Effective medium approximation theories

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... 

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%. 

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. 

Fig. 17 Intracrystalline self-diffusivity of methane ( 2 molecules per supercage, at 25 °C) as a function of the amount of co-adsorbed molecules per window . The solid lines are predictions based on the effective medium approximation of percolation theory with / denoting the ratio of the transition rates through blocked and open windows. From [158] with permission...

Porous silicon can be specified as an effective medium, whose optical properties depend on the relative volumes of silicon and pore-filling medium. Full theoretical solutions can be provided by different effective medium approximation methods such as Maxwell-Gamett s, Looyenga s, or Bruggeman s (Arrand 1997). Effective medium theory describes the effective refractive index, fieff, of porous silicon as a function of the complex refractive index of silicon, fisi, and that of the porefilling material, flair = 1, for air. The porosity P and the topology of the porous structure will also affect fleff (Theiss et al. 1995). 

Diffractive ARS are usually calculated either using the effective medium approximation, or the rigorous coupled-wave analysis [169]. Direct numerical solution of Maxwell equations and the approach using the plane wave theory were also used [174]. 

The various methods used by researchers to predict the permittivity of the mixture from the permittivity of the components are often broadly divided into the following bounding methods, effective medium approximation, percolation theory and numerical simulation. The various formulas obtained using many of these methods have been reviewed in a number of articles [2, 12, 15]. 

P. Chylek, G. Videen, Effective medium approximations for heterogeneous particles, in Light Scattering by Nonspherical Particles Theory, Measurements and Applications, ed. by M.I. Mishchenko, J.W. Hovenier, L.D. Travis (Academic, San Diego, 2000) pp. 273-308... 

In obtaining Eqs. (217)-(219), we have employed the preaveraging approximation and assumed that solvent motion is instantaneous in comparison to the motion of poly electrolytes. For a solution of polyelectrolytes, the effective medium theory for the equilibrium properties gives... 

The above considerations referred to the practically important examples of more or less ordered heterogeneities. If we face random distribution, usually effective medium and percolation theory have to be referred to in order to evaluate the inhomogeneous situations properly. However, attention has to be paid to the fact that they often require nonrealistic approximations. For more details see Ref.300 In such cases numerical calculations, e.g., via finite element methods are more reliable. 

Section IV is devoted to excitons in a disordered lattice. In the first subsection, restricted to the 2D radiant exciton, we study how the coherent emission is hampered by such disorder as thermal fluctuation, static disorder, or surface annihilation by surface-molecule photodimerization. A sharp transition is shown to take place between coherent emission at low temperature (or weak extended disorder) and incoherent emission of small excitonic coherence domains at high temperature (strong extended disorder). Whereas a mean-field theory correctly deals with the long-range forces involved in emission, these approximations are reviewed and tested on a simple model case the nondipolar triplet naphthalene exciton. The very strong disorder then makes the inclusion of aggregates in the theory compulsory. From all this study, our conclusion is that an effective-medium theory needs an effective interaction as well as an effective potential, as shown by the comparison of our theoretical results with exact numerical calculations, with very satisfactory agreement at all concentrations. Lastly, the 3D case of a dipolar exciton with disorder is discussed qualitatively. 

More recently Morse produced a complete microscopic tube theory for stiff polymers that successfully interpolates between the rigid-rod and flexible chain limits. This theory explains many features of semiflexible polymer rheology, including the two mechanisms for plateau moduli described above (which depend on a comparison of timescales), with the tube diameter being the sole fitting parameter as in the Doi-Edwards theory. More recently, Morse successfully computed a tube diameter from two different approaches (self-consistent binary collision and continuum effective medium) that give similar results, e.g. modulus G p and respectively). An elastic network approximation... 

Pellegrini and Barthelemy [128] studied effective medium theory approximations for linear composite media by means of a path integral formalism. They obtained the following values of the conductivity critical exponents s = 0, t = 2 in any spatial dimension d > 2. Perturbation theory for a 3D composite has been described in Refs. 116, 127, and 129. 

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