Maxwell-Garnett theory

Fig. 6. Maxwell-Garnett theory used for the prediction of dielectric constant containing dispersed regions of low dielectric polymer (e = 2.0,0) or air (e = 1.0, )...

The dielectric constant of the pure cyanurate network under dry nitrogen atmosphere at 20 °C is 3.0 (at 1 MHz). For the macroporous cyanurate networks, the dielectric constant decreases with the porosity as shown in Fig. 57, where the solid and dotted lines represent experimental dielectric results together with the prediction of the dielectric constant from Maxwell-Garnett theory (MGT) [189]. The small discrepancies between experimental results and MGT might be due to the error in estimated porosities, which are calculated from the density of the matrix material and cyclohexane assuming that the entire amount of cyclohexane is involved in the phase separation. It is supposed that a small level of miscibility after phase separation would result in closer agreement of dielectric constants measured and predicted. Dielectric constant values as low as 2.5 are measured for macroporous cyanurates prepared with 20 wt % cyclohexane. 

We can estimate the magnitude of the shift attributable to interaction between particles by appealing to the Maxwell Garnett theory (Section 8.5). This theory is strictly applicable only to a medium consisting of small particles distributed throughout a volume, whereas the slides consist of a single layer of particles on a surface. Nevertheless, for our limited purposes here the Maxwell Garnett theory is adequate. 

Treu, J. I., 1976. Mie scattering, Maxwell Garnett theory, and the Giaever immunology slide, Appl. Opt., 15, 2746-2750. 

Fig-1 Maxwell-Garnett theory used for the prediction of dielectric constant containing dispersed regions of air... 

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]. 

In 1906, J. C. Maxwell Garnett used the Maxwell Garnett theory, equation (12), for the first time to descibe the color of metal colloids glasses and of thin metal films. Equation (12) can be deviated from the Rayleigh scattering theory for spherical particles [21], or from the Lorentz-Lorenz assumption for the electrical field of a sphere and the Clausius-Mossotti Equation by using the polarizability of an metal particle if only dipole polarization is considered [22]. 

For the case where the particles do not have a spherical shape, various extensions of the Maxwell Garnett theory for nonspherical particles were introduced. The particles are spheroidal with the same shape (ratio of major-axis A and minor axis B), but with different sizes still in the wavelength limit. It remains only to choose between a parallel or a random orientation of the mean axis of the ellipsoids. 

At the extensions of the Maxwell Garnett theory for parallel-oriented ellipsoids, equation (13) [23, 24], all particles are ellipsoids with parallel the mean axis. Only one depolarization factor L is necessary. L describes the ratio between the axes of the ellipsoids, and values for L between 0spherical particles (L =i), equation (13) gives the same result as the Maxwell Garnett theory, equation (12). For particles embedded in one plane, the orientation of the ellipsoids in ratio to the substrate is parallel or perpendicular to the plane of the incident light. An orientation of the ellipsoidal particles diagonally to the substrate cannot be considered. 

The extensions of the Maxwell Garnett theory for random-oriented ellipsoids, equation (14) [25], needs three depolarization factors Li, Li, L3 with 2L, = 1 to describe the embedded ellipsoids. Frequently, ellipsoids with a symmetrical axis of rotation are assumed with Li = L3. Extreme geometries are rods with Li L2 = L3 and disks with Li L2 = L3. For Li =y, the extensions of the Maxwell Garnett theory for random-oriented ellipsoids, equation (14), give the same result as the Maxwell Garnett theory, equation (12). 

There is also another extension of the Maxwell Garnett theory for random-oriented ellipsoids [29] which is different from equation (14). Other extensions of the Maxwell Garnett theory are described for chiral aggregates of spheres [26], for thin hlms with columnar structures [27], and for embedded spherical particles of several metals [28],... 

For the following calculation, experimentally determined dielectric functions for silver [30] and for a plasma polymer [31] were taken. The effective dielectric functions e(v) were calculated with the Maxwell Garnett theory for parallel-oriented particles, equation (13). From the effective dielectric function, transmission or extinction spectra can be calculated by using the Fresnel formulas [10] for the optical system air-composite media-quartz substrate. As a further parameter, the thickness of the film with embedded particles and the thickness of other present layers that do not contain metal nanoparticles have to be included. The calculated extinction spectra can be compared with the experimental spectra. 

Figure 6.9. Calculated extinction spectra (Maxwell Garnett theory) for parallel-oriented ellipsoids before (solid curve) and after (dashed curve) thermal treatment.

As we mentioned above, metal deposition on semiconductor electrodes usually starts with three-dimensional nucleation, which leads to discontinuous (clustered) overlayers. Such films have apparent optical properties which can be distinctly difierent from those of the metal clusters proper and which are determined also by the refractive index of the embedding medium (in our case, the electrolyte), the shape of the particles, and their volume fraction (filling factor fm). According to the Maxwell-Garnett theory, for spherical particles the optical response is ... 

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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