Surface heterogeneity models

The importance of the solid-liquid interface in a host of applications has led to extensive study over the past 50 years. Certainly, the study of the solid-liquid interface is no easier than that of the solid-gas interface, and all the complexities noted in Section VIM are present. The surface structural and spectroscopic techniques presented in Chapter VIII are not generally applicable to liquids (note, however. Ref. 1). There is, perforce, some retreat to phenomenology, empirical rules, and semiempirical models. The central importance of the Young equation is evident even in its modification to treat surface heterogeneity or roughness. ... 

The first term on the right is the common inverse cube law, the second is taken to be the empirically more important form for moderate film thickness (and also conforms to the polarization model, Section XVII-7C), and the last term allows for structural perturbation in the adsorbed film relative to bulk liquid adsorbate. In effect, the vapor pressure of a thin multilayer film is taken to be P and to relax toward P as the film thickens. The equation has been useful in relating adsorption isotherms to contact angle behavior (see Section X-7). Roy and Halsey [73] have used a similar equation earlier, Halsey [74] allowed for surface heterogeneity by assuming a distribution of Uq values in Eq. XVII-79. Dubinin s equation (Eq. XVII-75) has been mentioned another variant has been used by Bonnetain and co-workers [7S]. 

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. 

On the other hand, as applied to the submonolayer region, the same comment can be made as for the localized model. That is, the two-dimensional non-ideal-gas equation of state is a perfectly acceptable concept, but one that, in practice, is remarkably difficult to distinguish from the localized adsorption picture. If there can be even a small amount of surface heterogeneity the distinction becomes virtually impossible (see Section XVll-14). Even the cases of phase change are susceptible to explanation on either basis. 

Many simple systems that could be expected to form ideal Hquid mixtures are reasonably predicted by extending pure-species adsorption equiUbrium data to a multicomponent equation. The potential theory has been extended to binary mixtures of several hydrocarbons on activated carbon by assuming an ideal mixture (99) and to hydrocarbons on activated carbon and carbon molecular sieves, and to O2 and N2 on 5A and lOX zeoHtes (100). Mixture isotherms predicted by lAST agree with experimental data for methane + ethane and for ethylene + CO2 on activated carbon, and for CO + O2 and for propane + propylene on siUca gel (36). A statistical thermodynamic model has been successfully appHed to equiUbrium isotherms of several nonpolar species on 5A zeoHte, to predict multicomponent sorption equiUbria from the Henry constants for the pure components (26). A set of equations that incorporate surface heterogeneity into the lAST model provides a means for predicting multicomponent equiUbria, but the agreement is only good up to 50% surface saturation (9). 

FIG. 19 Scheme of a simple fluid confined by a chemically heterogeneous model pore. Fluid modecules (grey spheres) are spherically symmetric. Each substrate consists of a sequence of crystallographic planes separated by a distance 8 along the z axis. The surface planes of the two opposite substrates are separated by a distance s,. Periodic boundary conditions are imposed in the x and y directions (see text) (from Ref. 77). 

Both extreme models of surface heterogeneity presented above can be readily used in computer simulation studies. Application of the patch wise model is amazingly simple, if one recalls that adsorption on each patch occurs independently of adsorption on any other patch and that boundary effects are neglected in this model. For simplicity let us assume here the so-called two-dimensional model of adsorption, which is based on the assumption that the adsorbed layer forms an individual thermodynamic phase, being in thermal equilibrium with the bulk uniform gas. In such a case, adsorption on a uniform surface (a single patch) can be represented as... 

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 mean field treatment of such a model has been presented by Forgacs et al. [172]. They have considered the particular problem of the effects of surface heterogeneity on the order of wetting transition. Using the replica trick and assuming a Gaussian distribution of 8 Vq with the variance A (A/kT < 1), they found that the prewetting transition critical point is a function of A and... 

Surface roughness is also expected to result in depression of the capacitance semi-circle. This phenomenon, which is indeed apparent in both Figures 1 and 2, is, however, unrelated to surface area. Rather, it is attributable to surface heterogeneity, i.e. the surface is characterized by a distribution of properties. Macdonald (16) recently reviewed techniques for representing distributed processes. A transmission line model containing an array of parallel R/C units with a distribution of values is physically attractive, but not practical. An alternative solution is introduction of an element which by its very nature is distributed. The Constant Phase Element (CPE) meets such a requirement. It has the form P = Y0 wn... 

The starting points for the continuity and energy equations are again 21.5-1 and 21.5-6 (adiabatic operation), respectively, but the rate quantity7 (—rA) must be properly interpreted. In 21.5-1 and 21.5-6, the implication is that the rate is the intrinsic surface reaction rate, ( rA)int. For a heterogeneous model, we interpret it as an overall observed rate, (—rA)obs, incorporating the transport effects responsible for the gradients in concentration and temperature. As developed in Section 8.5, these effects are lumped into a particle effectiveness factor, 77, or an overall effectiveness factor, r]0. Thus, equations 21.5-1 and 21.5-6 are rewritten as... 

Electrostatic vs. Chemical Interactions in Surface Phenomena. There are three phenomena to which these surface equilibrium models are applied regularly (i) adsorption reactions, (ii) electrokinetic phenomena (e.g., colloid stability, electrophoretic mobility), and (iii) chemical reactions at surfaces (precipitation, dissolution, heterogeneous catalysis). 

Surface-complexation models require a high degree of detail about the heterogeneous systems. Unfortunately, the chemical detail required to use surface-complexation models will often exceed our knowledge of interactions taking place in natural systems. Consequently, geochemists have often resorted to semi-empirical, macroscopic descriptions, which are more easily utilized. 

In surface-complexation models, the relationship between the proton and metal/surface-site complexes is explicitly defined in the formulation of the proposed (but hypothetical) microscopic subreactions. In contrast, in macroscopic models, the relationship between solute adsorption and the overall proton activity is chemically less direct there is no information given about the source of the proton other than a generic relationship between adsorption and changes in proton activity. The macroscopic solute adsorption/pH relationships correspond to the net proton release or consumption from all chemical interactions involved in proton tranfer. Since it is not possible to account for all of these contributions directly for many heterogeneous systems of interest, the objective of the macroscopic models is to establish and calibrate overall partitioning coefficients with respect to observed system variables. 

Cs-Ca selectivity modelling is well suited to detect surface heterogeneity in an extremely small fraction of the CEC, which is difficult to measure by differential calorimetry. However, differential calorimetry detects site heterogeneity covering the whole range of CEC. 

Hermance, C. E., A Model of Composite Propellant Combustion Including Surface Heterogeneity and Heat Generation, AIAA Journal, Vol. 4, No. 9,1965,... 

Van Riemsdijk.W.A. Bolt, G.H. Koopal, L.K. Blakemeer, J. (1986) Electrolyte adsorption on heterogeneous surfaces. Adsorption models. 

According to the ideal Langmuir model [156] the heats of adsorption should be independent of coverage, but this requirement is seldom fulfilled in real systems because the effects of surface heterogeneity and sorbate-sorbate interactions are generally significant. 

The multisite surface complexation model (MS-SCM) by Hiem-stra, Van Riemsdijk, and Bolt (27) was the first effort directed at understanding the reactivity of an oxide surface in terms of heterogeneous array individual surface functional groups. These authors... 

---