Inhomogeneous Particles

The transition matrix of an uniaxial anisotropic particle then becomes 

The expressions of the elements of the Qaiis matrix are similar but with M and 7V in place of M and iV, respectively. Using the properties of the vector quasi-spherical wave functions (cf. (1.47)) it is simple to show that for Siz = Si, the present approach leads to the T-matrix solution of an isotropic particle. 

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For homogeneous particles, it represents the number of distances within the particle. For inhomogeneous particles, it has to take into account the different electron density of the volume elements. Thus it represents the number of pairs of difference in electrons separated by the distance r. A qualitative description of shape and internal structure of the... 

A discussion of some theoretical approaches to scattering by randomly inhomogeneous particles is followed in the final section by an outline of recent progress in constructing solutions to problems of scattering by nonspherical particles, including those of arbitrary shape. 

The classical method of solving scattering problems, separation of variables, has been applied previously in this book to a homogeneous sphere, a coated sphere (a simple example of an inhomogeneous particle), and an infinite right circular cylinder. It is applicable to particles with boundaries coinciding with coordinate surfaces of coordinate systems in which the wave equation is separable. By this method Asano and Yamamoto (1975) obtained an exact solution to the problem of scattering by an arbitrary spheroid (prolate or oblate) and numerical results have been obtained for spheroids of various shape, orientation, and refractive index (Asano, 1979 Asano and Sato, 1980). 

Further work on the theory of backscattering by inhomogeneous particles and its comparison with measurements has been done by Bohren and Battan (1981). 

The obvious advantage of the microwave experiment is that oriented single particles of arbitrary shape and, within limits, arbitrary refractive index, can be studied easily. Multilayered and other inhomogeneous particles pose no particular problems. 

Yeh, C., and K. K. Mei, 1980. On the scattering from arbitrarily shaped inhomogeneous particles—exact solutions, in Light Scattering by Irregularly Shaped Particles, D. Schuerman (Ed.), Plenum, New York, pp. 201-206. 

We begin with the simplest case. A vacancy flux j° (driven, for example, by inhomogeneous particle radiation) flows across a multicomponent crystal (k = 1,2,.., n) and the component fluxes are restricted to one sublattice. We assume no other coupling between the fluxes except the lattice site conservation, which means that we neglect cross terms in the formulation of SE fluxes. (An example of coupling by cross terms is analyzed in Section 8.4.) The steady state condition requires then that the velocities of all the components are the same, independent of which frame of reference has been chosen, that is,... 

Inhomogeneous particles. For particles composed of a matrix and inclusions one approach for calculating optical properties is to assume an average dielectric coefficient (e) for the composed particle. A number of so-called mixing rules have been proposed a frequently used one is the Maxwell-Gamett mixing rule (cf. Bohren Huffman 1983). For a matrix with dielectric coefficient em, and a number of different kinds of inclusions with dielectric coefficients ej and volume fractions f) one uses... 

Depolarized scattering occurs because of various forms of particle anisotropy. Distinct classes of depolarizing scatterers include nonspherical particles with uniform isotropic (scalar) polarizabilities (sometimes called form anisotropy), inhomogeneous particles with nonuniform distributions of isotropic polarizability, and particles with anisotropic (tensor) polarizabilities. For each of these classes, the intensity of depolarized light scattered by a particle will change as the particle translates, rotates, or manifests internal rearrangement of its scattering elements. DDLS can provide information on the dynamics of each of these processes. 

Other errors, which could influence the results obtained, are, for example, wall effects ( slipping ), the dissipation of heat, and the increase in temperature due to shear. In a tube, the viscosity of a flowing medium is less near the tube walls compared to the center. This is due to the occurrence of shear stress and wall friction and has to be minimized by the correct choice of the tube diameter. In most cases, an increase in tube diameter reduces the influence of wall slip on the flow rate measured, but for Newtonian materials of low viscosity, a large tube diameter could be the cause of turbulent flow. ° When investigating suspensions with tube viscometers, constrictions can lead to inhomogeneous particle distributions and blockage. Due to the influence of temperature on viscosity (see Section Influence Factors on the Viscosity ), heat dissipated must be removed instantaneously, and temperature increase due to shear must be prevented under all circumstances. This is mainly a constructional problem of rheometers. Technically, the problem is easier to control in tube rheometers than in rotating instruments, in particular, the concentric cylinder viscometers. ... 

In the case of inhomogeneous particles, contrast variation techniques can be used to separate the shape and density fluctuation terms. D r) is now proportional to the number of pairs of scattering lengths separated by the distance r in the volume elements p and q, where each value of r is weighted by the product fp -f. Depending on the solvent scattering density, some volume elements can make a negative contribution to 7>(r). As previously, the contrast dependence of 7)(r) is readily derived in terms of three basic functions ... 

Kolokolova L, Gustafsonm BAS Scattering by inhomogeneous particles microwave analog experiments and comparison to eiiective medium theories, J Quant Spectrosc Radiat Transf 70(4-6) 611-625, 2001. 

Wall Shear Stress Measurements, Fig. 3 Schematic of in-plane velocity measurement. The inhomogeneous particle distribution and defocused particle images with different velocities due to out-of-plane gradients bias the results... 

Key words Emulsion polymerization - contrast variation - small-angle X-ray scattering - inhomogeneous particles... 

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