Langmuir 1993,9, 2569-2575
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Computer Simulation of Physical Adsorption on Stepped Surfaces Mary J. Bojan and W.A. Steele’ Department of Chemistry, Pennsylvania State University, University Park, Pennsylvania 16802 Received October 14, 1992. In Final Form: February 2, 1 9 9 9
A stepped surface has been modeled by taking stacks of graphite basal planes truncated to form an effectively infinite set of steps of constant width and height. The gas-solid interaction potential used exhibits a deep trough at the bottom of each step and a large barrier to free translation at the top. Four widths were studied, ranging from 25 to 8 A (step heights were held fixed at 3.4 A). The adsorption of krypton on these surfaces was simulated at temperatures of 110,130,and 150 K and coverages ranging ~ 2 nominal monolayers. It was found that the presence of the steps produces strong ordering from 1 / to in these layers. In addition to structural information,as measured by the local densities of these systems, the coverage dependence of the energy of adsorption is also reported. The pronounced heterogeneity in the gas-solid interaction does not give the expected decay in the adsorption energy curves. Reasons for this behavior (and other results) are suggested. In addition to the thermodynamic data, the molecular dynamics data have been used to extract diffusive fluxes of molecules jumping from one step to another. The coverage and temperature dependence of these fluxes are presented and discussed. Introduction Although the great majority of computer simulation studies of physical adsorption have been until recently concerned with surfaces that are either idealized homogeneous solids or exposed single crystal faces: there is a growing effort to use this technique to study the effects of heterogeneity upon the thermodynamic, structural, and dynamical properties of adsorbed phases. Conventionally, one speaks of chemical and geometrical heterogeneity. The former case, as expected, is pictured as a solid surface containing chemically distinct heteroatoms, often impurities such as surface oxidese2Geometrical heterogeneity, which is the area of interest in this paper, is caused by features such as cracks, pits, and steps or it can be associated with the presence of regions of imperfectly crystalline surfaces (often, amorphous). These types of adsorbent can in principle be modeled in a computer simulation and, indeed, a growingnumber of papers dealing with aspects of this problem have already appeared.3~~ Stepped surfaces produced by cutting a metal crystal a t a slight angle to one of its close-packed planes have been extensively investigated by workers in chemisorption and catalysis because of the reactivity associated with these steps.6 However, these surfaces should have several interesting features in physisorption as well. In the first place, if one imagines the terraces in such systems to be narrow homogenous strips, the effect of finite size in one dimension upon adsorbate properties can be investigated. Secondly, a reasonable model of the gas-solid physical interaction for such systems will produce a very deep potential well along a line at the foot of each step and, in Ahtractpublished in Advance ACSAbetracts, August 16,1993. (1) Nicholson, D.; Parsonage, N. G. Computer Simulation and the Statistical Mechanics of Adsorption; Academic Press: London, 1982. (2) Soto, J. L.;Myers, A. L. Mol. Phys. 1981,42,971. MacElroy, J. M. D.; Suh,5.-H. Mol. Simul. 1989,2, 313. Yashonath, 5.;Demontie, P.; Klein, M. L.Chem. Phys. Lett. 1988,163,551. Yashonath, S.; Thomas, J. M.; Nowak, A. K.; Chwtham, A. K. Nature, 1988,331,601. Leherte, L.;Lie, G. C.; Swamy, K. N.; Clementi, E.; Derouane, E. G.; Andre, J. M. Chem. Phys. Lett. 1988,146,237. Cohen De Lara, E.; Kahn, R.; Goulay, A.M. J. Chem.Phys. 1989,90,7482. Demontie,P.;Yashonath,S.;Klein, M. L.J. Phye. Chem. 1989,93,6016. Woods, G. B.; Rowlineon, J. 5.J. Chem. Soc., Faraday Tram. 2,1989,86,765. Van Tassel, P. R.; Davis, H.T.; McCormick,A. V. Mol. Phys. 1991,73,1107. Karaviae,.F.;Myers, A.L.Mol. Simul. 1991,8,23. Karavias,F.;Myers,A. L.Mol. S u ” . 1991, 8,51. 0
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Figure 1. Side view of the trajectories followed by the krypton atoms over a period of 71 ps on a stepped graphite surface at monolayer coverage. The atomic planes of the substrate are represented by the thin lines. Table I. Parameters of the Interaction Energy Functionm elk (K) (A) Q
Kr-Kr KI-C
170 75
3.60
3.42
addition, a barrier to translation at the top of each step (see below). Thus, these surfaces can be viewed as examples of well-characterized heterogeneity. This paper will be concerned with a molecular dynamics simulation of krypton adsorbed on stepped graphite-like surfaces. That is, the adsorbent was modeled as a semi(3) Riccardo,J. L.;Chade, M. A.;Pereyra,V. D.;Zgrablich,G. Langmuir 1992,8,1518. Bojan, M. J.; Vernov, A. V.; S h l e , W. A. Langmuir 1992, 8,901. Bakaev, V. A.; S h l e , W. A. Langmuir 1992,41379. Bakaev, V. A,; Steele, W. A. Langmuir 1992,8,148. Bakaev, V. A. Surf. Sci. Lett. 1992,264, L218. Diestler, D. J.; Schoen, M.; Heher, A. W.; Cushman, J. H. J. Chem. Phys. 1991,96,5432. Bowler, A.M.; Hood, E. S. J. Chem. Phye. 1992,97,1260. Bowler, A. M.; Hood,E. S. J. Chem. Phys. 1992, 97,1257. Patrykiejew, A. Thin Solid Film 1992,208,189. C o w , J.; Araya, P. J. Chem. Phys. 1991,96,7741. Demi,T. J. Chem. Phys. 1991, 96,9242. Demi, T.; Nicholson, D. Langmuir 1991, 7, 2342. Demi, T.; Nicholson, D. Mol. Simul. 1991, 7,121. Demi, T.; Nicholson, D. Mol. Simul. 1991,7,363. Brodka, A.; Zerda, T. W. J. Chem. Phys. 1991,96, 3710. Albano, E. V.; Binder, K.; Heermann, D. W.; Paul, W. Surf. Sci. 1989,223,151. Albano, E. V.; Binder, IC;Hwrmann, D. W.; Paul, W. Z . Phys. B 1989,77,446. Bakaev, V. A. Surf. Sci. 1988,198,571. Mak, C. H.;Andemen, H. C.;George, S. M. J. Chem. Phys. 1988,88,4052. Albano, E.V.; Martin, H.0. Phys. Rev. B 1987,36,7820. Bojan, M. J.; Steele, W. A. Ber. Bunsen-Ges. Phys. Chem. 1990,94,300. Steele, W. A; Bojan, M. J. Pure Appl. Chem. 1989,61,1927. Bojan, M. J.; S k l e , W. A. To be published in Proceedings of the Fourth International Conference on Adsorption, Kyoto (1992). Sokolmki, S. Mol. Phys., to be published. (4) Bojan, M. J.; Steele, W. A. Longmuir 1989,5,625. Bojan, M. J.; Steele, W. A. Surf.Sci. 1988,199, L395. (5) Wagner, H.In Physical and Chemical Properties of Stepped Surfaces; Springer Tracts in Modern Physics; H6hler, G., Ed.; Springer-Verlag: Berlin,1979; Vol. 85. Wandelt, K.;Hulse,J.; Krtippers, J. Surf. Sci. 1981,104,212. Davies, P. W.; Quinlan,M. A.; Somorjai, G. A. Surf. Sci. 1982,121,290. Miranda, R.; Daiser, S.; Wandelt, K.; Ertl, G. Surf, Sci. 1983,131,61. Fink, H. W.; Ehrlich, G. Surf.Sci. 1984,143, 125. Q 1993 American
Chemical Society
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Figure 2. Adsorption energies for krypton on a stepped graphite surface. The variation in the minimum gas-solid energy as an atom moves down the steps is shown for surfaces 102 A wide divided into 4,8, and 12 steps. The vertical axes show energy/R in dag K. T=l10K 4 Stem
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Figure 3. Local densities of krypton adsorbed at '/z monolayer coverage (144atoms) on the surfaces of Figure 1. Results for two temperatures and three surfaces are shown.
infiiite stack of featureless planes made up of carbon atoms with an intraplanar density and gas-solid interactions appropriate for Kr on the basal plane of graphite. However, the planes in this stack are truncated in such a way as to form an infinite series of steps. The usual periodic boundary conditions and minimum image convention are reformulated to allow one to simulate finite size samples of krypton adsorbed on these surfaces. The studies were carried out a t three temperatures, 110,130, and 150 K.,and nominal surface coverages were chosen that ranged from 1/2 to 2 monolayers. In previous work, simulations of krypton adsorbed on a similar solid made up of carbon planes but exhibiting a surface of parallel grooves were carried Since the groove walls were modeled in the same way as the steps
studied here, the potential energy variations near the groove wall were essentially the same as those found here for the region near a step. The differences in the two investigations were that terrace widths ranging from 25 A down to 8 A were studied here whereas groove widths were taken that ranged from 17 to 51 A with a single terrace width of 17A; also, only a single temperature of 110K was considered in the earlier work. The main qualitative finding that the presence of the groove produces very strong one-dimensional ordering in the direction perpendicular to the groove walls is also observed on the terraces studied here in spite of the fact that the deep potential minimum found on each side of a groove is present only on one side of a terrace on a stepped surface.
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Figure 4. Same as Figure 3, but the densities for the first layer atoms in systems with monolayer coverage are shown (288 atoms).
Simulations The molecular dynamics were performed using an isokinetic algorithm6 together with periodic boundary conditions and the minimum image convention. However, the implementation of these boundaries required some thought, since one naturally views this surface as a set of descending (or ascending) steps in the x-direction. In this case, the images on the right and left must be shifted vertically to match the steps. A better way to treat the problem is to rotate the coordinate axes by 8 where tan 8 = step heightjterrace width, Visually, one then has a sawtoothed surface where each tooth is identically positioned relative to the new x-axis. Conventionalimage conditions now apply. It was actually found to be most convenient to work with the original staircase picture, using the rotated frame only in those statements in the code involving the periodic boundary and/or minimum image calculations. The potential energy functions used were conventional (and identical to those in the previous paper). Kr-Kr interactions were taken to be Lennard-Jones 12-6,with well-depth and size parameters elk and u taken from gas phase studies.' Values of the parameters used are listed in Table I. The krypton-carbon interaction was basically an infinite sum of pairwise Kr-site 12-6 energies with parameters that are also given in Table 1.8 However, the summation within a carbon plane was replaced by an integral over a truncated plane to give an interaction that is invariant to translation parallelto the plane except when the krypton atom is near the edge of the plane. The (6) Hoover, W.G. Phys. Rev. A Gen. Phys. 1986, 31, 1696. Evans, D.3.;Morriss, G. P.Chem. Phya. 1983, 77,63.
(7) Hirechfelder, J. 0.; Curties, C. F.;Bud, R. B. Molecular Theory of Gases and Liquids; Wiley: New York, 1964, pp 22 and 23. (8) Steele, W.A. J. Phys. Chem. 1978, 82, 817.
integration for an inverse power law interaction can be done analytically-see ref 4 for the explicit form of the finalexpression. Of c o w , the totalinteraction is obtained by summing up krypton-plane energies. Because the energies decay steeply with increasing distance and because the planes are separated by a fixed distance equal to 3.4 A, only a few terms in the sum need be explicitly evaluated. Since the krypton atomic coordinates were evaluated as a function of time during the simulation, one convenient way to gain a visual picture of the system is to have the atomic trajectories plotted by the computer. For example, Figure 1shows a side view of trajectories of 288 atoms on a surface consisting of four steps with a total width (in the plane of the figure) of 102.3 A and with length (perpendicular to the plane of the figure) of 44.3 A. The resulting density of 0.0636 atom/A2corresponds closely to that for a monolayer on the flat surface, but the trajectories of Figure 1 show that at least a few atoms are in the second layer. These trajectories were generated by first equilibrating the system for 57 ps and then recording the computed atomic positions for a period of 71 ps, all at 110 K. One could tell more about the degree of ordering produced on this surface by a top view of the trajectories. However, a better way of presenting this information is to evaluate the local layer density as a function of position relative to the steps. (Layer densities are evaluated by discarding all atomic coordinates except those lying in the desired range of distance way from the surface and then summing those that are in bins defined by strips of width 0.25 A lying parallel to the step risers.) In addition to presenting these local densities in the following section, average potential energies of the adsorbed krypton atoms have been computed as the s u m of the component gas-gas
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2572 Langmuir, Vol. 9,No. 10, 1993 T = l 1OK 8 Steps
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Figure 5. Same as Figure 4, but first layer densities for a nominal coverage of two layers (576 atoms). and gas-solid terms; the coverage dependence of the the two extreme temperatures studied (110 and 150 K). Figure 3 shows the local densities for - l / 2 monolayer component parts and the total will be presented. coverage. For this system, a sharp peak is observed for In a separate calculation, the average energy of interthose atoms adsorbed in the deep potential energy troughs action of a single, isolated Kr atom with this surface was shown in Figure 2. In addition, the densities become very evaluated together with the Henry’s law constant for the small in the regionof the potential barriers. At 110K, the limiting low coverage isotherms. Attempts to evaluate adsorption isotherms for these systems using the Widom densities on the terraces away from the regions of varying particle insertion technique have not been successful to adsorption potential show several peaks for the widest date, presumably because the layers are too highly ordered terraces (25.5 A), two peaks for M = 8 (terrace width = and too tightly packed to allow for “successful” particle 12.8 A). For M = 12,where there is no region of flat surface insertions-this difficulty was first observed in attempts as far as the adsorption energy goes, there is one clearly to evaluate chemical potentials for bulk solid^.^ visible peak in the density. In fact, the numbers of molecules on each strip need not be the same and, as it Results turns out, the fluctuations from strip to strip are large enough to produce a second peak on only some of the First, the variation in the minimum gas-solid potential steps. Note that the terrace width is only 8.5 A for M = energy for a Kr atom moving down the steps is shown in 12; since the krypton atom diameter is 3.8 A (aeB)or 4.3 Figure 2 for three choices of the step width. A value of A (r-, the distance of minimum Kr-Kr energy), one UIR = -1450 K is characteristic of Kr on an infinite encounters packing problems on these narrow terraces. A perfectly flat approximation to the graphite basal plane.4 row of atoms adsorbs in the trough, which leaves only a It is evident that this energy is achieved only over a limited 4.7 A width for more rows. Furthermore, atoms in these range of distance for the terraces of width 25.5 A. As the rows will necessarily encounter a strong force pushing them terraces are made narrower, the very large variations in away from the next step and back toward the row of atoms energy associated with the two edges extend so far toward initially adsorbed. Consequently, there is no clear-cut the middle that the “flat” regions disappear. A deep pattern of ordered adsorption seen in this region at either potential well occurs at the bottom of each step where 110 or 150 K. Raising the temperature from 110 to 150 U/R = -1750 K and a high barrier (relative to the energy K affects the peak heights but not the qualitative patterns. on the “flat” part) occurs at the top of each step where The densities for a monolayer coverage shown in Figure UIR = -750 K. These energies change only slightly with 4 show a considerable increase in order relative to the ‘/z terrace width over the range of widths studied and they layer case. For the 25.5 A terraces, six equally spaced are the determining features for the structure of the peaks plus an empty region were observed. The peaks are adsorbed krypton observed in this work. 3.6 A apart, indicating fairly close-packing. In particular, In Figures 3-6, the local densities of krypton on the the rows of spheres must be interdigitated to some extent surfaces with 4,8,and 12 steps per 102 A are shown for (see below). Much of the order observed at 110 K for (9) Shing, K. S.;Gubbins, K.E.Mol. Phys. 1981, 43, 717. these wide terraces has disappeared at 150K. For terraces
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Figure 6. Same as Figure 5, but densities for atoms in the second layer are plotted.
of width 12.8A, the three peaks per terrace are separated by 3.6 A, but the vacant region (defined as the region between the last peak on each terrace and the first peak on the adjacent terrance to ita right) is now 5.6 A wide compared to 7.5 A for the 25.5 A terraces. Finally, two peaks 3.6 A apart are present when the terrace width is 8.5 A, leaving unoccupied regions of width 4.9 A. The conclusion is that any misfita of the rows that form on these terraces are taken up by a (nearly) vacant region of variable width that is present at the terrace edge. At 150 K, the same numbers of peaks and peak spacings are present as at 110 K, but the peak heights have obviously decreased. For coverage of two layers, the local densities for atoms in the first (Figure 5 ) and second layers (Figure 6) are shown separately. The addition of a second layer causes a significant increase in first layer ordering, as indicated by the increase in peak height and decrease in peak width relative to that for the monolayer coverage. However, the peak spacings are essentially unaffected by the coverage change. It is interesting to note that this first layer order propagates into the second layer, as shown in Figure 6. The two layers exhibit the same peak spacing but the second layer peaks are slightly off-set from those for the first layers. The position shift amounts to 1.8A, which of course means that each second layer row of atoms is equidistant between a pair of first layer rows. Clearly the degree of order in the second layers, which is measured by the peak heights, is less than that in the first layer at 110 K and much less at 150 K. A look at the trajectory plots for these system confirms the findings deduced from the local densities. For example, Figure 7 shows a top and a side view of the trajectories at 110 K and a coverage of two layers. The side view clearly shows the presence of the vibrating atoms in three
first layer rows per terrace, with another row nearby that is here classified as second layer; this plus further second layer rows lie above and essentially half way between the first layer rows. The top view shows that arough hexagonal pattern is formed by the first layer atoms on each terrace, meaning that successive rows are shifted slightly along the row so that closer packing can occur. In fact, the ordering in this structure propagates from one terrace to another. (For example,one can see diagonalrows of atoms inclined at a 60' angle to the terrace edges that extend for more than six terraces without deviating significantly from linearity.) Figure 8shows the simulatedvalues of the average molar energy of the krypton atoms with each other (gas-gas), with the surface (gas-solid), and the total. This total energy E is related to the isosteric heat of adsorption qat by
Curves of E versus coverage are shown only for the two extreme values of the terrace width at a single temperature. (These energies vary slightly with temperature over the range of this study.) It is evident that the Kr-Kr interactions can be a significant fraction of the total and actually produce a total energy that does not have the characteristic signature of a heterogeneous surface. In fact, the (negative) gas-solid part behaves as expected, decreasingwith increasing coverage as,first, the atoms fill the troughs in the potential function and, later at multilayer coverages, adsorb a t positions not on the surface. The (negative) gas-gas energies are insensitive to the terrace width and increase almost linearly with coverage up to and into the multilayer adsorption regime. (Plateaus
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terraces, there is also a s m d contribution due to adsorption on the flat regions of the terraces where the adsorption energy is only -2.8 kcal/mol. As one might guess from the height of the energy barrier to translation from one step to another, diffusion across the steps is highly hindered, at least a t low coverages. One can readily extract information concerning this jump rate from the molecular dynamics data. Positions of the molecules were checked every 0.07 ps (10 timesteps) to determine whether any had moved from one step to another. These data were averaged over all steps, divided by the total length of step (number of steps times 45 A) and by the observation time (71 ps). Values of the flux obtained in this way are plotted in Figure 9 for all three temperatures considered. At coverages greater than monolayer, nearly all the flux is due to molecules in the second and higher layers which do not encounter the energy barriers shown in Figure 1. The jump rates are low and coverage independent a t 110 K in this range of coverage-evidently, the multilayer films are essentially solid under these conditions. The change in jump rates that occurs somewhere between 110 and 130 K is most likely due to a solid liquid transition. At coverages of less than a monolayer, the diffusive fluxes become quite small. (Indeed, it is questionable whether one has true equilibrium at the lowest coverage studied. However, simulations carried out using single strips of varying width rather than multiple steps show no significant differences between the two cases, at least as far as the average energies are concerned.) It is not surprising to see that the fluxes increase with coverage in the submonolayer regime-in effect, there are more molecules available for jumping. No plots of the flux per molecule are shown, on the grounds that it is the molecular density near the steps that controls the flux, whereas the overall number a t any instant in time includes many that are too far from the step to take part in the jump process. Figure 9 shows that the flux does not depend very much on the number of steps or, in other words, on the distance between steps. This also supports the idea that it is the molecular density in the immediate vicinity of the step that primarily controls the jump flux, especiallyat coverages of less than a monolayer.
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Figure 7. Side and top views of the trajectory plots for 576 atoms adsorbed on eight steps at 110 K. The planes of the substrate are represented by thin lines.
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Figure 8. Average energies at 130 K are plotted for the 4- and la-step surfaces. The component Kr-Kr and Kr-solid energies are shown together with the total.
in these curves would be expected if the atoms were undergoing condensation into ordered phases.) The only significant difference in the curves for the two terrace widths appears at low coverage, where the (negative) interaction of the krypton with the surface is -0.16 kcaUmo1larger for the narrowest terrace than for the widest in the low coverage limit. Here, the average energy is that of a single isolated atom with the solid. This average is dominated by the adsorption energy in the troughs, which is equal to -3.5 kcal/mol plus the energy of (anharmonic) vibration perpendicular to the trough walls. For the wide
Discussion Most of the experimental data available for adsorption on stepped surface is for chemisorption on metals.6 However, it has been shown in diffraction studies from nitrogenloand from rare gas" overlayerson stepped metals that overlayer structures form that extend over several terraces. Even though the atomic nature of the steps has been replaced here by a potential which is smooth and featureless in the direction parallel to the steps, the ordering found a t high coverages does reproduce the experiment. However, this long-range ordering was not observed at monolayer coverage but a shift to even lower temperature might produce it. The two-dimensional melting point of krypton on this surface is 100 K.12 Of course, an important difference between physisorbed and chemisorbed systems is that it is believed that the changes in the adsorbate-adsorbate interactions due to adsorption can be much larger for chemisorption than for physisorp-
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(10)Kirby, R. E.; McKee, C. S.;Roberta, M. W. Surf. Sci. 1976,55, 725. Perdereau, J.; Rhead,G. E. Surf. Sci. 1971,24, 555. (11) Roberts, R. H.; Pritchard, J. Surf. Sei. 1976,54,687. Papp, H.; Pritchard, J. Surf. Sci. 19711, 53, 371. Cheaters, M. A.; Hussain, M.; Pritchard, J. Surf. Sci. 1973, 35, 161. (12) Shriipton,N.D.;Cole,M.W.;Steele,W.A.InSurfaceProperties of Layered Matenah; Benedek, G., Ed.; Kluwer Academic Publishers: Dordrecht, T h e Netherlands, 1991.
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Physical Adsorption on Stepped Surfaces 8 Steps
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Figure 9. Diffusive jump fluxes for different terrace widths. (Straight lines connect the actual data points.) The three curves in each panel show the temperature dependence of these fluxes.
tion. (This is the explanation put forth for systems where step structures appear to destroy the long range order present in layers adsorbed on the flat surface.) In fact, the average Kr-Kr interaction energy plotted in Figure 8 is not significantly different from the data obtained for a flat graphite surface,13 at least at coverages and temperatures where condensation does not occur. The behavior of the energies of adsorption for these systems is mildly surprising, since the very strong changes in the gas-solid interactions as an atom moves down the steps do not produce the anticipated decay in the energy (or heat) of adsorption with increasing coverage. In part, this is due to the compensation of the decay in the gassolid term by the growing gas-gas term. We speculate (13)Bethanabotla, V.; Steele, W.A. J. Phye. Chem. 1988,92, 3285.
that another contributing factor may be ascribed to the fact that the interaction energy changes are too large (rather than too small)-if atoms do not adsorb in the regions of weak gas-solid interaction, there will be no contribution to the average energy from this region of the potential. More study of this point is needed-for example, histogramsof the distribution of gas-solid energy for the molecules adsorbed a t a given coverage would be helpful in determining which specific interactions are actually contributing to the overall value of the average energy.
Acknowledgment. Support for this research from the Division of Materials Research of the NSF is gratefully acknowledged (Grant DMR 9022681).
