Local surface curvature

Because surface curvature depends on radius and different atoms have different sizes, and because the atomic surface tension depends on atomic number, the atomic surface tensions also include surface curvature effects, which has recently been studied as a separate effect.7 Local surface curvature may also correlate with nearest-neighbor proximity and thus may be implicitly included to some extent when semiempirical atomic surface tensions depend on interatomic distances in the solute. 

The local surface curvature is determined by construction of a vector normal to the surface and drawing of two orthogonal planes through the normal vector (Figure 9.4). The location of the planes is chosen according to a requirement that the principal radii, r, and r2, of curvature of lines formed by intersection of the planes with the surface have the minimum and the maximum values. In inverse proportion to them are the principal surface curvatures, g1 = Hrl and g2= l/r2. 

The mean local surface curvature is expressed by Equation 9.7, and the full (Gaussian) local surface curvature is determined as... 

The diffusion potential of an atom at the surface is proportional to local surface curvature as demonstrated in Section 3.4. The curvature can be determined from Eq. 14.1 and is a function of x. The local diffusion potential produces boundary conditions for diffusion through the bulk or transport via the vapor phase. For surface diffusion, gradients in the diffusion potential produce fluxes along the surface. 

As the SFM provides local surface curvatures at a given point on the surface, the curvature distributions of H and K can be obtained by conducting the above measurements at many points on the surface. A joint probability density of H and K, is calculated as... 

A drop of water that is placed on a hillside will roll down the slope, following the surface curvature, until it ends up in the valley at the bottom of the hill. This is a natural minimization process by which the drop minimizes its potential energy until it reaches a local minimum. Minimization algorithms are the analogous computational procedures that find minima for a given function. Because these procedures are downhill methods that are unable to cross energy barriers, they end up in local minima close to the point from which the minimization process started (Fig. 3a). It is very rare that a direct minimization method... 

The goal of all minimization algorithms is to find a local minimum of a given function. They differ in how closely they try to mimic the way a drop of water or a small ball would roll down the slope, following the surface curvature, until it ends up at the bottom. Consider a Taylor expansion around a minimum point Xq of the general one-dimensional function F(X), which can be written as... 

The Euler characteristic, %, of a closed surface is related to the local Gaussian curvature K r) via the Gauss-Bonnet theorem [Eq. (8)]. A number of different schemes have been proposed to calculate the local curvatures and the integral in Eq. (8). 

Similar analysis can be made for other types of materials. Thus, as a generalization, the curvature of a surface causes field intensification, which results in a higher current than that on a flat surface. Although the detailed current flow mechanism can be different for different types of materials under different potentials and illumination conditions, the effect of surface curvature on the field intensification at local areas is the same. The important point is that the order of magnitude for the radius of curvature that can cause a significant effect on field intensification is different for the substrates of different widths of the space charge layer. This is a principle factor that determines the dimensions of the pores. 

In this case, Einstein computed that light from a certain star passing close to the sun s surface would be deflected by the sun s local spacetime curvature by a factor of 1.7 seconds of arc, that is, as = 1.7". The effect of STC near massive bodies was verified during the eclipse of the sun in 1919 when the following values were measured as = 0.8" and as = 1.8" [10,24,25]. 

Remarkably, the connection between a force between spheres and an energy between planes holds for the fullest expression of Gpp(/). In the regime of large radii compared with minimal separation, this relation can also hold for other curved surfaces or protrusions having gradual local spherical curvature. 

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