Interactions lateral

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. 

It is convenient to illustrate the lateral interaction effect by plotting 6 versus... 

Fig. XVII-4. Langmuir plus lateral interaction isotherms.

This is useful since c can be estimated by means of the BET equation (see Section XVII-5). A number of more or less elaborate variants of the preceding treatment of lateral interaction have been proposed. Thus, Kiselev and co-workers, in their very extensive studies of physical adsorption, have proposed an equation of the form... 

It must be remembered that, in general, the constants a and b of the van der Waals equation depend on volume and on temperature. Thus a number of variants are possible, and some of these and the corresponding adsorption isotherms are given in Table XVII-2. All of them lead to rather complex adsorption equations, but the general appearance of the family of isotherms from any one of them is as illustrated in Fig. XVII-11. The dotted line in the figure represents the presumed actual course of that particular isotherm and corresponds to a two-dimensional condensation from gas to liquid. Notice the general similarity to the plots of the Langmuir plus the lateral interaction equation shown in Fig. XVII-4. 

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. 

Fig. XVII-25. Interaction energy distributions for N2 on BN (a) Langmuir b) Langmuir plus lateral interaction (c) van der Waals. (From Ref. 162.)...

If adsorption occurs via a physisorbed precursor, then the sticking probability at low coverages will be enhanced due to the ability of the precursor to diflfiise and find a lattice site [30]. The details depend on parameters such as strength of the lateral interactions between the adsorbates and the relative rates of desorption and reaction of the precursor. In figure Al.7,8 an example of a plot of S versus 0 for precursor mediated adsorption is presented. 

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 lecture deals with the advantages of IR spectroscopy at low or variable temperatures in the studies of molecule-surface interactions, lateral interactions between the adsorbed molecules and catalysis. 

Lateral interactions between the adsorbed molecules can affect dramatically the strength of surface sites. Coadsorption of weak acids with basic test molecules reveal the effect of induced Bronsted acidity, when in the presence of SO, or NO, protonation of such bases as NH, pyridine or 2,6-dimethylpyridine occurs on silanol groups that never manifest any Bronsted acidity. This suggests explanation of promotive action of gaseous acids in the reactions catalyzed by Bronsted sites. Just the same, presence of adsorbed bases leads to the increase of surface basicity, which can be detected by adsorption of CHF. ... 

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. 

Note that if sticking is controled by site-exclusion only, i.e., if S 6,T) = 5 o(P)(l — 0), this rate is that of a first-order reaction at low coverage. This simple picture breaks down when either the sticking coefficient depends dilferently on the coverage, as it does for instance for precursor-mediated adsorption, or when lateral interactions become important. It then does not make much physical sense to talk about the order of the desorption process. 

To gain some qualitative insight into the elfect of lateral interactions it is useful to employ simple analytical approximations in the calculation of the chemical potential, of which the quasichemical approximation is the best suited. We split the chemical potential into a non-interacting part, Eq. (10), and a term due to lateral interactions, and get for the... 

The constants rc, u ic, etc. are specified in terms of microscopic parameters and the functions fc, f, f c tc. account for the various lateral interactions between the particles in the adsorbed and precursor states. We have factored out an explicit dependence on the coverages so that in the absence of any lateral interactions these functions are all equal to one. 

Here is the energy gain or loss when a site reconstructs. The lateral interaction energies and V2s between nearest (a) and next nearest (b) (and further) neighbors are most likely attractive to favor the growth of domains that are either reconstructed or unreconstructed. If V2s were repulsive then a c(2 x 2) pattern of alternately reconstructed and unreconstructed cells would be favored. A gas phase particle can adsorb either on the unreconstructed ui = 0 or 1) or the reconstructed surface (r, = 0 or 1) subject to the constraints... 

The lateral interactions in the adsorbate can enhance or diminish the interaction energy in the surface. If the adsorption sites at the boundary between reconstructed and unreconstructed areas of surface are further distinguished from those inside these patches, we can introduce more interactions such as... 

We can include surface diffusion by adding to (90), neglecting again the effects of lateral interactions,... 

It shows that sticking is proportional to the availability of empty sites (because there are no lateral interactions in the adsorbate), and the sticking probabilities, Sy and S, are weighted by the fraction of the adsorbate-free surface that is reconstructed or not. This can obviously introduce a substantial temperature dependence in the sticking coefficient. 

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. 

None of the procedures outlined can claim any strict justification. Indeed, the deviations of experimental curves from the calculated ones based on simple assumptions can be due in general to a number of causes, some of which were dealt with in Section II.A. A principal ambiguity lies in the choice of whether to treat such departures in terms of either variable Ed or kd, and in the former case often whether the changes in Ed are to be attributed to nonequivalence of adsorption sites, or to lateral interactions between the adsorbed particles, or to yet some other factor (98). 

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