Geometry ellipsoidal

Interfacial polarization in biphasic dielectrics was first described by Maxwell (same Maxwell as the Maxwell model) in his monograph Electricity and Magnetism of 1892.12 Somewhat later the effect was described by Wagner in terms of the polarization of a two-layer dielectric in a capacitor and showed that the polarization of isolated spheres was similar. Other more complex geometries (ellipsoids, rods) were considered by Sillars as a result, interfacial polarization is often called the Maxwell-Wagner-Sillars (MWS) effect. 

The early Hartley model [2, 3] of a spherical micellar stmcture resulted, in later years, in some considerable debate. The self-consistency (inconsistency) of spherical symmetry witli molecular packing constraints was subsequently noted [4, 5 and 6]. There is now no serious question of tlie tenet tliat unswollen micelles may readily deviate from spherical geometry, and ellipsoidal geometries are now commonly reported. Many micelles are essentially spherical, however, as deduced from many light and neutron scattering studies. Even ellipsoidal objects will appear... 

A drawback of the SCRF method is its use of a spherical cavity molecules are rarely exac spherical in shape. However, a spherical representation can be a reasonable first apprc mation to the shape of many molecules. It is also possible to use an ellipsoidal cavity t may be a more appropriate shape for some molecules. For both the spherical and ellipsoi cavities analytical expressions for the first and second derivatives of the energy can derived, so enabling geometry optimisations to be performed efficiently. For these cavil it is necessary to define their size. In the case of a spherical cavity a value for the rad can be calculated from the molecular volume ... 

The spherical geometry assumed in the Stokes and Einstein derivations gives the highly symmetrical boundary conditions favored by theoreticians. For ellipsoids of revolution having an axial ratio a/b, friction factors have been derived by F. Perrin, and the coefficient of the first-order term in Eq. (9.9) has been derived by Simha. In both cases the calculated quantities increase as the axial ratio increases above unity. For spheres, a/b = 1. 

External-pressure failure of shells can result from overstress at one extreme or n om elastic instability at the other or at some intermediate loading. The code provides the solution for most shells by using a number of charts. One chart is used for cylinders where the shell diameter-to-thickness ratio and the length-to-diameter ratio are the variables. The rest of the charts depic t curves relating the geometry of cyhnders and spheres to allowable stress by cui ves which are determined from the modulus of elasticity, tangent modulus, and yield strength at temperatures for various materials or classes of materials. The text of this subsection explains how the allowable stress is determined from the charts for cylinders, spheres, and hemispherical, ellipsoidal, torispherical, and conical heads. 

Einstein coefficient b, in (5) for viscosity 2.5 by a value dependent on the ratio between the lengths of the axes of ellipsoids. However, for the flows of different geometry (for example, uniaxial extension) the situation is rather complicated. Due to different orientation of ellipsoids upon shear and other geometrical schemes of flow, the correspondence between the viscosity changed at shear and behavior of dispersions at stressed states of other types is completely lost. Indeed, due to anisotropy of dispersion properties of anisodiametrical particles, the viscosity ceases to be a scalar property of the material and must be treated as a tensor quantity. 

The latter allows us to determine the potential of the gravitational field outside the ellipsoid of rotation, including points of its surface, provided that the mass, geometry, and angular frequency are given. Making use of Equation (2.148), we can represent the potential of the gravitational field in the form kM 1 1 2 2 SiC/s) - 1... 

Fig. 15. Thermal ellipsoidal plot80 of r-Bu2GaP(Mes )SiPh, showing the flattened nature of phosphorus geometry and the alignment between the gallium p-orbital and the phosphorous lone pair. [Reprinted with permission from Petrie, Ruhlandt-Senge, and Power.79 Copyright 1992 American Chemical Society.]...

Fig. 22. Molecular geometry and thermal ellipsoids (75% probability) of D(+)-tartaric acid at 295, 160, 105, and 35 K.

Another complication is the demagnetization correction due to the geometry of the specimen. Demagnetization (or the equivalent depolarization problem for dielectric bodies in an electric field) can only be solved analytically for an ellipsoid of revolution (27X28). When He is applied parallel to one of the three axes of revolution, the magnetization is parallel to He, but the internal field H is given by (29) ... 

Figure 29 The 50% probability thermal ellipsoids of Co4(CO) 2Sb4. The molecule of idealized T -Aint geometry has...

This method relies on the exact solution of the elastic problem for an inclusion of known geometry (an ellipjsoid) surrounded by an infinite matrix. The composite problem to be solved is that in which the included phases are ellipsoidal in shap>e. Selecting one as the reference ellipwoid, the effect of the remainder is approximated by a continuum surrounding the reference ellip>soid, thus reducing the problem to one for which there is an... 

Any exact theory, unless the geometry is simple, involves hopelessly complicated calculations of stress distributions even if the elements are large enough for these to be valid (which is not the case for small assemblies of polymer chains). In principle (see e.g. Chen and Young91 ) any geometry may be treated, but ellipsoids and parallelepipeds are the most usual. 

Alternatively, since molecules often adopt ellipsoidal geometries both in the unfolded and the folded states, their shapes may be more accurately represented by the axial ratio (a/b) parameter (estimated from f using SEDNTERP), where a and b are the major and minor axes of the ellipsoids, respectively. However, in AUC studies on cation-mediated RNA conformational changes, both the RH and the axial ratio (a/b) have been found to decrease in an identical manner, as the RNA molecules become compactly folded with increasing concentration of divalent or monovalent ions (Takamoto et al, 2002). Therefore, SV experiments report, in multiple ways, on the equilibrium changes in the global dimensions of RNA molecules as they fold from the ensemble of unfolded states to the native state. 

X-ray crystallographic studies (Kannen et al., 1975) of the human enzyme have established that the molecule is roughly ellipsoidal (40 A x 45 A x 55 A) with the zinc lying near the bottom of a deep cleft near the centre of the molecule. The zinc is ligated by three histidine residues (His-117, His-93 and His-95) in a distorted tetrahedral geometry, with the fourth site occupied by a water molecule. 

---