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Lattice-gas model for calculating thermal desorption spectra: Comparison between analytical and Monte Carlo results
208
Surface
Science 209 (1989) 208-274 North-Holland, Amsterdam
LATTICE-GAS MODEL FOR CALCULATING THERMAL SPECTRA: COMPARISON BETWEEN ANALYT‘ICAL AND MONTE CARLO RESULTS
DESORPTION
J.L. SALES, G. ZGRABLICH Institute de Investigaciones en Technologia Quimica (INTEQUI), _ CONICET, Casilla de Correo 290, 5700 San Luis, Argentina
Universidad
Nacionul de San LUIS
and V.P. ZHDANOV Institute
of Catalysis, Novosibrrsk
630090, USSR
Received
27 June 1988; accepted
for publication
5 October
1988
Thermal desorption spectra calculated using analytical approximations (the quasi-chemical and the mean-field approximation) and Monte Carlo simulations, are compared. The results are in good agreement when no order-disorder transitions occur in the overlayer during thermal desorption. If lateral interactions are such that the system crosses through ordered phases, various differences arise in the spectra which are discussed in terms of the adsorbate configurations involved.
1. 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 amongst 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 [l]. 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 0039-6028/89/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
209
J. L. Sales et al. / Lattice-gas model for thermal desorption spectra
lattice-gas model is well known to be exactly resoluble only in exceptional cases [2]. The kinetics of real surface processes is usually studied at 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 neighbours and the mean-field, the quasi-chemical or the Bethe-Peierls approximation are used [1,3-61. More precise results may be derived using Monte Carlo simulations [7,8]. The Monte Carlo method is still not used widely for describing the kinetics of elementary surface processes because this method is rather laborious. However, further simulations of the surface rate processes will apparently be linked with more extensive use of the Monte Carlo method. The objective of the present paper is to verify the accuracy of analytical approximations for calculating thermal desorption spectra using Monte Carlo simulations.
2. General equations General equations for describing kinetics of various surface rate processes are derived in the framework of the lattice-gas model in ref. [4]. In particular, the kinetics of monomolecular desorption, A, + A,, is described as dO/dt
= - k,B,
(1)
exp[ -(Ed(O) +Ac,)/T],
k,= vCpA,,
where B is the coverage, v is the pre-exponential factor, Ed(O) is the activation energy for desorption at low coverages, PA,, is the probability that an adsorbed particle has the environment marked by index i, Aci = ef - c,, ei is the lateral interaction of molecule A and its environment (repulsive interactions are assigned positive values), e,* is the lateral interaction of the activated complex A* with the same environment; the Boltzmann constant is set to unity. Using eqs. (1) and (2) we assume that the surface is uniform and that a given cell is either vacant or occupied by a single adsorbed particle; the precursor states are neglected. Using the quasi-chemical approximation for nearest-neighbour (NN) pairs and the mean-field approximation for next-nearest-neighbour (NNN) pairs, and neglecting the lateral interaction of the activated complex with its environment, we have [4,5] k
=
d where
v
-CA exp(WT) +0.59ko .PAA + 0.59,, E, and cz are the lateral
““Pi-(Ed@) interactions
-262
(3)
%‘T],
for NN and NNN
pairs,
respec-
J.L. Sales et al. / Lattice-gas model for thermal desorption spectra
210
tively, .z is the number of neighbouring sites for one site, PAA and PA0 are the famous quasi-chemical probabilities for a pair with both sites occupied and a pair with one occupied and one empty site, respectively, given by [4] PAA = 8 - { 1 - [l - 2c&(lPA0 = 2{ 1 - [l - 2&(1
f?)]“‘}/a,
- e)]1’2}/n,
(5)
with (Y= 2[1 - exp( - e,/T)]. One the other hand, Monte Carlo simulations of thermal desorption spectra can be obtained [8] using directly eq. (2), i.e. using the following expression for the probability for a particle at site i to desorb in the interval (t, t + At) P(At)
= Arv exp( -&J/T),
(6)
with E;=&(O)-6,
C JNN
n,--e, to 1
C
n,,,
/NNN to i
(7)
where n, is the occupation number of site j. Here, the first sum is the contribution of lateral interactions with NN sites and the second sum gives the contribution of interactions with NNN sites. For the following discussion, it is of interest also to define the mean desorption activation energy E,(B)
= -T
The Arrhenius E,(d)
ln(k,/Y).
(8)
procedure,
d In k, = T2-gc
defines the desorption activation energy more accurately. Both definitions however yield about the same results. In our context, we prefer to use eq. (8) because this equation is more simple.
3. Result of the calculations
and discussion
To compare both approaches, a set of thermal desorption spectra for different initial coverages 0, = 0.2, 0.4, 0.6, 0.8, 1.0 were calculated with z = 4, y = 10’5 s-1 and and Ed(O) = 35 kcal/mol. The heating rate was assumed to be /? = 40 K/s. Different values of E, and c2 were chosen in the most interesting range of repulsive NN interactions combined with repulsive, zero and attractive NNN interactions. This stresses the test of approximations used in eq. (3) since under the chosen conditions order-disorder phase transitions may occur.
211
J. L. Sales et al. / Lattice-gus model/or thermal desorption spectra
400 Fig. 1. Thermal
desorption
400
500 spectra
500
calculated using eq. (3). The lateral presented in kcal/mol.
400
500
interactions
T(K) c1 and c2 are
All methodological aspects of Monte Carlo simulations were the same as in ref. [8]. Figs. 1 and 2 show thermal desorption spectra calculated using analytical approximations and Monte Carlo simulations, respectively, while fig. 3 shows the corresponding curves for the mean desorption activation energy as a function of coverage, E,(0). We see that the analytical approximations somewhat underestimate the mean activation energy at 8 < 0.5 and overestimate at 0 > 0.5. The general qualitative features of the spectra are seen to be similar in both models: the splitting of the thermal desorption peaks due to repulsive interactions is rather weak in the case e1 = 1.4, e2 = 0.6 kcal/mol and increases as the NN interactions become more repulsive and the NNN interactions more attractive. The main differences, which will be discussed below, are the height and position of the high-temperature peak, its temperature shift with decreasing initial coverage, and the existence of a weak in the Monte intermediate peak in the case E, = 2.6 and t2 = - 0.6 kcal/mol Carlo simulation.
r
400
400
500
500
400
500 T (K)
Fig. 2. Thermal
desorption
spectra
calculated
using Monte Carlo simulations.
212
J.L. Sales et al. / Lattice-gas
model for thermal desorption spectra
-
36-
0
0.2
04
0.6
0.6
9
_. ‘-‘\
0.2
0.4
0.6
0.6
Fig. 3. Mean desorption activation energy as a function of coverage calculated at T= 500 K (a), and by the Monte Carlo simulation (b), Lateral interactions figs. 1 and 2.
1.0 by eqs. (3) and (8) are the same as in
To analyze the effect of the quasi-chemical approximation, which makes the main contribution because it deals with NN interactions, let us consider first the case Ed = 2 kcal/mol and Ed = 0 where the mean-field approximation is absent. In this case, a phase diagram like that of fig. 4a is appropriate for the adsorbed layer [1,9]. We see that during thermal desorption, the system follows a trajectory (dashed line) which starts from a disordered “lattice fluid” at high coverage, enters the ordered c(2 X 2) structure and finally leaves it ending up in a disordered “lattice gas” at low coverages. The particles, located in the forbidden positions of the c(2 X 2) structure (see fig. 5a), are responsible for desorption in the region between the low- and the high-temperature peak, i.e.,
r,
\ I I
T
1.0
05
,
Lattice Gas
0
Fig. 4. Phase diagram of an overlayer on a square lattice for different lateral interactions: (a) Dashed lines represent the trajectories L, = 2 kcal/mol, c2 = 0; (b) z, = 2.6, ez = -0.6 kcal/mol. followed by the system during thermal desorption.
J.L. Sales et al. / Lattice-gas model for thermal desorption spectra
Of
J5
0
0
0
0
0
0
0
0
0
0
0
.
0
0
0
‘rdL
0
‘0
0
0
‘0
0
0
0
0
0
0
0
0
0
a
213
b
C
Fig. 5. Particles on the square lattice. Occupied sites are indicated by full circles, empty sites by empty circles. In case (a), particle 1 is located in the forbidden positions of the c(2 x 2) structure. In cases (b) and (c), configurations where two sites which are nearest-neighbour to particle 1 are empty.
for desorption at temperatures 400 < T < 500 K. In the framework of the quasi-chemical approximation, different pairs of sites are assumed to be independent. For this reason, the quasi-chemical approximation overestimates the probabilities of forbidden configurations. The result is that the high-temperature peak is located at lower temperatures, compared to the Monte Carlo results, moreover this peak is wider and lower and shifts to higher temperatures with decreasing initial coverage. The case ei = 2.6 and e 2 = - 0.6 kcal/mol can be analyzed in a similar way. The appropriate phase diagram is shown in fig. 4b, where five phases are present but the trajectory, followed by the system during thermal desorption, crosses through three of them: the lattice fluid, the c(2 X 2) ordered phase and the lattice gas. Accounting that now the trajectory of the system is always lower than in the previous case, the differences between the analytical approximation and the Monte Carlo simulation are also stronger. In particular, Monte Carlo results present a small intermediate peak that seems to correspond to configurations where two sites which are NN to the desorbing particle are empty (figs. 5b and 5~). Moreover, the high-temperature peak moves appreciably to the left with decreasing initial coverage, as expected. These effects are smeared out in the analytical approximations. Finally, for the case c1 = 1.4 and e2 = 0.6 kcal/mol, the trajectory of the system during thermal desorption is such that T > T, for all coverages, so that no phase transitions occur and thermal desorption spectra predicted by both methods are seen to be quite similar.
4. Summary The present analysis provides us with an idea of the conditions under which the analytical approximations may be used with a certain degree of confidence and the deviations which are to be expected when these conditions are not
.I .L. Sales et al. / Lattice-gas model for thermal desorption spectra
214
fulfilled. Analytical and Monte Carlo results are in good agreement when no phase transitions occur in the adsorbed layer during thermal desorption. If lateral interactions are such that the system crosses through ordered phases, various differences arise in the behaviour of thermal desorption spectra which can be discussed in terms of the adsorbate configurations involved. Finally, some comments on the applicability of the calculations should be made. In our simulations, we have used the parameters that are typical for CO desorption from close-packed faces of the Ni, Pd, Pt, Rh and Ru single crystals. The results, presented in figs. 1 and 2, reproduce qualitatively the various features of the real thermal desorption spectra (cf. figs. 37 i-37k in ref. [lo]). However, the experimental results as a whole are somewhat more complex than the theoretical ones. This fact is explained as follows. First, the real lateral interactions and locations of adsorbed molecules are more diverse than those considered in our paper. Second, the adsorbate-induced changes in the surface should be apparently incorporated into the theory to interpret experimental thermal desorption spectra in detail.
Acknowledgement The authors
thank
Dr. V. Pereyra
for helpful
discussions.
References [I] V.P. Zhdanov and K.I. Zamaraev, Usp. Fiz. Nauk 149 (1986) 635 [English Translation: Soviet Phys. Usp. 29 (1986) 7551. [2] R.J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, London, 1982). [3] D.A. King, CRC Crit. Rev. Solid State Mater. Sci. 7 (1978) 167; D.L. Adams, Surface Sci. 42 (1974) 12. [4] V.P. Zhdanov. Surface Sci. 111 (1981) 63. [5] V.P. Zhdanov. Surface Sci. 133 (1983) 469. [6] V.P. Zhdanov, Surface Sci. 123 (1982) 106; 137 (1984) 515; 169 (1981) 1; 179 (1987) L57; J.B. Benziger and G.R. Schoofs, J. Phys. Chem. 88 (1984) 4439; S. Sundaresan and K.R. Kaza, Surface Sci. 160 (1985) 103; Chem. Eng. Commun. 32 (1985) 333; 35 (1985) 1; H. Pak and J.W. Evans, Surface Sci. 186 (1987) 550; 199 (1988) 28; B. Hellsing and V.P. Zhdanov. Chem. Phys. Letters 147 (1988) 613. [7] M.E. Bridge and R.M. Lambert, Proc. Roy. Sot. (London) A 370 (1980) 545; Surface Sci. 94 (1980) 469; M. Silverberg, A. Ben-Shaul and F. Robentrost, J. Chem. Phys. 83 (1985) 6501; M. Silverberg and A. Ben-Shaul, Chem. Phys. Letters 134 (1987) 491; J. Chem. Phys. 87 (1987) 3178; M. Stiles and H. Metiu, Chem. Phys. Letters 128 (1986) 337. [8] J.L. Sales and G. Zgrablich, Surface Sci. 187 (1987) 1; Phys. Rev. B 35 (1987) 9520. [9] K. Binder and D.P. Landau. Surface Sci. 61 (1976) 577; K. Binder, W. Kinzel and D.P. Landau. Surface Sci. 117 (1982) 232. (IO] M.A. Morris. M. Bowker and D.A. King, Comprehensive Chem. Kinet. 19 (1984) 163.
Surface
Science 209 (1989) 208-274 North-Holland, Amsterdam
LATTICE-GAS MODEL FOR CALCULATING THERMAL SPECTRA: COMPARISON BETWEEN ANALYT‘ICAL AND MONTE CARLO RESULTS
DESORPTION
J.L. SALES, G. ZGRABLICH Institute de Investigaciones en Technologia Quimica (INTEQUI), _ CONICET, Casilla de Correo 290, 5700 San Luis, Argentina
Universidad
Nacionul de San LUIS
and V.P. ZHDANOV Institute
of Catalysis, Novosibrrsk
630090, USSR
Received
27 June 1988; accepted
for publication
5 October
1988
Thermal desorption spectra calculated using analytical approximations (the quasi-chemical and the mean-field approximation) and Monte Carlo simulations, are compared. The results are in good agreement when no order-disorder transitions occur in the overlayer during thermal desorption. If lateral interactions are such that the system crosses through ordered phases, various differences arise in the spectra which are discussed in terms of the adsorbate configurations involved.
1. 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 amongst 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 [l]. 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 0039-6028/89/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
209
J. L. Sales et al. / Lattice-gas model for thermal desorption spectra
lattice-gas model is well known to be exactly resoluble only in exceptional cases [2]. The kinetics of real surface processes is usually studied at 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 neighbours and the mean-field, the quasi-chemical or the Bethe-Peierls approximation are used [1,3-61. More precise results may be derived using Monte Carlo simulations [7,8]. The Monte Carlo method is still not used widely for describing the kinetics of elementary surface processes because this method is rather laborious. However, further simulations of the surface rate processes will apparently be linked with more extensive use of the Monte Carlo method. The objective of the present paper is to verify the accuracy of analytical approximations for calculating thermal desorption spectra using Monte Carlo simulations.
2. General equations General equations for describing kinetics of various surface rate processes are derived in the framework of the lattice-gas model in ref. [4]. In particular, the kinetics of monomolecular desorption, A, + A,, is described as dO/dt
= - k,B,
(1)
exp[ -(Ed(O) +Ac,)/T],
k,= vCpA,,
where B is the coverage, v is the pre-exponential factor, Ed(O) is the activation energy for desorption at low coverages, PA,, is the probability that an adsorbed particle has the environment marked by index i, Aci = ef - c,, ei is the lateral interaction of molecule A and its environment (repulsive interactions are assigned positive values), e,* is the lateral interaction of the activated complex A* with the same environment; the Boltzmann constant is set to unity. Using eqs. (1) and (2) we assume that the surface is uniform and that a given cell is either vacant or occupied by a single adsorbed particle; the precursor states are neglected. Using the quasi-chemical approximation for nearest-neighbour (NN) pairs and the mean-field approximation for next-nearest-neighbour (NNN) pairs, and neglecting the lateral interaction of the activated complex with its environment, we have [4,5] k
=
d where
v
-CA exp(WT) +0.59ko .PAA + 0.59,, E, and cz are the lateral
““Pi-(Ed@) interactions
-262
(3)
%‘T],
for NN and NNN
pairs,
respec-
J.L. Sales et al. / Lattice-gas model for thermal desorption spectra
210
tively, .z is the number of neighbouring sites for one site, PAA and PA0 are the famous quasi-chemical probabilities for a pair with both sites occupied and a pair with one occupied and one empty site, respectively, given by [4] PAA = 8 - { 1 - [l - 2c&(lPA0 = 2{ 1 - [l - 2&(1
f?)]“‘}/a,
- e)]1’2}/n,
(5)
with (Y= 2[1 - exp( - e,/T)]. One the other hand, Monte Carlo simulations of thermal desorption spectra can be obtained [8] using directly eq. (2), i.e. using the following expression for the probability for a particle at site i to desorb in the interval (t, t + At) P(At)
= Arv exp( -&J/T),
(6)
with E;=&(O)-6,
C JNN
n,--e, to 1
C
n,,,
/NNN to i
(7)
where n, is the occupation number of site j. Here, the first sum is the contribution of lateral interactions with NN sites and the second sum gives the contribution of interactions with NNN sites. For the following discussion, it is of interest also to define the mean desorption activation energy E,(B)
= -T
The Arrhenius E,(d)
ln(k,/Y).
(8)
procedure,
d In k, = T2-gc
defines the desorption activation energy more accurately. Both definitions however yield about the same results. In our context, we prefer to use eq. (8) because this equation is more simple.
3. Result of the calculations
and discussion
To compare both approaches, a set of thermal desorption spectra for different initial coverages 0, = 0.2, 0.4, 0.6, 0.8, 1.0 were calculated with z = 4, y = 10’5 s-1 and and Ed(O) = 35 kcal/mol. The heating rate was assumed to be /? = 40 K/s. Different values of E, and c2 were chosen in the most interesting range of repulsive NN interactions combined with repulsive, zero and attractive NNN interactions. This stresses the test of approximations used in eq. (3) since under the chosen conditions order-disorder phase transitions may occur.
211
J. L. Sales et al. / Lattice-gus model/or thermal desorption spectra
400 Fig. 1. Thermal
desorption
400
500 spectra
500
calculated using eq. (3). The lateral presented in kcal/mol.
400
500
interactions
T(K) c1 and c2 are
All methodological aspects of Monte Carlo simulations were the same as in ref. [8]. Figs. 1 and 2 show thermal desorption spectra calculated using analytical approximations and Monte Carlo simulations, respectively, while fig. 3 shows the corresponding curves for the mean desorption activation energy as a function of coverage, E,(0). We see that the analytical approximations somewhat underestimate the mean activation energy at 8 < 0.5 and overestimate at 0 > 0.5. The general qualitative features of the spectra are seen to be similar in both models: the splitting of the thermal desorption peaks due to repulsive interactions is rather weak in the case e1 = 1.4, e2 = 0.6 kcal/mol and increases as the NN interactions become more repulsive and the NNN interactions more attractive. The main differences, which will be discussed below, are the height and position of the high-temperature peak, its temperature shift with decreasing initial coverage, and the existence of a weak in the Monte intermediate peak in the case E, = 2.6 and t2 = - 0.6 kcal/mol Carlo simulation.
r
400
400
500
500
400
500 T (K)
Fig. 2. Thermal
desorption
spectra
calculated
using Monte Carlo simulations.
212
J.L. Sales et al. / Lattice-gas
model for thermal desorption spectra
-
36-
0
0.2
04
0.6
0.6
9
_. ‘-‘\
0.2
0.4
0.6
0.6
Fig. 3. Mean desorption activation energy as a function of coverage calculated at T= 500 K (a), and by the Monte Carlo simulation (b), Lateral interactions figs. 1 and 2.
1.0 by eqs. (3) and (8) are the same as in
To analyze the effect of the quasi-chemical approximation, which makes the main contribution because it deals with NN interactions, let us consider first the case Ed = 2 kcal/mol and Ed = 0 where the mean-field approximation is absent. In this case, a phase diagram like that of fig. 4a is appropriate for the adsorbed layer [1,9]. We see that during thermal desorption, the system follows a trajectory (dashed line) which starts from a disordered “lattice fluid” at high coverage, enters the ordered c(2 X 2) structure and finally leaves it ending up in a disordered “lattice gas” at low coverages. The particles, located in the forbidden positions of the c(2 X 2) structure (see fig. 5a), are responsible for desorption in the region between the low- and the high-temperature peak, i.e.,
r,
\ I I
T
1.0
05
,
Lattice Gas
0
Fig. 4. Phase diagram of an overlayer on a square lattice for different lateral interactions: (a) Dashed lines represent the trajectories L, = 2 kcal/mol, c2 = 0; (b) z, = 2.6, ez = -0.6 kcal/mol. followed by the system during thermal desorption.
J.L. Sales et al. / Lattice-gas model for thermal desorption spectra
Of
J5
0
0
0
0
0
0
0
0
0
0
0
.
0
0
0
‘rdL
0
‘0
0
0
‘0
0
0
0
0
0
0
0
0
0
a
213
b
C
Fig. 5. Particles on the square lattice. Occupied sites are indicated by full circles, empty sites by empty circles. In case (a), particle 1 is located in the forbidden positions of the c(2 x 2) structure. In cases (b) and (c), configurations where two sites which are nearest-neighbour to particle 1 are empty.
for desorption at temperatures 400 < T < 500 K. In the framework of the quasi-chemical approximation, different pairs of sites are assumed to be independent. For this reason, the quasi-chemical approximation overestimates the probabilities of forbidden configurations. The result is that the high-temperature peak is located at lower temperatures, compared to the Monte Carlo results, moreover this peak is wider and lower and shifts to higher temperatures with decreasing initial coverage. The case ei = 2.6 and e 2 = - 0.6 kcal/mol can be analyzed in a similar way. The appropriate phase diagram is shown in fig. 4b, where five phases are present but the trajectory, followed by the system during thermal desorption, crosses through three of them: the lattice fluid, the c(2 X 2) ordered phase and the lattice gas. Accounting that now the trajectory of the system is always lower than in the previous case, the differences between the analytical approximation and the Monte Carlo simulation are also stronger. In particular, Monte Carlo results present a small intermediate peak that seems to correspond to configurations where two sites which are NN to the desorbing particle are empty (figs. 5b and 5~). Moreover, the high-temperature peak moves appreciably to the left with decreasing initial coverage, as expected. These effects are smeared out in the analytical approximations. Finally, for the case c1 = 1.4 and e2 = 0.6 kcal/mol, the trajectory of the system during thermal desorption is such that T > T, for all coverages, so that no phase transitions occur and thermal desorption spectra predicted by both methods are seen to be quite similar.
4. Summary The present analysis provides us with an idea of the conditions under which the analytical approximations may be used with a certain degree of confidence and the deviations which are to be expected when these conditions are not
.I .L. Sales et al. / Lattice-gas model for thermal desorption spectra
214
fulfilled. Analytical and Monte Carlo results are in good agreement when no phase transitions occur in the adsorbed layer during thermal desorption. If lateral interactions are such that the system crosses through ordered phases, various differences arise in the behaviour of thermal desorption spectra which can be discussed in terms of the adsorbate configurations involved. Finally, some comments on the applicability of the calculations should be made. In our simulations, we have used the parameters that are typical for CO desorption from close-packed faces of the Ni, Pd, Pt, Rh and Ru single crystals. The results, presented in figs. 1 and 2, reproduce qualitatively the various features of the real thermal desorption spectra (cf. figs. 37 i-37k in ref. [lo]). However, the experimental results as a whole are somewhat more complex than the theoretical ones. This fact is explained as follows. First, the real lateral interactions and locations of adsorbed molecules are more diverse than those considered in our paper. Second, the adsorbate-induced changes in the surface should be apparently incorporated into the theory to interpret experimental thermal desorption spectra in detail.
Acknowledgement The authors
thank
Dr. V. Pereyra
for helpful
discussions.
References [I] V.P. Zhdanov and K.I. Zamaraev, Usp. Fiz. Nauk 149 (1986) 635 [English Translation: Soviet Phys. Usp. 29 (1986) 7551. [2] R.J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, London, 1982). [3] D.A. King, CRC Crit. Rev. Solid State Mater. Sci. 7 (1978) 167; D.L. Adams, Surface Sci. 42 (1974) 12. [4] V.P. Zhdanov. Surface Sci. 111 (1981) 63. [5] V.P. Zhdanov. Surface Sci. 133 (1983) 469. [6] V.P. Zhdanov, Surface Sci. 123 (1982) 106; 137 (1984) 515; 169 (1981) 1; 179 (1987) L57; J.B. Benziger and G.R. Schoofs, J. Phys. Chem. 88 (1984) 4439; S. Sundaresan and K.R. Kaza, Surface Sci. 160 (1985) 103; Chem. Eng. Commun. 32 (1985) 333; 35 (1985) 1; H. Pak and J.W. Evans, Surface Sci. 186 (1987) 550; 199 (1988) 28; B. Hellsing and V.P. Zhdanov. Chem. Phys. Letters 147 (1988) 613. [7] M.E. Bridge and R.M. Lambert, Proc. Roy. Sot. (London) A 370 (1980) 545; Surface Sci. 94 (1980) 469; M. Silverberg, A. Ben-Shaul and F. Robentrost, J. Chem. Phys. 83 (1985) 6501; M. Silverberg and A. Ben-Shaul, Chem. Phys. Letters 134 (1987) 491; J. Chem. Phys. 87 (1987) 3178; M. Stiles and H. Metiu, Chem. Phys. Letters 128 (1986) 337. [8] J.L. Sales and G. Zgrablich, Surface Sci. 187 (1987) 1; Phys. Rev. B 35 (1987) 9520. [9] K. Binder and D.P. Landau. Surface Sci. 61 (1976) 577; K. Binder, W. Kinzel and D.P. Landau. Surface Sci. 117 (1982) 232. (IO] M.A. Morris. M. Bowker and D.A. King, Comprehensive Chem. Kinet. 19 (1984) 163.