Effective medium approximation

In order to be useful in practice, the effective transport coefficients have to be determined for a porous medium of given morphology. For this purpose, a broad class of methods is available (for an overview, see [191]). A very straightforward approach is to assume a periodic structure of the porous medium and to compute numerically the flow, concentration or temperature field in a unit cell [117]. Two very general and powerful methods are the effective-medium approximation (EMA) and the position-space renormalization group method. 

The TEM data have been used to simulate, in the frame of the Mie theory and Maxwell-Garnett effective medium approximation [15], the optical absorption spectra of the sample implanted with 5 x lO Au /cm. The results are reported in Figure 8(c). In the first model used to describe... 

The physical properties of atoms and molecules embedded in polar liquids have usually been described in the frame of the effective medium approximation. Within this model, the solute-solvent interactions are accounted for by means of the RF theory [1-3], The basic quantity of this formalism is the RF potential. It is usually variationally derived from a model energy functional describing the effective energy of the solute in the field of an external electrostatic perturbation. For instance, if a singly negative or positive charged atomic system is considered, the RF potential is simply given by... 

The macroscopic dielectric constant of PS is very sensitive to the frequency range investigated. For very high frequencies the dielectric constant can be obtained by optical measurements. In this regime the effective medium approximation has been found to be an adequate description [Th3]. For lower frequencies the dielectric constant e is commonly obtained by capacitance measurements of metal/PS/ Si structures. AC conductivity measurements of such structures indicate a hop-... 

Bulk silicon constitutes an optical long-pass filter, as shown in Fig. 7.6. The same is true for micro- and mesoporous silicon, for which the effective medium approximation (EMA) is valid in the visible regime. The dimensions of macroporous silicon are in the visible regime and the EMA becomes invalid. 

The simplesf mefhod of solution of fhe Kirchhoff equations that correspond to the random network of conducfance elemenfs in three dimensions is in the single-bond effective medium approximation (SB-EMA), wherein a single effective bond between two pores is considered in an effective medium of surrounding bonds. The conductivify (7b, of fhe effective bond is obtained from the self-consistent solution of fhe equation ... 

This model, when applied to Nation as a function of water content, indicated a so-called quasi-percolation effect, which was verified by electrical impedance measurements. Quasi-percolation refers to the fact that the percolation threshold calculated using the single bond effective medium approximation (namely, x = 0.58, or 58% blue pore content) is quite larger than that issuing from a more accurate computer simulation. This number does not compare well with the threshold volume fraction calculated by Barkely and Meakin using their percolation approach, namely 0.10, which is less than the value for... 

Properties of substances, undergoing structural phase transitions at pressure P, are a complex functions of the extent of transformation [1], In the region of the phase transformation the sample of volume V appear to be a mixture of phases with the concentrations c, = Vi/V, where Vj - is the volume of i-phase (5Z K = F). In effective medium approximation the value of specific resistivity p of such mixture may be written as a normalized sum of the phases contributions [2]... 

Another refinement of the VRH model consists in assuming that the charges are delocalized over segments of length L, instead of being strictly localized on point sites [40]. This is indeed a more realistic picture, leading to better fits with the data, but it has the drawback that an extra parameter has been added. Note that the temperature dependence, log o- -T y, can be found by other approaches, such as the percolation model, the effective medium approximation (EMA), the extended pair approximation (EPA) [41], the random walk theory, and so on. 

The Effective Medium Approximation (EMA), based in some assumptions, allows us to employ linear regressions as an approximation of the behavior of a disordered system outside the critical range. Based on EMA theory, two linear regressions have been performed as an approximation for estimating the percolation threshold as the point of intersection between both regression lines (see Figures 43 15). The values of the excipient percolation thresholds estimated for all the batches studied, based on the behavior of the kinetic parameters, ranged from 25.99 to 26.77%. 

DEMA The iterative dynamic effective medium approximation... 

Fig. 6. Normalized time-dependent rate coefficient. Solid line corresponds to Effective Medium Approximation, and dotted line shows the usual Smoluchowski kinetics (without any obstruction). [Reproduced with permission from Ref. 41, Copyright 2003 American Chemical Society].

Elber, R. Karplus, M. Low-frequency modes in proteins Use of the effective-medium approximation to interpret the fractal dimension observed in electron-spin relaxation... 

A great number of studies have been published to deal with relation of transport properties to structural characteristics. Pore network models [12,13,14] are engaged in determination of pore network connectivity that is known to have a crucial influence on the transport properties of a porous material. McGreavy and co-workers [15] developed model based on the equivalent pore network conceptualisation to account for diffusion and reaction processes in catalytic pore structures. Percolation models [16,17] are based on the use of percolation theory to analyse sorption hysteresis also the application of the effective medium approximation (EMA) [18,19,20] is widely used. 

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... 

In order to estimate the e ct of metallic screening on the electrons on the same molecufc, we imagine embedding a molecule in a spherical metallic cavity. Ihc metallic cavity mimics the effect of the metallic electrons on the other molecules in an effective medium approximation. A simple calculation now leads to the following expression for the charging energy, c=... 

Christensen et al.103 studied bimetallic clusters using a Monte-Carlo method together with the effective-medium approximation for the description of the interatomic interactions (this method is closely related to the embedded-atom method). As initial structures they use ones that were derived from the fee... 

Fig. 2 Bruggeman effective medium approximation of a two-component film of a PMSSQ (fc = 2.85) and air (k =1).

Brinkman s equation represents a variant of the effective medium approximation, which does not describe explicitly the generation of non-laminar liquid motion and conversion of the in-plane surface motion into the normal-to-interface liquid motion. These effects result in additional channels of energy dissipation, which are effectively included in the model by introduction of the Darcy-like resistive force. 

The experiments on second- and third-harmonic generation in the reflection geometry demonstrate a giant enhancement of nonlinearities of the bireffingent PS films. The effect cannot be explained in terms of effective medium approximation and is apparently related to strong local fields in nanostructured solids (Fig. 3). 

Figure 4.9 Estimation of reduced diffusion coefficient by effective medium approximation. Reduced diffusion coefficient for solutes of radius 3.4 and 10 nm in a solution...

Figure 4.13 Estimation of reduced diffusion coefficient by effective medium approximation. Combination of steric and hydrodynamic effects on reduced diffusion coefficient. The solid lines represent hydrodynamic effect for probe radii of 3.4 and 10 A calculated using Brinkman s equation (see Figure 4.9). The dashed lines represent the combined steric and hydrodynamic effect using Equation 4-40 for the steric effect.

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