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The Supersite Approach to Adsorption on Heterogeneous Surfaces† William Steele Department of Chemistry, Penn State University, University Park, Pennsylvania 16802 Received October 22, 1998. In Final Form: March 15, 1999 The physical adsorption of a simple gas on a heterogeneous surface is reformulated in terms of a description of the adsorbent surface as a collection of supersites of finite sizes. Formal expressions for the thermodynamic properties of an adsorbate on such a surface are given. Computer simulations of the isotherms and local densities of xenon adsorbed on stepped surfaces with several step separations are presented and analyzed in terms of the supersite formulation. It is argued that these surfaces can be divided into two types of supersites: the terraces and the narrow regions of strong adsorption at the bottom of the steps. Local isotherms are evaluated for adsorption on both types of supersites, and the changes in the isotherms with changes in the surface dimensions are discussed. Finally, the intersite interaction between atoms on the “terrace” and the “step” supersite is approximately calculated, and the effect of this interaction on the local isotherms is presented. Future prospects for this novel approach to adsorption on a heterogeneous surface are briefly discussed.
1. Introduction and Basic Theory The development of a quantitative theory to describe physical adsorption on a heterogeneous surface still remains an incompletely solved problem even after many decades of intensive study.1,2 The most successful formulation for the calculation of the adsorption isotherm is based upon the following general expression:
Θtot(p,T) )
∫θloc(,p,T)f() d
(1.1)
Here, Θtot (p,T) is the experimental or simulated total coverage of the adsorbate at pressure and temperature p,T for a gas on a surface that is characterized by an adsorption energy distribution, f(). The coverage due to adsorption on the part of the surface with energy between and + d is given by θloc(,p,T). A possibly significant approximation is made in writing this general expression because lateral interactions are not handled well when they involve pairs of adsorbed molecules with different energies, ,′. Formally, to deal with this problem, one needs to extend the theory to include this interaction by evaluation of the average of exp[-u(r)/kT], where u(r) is the interaction energy of a pair of adsorbed molecules separated by a distance r on the surface. To do this, one must use the pair distribution of adsorption energies ,′ on pairs of sites that are separated by a distance r together with some estimate of the local coverages on these sites. The integral in eq 1.1 that relates the unknown f() to the known isotherm Θtot(p,T) becomes much more complicated if one takes the lateral interaction into account in this way and one almost always finds that lateral interactions † Presented at the Third International Symposium on Effects of Surface Heterogeneity in Adsorption and Catalysis on Solids, held in Poland, August 9-16, 1998.
(1) Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Surfaces; Elsevier: Amsterdam, 1988. Rudzinski, W.; Everett, D. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: London, 1992. (2) Recent work dealing with adsorption on heterogeneous surfaces can be found in the proceedings of a series meetings on The Effects of Surface Heterogeneity in Adsorption and Catalysis on Solids. The first of these is in Langmuir 1993, 9 (10); the second is in Langmuir 1997, 13 (5); the present paper is part of the proceedings published in Langmuir 1999, 16.
are dealt with by introducing a mean field approximation, if at all. Note also that this description is based on a sitewise description of physical adsorption, but computer simulations are now beginning to show that this is often not an adequate description of the adsorption process.3 For the moment, we will ignore this aspect of the problem in order to focus on some of the difficulties associated with f() and with the local isotherm which plays such an important role in the determination of f() from a known Θtot(p,T). Ordinarily, a simple and not necessarily realistic form is chosen for θloc(,p,T). Commonly used examples are the Langmuir or BET equation for submonolayer and multilayer adsorption, respectively. In both cases these equations describe adsorption on isolated sites of energy whose properties are independent of those of the neighboring sites. At the other extreme, one assumes that the heterogeneous surface is made up of a collection of patches of size sufficiently large that boundary effects on the local isotherms can be neglected.1,2 (In addition, it is often assumed that the adsorption on a patch is completely mobile which means that the energy is constant at all points within the patch. In the absence of edge effects, one can then use a local isotherm such as the Hill-deBoer for θloc(,p,T).) The problem is that the use of an oversimplified local isotherm, whether it be sitewise or mobile, is likely to lead to physically unrealistic f(). Thus, it would seem that an improved θloc(,p,T)4 would possibly give better (i.e., more believable) results for the adsorption energy distributions. In an attempt to do this, we will begin by suggesting a new and more flexible definition of an adsorption site for a heterogeneous surface5 which we will rather arbitrarily call “supersites” to distinguish them from the previous descriptions. Since the effects of heterogeneity are most important in submonolayer films, we will focus on this part of the adsorption for spherical atoms on a rigid substrate. Useful criteria for the definition of a supersite might be: (1) the variation in the adsorption energy (i.e., in the minimum gas-solid energy) within a supersite should be small; (2) the area of a supersite should (3) Bakaev, V. A.; Steele, W. A. Langmuir 1992, 8, 1372. (4) Riccardo, J.; Steele, W. Surf. Sci. 1996, 356, 257. (5) Bottani, E.; Steele, W. Adsorption, in press.
10.1021/la981483b CCC: $18.00 © 1999 American Chemical Society Published on Web 05/06/1999
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be as large as possible and the boundaries of such a supersite should be as simple as possible, consistent with requirement (1). One next needs local isotherms for the adsorption on each supersite. These would be for a mostly mobile layer, because of the requirement that the adsorption energy not vary greatly within the supersite, but it would also require that one deal with adsorption on a finite element of area rather than the current treatments of mobile adsorption which are based on the assumption of an infinitely large area even when used as local isotherms for a heterogeneous surface. It would also be desirable to treat the lateral interactions between atoms on neighboring supersites in some simple but approximate manner. Since the thermodynamics of particles on a supersite will now depend on the area and boundary conditions at the site edges, the distribution function f() should be generalized to include this information and thus should be written as F(,A), where the boldface A indicates the size and shape of a supersite with adsorption energy . One possible advantage of this approach is that the number of such supersites needed to describe the entire adsorbent surface may be small, at least when compared to the number of conventional sites. In this case, integrals over the continuous distribution of site energy would be replaced by sums over supersites. The equation relating the supersite distributions to thermodynamic properties such as the total adsorption Na(p,T), the total potential energy of the adsorbed atoms Ua(p,T), and the total area A (or pore volume Vp) become
Na(p,T) ) Ua(p,T) )
∑J Nloc(J,AJ,p,T)F(J,AJ)
∑J JNloc(J,AJ,p,T) + Ulat(J,p,T) A)
∑J AJ
(1.2) (1.3) (1.4)
where Ulat(J,p,T) is the energy due to the lateral interactions of the particles adsorbed on supersite J. (For sorption in pores, one should replace AJ by VJ, where VJ is the volume of the Jth element in the supersite representation of the interior volume of the pore.) In writing eqs 1.2-1.4, effects of lateral interactions between molecules on neighboring supersites have been neglected. One of the considerations in choosing a supersite geometry is to attempt to minimize such interactions, but their unavoidable presence means that one should attempt to take them into account, at least at the mean field level.6 In particular, if u(rIJ) is the interaction energy of a pair of molecules separated by rIJ but on supersites I,J, one can evaluate an average interaction umf(I,J) from
umf(I,J) )
∫A ∫A u(rIJ)FI(rI)FJ(rJ) drI drJ J
I
(1.5)
where rIJ ) rI - rJ, with rI,rJ equal to the two-dimensional vectors (for a planar surface) that define the positions of atoms adsorbed on supersites I and J, and FI(r),FJ(r) equal to the densities on supersites I and J. (One might suppose that these densities would be independent of r within a supersite, but it is well-known that the presence of boundaries induces strong oscillations in density in their vicinity.) (6) Hill, T. Introduction to Statistical Thermodynamics; AddisonWesley: Reading, MA, 1960; Section 14.4.
This formalism leads to slight alterations in the equations relating the supersite to global adsorption. In particular,
Ulat(p,T) )
1
∑J Uloc(J,p,T) + 2I*J ∑umf(I,J)
(1.6)
A mean-field correction should be made to the pressures in these equations, but we do not specify the actual form of the correction at this point. At this point, we can contrast the supersite formalism with the standard approach to adsorption on a heterogeneous surface which generally takes one of two forms, depending upon the choice of local isotherm: these are known as the patchwise and the sitewise theories. In the patchwise approach, the surface is assumed to be made up of a collection of patches each of which is characterized by a constant value of adsorption energy within the patch. Edge effects which might arise for a patch of finite size are neglected. This formalism is not too different from the supersite idea, but interactions between molecules on different patches are neglected, as is the effect of finite patch size upon the local isotherm. In the sitewise case, the surface is usually divided into sites with areas only large enough to hold a single molecule per layer, and lateral interactions between molecules on neighboring sites are often taken into account in a mean field way. It is wellknown that one can have major problems if the initial selection of sites is then used to describe the adsorption of larger molecules or of mixtures of molecules of different sizes. One can scale the site energies to account for the different adsorption energies of different molecules, but the problem of placing a molecule too large for the original site has not yet been solved. Furthermore, simulations indicate that atoms that adsorb on a specific set of energetic sites may actually move off these sites as the monolayer approaches completion and move to places on the surface that were originally points where the gas-solid adsorption energy was weak, but where this energy has been enhanced by extra interactions of an atom with previously adsorbed atoms in the vicinity of the “weak” site.3 Thus, it is envisioned that a supersite will be large enough to hold at least several molecules, which will greatly reduce the problem of fitting variably sized molecules on the site. There will be optimum choices for the size and shape of such sites, which will be hard to make if nothing is known about the surface. Indeed, the conventional approaches to this problem were developed at a time when this was generally the case, but nowadays a variety of surface probes are available that can yield useful information concerning the best choices for a representation of the solid surface under investigation. These could be low-energy electron diffraction (LEED) patterns if the surface is that of a single crystal or they could be atomic force microscopy (AFM) or scanning tunneling microscopy (STM) pictures of surfaces with defects. The structures of many porous solids such as zeolites are now quite well-known from X-ray diffraction. More speculatively, one might use measurements of the chemical composition of different areas of a surface to help define a useful set of supersites. One of the more interesting aspects of the supersite approach is that it attempts to make a close association between surface structure and adsorption energy distributionssa feature which is not emphasized for the energy distributions obtained via the conventional approaches to this problem.1,2
Adsorption on Heterogeneous Surfaces
The remainder of this paper will be devoted to the discussion of computer simulations of the adsorption thermodynamics of a rare gas on a specific model surface that appears to be well-described by the supersite formalism. The system is xenon on a stepped graphite surface. Here, previous simulations7 will be extended and subjected to a detailed analysis in light of the supersite formalism. 2. Computer Simulations Thermodynamic properties such as the adsorption isotherm and energies were obtained in the usual way by grand canonical Monte Carlo (GCMC) simulations.8,9 As is well-known, the GCMC simulation of the physical adsorption of a simple gas on a rigid model surface is based on the evaluation of Markov chains of three types of change: shifts in the positions of adsorbed atoms, creation of new atoms in the computer box, and destruction of previously created atoms. The rules for the acceptance of any of the changes are constructed such that the consequence of a long sequence of changes will eventually produce the atomic configurations that are characteristic of the adsorbed fluid in a grand canonical ensemble. This means a fluid held at constant T, V, and µads, where µads is the chemical potential of the adsorbed fluid. The volume V is the adsorption space with the adsorbing surface forming one wall of the box, which is extended to infinity by the use of periodic boundary conditions in the two directions parallel to the surface. Interaction energies of an adsorbate atom with the solid are evaluated with the help of the minimum image convention which means that atoms close to a wall of the box interact with the original surface of the box and with the image surfaces that surround it. Adsorbate-adsorbate interactions are calculated similarly. Since µads ) µgas, where gas refers to the bulk phase in equilibrium with the adsorbate, the assumption that this gas phase is ideal allows one to express µads as kT ln(p/p*), where p* is a reference pressure that is easily calculated from elementary statistical thermodynamics.10 The consequence is that such a simulation yields the averages 〈Nads〉 and 〈Uads〉 as a function of p at constant V,T. Of interest in the present context, the average number of adsorbed particles 〈Nads〉 can be partitioned into the numbers adsorbed on the various supersites chosen to represent a given heterogeneous surface, so that local supersite isotherms can be directly simulated. The average energy 〈Uads〉 can be similarly partitioned into the energy of the atoms on each supersite plus the energies involving an atom with the surface and with other adsorbate atoms on neighboring supersites. Molecular simulations rely upon accurate representations of the interatomic potentials for their success in adsorption or bulk-phase studies. In general, potentials for adsorption simulations rely upon the successful interaction laws used in the bulk-phase work. Thus, the pairwise interactions between adsorbate atoms are generally taken to be Lennard-Jones inverse 12-6 power functions with parameters for the well depth gg and size σgg that have been shown to give at least semiquantitative agreement between simulation and experiment for bulk phases. These functions have been used here with Xe-Xe parameters /k ) 221 K, σ ) 0.41 nm. A simple extension of the idea that the gas-gas interactions are pairwise additive will give gas-solid (7) Bojan, M. J.; Steele, W. A. Mol. Phys. 1998, 95, 431; Rabedeau, T. A.; Sullivan, T. S. Phys. Rev. B 1986, 34, 8118. (8) Allen, M. P.; Tildesley, D. J. Computer Simulation of Simple Liquids; Clarendon Press: Oxford, 1987. (9) Nicholson, D.; Parsonage, N. Computer Simulations and the Statistical Mechanics of Adsorption; Academic Press: New York, 1982. (10) Reference 6, Chapter 4.
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potentials as sums of pairwise interactions of the interaction of a gas atom with sites that are most often taken to be the solid atoms.11,12 This model is used in the present study. However, accurate estimates of the parameters of these atom-site potentials are difficult to come by and are often based on comparison between experiment and theory for the limiting low-coverage data where the adsorbed fluid behaves as if it were made up of isolated atoms (the “ideal adsorbed gas”).12 In fact, the values for these parameters used here are identical to those taken in the previous simulations;7 thus, for the Xe-C pair, /k ) 94 K, σ ) 0.378 nm. For the interaction of a xenon atom with the graphite basal plane, it is often adequate to replace the sums over carbon atom sites by twodimensional (2D) integrations over each basal plane and then sum over these planes.12 Thus, one writes ∑2D w F2D∫dA, where A is the area of the planar element and F2D is the density of sites in the plane. The result is that the sum of 12-6 functions over the sites in the plane is replaced by the 10-4 function that is given by the integration. This must still be summed over planes.12 If the solid surface is not infinite and planar but is made up of straight-edged steps and terraces extending to infinity in the x-direction, integration of the site-site potential from y ) +∞ to -∞ will be replaced by an integral from +∞ to ye, where ye is the position of the end of the plane and thus is the position of a step. The result of this integration gives the gasplane interaction us (z,ye) for an atom at a distance z from the plane as7,13
us(z,ye) ) 2πgsF2D(σgs12I6(z,ye) - σgs6I3(z,ye)) (2.1) where
In(z,ye) ) Qn(z,ye) )
2 (n - 1)z
if ye > 0 - Qn(z,ye) 2(n-1)
if ye < 0
(2.2)
Here, ye is negative if the atom is over the truncated plane and is positive if the atom is in the volume next to it. The functions Qn are
Q3(z,ye) ) Q6(z,ye) )
{
{ ( )}
1 z 3q 4z4
2
(2.3)
1 z2 z4 1008 - 1680 + 1080 10 q q 640q z6 z8 315 + 35 (2.4) q q
()
() ( ) ( )}
with q2 ) z2 +ye2 +ye (z2 +ye2)1/2. Qn will be zero and the In will be zero for ye . 0sthe case of an atom not over the plane which obviously gives zero interaction. For ye , 0, they will give an energy equal to the usual result for a single infinite planar surface which is an inverse 10-4 power law interaction. Finally, when the atom is directly over the end of the truncated plane, ye ) 0, q ) z, and the In will equal 1/2 their values for the infinite plane. If one constructs a model adsorbent from a stack of truncated planes that end at various values of ye, the adsorption energy changes that occur when an atom passes over the end of a plane will obviously create a heterogeneous surface, with the heterogeneity confined to one dimension that is denoted here by y. Thus, a stepped (11) Steele, W. A. The Interaction of Gases with Solid Surfaces; Pergamon Press: New York, 1974. (12) Steele, W. A. J. Phys. Chem. 1978, 82, 817.
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Steele Table 1. Adsorbent Dimensions effective effective step box area terrace terrace spacing (nm) length (nm) (nm2) width (nm) area (nm2) b bb 4b flat
Figure 1. The adsorption energy for xenon on a stepped surface made up of graphite basal planes is plotted here as a function of the position y of the xenon atom. The steps are spaced 8.5 nm apart and are straight-edged; i.e., independent of x.
surface can be constructed from such a stack of planes; the steps can be one or more planes in height (two were chosen in the work reported here) and the terrace width is defined by the difference in ye for successive steps. In the present study, two steps were defined in the computer box, but the periodic boundary conditions give an infinite staircase of steps. One must take care that the periodicity is such that the image atom entering the box on one side is at a distance z* from the (local) surface when a real atom leaves the box on the other side at a distance z* from the (local) surface. Thus, if the number of truncated planes within the box is m and their separation distance is d ()0.34 nm for graphite), the entrance and exit values of z will differ by md. Here, we take m ) 4 (two steps that are each two planes in height) and vary the terrace widths to determine the effect of finite size upon the isotherms for the atoms on these terraces. The adsorption energy for a xenon atom over such a surface is shown in Figure 1,7 where it can be seen that this energy is that for the infinite plane (-18.8 kJ/mol) on the terraces far from the steps at y ) 4.3 and 13.2 nm but that it varies greatly in the neighborhood of a step, showing a sharp, deep minimum for an atom at the bottom of the step and a maximum for an atom at the top. Clearly, this system exhibits heterogeneity in the y direction. The interaction potentials suggest that one might divide the surface into two supersites, the first of which includes the nearly one-dimensional regions of the deep minima found at the step bottoms and the second composed of the terraces that make up the remainder of the surface. It is this idea that we explore here. 3. Adsorption on Stepped Surfaces Simulations of xenon at 166 and 120 K have been carried out for several stepped surfaces that differ by their terrace widths. In all cases, the adsorbent was taken to be graphite basal planes with steps that are two planes in height (0.68 nm) that are set at 1/4 and 3/4 of the box length. The width of the computer box was fixed at 4.92 nm, so that the total length of steps in the x direction amounted to 9.84 nm. for surfaces composed of two steps (before periodic boundary conditions). The distances between steps were varied by varying the size of the y dimension of the box and amounted to 2.13, 4.26, and 8.52 nm. (The adsorption energies for a xenon atom over the largest of these surfaces are shown in Figure 1.) In addition, simulations were carried out for the flat surface at these temperatures to give reference data. Note first that 120 K is lower than the critical
2.13 4.26 8.52
4.26 8.52 17.04 8.20
20.9 41.9 83.8 67.2
1.3 3.4 7.6
13 34 75
temperature of 130 K for 2D xenon condensation on the flat surface,13 but 166 K is above it. (The 2D xenon melting point is 100 K.) Since the emphasis here is on the generation of the local isotherms for the two supersites that we call “step” and “terrace”, the total number of atoms adsorbed at any pressure is partitioned into the numbers on each of the supersites. In general, we will be interested in submonolayer adsorption only, which determines the coverage ranges for both local isotherms. Since the step supersites are essentially one-dimensional, their coverages will be given in terms of atoms/nm; on the terraces, the coverages are most conveniently given as atoms/nm2. Here and elsewhere in this paper, b, bb, and 4b denote the increasing step spacing and terrace widths that are listed in Table 1. (Of course, flat denotes an infinite terrace width. In this case, the adsorbing area in the computer box is defined to be 8.2 × 8.2 nm2.) Figure 2 shows the total isotherms for the four surfaces at two temperatures. The amount adsorbed is given as atoms per unit area to allow one to compare the adsorption on surfaces with different areas. The pressure scale is logarithmic to give a close relation to chemical potential and to emphasize the characteristics of the low-coverage portions of the isotherms. In fact, the variable used throughout this paper is pKH, where KH is Henry’s law constant for the system, which is defined by
KH )
1 AkT
∫A ∫z[exp(-ugs(r)/kT) - 1] dA dz
(3.1)
where KH has units of atm-1 A-1, A is surface area, and z is the coordinate perpendicular to the surface. (For the flat surface, ugs is a function of z only, but it depends on both z and y for all the stepped surfaces.) It is clear that one can also define KH for supersite J merely by integrating over the area of the supersite rather than the total area of the adsorbent. If such a KH is denoted by KH(J), then KH ) ∑JKH(J). The utility of these constants lie in their use in the low-coverage parts of the global and local isotherms, since
lim Nloc(J,AJ,p,T)/AJ ) pKH(J) pf0
(3.2)
Thus, the KH(J) can be calculated either from eq 3.1 or by fitting the low coverage simulations of Nloc to eq 3.2. These constants are used throughout this paper and their logarithms are listed in Table 2. The constants for the two supersites (terrace and step) hardly change as the fraction of the surface devoted to each is changed, but the constant for the entire surface changes quite significantly as the fraction of the surface devoted to the relatively weakly adsorbing terraces increases. Of course, the flat surface is completely lacking in step supersites and thus is a very weak adsorber compared to those with some steps. In fact, (13) Steele, W. A. In Surfaces of Nanoparticles and Porous Materials; Schwarz, J., Contescu, C., Eds.; M. Dekker: New York, 1999. (14) Lieb, E. H.; Mattis, D. C. Mathematical Physics in One Dimension; Academic Press: New York, 1966; Bakaev, V. A.; Steele, W. A. J. Chem. Phys. 1995, 103, 751 and references therein.
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Figure 2. Simulated adsorption isotherms for xenon on three stepped surfaces of varying step separation. The notation b, bb, and 4b indicates step separations of 2.2, 4.4, and 8.8 nm. Isotherms for the reference flat surface are also shown. The upper panel (A) shows isotherms for 120 K and the lower panel (B) for 166 K, with coverages in the monolayer regimes plotted as a function of the logarithm of pKH, where KH is Henry’s law constant for each surface. (Values are listed in Table 2.) Table 2. Natural Logarithms of Henry’s Law constants (In the Units Shown) 120 K
166 K
adsorbent
total (atm-1 nm-2)
terrace (atm-1 nm-2)
step (atm-1 nm-1)
total (atm-1 nm-2)
terrace (atm-1 nm-2)
step (atm-1 nm-1)
b bb 4b flat
16.1 15.5 15.0 11.9
13.1 13.1 13.1
16.8 16.7 16.6
9.5 8.6 8.1 6.8
7.5 7.5 7.5
9.9 9.8 9.7
the steps make a major contribution to the total KH, as one might expect. In Figure 2A, the isotherm for the flat surface exhibits 2D condensation, as expected, but those for the stepped surfaces are lacking this feature. Presumably, this is a manifestation of the finite size (in the y dimension of the flat portions) of the stepped surface adsorbents. Rough estimates of the monolayer capacity of these surfaces can be obtained from the positions of “point B”, which is the coverage at which the slopes of the isotherms flatten out (seen most clearly for the flat surface case). One thus estimates ≈4 atoms/nm2 (0.007 mmol/m2). Better estimates will be made below on the basis of a detailed analysis of the adsorption on the two supersites. The large KH values for the step supersites indicate that adsorption will occur at considerably lower pressure in these strongly adsorbing regions than on the terraces. This behavior can be plainly seen in plots of the local
Figure 3. Local densities for xenon layers on the 4b surface at 166 K are shown here. The natural logarithms of the adsorption pressures in atm for these densities are -7.60 (A) and -6.21 (B). The heterogeneity at the steps produces pronounced y-dependent order in these films. The adsorbed atom densities on the terraces (Nterr) and in the rows at the bottom of each step (Nstep) are shown on the figure. The approximate upper limits of the densities for the completed layer are 2.1 atoms/nm and 5.0 atoms/nm2 for the step and the terrace, respectively. The strong adsorption at the bottom of the steps is shown by the highest peaks; immediately adjacent to these at lower y, negligible adsorption occurs, as is indicated by the small region of zero density. This is due to the weak adsorption energies shown in Figure 1 for these regions.
densities of atoms adsorbed on the stepped surfaces. Figures 3 and 4 show the y dependence of the onedimensional (1D) local densities for the 4b stepped surface for two temperatures and several sets of supersite coverage. In all cases, a sharp spike appears in the density of the atoms that are adsorbed by the strong minima of the gas-solid potential energy at the bottom of each step. The densities on the terraces are much more variable, showing moderately strong spikes in the regions just adjacent to the rows adsorbed at the steps and then continuing peaks whose heights depend significantly on total terrace coverage and temperature. Local density plots such as these will help characterize the intersite interaction energy, to be discussed below. The local adsorption isotherms for the step and the terrace supersites can now be extracted from the simulation data for the local densities. One merely integrates the density over the area of the supersite to obtain Nstep and Nterr as a function of p at a given T and divides by the length of the step supersite, since it is essentially a 1D adsorbing element, and by the effective area of the terraces, for these supersites. Effective areas are used because the local density plots of Figures 3 and 4 show that the terrace adsorption is excluded from strips at the tops of the steps which are ≈0.4 nm wide (that occur in the regions of the
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Figure 4. Same as Figure 3, but for 120 K. The natural logarithms of the adsorption pressures in atm for these densities are -14.47 (A) and -13.46 (B).
peaks in the adsorption energy shown in Figure 1). Figure 5 shows the 1D adsorption isotherms for the step supersites on all three stepped surfaces at two temperatures. These isotherms are essentially identical in all three cases. This is expected because the nature of these steps is not changing, only the distance between them. It is interesting to note that the exact theoretical isotherm has been derived for a 1D gas with arbitrary pairwise nearest neighbor interactions.14 If one inserts the Xe-Xe Lennard-Jones potential in this theory, a straightforward calculation gives the isotherms denoted “1D LJ” in the figures. (For this theory, KH ) 1.) It is evident that there is excellent agreement between the simulations and the theory, thus leading to the pleasing conclusion that the local isotherm for 1D supersites such as those considered here can be calculated exactly. The only input data needed are the gas-gas interaction functions and the local Henry’s law constant. Figure 6 shows the supersite isotherms for the terraces. The simulations give the numbers of atoms adsorbed and these are converted to atoms/nm2 by dividing by the effective terrace areas listed in Table 1. The differences between the isotherms for the various terrace widths are primarily due to the effects of the finite widths of the terraces. At present, there seems to be no theory that will quantitatively account for the changing shapes of the isotherms. However, there is one factor that still must be taken into account, which is the intersite interaction and its effect on the isotherms. 4. Step-Terrace Interactions Since the step supersites are essentially completely covered before terrace adsorption becomes significant, it
Steele
Figure 5. Local isotherms for adsorption at the bottom of the steps for all three surfaces are shown here. Coverage in atoms/ nm of step length is plotted as a function of the logarithm of pKstep, where Kstep is the Henry’s law constant for the supersite at the step bottom. All systems give the same isotherm, which is in excellent agreement with the theoretical isotherm for a one-dimensional Lennard-Jones gas with nearest neighbor interactions. Panels A and B are for 120 and 166 K, respectively.
is the effect of step-terrace interactions upon the terrace supersite isotherms that must be taken into account. The local densities shown in Figures 3 and 4 are typical and indicate that the interactions are primarily due to the atoms in the rows at the steps interacting with the terrace atoms in the first rows next to the steps. These two rows are δy ≈ 0.45 nm apart. The interaction u(δy) for a single atom in the terrace row with those in the step row can readily be evaluated by integrating the Xe-Xe energy along the row. For 1D step density F1D ()Nstep), the energy is
{(
∫-∞+∞
u(δy) ) 4F1D
) (
σ2 δy2 + x2
6
-
{ ( ) - (δyσ ) }
21 σ 3 ) πF1Dσ 2 32 δy
11
5
)}
σ2 δy2 + x2
3
dx (4.1)
The limiting density obtained from the simulation gives F1D ) 2.1 atoms/nm; using /k and σ for the Xe-Xe interaction and the value of δy given above, one finds u(δy)/k ≈ -350 K per atom, a significant interaction compared to the temperatures of the simulations. To include this interaction in the isotherm calculations, we invoke first-order statistical mechanical perturbation theory (for example, see ref 15). In this case, the part of the total potential energy Ulat due to the step-terrace lateral interactions is assumed to be small compared to the interactions of the atoms on the steps and the terraces. A standard argument gives the total free energy A of the
Adsorption on Heterogeneous Surfaces
Langmuir, Vol. 15, No. 18, 1999 6089
Figure 6. Local isotherms for the terrace supersites on the three surfaces are shown here for 120 K (panel B) and 166 K (panel A). Coverage in atoms/nm2 of accessible terrace area is plotted against the logarithm of pKterr, where Kterr are Henry’s law constants listed in Table 2 for these supersites. The isotherms for the reference flat surface shows 2D condensation at 120 K, but not at 166 K. The terrace supersites fail to show this condensation, presumably because of finite size effects.
N adsorbed atoms as A0 + A(1) where A(1) ) -kT ln〈exp(-U(1)/kT〉0. The subscript 0 means that the average is to be taken over the configurations of the unperturbed reference state, which is here the atoms on the step and the terrace supersites. The quantity wanted is of course the chemical potential change δµ for the terrace atoms due to the perturbation and this is given by
δµ )
( ) (1)
∂A ∂n
(
) ( )
∂ ln 〈exp(-U(1)/kT〉0 ∂〈U(1)〉 ) -kT ≈ ∂n ∂n (4.2)
where n is the total number of atoms on the terrace supersite with area Aterr. Thus, dn )Aterr dNterr where Aterr is the effective terrace area. Furthermore, the entire U(1) ) Ulat is taken to be due to the atoms in the first row on the terrace interacting with those in the filled row at the step. If we denote the number of those terrace atoms by LNt1, where L is the length of this row and Nt1 is the density of atoms in this row, then
( ) ( )( )( ) ∂Nt1 ∂ 1 ∂ ) ∂n Aterr ∂Nterr ∂Nt1
(4.3)
The interaction energy divided by k is -175 × nt1 ) -1/2350Nnt1 Aterr (The factor of 1/2 is included because half of the energy is the energy of perturbation of the step atoms by those on the terrace.) One needs only the derivative (∂Nt1/∂Nterr) in eq 4.3 to complete the calcula-
Figure 7. Isotherms for the atoms adsorbed on the terraces in the rows adjacent to those at the step bottoms (see Figures 3 and 4). The amounts adsorbed in these rows in units of atoms/ nm of row length are plotted against the total coverages on the terraces in units of atom/nm2 of effective terrace area. The simulation points are compared to fitting curves denoted by Nt1 for 120 K (panel A) and 166 K (panel B).
tion. This is obtained by first evaluating Nt1 from local densities such as those shown in Figures 3 and 4. The area under the first peak adjacent to that for Nstep is that for the desired Nt1 . The results of this calculation are plotted as a function of Nterr, as shown in Figure 7. A simple function is fitted to the data and then differentiated. The function and its derivative are
Nt1 ) 2.1[1 - exp(-Nterr/η)]
(4.4)
( )
(4.5)
∂Nt1 2.1 exp(-Nterr/η) )∂Nterr η
where η is a best-fit parameter with units of atoms/nm2 and the factor 2.1 is the limiting density in a row with units of atoms/nm. (A more complicated fitting function than that of eq 4.5 would undoubtedly give better results, but here we will be satisfied with an approximate calculation that illustrates the general theory.) Thus, the final result for the change in chemical potential of the atoms on the terrace due to the intersite interaction can be written as
370 δµ )exp(-Nterr/η) kT WterrηT
(4.6)
The width of a terrace is Wterr )Aterr/L and the constant 370 has a unit of K/atom‚nm. Values of η are listed in Table 3. Isotherms for xenon on the terrace supersites corrected for the intersite interaction are plotted as Nterr
6090 Langmuir, Vol. 15, No. 18, 1999
Steele
Table 3. Best-Fit Values of η (Units of Atoms/nm2) adsorbent b bb 4b
120 K
166 K
1.4 0.8 0.7
2.7 2.2 1.9
Figure 8. Two isotherms for atoms on terrace supersites are shown here. In both cases, T ) 120 K and the isotherms are shown with and without the correction term δµ/kT due to interactions across the step-terrace boundary. The corrections to log(pKH) are largest at low T and for small terraces.
versus log(pKH) + δµ/kT in Figure 8. Also shown are isotherms without the δµ/kT term. It is evident that the correction shifts log(p) to more negative (i.e., lower p) values compared to the uncorrected cases. The shift is largest at low p (i.e., low coverage). Shifts for those systems not shown (which includes all isotherms at 166 K and the 4b isotherm at 120 K) are smaller than those shown in Figure 8. 5. Discussion and Conclusions The use of the supersite formalism to represent the local isotherms for a relatively simple example of surface heterogeneity shows that this approach can indeed give a detailed picture of the submonolayer physical adsorption in these stepped surfaces. The local isotherms for the two supersites used to represent these adsorbents were simulated, and it was shown that the isotherm for one of them (the “step” supersite) can be calculated to good accuracy using the exact 1D isotherms for an interacting Lennard-Jones gas. A problem for future work is the derivation of local isotherms for a 2D film on the almost uniform surfaces of the “terrace” supersites. It was shown that the effects of size restriction (in the y direction) on these isotherms are quite significant for the sizes considered here. Finite size effects on monolayer properties have of course been studied previously.16 Qualitatively, these effects become significant when the correlation lengths in a 2D fluid are of the order of the size of the finite dimension (and thus are most important near the 2D critical point). The present case of a fluid confined to a long strip of finite width on a perfectly flat surface has not been studied theoretically (to the author’s knowledge). Furthermore, the boundary conditions that generate the finite dimension are not as simple as implied by the description “perfectly flat surface”. (15) McQuarrie, D. A. Statistical Mechanics; Harper and Row: New York, 1976; Section 14.1. (16) Bruch, L. W.; Cole, M. W.; Zaremba, E. Physical Adsorption: Forces and Phenomena; Clarendon Press: Oxford, 1997; p 174.
One feature of this work that is somewhat obscured by the method of presentation of the simulated isotherms is the important role of Henry’s law constant in determining the pressure scales of the various local isotherms. For example, the strong interactions that produce the local “step” isotherms give rise to adsorption there that is essentially complete before significant adsorption begins on the “terrace” supersites. As a result, the only corrections needed for the local isotherms that arise from the intersite interactions at the boundary between the “step” and the “terrace” sites are those for the “terrace” sites. In the “step” case, the sites are already fully occupied before the stepterrace interaction is significant, so the effect of this perturbation energy upon the “step” isotherms is not observable. The stepped surface is not unique in its suitability for a supersite analysis. For example, the exposed (110) surface of TiO2 and similar solids is composed of rows where the adsorption energy is quite strong, separated by other rows where the energy is weak. This surface would seem to lend itself to a supersite treatment where the two types of row are essentially 1D supersites. Several simulation studies of these adsorption systems have already been reported,3,17,18 and work on reinterpreting and extending these simulations is now under way. Other systems can easily be thought of: for example, uniform surfaces with isolated defects, either chemical or physical. The characterization of the nature of such systems before (or during) adsorption studies should be very useful in providing a tentative assignment of supersites for use in an analysis that relates adsorption behavior to the surface structure. As noted above, this is one of the more interesting aspects of the present approach to the problem on adsorption on heterogeneous surfaces. One of the questions that could be raised concerning the nature of a supersite is whether the introduction of arbitrary walls enclosing the supersite will introduce artifacts in the local isotherms due to the finite size effects imposed in this way. As it happens, the supersite boundaries for the present stepped surfaces are physically real: atoms on the step supersite are confined to one dimension by the rapid increase in adsorption energy when such atoms undergo displacements in the y direction perpendicular to the row direction and, similarly, atoms on the terraces are confined in one y direction by the strong potential energy barriers shown in Figure 1 and, in the other, by the atoms adsorbed at low pressure in the rows associated with the steps. Similar arguments may be applicable to adsorption on the TiO2 (110) face or even to the case of isolated defect sites on otherwise homogeneous surfaces. Of course, when exceptions to this kind of potential surface do occur, the supersite approach must be used with caution. Acknowledgment. Support for this work was provided in part by Grant NSF 9803884. LA981483B (17) Rittner, F.; Paschek, D.; Boddenberg, B. Langmuir 1995, 11, 3097, Steele, W. A.; Rittner, F.; Boddenberg, B. In Adsorption by Porous Solids; Staudt, R., Ed.; Fortschritte-Bericht VDI, Reihe 3, Verfahrenstechnik 555, VDI Verlag: 1998; p 47. (18) Rittner, F.; Boddenberg, B.; Bojan, M. J.; Steele, W. A. Langmuir 1999, 11, 1456.
