A stochastic model and a Monte Carlo simulation for ...

A stochastic model and a Monte Carlo simulation for the description of CO oxidation on Pt/Sn alloys J. Mai Department of Chemistry and Institute of Nonlinear Science, University of California at San Diego, La Jolla, California 92093-0340

A. Casties and W. von Niessen Institut fu¨r Physikalische und Theoretische Chemie, Technische Universita¨t Braunschweig, D-38106 Braunschweig, Germany

V. N. Kuzovkov Institute of Theoretical Physics, University of Latvia, Rainis Boulevard 19, Riga, Latvia

~Received 16 June 1994; accepted 16 December 1994! In this paper we study CO oxidation on a catalyst consisting of a Pt/Sn alloy. On this catalyst the reaction can take place at room temperature. We use for the description two basically different methods: Monte Carlo simulations and a theoretical stochastic ansatz. The stochastic ansatz introduced recently @Mai, Kuzovkov, and von Niessen, Physica A 203, 298 ~1994!# is a general method for the description of surface reaction systems including mono- and bimolecular steps. Using the Markovian behavior of these systems we formulate this ansatz in terms of master equations. It turns out that the stochastic ansatz can be used as an interesting and advantageous alternative to the standard Monte Carlo simulations. The particles involved in the reaction system have different tendencies toward building structures on the surface. The coverages show a strong dependency, not only on the composition of the gas phase but also on the initial concentration of the reaction promotor OH and the concentration of Pt sites in the catalyst material. The reaction probability does not influence the qualitative trends of the coverages versus the gas phase concentration of CO. © 1995 American Institute of Physics.

I. INTRODUCTION

Heterogeneously catalyzed reaction systems are of great interest in theoretical and applied ~industrial! branches of research. Much effort has been undertaken to obtain a better understanding of the reaction steps. The behavior of these systems depends strongly on the structure of the surface on which the reaction takes place. Surface defects like holes, steps, and terraces influence the reaction.1,2 Another important influence arises from components added to the catalyst material. A pretreatment of the catalyst surface with substances that act as structural or electronic promotors is also important for the selectivity and the reactivity of the catalyst. In order to understand and to design new catalysts, the reaction mechanism and many details of the individual reaction steps have to be considered. Experimental information is often insufficiently available, and the interpretation of the data is normally very difficult because many different effects are measured at the same time. An alternative to experiments is the use of computer simulations and theoretical stochastic models if these can handle sufficiently complex reactions— which is one aim of research. Most of the computer simulations for surface reaction systems are Monte Carlo ~MC! simulations. In addition, cellular automata ~CA! have been studied. Some of these simulations are in reasonable agreement with some experimental observations of real surface reaction systems.3 The most prominent systems that have been studied are the CO oxidation on a Pt catalyst4,5 and the formation of NH3 .6 Dynamical studies of surface reactions were also performed. With the use of molecular dynamics simulations a subtle study of the individual steps of surface J. Chem. Phys. 102 (12), 22 March 1995

reactions and the energetic transfers is possible. Examples are careful studies of adsorption and desorption processes.7 In principle, the description of these systems by theoretical stochastic models has some advantages over the use of computer simulations, which require a large amount of computer time to obtain good statistics. But it has turned out that only very simple systems such as the A1A→0 reaction can be solved purely analytically, and this only in one spatial dimension8,9 so far. Therefore we introduced in Ref. 10 a general stochastic ansatz that is able to handle more complex systems by using a combination of analytical and numerical techniques. The temporal evolution is formulated with master equations for the one- and two-point densities taking structural correlations explicitly into account. The final equations are solved numerically on a lattice. In a small region of a reference point the equations are solved exactly. To take longer range correlations into account, we connect the solution to continuous functions that represent the system behavior for large distances. We applied this model to CO oxidation on a Pt surface12 and the formation of NH3 .13 It turned out that the results are in overall agreement with computer simulations. Small deviations arise due to differences in the reaction mechanism used in the different approaches and the finite lattice size in the computer simulations. The last aspect is of importance, especially near the phase transition points where large aggregates of adsorbates are formed. In this paper we want to study CO oxidation taking place on a very unusual catalyst. We use the techniques of MC simulation and the stochastic ansatz outlined above. The

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Mai et al.: CO oxidation on Pt/Sn alloys

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catalyst of interest consists of a Pt/Sn alloy, which is formed from Pt/SnOx by pretreatment with water vapor resulting in a catalyst of the type Pt/Sn/OH. It has been shown15,16 that this catalyst allows the CO oxidation at remarkably low temperatures. Therefore this catalyst is of great importance in technical applications, such as in closed-cycle CO2 lasers and for air purification. We have seven different states of the catalyst in our model. These are Sn, Pt, O/Sn, O/Pt, CO/Pt, OH/Sn, Pt, and H/Sn, Pt. Here Pt or Sn means a vacant Pt or Sn site, respectively. The notation X/Y for the other states denotes an X atom on a Y type surface site, e.g., O/Sn means a Sn site, which is occupied by an O atom. The symbol Pt, Sn means that either a Pt or a Sn site can be involved in the step. Isotopic experiments have given evidence that CO is oxidized by OH rather than by the adsorbed and dissociated O2 . A postulated reaction mechanism15 includes the following elementary reaction steps: CO~ g ! 1Pt→CO/Pt,

~1!

O2 ~ g ! 12Pt,Sn→2O/Pt,Sn,

~2!

CO/Pt1OH/Pt,Sn→H/Pt,Sn1CO2~ g ! 1Pt,

~3!

H/Pt,Sn1O/Pt,Sn→OH/Pt,Sn1Pt,Sn.

~4!

We assume that our catalyst is an ideal random alloy. Step one @Eq. ~1!# is the adsorption of CO from the gas phase. It should be noticed that CO can only adsorb on a Pt surface site, i.e., the state CO/Sn does not exist.15 The dissociative adsorption of O2 on a vacant pair of surface sites is shown in step two. This step seems to have no preference for a Pt or Sn site of the catalyst. Step three shows the oxidation of CO by OH, resulting in the formation of CO2(g), a free Pt site, and a hydrogen species H/Pt, Sn; step four is the regeneration of the OH species. In both simulations the product CO2 does not influence the system. In this paper we use this simplified reaction mechanism, which does not take the transformation of oxygen existing in the form of Pt/SnOx to the initial hydroxyl into account, i.e., the pretreatment of the catalyst to the Pt/Sn/OH type is of no concern in our model. We also exclude the possibility of bicarbonate formation15 as well as CO and H diffusion to keep the model somewhat simpler. We study the reaction system as a function of the normalized concentration of Pt and Sn sites in the catalyst ~y Pt , y Sn512y Pt!, the mole fraction of CO and O2 in the gas phase ~y CO , y O512y CO!, and as a function of different initial concentrations of OH on the surface @y OH~t 0!#. The following initial concentrations are determined by the ones listed above: y Pt~t 0!5y Pt@12y OH~t 0!#, this is the concentration of vacant Pt sites in the beginning; y Sn~t 0!5~1 2y Pt!@12y OH~t 0!#, the concentration of vacant Sn sites in the beginning. It is thus presumed that OH has no preference for either Pt or Sn. Another parameter in our model is the reaction probability R for steps ~3! and ~4!. We choose values of R between R510 and R5100. This complexity arising from the number of variables is the reason for excluding the diffusion in both models for the time being. We compare the results of the stochastic ansatz of this system with the corre-

sponding MC simulation. Moreover, comparisons are made to the CO oxidation taking place on a pure Pt catalyst. The paper is structured as follows: In Sec. II the MC simulation of the reaction system @Eqs. ~1!–~4!# is briefly introduced and remarks are made about the differences between CO oxidation on Pt and CO oxidation on OH/Pt,Sn. The stochastic model is introduced in a concise way in Sec. III, where we refer to recently published work containing the theoretical background of the model. In this section some numerical conditions for solving the equations are given, and the results are compared with those obtained from the MC simulation. II. THE MC SIMULATION

The reaction model for the MC simulation of the CO oxidation on a Pt/Sn alloy is based on a MC simulation that was introduced by Ziff, Gulari, and Barshad ~ZGB model!4 as a simplified model for the heterogeneously catalyzed oxidation of CO on Pt. The catalyst is represented by a twodimensional square lattice with periodic boundary conditions. A lattice of size 64364 sites has been used. A gas phase containing CO and O2 is above this surface. CO adsorption can occur if a randomly selected Pt site on the surface is vacant. In the case of O2 adsorption, two vacant neighbor sites are necessary. In our model the adsorption of CO and O2 @Eqs. ~1! and ~2!# is treated in the same way as in the CO oxidation on Pt ~also see Ref. 5!. A different situation occurs after a reaction event between OH and CO: not two previously occupied sites are empty, but only one site—the one from which the CO came. The other site is occupied by the remaining H species @Eq. ~3!#. This H can react with a neighboring O to form a new OH species @Eq. ~4!#. OH and H can either occupy a Pt or a Sn site. Each simulation is set up for one distinct value of each parameter and the variable y CO . For each value of y CO , the coverages ~i.e., the surface concentrations! of the reaction components are given after the system has reached the steady state. The generation of the catalyst surface is realized by the use of random numbers with the concentration of Pt sites as parameter. In the MC simulations reaction occurs between nearest neighbor sites. A discussion of the results of the MC simulation appears in Sec. III D in the form of a comparison with the results of the stochastic simulation. III. THE STOCHASTIC MODEL A. Definitions

In the present stochastic ansatz we make use of the assumption that the reaction system is of Markovian type. A description via master equations is therefore possible.10 The stochastic simulation takes place on a lattice with a given coordination number. Each lattice site is given a lattice vector l. The state of the site l is represented by the lattice variable sl . sl may depend on the site of the catalyst ~Pt or Sn! and on the occupation with a particle. A distinction between promoted and nonpromoted sites is not included in our model ~the other case is explained in Ref. 13!.

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Mai et al.: CO oxidation on Pt/Sn alloys

The most important properties of the system are the densities of the individual states. We call these densities i-point probabilities p (i) , where i stands for the number of considered sites in the state. B. The main equations

The reaction scheme @Eqs. ~1!–~4!# includes mono- and bimolecular steps. A monomolecular step needs only one lattice site, e.g., the CO adsorption. A bimolecular step needs two nearest neighbor sites, e.g., the O2 adsorption or the reaction between CO and OH. The stochastic description of such a model leads to an infinite chain of differential equations ~master equations! for the densities of each state. Beginning with the one-point probability ~i51! we get the form10 dC l 5A l ~ C,F,K ! 2B l ~ C,F,K ! •C l . dt

~5!

C l means the concentration of the state l on the lattice with l[sl . The one-point probabilities are the simple concentrations of the states, i.e., r~1!( s l )[C l . A l (C,F,K) and B l (C,F,K) are simple positive functions ~polynomials! of the densities C of the respective states, of the correlation functions F and of the transition probabilities K of the individual steps ~the correlation functions are related to the twopoint probabilities!. Equation ~5! is exact, because only twopoint probabilities are additionally needed to describe the dependencies of the one-point probabilities. In the equation of motion for the two-point probabilities two- and threepoint probabilities are involved, and so on. The infinite hierarchy of equations cannot be solved exactly because one would need the correlations of the different particles for all distances and configurations on the infinite lattice. The hierarchy of master equations must be truncated in a way that retains the main correlations in the system. With the help of the superposition approximation ~SA! of Kirkwood14 at the level of the three-point probabilities, we obtain a finite system of nonlinear equations. The three-point probabilities in the SA are expressed in terms of the one-point probabilities, i.e., the simple concentrations of the particles, and the correlation functions,

r ~ 3 ! ~ s l s n s m ! ⇒C l C n C m F l n ~ l2n! 3F nm ~ n2m! F m l ~ m2l! .

~6!

The arguments of the correlation functions are the vectors n, m, and l corresponding to the lattice sites of the states l, m, and n. Use of the SA leads to the following equation of motion for the correlation functions: d F ~ r! 5A l m ~ C,F,K ! 2B l m ~ C,F,K ! F l m ~ r! . dt l m

~7!

r means the difference vector between two lattice states. The sets of Eqs. ~5! and ~7! have to be solved simultaneously. C. Method of solution

The main problem in solving the obtained equation system is connected with the solution of an infinite system of

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nonlinear differential equations for a chosen type of lattice. To solve this problem in practice, the following approximation is used. A threshold value m 0 is introduced. For uru,m 0 the lattice equations are solved for all nonequivalent points of the lattice ~here we use m 055!. This first region determines several coordination spheres in which the lattice aspect of the problem is important. We choose m 055 because up to this distance the geometric factors in the densities and correlation functions can be calculated with an acceptable effort and all essential correlation effects should be included. In the region outside all properties change quasicontinuously with the distance uru. Therefore we can use a continuum approximation by introduction of the coordinates r5uru and substituting for the correlation function F~r! the radial one F(r). By this substitution the equations transform into nonlinear equations in partial derivatives. As the left ~or inner! boundary condition the solution within the first area at uru5m 0 ~circumference of the circle with radius m 0! is used. Because of the weakness of the correlation for large distances, we can use F~`!51 as the right ~or external! boundary condition. More details of the method of solution are given in Ref. 10. D. Results

We want to focus the discussion of the results mainly on the data obtained from the stochastic approach. Only in a few cases will we refer to the results of the MC simulation explicitly. On the whole, all studied properties in the present scheme are similar in the stochastic and the MC models. We may obtain very different functional forms of the coverages vs y CO depending on the parameters of the system, but they are common to both types of approaches. When we compare the results for the CO oxidation on a Pt/Sn catalyst with the ones for the simple oxidation of CO on Pt, we immediately note that they are very different, even at a qualitative level. As a survey we give below some of the main results and trends of the reaction systems. ~a! A principal result of the simulation of CO oxidation on a Pt/Sn catalyst is the disappearance of the phase transitions of the CO and O poisoning on the surface. For the simple CO oxidation on Pt two poisoning phase transitions were found: a poisoning by O at lower values of y CO and a poisoning by CO at higher values of y CO . This studied behavior is shown in Fig. 1 from Ref. 11, where the coverages of the reaction components are given as a function of y CO . The disappearance of the poisoning transitions is a consequence of the more complex reaction scheme, whereby the species H or OH remain on the surface, even at the highest concentrations of CO or O, respectively ~also see Figs. 2, 3, and 5, which we discuss in detail later in this section!. The coverages, Q, of the individual components in the present model lie in the range 0.0–0.6. The maximum value of each coverage is unity, i.e., the situation of complete occupation of the surface by the corresponding species. This maximum value is never encountered in the present model. It is clear that the sum of all coverages is also limited to unity. ~b! H always remains on the surface for y CO→1. This fact is a consequence of Eq. ~3!. On the other hand, OH only remains on the surface in a larger amount if y OH(t 0 )>y Pt . O

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Mai et al.: CO oxidation on Pt/Sn alloys

FIG. 2. The coverages of CO, O2 , H, and OH, and the reaction rate for the creation of CO2 , R CO2 vs y CO , the mole fraction of CO in the gas phase, in the stochastic approach. The parameters are as follows: y Pt50.8, y OH~t 0!50.6, R510. The symbols denote calculated values, the curves are drawn as a guide for the eyes. FIG. 1. The coverages of CO and O and the reaction rate for the CO2 production versus y CO , the mole fraction of CO in the gas phase, in a cellular automaton simulation of the ZGB model ~from Ref. 11!. The reaction was realized on a square lattice without diffusion. Error bars are given for QO . QCO is zero up to the phase transition of CO poisoning.

remains on the surface in a larger amount for y CO→1 if y OH~t 0! and especially y Pt are small. The details will be given below. ~c! We observe some general trends for the coverages of the species as a function of y CO : QOH shows a trend opposite to QH ; if one quantity increases, the other one decreases, and vice versa. The same is true for the pair QCO and QO . QOH shows a trend parallel to QO and QCO is parallel to QH ~we refer to Figs. 2, 3, and 5!. These rough tendencies are easy to explain with the reaction mechanism @Eqs. ~3! and ~4!#. The detailed quantitative trends depend strongly on the parameters y OH~t 0! and y Pt . The reaction probability for the steps ~3! and ~4! have a much weaker influence. Only for large values of y Pt do we get large changes in the coverages. This result is understandable given the fact that CO can only adsorb on Pt sites. The strong changes of the coverages can be interpreted as phase transitions without poisoning. Such a transition occurs near y CO50.3 in Fig. 2. There the dependence on y CO of the coverages of the particles and the reaction rate R CO2 of the CO2 creation for y Pt50.8 and y OH~t 0!50.6 in the stochastic approach is shown. Poisoning means a complete occupation of the lattice by one species. This poisoning does not occur in the present model. The strength of the influence of y OH~t 0! is connected to the value of y Pt . It is frequently found that in the cases where y OH(t 0 ),y Pt the coverage by H particles increases to its maximally possible value QH5y OH~t 0! for y CO→1 because here the influence of step ~3! is dominant and step ~4! has little influence. The reaction in step ~3! then comes to a stop after the stoichiometric amount of the components CO and OH has reacted to the products H and CO2 . Because O does not exist for y O→0 on the surface for the given parameters, no regeneration of OH and therefore no decrease of the number of H particles occurs. But there may be other situa-

tions. Such a situation is seen in Fig. 2. Here the maximum value QH'0.6 is reached for y CO,1. QOH shows as usual the opposite trend. The appearance of extrema is a special phenomenon of this system, and will be discussed later in this section. The coverage by O decreases with increasing y OH~t 0!, which is easy to understand with the reaction scheme. The trend of QCO also depends on the ratio of y Pt to y OH~t 0!: For y OH(t 0 )>y Pt , QCO starts with a value smaller than y CO and for y OH(t 0 ),y Pt , QCO starts with a value larger than y CO . This result can be deduced from the influence of reaction ~3! and shows a behavior different from that seen in the ZGB model. There we have QCO'0 up to y CO,y 2 , the phase transition point for CO poisoning. If we do not consider energetic parameters, as in the present model, it seems that the present catalyst system cannot reach the reactivity of a simple Pt catalyst. This conclusion is fairly obvious, because in our model there is no value of y CO , where the sum of the coverages of the particles is very small. A small coverage is caused by high reaction rates. Our system shows relatively large coverages and rather small reaction rates. To come to definite conclusions, we must consider the influence especially of the temperature in the description of the reaction system. For small values of y Pt , the trends in all coverages are quite different from the case above. All coverages vary smoothly and no phase-transition-like behavior is observed. In Fig. 3 the coverages versus y CO are shown for y Pt50.2, y OH~t 0!50.2, and R510. y OH~t 0! plays a minor role compared to y Pt . This result can be concluded from the results for y Pt50.2 and y OH(t 0 ).y Pt , where we obtain similar trends ~not shown!. In this regime the reactions @~3! and ~4!# are of secondary importance compared with the adsorption @~1! and ~2!#. The reaction is strongly inhibited because of the few places available for the CO adsorption. The absence of CO diffusion in our model strengthens this restriction. QO is influenced by y OH~t 0! in the same way as we found in the case where y Pt is large. For larger values of y OH~t 0!, the coverage by O is reduced over the whole range of y CO com-

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Mai et al.: CO oxidation on Pt/Sn alloys

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FIG. 3. The coverages of CO, O2 , H, and OH, and the reaction rate for the creation of CO2 , R CO2 vs y CO in the stochastic approach. The parameters are as follows: y Pt50.2, y OH~t 0!50.2, R510. The symbols denote calculated values, the curves are drawn as a guide for the eyes.

FIG. 4. Correlation function of the pairs Op–Op at y CO50.1, 0.2, 0.25 versus the distance r in the stochastic approach. The symbol p means the occupation of a Pt site. The parameters are as follows: y Pt50.8, y OH~t 0!50.6, R510. Calculations were made at specific distances and the resulting values were connected with lines.

pared to the case of smaller values of y OH~t 0!. But here we have the extreme situation that even for large y OH~t 0! the coverage by O remains larger than zero when y CO approaches unity. Also, QCO is influenced by y OH~t 0! in the same way as we found in the case where y Pt is large. Larger values of y OH~t 0! reduce the coverage of CO. We thus note that both QO and QCO are reduced by large values of y OH~t 0!. This reduction is an interesting phenomenon, in view of the fact that QO and QCO have opposite trends as a function of y CO . The maximum value of QH is far below the initial concentration of OH @y OH~t 0!#, i.e., less reaction with CO occurs. Even for y OH~t 0!50.6, the maximal concentration of H particles is smaller than 0.2. This fact shows once more the dominant influence of y Pt in our model. The behavior of the coverages for y CO→1 in this model is different from that in the ZGB model. Both QCO~y CO! and the coverages of other reaction components are larger than zero. This behavior depends strongly on the parameters that we want to discuss here. H always remains on the surface for y CO→1. This fact is a consequence of Eq. ~3!, but QH cannot exceed the value given by y OH~t 0!. OH only remains on the surface in a larger amount if y OH(t 0 )>y Pt . The explanation is as follows. OH particles randomly distributed at t 0 on the surface can survive in a larger amount when they occupy Sn islands, where CO can only reach the border @CO can only occupy Pt sites ~also see ~3!#. The appearance of islands of sites of Sn or Pt is possible because of the random distribution of these sites in both models. So the formation especially of Sn islands restricts the reaction in both models to the same degree. For y CO approaching unity, O remains on the surface in a larger amount if y OH~t 0!, and especially y Pt are small. Even with the largest reaction probability ~R5100!, QO remains finite ~we refer to Figs. 3 and 4!. This result is not possible in the ZGB model. A look at Eq. ~3! shows that less H is formed if y Pt is small, and only a small amount of OH particles is placed on the surface. Consequently, there is only a small chance of the existence of a CO/OH pair. This chance is further reduced, due to the fact that OH particles can oc-

cupy sites of Sn islands, which are abundant for small y Pt . Physisorption is neglected in the present model, and therefore CO cannot reach these OH particles. Complete reaction of the small amount of created H particles with the O particles coming from the gas phase is hardly probable. This reaction becomes exceedingly more difficult for y O→0 ~y CO→1!. If, on the other hand, the conditions hold that y Pt is large, y Pt.y OH~t 0! and y CO→1, then the coverage of O approaches zero. One distinct difference between the trends observed for large and small values of y Pt is the appearance of extrema in all coverages when y Pt is large and y OH~t 0!.0.2, or, for the case of a large value of y Pt and R5100. The corresponding pairs of opposite extrema in QH~y CO! and QOH~y CO! appear always at the same value of y CO . This result is different from the trends of QCO and QO , where the extrema can appear at slightly differing values of y CO . These extrema are a result of the competition between the creation and annihilation of each of the species. The creation of CO and O is realized by the adsorption. In Fig. 2 we see that for small values of y CO the reaction between CO and OH reduces QOH from the value y OH~t 0! to somewhat above half of this value. This value might appear to be rather small. But one has to consider that all results discussed here are obtained after the system has reached the steady state. The relatively large value of QH ~and therefore the small value of QOH! for y CO50.1 is caused by the inhibition of the reaction between O and H, due to the fact that O forms large islands. With increasing y CO , the O islands become less compact, and more vacant sites occur because the influence of reaction ~3! grows. The length of the reaction front increases. More and more H is produced via Eq. ~3!. But y O is still large, so that a large amount of H is converted to OH @Eq. ~4!#. The effect is a decrease of QH . The OH that is regenerated will not entirely react via Eq. ~3!. That is because the free Pt site after reaction ~3! is not always occupied by a new CO species out of the gas phase, and because the regenerated OH species might now be placed inside small Sn islands, where CO cannot adsorb. Thus, QOH increases. If y CO is increasing further

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Mai et al.: CO oxidation on Pt/Sn alloys

~y O is decreasing! the importance of reaction ~3! increases, but the increasing amount of H particles cannot react proportionately with the decreasing number of O particles on the surface. QH now increases and QOH decreases. This change happens at y CO50.2. The extrema are passed and now the coverages show stronger changes, as in a phase transition of second order. For QCO and QO , the appearance of the extrema is directly related to those of QH and QOH ~parallel trends; see above in this section!. The values of the correlation function for O–O pairs confirm this argument. Figure 4 shows the trends of the Op–Op pair correlation functions versus the distance between two sites, r, at y CO50.1,0.2,0.25. The symbol Op denotes an O atom on a Pt site. The Op–Op correlation is the most convincing one for our discussion of structural effects. Up to r'5, F OpOp(r) is minimal for y CO50.2 on the average. F OpOp(r) is oscillating around the lowest value at y CO50.2. This minimum appears exactly at the local maximum of QO~y CO!, where we also find the local extrema of the other coverages. For the O structure this result means a perturbation of the order that occurred at y CO,0.2 ~alternating occupation of the Pt sites!. At y CO50.2, we have the situation where for the first time CO can be adsorbed in a larger amount, so that more connected free sites appear as a result of step ~3!. O2 molecules now have a better chance to adsorb because of the temporary appearance of more free sites. This effect is confirmed by the larger value of the Op–Op correlation function at r51 for y CO.0.1 ~also see Fig. 4!. Pair formation is smoothing the oscillations in the correlation functions. A consequence is the minimum found for QH . For y CO.0.2, F OpOp(r) increases again, but the coverage of O decreases. For the formation of O islands, this change means the creation of smaller structures. These are more compact so that the reaction probability between O and H decreases. This result is confirmed by the strong increase of QH . A second extremum in QH~y CO! and QOH~y CO! appears for larger values of y OH~t 0! in the region of larger values of y CO ~also see Fig. 2!. QH begins to increase for y CO.0.2. The reaction between O and H particles must be inhibited, because the O structures become smaller and more compact. This result is seen from F OpOp(r), which is increasing while QO is decreasing with increasing y CO . The flat extrema in the curves appear subsequent to the maximal change of QH~y CO! and QOH~y CO!. For y CO>0.5 QCO~y CO! and QO~y CO! are nearly constant. This behavior means that the production of CO and O, i.e., their adsorption, and their reaction are nearly equal. How is it possible that the coverage by O remains constant while the size of the O islands is shrinking, and the concentration of O2 in the gas phase is decreasing? The answer is that we have more free sites for O2 adsorption on the lattice, though the sum of the coverages is nearly constant in that range of y CO . The destruction of the O islands gives more connected free sites, and leads to an increase of the reaction rate of step ~4!, and consequently also of step ~3!. With increasing y CO , especially F OpOp~1!, strongly increases. This increase means that the coverage of O particles is realized more and more in small clusters. Especially pairs of nearest neighbors exist. The reactivity between O and H must increase again, as must the adsorption rate of O2 . QH

must decrease again and consequently it is going through a maximum and correspondingly QOH shows a minimum value. The MC simulations do not show the appearance of clear extrema. Due to the large amount of computing time needed for one parameter run over the whole range of y CO and the relatively small lattice with respect to the relatively large number of states, large fluctuations in the coverages appear. Therefore it is not possible to give a detailed discussion of these properties. max in the The reaction rate R CO2 (y CO) has a maximum R CO 2 region where the largest changes of QO and QCO occur, but this situation is only the case if y Pt is large enough. Then the model is in agreement with the ZGB model in this point. The max , is shifted significantly on position of the maximum, R CO 2 the y CO axis by the variation of y OH~t 0! and y Pt . If all parameters, i.e., y Pt , y OH~t 0! and R @the reaction probability for max reaches the value 0.1. the steps ~3! and ~4!# are large, R CO 2 R CO2 is defined as the change in the amount of CO2 in step ~3! and is normalized to unity. In the MC model we have an equivalent definition. This value also shows that the present catalyst does not have the highest possible reactivity. For an ideal distribution and concentration of Pt and Sn sites and the possibility of CO and H diffusion, we would obtain larger values of R CO2 . Nevertheless, it should be noticed that this catalyst works at a low temperature, which is for special applications a great advantage compared to other catalysts. If R is small and the other two parameters are large enough, we get a maximum in R CO2 , which is less distinct. This result is easy to understand because both steps ~3! and ~4! in the reaction scheme depend on R. For small y Pt ,R CO2 (y CO) is flat and has a small value over the whole range of y CO . No strong mutual influence between ~3! and ~4! occurs, because over the whole range of y CO the concentration of CO molecules on the surface is too small. The position of the phase transition points ~or their remmax are influenced by the paramnants! and the position of R CO 2 eters. A large value of y Pt shifts the phase transitions and the corresponding maximum of R CO2 to smaller values of y CO . Larger values of y OH~t 0! also shift the points of the phase transition to smaller values of y CO . The strong shift from y CO'0.3 for y Pt50.8, y OH~t 0!50.6 to y CO'0.6 for y Pt50.4, max shows directly the necessity to have a y OH~t 0!50.4 in R CO 2 certain amount of CO for reaching the critical state. The latter case is shown in Fig. 5. This extreme shift was not discussed in the former publication ~Ref. 10!, where we only gave a short overview of the dependencies. R plays a secondary role in influencing the position of the phase transition points. The width of the reactive interval, i.e., the range of y CO between the phase transition points or, as in our case, the range of y CO , where the largest changes in the coverages occur, remains rather small ~10% of the whole range of y CO! for all parameter runs ~larger values of y Pt are necessary; see above!. This behavior can be explained by the limited mobility of the species ~no diffusion! and the relatively complex

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Mai et al.: CO oxidation on Pt/Sn alloys

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FIG. 7. The same plots as in Fig. 6, but at y CO50.6. FIG. 5. The coverages of CO, O2 , H, and OH, and the reaction rate for the creation of CO2 , R CO2 vs y CO in the stochastic approach. The parameters are as follows: y Pt50.4, y OH~t 0!50.4, R5100. The symbols denote calculated values, the curves are drawn as a guide for the eyes.

reaction mechanism compared to the simple CO oxidation ~ZGB model!. Taking a look again at Figs. 2, 3, and 5 ~stochastic simulations!, we see that points of intersection in QO~y CO! and QCO~y CO!, as well as in QH~y CO! and QOH~y CO! appear for high concentrations of Pt sites and for larger values of y OH~t 0!. This result is a simple consequence of the appearance of the opposite extrema, and, in the case of QH and QOH , of the simple dependency, QH~y CO!5y OH~t 0! 2QOH~y CO!. As a significant property of the discussed reaction system, we get a phase-transition-like behavior of the coverages for large values of y OH~t 0! and y Pt . A brief comparison of the trends of some correlation functions may illustrate this phenomenon. At y CO50.2, an oscillating behavior of F OO(r) and F OsH(r) is observed so that a sieve structure of O and H on the surface can be concluded. This result is shown in Fig. 6, where the pair correlation functions F OpOp(r), F OsOp(r),

F OsOs(r), F OsH(r), and F OpH(r) are given for y CO50.2. The indices s and p denote the occupied lattice sites ~Sn or Pt, respectively!. For larger y CO , this geometrical order becomes weaker and the correlation functions are essentially monotonic. This situation is shown in Fig. 7, where the same plots as in Fig. 6 are given for y CO50.6. All O–O correlations have much larger values in this case. Especially the values for nearest neighbors ~r51! are very large. At y CO50.6 small O clusters with an average radius of three lattice units are the most favored structures. The O–H correlation is smaller than unity, a result that means that O and H show a repulsion behavior due to structure formation. This conclusion supports a stronger separation of the O and H regimes, which is also seen as a phase-transition-like behavior. The trends in the coverages given above are also found for the MC simulation. The positions of the phase transitions differ between the models, as do the values of the coverages. One reason is an insufficient averaging in the MC simulation. A better agreement in the trends is obtained for small y Pt and small y OH~t 0!. Figure 8 shows the coverages of the components as a function of y CO in a MC simulation run, with the parameters y Pt50.2 and y OH~t 0!50.2. Comparing Fig. 3 and Fig. 8, we see that the values of the coverages as well as the shapes of the flat curves are very similar in the two models. In this case the CO–OH and H–O correlations are small for small distances, and the O–O correlations are close to unity over the whole range of y CO for all considered distances. Deviations arising from different procedures of including correlations in the two models should cause only a minor effect in the coverages because the correlations are weak. IV. CONCLUSIONS

FIG. 6. Correlation function of the pairs Op–Op, Os–Os, Os–Op, Op–H, and Os–H at y CO50.2 versus the distance r in the stochastic approach. The parameters are as follows: y Pt50.8, y OH~t 0!50.6, R510. The symbols p and s mean the occupation of a Pt and Sn site, respectively. For H there is no differentiation made between the lattice sites. Calculations were made at specific distances and the resulting values were connected with lines.

We have introduced a stochastic ansatz, which is useful to describe CO oxidation on a Pt/Sn alloy. Some special properties of the reaction system could be explained: the appearance of extrema in the coverages, the disappearance of phase transitions, and the symmetric behavior of some coverages. We have characterized the system by the parameters y Pt ~concentration of Pt sites on the lattice!, y OH~t 0! ~initial concentration of OH particles on the lattice! and R ~reaction

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Mai et al.: CO oxidation on Pt/Sn alloys

With the present stochastic ansatz we wanted to characterize the spatial behavior of the components of the reaction system without taking energetic parameters into account. A later publication will bring detailed information about the influence of the temperature on surface reaction systems. The stochastic ansatz offers the possibility to study complex systems, including the distribution of special surface sites and correlated initial conditions for the surface and the coverages of particles. ACKNOWLEDGMENTS

This research was supported by the Deutsche Forschungsgemeinschaft via a fellowship for J. Mai and V. N. Kuzovkov, by a Grant No. LB2000 from the International Science Foundation via a fellowship for V. N. Kuzovkov, and in part by the Fonds der Chemischen Industrie. H. Hopster and H. Ibach, Surf. Sci. 77, 109 ~1978!. M. R. McClellan, J. L. Gland, and F. R. McFeeley, Surf. Sci. 112, 63 ~1981!. 3 M. Ehsasi, M. Matloch, O. Frank, J. H. Block, K. Christmann, F. S. Rys, and W. Hirschwald, J. Chem. Phys. 91, 4949 ~1989!. 4 M. Ziff, E. Gulari, and Y. Barshad, Phys. Rev. Lett. 56, 2553 ~1986!. 5 J. Mai, W. von Niessen, and A. Blumen, J. Chem. Phys. 93, 3685 ~1990!. 6 J. Mai and W. von Niessen, Chem. Phys. 165, 65 ~1992!. 7 J. C. Tully, Surf. Sci. 299/300, 667 ~1994!. 8 Z. Racz, Phys. Rev. Lett. 55, 1707 ~1985!. 9 D. Ben-Avraham and S. Redner, Phys. Rev. A Gen. Phys. 34, 501 ~1986!; Z. Y. Shi and R. Kopelman, Chem. Phys. 167, 149 ~1992!; I. M. Sokolov, H. Schnoerer, and A. Blumen, Phys. Rev. A 44, 2388 ~1991!; K. Lindenberg and B. J. West, ibid. 42, 890 ~1990!. 10 J. Mai, V. N. Kuzovkov, and W. von Niessen, Physica A 203, 298 ~1994!. 11 J. Mai and W. von Niessen, Phys. Rev. A 44, R6165 ~1991!. 12 J. Mai, V. N. Kuzovkov, and W. von Niessen, J. Chem. Phys. 98, 10017 ~1993!. 13 J. Mai, V. N. Kuzovkov, and W. von Niessen, Phys. Rev. E 48, 1700 ~1993!. 14 J. G. Kirkwood, J. Chem. Phys. 76, 479 ~1935!. 15 D. R. Schryer, B. T. Upchurch, B. D. Sidney, K. G. Brown, G. B. Hoflund, and R. K. Herz, J. Catal. 130, 314 ~1991!. 16 D. R. Schryer, B. T. Upchurch, J. D. Van Norman, K. G. Brown, and J. Schryer, J. Catal. 122, 193 ~1990!. 1 2

FIG. 8. The coverages of CO, O2 , H, and OH vs y CO in the MC simulation. The parameters are the same as in Fig. 3. The symbols in the curves represent the values obtained in the MC simulation.

probability of the reaction steps!. Large values of y Pt and y OH~t 0! lead to greater changes in the coverages than are found for small values of these parameters. The reaction rate of CO2 production then increases, but never reaches large values. This result is explainable, given the fact that the reaction steps are only coupled in a limited way, because all components ~except CO2! remain bound on the surface after creation and can hinder further reaction by forming islands. A pure Pt catalyst would yield a maximum reactivity in our model, because CO is favored for adsorption. But this simplified conclusion holds only without considering energetic parameters and the creation of OH, where Sn sites are needed ~they bind the oxygen for the initial OH creation!.

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