Surface Curvature Model

The model was considered to also be applicable for the PS formed on other types of silicon substrates. As a generalization, Zhang stated that it is the sensitivity of the semiconductor interface reactions to the curvature of the interface that enhances the preferential dissolution and leads to the formation of pores. For p-Si and heavily doped -Si which have much thinner space charge layers than does -Si, the radius of curvature must be small to affect the width of the space charge layer and, as a result, much 

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It may be assumed that the penetration model may be used to represent the mass transfer process. The depth of penetration is small compared with the radius of the droplets and the effects of surface curvature may he neglected. From the penetration theory, the concentration C, at a depth y below the surface at time r is given by ... 

In all the models discussed so far, the support surface was assumed to be flat. In fact, very many case studies, particularly of supported metals, have used flat, low-surface-area substrates. However, Wynblatt and Ahn [36] have demonstrated that surface curvature does affect the surface free energy, the growth of particles (sintering) via particle migration and interparticle transport. Therefore, the sintering process of practical supported catalysts which frequently use high-surface-area, porous supports must be significantly more complex than described by the simple models. 

In order to correct these discrepancies, Broekhoff and de Boer [5] generalized Kelvin s equation by taking pore shape into account, as well as the influence of surface curvature on the thickness of the adsorbed layer. The BdB method can be applied to both adsorption and desorption isotherms using four different pore models defined by a shape factor. Unfortunately, owing to computational difficulties, this last method, although more general, has been far less applied than the first two. 

These models will rarely give a perfect description of the response surface (curvature is not described), but they will give estimates of the slopes, b along each variable axis, and the twists, hy, to describe interactions. This is exactly what is needed to assess the influence of the variables. 

These steric problems with their consequences should affect all vinylic surfactant polymers, independent of the inherent surface curvatures of the different models of polymeric micelles (see Sect. 4.2). Thus, vinylic surfactant homopolymers of other than tail end geometry should be of very limited use as polysoaps. In fact, very few exceptions to the geometry controlled model of solubility have been reported [82, 106, 128, 225, 289-291]. In these examples, the chemical integrity of the polymers prepared, the attributed structures, or the polymeric nature may be questioned considering the results obtained for very similar compounds [115,232,309], But even if these exceptions are real, this rule will help to design new monomers and polymers. 

Although PL liposomes are favored systems for the study of apolipoprotein binding to PL surfaces, vesicle-apolipoprotein complexes are not the ideal models for lipoproteins. Vesicles have an interior water compartment not present in lipoproteins, are incapable of solubilizing large amounts of neutral lipids within the PL bilayer, and are too large to mimic the surface curvature of HDL. Thus, methods have been developed to prepare small, micellar complexes of exchangeable apolipoproteins (in particular apo Al) with lipids that mimic discoidal and spherical HDL in shape, composition, and functional properties. For LDL and VLDL, microemulsions and emulsions of lipids of selected diameter and composition, with added apo B 100, make good models of the native lipoproteins. 

Gibbs main idea was to introduce a dividing (mathematical ) surface its position (inside of the s-phase of Bakker s model) is defined in such a way that the adsorption of one of the components is zero and with a surface energy independent of surface curvature. This surface is noted as the surface of tension. In essence we can restrict Gibbs thermodynamics to the lUPAC recommendation for colloid and surface chemistry, prepared by Everett (1971). Gibbs idea of a dividing surface agrees with the definition by Rusanov (1981) mention above. 

To recap Equation (10.15) was derived by assuming a representative shape (Fig. 10.12a), from which the surface curvature was calculated as a function of geometry [Eq. (10.13)]. A flux equation [Eq. (10.11)] was then assumed and integrated to yield the final result. By using essentially the same procedure, the following results for other models are obtained. 

Local geometrical features are essential for understanding binding properties, catalytic behavior, and molecular recognition. Many of the descriptors used for global analysis can be adapted to study local features. For instance, mean and Gaussian curvature distributions of a surface, curvature and torsion of a molecular space curves, and the variation of the fractal index Df(r) over a molecular model serve this purpose. 

As mentioned above, a parallel line of research has been carried out by Dzubiella, Hansen, McCammon, and Li. Early work by Dzubiella and Hansen demonstrated the importance of the self-consistent treatment of polar and nonpolar interactions in solvation models [137, 138]. These observations were then incorporated into a self-consistent variational framework for polar and nonpolar solvation behavior by Dzubiella, Swanson, and McCammon [131, 139] which shared many common elements with our earlier geometric flow approach but included an additional term to represent nonpolar energetic contributions from surface curvature. Li and co-workers then developed several mathematical methods for this variational framework based on level-set methods and related approaches [140-142] which they demonstrated and tested on a... 

In Coble s initial stage hot pressing model, the effective stress applied to the grain boundaries, p, is related to the externally applied stress, p, which is by Eq. (5.180). In that model, the total driving force (DF) is a linear combination of two effects (i) external applied stress and (ii) surface curvature, which determines the densification rate, and it is... 

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