Model effective medium

The optical properties of heterogeneous materials are of considerable interest in atmospheric science, astronomy and optical particle sizing. Random dispersions of inclusions in a homogeneous host medium can be equivalently considered as effectively homogeneous after using various homogenization formalisms [36,210]. The effective permittivity of a heterogeneous material relates 

The main issue in the effective-medium approximations is to relate the polarizability a. to the relative permittivity r, where = i/ s, and E[ is the relative permittivity of the inclusion. For spherical particles and at frequencies for which the inclusions can be considered very small, the Maxwell-Garnett formula uses the relation 

Other contributions which are related to the present analysis includes the work of Foldy [69], Waterman and Truell [258], Fikioris and Waterman [65], Twerski [230,231], Tsang and Kong [223,224] and Tsang et al. [227]. The case of a plane electromagnetic wave obliquely incident on a half-space with densely distributed particles and the computation of the incoherent field with the distorted Born approximation have been considered Tsang and Kong [225, 226]. 

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Johnson, EM Berk, DA Jain, RK Deen, WM, Hindered Diffusion in Agarose Gels Test of Effective Medium Model, Biophysical Journal 70,1017, 1996. 

Figure 3.68 Total reduction charge (+ ) for the outer film and water content ( ) of the outer film (as estimated from the effective medium model and the data at 633 nm (see text for details). Reprinted from Corrosion Science, 28, P. Southworth, A. Hamnett, A.M. Riley and J.M. Sykes, An Ellipsometric and RRDE Study of Iron Passivation and Depassivation in Carbonate BufTer, pp. 1139-1161 (1988), with kind permission from Pergamon Press Ltd., Headington Hill Hall,...

Niklasson, G. A., C. G. Granqvist, and O. Hunderi, 1981. Effective medium models for the optical properties of inhomogeneous media, Appl. Opt., 20, 26-30. 

Two models are available for interpreting attenuation spectra as a PSD in suspensions with chemically distinct, dispersed phases using the extended coupled phase theory.68 Both models assume that the attenuation spectrum of a mixture is composed of a superposition of component spectra. In the multiphase model, the PSD is represented as the sum of two log-normal distributions with the same standard deviation, that is, a bimodal distribution. The appearance of multiple solutions is avoided by setting a common standard deviation to the mean size of each distribution. This may be a poor assumption for the PSD (see section 11.3.2). The effective medium model assumes that only one target phase of a multidisperse system needs to be determined, while all other phases contribute to a homogeneous system, the so-called effective medium. Although not complicated by the possibility of multiple solutions, this model requires additional measurements to determine the density, viscosity, and acoustic attenuation of the effective medium. The attenuation spectrum of the effective medium is modeled via a polynomial fit, while the target phase is assumed to have a log-normal PSD.68 This model allows the PSD for mixtures of more than two phases to be determined. 

The effective medium model has been described in [84,86]. It constitutes an isolated spherical insertion (component 1) in a continuous medium with effective (to be determined) properties. Thus, the following formula was obtained ... 

Figure 30 shows a comparison between the results for effective conductivity obtained by means of the iterative method (continuous) and a calculation using the formula (240) obtained by the effective medium theory model (dashed). The figure compares the results of the calculation of the effective conductivity using the iteration method (the continuous line) to the calculation by formula (240) (the dotted line) obtained from the effective medium model. The comparison... 

We apply simple effective medium models in an attempt to understand the diffusion process in the complex pore network of a porous SiC sample. There is an analogy between the quantities involved in the electrostatics problem and the steady state diffusion problem for a uniform external diffusion flux impinging on a coated sphere. Kalnin etal. [17] provide the details of such a calculation for the Maxwell Garnett (MG) model [18]. The quantity involved in the averaging is the product of the diffusion constant and the porosity for each component of the composite medium. The effective medium approach does not take into account possible effects due to charges on the molecules and/or pore surfaces, details in the size and shape of the protein molecules, fouling (shown to be negligible in porous SiC), and potentially important features of the microstructure such as bottlenecks. 

We consider two effective medium models, corresponding to distinct morphologies on the micro- or nanoscale. The resulting quantity (pD)e replaces D in Equation (12.1). Subscripts 1 and 2 refer to the two... 

In these expressions, p is the porosity of the SiC, i.e. the volume fraction of empty space. In the asymmetric MG model, we have chosen the coating to be the solution, since the opposite choice of SiC-encapsulated liquid spheres will not permit diffusion through the medium. With this choice, the SiC does not percolate and hence there is no structural support. The selectivity of the membrane is based in part on the size and shape of the protein molecules. The expressions for (pD)eff in the effective medium models [Equations (12.2) and (12.3)] do not contain a size scale, but it is necessary to introduce a scale in order to account for the size of a protein molecule. For simplicity, we assume that the proteins are spherical with effective (hydration) radius r. The excluded volume within the pores due to nonzero size is taken into account by replacing the porosity p with an effective porosity p. For the columnar... 

Johnson, E.M., et al. Hindered diffusion in agarose gels test of the effective medium model. Biophysical Journal, 1996, 70, 1017-1026. 

The understanding of factors that lead to enhanced band intensities and dispersive band shapes is of central interest in studies with nanostructured electrodes. Effective medium theory has often been employed to identify mechanisms for enhanced infrared absorption [28, 128, 172, 174, 175]. Osawa and coworkers applied Maxwell-Garnett and Bruggeman effective medium models in early SEIRAS work [28, 128]. Recently, Ross and Aroca overviewed effective medium theory and discussed the advantages and disadvantages of different models for predicting characteristics of SEIRAS spectra [174]. When infrared measurements on nanostructured electrodes are performed by ATR sampling, as is typically the case in SEIRAS experiments, band intensity enhancements occur, but the band shapes are usually not obviously distorted. In contrast, external... 

Effective medium theories characterize the frequency-dependent transport in systems with large-scale inhomogeneities such as metal particles dispersed in an insulating matrix [118,119]. An IMT in the effective medium model represents a percolation problem where a finite a c as T 0 is not achieved until metallic grains in contact span the sample. To understand the frequency dependence of the macroscopic material, an effective medium is built up from a composite of volume fraction /of metallic grains and volume fraction 1 — / of insulator grains. The effective dielectric function semaCw) and conductivity function (Tema(w) are solved self-consistently. 

Figure 4.1.8. Hypothetical microstmctures that may be described by the effective medium model (a) Continuous matrix of phase 1 containing a dilute dispersion of spheres of phase 2. (f>) A grain boundary shell of phase 1 surrounding a spherical grain of phase 2.

Brailsford and Hohnke [1983] have applied the Maxwell-Wagner model to grain boundaries in two-phase systems. Their microstructural model, shown in Figure 4.1.8fc, consists of a spherical grain of radius r2 surrounded by a shell of outer radius a, which represents the grain boundary and has a volume fraction Xi = 1 - (jjrif. The authors observe that for Xi —> 0 and 1/ 2 V i the effective medium model becomes identical to case (i) of the brick layer model, namely Eq. (6). Further, we have found that for x —> 0 and y/i xffz, it reduces to case (ii) of the brick layer model, namely Eq. (7). ... 

Figure 4.1.9. Simulated impedance and modulus spectra for a two-phase microstructure, based on the effective medium model. Values of the input parameters are given in Table 4.1.1. (a, b) Spectra for a matrix of phase 1 containing 25% by volume of spheres of phase 2. Resolution is achieved in the modulus spectrum (b) but not the impedance spectrum (a), (c, d) Spectra for a spherical grain of phase 2 surrounded by a grain boundary shell of phase 1. The ratio of shell thickness to sphere radius is 10" Resolution is achieved in the impedance spectrum (c) but not the modulus spectrum (d).

These reactions generate electrochemical impedances due to charge transfer, gas or solid state diffusion, etc. Since these impedances appear specifically at the boundaries between dissimilar phases, the composites cannot be fully described by simple effective medium models, even if these impedances are approximated by linear resistive elements. As pointed out by several authors, in the mixture of electronic and ionic phases there are clusters connected to (i) both current collector and electrolyte, (ii) only to the electrode, and (iii) isolated clusters. Clusters of all three types are visible in Figure 4.1.14. 

There are two important questions arising from the present model discussions. First, a microscopic model needs to be developed that leads to Pic = 1/3 and is less approximate than the Scher-Lax one and second, a microscopic model is also needed that yields response like the present effective medium model and takes explicit account of the detailed interactions, electromagnetic and otherwise, between vibrating ions and bulk dipoles. 

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