Langmuir 1992,8, 1757-1761
1757
Thermal Desorption of Interacting Particles from Honeycomb and Triangular Lattices L. V. Lutsevich,+0. A. Tkachenko,?and V. P. Zhdanov'J Computer Center, Novosibirsk 630090,Russia, and Institute of Catalysis, Novosibirsk 630090,Russia Received December 10, 1991
The lattice-gas model is employed to study by Monte Carlo simulations associative desorption from a honeycomb lattice and monomolecular desorption from a triangular lattice. In the former case, the nearest-neighbor lateral interaction is assumed to be so strong that the nearest-neighbor pairs of adparticles are absent. In the latter case, the occurrence of adsorption at both top and bridge sites is taken into account. General results are used to interpret thermal desorption data for the H/Ni(lll) and CO/Pt(lll) systems. These approximations are rather rough at T < T,,where T,is a critical temperature of phase transitions in the In the case of chemisorption of atoms and simple adsorbed overlayer. More precise results may be derived molecules on close-packed faces of single crystals, the by employing Monte Carlo simulationss or the transferassumption of surface uniformity is often justified; i.e. it matrix technique.' may be assumed that adsorbed particles are distributed Almost all the author^^-^ to date have analyzed the among equivalent elementary cells. The nonideality of kinetics of various rate processes on a square lattice. the adsorbed layer in this case is due to lateral interactions Recently, we8 have studied thermal desorption spectra between adsorbed particles. In statistical physics, a system for adsorption on the vertex sites of a triangular lattice. of interacting particles distributed among equivalent cells The results obtained have been used to interpret thermal is called a lattice gas. It turns out that many phenomena desorption (TD) spectra for CO on Ru(001). The objective occurring on the surfaces of solids (kinetics of adsorption of the present Monte Carlo simulations is to explore the and desorption, kinetics of chemical reactions, surface diffusion, phase diagrams, surface reconstruction induced effect of lateral interactions on TD spectra for dissociative by adsorption) can be described in the framework of the adsorption on a honeycomb lattice and monomolecular lattice-gas model.' adsorption on a triangular lattice. In the latter case, we The general formulas for describing various phenomena will assume the occurrence of adsorption a t both top and in the lattice-gas model have, as a rule, a simple form. bridge sites. General results will be applied to hydrogen However, these formulas are not so much a solution as a adsorption on Ni(ll1) and CO adsorption on Pt(ll1). formulation of the problem, since the main difficulty lies To simulate thermal desorption, we have employed the in calculating the various probabilities appearing in these algorithm described in detail by Sales and Zgrablichs and formulas. Indeed, the lattice-gas model is well-known to also our own algorithm.8 Both approaches have been be exactly resoluble only in exceptional cases.2 The shown to yield the same results. We discuss only briefly kinetics of real surface processes is usually studied at the technical details of calculations because the setup for comparatively high temperatures. In this temperature simulations in the present paper is in fact the same as range, the cluster method is suitable for calculating the described in ref 8. rates of elementary processes. As a rule, the practical All the results presented below are obtained assuming calculations take into account lateral interactions only that surface diffusion is rapid in comparison with debetween nearest neighbors and the mean-field, the quasichemical, or the Bethe-Peierls approximations are ~ s e d . ~ - ~sorption, and consequently, the adsorbed overlayer is considered to be in an equilibrium state. This assumption is usually justified because in real systems the activation * To whom correspondence should be sent. energy for surface diffusion is customarily low in comt Computer Center, Novosibirsk. t Institute of Catalysis, Novosibirsk. parison with that for desorption. 1. Introduction
(1)Zhdanov, V. P. Elementary Physicochemical Processes on Solid Surfaces;Plenum: New York, 1991. Lombardo, S.J.; Bell, A. T. Surf. Sci. Rep. 1991,13,1. (2)Baxter, R. J. Exactly Solued Models in Statistical Mechanics; Academic Press: London, 1982. (3)King, D.A. Crit.Reu. Solid State Mater. Sci. 1978,7,167.Adams, D. L. Surf. Sci. 1974,42,12. (4)Zhdanov, V. P. Surf. Sci. 1981,111, 63. (5)Zhdanov, V. P. Surf. Sci. 1982,123,106;1983;133,469;1984,137, 515;1986,f69,l;1987,179,L57. Benziger, J. B.; Schoofs, G. R. J. Phys. Chem. 1984,88,4439.Sundaresan, S.;Kaza, K. R. Surf. Sci. 1985,160, 103;Chem. Eng. Commun. 1985,32,333;1985,35,1.Pak, H.; Evans, J. W. Surf.Sci. 1987,186,550. Evans, J. W.;Hoffman, K. K.;Pak, H. Surf. Sci. 1987,192,475.Kreuzer, H. J.; Payne, H. S. Surf. Sci. 1988,198,235; 1988,200,L433;1988,205,153;1989,222,404.Hellsing, B., Zhdanov, V. P. Chem. Phys.Lett. 1988,147,613.Asada, H.; Masuda,M. Surf. Sci. 1989,207,517. Surda, A.; Karasova, I. Surf. Sci. 1981,109,605.Surda, A. Surf. Sci. 1989,220,295.
(6)Bridge, M.E.;Lambert, R. M. Proc. R. Soc. London, A 1980,370, 545;Surf. Sci. 1980,94,469.Silverberg, M.; Ben-Shad, A.; Robentrost, F. J . Chem. Phys. 1985,83,6501.Silverberg, M.; Ben-Shad, A. Chem. Phys.Lett. 1987,134,491; J. Chem.Phys. 1987,87,3178;J. Stat. Phys. 1988,52,1179; Surf.Sci. 1989,214,17.Stiles, M.;Metiu, H. Chem. Phys. Lett. 1986,128,337.Gupta, D.;Hirtzel, C. S. Chem. Phys. Lett. 1988, 149,527;Surf. Sci. 1989,210,208;Mol. Phys. 1989,68, 583. Evans, J. W.;Pak, H. Surf. Sci. 1988,199,28. Lombardo, S.J.; Bell, A. T. Surf. Sci. 1988,206,101;1989,224,451.Sales, J. L.;Zgrablich, G. Surf. Sci. 1987,187,1; Phys. Rev. B: Condens. Matter 1987,35, 9520. Sales, J. L.;Zgrablich, G.; Zhdanov, V. P. Surf. Sci. 1989,209,208. (7)Myshlyavtaev, A. V.;Zhdanov, V. P. Chem. Phys. Lett. 1989,162, 43. Myshlyavtaev, A. V.;Sales, J. L.; Zgrablich, G.; Zhdanov, V. P. J. Stat. Phys. 1990,58,1029. (8)Lutaevich,L.V.;Tkachenko, 0.A.; Zhdanov,V. P. Langmuir 1991, 7.1225.
0 1992 American Chemical Society
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1758 Langmuir, Vol. 8, No. 7, 1992
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Figure 1. Particles on a honeycomb lattice formed by %fold hollows on a close-packedface. Panel a shows lateral interactions between adparticles. Panels b-d display the ordered structures at 0 = l / ~ ,2/3, and 1. Location of particles in nearest-neighbor sites is forbidden,
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TEIIPKRATUUE ( I ) Figure 2. Thermal desorption spectra for adsorption on a honeycomb lattice at c2 = 1.5 kcal/mol. The third-neighbor interaction (cg, in kcal/mol) is shown in the upper right-hand part of the panels. The initial coverages are 0.2,0.4,0.6, and 0.9. The results presented in Figures 2-4 were obtained by employing a (100X 100)lattice with periodic boundary conditions. The number of diffusion steps per desorption step is 10.
2. Associative Desorption from a Honeycomb Lattice In this section, we will simulate associative desorption from the vertex sites of a honeycomb lattice (Figure 1) taking into account the second-neighbor and thirdneighbor pair interactions, €2 and €3. The nearest-neighbor interaction €1 is assumed to be repulsive and so strong
that the nearest-neighbor pairs of adparticles are absent. The adsorbate coverage will be normalized so that B = 1 corresponds to the (1 X 1) structure (Figure Id). The model Presented is applicable, for example, to the WNi(111)system, where hydrogen occupies both the fcc and hcp 3-fold hollows and the binding energy difference between the two kinds of sites is insignificant.9-11
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TEHPERATURE (I) Figure 3. Thermal desorption spectra for adsorption on a honeycomb lattice at e2 = 1 kcal/mol. The interaction (3 (in kcal/mol) is shown in the upper right-hand part of panels. The initial coverages are 0.2, 0.4, 0.6, and 0.9.
Assuming that associative desorption occurs only from next-nearest-neighbor sites, we can represent the desorption rate as ~ s u a l l * ~ I
where v is the preexponential factor, E d o the energy difference between the activated complex and the nextnearest-neighbor pair AA provided that the adjacent sites are empty, pM,ithe probability that a pair of next-nearestneighbor sites is occupied by particles A (with the environment marked by index i), and At; = ti* - ti, where ti is the lateral interaction between pair AA and the environment (repulsiveinteractions are assigned positive values) and ti* is the lateral interaction of the activated complex with the same environment; the Boltzmann constant is set to unity. The interaction ti* is customarily believed to be weak compared to the other interactions and we will assume ti* = 0. The results of Monte Carlo simulationsof TD spectra carried out in accordance with eq 1with v = 4 X 1013s-l, Edo = 26 kcal/mol, and /3 = 10 K/s (/3 is the heating rate) are shown in Figures 2 and 3. If the repulsive next-nearestneighbor interaction is rather strong, €2 = 1.5 kcal/mol, and the third-neighbor interaction is absent, €3 = 0, the spectrum contains three peaks (see the central panel in Figure 2). This spectrum can at least partly be rationalized by employingsimple qualitative suggestions. In particular, at coverages up to B = l/2, the adsorbed particles can be ordered so that next-nearest-neighborsites are unoccupied (see, e.g., the (2 X 2) structure at B = 1/2 in Figure lb). This results in formation of the high-temperature peak with the integral intensity of N V 2 . With increasing coverge, (9) Christmann,K.; Behm, R. J.;Ertl, G.; Van Hove, M. A.; Weinberg, W. H. J. Chem. Phys. 1979, 70,4168. (10) Van Hove, M. A.; Weinberg, W. H.; Chan, C.-M. Low-Energy Electron Diffraction; Springer: Berlin, 1986. (11) Roelofs, L. D.; Einstein, T. L.; Bartelt, N. C.; Shore, 3. D. Surf. Sci. 1986,176,295. Einstein, T. L.; Daw, M. S.; Foiles, S. M. Surf. Sci. 1990,227, 114.
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Figure 4. Thermal desorption spectra for hydrogen adsorption on Ni(ll1): (a) theoretical spectra at Y = 4 X 1013s-l, E d o = 26 kcal/mol, e2 = e3 = 0.4 kcal/mol, @ = 10 K/s, and the initial coverages of 0.2,0.4,0.6,and 0.9;(b)experimental spectraewith exposuresof 1,20,150,and 4oooO langmuirs (the latter exposure corresponds'to 8 N 0.9).
the other ordered structures can be formed (see, e.g., Figure IC)and we can expect the appearance of low-temperature peaks with the integral intensities less than '/2. If we take into account the attractive third-neighbor interaction of -0.5 kcal/mol (the left-hand panel in Figure 2), the spectrum is qualitatively the same but all the peaks are more pronounced. The repulsive third-neighbor interaction of 0.5 kcal/mol (the right-hand panel in Figure 2) results in the smearing of the intermediate and high-temperature peaks. The same effect occurs if we decrease the next-nearest-neighborinteraction (see,e.g., Figure 3, where e2 = 1 kcal/mol). Finally, it is of interest to discuss TD spectra for hydrogen adsorption on Ni(ll1) (Figure 4). Lateral interactions for this system have been calculated in ref 11. Unfortunately, the theory yields the interactions that are
1760 Langmuir, Vol. 8, No. 7,1992
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TEHPBRATURE ( I ) Figure 6. Thermal desorption spectra (the upper panel) for adsorption on a triangular lattice at €2 = 1 kcal/mol and cg = 0.5 kcal/mol. The lower panel shows the total coverageand coverages corresponding to molecules located in top and bridge sites. The results presented in Figures 6-8 are obtained by employing a (100 X 100) lattice with periodic boundary conditions. The number of diffusion steps per desorption step is 100.
e = 1/2
Figure 5. Particles on a triangular lattice. Panel a indicates various lateral interactions. Panels b and c show the ( 4 3 x 4 3 ) R30° and ~ ( 4 x 2 structures ) at B = and 0 = '12. Location of particles at R < a is forbidden (a is the lattice spacing).
adsorption takes place above f3 = l / 3 (see the analysis of the IRS, HREELS, and LEED data12). In the framework of the lattice-gas approximation, the kinetics of CO desorption from Pt(ll1) can be described as
+ Aer)/Tlf3-
df3/dt = - V ~ C P :exp[-(& ,~ 1
an order of magnitude lower than are needed to account for the transition temperature of the (2 X 2) structure or to simulate the splitting of TD peaks. For this reason, we have considered interactions €2 and €3 merely as free parameters. The best set has been found to be €2 = €3 = 0.4 kcal/mol. In this case, our simulations are in good agreement with the experimental data (Figure 4). 3. Monomolecular Desorption from a Triangular Lattice
TD spectra for adsorption on the vertex sites of a triangular lattice have been studied in detail in ref 8. In this section, we will consider a more complex case of desorption. In particular, we will assume the occurrence of adsorption a t both top and bridge sites of the lattice (Figure 5). This model is applicable, for example, to the CO/Pt(111)system, where at lowtemperatures top site adsorption occurs at all coverages and additionally bridge site
I
where v and E d are the Arrhenius parameters at low coverages, PAi is the probability that particle A is in the top or bridge state with the environment marked by index i, and Aei = ci* - ti, where ti* and ti are lateral interactions in the activated and nonactivated states. The indexes t and b in eq 2 mark symbols corresponding to CO molecules located respectively in top and bridge sites. In our calculations, we assume that the activated complex does not interact with neighbors, i.e. ti* = 0. The lateral interactions between nonactivated CO molecules are assumed to be about the same as that obtained by Perssonl3 in Monte Carlo simulations of adsorbate structures at low temperatures. In particular, the nearestneighbor repulsive interaction is t1 = 8 kcal/mol. The (12)Biberian, J. P.; Van Hove, M. A. Surf. Sci. 1984, 138, 361. (13)Persson, B. N. J. Phys. Reo. B Condens. Mutter 1989,40,7115.
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TEHPERATURE ( I ) Figure 7. Same as in Figure 6, except that cz = €3 = 1 kcal/mol.
other interactions (€2 and €3) are repulsive too (both in the range from 0.5 to 2 kcal/mol). We employ also ut = vb = 1017 s-1, El = 37 kcal/mol, E: = 35 kcal/mol, and 0 = 10 K/s. The initial coverages are 0.5, 0.3, and 0.1 (0 = 0.5 corresponds to the c(4 X 2) structure a t low temperatures). If the second-and third-neighbor interactions are rather weak, €2 = 1 kcal/mol and €3 = 0.5 kcal/mol, the TD spectra (Figure 6) contain a single major peak and a weak lowtemperature shoulder. The major peak can formally be represented as a sum of two overlapped peaks. However, this representation seems to have no sense because the overlapping is too strong. The intensity of the low-temperature shoulder increases with increasing repulsive interactions (see, e.g., Figure 7, where €2 = €3 = 1 kcal/mol). A further increase in interactions leads to an appearance of a pronounced lowtemperature peak at high coverages (see, e.g., Figure 7, where €2 = 2 kcal/mol and €3 = 1kcal/mol). However, the intensity of the latter peak is very low. CO desorption from Pt(111)was experimentally studied in a variety of laboratories (see, e.g., refs 14-18). Typical experimental data shown in Figure 9 are in good agreement with the resulta of our calculations (cf., e.g., Figures 8 and (14) Ertl, G.; Neumann, M.; Streit, K. M. Surf. Sci. 1977,64, 393. (15) McCabe, R. W.; Schmidt, L. D. Surf,. Sci. 1977,65, 189. (16) Collins, D. M.; Spicer, W. E. Surf. SIci. 1977, 69, 85. (17) Norton, P. R.; Goodale, J. W.; SelkirlL, E. B. Surf. Sci. 1979,83, 1 na &"-. (18) Morris, M. A,; Bowker, M.; King, D. A. Compr. Chem. Kinet. 1984, 19, 1.
(4
Figure 8. Same as in Figure 6, except that €2 = 2 kcal/mol and €3 = 1 kcal/mol.
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Figure 9. Thermal desorption spectra for CO adsorption on Pt(ll1): (a) data obtained by McCabe and Schmidtl6 for exposures up to 3 X 1V Torr s (that correspondsto 8 = 0.5); (b) data obtained by Norton et al.1' (the maximum initial coverage is assumed to be 0.43). The small high-temperature peak (at T > 530 K) observed in both studies above the major peak is associated with defects on the crystal or its edges.
9). A slight difference in the shape of major peak seems to be explained by ignoring in our simulations the shifta of equilibrium positions of CO molecules due to lateral interactions between them (thiseffect is discussed in detail in ref 19). Registry No. Hz,1333-74-0;Ni, 7440-02-0; CO, 630-08-0; Pt, 7440-06-4.
(19) Zhdanov, V. P. J. Chem. Phys. 1991,962162.
