Effective-medium approach, for

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... 

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. 

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... 

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... 

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... 

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. ... 

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. 

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). 

It has been argued that the metals do not act as a collection of atoms, but rather as a uniform entity. Thus various properties of the metal, such as elastic constants, are not accurately described by pair potentials [146]. If that is so, a model which assumes a pairwise atom-atom interaction will perhaps have to be replaced with one which incorporates the influence of electron density from a distributed electron gas rather than from localized ones. Such theories can furthermore be based on the Hohenberg Kohn theorem [211], namely that the energy is a functional of the electron density. Thus several approximate theories are based on the electron gas concepts, for example, the embedded atom and diatom methods, or the effective medium approach [210]. 

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.

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. 

Many approaches have been used to correlate solvent effects. The approach used most often is based on the electrostatic theory, the theoretical development of which has been described in detail by Amis [114]. The reaction rate is correlated with some bulk parameter of the solvent, such as the dielectric constant or its various algebraic functions. The search for empirical parameters of solvent polarity and their applications in multiparameter equations has recently been intensified, and this approach is described in the book by Reich-ardt [115] and more recently in the chapter on medium effects in Connor s text on chemical kinetics [110]. 

Fermi (1940) pointed out that as /)—-1 the stopping power would power would approach °° were it not for the fact that polarization screening of one medium electron by another reduced the interaction slightly. This effect is important for the condensed phase and is therefore called the density correction it is denoted by adding -Z<5/2 to the stopping number. Fano s (1963) expression for 8 reduces at high velocities to... 

The improvement came in the form of the coherent-potential approximation (CPA) (Soven 1967, Taylor 1967, Velicky et al 1968), which remedied the lack of self-consistency exhibited by the ATA. The crux of this approach is that each lattice site has associated with it a complex self-consistent potential, called a coherent potential (CP). The CP gives rise to an effective medium with the important property that removing that part of the medium belonging to a particular site, and replacing it by the true potential, produces, on average, no further scattering. Because the CPA is used for our discussion of chemisorption on DBA s, its mathematical formulation is given below. 

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