Ellipsoid in the Electrostatics Approximation

The most general smooth particle—one without edges or comers—of regular shape is an ellipsoid with semiaxes a b c (Fig. 5.5), the surface of which is specified by 

The natural coordinates, albeit unfamiliar and not without their disagreeable features, for formulating the problem of determining the dipole moment of an ellipsoidal particle induced by a uniform electrostatic field are the ellipsoidal coordinates ( , rj, f) defined by 

The surfaces = constant are confocal ellipsoids, and the particular ellipsoid = 0 coincides with the boundary of the particle. The surfaces rj = constant 

This ambiguity may be removed by introducing the Weierstrassian elliptic function (Jones, 1964, p. 32). Fortunately, such a drastic step is not necessary in the problem at hand, a homogeneous ellipsoid in a uniform electrostatic field aligned along the z axis. In this instance the potential has the symmetry properties 

Let us consider the octant in which x, y, z are positive. We denote by 3 t the potential inside the particle outside the particle the potential 02 may be written as the superposition of the potential 3 0 of the external field 

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Notable progress in analyzing nonspherical particles has been made by Fuchs (1975), who calculated absorption by cubes in the electrostatics approximation and applied the results to experimental data for MgO and NaCl. We shall discuss Fuchs s results at the end of Section 12.3. Langbein (1976) also did calculations for rectangular parallelepipeds, including cubes, which give valuable insights into nonspherical shape effects. Because the cube is a common shape of microcrystals, such as MgO and the alkali halides, these theoretical predictions have been used several times to interpret experimental data. We shall do the same for MgO. Our theoretical treatment of nonsphericity, however, is based on ellipsoids. Despite its simplicity, this method predicts correctly many of the nonspherical effects. 

Dipole moment of each BNN particle is calctrlated using Claussis-Mossotti equations. Clausius-Mossotti approximation is one of the most commonly used equations for calculating the bulk dielectric properties of inhomogeneous materials (Ohad and David, 1997). It is useful when one of the components can be considered as a host in which, inclusions of the other components are embedded. It involves an exact calculation of the field induced in the tmiform host by a single spherical or ellipsoidal inclusion and an approximate treatment of its distortion by the electrostatic interaction between the different inclusiorts. The Clausius-Mossotti equation itself does not consider any interaction between filler and matrix. This approach has been extensively used for studying the properties of two-component mixtures in which both the host and the inclusions possess different dielectric properties. In recent years, this approximation has been extensively applied to composites involving ceramics and polymers. 

The computation was improved by Westheimer and Kirkwood, who assumed a dielectric constant of 2.0 within the molecule. By approximating the molecule as an ellipsoid of revolution, they were able to make reasonably accurate calculations of electrostatic effects on pKa values.15 Thus, for malonic acid Westheimer and Shookhoff16 predicted r = 0.41 nm for malonic acid dianion. Recently more sophisticated calculations17 have been used to predict pKa values for the compounds in Table 7-1 and others.18... 

Mg. 2. The experimental and theoretically predicted dependence of electrooptic coefficient for the FTC chromophore (see Pig. 1) in poly(methyl methacrylate) (PIVOVIA) upon chromophore number density (concentration in PMMA) is shown. Theoretical results are shown for various shape approximations and for the neglect of intermolecular electrostatic interactions (the ideal gas model)...Gas model — Sphere —Prolate ellipsoid ... 

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