Heterogeneity surfaces

135 Experimental (symbols) and calculated (lines) surface charge density of alumina as the function of the pH and ionic strength 0.1 mol dm squares, solid 0.01 mol dm circles, dash 0.001 mol dm triangles, dot. For model parameters see text. 

Surface charging of rutile at one ionic strength was modeled using 3 kinds of surface sites (discrete distribution) having different proton binding constants [114]. 

The calculated distribution is again conditional , i.e, it applies to one ionic strength. 

The above results refer to metal oxides, but similar model exercises (discrete distributions of proton binding constants derived from one potentiometric curve) were reported for sparingly soluble phosphates [115] and for clay minerals [116], 

Charmas [117] discussed homogeneous and heterogeneous version of TLM with different C+ and C. The calculated charging curves did not significantly differ in these two versions, but a substantial difference in the calculated potentials and adsorption of ions of inert electrolyte was reported. 

A sorbent, especially in natural environments like soils, is usually not composed of equal and equivalent sites but of a number of different types, each with its own characteristics. For the case of a relatively simple patchwise surface (a few site types), expressions like Equation 4.52 will be applicable. However, for highly heterogeneous surfaces, a high number of different adsorption energies will be present in these cases, it is better to treat the surface as composed of sites belonging to a continuous distribution of adsorption energies. The general expression for the amount adsorbed becomes 

For the determination of the adsorption energy distribution, F e), from Eq. (16.32), and assumption on the shape of the local isotherms has to be made and usually complex numerical analysis is required [39, 40]  

Concerning the adsorption potential distribution, it can be obtained easily form the adsorption isotherm. Furthermore, is less affected by experimental noise and produce reliable results. Once the adsorption isotherm is derived from IGC, the adsorption potential, AP, is calculated according to  

The distribution parameter, 4 , represents the first derivation of the adsorbed amount of molecules, n, with the adsorption potential  

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SHEED Scanning high-energy electron diffraction [106] Scanning version of HEED Surface heterogeneity... 

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. ... 

Surface heterogeneity may be inferred from emission studies such as those studies by de Schrijver and co-workers on P and on R adsorbed on clay minerals [197,198]. In the case of adsorbed pyrene and its derivatives, there is considerable evidence for surface mobility (on clays, metal oxides, sulfides), as from the work of Thomas [199], de Mayo and co-workers [200], Singer [201] and Stahlberg et al. [202]. There has also been evidence for ground-state bimolecular association of adsorbed pyrene [66,203]. The sensitivity of pyrene to the polarity of its environment allows its use as a probe of surface polarity [204,205]. Pyrene or ofter emitters may be used as probes to study the structure of an adsorbate film, as in the case of Triton X-100 on silica [206], sodium dodecyl sulfate at the alumina surface [207] and hexadecyltrimethylammonium chloride adsorbed onto silver electrodes from water and dimethylformamide [208]. In all cases progressive structural changes were concluded to occur with increasing surfactant adsorption. 

Finally, in the case of solids, there is the difficulty that surface atoms and molecules differ in their properties from one location to another. The discussion in Section VII-4 made clear the variety of surface heterogeneities possible in the case of a solid. Those measurements that depend on the state of surface atoms or molecules will generally be influenced differently by such heterogeneities. Different methods of measuring surface area will thus often not only give different absolute values, but may also give different relative values for a series of solids. 

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. 

Surface heterogeneity may merely be a reflection of different types of chemisorption and chemisorption sites, as in the examples of Figs. XVIII-9 and XVIII-10. The presence of various crystal planes, as in powders, leads to heterogeneous adsorption behavior the effect may vary with particle size, as in the case of O2 on Pd [107]. Heterogeneity may be deliberate many catalysts consist of combinations of active surfaces, such as bimetallic alloys. In this last case, the surface properties may be intermediate between those of the pure metals (but one component may be in surface excess as with any solution) or they may be distinctly different. In this last case, one speaks of various effects ensemble, dilution, ligand, and kinetic (see Ref. 108 for details). 

Surface heterogeneity tends to mask the effect of monolayer completion. 

Surface heterogeneity is difficult to remove from crystalline inorganic substances, such as metal oxides, without causing large loss of surface areas by sintering. Thus in Fig. 2.14 in which the adsorbent was rutile (TiO ) all three adsorbates show a continuous diminution in the heat of adsorption as the surface coverage increases, but with an accelerated rate of fall as monolayer completion is approached. 

The way in which these factors operate to produce Type III isotherms is best appreciated by reference to actual examples. Perhaps the most straightforward case is given by organic high polymers (e.g. polytetra-fluoroethylene, polyethylene, polymethylmethacrylate or polyacrylonitrile) which give rise to well defined Type III isotherms with water or with alkanes, in consequence of the weak dispersion interactions (Fig. S.2). In some cases the isotherms have been measured at several temperatures so that (f could be calculated in Fig. 5.2(c) the value is initially somewhat below the molar enthalpy of condensation and rises to qi as adsorption proceeds. In Fig. 5.2(d) the higher initial values of q" are ascribed to surface heterogeneity. 

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). 

Other reactions at surfaces (heterogeneous catalysis and reduction reactions)... 

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... 

III. SURFACE HETEROGENEITY AND PHASE TRANSITIONS IN MONOLAYER FILMS... 

From the above argument and Eq. (16) we instantaneously find that the isosteric heat of adsorption cannot be constant within the two-phase region but must also show changes with the surface coverage. In the case of heat capacity we also observe important effects due to the surface heterogeneity. 

The question we ask here is whether the surface heterogeneity, either energetical or geometrical, changes the properties of the critical region and whether the transition retains its second-order character. 

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... 

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