RATES OF DESORPTION FROM SOLID SURFACES: COVERAGE DEPENDENCE · RATES OF DESORPTION FROM SOLID...

Surface Science 136 (1984) 41-58

North-Holland, Amsterdam

41

RATES OF DESORPTION FROM SOLID SURFACES: COVERAGE DEPENDENCE

Antonio REDONDO *, Yehuda ZEIRI ** and William A. GODDARD III

Arthur Amos Noyes Laboratory of Chemical Physics ***, California Institute of Technology, Pasadena,

California 91125, USA

Received 24 February 1983; accepted for publication 2 August 1983

The recently developed Classical Stochastic Diffusion Theory is applied to obtain the coverage

dependence of desorption rates for Xe on W(111). Using the attractive Xe-Xe potential from gas

phase experiments, we find a strong coverage dependence in the desorption rates and calculate

Temperature Programmed Desorption Spectra (for a potential with reduced attractiveness) that are

in excellent qualitative agreement with experimental results. We also investigated the effect of

purely repulsive adsorbate-adsorbate interactions where we find, for some coverage ranges, that

two different adsorption configurations can be stable (at different temperatures) leading to a

marked change in the corresponding desorption rates and to distinct non-Arrhenius behavior.

1. Introduction

Recently we reported a new Classical Stochastic Diffusion Theory (CSDT) [l] to calculate rates of desorption of atoms and molecules from solid surfaces. This method relates the rates of desorption to the microscopic parameters of the system (i.e. bond energies and vibrational frequencies). In the original presentation [l] the CSDT formulation was applied to the study of desorption of atoms and molecules in the limit of low coverage (no adsorbate-adsorbate interactions). Of course, most experimental studies of desorption processes involve the effects of adsorbate-adsorbate interactions (leading to coverage dependent desorption rates [2,3]), and in this paper we extend the theoretical studies to include this more general situation. A particularly important type of experiment in which adsorbate-adsorbate effects are often observed is Tem- perature Programmed Desorption (TPD) [2,3]. In a typical TPD experiment the surface is prepared with a given adsorbate coverage and the gas pressure is monitored as the temperature is increased (usually linearly with time). From

* Present address: Group E-11, MS/D429, Los Alamos National Laboratory, Los Alamos,

New Mexico 87545, USA. ** Chaim Weizmann Postdoctoral Fellow.

*** Contribution No. 6778.

0039-6028/84/$03.00 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

42 A. Redondo et al. / Rates of desorption from solid surfaces

this type of experiment one can extract the strength of the adsorbate-surface interactions and some information about the distribution of adsorption sites on the surface. We have used our theory to predict these TPD spectra.

Experimental rates of desorption are usually expressed in an Arrhenius form,

R = A exp( - E,/kT), (1.1)

where A is referred to as the preexponential factor and Ed is the desorption enthalpy. In general both A and Ed are functions of coverage [3], reflecting thereby the adsorbate-adsorbate interactions. Although the experimental results can usually be fitted to the form of eq. (1.1) it is desirable to have a theoretical expression for the rate whereby the A and E,, are related to the specific parameters involved in the adsorbate-surface and adsorbate-adsorbate interactions. This would allow these interaction parameters to be extracted from a fit to experimental TPD rates,

In the present paper we shall describe an extension of the CSDT formulation to include surface coverage dependence of the desorption rate. In section 2 we describe the theoretical details of the calculations. We start with a brief outline of the CSDT theory which is followed by a detailed discussion of the incorporation of coverage dependence into the formalism. In section 3 we present the results of model calculations of Xe atoms desorbing from W(lll) surfaces using both attractive and repulsive adsorbate-adsorbate interactions.

2. Theory

2.1. CSDT formulation

In this section we shall give a brief description of the CSDT formulation since a detailed derivation was presented in ref. [l]. We start by considering a system consisting of a single particle (atom or molecule) adsorbed on a solid surface. Following the work of Adelman and Doll [4] and Tully and coworkers [5], we divide the crystal into two groups of atoms a primary zone and a heat bath. The primary zone consists of a small number of surface atoms (l-6) which interact strongly with the adparticle, while the heat bath contains the rest of the crystal atoms. For simplicity, we shall consider a primary zone which contains only one surface atom. In order to simplify the problem we assume a one-dimensional system in which both the adparticle and the surface atom are restricted to move in a direction normal to the surface.

The motion of the adparticle is described in terms of a Newtonian equation of motion,

mjz - av(z - -5) az ’ (2.1)

A. Redondo et al. / Rates of desorption from solid surfaces 43

where z and [ are the positions of the adparticle and the surface atom,

respectively, m is the mass of adparticle, and V(z - [) is the interaction potential between the adparticle and the surface. The motion of the surface atom is described by a generalized Langevin equation [4,5],

m I= _ wz-0 s at - rn,wf,$ - m,

J ‘O( t - t’) .$( t’) dt’ +f( t),

0 (2.2)

where m, represents the mass of the surface atom, o, is the characteristic frequency of the solid, and O( t - t’) and i(t) correspond to a memory kernel and random force which include the influence of the heat bath on the motion of the surface atom.

Next, we solve formally (using the Laplace transform method [l]) eq. (2.2) and substitute the solution into eq. (2.1). As a result we obtain a generalized Langevin equation which describes the motion of the adparticle subject to the influence of a heat bath (which consists of the whole crystal). We can now take advantage of the fact that typical desorption times (lop3 to lo3 s) are much longer than the characteristic periods for molecular vibrations (- lo-l3 s) to apply the Markovian approximation. Thus, the generalized Langevin equation is reduced to a Langevin equation of the form

(2.3)

where /I is a friction coefficient and V,,, is the adparticle-surface interaction potential modified by the motion of the primary zone atoms. Eq. (2.3) describes the motion of the adparticle in terms of a Brownian oscillator in an external field.

Eq. (2.3) is subsequently transformed into the corresponding generalized Liouville (Fokker-Planck) equation [6] for the probability distribution function, W(z, u, t). Here, W is the probability of finding the adparticle at a position z with a velocity u at time t. The flux of desorbing particles at any given position can be obtained [6] by solving the generalized Liouville equation

under steady state conditions. We evaluate the flux at a distance z,,, which is much larger than the adsorbate-surface equilibrium position, and count only those particles whose kinetic energy is large enough to desorb. Thus, the limiting velocity (below which the particle will not desorb), u0 is given by

D, = V,,,(z,) +:mui.

Once the flux is known, the rate of desorption can be obtained directly [1,6]. The final expression for the desorption rate is

R=(E,“‘i/_“_exp[-w]dz)-‘F(T)Z’(T)exp(-2). (2.4)

where D, is the dissociation energy associated with the adsorbate-surface

44 A. Redondo et al. / Rates of desorptron from solid surfaces

interaction and

with

(Y, = [ m~~~)]1’2[uu-(Uzu+b)]. (Y*= [~]“*[-iv(ol,,+b)+~~], and where

9= 1 --P/a, 9 = m9/3(az, + b)2/2kTp,

and

a=+[P-(P2-4fi2)1’2], b= _ 28

_P+(P*_~@)‘/~’

B and 8* being the first and second derivatives of V,,, evaluated at z,,, while the symbol @(a) stands for the error function [7].

The function F(T) is given by

F(T)=1 (2.5a)

for atomic desorption, while for molecular desorption it is a function of the temperature and the microscopic parameters of the adsorbate-surface system. In particular, for a diatomic molecule (e.g. CO) on metal surfaces F(T) takes the form [l]

F(T) = 2j.d2L?fy;/rkT, (2.5b)

where $*L?F is the force constant for the bending motion of the CO about the surface-carbon axis (frustrated rotation), and y0 is the maximum bending

angle. Eq. (2.4) constitutes the expression for the desorption rate obtained by the CSDT formalism. For temperatures relevant for TPD (kT +=s De), eq. (2,4)

reduces to

R = (L?,/27r) F(T) ‘7(T) exp( -D,/kT), (2.5~)

where 52, is the (stretch) vibrational frequency at the minimum of V,,,(z).

The magnitude of the function T(T) is determined by the microscopic parameters of the system, in particular the friction constant, p. As discussed in ref. [l], ‘I’(T) = 1 for most systems in which the the adparticle desorbs from a solid surface to the vacuum, thus, for such systems the rate expressions (eqs. (2.5)) are equivalent to those obtained using transition state theory [8]. How- ever, when the friction felt by the adsorbate is large, as is the case for desorption from a solid surface to a liquid [9], Y(T) is markedly different from unity.


2.2. Coverage dependence

Adsorbate-adsorbate interactions influence the desorption rate via a mod- ification of the interaction potential, V,,,(z) (and hence 0,). In the following we shall assume that the direct adsorbate-surface interaction is not changed by the adsorbate-adsorbate interaction, so that the net potential seen by the desorbing particle is the superposition of its interaction with the surface and with the other adparticles. Although this assumption may not be valid in all cases (e.g. when three-body or charge transfer effects are important), it is likely to be fairly accurate for noble gas adsorbates. However, this assumption is not essential since the present method can be implemented to include more general potentials.

In order to incorporate coverage effects into D, and V,,,(z), we must determine the optimum (minimum free energy) arrangement of the adparticles on the surface. To do this we shall first assume that the particles adsorb on well defined (periodic) sites on the infinite surface. Next, we choose a relatively small region of the surface, referred to as the desorption region (in the results described in section 3 this area contained 19 adsorption sites), which is used to generate all possible arrangements of the adparticles for each coverage, 9. In order to determine the configuration(s) exhibiting the minimum free energy, one must calculate the total energy associated with each configuration. Since the adparticle-surface interactions are the same for all arrangements (for a given S), their sum over all the adparticles in the system can be omitted from the calculation of the total energy, hence the total energy consists of a pairwise sum of adsorbate-adsorbate interaction energies. We divide this pairwise sum into two terms,

E total = c (T/in-in + vi”-out)? (2.6) desorption region

where Etotal is the total adsorbate-adsorbate interaction energy of the desorption region, Vi,_i, represents the adsorbate interactions within the desorption region, and Vin_OUt corresponds to the interaction energy of an adparticle inside the desorption region and all the other adparticles outside this zone. To evaluate the second term in eq. (2.6) we use a mean field approximation in

which each site outside the desorption region is assumed to contain 9 adparticles (where 6 is the coverage). The value of E,ota,, for the different configurations, is then calculated by combining an appropriate adsorbate-adsorbate interaction potential with the arrangement of the adparticles in the desorption region.

Once the total energies for all the possible arrangements (at a given S) have been evaluated, the configurations are sub-divided into groups according their E tota, values. The free energy associated with each group of configurations is

46 A. Redondo et al. / Rates of desorption from soled surfaces

then obtained, as a function of temperature, using

F = E,,,,, - TS, (2.7a)

S = k log(r), (2.7b)

where r is the number of configurations with energy El_,. The arrangement of adparticles (inside the desorption region) used to evaluate the desorption rate is chosen from the group of configurations corresponding to the lowest free energy. The validity of this use of only the minimum free energy configuration in calculating the rate of desorption is established in appendix A.

After identifying the minimum free energy configuration(s) (for a given coverage and temperature range), we calculate the net potential felt by the desorbing particle as

adparticles

v,&> = v,,,(z) + c b,(z), (2.8)

j#O

where V,,, is the effective adsorbate-surface interaction potential for 19 = 0 and VO, is the pairwise interaction between the desorbing particle and the jth

adparticle on the surface. Finally, the potential energy in eq. (2.4) is replaced by Vn,,(z) and the D, by the corresponding well depth of the net potential to obtain the temperature dependence of the rate of desorption for a given

coverage.

2.3. Temperature programmed desorption (TPD) spectra

From the coverage dependence of the desorption rate, R( IY), we can extract the coverage dependence of the preexponential factor, A( 9) and the effective dissociation energy, D,(8). From these relationships we can calculate TPD spectra by solving the Redhead equation [lo]

d8/dt = --9A(19) exp[ -D,(b)/kT]. (2.9)

We have solved this equation numerically for a linear time dependence of the temperature.

3. Results and discussion

To study the effect of coverage on the desorption rate, within the framework described in the previous section, a model system was chosen. A difficulty associated with this type of study stems from the lack of quantitative experimental and theoretical data on the modifications to the adsorbate-surface and adsorbate-adsorbate interaction potentials due to changes in coverage. Such changes should become evident in systems where three-body interactions


and/or charge transfer effects are important in the present study we are mainly concerned with the qualitative behavior of the desorption rates as a function of coverage.

We have studied, as a model system, the desorption of Xe atoms from W(lll) surfaces. Because of the weak bonding (De = 5 kcal), we expect only a small overlap between the adsorbate and surface electronic wavefunctions and hence the three-body and charge transfer effects should be negligible. Conse- quently, the adsorbate-surface interaction can be obtained from low coverage experimental results [2]; however, the nature of the adsorbate-adsorbate interaction is not well known and we shall consider two limiting cases: (i) an attractive Xe-Xe interaction obtained from gas phase scattering experiments and (ii) a purely repulsive Xe-Xe interaction based also on experimental gas phase studies. The presence of the surface will probably make the effective Xe-Xe interaction less attractive than in the gas phase so that the actual potential should being between the above two cases.

In calculating the desorption rate we always focus on a site (site 1 in fig. 1) that happens to be occupied, and we calculate the rate of desorption from this site. We will assume that the other possible adsorption sites are arranged in a periodic lattice forming a hexagonal pattern commensurate with the substrate (see fig. 1). The specific arrangement of occupied sites used in calculating the TPD spectra is the most probable one (the lowest free energy) for a given coverage, 9, and temperature, T. In order to calculate this most probable configuration we consider the desorption region of fig. 1 containing 18 additional binding sites. This includes all sites within 10 A of site 1, which should be sufficient since the range of the Xe-Xe potential is only 8 A (the

interaction with adsorbates outside the desorption region, shown in fig. 1, is

.8

‘9. .‘4

13. 02 09

l 7 03

18. 91 l I5

69 04

Fig. 1. Arrangement of adsorption sites of Xe on W(lll) surface. The figure shows all the sites

located within the desorption region.


included in the calculations via a mean field description). The desorption rate is then calculated assuming that the configuration of the adsorbates does not change during the process of desorbing the adsorbate from site 1 (this assumption should be valid for Xe on W(111) because the diffusion barrier is much larger than kT, see ref. [2g]).

3. I. Attractive Xe-Xe interaction

For the attractive Xe-Xe potential we have used the form determined experimentally by Barker et al. [ll] from gas phase molecular beam scattering studies. This potential has a well depth of 281 K (0.024 ev> and an equilibrium distance of 4.35 A, slightly shorter than the nearest neighbor site distance (4.47 A) on W(111). In appendix B we show the most probable (minimum free energy) configurations for coverages ranging from 0.05 to 1.00. A typical arrangement 9 = 0.42) is shown in fig. 9a. A common characteristic is that for all the coverages studied the optimum arrangement (configuration) is independent of temperature over the range considered (50 to 150 K). Essentially, at these temperatures, the system has condensed to the lowest enthalpy configuration (see fig. 9a). At higher temperatures the entropy would lead to a more dispersed configuration.

In fig. 2 we present the calculated rates of desorption as a fuanction of temperature for different coverages. At a given temperature the desorption rate increases with decreasing coverage. This behavior is a consequence of the attractive Xe-Xe interactions that lead to an increase of the effective bond energy as the coverage is increased. This trend is apparent from an examination of fig. 3, where the variation in bond energy is given as a function of coverage. The dependence of the preexponential factor on the coverage is also

T [Kl 167.0 114.0 07.0 70.0 59.0

IO-4 6.0 6.75 Il.5 14.25 17

1000/T [ K]

Fig. 2. Rates of desorption of Xe from the W(lll) surface (using the attractive Xe-Xe interaction

[S]) as a function of temperature and coverage.


0

c 0 0

25

9.0 -

0.0 -

7.0 -

A-D, values at 125 K

0 - preexponentiol factor values at 125 K

;:;t I/P-F , , , jyoig

0 0.2 0.4 0.6 0.6 1.0 ’ COVERAGE

Fig. 3. Variation of the effective dissociation energy and the preexponential factor as a function of

coverage for the desorption of Xe from W(111) (using the attractive Xe-Xe interaction) at 125 K.

shown in fig. 3, where we find that A(9) is essentially independent of 9. Thus one can assume a constant preexponential factor throughout the range of coverages studied. This behavior is somewhat surprising, since (from eq. (2.4)) it suggests that the vibrational frequency and the anharmonicity of the net potential, eq. (2.8), are independent of coverage. Hence, the most important

factor governing the coverage dependence of the overall rate of desorption (for an attractive Xe-Xe interaction) is the variation of the well depth.

3.2. Repulsive Xe-Xe interaction

The behavior of the desorption rate as a function of coverage for a repulsive adsorbate-adsorbate interaction was studied using the Xe-Xe interaction potential reported by Leonas [12] as obtained from gas phase molecular beam experiments. This potential includes only the repulsive interactions between the electron clouds of two xenon atoms as they are brought close together. As in the attractive interaction case, this potential is short ranged (at 8 A the interaction energy is lop6 ev), hence the same desorption region was used (fig.

1). The minimum free energy configurations corresponding to a repulsive

Xe-XE interaction are summarized in appendix B. In contrast with the results obtained for the attractive potential case, we find that for some coverages the optimum configuration depends strongly on the temperature. In particular, for some coverages, it is possible to find two different optimal arrangements of adparticles in the desorption region, depending on the temperature. This

50 A. Redondo et al. / Rates of dmorption from solid surfaces

phenomenon is due to the increasing importance of the entropy term in the

expression for the free energy, eq. (2.7). For all of the coverages where two optimum configurations are found, the one corresponding to the higher temperatures has an additional Xe atom in one of the nearest neighbor sites (labeled 2 to 7 in fig. 1).

The variation of the desorption rate as a function of temperature (for different coverages) is shown in fig. 4. In this case we find that, for a given

temperature, the ~es~rpt~~n rate increases with jncre~sing coverage. This behavior is opposite to that found for the attractive potential (fig. 2). As one would expect, the difference between the attractive and repulsive potentials is most marked for high coverages (e.g. for 9 = 0.95 the rates of desorption differ by at least 6 orders of magnitude, whereas for 9 = 0.1 they differ by - 0.5 order of magnitude, see figs. 2 and 4). Fig. 5 shows the variation of the effective dissociation energies and preexponential factors as a function of coverage. Again we find that the dissociation energy has a strong dependence on the coverage, while the preexponential factor is practically constant. As a result the dominant factor in the behavior of the total rate of desorption is the decrease of the effective dissociation energy with increasing coverage.

The behavior of the desorption rate as a function of temperature (fig. 4) suggests that for some coverage the observed rate will depend on the temperature at which the adsorbate was deposited on the surface. For example, for a coverage of 0.42, curve C in fig. 4, if the system is prepared at a temperature of 125 K, we expect to observe a desorption rate consistent with a De = 4.17

kcal/mol and a prefactor of 6.48 X 10 ” S-‘. Hiowever, if the adsorbate is

deposited on the surface at - 80 K, one expects (if surface diffusion is slow) to observe desorption rates which corresponds to an effective dissociation energy of 4.45 kcal/mol and a preexponential factor of 6.78 X 10” s-‘. On the other

T [Kl 167.0 114.0 87.0 70.0 59.0

/ 1 , I

1000/T [ K]

Fig. 4. Rates of desorption of Xe from the W(111) surface (using the repulsive Xe-Xe interaction [9]) as a function of temperature and coverage (the lower part of curve D and its dashed Iine continuation nearly coincide with curve E).


6.0 - X X

A D, values at 125 K

q De values at 80 K

0 preexpanentia factor values at 125 K

x preexponentia factor values ot 80 K

4.0 -

3.5 -

I

II ’

,

,

, 1.0

7

-i 0

%

7.2 =

b

x

7.0 -

6.6 2

F Z

6.4 W Z

2

6.2 2

E 6.0 a

Fig. 5. Variation of the effective dissociation energy and the preexponential factor as a function of

coverage for the desorption of Xe from W(111) (using a repulsive Xe-Xe interaction) at 125 K and

80 K. The preexponential factors and dissociation energies corresponding to the optimum

configuration at 125 K correspond to the circles and triangles, respectively, and are connected by

solid lines. The corresponding quantities at 80 K are denoted by crosses and squares, respectively

(these are only shown when they differ from the respective values at 125 K).

hand, if surface diffusion is fast cornpaired to desorption, the adparticles will diffuse so as to form the minimum free energy arrangement for each temperature leading to a non-Arrhenius desorption behavior resembling the solid line in curve C (fig. 4). The optimum arrangement for these cases are shown in figs. 9b and 9c.

3.3. TPD spectra

As discussed above, our calculations show (figs. 3 and 5) that the preexponential factors for both the attractive and repulsive Xe-Xe interactions are practically coverage independent. Thus, in the calculations reported below, we have taken A(6) in eq. (2.9) to be constant and equal to 7.0 x 10” s-l for the attractive potential and 6.5 X 10” s-l for the repulsive potential. Figs. 3 and 5 show that the variation of D with the coverage can be approximated by a linear

52 A. Redondo et al. / Rates of desorption from solid sur&xs

relationship of the form

Q.(s)=D,“+j3s,

where I$” = 0.217 eV and j3attractive = 0.152 eV; &,,pu,sive = - 0.061 eV.

(1)

The TPD spectra corresponding to the attractive and repulsive potentials (with values of /3 obtained from the results presented in figs. 3 and 5) are shown in fig. 6. The only available TPD spectra, that we know of, for Xe desorbing from W(l11) is due to the work of Dresser et al. [13a] and Yates et

Temperature ramp = 10 K/see p= +0.152

B = 1.00

105.0 130.0

T[ KI

50.0 70.0 90.0 110.0

T[ Kl

Fig. 6. Temperature Programmed Desorption spectra calculated for Xe on W(I11): (a) attractive Xe-Xe interaction; (b) repulsive Xe-Xe interaction.


al. [13b]. However, the authors of these papers believe that the single crystal surface used in their work contained non-(111) planes some of which are known to exhibit higher adsorption energies than the W(111) planes. For this reason we have not used these experimental data to compare with the calcu-

lated results obtained in this work. Opila and Gomer [14] have reported TPD curves of XE on W(ll0); because of the significant differences in the adsorption configurations of these surfaces, one can only make qualitative compari-

sons between the theory and experiment. Comparison of our results for Xe/W(lll) with the experiments of Opila and Gomer [14] for Xe/W(llO) indicates that the effective Xe-Xe potentials should be intermediate between the attractive and repulsive functions we have used. An examination of fig. 6 shows that for an attractive adsorbate-adsorbate interaction the position of the maxima of the TPD curves occurs at higher temperatures for increasing coverages; for the repulsive interaction the maxima shift to lower temperatures as the coverage is increased. The experimental TPD curves presented in fig. 3 of ref. [14] show a shift of the maxima towards higher temperatures (with increasing coverage) indicating that the actual interaction between two physisorbed Xe atoms is attractive.

Using the gas phase Xe-Xe (attractive) interaction [ll] to describe the adsorbate-adsorbate interaction on the W(lll) surface led to a variation in De of - 3.5 kcal/mol between 9 = 0 and I? = 1 (fig. 3). It is evident from the results of Opila and Gomer [14f that for Xe/W(llO) the shift is much smaller. In fig. 7 we show the TPD spectra calculated assuming a total variation in De of 0.5 kcal/mol between the limits of 6 = 0 and 6 = 1. In good qualitative

Temperature ramp = IO Kfsec

75.0 92.5 118.0

T[ KI Fig. 7. Temperature Programmed Desorption spectra calculated for Xe on W(111) using a

modified attractive Xe-Xe interaction potential.


agreement with the experimental results (for Xe/W(llO)), fig. 7 shows that the position of the TPD spectra peaks shift towards higher temperatures for increasing coverage within a range of - 6 K. Moreover, the different curves rise at approximately the same rate, whereas they decay faster as the coverage increases (the opposite trend is observed for a repulsive adsorbate-adsorbate interaction, fig. 6b). In order to illustrate how one might combine theoretical and experimental desorption results to extract adsorbate-adsorbate interaction parameters, we will assume for the moment that experiments on Xe/W(lll) would yield similar results as Xe/W(llO). In this case the theory (fig. 7) suggests that De increases by only 0.5 kcal for 9 = 0.0 to 9 = 1.0. Since at I!+ = 1.0 each physisorbed Xe atom has six nearest neighbor xenon atoms, the Xe-Xe potential well depth corresponds to l/6 x 0.5 kcal = 0.083 kcal = 50 K (as compared to 288 K for the gas phase potential). A knowledge of the experimental TPD spectra of Xe desorbing from the W(111) plane would allow us to obtain a more accurate value for the Xe-Xe interaction potential well on this surface.

4. Conclusions

We have described an extension of the CSDT approach [l] to include coverage dependence of the rate of desorption. This method is very easily implemented once the microscopic parameters of the system are known (experimentally or from theoretical calculations). Although the calculations presented in this work refer to atomic desorption, the method is applicable to molecular systems. The results obtained in these calculations suggest that for

some systems it is possible to observe non-Arrhenius behavior of the desorption rate in experimentally accessible temperature ranges.

Acknowledgment

This work was supported in part by the Department of Energy (contract No. DE-AM03-76SF00767; Project Agreement No. DE-AT03-80ER10608).

Appendix A

In the calculation of the rates of desorption for different coverages, we have used the minimum free energy configuration to determine the net potential that the desorbing particle feels. In order to justify this approximation we have carried out similar calculations using an average potential obtained from the statistical distribution of the system. In particular, since we calculated the total


energy for each possible configuration (for a given coverage), the Boltzmann factor, exp( - E,,,,/kT), was used to weight the contribution of each configuration to the net interaction potential felt by the desorbing particle. The results of this calculations are shown in fig. 8. Here we have plotted the effective dissociation energy as a function of temperature for the different coverages (for the attractive Xe-Xe interaction). Although the average number of particles at a given distance from the desorbing site varies as a function of temperature, these changes are sufficiently small so that the effective De is independent of temperature and only varies with coverage. The only apprecia- ble variation of De versus T occurs for the lowest coverages. Even in these cases the total change in the desorption rate corresponds to less than 5% throughout the temperature range studied, well within the experimental error. In addition, the average number of particles at a given distance from the desorption site,

De [kcal/mole]

8.40

8.15

6.90

6.65

6.40

6.15

5.65

0 . .z _*-* 0.32 .

- * 0.26 l . . 0.21

- k~ -w

. I I I I I I I

70 80 90 100 I IO 120 130 140

T 1 Kl

Fig. 8. Variation of the effective dissociation energy as a fuanction of temperature for a statistical

average over all possible configurations for different coverages (attractive Xe-Xe interaction).


for any temperature and coverage, is practically constant and almost equal to the corresponding value for the minimum free energy configuration. For this reason the use of the minimum free energy configuration to determine the rate of desorption is justified even in the case of high diffusion rates. This approximation is expected to be valid for systems in which the adsorbate-adsorbate interaction is short-ranged and weak. For other systems one should check its validity.

Appendix B

In this appendix we present the minimum free energy configurations obtained for the system Xe on W(111) for different coverage values. Each of the configurations shown in tables 1 and 2 represent a typical arrangement of adsorbed particles inside the desorption region corresponding to a given optimum free energy. Table 1 corresponds to an attractive adsorbate-adsorbate interaction and table 2 to a repulsive adsorbate-adsorbate potential. A value of 1 at a given site represents the presence of a Xe atom at that site, while a zero corresponds to an empty site (the site numbering is the same as in fig. 1). For all cases the temperature range examined was 50 < T < 150 K. Note that sites 2-7 are nearest neighbor sites (4.476 A), 14-19 are second nearest neighbor sites (7.74 A), and 8-13 are third nearest neighbors (8.952 A). The optimum configurations for 9 = 0.42 are shown in fig. 9. We have defined the coverage here as the average occupation for the 19 sites in the desorption

region. The lowest coverage (one adatom at site 1) then corresponds to 9 = 0.05.

Table 1

Attractive Xe-Xe interaction

I9 Site Temp.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 range

0.05 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 All T 0.11 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 All T

0.16 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 All T 0.21 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 All T 0.26 1 1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 All T 0.42 1 1 1 1 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 All T 0.53 1 1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 All T 0.63 1 1 1 1 1 0 0 1 1 1 1 0 0 1 1 1 0 0 0 All T 0.79 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 0 0 All T 0.84 11 11 11 0 11 1 1 1 0 1 1 1 1 1 0 All T 0.90 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 All T 0.95 1111111111 1 1 0 1 1 1 1 1 1 All T 1.00 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 All T

A. Redondo et al. / Rates of desorption from solid surfaces

Table 2 Repulsive Xe-Xe interaction

8 Site Temp.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 range

0.05 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 AIIT 0.1110 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 All T

0.16 10 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 All T

0.2110 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 T<120K 1001000000 10 0 10 0 0 0 0 T>120K

0.26 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 T<lOOK 1001000000 10 0 000110 T>lOOK

0.42 1 1 0 1 0 0 0 1 1 0 1 0 1 0 0 0 1 1 0 T-=lOOK 1110000110 0 0 0 0 0 11 10 T>lOOK

0.53 1 1 1 0 1 0 0 0 1 1 1 1 0000011 All T 0.63 1 1 1 1 0 1 0 1 1 1 1 0 1 001100 Tx130K

1111000111 10 0 0 0 11 11 T>130K 0.79 1 1 1 1 0 1 0 0 1 1 1 0 1 1 1 1 1 1 1 T<lOOK

1111010111 10 11 10 11 1 T>lOOK 0.84 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 All T 0.90 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 All T 0.95 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 All T 1.00 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 All T

a) Attractive Xe-Xe \nteraction b) Repulwe Xe-Xe mteractlon c) Repuhve Xe-Xe lnteractlon 70~TSl40 K 70~T<l00 K IOOsT~140 K

@P

@I4 ‘9. @a* C&P

19. ‘9.

13. aa2 @a9 ‘3. @2 .I4 @9 13@

.7 @3 .7 .3 .7 @I2 1’1 @a9

18. @’

B4 ;,I

‘*@ES @I .I5 ‘8. @I .‘5

6. .6 @4 @a6 .4

12. .5 12. .5 @IO 12 . 05 @IO 17. .‘6 ‘7@ .I6 .I7 .I6

1 I . .” . ‘1

Fig. 9. Optimum configurations for d = 0.42 (circle represents an occupied site).

[l] (a) A. Redondo, Y. Zeiri and W.a. Goddard III, Phys. Rev. Letters 49 (1982) 1847; (b) Y. Zeiri, A. Redondo and W.A. Goddard III, Surface Sci. 131 (1983) 221.

[2] See for example: (a) M.J. Dresser, T.E. Madey and J.T. Yates, Surface Sci. 42 (1974) 533; (b) J.T. Yates, Jr. and N.E. Erickson, Surface Sci. 44 (1974) 489; (c) G. Erhch, Advan. Catalysis 14 (1963) 255; (d) T. Engel and R. Comer, J. Chem. Phys. 52 (1970) 5572; (e) L. Schmidt and R. Comer, J. Chem. Phys. 42 (1965) 3573.


[3] See for example:

(a) H. Ibach, W. Erley and H. Wagner, Surface Sci. 91 (1980) 29;

(b) J.L. Taylor and W.H. Weinberg, Surface Sci. 78 (1978) 259;

(c) W. Erley, K. Besocke and H. Wagner, J. Chem. Phys. 66 (1977) 5269;

(d) C. Kohrt and R. Gomer, Surface Sci. 24 (1971) 77;

(e) G.E. Gdowski and R.J. Madix, Surface Sci. 115 (1982) 524.

[4] (a) S.A. Adelman and J.D. Doll, J. Chem. Phys. 61 (1974) 4242;

(b) S.A. Adelman and D.J. Doll, J. Chem. Phys. 63 (1975) 4908;

(c) S.A. Adelman and J.D. Doll, J. Chem. Phys. 64 (1976) 2375;

(d) S.A. Adelman, J. Chem. Phys. 71 (1979) 4471.

[5] (a) J.C. Tully, J. Chem. Phys. 73 (1980) 1975;

(b) E.K. Grimmelmann, J.C. Tully and E. Helfand, J. Chem. Phys. 74 (1981) 5300;

(c) J.C. Tully, Ann. Rev. Phys. Chem. 31 (1980) 319;

(d) J.C. Tully, Surface Sci. 111 (1981) 461.

[6] (a) H.A. Kramers, Physica 7 (1940) 284;

(b) S. Chandrasekhar, Rev. Mod. Phys. 15 (1943) 1.

[7] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions (Dover, New York,

1968). The error function is defined as a(z)-(l/h)/< exp(- t2)dr.

(81 See, for example:

(a) B.J. Garrison and S.A. Adelman, J. Chem. Phys. 67 (1977) 2379:

(b) J.E. Adams and J.D. Doll, J. Chem. Phys. 74 (1981) 1467;

(c) J.E. Adams and J.D. Doll, J. Chem. Phys. 77 (1982) 2964;

(d) A. Redondo, Y. Zeiri, J.J. Low and W.A. Goddard III, J. Chem. Phys., in press.

[9] Y. Zeiri, A. Redondo and W.A. Goddard III, J. Electrochem. Sot., submitted.

[lo] P.A. Redhead, Vacuum 12 (1962) 203.

[ll] J.A. Barker, R.O. Watts, J.K. Lee, T.P. Schaefer and Y.T. Lee, J. Chem. Phys. 61 (1974) 3081.

[12] V.B. Leonas, Soviet Phys.-Usp. 15 (1973) 266.

[13] (a) M.J. Dresser, T.E. Madey and J.T. Yates, Jr., Surface Sci. 42 (1974) 533;

(b) J.T. Yates, Jr. and N.E. Erickson, Surface Sci. 44 (1974) 489.

[14] R. Opila and R. Gomer, Surface Sci. 112 (1981) 1.

RATES OF DESORPTION FROM SOLID SURFACES: COVERAGE DEPENDENCE · RATES OF DESORPTION FROM SOLID...

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