Curvature spherical

There are a number of relatively simple experiments with soap films that illustrate beautifully some of the implications of the Young-Laplace equation. Two of these have already been mentioned. Neglecting gravitational effects, a film stretched across a frame as in Fig. II-1 will be planar because the pressure is the same as both sides of the film. The experiment depicted in Fig. II-2 illustrates the relation between the pressure inside a spherical soap bubble and its radius of curvature by attaching a manometer, AP could be measured directly. 

The exact treatment of capillary rise must take into account the deviation of the meniscus from sphericity, that is, the curvature must correspond to the AP = Ap gy at each point on the meniscus, where y is the elevation of that point above the flat liquid surface. The formal statement of the condition is obtained by writing the Young-Laplace equation for a general point (x, y) on the meniscus, with R and R2 replaced by the expressions from analytical geometry given in... 

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... 

In a cylindrical pore the meniscus will be spherical in form, so that the two radii of curvature are equal to one another and therefore to r (Equation (3.8)). From simple geometry (Fig. 3.8) the radius r of the core is related to r by the equation... 

Fig. 3. Two-dimensional schematic illustrating the distribution of Hquid between the Plateau borders and the films separating three adjacent gas bubbles. The radius of curvature r of the interface at the Plateau border depends on the Hquid content and the competition between surface tension and interfacial forces, (a) Flat films and highly curved borders occur for dry foams with strong interfacial forces, (b) Nearly spherical bubbles occur for wet foams where...

Implicit in all these solutions is the fact that, when two spherical indentors are made to approach one another, the resulting deformed surface is also spherical and is intermediate in curvature between the shape of the two surfaces. Hertz [27] recognized this concept and used it in the development of his theory, yet the concept is a natural consequence of the superposition method based on Boussinesq and Cerutti s formalisms for integration of points loads. A corollary to this concept is that the displacements are additive so that the compliances can be added for materials of differing elastic properties producing the following expressions common to many solutions... 

A similar analysis can be done for the curved surface of an essentially spherical particle that contains asperities. Let us assume that all the asperities are the same size. Initially, no more than three asperities on the particle can contact the presumably smooth surface. As the asperities compress under the applied load, more asperities, that are situated further away from the substrate due to the curvature of the particle s surface, come into contact. These are the first to separate from the substrate upon application of a detachment force. In essence, detachment occurs by breaking the contacts between the asperities and the contacting surface, one at a time. 

For a spherical mirror the conjugate points lie on top of each other at the center of curvature (COC). Light emitted from the COC of the spherical mirror will focus back onto the COC without any additional system aberration. 

In many cases the potential P(z) is small compared with the surface energy of the liquid and the droplet shape is very close to a spherical cap. If the height e and the radius of curvature R at the top of the droplets can be measured, an effective contact angle can be defined through the expression ... 

The present review intends to cover the work reported on nonlinear optics at liquid-liquid interface since the first report of S. G. Grubb et al. [18]. The theoretical aspects of nonlinear optics are first introduced in Section II. The experimental results covering the molecular structure of liquid interfaces are presented in Section III, followed by a section devoted to the dynamics and the reactivity at these interfaces. Section V focuses on new aspects where spherical interfaces with radii of curvature of the order of the wavelength of light are investigated. Section VI presents the field of SFG. 

In the theoretical section above, the nonlinear polarization induced by the fundamental wave incident on a planar interface for a system made of two centrosymmetrical materials in contact was described. However, if one considers small spheres of a centrosymmetrical material embedded in another centrosymmetrical material, like bubbles of a liquid in another liquid, the nonlinear polarization at the interface of a single sphere is a spherical sheet instead of the planar one obtained at planar surfaces. When the radius of curvature is much smaller than the wavelength of light, the electric field amplitude of the fundamental electromagnetic wave can be taken as constant over the whole sphere (see Fig. 7). Hence, one can always find for any infinitely small surface element of the surface... 

The four maxima and the saddle point are critical points in the function L(r) analogous to the maxima and saddle points in p(r) discussed in Chapter 6. Every point on the sphere of maximum charge concentration of a spherical atom is a maximum in only one direction, namely, the radial direction. In any direction in a plane tangent to the sphere, the function L does not change therefore the corresponding curvatures are zero. When an atom is part of... 

Comparing this expression with eq. (6.27), we see that yv/hv for each crystal face represents cr divided by the radius of curvature for an isotropic spherical phase. As a first approximation we may replace yv/hv with y/r for near-spherical crystals. In this case /represents an average surface energy of all possible crystal faces. 

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