Adsorbate lateral interaction

Phase separation or, more broadly, island formation in HCR is possible due to attractive adsorbate-adsorbate lateral interactions (thermodynamic mechanism)... 

If we return now to the question of the uniqueness of the rate parameters determined from thermal desorption measurements, we see that all of the analytical methods depend on the assumption of a rate equation whose validity, in general, is not tested. In particular, when there are adsorbate—adsorbate (lateral) interactions, or where desorption occurs via a precursor state, the coverage dependence in the pre-exponential term is not a simple function and the concept of reaction order is not meaningful. 

This method was successful in the first attempts to solve equation (15) using a local isotherm equation that allowed adsorbate lateral interactions within patches. The ingenious graphical method, now referred to as the Ross and Olivier method of analysis, has been described in their monograph and in a condensed account by Ross. The technique used by Ross and Olivier employs a Gaussian probability function for the distribution of adsorption energies, i.e. a two-parameter generalized distribution of the form ... 

Although this method has only been applied using the Langmuir local isotherm equation, there is no reason why the coefficients Aj could not be obtained for a more sophisticated local isotherm, particularly one that accounts for adsorbate lateral interactions. 

General Analytical Metiwds of Solution.— The solutions obtained using these methods generally depend on finding an analytical form for the dataset g(x) and have been successful for local isotherms that ignore the effects of adsorbate lateral interactions, i.e. it must be possible to express the adsorbate coverage per patch in terms of the bulk gas pressure and the adsorption energy of the patch. 

Applications of Hobson s Method.—Zeldowich s kernel approximation [equation (6)] discussed in Section 2 has been generalized by Hobson to account for adsorbate lateral interactions. The major problem associated with the Langmuir equation is its inability to predict phase transitions. Rather than use the step approximation to the local isotherm, Hobson chooses a combination of a Henry s law isotherm and a step approximation , as shown in Figure 1(c) ... 

In order to distinguish between the static and dynamic contributions to adsorbate— adsorbate lateral interactions at maximum coverage, experiments with diluted isotopic mixtures of CO in CO were performed [31]. At low surface coverage, where 6 —-O, flie CO stretching peak occurs at 2152 cm in Scheme 6, independent of whether the CO CO ratio is 99 1 or 12 88. At infinite surface dilution, neighboring CO molecules are too far apart to incur either static or dynamic... 

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. 

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. 

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. 

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. 

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... 

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). 

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. 

Figure 8 shows an example of the most common behavior of AEam/0 as a function of adsorbate coverage. Linear behavior, if ever observed, is seen at the air/solution interface.93 At metal/solution interfaces, if chemical interactions with the metal can be ruled out, electrostatic interactions cannot be avoided, and these are responsible for the downward curvature.91 Upward curvatures are often observed at air/solution interfaces as a consequence of lateral interactions.95... 

This type of isotherm is more realistic for describing chemisorption at intermediate 0a values but quickly leads to mathematically cumbersome or intractable expressions with many unknown parameters when one considers coadsorption of two gases. One needs to know how -AHa is affected both by 0A and by the coverages of all other adsorbates. Thus for all practical purposes the LHHW kinetics represent even today the only viable approach for formulating mathematically tractable, albeit usually highly inaccurate, rate expressions for catalytic kinetics. In Chapter 6 we will see a new, medium field type, approach which generalizes the LHHW kinetics by accounting also for lateral interactions. 

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