Geometry surface curvature

Usually, the zeolite inner surface characteristics are rather complex as a consequence of the (3D) character of the porous topologies of most of the zeolite types. The porous framework is a (3D) organization of cavities connected by channels. Inner surfaces are composed of several sorption sites characterized by their local geometry and curvature. Illustrative examples of such inner surface complexity are represented on Figures 1 and 2 they concern the Faujasite and Silicalite-I inner surfaces respectively. 

It was found that nanosize Ti02 particles experience an adjustment in the coordination geometry of the Ti atoms near the particle surface from octahedral to square-pyramidal in order to accommodate the large surface curvature [57]. X-ray absorption near edge structure reveals that surface modification with enediol ligands (ascorbate, ortho-hydroxy cyclobutene dione, catechol, etc.) restores the pre-edge features of octahedrally coordinated Ti in the anatase crystal environment. Specific binding of the enediol modifiers to surface comer defects ... 

Another entity that we shall need belongs to the realm of intrinsic geometry geodesic curvature. Consider a surface x, a point P on x and a curve on x passing through P. The curvature vector of at P joins P to the centre of curvature of This curvature vector may be decomposed into mutually orthogonal components. These components are given by projection of the... 

We have seen that the local constraint on the surface curvatures, set by the surfactant parameter, can be treated within the context of differential geometry, which deals with the intrinsic geometry of the surface. In contrast, the global constraint, set by the composition of the mixture, is dependent upon the extrinsic properties of the surface, which need not be related to its intrinsic characteristics. (For example, the surface to volume ratio of a set of parallel planes can assume any value by suitably tuning the spacing bebveen the planes. Similarly, the ratio of surface area to external volume i.e. the volume of space outside each sphere closer to that sphere than any other) of a lattice of spheres depends upon the separation between the spheres.)... 

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. 

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. 

Hence, it is not possible to redefine the characteristic length such that the critical value of the intrapellet Damkohler number is the same for all catalyst geometries when the kinetics can be described by a zeroth-order rate law. However, if the characteristic length scale is chosen to be V cataiyst/ extemai, then the effectiveness factor is approximately A for any catalyst shape and rate law in the diffusion-limited regime (A oo). This claim is based on the fact that reactants don t penetrate very deeply into the catalytic pores at large intrapellet Damkohler numbers and the mass transfer/chemical reaction problem is well described by a boundary layer solution in a very thin region near the external surface. Curvature is not important when reactants exist only in a thin shell near T] = I, and consequently, a locally flat description of the problem is appropriate for any geometry. These comments apply equally well to other types of kinetic rate laws. 

In addition to optimized geometries, which are minima on the potential energy surface, curvature on the free energy surface is also of interest in many QM/MM studies. Therefore, efficient calculations of free energy are required. One of the most widely used techniques to calculate the free energy in QM/MM methods is free energy perturbation (FEP) theory. Originally developed in MM, this FEP... 

FIGURE 2.58 Capacitance as a function of electrode surface curvature radius and geometry for various open structure electrode geometries as obtained from MD simulations at the 4 V potential difference between electrodes. (Reprinted with permission from Vatamanu, J. et al., 2013. Increasing energy storage in electrochemical capacitors with ionic liquid electrolytes and nanostructured carbon electrodes. Journal of Physical Chemistry Letters 4 2829-2837. Copyright 2013 American Chemical Society.)... 

The driving force of sintering appears as differences in bulk pressure, vacancy concentration and vapour pressure — parallel phenomena — due to differences in surface curvature of the particles. For the geometry in Figure 4.2, the pressure difference AP is... 

Figure 11.2 Dihedral angle dependence of X.mm, the minimum wavelength perturbation capable of increasing in magnitude, compared with values obtained assuming a cylindrical pore geometry. Cylindrical pores of radius rv and n have respective volumes and surface curvatures equivalent to those of pore characterized by a dihedral angle of i j. (From Ref. 7.)...

From sorption experiments the efficacy of a sorbate has been measured as heat of adsorption and described as nest effect , relating size and shape of the sorbate with the surface curvature of the pore [48]. Recently, host-guest complexes have been formulated quantitatively in terms of van der Waals interactions. Lewis et al. [47] calculated the nonbonded interactions energy of the SDA within the cavities of different silica zeoHtes, which was in good agreement with the experimental synthesis experience. The computational strategy developed in this study should stimulate the systematic search for new effective SDAs for the synthesis of new porosil structures with tailored pore geometry [49]. 

While the efficiency of nanoparticles for enhancing nucleation has been widely reported and superior to micron-sized particles, the effects of particle size and geometry in general (shape, aspect ratio, and surface curvature) require further elucidation. 

To account for this effect Buist et ah ( ) pointed out that in an assemblage of spherical grains with finite contact areas the surface curvature would tend to decrease as 0 increased, and it was assumed that the diameters of the necks (contact areas) between the grains remained constant they showed that if a suitable geometry was assumed the observed effects on grain growth could be accounted for. 

In some systems, the local shape of surfaces is significant in understanding the formation and stability of the morphologies and thus it can be a key structural measure to clarify the underlying physics. Surface curvatures are fundamental parameters characterizing the shape of surfaces in the differential geometry. From the volume data taken by the 3D microscopy, it is possible to determine the smface curvatures/or the first time. 

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