreases seen in the rate of the CO oxidation reaction on several single­crystal metal surfaces. 16

ACKNOWLEDGMENTS

This research was supported by the National Science Foundation under Grant No. CTS-9058013. The computer

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K. A. Fichthorn and P. G. Balan: Rate of a catalytic-surface reaction 10037

workstations were provided by a NSF equipment grant. K.A.F. thanks Dr. Themis Matsoukas for helpful discussions.

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