Anisotropic Sphere

Another important application of perturbation theory is to molecules with anisotropic interactions. Examples are dipolar hard spheres, in which the anisotropy is due to the polarity of tlie molecule, and liquid crystals in which the anisotropy is due also to the shape of the molecules. The use of an anisotropic reference system is more natural in accounting for molecular shape, but presents difficulties. Hence, we will consider only... 

In addition to tire standard model systems described above, more exotic particles have been prepared witli certain unusual properties, of which we will mention a few. For instance, using seeded growtli teclmiques, particles have been developed witli a silica shell which surrounds a core of a different composition, such as particles witli magnetic [12], fluorescent [13] or gold cores [14]. Anotlier example is tliat of spheres of polytetrafluoroetliylene (PTFE), which are optically anisotropic because tire core is crystalline [15]. 

It is now well-established that for atomic fluids, far from the critical point, the atomic organisation is dictated by the repulsive forces while the longer range attractive forces serve to maintain the high density [34]. The investigation of systems of hard spheres can therefore be used as simple models for atomic systems they also serve as a basis for a thermodynamic perturbation analysis to introduce the attractive forces in a van der Waals-like approach [35]. In consequence it is to be expected that the anisotropic repulsive forces would be responsible for the structure of liquid crystal phases and numerous simulation studies of hard objects have been undertaken to explore this possibility [36]. 

There have been several simulations of discotic liquid crystals based on hard ellipsoids [41], infinitely thin platelets [59, 60] and cut-spheres [40]. The Gay-Berne potential model was then used to simulate the behaviour of discotic systems by Emerson et al. [16] in order to introduce anisotropic attractive forces. In this model the scaled and shifted separation R (see Eq. 5) was given by... 

History. Wilke [129] considers the case that different orders of a reflection are observed and that the orientation distribution can be analytically described by a Gaussian on the orientation sphere. He shows how the apparent increase of the integral breadth with the order of the reflection can be used to separate misorientation effects from size effects. Ruland [30-34] generalizes this concept. He considers various analytical orientation distribution functions [9,84,124] and deduces that the method can be used if only a single reflection is sufficiently extended in radial direction, as is frequently the case with the streak-shaped reflections of the anisotropic... 

In the previous discussion, the electron-nucleus spin system was assumed to be rigidly held within a molecule isotropically rotating in solution. If the molecule cannot be treated as a rigid sphere, its motion is in general anisotropic, and three or five different reorientational correlation times have to be considered 79). Furthermore, it was calculated that free rotation of water protons about the metal ion-oxygen bond decreases the proton relaxation time in aqua ions of about 20% 79). A general treatment for considering the presence of internal motions faster than the reorientational correlation time of the whole molecule is the Lipari Szabo model free treatment 80). Relaxation is calculated as the sum of two terms 8J), of the type... 

Relationship Between Nodular and Rejecting Layers. Nodular formation was conceived by Maler and Scheuerman (14) and was shown to exist in the skin structure of anisotropic cellulose acetate membranes by Schultz and Asunmaa ( ), who ion etched the skin to discover an assembly of close-packed, 188 A in diameter spheres. Resting (15) has identified this kind of micellar structure in dry cellulose ester reverse osmosis membranes, and Panar, et al. (16) has identified their existence in the polyamide derivatives. Our work has shown that nodules exist in most polymeric membranes cast into a nonsolvent bath, where gelation at the interface is caused by initial depletion of solvent, as shown in Case B, which follows restricted Inward contraction of the interfacial zone. This leads to a dispersed phase of micelles within a continuous phase (designated as "polymer-poor phase") composed of a mixture of solvents, coagulant, and a dissolved fraction of the polymer. The formation of such a skin is delineated in the scheme shown in Figure 11. 

Generally, the thermal interaction of spheres that can occur at high volume fractions is negligible, such that the simpler Eq. (4.48) is often used. For nonspherical reinforcement, such as aligned, continuous hbers, the thermal conductivity is anisotropic—that is, not the same in all directions. Rayleigh showed that the effective thermal conductivity along the direction of the fiber axis is... 

It is interesting to compare and contrast an isotropic ellipsoid and an anisotropic sphere the polarizability of both particles is a tensor, the principal values of which are... 

Turner (1973) and McKellar (1976) applied RG theory to ensembles of randomly oriented particles of arbitrary shape the former author included spheres with anisotropic optical constants. Optically active particles have been treated within the framework of the RG approximation by Bohren (1977). 

In this chapter we consider theories of scattering by particles that are either inhomogeneous, anisotropic, or nonspherical. No attempt will be made to be comprehensive our choice of examples is guided solely by personal taste. First we consider a special example of inhomogeneity, a layered sphere. Then we briefly discuss anisotropic spheres, including an exactly soluble problem. Isotropic optically active particles, ones with mirror asymmetry, are then considered. Cylindrical particles are not uncommon in nature—spider webs, viruses, various fibers—and we therefore devote considerable space to scattering by a right circular cylinder. 

We have discussed intrinsically anisotropic particles—ones with anisotropy originating in their optical constants rather than their shape—in previous chapters. In Section 5.6 we gave the solution to the problem of scattering by an anisotropic sphere in the Rayleigh approximation. From the results of that section and Section 5.5 it follows that the average cross section (C) (scattering or absorption) of a collection of randomly oriented, sufficiently small, anisotropic spheres is... 

The reason for the intractability of the anisotropic sphere scattering problem is the fundamental mismatch between the symmetry of the optical constants and the shape of the particle. For example, the vector wave equation for a uniaxial material is separable in cylindrical coordinates that is, the solutions to the field equations are cylindrical waves. But the bounding surface of the... 

A special anisotropic particle scattering problem has been treated by Roth and Dignam (1973), who considered an isotropic sphere coated with a uniform film with constitutive relations... 

Scattering media to which this matrix applies include randomly oriented anisotropic spheres of substances such as calcite or crystalline quartz (uniaxial) or olivine (biaxial). Also included are isotropic cylinders and ellipsoids of substances such as glass and cubic crystals. An example of an exactly soluble system to which (13.21) applies is scattering by randomly oriented isotropic spheroids (Asano and Sato, 1980). Elements of (13.21) off the block diagonal vanish. Some degree of alignment is implied, therefore, if these matrix elements... 

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