Maxwell-Garnett effective medium

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 agreement between the experimental results and the calculated spectra demonstrates that Maxwell-Garnett effective medium theory effectively accounts for such dipole-dipole interactions. Similarly, the reflectance properties of the films are also affected by the interparticle spacing. This is supported by the data shown in Fig. 3, where the specular reflectance for films with different interparticle separation are compared to the spectra calculated by means of Eqs. (11), (18) and (19) [13]. Again, as the particles approach each other, the reflectance red-shifts and broadens. The effect is not as dramatic as seen for absorption, but the film changes from a green reflected hue to a metallic gold as (j) increases. 

Figure 3.60. The TO and LO energy loss functions and real part of effective dielectric function, obtained using Maxwell-Garnett effective medium theory, are compared with corresponding experimental functions. Filling factor is 0.3 and pores are regarded as cylinders perpendicular to surface. Reprinted, by permission, from E. Wackelgard, J. Phys. Condens, Matter 8, 4289 (1996), p. 4297, Fig. 4. Copyright 1996 lOP Publishing Ltd.

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 referred topology, known as Maxwell Garnett geometry [21], is considered to be isotropic macroscopically, and an effective-medium dielectric coistant can be written as... 

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. 

In the statistic route, an effechve dielectric function e(v) is calculated from the dielectric funchon of the metal Enie(v) and the of polymer material po(v) by using a formula, the effective medium theory. The most general effective medium theory is the Bergman theory in which the nanostructure of the composite material can be considered by a spectral density function. The Bergman theory includes the soluhons from the Bruggeman theory and the Maxwell Garnett theory for spherical, parallel-oriented, and random-oriented ellipsoidal parhcles. 

The most popular effective medium theories are the Maxwell Garnett theory [18], which was derived from the classical scattering theory, and the Bruggeman theory [19]. With these theories, an effective dielectric function is calculated from the dielectric functions of both basic materials by using the volume filling factor. At some extensions of these theories, a unique particle shape for all particles is assumed. There is also an other concept based on borders for the effective dielectric functions. The borders are valid for a special nanostructure. Between these borders, the effective dielectric function varies depending on the nanostructure of the material. The Bergman theory includes a spectral density function g(x) that is used as fit function and correlates with the nanostructure of the material [20]. 

Figure 7.2. Random units 1 and 2 in the effective medium (symmetrized Maxwell Garnett approximation).

The understanding of factors that lead to enhanced band intensities and dispersive band shapes is of central interest in studies with nanostructured electrodes. Effective medium theory has often been employed to identify mechanisms for enhanced infrared absorption [28, 128, 172, 174, 175]. Osawa and coworkers applied Maxwell-Garnett and Bruggeman effective medium models in early SEIRAS work [28, 128]. Recently, Ross and Aroca overviewed effective medium theory and discussed the advantages and disadvantages of different models for predicting characteristics of SEIRAS spectra [174]. When infrared measurements on nanostructured electrodes are performed by ATR sampling, as is typically the case in SEIRAS experiments, band intensity enhancements occur, but the band shapes are usually not obviously distorted. In contrast, external... 

By far the most convincing explanation of how effective medium theory could be used to predict the enhancement and shape of adsorbate bands in SEIRA spectra was made by Su et al. who studied both the Maxwell-Garnett (MG) and Bruggeman representations of EMT. The MG representation of is by far the simplest of the Bergman, Maxwell-Garnett and Bruggeman formalisms... 

This possibility is extremely useful for the structural characterization of polymer blends. The optical properties of the polymer layer can be described by a so-caUed effective dielectric function, which is a suitable average of the dielectric functions of the two components. Three averaging effective medium approximation (EMA)s -the linear, Maxwell-Garnett and Bruggeman EMAs - are widely used for this purpose [9]. These approximations differ in their spectral densities for a given volume fraction. 

Gittleman, J. I., Abeles, B. (1977). Comparison of the effective medium and the Maxwell-Garnett predictions for the dielectric constants of granular metals. Physical Review B, 15, 3273-3275. 

The main issue in the effective-medium approximations is to relate the polarizability a. to the relative permittivity r, where = i/ s, and E[ is the relative permittivity of the inclusion. For spherical particles and at frequencies for which the inclusions can be considered very small, the Maxwell-Garnett formula uses the relation... 

A collection of nanoparticles embedded in a dielectric medium is modelled by effective medium theories such as the Maxwell-Garnett theory where each nanoparticle is treated as a dipole, and the medium is treated as homogeneous with effective dielectric properties. This model provides qualitative agreement with experimental absorption spectra, but applications such as sensing and catalysis demand greater agreement between theoretical predictions and experimental results. 

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