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Since then, pattern formation in chemical reactions has attracted considerable attention of chemists, physicists, and biologists.2-4 Traditionally, th...
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Langmuir 2001, 17, 1793-1799

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Pattern Formation in Catalytic Reactions Due to Lateral Adsorbate-Adsorbate Interactions V. P. Zhdanov* Department of Applied Physics, Chalmers University of Technology, S-412 96 Go¨ teborg, Sweden, and Boreskov Institute of Catalysis, Russian Academy of Sciences, Novosibirsk 630090, Russia Received August 11, 2000. In Final Form: November 1, 2000 We present Monte Carlo simulations of the formation of islands in an overlayer on a single-crystal surface and on nanometer-size catalyst particles during the A + B f AB reaction, occurring under the steady-state conditions via the Eley-Rideal mechanism, including A adsorption on vacant sites and further conversion to AB in collisions with gas-phase B particles. The model employed takes into account the effect of attractive lateral adsorbate-adsorbate interactions on the rates of A diffusion and elementary reaction events. The results obtained indicate that the lateral interactions influencing the reaction may facilitate the formation of islands. This is possible provided that the interactions partly suppress the reaction inside islands. The size of islands is however rather small even if diffusion is much faster than reaction.

1. Introduction In chemically reactive systems, spontaneous spatial organization is possible if reactant diffusion is coupled with chemical feedback. This seminal conclusion was drawn by Turing1 in 1952. Since then, pattern formation in chemical reactions has attracted considerable attention of chemists, physicists, and biologists.2-4 Traditionally, this phenomenon is described by using the mean-field reaction-diffusion (MFRD) equations based on the massaction law. This approach is applicable if the average distance between reactants is large compared to their size. In heterogeneous catalytic reactions, this condition often does not hold even if the chemical conversion occurs on a uniform single-crystal surface, because during reaction the adsorbate coverages are as a rule appreciable, and accordingly the probability of finding reactants in nearestneighbor (NN) or next-nearest-neighbor (NNN) adsorption sites is high. In this case, the kinetics of elementary reaction steps is usually affected by NN and NNN adsorbate-adsorbate lateral interactions (ordering of adsorbed atoms or molecules due to such interactions is a common phenomenon observed during the formative years of surface science (1965-1990) in thousands of lowenergy electron diffraction (LEED)-based studies5). On supported nanometer-size catalyst particles, widely employed in practice,6 the reaction kinetics can in addition be complicated by various effects inherent to nanometersize chemistry7 (e.g., by the interplay of the reaction kinetics on different facets due to reactant diffusion via the facet boundaries). Scrutinizing phase diagrams of chemisorbed particles and thermal desorption spectra, one can conclude that * E-mail: [email protected] and [email protected]. (1) Turing, A. M. Philos. Trans. R. Soc. London, Ser. B 1952, 327, 37. (2) Chemical Waves and Patterns; Kapral, R., Showalter, K., Eds.; Kluwer: Dordrecht, 1994. (3) De Wit, A. Adv. Chem. Phys. 1999, 109, 435. (4) Nonlinear Dynamics in Chemical and Biochemical Processes. Drahos, J., Marek, M., Schreiber, I., Eds. Chem. Eng. Sci. (Special issue) 2000, 55, 207-470. (5) Somorjai, G. A. Introduction to Surface Chemistry and Catalysis; Wiley: New York, 1994. (6) Thomas, J. M.; Thomas, W. J. Principles and Practice of Heterogeneous Catalysis; VCH: Weinheim, 1997. (7) Zhdanov, V. P.; Kasemo, B. Surf. Sci. Rep. 2000, 39, 25.

often the NN lateral interactions are relatively strong (2-3 kcal/mol) and repulsive and the NNN interactions are weak.8 The situations when the NN and/or NNN interactions are attractive and relatively strong are however possible as well. From the point of view of chemical kinetics and statistical physics, this case discussed in our present paper is especially interesting because the island growth, occurring due to attractive adsorbate-adsorbate interactions (for general theory, see the review by Bray9), can be terminated by reaction. Under such circumstances, one can observe frozen or traveling patterns in the adsorbed overlayer under the steady-state conditions in the gas phase. Our presentation below is focused on the steady-state regimes because such regimes are practically important. First, it makes sense to discuss briefly transient reaction regimes. The idea that the island formation caused by adsorbate-adsorbate interaction and/or limited mobility of reactants may affect such regimes has been widely used for 2 decades (see, e.g., experimental studies of CO10 and H211 oxidation on Pt and Pd and related simulations12). During the past 10 years, the nanometer-sized oxygen islands have been explicitly observed (by using scanning tunneling microscopy (STM)) during titration of preadsorbed oxygen, e.g., by CO on Rh(110),13 Cu(110),14 and Pt(111),15 hydrogen on Ni(110)16 and Pt(111),17 methanol (8) Zhdanov, V. P. Elementary Physicochemical Processes on Solid Surfaces; Plenum: New York, 1991. (9) Bray, A. J. Adv. Phys. 1994, 43, 357. (10) Conrad, H.; Ertl, G.; Kuppers, J. Surf. Sci. 1978, 76, 323. Gland, J. L.; Kollin, E. B. J. Chem. Phys. 1983, 78, 963. Matsushima, T.; Matsui, T.; Hashimoto, J. J. Chem. Phys. 1984, 81, 5151. (11) Gland, J. L.; Fisher, G. B.; Kollin, E. B. J. Catal. 1982, 77, 263. (12) Silverberg, M.; Ben-Shaul, A.; Rebentrost, F. J. Chem. Phys. 1985, 83, 6501. Silverberg, M.; Ben-Shaul, A. J. Chem. Phys. 1987, 87, 3178. Hellsing, B.; Zhdanov, V. P. Chem. Phys. Lett. 1988, 147, 613. Zgrablich, G.; Sales, J. L.; Unac, R.; Zhdanov, V. P. Surf. Sci. 1993, 290, 163. Zhdanov, V. P. Phys. Rev. Lett. 1996, 77, 2109. (13) Leibsle, F. M.; Murray, P. W.; Francis, S. M.; Thornton, G.; Bowker, M. Nature 1993, 363, 706. (14) Crew, W. W.; Madix, R. J. 1996, 349, 275; 1996, 356, 1. (15) Wintterlin, J.; Vo¨lkening, S.; Janssens, T. V. W.; Zambelli, T.; Ertl. G. Science 1997, 278, 1931. (16) Sprunger, P. T.; Okawa, Y.; Besenbacher, F.; Stensgaard, I.; Tanaka, K. Surf. Sci. 1995, 344, 98. (17) Renisch, S.; Schuster, R.; Wintterlin, J.; Ertl. G. Phys. Rev. Lett. 1999, 82, 3839.

10.1021/la001161j CCC: $20.00 © 2001 American Chemical Society Published on Web 01/31/2001

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on Cu(110),18 and ammonia on Ni(110),19 Cu(110),20 and Rh(110)21 (a good review of recent STM studies of catalytic reactions has recently been published by Wintterlin22). Experimental reports on observation of pattern formation on the micrometer scale in catalytic reactions exhibiting kinetic oscillations (e.g., in CO oxidation on Pt(110)) are numerous.23,24 Such parterns related to adsorbate-induced surface restructuring are however of the Turing-like type; i.e., they can be described by using the conventional MFRD equations25 (there is no need in attractive adsorbate-adsorbate interactions). There are numerous indirect experimental data26 indicating that kinetic oscillations observed in such reactions as CO oxidation or NO reduction by CO or H2 on Pt(100) should be accompanied by phase separation on the nanometer scale (see also recent Monte Carlo (MC) simulations27), but this effect is again connected primarily with adsorbateinduced surface restructuring. Direct experimental data on the pattern formation in catalytic reactions on the nanometer scale under the steady-state conditions are scarce, because such experiments are difficult in realization.22 To our knowledge, there is only one relevant report14 on observation of islands by STM (during CO oxidation on Cu(110)) and there are a few reports28 on observations of patterns in reactions (e.g., H2 and CO oxidation on Pt) occurring on a tip of the field ion microscope. Despite this state of the art, one can expect that with the development of experimental methods the problem under consideration will attract more attention. Theoretical studies of 2D first-order phase separations caused by attractive adsorbate-adsorbate under the steady-state reactive conditions are also not numerous. To clarify the situation in this field, we have executed29 MC simulations of the simplest catalytic reaction occurring under the steady-state conditions on a square lattice via the Eley-Rideal (ER) mechanism, including A adsorption on vacant sites

Agas f Aads

(1)

and A consumption in collisions with gas-phase B particles

Aads + Bgas f (AB)gas

(2)

To incorporate the island formation into this scheme, the lateral interaction between NN A particles was considered to be attractive, 1 < 0, and the temperature was put below (18) Leibsle, F. M.; Francis, S. M.; Davis, R.; Xiang, N.; Haq, S.; Bowker, M. Phys. Rev. Lett. 1994, 72, 2569. (19) Ruan, L.; Stensgaard, I.; Lagsgaard, E.; Besenbacher, F. Surf. Sci. 1994, 314, L873. (20) Guo, X.-C.; Madix, R. J. Surf. Sci. 1996, 367, L95. (21) Kiskinova, M.; Baraldi, A.; Rosei, R.; Dhanak, V. R.; Thornton, G.; Leibsle, F. M.; Bowker, M. Phys. Rev. B 1995, 52, 1532. (22) Wintterlin, J. Adv. Catal. 2000, 45, 131. (23) Imbihl, R.; Ertl, G. Chem. Rev. 1995, 95, 697. Ertl, G. Adv. Catal. 2000, 45, 1. (24) Rotermund, H. H. Surf. Sci. Rep. 1997, 29, 265. (25) Eiswirth, M.; Ertl, G. In Chemical Waves and Patterns; Kapral, R., Showalter, K., Eds.; Kluwer: Dordrecht, 1994; p 447. (26) Crossley, A.; King, D. A. Surf. Sci. 1980, 95, 131. Behm, R. J.; Thiel, P. A.; Norton, P. R.; Ertl, G. J. Chem. Phys. 1983, 78, 7437. Gardner, P.; Martin, R.; Tu¨shaus, M.; Bradshaw, A. M. J. Electron. Spectrosc. Relat. Phenom. 1990, 54, 619. (27) Zhdanov, V. P. Surf. Sci. 1999, 426, 345. J. Chem. Phys. 1999, 110, 8748. Phys. Rev. E 1999, 59, 6292. Zhdanov, V. P.; Kasemo, B. Phys. Rev. E 2000, 61, R2184. (28) Gorodetskii, V.; Lauterbach, J.; Rotermund, H. H.; Block, J. H.; Ertl, G. Nature 1994, 370, 276. Gorodetskii, V. V.; Drachesel, W. Appl. Catal. A 1999, 188, 267. Suchorski, Y.; Imbihl, R.; Medvedev, V. K. Surf. Sci. 1998, 401, 392. Cobden, P. D.; van Breugel, J.; Nieuwenhuys, B. E. Surf. Sci. 1998, 404, 1998. (29) Zhdanov, V. P. Surf. Sci. 1997, 392, 185.

the critical one, T < Tc ) 0.567|1|. The rates of the adsorption and reaction steps ((1) and (2)) were assumed to be independent of the arrangement of particles in adjacent sites. Diffusion of adsorbed A particles was considered to occur via jumps to NN vacant sites. The dimensionless jump probabilities were calculated by employing the standard Metropolis rule (we put kB ) 1)

Pidif )

{

1 for f e i exp[-(f - i)/T] for f > i

(3)

and also a more realistic rule

Pidif ) (i/T)

(4)

often used in simulations of surface diffusion.31 In the former case, the jump rate of a given particle is dependent on the difference of the adsorbate-adsorbate interactions in the final and initial states (the subscripts f and i correspond to these states). In the latter case, the jump rate is defined via the lateral interaction in the initial state. The main qualitative difference between eqs 3 and 4 is that the former one predicts the same rates of diffusion at low and high coverages while the latter one predicts different rates. In reality, the rates of diffusion at low and high coverage are almost never equal.31,32 In this sense, the “initial state” diffusion dynamics defined by eq 4 is more realistic than that given by eq 3. With the rules described above, the formation of large frozen (1 × 1) islands is easily observed29 only for the Metropolis dynamics of surface diffusion (these islands are similar to those found earlier30 in MC simulations of the reversible noncatalytic A h B reaction in a binary mixture during spinodal decomposition with the Kawasaki dynamics of exchange of NN molecules). For a more realistic dynamics (eq 4), the traveling islands with an appreciable size were obtained only if the reaction was several orders of magnitude slower than diffusion. The difference in the results predicted by the two dynamics is connected with the ratio of the rates of diffusion jumps of monomers in the dilute phase and holes in the dense phase. For the Metropolis dynamics, as already noted, the jump rates of monomers and holes are equal. With this ratio of jump rates, diffusion (toward the island boundaries) of monomers, arriving from the gas phase to the dilute surface phase, can easily be compensated by diffusion of holes, produced in the dense surface phase due to reaction, and accordingly one may obtain large frozen patterns even at moderate rate of diffusion. For more realistic diffusion (e.g., for eq 4), the formation of large islands is effectively suppressed because the jump rate of monomers is much higher than that of holes and accordingly the balance of diffusion fluxes can hardly be established. The island formation related to adsorbate-adsorbate interactions has also been simulated33 in the A + B reaction occurring under the steady-state conditions via the Langmuir-Hinshelwood (LH) mechanism. The analysis was based on the MFRD equations with long-range attractive A-A and A-B interactions influencing the rate of A diffusion (the B-B interaction was considered to be irrelevant, because B particles were assumed to be (30) Glotzer, S. C.; Stauffer, D.; Jan, N. Phys. Rev. Lett. 1994, 72, 4109. (31) Nieto, F.; Tarasenko, A. A.; Uebing, C. Defect Diffus. Forum 1998, 162, 59. (32) Gomer, R. Rep. Prog. Phys. 1990, 53, 917. (33) Hildebrand, M.; Mikhailov, A. S.; Ertl, G. Phys. Rev. Lett. 1998, 81, 2602.

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immobile). In reality, the long-range attractive adsorbateadsorbate interactions do not seem to exist (see, e.g., ref 34 for a review). Another shortcoming of simulations33 is connected with the employed diffusion dynamics which predicts the same value of the chemical diffusion coefficient for dilute and dense phases. The simulations29,33 were focused on the case when the adsorbate-adsorbate interactions affect only the rate of diffusion jumps. Elementary reaction acts were assumed to be independent of surrounding particles. In reality, the adsorbate-adsorbate interactions may modify the reaction rate as well. Our recent Monte Carlo simulations35 of the 2A + B2 reaction occurring via the LH mechanism indicate that the latter effect may be important for the understanding of pattern formation in chemically reactive adsorbed overlayers. In the present study, we confirm this conclusion by analyzing in detail the kinetics of the ER A + B reaction. The latter reaction is simpler compared to that treated in ref 35. For this reason, the results presented below are more transparent than those obtained earlier.35 Employing our model, we show patterns which can be observed on a uniform single-crystal surface (sections 2-4) and also on the surface of nanometer-size supported catalyst particles (section 5). The latter case not treated before is especially interesting, because it contributes to the formation of the conceptual basis for the understanding of the reaction kinetics occurring on practically important supported catalysts.

Figure 1. Schematic potential-energy surface along the reaction coordinate. The thick and thin solid lines correspond to the cases when the reacting particles have no neighbors and have neighbors, respectively. i and i* are the lateral adsorbateadsorbate interactions in the initial and activated states. The subscript i characterizes the arrangement of particles.

2. Model In our simulations, as already noted, the A + B reaction is considered to occur via steps 1 and 2. To mimic the reaction kinetics on a single-crystal surface, we use a L × L square lattice with periodic boundary conditions (the specifications corresponding to supported catalysts are described in section 5). The lateral interaction between NN A particles is assumed to be attractive, 1 < 0. The interactions beyond the NN sites are ignored. To incorporate the effect of lateral interactions on the rates of reaction and A diffusion, we recall that according to the transition state theory the rate constant of an elementary rate process for a given arrangement of adjacent particles is represented as8

Figure 2. Reaction rate (AB molecules per site per 1000 MCS) as a function of p for P ) 0.001 and 1* ) 1. The data points were obtained starting with a clean lattice and p ) 0.1. The reaction rate for this value of p was calculated after 104 MCS. The configuration obtained in the end was then used as the initial condition for simulating the reaction kinetics at p ) 0.2. The reaction rate for the latter value of p was calculated after additional 104 MCS. This procedure was repeated step by step up to p ) 0.9.

ki ) ko exp[-(i* - i)/T]

i ) n1

(5)

where ko is the pre-exponential factor, i and i* are the lateral interactions in the initial and activated states (Figure 1), and i is the subscript characterizing the arrangement. (Note that at adsorption equilibrium the probabilities of arrangements of adsorbed particles depend only on the former interactions. Thus, in real systems, the interactions i can be obtained, e.g., by analyzing a phase diagram of the overlayer. In contrast, the interactions i* are manifested primarily in the dynamics of rate processes. For example, due to the former interactions, the energy distribution of the reaction products directly desorbing to the gas phase may depend on coverage.36,37) Equation 5 is applicable to both diffusion and reaction. In particular, taking into account that the initial state for diffusion and reaction is the same, we have in both cases (34) Norskov, J. K. In Coadsorption, Promoters and Poisons; King, D. A., Woodruff, D. P., Eds.; Elsevier: Amsterdam, 1993; p 1. (35) Zhdanov, V. P.; Kasemo, B. Surf. Sci. 1998, 412, 527. (36) Zhdanov, V. P. Surf. Sci. 1986, 165, L31. (37) Bald, D. J.; Bernasek, S. L. J. Chem. Phys. 1998, 109, 746.

(6)

where n is the number of NN particles. On the other hand, the pre-exponential factor and the interaction in the activated state for diffusion and reaction are expected to be different. For diffusion, the lateral interaction in the transition state is often weaker than that in the ground (initial) state. For this reason, we neglect the interaction i* (in order to reduce the number of model parameters). In this case, eq 5 becomes equivalent to eq 4. Substituting expression 6 into eq 4, we have

Pndif ) exp(n1/T)

(7)

If necessary, one can easily take into account the interaction i*. It will not change the final results qualitatively provided that the rate of diffusion of monomers is higher that that of holes. For the ER reaction, we employ i* ) n1*, where 1* is the NN lateral interaction in the activated state. Substituting this expression and eq 6 into eq 5 yields the normalized reaction probability

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Figure 3. Typical arrangements of A particles on a 50 × 70 fragment of the 200 × 200 lattice after reaching the steady-state reaction regime (Figure 2) at p ) 0.25 (a) and 0.5 (b). At these p values, the average coverage is one-quarter and one-half mononalyer (ML), respectively.

Pnrea ) exp[-n(1* - 1)/T)

(8)

In our previous simulations,29 the reaction probability was assumed to be independent of the arrangement of particles in adjacent sites. Physically, this corresponds to the situation when 1* ) 1. In the present work, we study a more general case when |1*| < |1|. To characterize the relative rates of adsorption and reaction, we use the dimensionless parameter p (0 e p e 1). The rates of these processes are considered to be proportional to p and (1 - p), respectively. Physically, the parameter p is defined as p ) kadPA/(kadPA + krPB), where kad is the adsorption rate constant, kr is the reaction rate constant at low coverages, and PA and PB are the reactant gas-phase pressures. In addition, we introduce the dimensionless parameter P characterizing the relative role of the adsorption/reaction channels in the whole adsorption/reaction/diffusion process. 3. Algorithm of Simulations After the specification above, the MC algorithm for describing the reaction is as follows: (i) A site for one of the elementary processes is chosen at random. (ii) The adsorption/reaction channels are selected with the probability P and the diffusion channel with the probability 1 - P, respectively. (iii) In the former case, the A adsorption step and the ER reaction are selected with the probabilities p and 1 p, respectively. An adsorption attempt is considered to be successful if the site chosen is vacant (if the site is occupied, the trial ends). A reaction attempt is successful if (a) the site chosen is occupied and (b) F e Pnrea, where F is a random number (0 e F e 1), and Pnrea is the normalized reaction probability defined by eq 8. If the site is vacant or F > Pnrea, the trial ends. (iv) In the case of diffusion, the trial ends if the site is vacant. Otherwise, an A particle located in this site tries to diffuse. In particular, an adjacent site is randomly selected, and if this site is vacant, the A particle jumps to it with the normalized probability prescribed by eq 7. If the adjacent site is occupied, the trial ends. Time is calculated in MC steps (MCS); 1 MCS corresponds to L × L attempts to realize one of the rate processes. All the simulations presented below in section 4 were executed on a 200 × 200 lattice at T ) 0.5Tc. The durations of MC runs were chosen to guarantee establishing the steady-state regime. Specifically, the runs were

so long that the average size of islands, defined as the position of the first zero of the occupation-number correlation function, was independent of the initial conditions. 4. Patterns on a Uniform Surface To illustrate the effect of adsorbate-adsorbate interactions on the reaction kinetics, it is instructive to start from the case when the rate of elementary reaction acts is independent of surrounding particles. This case treated in detail earlier29 is realized at 1* ) 1. Under this condition, the coverage and reaction rate are given by29 θA ) p and W ) p(1 - p). MC simulations reproduce this dependence of W on p (Figure 2) and indicate as noted in the Introduction that the size of A islands is rather small (see, e.g., Figure 3 for P ) 0.001). The reaction kinetics and typical arrangements of adsorbed particles for 1* ) 0.81 and P ) 0.001 are shown in Figures 4 and 5. In this case, the influence of surrounding particles on elementary reaction acts is relatively weak. Nevertheless, the size of islands is appreciably larger than for 1* ) 1 (cf. Figures 3 and 5). Figures 6 and 7 exhibit the results of simulations for 1* ) 0.51 and P ) 0.001. In this case, the size of islands is much larger than that for 1* ) 1 (cf. Figures 3 and 7). The results obtained show that the influence of adsorbate-adsorbate interaction in the activated state on the island formation may be significant. Even if this influence is favorable, the conditions for the formation of large islands are however rather tough. In particular, the islands obtained in our simulations are relatively small even if diffusion is rapid. One of the reasons why this is the case is connected with the difference of the rates of diffusion at low and high coverages (see the Introduction). The other reason is related with rapid decrease of the driving force for the island growth with increasing island size (as in the case of the island growth without reaction9). 5. Patterns on Nanometer-Size Supported Catalyst Particles Small supported catalyst particles (∼10 nm) usually contain a set of facets with the largest area belonging to close-packed facets.39 In the simplest case, accepted in our treatment, the supported particle is shaped as a pyramid with small top and large bottom (100) facets (the bottom facet contacts the support) and side (111) facets (38) Yates, J. T. J. Vac. Sci. Technol., A 1995, 13, 1359. (39) Marks, L. D. Rep. Prog. Phys. 1994, 57, 603.

Island Formation in Catalytic Reactions

Figure 4. (a) Coverage and (b) reaction rate (AB molecules per site per 1000 MCS) as a function of p for P ) 0.001 and 1* ) 0.81. The data points were obtained starting with a clean lattice and p ) 0.02. The reaction rate for this value of p was calculated after 105 MCS. The configuration obtained in the end was then used as the initial condition for simulating the reaction kinetics at p ) 0.04. The reaction rate for the latter value of p was calculated after additional 105 MCS. This procedure was repeated up to p ) 0.2.

(supported particles of this shape are often observed in real systems7,39). The global shape of such catalyst particles and the minimum-energy arrangement of substrate atoms on the (001) facet may be modified by adsorbates. In our present simulations, focused on the pattern formation, we assume that the same facet arrangement is stable, independent of variations in surface coverage. To mimic a catalyst particle, we use a 100 × 100 square lattice, where the central 50 × 50 array of sites mimics the top (001) facet, and the periphery represents the side (111)

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facets. In the framework of this model, one can take into account different catalytic and adsorption properties of the top (central) and side (peripheral) facets. In our treatment of the A + B reaction (steps 1 and 2), we assume that only the top facet is catalytically active (no reaction on the periphery). Attractive NN lateral A-A interaction is also introduced only for the top facet (note that in real systems the adsorbate-adsorbate lateral interaction on the (100) face is often stronger than that on the (111) face). In addition, we take into account that the sticking coefficients for A adsorption and the A adsorption energies on the top and side facets may be different. Many other effects can also be easily incorporated into the model, but are ignored in the present treatment, to not obscure the main message. In reality, for example, the arrangements of sites on the (001) and (111) facets have respectively a square and triangular symmetry. This effect is however relatively weak compared to that resulting from the difference in the catalytic activity, and accordingly it has been omitted. The edge sites are omitted from the analysis as well, although they might certainly be important in a full treatment (in analogy with step sites,38 the edge sites may sometimes have high catalytic activity). The algorithm of simulations of the reaction kinetics on a supported catalyst particle is basically the same as that described in section 3. The only significant difference is that now we use the open (“no flux”) boundary conditions and introduce somewhat more complex rules for A diffusion jumps between NN sites. Specifically, we employ the initial-state diffusion dynamics taking into account the difference of the A adsorption energies on the top and side facets. In particular, Pndif ) 1, if the initial (filled) and final (vacant) sites are located on the side facets. Pndif ) exp(n1/T), if the sites are located on the central facet (this rule is the same as that used in sections 2-4 (cf. eq 7)). If A adsorption is energetically preferable on the top facet, Pndif ) 1 for the side f top jumps, and Pndif ) exp[(n1 - ∆E)/T] for the top f side jumps (∆E g 0 is the difference of the adsorption energies). If A adsorption is preferable on the side facet, the corresponding probabilities are Pndif ) exp(-∆E/T) and Pndif ) exp(n1/T) (here we also have ∆E g 0, because ∆E is defined as the difference of the larger and smaller adsorption energies). Typical patterns predicted at T ) 0.5Tc are shown in Figures 8 and 9 for the cases when the A sticking coefficients and adsorption energies on the top and side facets are equal, and 1* ) 1 and 1* ) 0.81, respectively.

Figure 5. Arrangements of A particles on a 50 × 70 fragment of the 200 × 200 lattice after reaching the steady-state reaction regime (Figure 4) at p ) 0.06 (a) and 0.13 (b). At these p values, the average coverage is 0.26 and 0.50 ML, respectively.

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Figure 6. (a) Coverage and (b) reaction rate (AB molecules per site per 100 MCS) as a function of p for P ) 0.01 and 1* ) 0.51. The data points were obtained starting with a clean lattice and p ) 0.002. The reaction rate for this value of p was calculated after 105 MCS. The configuration obtained in the end was then used as the initial condition for simulating the reaction kinetics at p ) 0.004. The reaction rate for the latter value of p was calculated after additional 105 MCS. This procedure was repeated up to p ) 0.02.

In both cases, the side facets are completely covered by A. On the top facet, A molecules completely fill the sites located near the boundary between the facets. Blocking of these sites suppresses the A supply from the periphery to the central facet. The size of islands on the central facet is relatively small despite the fact that the A diffusion is 3 or 4 orders of magnitude faster compared to reaction. Keeping the main kinetic parameters the same as in Figures 8 and 9, we have also executed simulations with a lower A sticking probability for the side facets and/or unequal adsorption energies on the top and side facets (e.g., for ∆E/T ) 2). The effect of these parameters on the patterns was found to be nearly negligible. This finding can be rationalized if one takes into account that the A supply from the side facets to the center is limited not by A adsorption on these facets or by A jumps to the top facet but rather by A 2D evaporation from the site located on the top facet near the boundary with the side facets. With increasing the driving force for the island formation by decreasing P or 1* (e.g., for 1* ) 0.51), the top facet was found to be covered by A almost completely. 6. Conclusion In summary, our earlier simulations35 of the 2A + B2 reaction occurring via the LH mechanism and present simulations of the ER A + B reaction clearly indicate that the adsorbate-adsorbate lateral interactions suppressing the rate of elementary reaction acts inside islands may facilitate the island formation in catalytic reactions

Figure 7. Arrangements of A particles on a 50 × 70 fragment of the 200 × 200 lattice after reaching the steady-state reaction regime (Figure 6) at p ) 0.008 (a) and 0.012 (b). At these p values, the average coverage is 0.25 and 0.49 ML, respectively. The data were obtained by depositing 0.25 or 0.49 ML of A particles at random and then running the reaction up to t ) 105 MCS. Panel c shows the arrangement of particles for the same run as for (b) but for t ) 1.1 × 105 MCS. The additional 104 MCS, corresponding to exposition of 1.2 ML of A particles, result in appreciable changes in the position of island boundaries (cf. panels b and c). This indicates that the islands are traveling.

occurring on single-crystal surfaces and nanometer-size supported particles. For the model under consideration, the average island size is however rather small even if adsorbate diffusion is much faster than reaction. The latter

Island Formation in Catalytic Reactions

Figure 8. Typical arrangement of A molecules after reaching the steady-state reaction regime (at t ) 107 MCS) on the 100 × 100 lattice mimicking a nanometer catalyst particle. A molecules, located on the central and peripheral areas representing the top and side facets, are shown by filled and open circles, respectively. The parameters employed in simulations are as follows: p ) 0.05, P ) 10-4, ∆E ) 0, and 1* ) 1.

is connected with different rates of diffusion in the dilute and dense phases and also with a decrease of the driving force for the island formation with increasing the average island size. Albeit our present study is focused on attractive adsorbate-adsorbate interactions, we should note that the island formation in catalytic reactions is possible for repulsive interactions as well, because the reaction rate depends on the difference of lateral interactions in the activated and ground states (see, e.g., eqs 5 or 8) and the latter may be positive for both types of interactions. The specifics of the two cases are in the driving force for the island formation. For attractive interactions, the driving force is primarily thermodynamical and accordingly the size of islands increases with increasing the rate of adsorbate diffusion. For repulsive interactions, in contrast, the driving force is purely kinetic. In the latter case, even relatively slow surface diffusion may easily prevent the island formation. Finally, it is appropriate to notice that in real catalytic reaction the LH reaction steps usually dominate. The examples when reactions run via the ER or ER-like mechanism are however also available. For instance, C3H8

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Figure 9. Arrangement of A molecules after reaching the steady-state reaction regime (at t ) 106 MCS) on the 100 × 100 lattice for p ) 0.01, P ) 0.001, ∆E ) 0, and 1* ) 0.81. A molecules, located on the central and peripheral areas representing the top and side facets of a catalyst particle, are shown by filled and open circles, respectively.

oxidation on Pt/Al2O3 under lean-burn conditions40 seems to be limited by the ER-type dissociative C3H8 chemisorption occurring with breaking one of the C-H bonds formed by the central carbon atom

(CH3-CH2-CH3)gas + Oad f (CH3-CH-CH3)ads + (OH)ad Subsequent reaction steps (after breaking the first C-H bond) are believed to be rapid. Thus, adsorbed atomic oxygen is expected to be the dominant species under reaction conditions. Under such circumstances, one basically has here the ER-like A + B reaction. Such examples show that despite the simplicity of our A + B model the corresponding results may be relevant for interpretation of the kinetics of specific practically important reactions. Acknowledgment. The author thanks B. Kasemo for useful discussions. The financial support for this work was obtained from the Waernska Guest Professorship Fund at Go¨teborg University. LA001161J (40) Burch R.; Watling, T. C. J. Catal. 1997, 169, 45.