Surface patchwise model

The effects due to the finite size of crystallites (in both lateral directions) and the resulting effects due to boundary fields have been studied by Patrykiejew [57], with help of Monte Carlo simulation. A solid surface has been modeled as a collection of finite, two-dimensional, homogeneous regions and each region has been assumed to be a square lattice of the size Lx L (measured in lattice constants). Patches of different size contribute to the total surface with different weights described by a certain size distribution function C L). Following the basic assumption of the patchwise model of surface heterogeneity [6], the patches have been assumed to be independent one of another. 

The analysis we have dealt with in Sections 11.4 and 11.5 are applicable to energetic homogeneous surfaces. In reality, pores are heterogeneous and it should be considered in the modeling. The surface heterogeneity is modeled either by a periodic spatial variation of the solid—fluid potential [91—93], or in the framework of a patchwise model [94—96]. The latter is more general, and is often considered in the literature. 

The extension of the lattice gas formalism from monolayer adsorption to three-dimensional systems is straightforward. In this case, the adsorbate-adsorbent interaction energy depends not only a kind of site but also on distance from the surface. The Hamiltonian for such a system is calculated by replacing the constant value , by appropriate functions in Eq. (72). In consequence, either for the patchwise model or for randomly distributed sites, for successive layers we obtain the equations analogous to those previously discussed. A munerical solution of such a set of equations is not a difficult problem, but to achieve the stable results, a rather large number of layers in the system should be assumed. 

Gibb, A.W.M. and Koopal, L.K., Electrochemistry of a model for patchwise heterogeneous surfaces The rutile-hematite system, J. Colloid Interf. Sci., 134, 122, 1990. 

We consider a solid particle exposing to an environment containing N adsorbates of constant concentrations. The surface topography is assumed to have a patchwise configuration. For the development of the model we make the following assumptions ... 

The work discussed in this report is concerned with these two groups. The surface of type (a) will be referred to as patchwise heterogeneous after Ross and Olivier and type (b) as random heterogeneous after Hill. The development of the patchwise heterogeneous model followed an extension of the concepts discussed by Langmuir. It was realized that equation (5) could be written in a more general form ... 

Adsorption Isotherms on Homogeneous Surfaces.—The integral equation describing adsorption on a patchwise heterogeneous surface requires the choice of a local or model isotherm to describe the adsorption on individual patches... 

The complexity of equation (60) forbids its incorporation as a rigorous and general local isotherm in the integral equation for a patchwise heterogeneous surface. The majority of work published has chosen between equations that allow the adsorbate complete free translational motion over the surface or localized vibrational motion. For some systems, numerical estimates of the ratio of the translational barrier to adsorption energy, can lead to the choice of a realistic local isotherm equation. Alternatively isosteric enthalpy data may offer some guidance. In the absence of this information, there is no justification for the choice of either model and the only course is to employ both extremes in turn and assess the effects on the calculated adsorption energy distribution. 

Very little work has been published on this subject, although it is well known that multilayer adsorption may begin at low adsorbate surface coverages. The majority of treatments have adopted the patchwise heterogeneity model or have not accounted for adsorbate self-interactions and extended the analysis by using a multilayer adsorption isotherm as the local isotherm in the integral equation. The effects of multilayer adsorption in the patchwise heterogeneity analysis with a monolayer local isotherm have been demonstrated by van... 

A simplified model of equilibrium surface suggests that the DR behaviour is observed in low-pressure adsorption on patchwise, weakly heterogeneous surfaces which were grown in equilibrium conditions and hence were quenched at the adsorption temperature. At higher pressures, these surfaces should exhibit the Freundlich behaviour, while in the case of strong heterogeneity adsorption should be described by the Temkin isotherm. The three classic empirical isotherms, Freundlich, Dubinin-Radushkevich, Temkin, seem therefore to be related to adsorption on equilibrium surfaces, and the explanation of these experimental behaviours can be seen as a new chapter of the theory of adsorption the theory of physical adsorption on equilibrium surfaces. 

It is well known that in the hterature there are more than 100 isotherm equations derived based on various physical, mathematical, and experimental considerations. These variances are justified by the fact that the different types of adsorption, solid/gas (S/G), solid/liquid (S/L), and liquid/gas (L/G), have, apparently, various properties and, therefore, these different phenomena should be discussed and explained with different physical pictures and mathematical treatments. For example, the gas/solid adsorption on heterogeneous surfaces have been discussed with different surface topographies such are arbitrary, patchwise, and random ones. These models are very useful and important for the calculation of the energy distribution functions (Gaussian, multi-Gaussian, quasi-Gaussian, exponential) and so we are able to characterize the solid adsorbents. Evidently, for these calculations, one must apply different isotherm equations based on various theoretical and mathematical treatments. However, as far as we know, nobody had taken into account that aU of these different isotherm equations have a common thermodynamical base which makes possible a common mathematical treatment of physical adsorption. Thus, the main aim of the following parts of this chapter is to prove these common features of adsorption isotherms. 

FIG. 3 Comparison of the adsorption isotherms for the patchwise (short-dashed line), random (solid line), and regular crystalline (long-dashed line) models of the surface. Parameters = 114,798 atm ... 

Similar calculations have been performed for the dimer-monomer solutions [245]. The classical theory of adsorption of polymers has been extended to adsorption on heterogeneous surfaces of the patchwise topography. The adsorbed dimers could be parallel as well as normal to the surface. The three-dimensional model, thought somewhat more difficult from a mathematical point of view, is much more realistic than the parallel model used in the previously discussed articles. The theory has been compared with Monte Carlo simulations. 

---