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Image theory of lateral interactions applied to desorption rate of Xe from W and Ag
L141
Surface Science 181 (1987) L141-L146 North-Holland. Amsterdam
SURFACE
SCIENCE
IMAGE THEORY DESORPTION
K. JOYCE,
LETTERS
OF LATERAL
P.J. GROUT
and
Theoreticcrl Chemistry Department, OxJord OXI 3TG, UK
Received
INTERACTIONS
RATE OF Xe FROM
28 July 1986; accepted
APPLIED TO
W AND Ag
N.H. MARCH
University of Oxford, I South Parks Road,
for publication
4 November
1986
Th#e image theory of lateral interactions between rare gas atoms given by Majxmty and March [J. Phys. C9 (1976) 29051 is shown to be consistent with the mean feaiures of desorption rate measurements for Xe on W(111) and W(110). Some results are also presented for both Xe and Ar on A&Ill).
In recent work, Redondo et al. [l] have studied the effect of lateral interactions between Xe atoms physisorbed on the surface of W, on the rate of desorption. Their main conclusion was that the attractiveness of the free space Xe-Xe interaction had to be weakened appreciably to account for the experimental data. The purpose of this present work is to investigate how well the lateral Xe-Xe interactions in the presence of the metal surface can be accounted for by the image theory of such interactions presented by Mahanty and March [2]. These workers, starting from a Lennard-Jones 6-12 potential, argued that the coefficient of the dispersion term could be calculated from first principles by invoking image theory in conjunction with the assumption that the metal is a perfect conductor. Their result for the lateral interaction in the presence of the surface takes the explicit form
v(p,
z)=“-s P
1+ P6 i
1 (1 + 9)’
4(1 + s2/4) - 3(1 + s2)5’2 I ’
(1)
being the distance between adsorbate atoms, while s = 2z/p, with z the distance from the surface. The constants C, and B characterize the free space interaction for the 6-12 potential assumed. Since Redondo et al. [l] used the free space Xe-Xe potential of Barker et al. [3] and concluded for Xe on W(111) that the true potential was less attractive than this, it seemed clear that it should be possible to obtain p
0039-6028/87/$03.50
(North-Holland
0 Elsevier
Physics
Publishing
Science Publishers Division)
B.V.
Table 1 Values of average
preexponential
Xe/W(lll) Ar/Ag(lll) Xe/Ag(lll)
factors
and of p in eq. (3) for Xe and Ar on W and Ag surfaces
p (kJ mol-‘)
A,, (s-l)
7.46 4.50 7.18
5.9 x10” 1.15 x 10” 1.02 x 10’2
improved agreement with the experimental temperature programmed desorption (TPD) curves using eq. (1). We have therefore carried out calculations paralleling those of Redondo et al., but using eq. (1) for the lateral interaction. For each coverage 0, one can thereby determine the preexponential factor A(B) and desorption energy o,(e) in the Redhead equation [4] dO/dt
= -&f(B)
exp[ -D,(e)/RT].
Using the assumptions that A(B) is independent desorption energy can be approximated as Q”(e)
= Q(O)
+ N,
of coverage
and
that
the
(3)
table 1 records the value of p and the average values of the preexponential factor A for the systems for which we have carried out calculations. The Redhead equation (2) was solved numerically and the TPD spectra calculated, using a linear time dependence of the temperature dT/d t = 10 K s Al The only TPD spectra known to us for Xe desorbing from W(111) are due to Dresser et al. [5] and to Yates and Erickson [6]. However, these workers concluded that the single crystal surface used in their work contained planes other than (111); some of which are known to exhibit higher energies than the W(lll) planes. Thus, these experiments cannot be compared directly with our calculated spectra. Redondo et al. [l], using the free space Barker potential to model the lateral interaction, concluded that this potential gave too great a variation with coverage of the desorption energy. The Mahanty-March potential substantially reduces this variation, as can be seen by comparing the two curves in fig. 1, and this is certainly the direction in which improved agreement with experiment will result. In fact, the theory predicts that the desorption energy increases by - 7500 J mol -I as 0 increases from 0 to 1. Since at 13= 1 each physisorbed Xe atom has six nearest-neighbour Xe atoms, the Xe-Xe potential well depth corresponds to i x 7500 J mol-’ z 150 K. The gas phase potential has a well depth of 281 K. Therefore the Xe-Xe well depth on the surface is 0.53 of that of the gas phase potential which is some 20% above the maximum reduction predicted by Mahanty and March [2].
K. Joyce et al. / Image theon, of lamal
zo,A Fig. 1. Variation
interactions
L143
COVERAGE
of desorption
energy with coverage.
Unfortunately, for desorption from the Ag(lll) surface, no experimental data is presently known to us. Some attention has therefore been given to Xe on the W(110) surface, for which case Engel et al. [7] have performed a LEED investigation. Four ordered LEED patterns were observed for temperatures between 77 and 90 K. It was thus felt worthwhile to carry out calculations on this system. The desorption region is as shown in fig. 2, with 17 sites, and an additional 108 sites out to a maximum radius of approximately 27 A from the central site. A Morse function was used to model the surface-adsorbate interaction, a value of 38 493 J mol-’ [8] being used for the zero coverage desorption energy.
5 3 0
region,
Q +L
1 + I-----I
Fig. 2. The desorption
ii
i
1: 73
+17 5 +
;s +13
9 +
IL +
the numbers merely identify sites for use with table 2. (Site 17 is at the origin and the distances are in A.)
K. Joyce et 01. / Image theor), oflrterul
L144
rnterwtiom
Table 2 Minimum free energy configurations for Xe on the W(110) surface at 84 K (sites are grouped according to the distance from the central site, 17; 1 denotes site occupied. 0 unoccupied) 9
0.0588 0.1176 0.1765 0.2353 0.2941 0.3529 0.4706 0.5882 0.7059 0.8235 0.8824 0.9412 1.0000
Site
4.2215 A
17
5
8
9
12
1 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1000 1 0 1 1 1 1 1 1
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
1 0 1 1 1 1 1 1
5.53 I\ 6.38 A -~ 4 13 6 11
2
7
1015
1
3
14 16
00 10 10 10 10 11 11 11 11 11 11 11 11
0 0 0000 1000 1 0 1 0 1 0 1 1 1 1 I 1 1 1 1 1 1 1 1 1
0 0
0 0 0 0 0 0 0 0 1 1 1 1 1
0 0 0 0 0 0 0 0 1 1 1 1 I
0 0 0 0 0 0 0 0 0 1 1 1 1
00 0 0 00 10 10 10 11 11 11 11 10 11 11
8.443 .A
0 1 1 1 1 1 1 10 10 1
8.887 A
0 0 0 0 1 1 1
1
0 0 0 0 0 0 0 1 0 0 1 1 1
Hood et al. [9] quote a value of the Morse parameter 01 of 1.34 k’ without specifying the face of W. It was found that the coverage dependence of the desorption energy was not sensitive to the value of (Y. Table 2 shows the minimum free energy configurations, calculated following ref. [l]. We find the approximate linear relation for o,(e): 0, (6) = 38 493 + 28078 J mol-’ Fig. 3 shows the rates of desorption as a function of temperature and coverage, giving a preexponential factor of 1.6 x 1012 SC'; again essentially coverage independent. Opila and Gomer [lo] report a value of 1 X 10" s-l in good accord with our calculated value. Finally the associated TPD spectra are shown in fig. 4, the assumptions being again that A( 0) is constant and that O,( 19) is linear in 8. Experimental TPD curves are available for Xe/W(llO); the shape of the calculated TPD spectra is found to be in good agreement with the spectra of Opila and Gomer, in spite of the fact that there is a temperature difference of approximately 90 K between theory and experiment. This is in large part due to our use of 38493 J mall’ for the zero coverage desorption energy. Opila and Gomer conclude that the activation energy for Xe desorption from W(110) is less than the corresponding enthalpy and equal to 17 991 J mall’; we refer to their paper for an explanation of this decrease. Fig. 5 shows the TPD spectra calculated using o,(e)
= 17991 + 28070 J mol-‘,
(4)
K. Joyce et al. / Image themy of lateral interactions
-16
L145
H ;G CD L INVERSE TEMPERATURE hlOOK!‘e
Fig. 3. Rate of desorption of Xe from W(110) as a function of temperature and coverage. B = 0.94, B: B=O.82, C: B=O.59, D: 0 = 0.47, E: B=O.35, F: 0 =0.29, G: B =0.24, 6’ = 0.12. I and J respectively experimental and calculated lines with Do = 4.3 kcal mol-’ 8 = 0.235.)
(A: H: and
together with the peaks of the experimental curves. These latter points have been translated by + 7 K which is equivalent to a change in the zero coverage desorption energy of - 1% - well within any experimental uncertainty. 5
0 70 Fig 4. Thermal
desorption
spectra
80 90 TEMPERATURE (K)
for Xe on W(110) surface. e = 0.4, E: e = 0.2).
(A: 19=I, B: B = 0.8, C: B = 0.6. D:
L146
K. Jovce et al. / Image theoty of herd
0 70 Fig. 5. Thermal
desorption
spectra
TEWE!?%"FE
calculated
using fig. 4.)
interactions
90 (K]
eq. (4) for Xe on W(110).
(Labelling
as in
Since the experimental TPD curves are given by Opila and Gomer it is possible to obtain the variation with coverage (6) given the functional form Q(8)
= 17991+
pe.
TPD spectra were fitted most closely with the choice p = 1255 J mall’, which considering the simplicity of the model is not too different from the calculated value in eq. (4). Opila and Gomer also report the rate constant as a function of temperature at a coverage of approximately 0.25. The calculated temperature dependence of the rate constant has been compared with the experimental data of Opila and Gomer: there is excellent agreement between theory and experiment. In conclusion, available evidence from desorption measurements supports the Mahanty-March theory of the lateral interactions between rare gas atoms physisorbed on W surfaces. References [l] [2] [3] [4] [5] [6] [7] [E] [9] [lo]
A. Redondo, Y. Zeiri and W.A. Goddard III, Surface Sci. 136 (1984) 41. J. Mahanty and N.H. March, J. Phys. C9 (1976) 2905. J.A. Barker, R.O. Watts and J.K. Lee, J. Chem. Phys. 61 (1974) 3081. P.A. Redhead, Vacuum 12 (1962) 203. M.J. Dresser, T.E. Madey and J.T. Yates, Jr., Surface Sci. 42 (1974) 533. J.T. Yates, Jr. and N.E. Erickson, Surface Sci. 44 (1974) 489. T. Engel, P. Bornemann and E. Bauer, Surface Sci. 81 (1979) 252. T. Engel and R. Gomer, J. Chem. Phys. 52 (1970) 5572. E. Hood, C. JedrzeJek, K.F. Freed and H. Metiu, J. Chem. Phys. 81 (1984) 3277. R. Opila and R. Gomer, Surface Sci. 112 (1981) 1.
Surface Science 181 (1987) L141-L146 North-Holland. Amsterdam
SURFACE
SCIENCE
IMAGE THEORY DESORPTION
K. JOYCE,
LETTERS
OF LATERAL
P.J. GROUT
and
Theoreticcrl Chemistry Department, OxJord OXI 3TG, UK
Received
INTERACTIONS
RATE OF Xe FROM
28 July 1986; accepted
APPLIED TO
W AND Ag
N.H. MARCH
University of Oxford, I South Parks Road,
for publication
4 November
1986
Th#e image theory of lateral interactions between rare gas atoms given by Majxmty and March [J. Phys. C9 (1976) 29051 is shown to be consistent with the mean feaiures of desorption rate measurements for Xe on W(111) and W(110). Some results are also presented for both Xe and Ar on A&Ill).
In recent work, Redondo et al. [l] have studied the effect of lateral interactions between Xe atoms physisorbed on the surface of W, on the rate of desorption. Their main conclusion was that the attractiveness of the free space Xe-Xe interaction had to be weakened appreciably to account for the experimental data. The purpose of this present work is to investigate how well the lateral Xe-Xe interactions in the presence of the metal surface can be accounted for by the image theory of such interactions presented by Mahanty and March [2]. These workers, starting from a Lennard-Jones 6-12 potential, argued that the coefficient of the dispersion term could be calculated from first principles by invoking image theory in conjunction with the assumption that the metal is a perfect conductor. Their result for the lateral interaction in the presence of the surface takes the explicit form
v(p,
z)=“-s P
1+ P6 i
1 (1 + 9)’
4(1 + s2/4) - 3(1 + s2)5’2 I ’
(1)
being the distance between adsorbate atoms, while s = 2z/p, with z the distance from the surface. The constants C, and B characterize the free space interaction for the 6-12 potential assumed. Since Redondo et al. [l] used the free space Xe-Xe potential of Barker et al. [3] and concluded for Xe on W(111) that the true potential was less attractive than this, it seemed clear that it should be possible to obtain p
0039-6028/87/$03.50
(North-Holland
0 Elsevier
Physics
Publishing
Science Publishers Division)
B.V.
Table 1 Values of average
preexponential
Xe/W(lll) Ar/Ag(lll) Xe/Ag(lll)
factors
and of p in eq. (3) for Xe and Ar on W and Ag surfaces
p (kJ mol-‘)
A,, (s-l)
7.46 4.50 7.18
5.9 x10” 1.15 x 10” 1.02 x 10’2
improved agreement with the experimental temperature programmed desorption (TPD) curves using eq. (1). We have therefore carried out calculations paralleling those of Redondo et al., but using eq. (1) for the lateral interaction. For each coverage 0, one can thereby determine the preexponential factor A(B) and desorption energy o,(e) in the Redhead equation [4] dO/dt
= -&f(B)
exp[ -D,(e)/RT].
Using the assumptions that A(B) is independent desorption energy can be approximated as Q”(e)
= Q(O)
+ N,
of coverage
and
that
the
(3)
table 1 records the value of p and the average values of the preexponential factor A for the systems for which we have carried out calculations. The Redhead equation (2) was solved numerically and the TPD spectra calculated, using a linear time dependence of the temperature dT/d t = 10 K s Al The only TPD spectra known to us for Xe desorbing from W(111) are due to Dresser et al. [5] and to Yates and Erickson [6]. However, these workers concluded that the single crystal surface used in their work contained planes other than (111); some of which are known to exhibit higher energies than the W(lll) planes. Thus, these experiments cannot be compared directly with our calculated spectra. Redondo et al. [l], using the free space Barker potential to model the lateral interaction, concluded that this potential gave too great a variation with coverage of the desorption energy. The Mahanty-March potential substantially reduces this variation, as can be seen by comparing the two curves in fig. 1, and this is certainly the direction in which improved agreement with experiment will result. In fact, the theory predicts that the desorption energy increases by - 7500 J mol -I as 0 increases from 0 to 1. Since at 13= 1 each physisorbed Xe atom has six nearest-neighbour Xe atoms, the Xe-Xe potential well depth corresponds to i x 7500 J mol-’ z 150 K. The gas phase potential has a well depth of 281 K. Therefore the Xe-Xe well depth on the surface is 0.53 of that of the gas phase potential which is some 20% above the maximum reduction predicted by Mahanty and March [2].
K. Joyce et al. / Image theon, of lamal
zo,A Fig. 1. Variation
interactions
L143
COVERAGE
of desorption
energy with coverage.
Unfortunately, for desorption from the Ag(lll) surface, no experimental data is presently known to us. Some attention has therefore been given to Xe on the W(110) surface, for which case Engel et al. [7] have performed a LEED investigation. Four ordered LEED patterns were observed for temperatures between 77 and 90 K. It was thus felt worthwhile to carry out calculations on this system. The desorption region is as shown in fig. 2, with 17 sites, and an additional 108 sites out to a maximum radius of approximately 27 A from the central site. A Morse function was used to model the surface-adsorbate interaction, a value of 38 493 J mol-’ [8] being used for the zero coverage desorption energy.
5 3 0
region,
Q +L
1 + I-----I
Fig. 2. The desorption
ii
i
1: 73
+17 5 +
;s +13
9 +
IL +
the numbers merely identify sites for use with table 2. (Site 17 is at the origin and the distances are in A.)
K. Joyce et 01. / Image theor), oflrterul
L144
rnterwtiom
Table 2 Minimum free energy configurations for Xe on the W(110) surface at 84 K (sites are grouped according to the distance from the central site, 17; 1 denotes site occupied. 0 unoccupied) 9
0.0588 0.1176 0.1765 0.2353 0.2941 0.3529 0.4706 0.5882 0.7059 0.8235 0.8824 0.9412 1.0000
Site
4.2215 A
17
5
8
9
12
1 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1000 1 0 1 1 1 1 1 1
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
1 0 1 1 1 1 1 1
5.53 I\ 6.38 A -~ 4 13 6 11
2
7
1015
1
3
14 16
00 10 10 10 10 11 11 11 11 11 11 11 11
0 0 0000 1000 1 0 1 0 1 0 1 1 1 1 I 1 1 1 1 1 1 1 1 1
0 0
0 0 0 0 0 0 0 0 1 1 1 1 1
0 0 0 0 0 0 0 0 1 1 1 1 I
0 0 0 0 0 0 0 0 0 1 1 1 1
00 0 0 00 10 10 10 11 11 11 11 10 11 11
8.443 .A
0 1 1 1 1 1 1 10 10 1
8.887 A
0 0 0 0 1 1 1
1
0 0 0 0 0 0 0 1 0 0 1 1 1
Hood et al. [9] quote a value of the Morse parameter 01 of 1.34 k’ without specifying the face of W. It was found that the coverage dependence of the desorption energy was not sensitive to the value of (Y. Table 2 shows the minimum free energy configurations, calculated following ref. [l]. We find the approximate linear relation for o,(e): 0, (6) = 38 493 + 28078 J mol-’ Fig. 3 shows the rates of desorption as a function of temperature and coverage, giving a preexponential factor of 1.6 x 1012 SC'; again essentially coverage independent. Opila and Gomer [lo] report a value of 1 X 10" s-l in good accord with our calculated value. Finally the associated TPD spectra are shown in fig. 4, the assumptions being again that A( 0) is constant and that O,( 19) is linear in 8. Experimental TPD curves are available for Xe/W(llO); the shape of the calculated TPD spectra is found to be in good agreement with the spectra of Opila and Gomer, in spite of the fact that there is a temperature difference of approximately 90 K between theory and experiment. This is in large part due to our use of 38493 J mall’ for the zero coverage desorption energy. Opila and Gomer conclude that the activation energy for Xe desorption from W(110) is less than the corresponding enthalpy and equal to 17 991 J mall’; we refer to their paper for an explanation of this decrease. Fig. 5 shows the TPD spectra calculated using o,(e)
= 17991 + 28070 J mol-‘,
(4)
K. Joyce et al. / Image themy of lateral interactions
-16
L145
H ;G CD L INVERSE TEMPERATURE hlOOK!‘e
Fig. 3. Rate of desorption of Xe from W(110) as a function of temperature and coverage. B = 0.94, B: B=O.82, C: B=O.59, D: 0 = 0.47, E: B=O.35, F: 0 =0.29, G: B =0.24, 6’ = 0.12. I and J respectively experimental and calculated lines with Do = 4.3 kcal mol-’ 8 = 0.235.)
(A: H: and
together with the peaks of the experimental curves. These latter points have been translated by + 7 K which is equivalent to a change in the zero coverage desorption energy of - 1% - well within any experimental uncertainty. 5
0 70 Fig 4. Thermal
desorption
spectra
80 90 TEMPERATURE (K)
for Xe on W(110) surface. e = 0.4, E: e = 0.2).
(A: 19=I, B: B = 0.8, C: B = 0.6. D:
L146
K. Jovce et al. / Image theoty of herd
0 70 Fig. 5. Thermal
desorption
spectra
TEWE!?%"FE
calculated
using fig. 4.)
interactions
90 (K]
eq. (4) for Xe on W(110).
(Labelling
as in
Since the experimental TPD curves are given by Opila and Gomer it is possible to obtain the variation with coverage (6) given the functional form Q(8)
= 17991+
pe.
TPD spectra were fitted most closely with the choice p = 1255 J mall’, which considering the simplicity of the model is not too different from the calculated value in eq. (4). Opila and Gomer also report the rate constant as a function of temperature at a coverage of approximately 0.25. The calculated temperature dependence of the rate constant has been compared with the experimental data of Opila and Gomer: there is excellent agreement between theory and experiment. In conclusion, available evidence from desorption measurements supports the Mahanty-March theory of the lateral interactions between rare gas atoms physisorbed on W surfaces. References [l] [2] [3] [4] [5] [6] [7] [E] [9] [lo]
A. Redondo, Y. Zeiri and W.A. Goddard III, Surface Sci. 136 (1984) 41. J. Mahanty and N.H. March, J. Phys. C9 (1976) 2905. J.A. Barker, R.O. Watts and J.K. Lee, J. Chem. Phys. 61 (1974) 3081. P.A. Redhead, Vacuum 12 (1962) 203. M.J. Dresser, T.E. Madey and J.T. Yates, Jr., Surface Sci. 42 (1974) 533. J.T. Yates, Jr. and N.E. Erickson, Surface Sci. 44 (1974) 489. T. Engel, P. Bornemann and E. Bauer, Surface Sci. 81 (1979) 252. T. Engel and R. Gomer, J. Chem. Phys. 52 (1970) 5572. E. Hood, C. JedrzeJek, K.F. Freed and H. Metiu, J. Chem. Phys. 81 (1984) 3277. R. Opila and R. Gomer, Surface Sci. 112 (1981) 1.