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Langmuir 1991, 7, 1225-1228
Thermal Desorption of Interacting Particles from a Triangular Lattice L. V. Lutsevich,? 0. A. Tkachenko,?and V. P. Zhdanov'J Computer Center, Novosibirsk 630090, USSR, and Institute of Catalysis, Novosibirsk 630090, USSR Received July 6, 1990. In Final Form: October 9, 1990 Monomolecular desorption of particles from a triangular lattice is analyzed by employinga Monte Carlo simulation. The effect of the nearest-neighbor,next-nearest-neighbor,and three-body adsorbate-adsorbate interactions on thermal desorption spectra is demonstrated in detail. General results are used to interpret thermal desorption data for the CO/Ru(001) system. Introduction
In the case of chemisorption of atoms and simple molecules on close-packed faces of single crystals, the assumption of surface uniformity is often justified; i.e. it may be assumed that adsorbed particles are distributed among equivalent elementary cells. The nonideality of the adsorbed layer in this case is due to lateral interactions between adsorbed particles. In statistical physics, a system of interacting particles distributed among equivalent cells is called a lattice gas. It turns out that many phenomena occurring on the surfaces of solids (kinetics of adsorption and desorption, kinetics of chemical reactions, surface diffusion, phase diagrams, surface reconstruction induced by adsorption) can be described in the framework of the lattice-gas model.' The general formulas for describing various phenomena in the lattice-gas model have, as a rule, a simple form. However, these formulas are not so much a solution as a formulation of the problem, since the main difficulty lies in calculating the various probabilities appearing in these formulas. Indeed, the lattice-gas model is well-known to be exactly resoluble only in exceptional cases.2 The kinetics of real surface processes is usually studied a t comparatively high temperatures. In this temperature range, the cluster method is suitable for calculating the rates of elementary processes. As a rule, the practical calculations take into account lateral interactions only between nearest neighbors and the mean-field, the quasichemical, or the Bethe-Peierls approximations are used.3-5 These approximations are rather rough a t T S Tc,where Tc is a critical temperature of phase transitions in the adsorbed overlayer. More precise results may be derived
* To whom correspondence should be addressed. + Computer Center. t Institute of Catalysis.
(1)Zhdanov, V.P.;Zkaraev, K. I. Usp.Fiz. Nauk 1986,149,635; Sou. Phys. Usp. (Engl. Tronel.) 1986,29,755. (2)Baxter, R. J. Eractly Solued Models in Statistical Mechanics; Academic Prese: 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,Ill, 63. (5)Zhdanov, V. P.Surf. Sci. 1982,123,106,1983133,469;1984,137, 515; 1986,169,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. 1986,32,333;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.Hellaing, B.; Zhdanov, V. P. Chem. Phys. Lett. 1988,147,613.Aaada, H.; Maauda, M. Surf. Sci. 1989,207,517.Surda, A,; Karasova,I. Surf. Sci. 1981,109,605.Surda, A. Surf. Sci. 1989,220,295.
by employing Monte Carlo simulations6 or the transfermatrix technique.' Almost all the author^^-^ have analyzed the kinetics of various rate processes on a square lattice. The objective of the present paper is to study the effect of lateral interactions on thermal desorption spectra in the case of adsorption on a triangular lattice. The phase diagram of an overlayer adsorbed on this lattice has been discussed earlier in ref 8. Model and Algorithm of Calculations We will analyze monomolecular desorption of particles adsorbed on the vertex sites of a triangular lattice taking into account nearest-neighbor, next-nearest-neighbor,and three-body lateral interactions (Figure 1). The kinetics of desorption is described as usual4
d6f dt = -kd6
(1)
where 6 is coverage, u is the preexponentid factor, Ed@) is the activation energy for desorption at low coverages, and PA,^ is the probability that an adsorbed particle has the environment marked by index i, Aci = ci* - ci, where ei is the lateral interaction of molecule A and its environment (repulsive interactions are assigned positive values) and ti* is the lateral interaction of the activated complex A* with the same environment; the Boltzmann constant is set to unity. The interaction ti* is customarily believed to be weak compared to other interactions and we will assume Ei* = 0. In addition, surface diffusion is assumed to be rapid in comparison with desorption and consequently the adsorbed overlayer is considered to be in an equilibrium state. This assumption is usually justified because in real systems the activation energy for surface diffusion is customarily low in comparison with that for desorption. To achieve (6)Bridge, M. E.; Lambert, R. M. h o c . R. SOC.London, A 1980,370, 545;Surf. Sci. 1980,94,469;Silverberg, M.; Ben-Shaul, A.; Robentroet, F. J. Chem. Phys. 1985,83,6501.Silverberg, M.; Ben-Shaul, 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,l;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)Landau, D.P. Phys. Rev.: B Condens. Matter 1983,27,5604.
0743-7463/91/2407-1225$02.50/0 0 1991 American Chemical Society
1226 Langmuir, Vol. 7,No. 6,1991 0
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Figure 1. Particleson a triangularlattice. Occupied and empty sites are indicated respectively by filled and open circles. E1 is the energy of particle 1. cl, ea, and et are the nearest-neighbor, next-nearest neighbor, and three-body interactions. an equilibrium state, we will use the simplest scheme of diffusion (see below). To simulate thermal desorption from a triangular lattice (a (100 X 100) array with periodic boundary conditions), we have employed the following Monte Carlo algorithm: (1)Fill randomly N sites with adparticles and fix an initial temperature TO.The initial coverage is 00 = N/No, where NO= lo4 is the total number of sites. (2) Choose randomly a site. If the site is occupied, calculate the probability of desorption in the time interval At (cf eq 2)
P ( A t ) = Atu exp[-(Ed(O) + Ati)/T] (3) Generate a random number 0 < 5 < 1. If 5 < P ( A t ) , the particle is desorbed, otherwise it is not. (3) Choose randomly a site. If the site is occupied, choose randomly the nearest-neighbor site. If the latter site is empty, calculate the initial and final energies for a diffusional jump and then calculate the quantity Q = exp[(Ei - E f ) / q . Generate a random number 0 < 5 < 1. If 5 < Q, the jump is allowed, otherwise it is not. Repeat all the procedures presented in this step Nd times. (4) Repeat steps 2 and 3 NoAT//3At times (/3 is the heating rate). Increase the temperature by A T and then again go to step 2. As a rule, we have used in our simulations Nd = 12, AT = 3 K, and At = 6 X s. The selected values Of Nd and A T accurately reproduce an equilibrium state of the adlayer and simulate the continuous increase in temperature. The motivation of our choice of the value of At is as follows. The algorithm presented above is correct if P(At) Q 1 (4) for all the possible arrangements of adsorbed particle^.^ Condition 4 can be easily realized if lateral interactions are absent or are weak. However, if lateral interactions are strong, the distribution of P(At) for different arrangements of adparticles is very wide and, consequently, At should be very low in order to fulfill condition 4 for all the configurations. In turn, the computer time is limited, and we have no possibility of calculating a thermal desorption spectrum if At is too low. In practice, this controversy may be solved by selecting At so that condition 4 is fulfilled for the most important arrangements. The experience has shown that at our values of various parameters, a reasonable value of At is 6 X 10-2s. For example, let us consider the most difficult case when the nearest neighbor repulsion is very strong, €1 = 4 kcal/mol, and the other interactions are absent. The probability P ( A t ) for the arrangements when a particle has zero, three, and six neighbors, is shown in Figure 2. If a particle has six neighbors, condition 4 is not fulfilled a t T > 175 K. (9)Binder, K. In Monte Carlo Methods in StatisticalPhysics; Binder, K., Ed.; Springer: Berlin, 1979.
2 00
400
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Figure 2. (a) Coverage during the thermal desorption course and (b) the probability P(At) as a function of temperature at v = loi5 s-l, E&) = 35 kcal/mol, e1 = 4 kcal/mol, 8 = 50 K/s, and At = 6 X s. Curves 1-3 are constructed for the arrangementa when a particle has zero, three, and six neighbors. However, the configurations with six neighbors do not make a significant contribution to desorption a t T > 175 K, because in this temperature range the coverage is lower than 0.66 and the probability that a particle has six neighbors is low. The same is correct for the other configurations. We have verified our results by using At
