Langmuir 1996, 12, 95-100
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Monte Carlo Simulation of Temperature Programmed Desorption Spectra: A Guide through the Forest for Monomolecular Adsorption on a Square Lattice† J. L. Sales,‡ R. O. Un˜ac,‡ M. V. Gargiulo,‡ V. Bustos,§ and G. Zgrablich*,§,| Departamento de Geofı´sica y CISA J (CONICET), Universidad Nacional de San Juan, San Juan, Argentina, Departamento de Fı´sica, Universidad Nacional de San Luis, Chacabuco y Pedernera, 5700 San Luis, Argentina, and Centro Regional de Estudios Avanzados, Gobierno de la Provincia de San Luis, Casilla de Correo 256, 5700 San Luis, Argentina Received November 1, 1994. In Final Form: May 1, 1995X A comparative study of thermal desorption spectra obtained through Monte Carlo simulation for monomers, symmetric dimers, and asymmetric dimers, on heterogeneous surfaces presenting different topographies of two kind of sites, is presented in such a way as to serve as a guide for the interpretation of experimental data.
1. Introduction The use of Monte Carlo simulations for analizing molecular processes on solid surfaces has explosively increased during the last decade, contributing to the understanding of thermal desorption,1-19 mechanism and kinetics of heterogeneous reactions,20-33 and phase tran* To whom correspondence should be addressed. † Presented at the Symposium on Advances in the Measurement and Modeling of Surface Phenomena, San Luis, Argentina, August 24-30, 1994. ‡ Universidad Nacional de San Juan. § Universidad Nacional de San Luis. | Centro Regional de Estudios Avanzados, Gobierno de la Provincia de San Luis. X Abstract published in Advance ACS Abstracts, January 1, 1996. (1) Lambert, R. M.; Bridge, M. E. Proc. R. Soc. London 1980, A370, 345; Surf. Sci. 1980, 94, 469. (2) Hood, S. E.; Toby, B. H.; Weinberg, W. H. Phys. Rev. Lett. 1985, 55, 2437. (3) Stiles, M.; Metiu, H. Chem. Phys. Lett. 1986, 128, 337. (4) Weinberg, W. H. Kinetics of Interface Reactions; Grunze, M., Kreuzer, H. J., Eds.; Springer Series in Surface Science; Springer: Berlin, 1987; Vol. 8. (5) Sales, J. L.; Zgrablich, G. Surf. Sci. 1987, 187, 1. Phys. Rev. 1987, B35, 9520. (6) 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. (7) Gupta D.; Hirtzel, C. S. Chem. Phys. Lett. 1988, 149, 527; Surf. Sci. 1989, 210, 322; Mol. Phys. 1989, 68, 583. (8) Evans, J. W.; Pak, H. Surf. Sci. 1988, 199, 28. (9) Lombardo, S. J.; Bell, A. T. Surf. Sci. 1988, 206, 101; Surf. Sci., 1989, 224, 451. (10) Sales, J. L.; Zgrablich, G.; Zhdanov, V. P. Surf. Sci. 1989, 209, 208. (11) Sales, J. L.; Zgrablich, G.; Myshlyavtsev, V.; Zhdanov, V. P. J. Stat. Phys. 1990, 58, 1029. (12) Fichthorn, K. A.; Weinberg, W. H. Langmuir 1991, 7, 2539. (13) Ramirez Cuesta, A.; Zgrablich, G. Surf. Sci. 1992, 275, L636. (14) Kang, H. C.; Weinberg, H. Chem. Res. 1992, 25, 253. (15) Nagai, K.; Bennemann, K. H. Surf. Sci. 1992, 260, 286. (16) Un˜ac, R. O.; Sales, J. L.; Zgrablich, G. J. Phys.: Condens. Matter 1993, 5, A143. (17) Nieto, F. D.; Valladares, D. L.; Velasco, P. A.; Zgrablich, G. J. Phys.: Condens. Matter 1993, 5, A147. (18) Tysoe, W. T.; Ormerod, R. M.; Lambert, R. M.; Zgrablich, G.; Ramirez Cuesta A. Phys. Chem. 1993, 97, 3365. (19) Meng, B.; Weinberg, W. H. Chem. Phys. 1994, 100, 5280. (20) Ziff, R. M.; Gulari, E.; Barshad, Y. Phys. Rev. Lett. 1986, 50, 2553. (21) Zgrablich, G.; Sales, J. L.; Velasco, P. A. Proc. XI Simposio Iberoamericano de Cata´ lisis Vol. I 59, Guanajuato, Me´xico, Invited Paper, 1988. (22) Fichthorn, K.; Gulari, E.; Ziff, R. Phys. Rev. Lett. 1989, 63, 1527. (23) Heras, J. M.; Velasco, P. A.; Viscido, L.; Zgrablich G. Langmuir 1991, 7, 1124. (24) Kohler, J.; Ben-Avraham, D. J. Phys. (Paris) 1991, A24, L621.
0743-7463/96/2412-0095$12.00/0
sitions in adsorbed overlayers.34-38 Simulations of temperature programmed desorption (TPD) spectra were particularly useful because the data obtained by TPD spectroscopy (this experimental technique is the most popular for the study of desorption) are very sensitive with respect to the Arrhenius parameters for desorption, types of adsorption sites, and adsorbate-adsorbate lateral interactions. All this information characterizing desorption can be obtained by comparing the measured and calculated TPD spectra. General equations for describing the desorption kinetics are in principle simple. For example, in the framework of the lattice-gas model and transition state theory, the rate equation for monomolecular desorption can be represented as39
kBT FA*
dθ )dt
h FA 〈
×
∑R PA,R exp[-(Ed(0) + R* - R)/kBT]〉
(1)
where FA* and FA are the nonconfigurational partition functions of the activated complex and the adsorbate, respectively, PA,R is the probability that a site with environment marked by index R be occupied, Ed(0) is the desorption energy at low coverages, R* and R are the lateral interaction of the activated complex and the adsorbate, respectively, with other adsorbed particles (25) Luque, J. J.; Jimenez-Morales, F.; Lemos, C. J. Chem. Phys. 1992, 96, 8535. (26) Albano, E. V. J. Phys. (Paris) 1992, A25, 2557. (27) Maltz, A.; Albano, E. V. Surf. Sci. 1992, 277, 414. (28) Sales, J. L.; Un˜ac, R.; Zhdanov, V. P.; Zgrablich, G. Surf. Sci. 1993, 290, 160. (29) Nieto, F.; Riccardo, J. L.; Zgrablich, G. Langmuir 1993, 9, 2504. (30) Albano, E. V.; Pereyra, V. D. J. Chem. Phys. 1993, 98, 10044. (31) Pereyra, V. D.; Albano, E. V. J. Phys. (Paris) 1993, A26, 4175. (32) Gonzalez, A. P.; Pereyra, V. D.; Riccardo, J. L.; Zgrablich, G. J. Phys. C: Condens. Matter, in press. (33) Nieto, F.; Velasco, P. A.; Riccardo, J. L.; Zgrablich, G. Surf. Sci., in press. (34) Landau, D. P.; Binder, K. Phys. Rev. 1985, B31, 5946. (35) Landau, D. P. Applications of the Monte Carlo Method Statistical Physics, 2nd ed.; Binder, K., Ed.; Springer: Berlin, 1987. (36) Bustos, V.; Zgrablich, G.; Zhdanov, V. P. Appl. Phys. 1993, A56, 73. (37) Zuppa, C.; Bustos, V.; Zgrablich, G. Phys. Rev. 1994, B51, 2618. (38) Lombardo, S. J.; Bell, A. T. Surf. Sci. Rep. 1991, 13, 1. (39) Zhdanov, V. P. Elementary Physicochemical Processes on Solid Surfaces; Plenum: New York, 1991.
© 1996 American Chemical Society
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distributed according to the arrangement R, and the average 〈...〉 is taken over an ensemble of statistically equivalent system. The environment designed by index R not only refers to a particular configuration of admolecules but also to a particular space distribution of adsorption site energies. Exact analytical calculation of the sum in eq 1 is possible only in very simple cases. Approximate solutions can be obtained by employing the mean-field and quasi-chemical approximation or the transfer-matrix technique. Compared to the approachs above, Monte Carlo simulations are much more universal and powerful because such simulations allow one to describe the system with considerable physical detail including, in particular, the exact description of spatial correlations among adsorbed species and to obtain exact predictions of the model (exact within statistical fluctuations). Formally, the desorption kinetics can also be described by using the phenomenological equation
-
dθ ) νd (θ) exp[-Ed(θ)/kBT]θn dt
(2)
where νd(θ) and Ed(θ) are the “apparent” Arrhenius parameters and n is the order of desorption. Compared to eq 1, eq 2 has no deep physical sense because it does not contain explicitly the microscopic parameters characterizing the adsorbate/substrate system. We conclude that the best method available at present for the analysis of TPD spectra is Monte Carlo simulation. The results obtained by employing this technique are extensive.1-19 However, due to the fact that heterogeneity and adsorbate-adsorbate interactions effects overlap giving rise to complex behaviors, a systematic study is necessary in order to facilitate the analysis of experimental data. In the present work we produce such a systematic study for the case of monomolecular adsorption on well-determined homogeneous and heterogeneous surfaces. Moreover, since many different algorithms have been used in the literature to perform Monte Carlo simulations, some of them being valid under particular conditions, we also discuss the appropriate algorithm that should be used universally in order to be able to make useful comparisons between the results of different authors. 2. Model and Algorithm of Simulations The solid surface is represented by a regular twodimensional lattice of N sites, with periodic boundary conditions, whose geometry (square, triangular, hexagonal, etc.) corresponds to the system under consideration. Adsorption energies Ei are assigned to each site (i ) 1, ..., N) according to the desired distribution and topography to describe the energetic characteristics of the surface (Ei ) E ) constant for homogeneous surfaces). In order to develop a general model for describing monomolecular processes on such a surface, let us define elementary transition probabilities per unit time, wij, for a given molecule to make a transition from the initial state representing the molecule at site i to the final state j. To be specific, for monomolecular processes, the final state j could be the molecule in the gaseous state for desorption, or the molecule occupying one of the vacant nearest-neighbor (NN) sites for surface diffusion (the consideration of longer jumps being irrelevant39), or the molecule occupying an interstitial site below the surface layer for diffusion into the bulk. Let
ri )
wij ∑ j*i
and R )
∑i ri
(3)
be the total probabilities for particle at site “i” and for the whole system, respectively, to change its state per unit time. Then, the probability for the system to change its state at a time in the interval (t, t + dt) is given by (Poisson process):
P(t) dt ) R e-Rt dt
(4)
This means that the time elapsed before the system makes a transition should be obtained as
∆t ) -ln(ξ)/R
(5)
where ξ is a random number uniformly distributed between 0 and 1. Note that eq 5 is applicable even if the system under consideration contains only a few particles. In the case of desorption, however we always have a lot of molecules. Under such circumstances, one can in principle employ the average time increment
〈∆t〉 ) -
∫01 ln(ξ)dξ/R ) 1/R
The latter approach has been used in refs 5-9 and 1719. It is easy to show40 that if an arbitrarily large probability R′ is used instead of R, then the time elapsed before a transition has the correct distribution as long as each event is accepted with probability R/R′. Then the following algorithm is valid (it is assumed that initially we have an equilibrium distribution with coverage θ): Algorithm A. (i) Let r g max ri; R′ ) Nar, where Na is the number of adsorbed molecules (θ ) Na/N); t ) t0. (ii) Obtain a random number ξ; ∆t ) -ln ξ/R′; t ) t + ∆t. (iii) Select at random an occupied site “i” (i ) 1, ..., Na). (iv) Obtain a random number ξ′; if ξ′ < ri/r, then accept the event at time t and select the final state j with probability wij/ri,.41 (v) Repeat from step ii. In most practical cases, however, surface diffusion is much faster than desorption (the activation energy for diffusion of molecules on metal surfaces is often within 10%-15% of desorption energy) and diffusion into the bulk does not occur. In such a case we can consider a unique desorption probability wid for a molecule located in site i and apply a separate relaxation procedure to adsorbed molecules to keep the equilibrium distribution. The appropriate algorithm is now much simpler and faster: Algorithm B. (i) Let r g max wid; R′ ) Nar; t ) t0. (ii) Obtain a random mumber ξ; ∆t ) -ln(ξ)/R′; t ) t + ∆t. (iii) Select at random an occupied site “i”. (iv) Obtain a random number ξ′; if ξ′ < wid/r, then accept the desorption step at time t. (v) Relax the adsorbate through a Monte Carlo exchange process between adsorbed molecules and empty sites until equilibrium is reestablished. (vi) Repeat from step ii. Since we are interested in describing the general behavior of TPD spectra, it is sufficient to consider only NN interactions between adsorbed molecules and no (40) Binder, K. Monte Carlo Methods in Statistical Physics, Topics in Current Physics; Springer: Berlin, 1978; Vol. 7. (41) For example, if particles at site i can only undergo one of these final states: 1, jump into the bulk; 2, jump to the right; 3, desorb; with probabilities wi1, wi2, wi3, respectively, then a random number ξ′′ is selected and: if ξ′′ < wi1/ri, then final state 1 is accepted; if wi1/ri < ξ′′ < (wi1 + wi2)/ri, then final state 2 is accepted; if ξ′′ > (wi1 + wi2)/ri, the final state 3 is accepted.
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interactions for the activated complex, so that
wid ) ν exp[-(Ei0 - nNN)kBT]
(6)
where ν is a preexponential factor, Ei0 the desorption energy at site i (at zero coverage), nNN the number of occupied NN sites, and the interaction energy between NN particles. The relation between time and temperature is given through the heating rate β ) dT/dt. 3. Discussion of Results We have obtained TPD spectra for monomolecular desorption of monomers and dimers on homogeneous and heterogeneous surfaces, the latter being defined by mixtures of two kind of sites with different topographies. To be specific, we have considered thermal desorption of (i) monomers A, (ii) symmetric dimers AA, and (iii) asymmetric dimers AB on (a) homogeneous surface, (b) heterogeneous surface with random mixture of sites S and S′, and (c) patchwise surface with spots of sites S and S′ of different sizes. Values of ν ) 1 × 1013 s-1 and β ) 10 K/s were used all over and in each case initial coverages from θ ) 1 to θ ) 0.2, in steps of 0.2 were considered. The surface was represented by a 100 × 100 square lattice. Calculated spectra cover a great variety of situations of practical interest and our discussion of the general characteristics of these spectra and the comparison between them pretends to serve as a guide for the analysis of experimental results. In what follows, each set of nine spectra represented in each figure is organized in such a way that spectra a and b correspond to homogeneous surfaces of sites S and sites S′, respectively, spectra c, d, e, and f correspond to surfaces of alternate patches of sites S and S′ with sizes of 1 × 1, 2 × 2, 5 × 5, and 25 × 25 sites, respectively, and spectra g, h, and i correspond to surfaces with random mixtures of sites S and S′ with concentrations (nS ) 1/3, nS′ ) 2/3) (nS ) 1/2, nS′ ) 1/2) and (nS ) 2/3, nS′ ) 1/3), respectively. Monomers A. In describing desorption of monomers A, we used the activation energies for sites S and S′ of EAS ) 35 kcal/mol, and EAS′ ) 40 kcal/mol, respectively, lateral interaction energy for NN pairs was taken as AA ) 0, Figure 1, AA ) -1.5 kcal/mol (attractive), Figure 2, and AA ) 1.5 kcal/mol (repulsive), Figure 3. The general behavior of these TPD spectra in this case is quite familiar, as they have been discussed extensively before, but we include them here for completeness and for comparative purposes. In addition, we want to recall the following principal characteristics of the spectra: (i) The absence of lateral interactions, Figure 1, is directly related to the fact that there is no peak shift as the initial coverage is changed. (ii) The shape of spectra is insensitive to topography of adsorption sites for noninteracting adsorbates; i.e., there is no variation with the patch size, spectra c-f, while in spectra g-i the only variation is in the relative height of peaks due to different proportions of sites S and S′ (Figure 1). (iii) Attractive lateral interactions result in a peak shift to higher temperatures as the initial coverage is increased (Figure 2). (iv) Attractive lateral interactions produce a mixing of peaks corresponding to sites S and S′ and the degree of mixing depends on the topography of adsorption sites (Figure 2). (v) Repulsive lateral interactions result in a peak shift to lower temperatures as the initial coverage is increased (Figure 3).
Figure 1. TPD spectra of noninteracting monomers, AA ) 0.
Figure 2. TPD spectra of attractive monomers, AA ) -1.5 kcal/mol.
(vi) Repulsive lateral interactions produce a peak splitting, the strength of this effect being sensitive to the surface topography and, in a very peculiar way, to the size of the interface between sites S and S′. In fact, the normally expected behavior observed in (d), (Figure 3), 2 × 2 patches, where each peak corresponding to sites S and S′ is split into two by repulsive interactions, changes to a three-peak structure (e) when the interface decreases, reaching a stable structure which is the superposition of spectra a and b for large patches, (f). On the other hand, when the interface increases, the central peaks separate more and more giving a double-peaked structure in (c) where it is maximum (1 × 1 patches). This behavior can
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Figure 3. TPD spectra of repulsive monomers, AA ) +1.5 kcal/mol.
be understood by considering that for considerable patches, small interface, the resulting spectra should be a superposition of (a) and (b) with an important overlapping of the two central peaks, producing three-peak spectra, while if the interface between sites S and S′ increases, and due to the fact that particles on S sites desorb preferently first, each S site will have more occupied NN sites than the normal (which is like having a shift to the left of the second peak in (a)) and when S′ sites desorb these will have less occupied NN than the normal (which is like having a shift to the right of the first peak in (b)). The situation of 1 × 1 patches (maximum interface) represents the extreme case where the two central peaks in (d) have joined the two lateral peaks producing a two-peak structure. For random surfaces (g), (h), and (i), the interface remains sufficiently small to produce the threepeak structure. Symmetric Dimers AA. In this case three desorption energies are considered: one corresponding to the adsorption of the dimer AA on a pair of sites SS, EAA,SS ) 35 kcal/mol; one corresponding to adsorption on a pair S′S′, EAA,S′S′ ) 40 kcal/mol; and the one corresponding to adsorption on a pair SS′, EAA,SS′ ) 37.5 kcal/mol. Another difference with monomer adsorption is that each dimer has six NN sites instead of four. TPD spectra are shown in Figures 4-6 for noninteracting, attractive and repulsive dimers, respectively. The spectra are seen to be sensitive to the adsorptive sites topography even for noninteracting dimers. For large patches, Figure 4f, we have a superposition of (a) and (b) spectra, since the interface has almost no influence. As the interface increases, the expected third (central) peak corresponding to pairs SS′ develops but shows up as a shoulder in (e), and finally, when the interface is maximum, (c), the structure reduces to a single peak. The desorption kinetics on random surfaces is not like that on a small interface now but rather like one intermediate between (c) and (d). Attractive interactions, Figure 5, produce a strong overlapping of peaks and only in the case of large patches do we get a two-peak structure, while for all other topographies the spectra are quite similar.
Sales et al.
Figure 4. TPD spectra of noninteracting symmetric dimers, AA ) 0.
Figure 5. TPD spectra of attractive symmetric dimers, AA ) -1.5 kcal/mol.
Repulsive interactions, Figure 6, produce a peak-splitting which depends on the topography and can be understood in terms of the size of the interface, as in the monomer case. Asymmetric Dimers AB. We must now take into account four different desorption energies, depending on whether the AB dimer is adsorbed on an SS, SS′, S′S, or S′S′ pair of sites. We have chosen: EAB,SS ) 30 kcal/mol, EAB,SS′ ) 42, 5 kcal/mol, EAB,S′S ) 35 kcal/mol, and EAB,S′S′ ) 47.5 kcal/mol. For the sake of simplicity the lateral interaction energies between mono-
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Figure 6. TPD spectra of repulsive symmetric dimers, AA ) +1.5 kcal/mol.
Figure 8. TPD spectra of attractive asymmetric dimers AA ) AB ) BB ) -1.5 kcal/mol.
Figure 7. TPD spectra of noninteracting asymmetric dimers AA ) AB ) BB ) 0.
Figure 9. TPD spectra of repulsive asymmetric dimers AA ) AB ) BB ) +1.5 kcal/mol.
mers are assumed to be the same, irrespectively of being A or B, AA ) AB ) BB. Figures 7-9 show the behavior of the spectra for noninteracting attractive and repulsive lateral interactions, respectively, which is similar to that of symmetric dimers with the difference that from a two-peak structure for large patches (f), where the superposition of spectra a and b is observed, Figure 7, a four-peak structure develops as the interface between S and S′ sites increases and this reduces to an almost one-peak structure when the interface is maximum (c). The small peak to the left in (c) is due to few dimers AB/S′S which desorb when the
surface is saturated. As soon as some room is left, dimers accommodate into an AB/SS′ position, which corresponds to the peaks on the right. Attractive, Figure 8, and repulsive, Figure 9, interactions produce the expected peak-overlapping and peaksplitting effects, respectively. Notice that if the strength of bonding energy of strong sites S′ were similar for A and B monomers, then EAB,SS′ would be very similar to EAB,S′S and the behavior would be the same as in the case of symmetric dimers. It should also be stressed that different
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lateral interactions AA * AB * BB would produce a much greater variety of TPD spectra, specially if sign (AA) * sign (BB). 4. Conclusions We have obtained a systematized set of TPD spectra for monomolecular desorption from homogeneous and heterogeneous surfaces which can be used as a guide for the analysis of experimental thermal desorption data. We have also given the appropriate algorithm by which the
Sales et al.
necessary spectra can be obtained for new situations. This kind of computer simulation can be easily performed now on the existing fast personal computers and for this reason the proposed method of analysis is expected to become increasingly popular among experimentalists. Acknowledgment. We thank Dr. V. P. Zhdanov for interesting and helpful discussions and for critically reading the manuscript. LA940859S
