Effective-medium theory

Effective medium theory was originally introduced in the early 1980s to describe chemisorption of gas atoms on metal surfaces. It has since been developed as a relatively efficient method for describing bonding in solids, particularly metals, and therefore has found considerable use in materials model-ing. It also forms the quantum mechanical basis for the more empirical and widely used embedded-atom method discussed below. Specific implementations of effective medium theory for materials simulation have been developed by Norskov, Jacobsen, and co-workers and by DePristo and co-workers.  

The following discusses the former implementation the reader is referred to other sources for a discussion of the latter.  

Because of the change in electrostatic potential, an atom embedded in a jellium alters the initially homogeneous electron density. This difference in electron density with and without the embedded atom can be calculated within the local density approximation of density functional theory and expressed as a spherical function Ap(r) about the atom 

In effective medium theory, the overall electron density of a solid, p(r), is constructed as a superposition of the perturbed electron densities  

Among the assumptions of effective medium theory, for an imperfect crystal the average embedding electron density is constructed, and the total electron density in turn depends on these local densities through the Ap(r) s. Fience a self-consistent problem is at hand. Because the Ap(r) s need be calculated only 

The derivation of the effective dielectric function of a medium composed of small spheres and a vacuum will be presented [183], From elementary electrostatics [12], the polarization P of a sphere in a constant and uniform far field is uniform, and its polarizability a (1.3.2°) is given by 

The frequencies at which the condition (1.117) is met are referred to as Frohlich frequencies cof, and the corresponding modes of a sphere that is small compared to k are referred to as the Frohlich modes, giving credit to the pioneering work of Frohlich [13], who determined theoretically that an ensemble of small spherical particles absorbs at (see the discussion in Section 3.9). When the imaginary part of s o)) vanishes, Eq. (1.117) is reduced to the Mie condition 

Accounting for the CMLL local field Clausius-Mossotti relation [9-13, 22, 43] 

8 denotes the dielectric function of the sphere material and / = NV is the volume fraction of these spheres, known as the filling fraction (V = is the 

ABSORPTION AND REFLECTION OF INFRARED RADIATION BY ULTRATHIN FILMS 

First let us consider how the velocity profile caused by a point force is affected when a small number of polymers are present in solution. For simplicity we consider the steady state in the velocity field, though this assumption is not essential. Let be the point force acting at the origin. For pure solvent, the velocity perturbance is given by 

Our problem is how this field is affected by the polymers. In principle the answer is given by solving the Smoluchowski equation 

Now it is difBcult to solve the diffusion equation (5.153) which includes pdymer-polymer interactions. So we first focus our attention on a particular pdymer, diich we shall call the test polymer, assuming that the veiodty field oeated by the othar polymers is known. Let v r) be the veiodty field created by the external force and the polymers occluding tte test polymer, i.e., 

The effect of the test polymer is solved by the Smoluchowski equation for the single chain  

The veloc perturbation created by the test polymer is then calculated by 

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Bonig L, Liu S and Metiu FI 1996 An effective medium theory study of Au islands on the Au(IOO) surface reconstruction, adatom diffusion, and island formation Surf. Sot 365 87... 

D. E. Aspnes. Thin Solid Films. 89, 249, 1982. A detailed review of effective medium theory and its use in studies of optical properties of solids. 

We examined the role of vector percolation in the fracture of model nets at constant strain and subjected to random bond scission, as shown in Fig. 11 [1,2]. In this experiment, a metal net of modulus Eo containing No = 10" bonds was stressed and held at constant strain (ca. 2%) on a tensile tester. A computer randomly selected a bond, which was manually cut, and the relaxation of the net modulus was measured. The initial relaxation process as a function of the number of bonds cut N, could be well described by the effective medium theory (EMT) via... 

The idea of adding smaller and smaller particles to fill in the interstices left by the larger particles can be continued. The viscosity of a multimodal suspension may be predicted from unimodal data based on the premise that the viscosity of the mixture of smaller fractions is the medium viscosity for the next largest fraction. This is an effective medium theory and basically assumes that the smaller particles act as a medium toward the larger particles. This was assuming at least an order of magnitude difference in size between successive fractions [26]. Thus, the viscosity of the ith component is ... 

Ritz G, Schmid M, Varga P, Borg A, Ronning M. 1997. Pt(lOO) quasihexagonal reconstruction A comparison between scanning tunneling microscopy data and effective medium theory simulation calculations. Phys Rev B 56 10518-10525. 

Here E(, an energy cut-off parameter, and have been adjusted to be -0.27 and 0.26 eV respectively for all hydrocarbon liquids. Application of the semiclassi-cal effective medium theory then gives the average mobility as... 

Bergman effective medium theory, in conjunction with a model dielectric function for the particles, has been used. For the other layers the Bruggeman effective medium theory was used. After [Bel6]. 

Effective medium theory, 37 154 Eggshell catalysts, 39 231 EH method, 37 153 EHT, see Extended Hiickel treatment Eigenberger model, oscillatory reactions, 39 80-81, 83... 

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... 

Within the effective medium theory, the potential functional can be written as [3]... 

Proton conductivity (23°C, 98% RH) of SPEKK/PEI blends as a function of SPEKK weight fraction (SPEKK lEC = 2.2 meq/g). Solid line is derived from an effective-medium theory (equation given). (From Gasa, J. V. et al. 2006. Journal of Polymer Science Part B 44 2253-2266.)... 

Hsu and Berzins used effective medium theories to model transport and elastic properties of these ionomers, with a view toward their composite nature, and compared this approach to that of percolation theory. ... 

Figure 5.26 Hydrogen absorption in the effective medium theory, (a) Hydrogen atom on a path to the surface of a metal and through the lattice (b) electron density along the path of the hydrogen atom (c) potential energy of the hydrogen atom along the path.

Monte Carlo simulations [54], analytical effective medium theory [64], and stochastic hopping theory [46] predict a dependence of the charge carrier mobility as a function of temperature and electric field given in (3) ... 

Quantitative simulation of spectra as outlined above is complicated for particle films. The material within the volume probed by the evanescent field is heterogeneous, composed of solvent entrapped in the void space, support material, and active catalyst, for example a metal. If the particles involved are considerably smaller than the penetration depth of the IR radiation, the radiation probes an effective medium. Still, in such a situation the formalism outlined above can be applied. The challenge is associated with the determination of the effective optical constants of the composite layer. Effective medium theories have been developed, such as Maxwell-Garnett 61, Bruggeman 62, and other effective medium theories 63, which predict the optical constants of a composite layer. Such theories were applied to metal-particle thin films on IREs to predict enhanced IR absorption within such films. The results were in qualitative agreement with experiment 30. However, quantitative results of these theories depend not only on the bulk optical constants of the materials (which in most cases are known precisely), but also critically on the size and shape (aspect ratio) of the metal particles and the distance between them. Accurate information of this kind is seldom available for powder catalysts. 

The effective medium theory consists in considering the real medium, which is quite complex, as a fictitious model medium (the effective medium) of identical properties. Bruggeman [29] had proposed a relation linking the dielectric permittivity of the medium to the volumetric proportions of each component of the medium, including the air through the porosity of the powder mixture. This formula has been rearranged under a symmetrical form by Landauer (see Eq. (8), where e, is the permittivity of powder / at a dense state, em is the permittivity of the mixture and Pi the volumetric proportion of powder / ) and cited by Guillot [30] as one of the most powerful model. 

The activation barriers AE for dissociation and recombination belong to the same realm of relative energies as AQAB. For this reason, we shall not discuss here purely numerical calculations of AE. Remarkably, many authors tried to conceptualize their computational results in terms of simple analytic models, which have no direct relation to the computations. For example, the effective medium theory (EMT) is a band-structure model with a complex and elaborated formalism including many parameters (154). Nevertheless, while reviewing the numerical EMT applications to surface reactions, Norskov and Stoltze (155) discussed the calculated trends in the activation energies for AB dissociation in terms of a one-parameter model (unfortunately, no details were provided) projecting A b to vary as NJ, 10 - Nd), where Nd is the d band occupancy [cf. Eqs. (21a)—(21c) of the BOC-MP theory]. 

Foss, C. A., Jr., Tierney, M. J., and Martin, C. R., Template synthesis of infrared-transparent metal microcylinders—comparison of optical properties with the predictions of effective medium theory. /. Phys. Chem. 96, 9001 (1992). 

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. 

Conductance of such a random network constitutes the simplest transport problem that can be studied by percolation ideas. A crude effective-medium theory shows that the bulk conductance is proportional to the number of active bonds ... 

Theoretical studies of the flow of concentrated dispersions in terms of statistical theories are also reported in the works of Lundgren [64], Beenakker [65], and others. On the other hand, using expressions obtained in describing the viscosity of concentrated emulsions (for example, equation (54)), on the basis of the effective medium theory, Bedeaux derival the following equation ... 

The methodology focuses, as many density-functional schemes do, on the key role of the electron density. The Schrodinger equation is then solved self-consistently in the Kohn-Sham scheme.86 Initial approaches dealt with a jellium-adatom system, which would at first sight seem rather unchemical, lacking microscopic detail. But there is much physics in such an effective medium theory, and with time the atomic details at the surface have come to be modeled with greater accuracy. 

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