Lateral Interaction Models

Attractive or repulsive through-surface interactions are readily understood in terms of the Bond Order Conservation principles. When an adatom binds to a neighboring surface metal atom, the metal-metal bonds that form to the surface metal atom of interest are weakened. This increases the potential reactivity of the neighboring metal atoms since less of its electron density is tied to the metal atom involved in the surface-adatom bond. Thus, another adatom bound to the neighboring surface metal atom would have an increased interaction energy. Through-surface interactions are repulsive when two or more adsorbates share a metal atom, but attractive when the adsorbates sit at neighboring metal atom sites. These effects are illustrated in Fig. 3.54. 

Although first-principle calculations offer quantitative estimates for specific configurations, the shear number of different scenarios which arise in any kinetic or dynamic Monte Carlo simulation make it impossible to compute all of the possible configurations 

Various models have been proposed in the literature to model adsorbate-adsorbate interaction At the simplest level, the interactions can be described by a single parameter ujaa which treats the repulsion (co 0) and attractive (w 0) interactions between two species labeled A  

This approach has been adapted for sterns whereby the lateral interactions are lumped into a single parameter. More advanced treatments add a second parameter  

A second approach which may be attractive for more complex surface systems involves the application of the Bond Order Conservation model that was developed by Shustorovich and co-workers . The BOC model treats the interaction between the adsorbate and the surface atom through the use of a Morse potential. The total heat of adsorption is then described by summing all interactions. The BOC model is based on the concept that the bonding potential for every atom in the system is conserved. The heat of adsorption for an atomic species A is described by the following expression  

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Recently, a quantitative lateral interaction model for desorption kinetics has been suggested (103). It is based on a statistical derivation of a kinetic equation for the associative desorption of a heteronuclear diatomic molecule, taking into account lateral interactions between nearest-neighbor adatoms in the adsorbed layer. Thereby a link between structural and kinetic studies of chemisorption has been suggested. 

From a survey of the literature in chemically modified electrodes [13], one can identify simple phenomenological models that have been very successful for the analysis of a particular aspect of the experimental data. Such models are, for instance, the Dorman partition model [24, 122], the Laviron [158], Albery [159] and Anson models [127] to account for the nonideal peak width, the Smith and White model for the interfacial potential distribution [129], and so on. Most of these models contain one or more adjustable parameters that give some partial information about the system. For example, the lateral interaction model proposed by Anson [127] provides a value for the lateral interactions between oxidized and reduced sites, but does not explain the origin of the interactions, neither does it predict how they depend on the experimental conditions or the polymer structure. In addition, none of these models provide information on the interfacial structure. 

Figure 10 Lateral interaction model on an fcc(lll) lattice. The adsorbed sulfate anion binds to two surface atoms in a bridged fashion (black atoms), making bonding to the first shell of neighboring sites (white) impossible. There is a finite attractive or repulsive interaction with sulfate anions binding to the second neighbor shell (grey)...

Various functional forms for / have been proposed either as a result of empirical observation or in terms of specific models. A particularly important example of the latter is that known as the Langmuir adsorption equation [2]. By analogy with the derivation for gas adsorption (see Section XVII-3), the Langmuir model assumes the surface to consist of adsorption sites, each having an area a. All adsorbed species interact only with a site and not with each other, and adsorption is thus limited to a monolayer. Related lattice models reduce to the Langmuir model under these assumptions [3,4]. In the case of adsorption from solution, however, it seems more plausible to consider an alternative phrasing of the model. Adsorption is still limited to a monolayer, but this layer is now regarded as an ideal two-dimensional solution of equal-size solute and solvent molecules of area a. Thus lateral interactions, absent in the site picture, cancel out in the ideal solution however, in the first version is a properly of the solid lattice, while in the second it is a properly of the adsorbed species. Both models attribute differences in adsorption behavior entirely to differences in adsorbate-solid interactions. Both present adsorption as a competition between solute and solvent. 

Brunauer (see Refs. 136-138) defended these defects as deliberate approximations needed to obtain a practical two-constant equation. The assumption of a constant heat of adsorption in the first layer represents a balance between the effects of surface heterogeneity and of lateral interaction, and the assumption of a constant instead of a decreasing heat of adsorption for the succeeding layers balances the overestimate of the entropy of adsorption. These comments do help to explain why the model works as well as it does. However, since these approximations are inherent in the treatment, one can see why the BET model does not lend itself readily to any detailed insight into the real physical nature of multilayers. In summary, the BET equation will undoubtedly maintain its usefulness in surface area determinations, and it does provide some physical information about the nature of the adsorbed film, but only at the level of approximation inherent in the model. Mainly, the c value provides an estimate of the first layer heat of adsorption, averaged over the region of fit. 

An alternative way of deriving the BET equation is to express the problem in statistical-mechanical rather than kinetic terms. Adsorption is explicitly assumed to be localized the surface is regarded as an array of identical adsorption sites, and each of these sites is assumed to form the base of a stack of sites extending out from the surface each stack is treated as a separate system, i.e. the occupancy of any site is independent of the occupancy of sites in neighbouring stacks—a condition which corresponds to the neglect of lateral interactions in the BET model. The further postulate that in any stack the site in the ith layer can be occupied only if all the underlying sites are already occupied, corresponds to the BET picture in which condensation of molecules to form the ith layer can only take place on to molecules which are present in the (i — l)th layer. 

The model is intrinsically irreversible. It is assumed that both dissociation of the dimer and reaction between a pair of adjacent species of different type are instantaneous. The ZGB model basically retains the adsorption-desorption selectivity rules of the Langmuir-Hinshelwood mechanism, it has no energy parameters, and the only independent parameter is Fa. Obviously, these crude assumptions imply that, for example, diffusion of adsorbed species is neglected, desorption of the reactants is not considered, lateral interactions are ignored, adsorbate-induced reconstructions of the surface are not considered, etc. Efforts to overcome these shortcomings will be briefly discussed below. 

J. Satulovsky, E. V. Albano. The influence of lateral interactions on the critical behavior of a dimer-monomer surface reaction model. J Chem Phys 97 9440-9446, 1992. 

In a recent paper [11] this approach has been generalized to deal with reactions at surfaces, notably dissociation of molecules. A lattice gas model is employed for homonuclear molecules with both atoms and molecules present on the surface, also accounting for lateral interactions between all species. In a series of model calculations equilibrium properties, such as heats of adsorption, are discussed, and the role of dissociation disequilibrium on the time evolution of an adsorbate during temperature-programmed desorption is examined. This approach is adaptable to more complicated systems, provided the individual species remain in local equilibrium, allowing of course for dissociation and reaction disequilibria. 

Finally, the probability factor rj, which is taken to be coverage-independent in the model of a homogeneous surface with no lateral interactions between adsorbed particles, will be expressed by means of the Arrhenius formalism based on the Boltzmann distribution, viz. 

A macroscopic model for regular air/solution interfaces has been proposed by Koczorowski et al 1 The model is based on the Helmholtz formula for dipole layers using macroscopic quantities such as dielectric constants and dipole moments. The model quantitatively reproduces Ax values [Eq. (37)], but it needs improvement to account for lateral interaction effects. 

Contrary to the last two isotherms, which take into the account interactions between the neighboring molecities ortiy, the Kiselev model assumes the singlecomponent localized adsorption, with the specific lateral interactions among all the adsorbed molecules in the monolayer [4—6]. The equation of the Kiselev isotherm is given below ... 

From the asymmetrical concentration profile with front tailing (see Figure 2.4b), it can correctly be deduced that (1) the adsorbent layer is already overloaded by the analyte (i.e., the analysis is being run in the nonlinear range of the adsorption isotherm) and (2) the lateral interactions (i.e., those of the self-associative type) among the analyte molecules take place. The easiest way to approximate this type of concentration profile is by using the anti-Langmuir isotherm (which has no physicochemical explanation yet models the cases with lateral interactions in a fairly accurate manner). 

In this chapter, we are going to show that using the one- and the two-component multilayer adsorption isotherm models or the models taking into the account lateral interactions among the molecules in the monolayer (discussed in Section 2.1), the overload peak profiles presented in Section 2.4 can be qualitatively modeled. 

It is known that the p values derived from experimental Ax" data, e.g., for insoluble monolayers, with the assumption e = 1 are substantially different from the dipole moment for the same molecule in the bulk of the solution [17]. The reasons offered to explain this difference are manifold e.g. (1) inappropriate value of e, (2) reorientation of water molecules around the adsorbate, and (3) lateral interaction between adsorbed molecules in the monolayer. The various models of p have been described in a number of papers [61-67]. 

In a few instances, quantum mechanical calculations on the stability and reactivity of adsorbates have been combined with Monte Carlo simulations of dynamic or kinetic processes. In one example, both the ordering of NO on Rh(lll) during adsorption and its TPD under UHV conditions were reproduced using a dynamic Monte Carlo model involving lateral interactions derived from DFT calculations and different adsorption... 

Monte Carlo simulations have been also used to reproduce the dynamics of adsorbates associated with NO reduction reactions. As mentioned above, complex desorption dynamics have been observed experimentally in some instances. For example, the N2 produced from decomposition of N20 on Rh(110) leaves the surface in five peaks associated with both the N20 dissociation events and the desorption of the adsorbed products. Monte Carlo simulations of those spectra was possible by using a model that takes into account both channels of N2 desorption and also N20 O lateral interactions to stabilize N20 adsorption [18],... 

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