Effective-Medium Approach

The effective-medium approach is a phenomenological approach that assumes that, in a suspension, there will be a well-defined phase velocity V, which 

The effective compressibility commonly used is a simple averaging, 

Ament (1953) considered the effect of fluid viscosity and particle size a, and obtained 

FIGURE 5-27 Phase velocity vs. volume fraction for glass beads, obtained from various models. 

---

The model has been treated analytically employing the effective medium approach [58] and by Monte Carlo simulation. It makes the following predictions A dilute ensemble of non-interacting charge carriers, initially generated at random within the DOS, lends to relax toward the tail slates and ultimately equilibrates at... 

A quantitative analysis of scattering data, originating from the crossover regime between short-time Rouse motion and local reptation, needs explicit consideration of the initial Rouse motion neglected by de Gennes. Ronca [50] proposed an effective medium approach, where he describes the time-dependent... 

Note that err = y (crr)a3/k Tand recall that in a concentrated dispersion the Peclet number is Pe = 67ry (crr)a3/k T. The use of the suspension viscosity implies that the particle diffusion can be estimated from an effective medium approach. Both Krieger and Cross gave the power law indices (n and m) as 1 for monodisperse spherical particles. In this formulation, the subscript c indicates the characteristic value of the reduced stress or Peclet number at the mid-point of the viscosity curve. The expected value of Pec is 1, as this is the point at which diffusional and convective timescales are equal. This will give a value of ac 5 x 10 2. Figure 3.15 shows a plot of Equation (3.57a) with this value and n = 1... 

An analytical theory based upon the effective medium approach (EMA) has been developed by Fishchuk et al. [70]. They consider the superposition of disorder and polaron effects and treat the elementary charge transfer process at moderate to high temperatures in terms of symmetric Marcus rates instead of Miller-Abrahams rates (see below). The predicted temperature and field dependence of the mobility is... 

Stroud, D., 1975. Generalized effective-medium approach to the conductivity of an inhomogeneous medium, Phys. Rev., B12, 3368-3373. 

Moreover, v is the fluid velocity, pf and p, are the density and the viscosity of the fluid, respectively (f) and K are porosity and permeability of the core c/.,s are specific heat of the fluid and of the solid respectively Cfast>siow are the sound propagation speed of the fast and slow waves I fast,slow are the intensities of the fast and slow waves, while a.fast,siow are their damping coefficients. We use an effective medium approach for the liquid, describing the effect of the acoustic waves as source terms. There are two source terms. First there... 

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 ... 

A fully microscopic interpretation of the temperature dependence of the absorption maximum, even well above any order-disorder transition temperature, is a formidable task because of the potential importance of many complicated physical factors (27-30). As a first attack on this problem, we have adopted a simple mean-field (or effective-medium) approach (28-30) with the assumption that the absorption peak (to) is linearly perturbed from its limiting dl -trans value ((Orod) by the presence of bond rotational defects (free energy of formation, e)... 

Oldroyd (1953, 1955) derived expressions for the linear viscoelasticity of suspensions of one Newtonian fluid in another. By using an effective-medium approach, he was able to relax the requirement of diluteness. For an ordinary interface whose interfacial tension r remains constant during the deformation, Oldroyd s result gives the following for the complex modulus G = G + iG" ... 

The theory described above has been developed for sample surfaces that are uniform over areas of the tip size. However, some substrates, such as self-assembled monolayers (SAM), reveal even smaller details that cannot be resolved by SECM, but for which a mean effective rate constant keff can be determined. The theory has been developed for a blocking film with diskshaped defects (18) for the kinetically-controlled regime (20) and irreversible kinetics (21) and is based on an effective medium approach of Szabo et al. (19) ... 

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. 

An effective medium approach, which uses hydraulic permeabilities to define the resistance of the fiber network to diffusion, has been used to estimate reduced diffusion coefficients in gels [77]. For a particle diffusing within a fiber matrix, the rate of particle diffusion is influenced by steric effects (due to the volume excluded by the fibers in the gel, which is inaccessible to the diffusing particle) and hydrodynamic effects (due to increased hydrodynamic drag on the diffusing particle caused by the presence of fibers). Recently, it was proposed that these two effects are multiplicative [78], so that the diffusion coefficient observed for particles in a fiber mesh can be predicted from ... 

The effective medium approach, including both hydrodynamic and steric interactions (Figure 4.13), compares favorably with experimental results for the diffusion of proteins and polysaccharides through agarose gels [81, 82]. 

Thus, if the wavelength of the light is much greater than the grain size, the long wavelength approximation and effective medium theory can be applied to determine the effective value of the composite dielectric constant and, consequently, describe composites optical properties. However, if the size of the structure is of the order of tens and even units of nanometers, then the effective medium approach is not applicable. Indeed, within this approximation, the effective permittivity of a composite is determined as a function of the permittivity for each composite component and, in turn, the nanocomposite components are characterized by the same tensor of permittivity as those used for bulk media. ... 

Bonding in Metallic Systems An Effective Medium Approach. 

There are two recent developments in the theory of acoustics which deserve to be mentioned here. The first one is a theory of acoustics for flocculated emulsions (21). It is based on EC AH theory, but it uses an addition an effective medium approach for calculating thermal properties of the floes. The success of this idea is related to the feature of the thermal losses that allows for insignifieant partiele -particle interactions even at high volume fractions. This mechanism of acoustic energy dissipation does not require relative motion of the particle and liquid. Spherical symmetrical oscillation is the major term in these kinds of losses. This provides the opportunity to replace the floe with an imaginary particle, assuming a proper choice of the thermal properties. 

One dimensional geometry Because the scale of phase separation in the junction is small compared to the device size, an effective medium approach is often applied, and one-dimensional symmetry assumed. This means neglecting the effect of local electric fields at the interface, and differences in behavior between electrons or ions that are near or further from the interface or grain boundaries. The effect of electrolyte on electron transport can be included through an effective diffusion coefficient. 

It has been proposed that sub-wavelength dielectric structures alter the effective medium without affecting the diffraction pattern in an optical system [35]. An expansion of the effective medium approach was also suggested for metals [36]. 

The effective-medium approach is valid only for the random-dispersion structure including the cases in which phase B disperses in matrix phase A and phase A conversely disperses in matrix phase B. However, for the percolationlike structure, in which the identification of dispersion phase and matrix phase is difficult to determine, the effective-medium theory cannot be used directly. To deal with such a transition area, a newly developed type of fuzzy logic [19, 20] may be useful for describing the complex microstructure and thermophysical properties. 

Kochergin V, Christophersen M, Foil H (2004) Effective medium approach for calculations of optical anisotropy in porous materials. Appl Phys B 79(6) 731-739 Kochergin V, Christophersen M, Foil H (2005) Surface plasmon enhancement of an optical anisotropy in porous silicon/metal composite. Appl Phys B 80(l) 81-87 Kovalev D et al (1995) Porous Si anisotropy from photoluminescence polarization. Appl Phys Lett 67(11) 1585-1587... 

Figure 1 shows the refractive index of bulk silicon. The application of different effective medium theories leads to different results. Figure 2 shows the comparison of the dependency of refractive index on porosity, determined by different effective medium approaches. A value of 3.4 for the infrared bulk value has been taken for the solid pore wall phase. The refractive index of porous silicon is expected to be lower than that of silicon, as porous silicon is a two-phase composite, being a mixture of air and solid phase (Theiss and Hilbrich 1997). 

Fig. 2 Refractive index (reai part) versus porosity, given by some effective medium approaches (Data from Theiss and Hilbrich 1997)...

Khardani M, Bouaicha M, Bessais B (2007) Bruggeman effective medium approach for modelling optical properties of porous silicon comparison with experiment. Phys Status Sohdi C 4 1986-1990... 

---