612
Langmuir 1989,5, 612-615
concurrently reducing the size of the smaller pores. The acid treatment may effect these changes by the transport of silica from the walls of larger pores to those of smaller ones.
Acknowledgment. We acknowledge the assistance of
R. Bartholomew, who determined the butane isotherms. Financial support by the Natural Sciences and Engineering Research Council of Canada is acknowledged. Registry No. "Os,
7697-37-2; Ar, 7440-37-1; Kr, 7439-90-9;
Xe, 7440-63-3.
Interaction of Rare Gases and H2with Surfaces of Magnesium Oxide and Other Insulatorst G. Vidali* Physics Department, Syracuse University, Syracuse, New York 13244-1130
M. Karimi Physics Department, Utica College, Utica, New York 13502 Received October 27, 1988. In Final Form: February 6, 1989 We propose a new model potential to describe the interaction between rare-gas atoms and Hz with surfaces of insulators. The potential consists of a repulsive part calculated by using the effective medium theory and an attractive part due to dispersion forces. We find that our potential gives shallower well depths than given by experiments. However, if we let c6 range within reasonable bounds, our potential fits the corrugation of the surface and the bound-state energies very well. We discuss in particular the He/MgO case, for which contradictory experimental results exist. Our model supports the recent data of Frankl's group, which predicts a shallower well depth than previously measured. We also comment on which parameters of the potentials should be known more accurately in order to bring calculations and experimental data to a closer agreement.
Introduction From the first experiments in the early 1930s by Stern and collaboratorson the scattering of helium and hydrogen atoms from surfaces of single crystals to today's molecular dynamics simulations, the need to model the atom-surface potential accurately has guided the work of many theoreticians throughout these years.'" While there has been a tremendous improvement from the early 1930s in experimental techniques designed to probe the atom-surface interaction, theoretical developments have somewhat lagged behind, both in the description of the atom-surface collision process and in representing the various physical aspects of the interaction. The technique of atom beam scattering (ABS)has been proven to be the most effective tool to study the atomsurface interaction.2f+11 The availability of accurate ABS data (bound-state levels of the atom-surface potential as well as diffraction patterns) for many atom-surface combinations has spurred theoreticians to come up with better models of the atom-surface potential. Other techniques, such as calorimetry and neutron scattering, provide complementary information, such as diffusion coefficients of adatoms on surfaces, adatom-substrate equilibrium distances, and strength of the adatom-adatom interaction. Another area in which an accurate description of the atom-surface potential is required is the interpretation of diffraction patterns from helium beam scattering to obtain information on the structure of the substrate. Great strides have been done recently with the application of density functional methods to relate the atom-surface potential Presented at the symposium on "Adsorption on Solid Surfaces", 62nd Colloid and Surface Science Symposium, Pennsylvania State University, State College, PA, June 19-22, 1988; W. A. Steele, Chairman.
0743-7463/89/2405-0612$01.50/0
to the charge density distribution of the surface a t o m ~ 4 . 8 ' ~ ~ ~ ,16 which calculates The effective medium theory (EMT) the cost of embedding a foreign atom in a homogeneous electron gas, has been applied successfully to describe the interaction of a helium atom with metal surfaces (described by the jellium model). In recent publi~ations,'~J~ we have explored the possibility of extending this approach to the interaction of light and rare-gas atoms with the (001) surfaces of LiF, NaC1, and MgO. Theoretical interest in these systems has been spurred by the availability of copious and accurate data of helium and hydrogen beam scattering'"22 and by the (1) Zaremba, E.; Kohn, W. Phys. Reu. 1976, B13, 2270. (2)Hoinkes, H. Reu. Mod. Phys. 1980,52,933. (3)Vidali, G.; Cole, M. W.; Klein, J. Phys. Reu. 1983,B28, 3064. (4)Hutaon, J. M.; Fowler, P, W. Surf. Sci. 1986,173, 337. (5)Celli, V.; Eichenauer, D.; Kaufhold, A.; Toennies, J. P. J . Chem. Phys. 1985,83,2504. (6) Bruch, L. W. Surf. Sci. 1983,125,194. (7)Cole, M. W.; Toigo, F.; Tosatti, E. In Surf. Sci. 1983,125, 1. (8)Liebsch, A.; Harris, J. Surf. Sci. 1982, f23, 355. (9)Frankl, D.R. Progr. Surf. Sci. 1983,13,285. (10)Engel, T.; Rieder, K. H. In "Structural Studies of Surfaces"; SDrinner Tracts Mod. Phvs. 1978.91. (117Cole, M. W.;Frankl, D. R.; Goodstein, D. L. Reu. Mod. Phys. 1981.53. 199. (12)Toigo, F.; Cole, M. W. Phys. Rev. 1986,B32,6989. (13)Karikorpi, M.; Manninen, M.; Umrigar, C. Surf. Sci. 1986,169, 299. (14)Terroff, J.; Cardillo, M. J.; Hamann, D. R. Phys. Rev. 1985,B32, 5044. (15)Karimi, M.; Vidali, G. Phys. Rev. 1988,B38,7759. (16)Norskov, J. K.Phys. Rev. 1982,B26,2875. Lang, N. D.;Norskov, J. K. Phys. Reu. 1983,827,4612. (17)Karimi, M.;Vidali, G. In Diffusionat Interfaces: Microscopic Concepts;Springer Verlag Seriea in Surface Science; Kreuzer, J., Weimer, J., Grunze, M., Eds.; 1988;Vol. 12,p 43. Karimi, M.; Vidali, G., unpublished results. (18) See references cited in ref 2,9,and 10.
0 1989 American Chemical Society
Interaction of Rare Gases and H2 use of MgO(001) as a substrate for studies of adsorption of rare-gas atoms and m o l e ~ u l e s . There ~ ~ ~ ~are several reasons why theoretical interpretation of data for these systems has somewhat lagged behind with respect to He/metal systems. First, the He/LiF and NaCl (001) interactions have been mostly described with the use of pair potentials between helium and each ion. While these models have been generally successful in reproducing or fitting the available ABS data:& they required complicated extensive calculations to produce He-ion pair potentials in the first place. Second, the EMT approach has not worked very well either, since an accurate knowledge of the ion charge densities is required. It was shown13@that the use of free ion charge densities in EMT leads to an overestimation of the corrugation of the potential across the unit cell. Third, it is known that some of these ionic surfaces relax or rumple (27); the extent of these effects for the surfaces used in the ABS experiments is not very well known. Fourth, in both the pair potential or our models, the attractive part of the interaction is described by damped dipole and quadrupole terms between the helium atom and the ions in the solid. The coefficients in front of these terms are calculated for helium interacting with free ions; in the case of He-LiF, there is evidence that indicates that the C6coefficients are inadequately Notwithstanding these difficulties, we have decided to apply a modified EMT approach to the systems outlined above. The discrepancies between our model (without fitting parameters) and experimental data might serve as a clue where and how to improve the proposed model. If we let C [ fluctuate around reasonable bounds (+/- 20%), we obtain a good fit to the bound states and the corrugation of the potential except for the case of H2. A rather complete description of our model has appeared el~ewhere.'~J'Here we will discuss in detail the He-MgO, H2-MgO, and He-rare-gas atom plated MgO. Results for other rare gases are given in ref 17.
The Model In recent publi~ations,'~J~ we have described our model a t length. Here we will outline the main characteristics. The interaction between a rare-gas atom and a surface is written as = vA(?) + vR(?) + VIn(?) where VAis the attractive part and V , the repulsive part, respectively. P gives the interaction due to the surface dipoles; it is a small contribution and short ranged. VR is obtained by using the EMT approach: VR(?) = .effP(?) ffeff = ffo - ffat
v(?)
(19) Cantini, P.; Cevasco, E. Surf. Sci. 1984,148,37. (20) Brusdeylins, G.; Doak,R. B.; Skofronick, J. G.; Toennies, J. P. Surf. Scc. 1983, 128, 191. (21) Rieder, K. H. Surf. Sci. 1982, 118, 57. (22) Jung, D. R.; Mahgerefteh, M.; Framkl, D. R., preprint. Mahgerefteh, M.; Jung, D. R.; Frankl, D. R. Phys. Reu. 1989,39,3900. (23)'Sullivan,S.; Migone, A. D.; Viches, 0. E. Surf. Sci. 1985,162,461. (24) Bienfait, M.; Coulomb, J. P.; Palmari, J. P. Surf. Sci. 1987,182, 551. (25) See other articles in this symposium collection in this issue. (26) Frigo, A.; Toigo, F.; Cole, M. W.; Goodman, F. 0.Phys. Rev. 1986, B3,4184. (27) See ref 21 for a discussion on this point. (28) Vidal-Valat, G.; Vidal, J. P.; Kurki-Suonio, K. Acta Crystallogr. 1978, AM, 594.
Langmuir, Vol. 5, No. 3, 1989 613 where p is the electron charge density, 4ais the electrostatic potential of the adatom, and a0 is a coefficient that depends only on the adatom. This description is thought to hold in the region of interest to us, i.e., for very low sampled charge densities and far from the ion cores. EMT gives the wrong functional dependence of the interaction at large distances; this is not surprising since exchange and correlation terms in EMT are calculated in the local density approximation. We have therefore used van der Waals dispersion terms to describe the long-range part of the potential. Damping has been included to turn off these terms where adatom and surface orbitals start to overlap. The complete formulas describing these terms have already been p~b1ished.l~ We do not expect that Vn will give an accurate description of the interaction at short distances. For z < 2.5 A, we kept P equal to the value at 2.5 A. Near the well, Vn gives very little contribution to the total potential. In the case of MgO, we took the in-crystal ion charge densities from ref 28. The total charge density was then given by a summation of the ionic charge densities. For LiF and NaC1, we were not able to find similar information. We decided to use available repulsive atom-ion potentials and reinterpret them in terms of charge densities, i.e. from which we obtain p. We stress that this procedure has been used here for comparison only; we still wait for better calculations of in-crystal ion charge densities. On the other hand, this procedure is not as outrageous as one might think at first. In fact, if EMT describes the major aspects of the physics of the interaction correctly, then VR
P(F)
is true; if one finds VR by other means, as was done in ref 4 and 5, then one can extract the charge density p from the above relation. One is "justified" to do so if the attractive part is treated in the same way, i.e., in ref 4 and 5 and in this work, or otherwise a stronger attraction balanced by a stronger repulsion might give misleading results. We indeed adopt a very similar formalism for treating the attractive part as it is in ref 4 and 5. In writing down the interaction between a helium atom and a rare-gas-plated MgO(001) surface, we have essentially followed the method of Chung et who addressed the problem of calcualting the interaction between He and a rare-gas-plated graphite surface. The method consists of writing the interaction as the s u m of the long-range part of the He-MgO potential plus the pairwise interaction of He with each adatom of the overlayer.30 Three-body terms are also accounted for.
Results and Discussion The results of our calculations are presented in Tables 1-111. By inspecting Table I, we note that the model presented here gives a ground-state energy that is usually lower, i.e., shallower,than what is obtained experimentally. If we let c6- assume values within 10-20% of the calculated values, we can get a very good fit to the experimental data for most of the cases. We should add that the corrugation, or modulation of the potential across the unit cell, changes little by changing CG-. This fact gives us some confidence that this model is essentially working. (29) Chung, S.; Holter, N.; Cole, M. W. Surf. Sci. 1986,165,466;Phys. Reu. 1985, B31,6660. (30) Aziz, R. A. In Inert Gases; Klein, M. L., Ed.; Springer Series in Chemical Physics; 1984; Vol. 34. Maitland, G. C.; Smith, E. B. Chem. Phys. Lett 1973,22, 443.
614 Langmuir, Vol. 5, No. 3, 1989
Vidali and Karimi
Table I. Comparison between Experimental and Theoretical Predictions of Mean Distance (z),Well Depth D , Corrugation [, Fitting Parameter q , and Energy Level E,, with the Calculations of This Paperc system H2-LiF
(2)
D
E
4
-E0
-E,
-E2
-E3
3.36
17.26
8.39
3.59
1.32
3.43
0.50 (0.40)b 1.0
1.45
H2-NaC1
1.68
29.48
17.42
9.49
0.82
HZ-MgO
3.35
0.30
1.51
[26.28]
15.83
8.84
0.88
He-LiF
3.26
(23.6)b (14.23)' (37.31)b (18.3)' 33.00 (18.22)" 8.70 (7.03)'
0.61 (0.52)b
1.17 1.0
3.76
6.19 (7.91)
1.20 (1.05)b
0.82 1.0
7.58 (6.90)"
0.42 (0.47)b
1.07 1.0
2.49 (2.15)"
0.80 (0.48)' [0.78] 0.45 (0.22)" [0.31] 1.00 (0.81)"
0.20 (0.08)' [0.21]
He-NaC1
2.40 (1.66)' [2.46] 1.51 (2.32)'
(2.62)
(1.17)
(0.49)
He-MgO
3.77
5.91 (4.58)" [5.90] 4.07 (5.49)" [4.11 5.41 (4.85)" [4.82] (5.32)
(-)
0.33 (0.25)'
"Predictions of our model with no adjustable parameters. models. 'Energies are in meV; lengths are in angstroms. [ 1, experimental values from ref 4, 5, and 33; ( ), experimental values from ref 22.
Table 11. Matrix Elements (mlVQJn)for He/MgO and H2/Mg0 (in meV) n
m
0
1
2
3
He/MgO Vo,
0 0.230 1
-0.170 0.148
2 3 V,I
0 0.349
X
1
VO,
v,
IO-' -0.269 x IO-' 0.239 X IO-'
2 3 0 0.150 X lo-' -0,125 1 0.114 2 0 o 0.327 x 10-3 -0.276 1 0.254 2 3
Vm 0 0.104 X 10' -*
1
-0.771 0.805
X X
x 10-3 x 10-3
H2/MgO
2 3 Vi,
0 0.186 1 2 3
Vo2 0 0.890 X 1 2 3 VI, 0 0.126 X 1 2
3
-0.146 0.155
lo-'
-0.771 X IO-' 0.823 X -0.108 X 0.115 X
0.112 -0.694 x IO-' -0.104 0.661 X lo-' 0.751 x IO-' -0.482 x 10-1 0.311 X IO-' 0.181 x lo-' -0.113 x lo-' -0.170 X IO-' 0.108 X IO-' 0.124 x lo-' -0.798 x 0.516 X 0.872 X -0.554 X -0.829 X 0.534 X 10" 0.612 X -0.398 X 0.259 X 0.195 x 10-3 -0.124 x 10-3 -0.185 x 10-3 0.120 x 10-3 0.137 X 10" -0.895 X IO-' 0.584 X IO4 0.531 -0.639 0.552
-0.171 0.229 -0.216 0.923 0.106 -0.361 -0.126 0.469 0.110 -0.441 0.190 0.590 X IO-' -0.213 -0.690 X IO-, 0.269 0.612 X IO-, -0.252 0.110 0.817 X -0.292 -0.956 X 0.370 0.846 x 10-3 -0.347 0.151
x lo-' x lo-' X X
lo-' lo-'
x 10-1 X IO-' X
X
lo-,
x 10-2 X X
x 10-3 X
There have been several experiments geared to probe the He-MgO interaction. Rieder?l Cevasco and Cantini,lg and Brusdelins et a1.20studied the diffraction features of He scattered from MgO. From the analysis of their data obtained by using a hard-wall model, they found a surface corrugation, determined as the peak-to-trough height at 10-meV energy along the direction of maximum corrugation, of 0.36,0.47, and 0.46 A, respectively. The well depth was estimated to be around 8 meV for both ref 21 and 19. Brusdeylans et a1.20measured the following bound-state energies: 10.2, 6.0, 2.6, and 1.2 meV. Very recently, Frankl's group22performed very careful measurements of bound-state energies and diffraction peaks. Their
Table 111. Properties of the He Potential with Various Adsorption Systems" Ar/MgO Kr/MgO Xe/MgO 5.87 3.08 3.62 1.11 0.23
-
6.41 3.21 4.16 1.45 0.36
-
6.35 3.60 4.26 1.62 0.46
-
4.04 (3.51) 0.396 3.39 0.34 3.2 0.90 2.3 0.8 0.056
5.10 (4.20) 0.325 3.56 0.375 3.3 0.90 2.4 0.8 0.056
6.12 (4.56) 0.232 3.88 0.340 3.5 0.90 2.6 0.8 0.056
1.15
1.15
1.15
'Units of all energies and lesgths are meV and A, respectively. D, t,, and E, are the well depth, minimum position, and eigenvalues of the laterally averaged potentials. 6, b, ( z ) , (nlV&), and 5 are coverage, length parameter, mean distance, matrix element, and corrugation of the potential. The coverage is defined in units of N,, the number density for a commensurate overlayer with a 4 2 x 2 ) well. Other symbols are defined in ref 29. Values in parentheses are without three-body interactions. bound-state energies are 5.52, 2.57, 1.16,0.54, 0.26 meV; a preliminary analysis of the diffraction features gives 0.34 %, for the corrugation of the surface. Vilches' deduced a ground-state energy of 4.8 meV from analysis of calorimetry data of He/MgO. Our potential, without fitting parameters (see Table I), supports the measurements that give a shallower well depth.19121-23With a correction of 7% in C6-, we can have a good fit to the experimental bound states. We also get a surface corrugation of 0.42 8;this should be considered to be in good agreement with the determinations using a hard-wall model. A better comparison can be made as detailed diffraction data become available. We plan to do close coupling calculations using our interaction potential to further test its ability to describe the He-MgO interaction. In Table I1 we report the matrix elements of the Fourier components of the potential ('$'rn(Z)IVGl'$'n(Z)) where VGis the Fourier component of the atom-surface potential and '$'mand '$',, are the wave functions satisfying
Interaction of Rare Gases and H2 the following Schrodinger equation:
Nonvanishing matrix elements will cause a splitting into energy bands for a single atom confined on a surface, analogous with what happens for an electron in a threedimensional crystal. These matrix elements can be actually measured by ABS as avoided crossing of bound-state resonances." Preliminary data for He-MgOM suggest that these matrix elements are smaller than for the He-graphite case. However, our calculation shows that the matrix elements for He/MgO are comparable in magnitude to the ones for Hefgraphite. Moreover, the corrugation of the potential is higher for He-MgO (0.34-0.46 A according to different sets of data) than for He-graphite (0.21 A). A higher corrugation implies larger Fourier coefficients and therefore larger matrix elements. At present we do not know how to resolve this contradiction. Rieder21has related the corrugation of the surface, as obtained by his ABS diffraction data, to the difference in ionic radii of the cation and anion. He observed that this difference, AR, is quantitatively very similar to the surface corrugation, obtained by ABS,for LiF and NaC1, but much different for NiO and MgO. He proposed that one should take in-crystal ionic radii by choosing the Fumi-Tosi3' crystal radii; he finds that AR becomes quite similar to the surface corrugation for MgO. For LiF and NaC1, AR is now much smaller than the ABS determination. In the case of MgO, there is some c o n t r ~ v e r s yon ~~ whether MgO can be described as a fully ionized solid, i.e., Mg2+and 02-,or rather as a partially ionized solid28resembling a Mg+ 0- configuration. We used charge densities as calculated in ref 28. The reason for this choice is twofold. First, the charge densities reported in ref 28 could be easily used in our model. Second, the lesser ionicity of the Vidal-Valat et al. model might be more suited for describing the surface layer where the crystal field is weaker, and therefore a relaxation from the fully doubly ionized oxygen atoms might be more likely.32 The charge densities that we used for MgOZ8tend to support Rieder's conjecture. Reference 28 shows that, if we take as the boundary of the ion the minimum in the electron charge density between two nearest-neighbour sites, MgO is best described by Mg+ and 0-. AR in this case is about 0.3 A. We should add that the small differences in the surface corrugation determined by various groups could be due to different treatments of MgO samples. The fact that our potential can fit bound-state energies and surface corrugations at the same time should be considered encouraging, especially if one keeps in mind that traditionally potentials modeling the atom-surface interaction have achieved one goal or the other, but not both. On the other hand, with no fitting parameter we are not able to obtain the experimental values for the energy levels. One can speculate on why this is so. One reason is that our charge densities are still inadequate to represent the surface of MgO. Relaxation, rumpling, or modified surface composition can all play a role, although we think that such (31) Fumi-Tosi radii are reported in: Shannon, R. D. Acta Crystallogr. 1976, A32, 751. (32) Causa, M.;Dovesi, R.; Pisani, C.; Roetti, C. Acta Crystallogr. 1986, B42, 247.
Langmuir, Vol. 5, No. 3, 1989 615 effects should be minor and mostly affect the repulsive part of the interaction and to a lesser extent the bound-state energy values. It would be nice if our theoretical understanding of the interaction were so developed that we could use ABS data to show these effects. We are not there yet since other factors are probably masking these subtler effects. First, c6 values are still somewhat uncertain. Second, we calculate the quadrupole term in a local approximation. Third, EMT, which establishes a proportionality between the electron surface charge density and the repulsive part of the potential (at least for rare-gas atoms), might necessitate further corrections for taking into account stronger gradients in the charge density than for the systems it was initially applied to, such as metals in the jellium approximation. For comparison purposes, we also report the results of our calculations for He/LiF and NaC1. The same comments written above for He/MgO apply also here. We should add that previous attempts26 to use EMT for He/LiF have yielded a much too large corrugation; in that case, free ion charge densities were used. For H2/LiF and NaC1, we do not have reliable data to make any meaningful comparison. For H2/Mg0, for which the isosteric heat has become available recentlyF3 the correction factor q (see Table I) is larger than for He/surfaces of insulators. This is probably due to the simplifications used here to treat the H2/Mg0 interaction. Finally, in Table I11 we report the results of our calculation for the interaction of He with rare-gas-plated MgO(001). We note that the well depths are similar to the one for He-MgO, while the corrugation of the potential is greatly increased.
Summary We have presented a new model to describe the interaction of rare-gas atoms with surfaces of insulators. Our approach combines EMT and dispersion-type interaction. We showed that when c6- is allowed to relax within reasonable bounds our potential describes well the corrugation of the surface and the bound-state energy values determined experimentally. We also considered the direction in which more work should be done in order for theoretical calculations to reach the level of accuracy needed to interpret experimental data. Eventually, we will be able to predict all these surface structure effects (rumpling, relaxation, ek.) that atom beam scattering has the potential to measure." Acknowledgment. We thank one of our referees for calling our attention to the paper by Fowles and Tole%on surface charge densities for MgO and LiF. We expect that the qualitative aspects of our model for the physisorption interaction will suffer little damage. Detailed calculations are in progress. We thank Professors Frank1 and Vilches for sharing their data prior publication. We also thank the many questioners for their comments when part of this work was presented at the 62nd Colloid and Surface Science Symposium. This work was supported in part by an Alfred P. Sloan Fellowship (to G.V.) and by the donors of the Petroleum Research Fund, administered by the American Chemical Society. Registry No. H2, 1333-74-0;MgO, 1309-484;LiF,7789-24-4; NaC1, 7647-14-5; He, 7440-59-7. (33) Vilches, 0.E.,privata communication. (34) Toennies, J. P. J. Vac. Sci. Technol. 1988, A5, 440. (35) Fowles, P. W.; Tole, P. Surf. Sei. 1988, 197, 457.
