Ellipsoids randomly oriented

Figure 12.6 Calculated absorption spectra of aluminum spheres, randomly oriented ellipsoids (geometrical factors 0.01, 0.3, and 0.69), and a continuous distribution of ellipsoidal shapes (CDE). Below this is the real part of the Drude dielectric function.

There are several interesting observations that can be made about (12.32). Integrated absorption is independent of the damping constant y the only bulk parameter that affects it is the plasma frequency. If the particles are in air, then integrated absorption is independent of the shape this is true not only for a single oriented ellipsoid but also for a collection of randomly oriented ellipsoids. It is instructive to rewrite (12.32) using (12.29) ... 

The average cross section of identical, but randomly oriented ellipsoids will, in general, exhibit three peaks in the frequency range between co, and o)t. An example of this is given in Fig. 12.7, where Cabs for a silicon carbide ellipsoid is shown as a function of frequency. 

The average absorption cross section of a randomly oriented collection of identical homogeneous ellipsoids (12.30) may be written... 

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... 

A necessary condition for the correctness of the multiple-scattering explanation of the observed circular polarization is that scattering by noctilucent cloud particles does not appreciably reduce the degree of circular polarization of the incident light. That this is so for randomly oriented Rayleigh ellipsoids is readily shown. M in (5.52) is nearly unity for ice ellipsoids, so to good approximation... 

This critical field called coercivity ff. or switching field Ff., is also equal to FF. If a field is applied in between 0 and 90° the coercivity varies from maximum to zero. In the case of this special example the applied field Ha = Hs = Hc = Hk. Based on the classical theory, Stoner-Wohlfarth (33) considered the rotation unison for noninteracted, randomly oriented, elongated particles. The anisotropic axis can be due to the shape anisotropy (depending on the size and shape of the particle) or to the crystalline anisotropy. In the prolate ellipsoids b is the short axis and a the longest axis. The demagnetizing factors are IV (in the easy direction) and The demagnetizing fields can then be calculated by Hda = — Na Ms, and Hdb = — Nb Ms. The shape anisotropy field is Hd = (Na — Nb)Ms. Then the switching field Hs = Hd = (Na — Nb)Ms. 

In addition to translational Brownian motion, suspended molecules or particles undergo random rotational motion about their axes, so that, in the absence of aligning forces, they are in a state of random orientation. Rotary diffusion coefficients can be defined (ellipsoids of revolution have two such coefficients representing rotation about each principal axis) which depend on the size and shape of the molecules or particles in question28. 

Particle asymmetry has a marked effect on viscosity and a number of complex expressions relating intrinsic viscosity (usually extrapolated to zero velocity gradient to eliminate the effect of orientation) to axial ratio for rods, ellipsoids, flexible chains, etc., have been proposed. For randomly orientated, rigid, elongated particles, the intrinsic viscosity is approximately proportional to the square of the axial ratio. 

Suppose that a tetrahedral molecule such as CCI4 is irradiated by plane polarized light (E ). Then, the induced dipole (Section 1.7) also oscillates in the same yz-plane. If the molecule is performing the totally symmetric vibration, the polarizability ellipsoid is always sphere-like namely, the molecule is polarized equally in every direction. Under such a circumstance, I (Iy) = 0 since the oscillating dipole emitting the radiation is confined to the xz-plane. Thus, pp = 0. Such a vibration is called polarized (abbreviated as p). In liquids and solutions, molecules take random orientations. Yet this conclusion holds since the polarizability ellipsoid is spherical throughout the totally symmetric vibration. 

There are a number of models for polarization of heterogeneous systems, many of which are reviewed by van Beek (23). Brown has derived an exact, though unwieldly, series solution using point probability functions (24). For comparison to spectra for the thermoplastic elastomers of interest here, the most useful model seems to be the one derived by Sillars (25) and, in a slightly different form, by Fricke (26). The model assumes a distribution of geometrically similar ellipsoids with major radii, r-p and rj which are randomly oriented and randomly distributed in a dissimilar matrix phase. Only non-specific interactions between neighboring ellipsoids are included in the model. This model includes no contribution from the polarization of mobile charge carriers trapped on the interfacial surfaces. 

Abbreviations EPR - electron paramagnetic resonance HE -high-energy IPS-isopentane solution LB-Langmuir-Blodgett LE - low-energy MCD - microcrystals dispersed in a KBr disk ORTEP - Oak Ridge thermal ellipsoid plot ROC - randomly oriented crystals SC-spin-coated SCE-spin-coated film... 

The corresponding problem for an assembly of randomly oriented ellipsoids is somewhat more difficult and no simple closed expression has been found for P( ). For discussions see Guinier (1939) and Roess and Shull (1947). The first three terms of the solution have been derived... 

In the statistic route, an effechve dielectric function e(v) is calculated from the dielectric funchon of the metal Enie(v) and the of polymer material po(v) by using a formula, the effective medium theory. The most general effective medium theory is the Bergman theory in which the nanostructure of the composite material can be considered by a spectral density function. The Bergman theory includes the soluhons from the Bruggeman theory and the Maxwell Garnett theory for spherical, parallel-oriented, and random-oriented ellipsoidal parhcles. 

For the case where the particles do not have a spherical shape, various extensions of the Maxwell Garnett theory for nonspherical particles were introduced. The particles are spheroidal with the same shape (ratio of major-axis A and minor axis B), but with different sizes still in the wavelength limit. It remains only to choose between a parallel or a random orientation of the mean axis of the ellipsoids. 

The extensions of the Maxwell Garnett theory for random-oriented ellipsoids, equation (14) [25], needs three depolarization factors Li, Li, L3 with 2L, = 1 to describe the embedded ellipsoids. Frequently, ellipsoids with a symmetrical axis of rotation are assumed with Li = L3. Extreme geometries are rods with Li L2 = L3 and disks with Li L2 = L3. For Li =y, the extensions of the Maxwell Garnett theory for random-oriented ellipsoids, equation (14), give the same result as the Maxwell Garnett theory, equation (12). 

There is also another extension of the Maxwell Garnett theory for random-oriented ellipsoids [29] which is different from equation (14). Other extensions of the Maxwell Garnett theory are described for chiral aggregates of spheres [26], for thin hlms with columnar structures [27], and for embedded spherical particles of several metals [28],... 

Estimation of effective elastic moduli of nanocomposites was performed by the version of effective field method developed in the framework of qirasi-crystalline approximation when the spatial correlatiorts of inclrrsion location take particirlar ellipsoidal forms [71]. The independent justified choice of shapes of inclirsiorts arrd correlation holes provide the formttlae of effective modirli which are syrrrmetric, corrrpletely explicit, and easily to use. The parametric numerical analyses revealed the most sensitive parameters influencing the effective moduli which are defined by the axial elastic moduli of nanofibers rather than their transversal moduli as well as by the justified choice of correlation holes, concentration, and prescribed random orientation of nanofibers [72]. 

The limitation when modeling erythrocytes using spheroids was considered by Gheorghiu (1999). He found that the ellipsoidal model is fairly good when analyzing the impedance spectra of randomly oriented erythrocjrtes but rather poor whenever oriented erythrocytes are considered. For the latter, microscopic models coping with the actual shape should be considered. 

The quantities g and gc describe the shapes of the inclusion and the Lorentz cavity, respectively. For the general case of randomly oriented ellipsoids, Polder... 

For highly diluted randomly oriented ellipsoids, van de Hulst [233] obtained the following expressions for the absorption and scattering cross sections ... 

The Fricke Model for Two-Phase Dispersions. Expressions similar to those of Maxwell have been derived for ellipsoidal particles of random orientation (Fricke [1932]) and for aligned ellipsoidal particles (Fricke [1953]). The expressions contain form factors which depend on the axial ratio of the ellipsoids and their orientation with respect to the electric field. The case of random orientation is the most interesting, as it describes a realistic ceramic microstructure and results in the following equation ... 

Figure 4.1.10. Simulated impedance and modulus spectra for a two-phase microstructure comprising a matrix of phase 1 with 25% by volume of randomly oriented ellipsoids of j ase 2.

Figure 4.1.11. Two equivalent circuits, for a matrix of phase 1 containing randomly oriented ellipsoids of phase 2 according to model proposed by Fricke [1953] (a) parallel circuit (b) series circuit.

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