Adsorption of Spherical Molecules in Probing the Surface Topography. 1. Patchwise Heterogeneity Model

Abstract

The Fowler−Guggenheim local isotherm corresponding to the lattice model of localized adsorption on a patchwise surface was used to evaluate the statistical distribution of adsorption sites with respect to their adsorption energies from an experimental adsorption isotherm. An accurate low-pressure argon adsorption isotherm at 77 K on muscovite was used as initial information. The calculations were carried out in two ways that imply different surface constructions. The first one is based on the classical hypothesis of Langmuir, who assumed a finite manifold of adsorption sites and wrote an overall isotherm as a series summing the additive contributions of the different sites. The second is derived on the assumption of an infinite manifold of sites, when the sum turns into an integral. Both representations are equivalent only with the exact adsorption model. However, as shown, in a number of cases they can give similar results even with the approximate model. The influence of the ω parameter expressing the reduced energy of lateral interactions was tested for different ω values ranging between 0 and 4 in units of RT. Muscovite possesses crystallinity. Therefore, one may expect that its surface consists of recurring adsorption sites. Peaks on the energy distribution could mean at least a short range ordering (SRO) on this surface. A derivative of a local isotherm for sites of each kind is known to be a bell-shaped curve. The greater the energy of lateral attraction is, the narrower is the bell; that is, the Langmuir local isotherm (the case of ω = 0) corresponds to the widest bell. Hence, if the hypothesis concerning SRO holds true, an energy distribution obtained with ω = 0 must be discrete, inasmuch as a broadening of the calculated partial distributions can lead only to a deteriorating of the approximation accuracy of local derivatives and, correspondingly, of the overall derivative and the overall isotherm. Indeed, the distributions calculated with a small ω represent sums of δ-like peaks. With a small ω, both algorithms show the same results; therefore, any numerical artifacts are excluded. However, SRO on the surface is not proven yet, because with a large ω the algorithms lead to continuous distributions rather than to discrete ones. The relative approximation error of the overall isotherm with ω = 0 is not too large (≅2−3%). However, the large approximation error of the overall derivative (≅15%) unambiguously indicates the unacceptability of the assumption ω = 0 (local Langmuir isotherm). Thus, to make final conclusions about the structure of the surface and validate such an analysis, it is necessary to improve the patchwise model, to test the method by using an independent molecular simulation, and to determine realistic values of the energies of lateral interactions being different and not equal for adsorption sites of each kind.