Conductivity effective medium model

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... 

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. 

Fig. 7. Model calculations for the reflectivity (a) and the optical conductivity (b) for a simple (bulk) Drude metal and an effective medium of small metallic spherical particles in a dielectric host within the MG approach. The (bulk) Drude and the metallic particles are defined by the same parameters set the plasma frequency = 2 eV, the scattering rate hr = 0.2 eV. A filling factor/ = 0.5 and a dielectric host-medium represented by a Lorentz harmonic oscillator with mode strength fttOy, 1 = 10 eV, damping ftF] = I eV and resonance frequency h(H = 15 eV were considered for the calculations.

Two other approaches have been taken to modelling the conductivity of composites, effective medium theories (Landauer, 1978) and computer simulation. In the effective medium approach the properties of the composite are determined by a combination of the properties of the two components. Treating a composite containing spherical inclusions as a series combination of slabs of the component materials leads to the Maxwell-Wagner relations, see Section 3.6.1. Treating the composite as a mixture of spherical particles with a broad size distribution in order to minimise voids leads to the equation ... 

Gas relative permeability, Pk, is defined as the permeability of a fluid through a porous medium partially blocked by a second fluid, normalized by the permeability when the pore space is free of this second fluid. This property diminishes at the percolation threshold , at which a significant portion of the pores are still conducting but they do not form a continuous path along the flow direction. It is obvious that only the network model, can provide a satisfactory analysis of the percolation threshold problem. Nicholson et al. [3] introduced a simple network model, and applied it on gas relative permeability [4]. For the gas relative permeability, an explicit approximate analytical relation between the relative permeability and the two network parameters, namely z and the first four moments of, f(r), has been developed, based on the Effective Medium Approximation (EMA) [5]. If a porous... 

The Calculation Results. The calculations were made for a two-component medium. Calculations were executed for a two-component 3D composite with random structure. First we shall consider a comparison of the outcome for the effective conductivity calculated by means of the iterative method with the calculation using formulas (240) obtained on the basis of the effective medium theory model. 

Figure 31. Comparison of the calculation of the effective conductivity of a composite based on the iterative method (continuous), on the effective medium method (dotted line) and numerical modeling (dots).

In the two-medium treatment of the single-phase flow and heat transfer through porous media, no local thermal equilibrium is assumed between the fluid and solid phases, but it is assumed that each phase is continuous and represented with an appropriate effective total thermal conductivity. Then the thermal coupling between the phases is approached either by the examination of the microstructure (for simple geometries) or by empiricism. When empiricism is applied, simple two-equation (or two-medium) models that contain a modeling parameter hsf (called the interfacial convective heat transfer coefficient) are used. As is shown in the following sections, only those empirical treatments that contain not only As/but also the appropriate effective thermal conductivity tensors (for both phases) and the dispersion tensor (in the fluid-phase equation) are expected to give reasonably accurate predictions. 

In these systems, it is possible to obtain low percolation thresholds if a double percolation is present, that is, particle and phase percolation. This effect may be observed when the conductive particles, localized preferentially in one polymer phase, have a concentration equal or larger than the electric percolation threshold, and when the host polymer phase is the matrix or continuous phase of the polymer blend [155]. There are several models that describe the electroconductivity of these systems the effective medium theory, the onset for percolation theory, and thermodynamic models. Sumita s model considers the formation of chainlike conductive structures [151, 156]. 

Impedance spectra of such crystals, reported by Bonanos and Lilley [1981], displayed only bulk and electrode arcs. The same data plotted in the modulus plane (Figure 4.1.45) revealed two overlapping arcs a low frequency arc ascribed to the matrix and a high frequency one ascribed to the dispersed phase. Using the Maxwell-Wagner effective medium relation (Eq. 20), the modulus spectra were modeled and the microscopic conductivities of the two phases were evaluated for... 

The thermal conductivity of a porous material is generally modelled using effective medium theory, which is successful for predicting the thermal conductivity of macroporous Si but less adequate for the mesoporous material. 

To explain the relationship between the electrical conductivity and the infrared absorption parameters, a model including the semiconducting phase for the structure of carbons heat treated at lower temperatures was proposed using the effective medium theory to simulate conductivity [103] (see Fig. 30). 

The interactions of the RE-ions with phonons and conduction electrons cause indirect interactions between different RE-ions. Phonons and conduction electrons serve as medium for these interactions. In the effective-interaction model they are eliminated and replaced by an ion - ion interaction. Since the latter is an instantaneous interaction all retardation effects are lost which are connected with the dynamical properties of the medium. This shows the limitations of the effective interaction model. The isotropic exchange interaction as described by eq. (17.40) leads to an effective ion-ion interaction of the form... 

In the case of not-too-diiferent conductivities dF compoi ts (3.10 A2/A1 < 1), the percolation effects were contidered insignificant, and Eq. (25) for dfective medium model was recommended throughout... 

A discrete model for magnetotransport in percolating systems has been proposed 1220,221]. This model, which assumes that the conducting component has a closed Fermi surface and that the MR saturates at high fields, predicts a large MR in the vicinity of the percolation threshold. This is contrary to the predictions of effective medium theory in which there is no MR near the percolation threshold [235,236]. For insulating PANI-CSA (100%), the MR tends to saturate at 8 T, but it is not known whether the Fermi surface of PANI-CSA is open or closed. In order to address this question, we have carried out MR measurements in many PANI-CSA/ PMMA samples (0.4-1.5 vol %) near the percolation threshold in which the volume fraction of PANI-CSA varied by 0.1% from sample to sample. At 4.2 K, the MR increases systematically upon dilution from 100% to 1.5% PANI-CSA, whereas below 1.5% the MR de-... 

The brief discussion above shows that the structure of a polymer electrolyte and the ion conduction mechanism are complex. Furthermore, the polymer is a weak electrolyte, whose ions form ion pairs, triple ions, and multidentate ions after its ionic dissociation. Currently, there are several important models that attempt to describe the ion conduction mechanisms in polymer electrolytes Arrhenius theory, the Vogel-Tammann-Fulcher (VTF) equation, the Williams-Landel-Ferry (WLF) equation, free volume model, dynamic bond percolation model (DBPM), the Meyer-Neldel (MN) law, effective medium theory (EMT), and the Nernst-Einstein equation [1]. 

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