Curvature spherical surface

For the case where the curvature is small compared to the thickness of the surface region, d(c - C2) = 0 (this will be exactly true for a plane or for a spherical surface), and Eq. III-28 reduces to... 

Retaining the approximations of an incompressible liquid phase, a discontinuous density profile and curvature independent surface tension the conditions are those studied by Rao, Berne and Kalos (2). The essential physics was unchanged from the usual treatment in an open system, except that a minimum in the free energy of formation is found which corresponds to the unique equilibrium phase separated state whose symmetry, in the absence of an external field, is spherical. 

The Laplace equation in this form is general and applies equally well to geometrical bodies whose radii of curvature are constant over the entire surface to more intricate shapes for which the Rs, are a function of surface position. In the instance of constant radii of curvature across the surface, Eq. (2.68) reduces for several common cases. For spherical surfaces, R = R2 = R, where R is the radius of the sphere, and Eq. (2.68) becomes ... 

In order to understand and interpret the many details that can be observed, it is necessary to examine how the metal atoms can arrange themselves on a spherical surface. The best way to do this is to make a model of a body-centered cubic crystal, such as tungsten, whose surface is as close to a mathematical sphere as the size of its atoms permits. We have constructed such a model in which marbles represent tungsten atoms. The radius of curvature of the model is 25 atom (or marble) diameters. This is 40 to 100 times smaller than the metal points used in the microscope but does not change any of the essential features which we wish to bring out. 

Here r and ft are related to the radius of curvature of the spherical surface, R, by the relation... 

Because of the proximity effect of surface diffusion, the flux from the regions adjacent to the neck leaves an undercut region in the neck vicinity.7 Diffusion along the uniformly curved spherical surfaces is small because curvature gradients are small and therefore the undercut neck region fills in slowly. This undercutting is illustrated in Fig. 16.3a. Because mass is conserved, the undercut volume is equal to the overcut volume. Conservation of volume provides an approximate relation between the radius of curvature, p, and the neck radius, x ... 

A grating ruled on a spherical surface combines the properties of the diffraction grating with the focusing ability of the optical surface. Such a device, with radius of curvature R, focuses spectra as images of the entrance (primaiy) slit on the circumference of a circle of diameter R, when the entrance slit is also located on the circumference of the circle.z... 

Two cases of sintering Transport of material from the spherical surfaces to the neck (top) does not contribute to densifi-cation. Transport of material from the interface between the particles to the neck (bottom) does contribute to densification. p is the neck s radius of curvature, r is the particle radius, 2h is the decrease of distance between particle centers, and x is the radius of contact. 

As a consequence of surface tension, there is a balancing pressure difference across any curved surface, the pressure being greater on the concave side. For a curved surface with principal radii of curvature rj and r2 this pressure difference is given by the Young-Laplace equation, Ap = y(llrx + l/r2), which reduces to Ap = 2y/r for a spherical surface. 

The form of the coherence function of a near-field imaging array approaches the far-field planar array limit when R0 -> 00 because at very large radii of curvature, the surface of the spherical array now becomes flat. When Ra —r 00, exp(/<5) = exp[ikz (cos2 — cosazimuthal angles are virtually zero. As Z0 00, exp(/<5) =... 

If instead of a spherical surface, any interface having two principal radii of curvature r, and r2 is considered, then... 

Ostwald (1900), using the thermodynamic relationship between vapor pressure of spherical drops and the curvature of surface, derived an equation giving the relation between solubility and particle size. Ostwald s equation was later modified by Freundlich (1909), and is as follows ... 

Curvature of field is an off-axis aberration. It occurs because the focal plane of an image is not flat but has a concave spherical surface, as shown in Figure 1.10. This aberration is especially troublesome with a high magnification lens with a short focal length. It may cause unsatisfactory photography. 

This entry is organized in the following paragraphs First, the advanced determination of van der Waals interaction between spherical particles is described. Second, the relevant approximate expressions and direct numerical solutions for the double-layer interaction between spherical surfaces are reviewed. Third, the experimental data obtained for AFM tips having nano-sized radii of curvature and the DLVO forces predicted by the Derjaguin approximation and improved predictions are compared. Finally, a summary of the review and recommended equations for determining the DLVO interaction force and energy between colloid and nano-sized particles is included. 

This is the Young-Laplace equation applied to a spherical surface. A more general form of this equation is used when the curvature of the interface is not spherical [Gl]. 

In this equation k is the linear parameter related to geometry and determined by the curvature of surfaces in contact. For two spherical particles of different radii, r and r", k = 2r r" (r + /) for two cylindrical surfaces positioned at a right angle with respect to each other, k = 2 (r r")m. 

The generating mechanism of the spherical surface ground with cup wheels was first introduced in 1920 by W. Taylor, an English scholar. As shown in Figure 6.17, the workpiece is mounted on a work spindle, and the inclination angle a between the axis of rotation of the workpiece and that of the wheel spindle is properly adjusted. Theoretically, the radius of curvature of the lenses that are produced can be calculated using the following equations ... 

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