In materials science, effective medium approximations (EMA) or effective medium theory (EMT) pertain to analytical or theoretical modeling that describes the macroscopic properties of composite materials.
EMAs or EMTs are developed from averaging the multiple values of the constituents that directly make up the composite material.
However, theories have been developed that can produce acceptable approximations which in turn describe useful parameters including the effective permittivity and permeability of the materials as a whole.
[4] Effective permittivity and permeability are averaged dielectric and magnetic characteristics of a microinhomogeneous medium.
They both were derived in quasi-static approximation when the electric field inside a mixture particle may be considered as homogeneous.
Nevertheless, they all assume that the macroscopic system is homogeneous and, typical of all mean field theories, they fail to predict the properties of a multiphase medium close to the percolation threshold due to the absence of long-range correlations or critical fluctuations in the theory.
These parameters are interchangeable in the formulas in a whole range of models due to the wide applicability of the Laplace equation.
The problems that fall outside of this class are mainly in the field of elasticity and hydrodynamics, due to the higher order tensorial character of the effective medium constants.
Indeed, among the numerous effective medium approximations, only Bruggeman's symmetrical theory is able to predict a threshold.
Bruggeman proposed a formula of the following form:[7] Here the positive sign before the square root must be altered to a negative sign in some cases in order to get the correct imaginary part of effective complex permittivity which is related with electromagnetic wave attenuation.
Formula (3) gives a reasonable resonant curve for plasmon excitations in metal nanoparticles if their size is 10 nm or smaller.
However, it is unable to describe the size dependence for the resonant frequency of plasmon excitations that are observed in experiments [8] Without any loss of generality, we shall consider the study of the effective conductivity (which can be either dc or ac) for a system made up of spherical multicomponent inclusions with different arbitrary conductivities.
Then the Bruggeman formula takes the form: In a system of Euclidean spatial dimension
The most general case to which the Bruggeman approach has been applied involves bianisotropic ellipsoidal inclusions.
then elementary considerations lead to a dipole moment associated with the volume This polarization produces a deviation from
[12] [13] It has been derived here from the assumption that the inclusions are spherical and it can be modified for shapes with other depolarization factors; leading to Eq.
Unfortunately, it is not the case close to the percolation threshold where the system is governed by the largest cluster of conductors, which is a fractal, and long-range correlations that are totally absent from Bruggeman's simple formula.
He proposed his formula to explain colored pictures that are observed in glasses doped with metal nanoparticles.
Because of these circumstances, formula (1) gives too narrow and too high resonant curve for plasmon excitations in metal nanoparticles of the mixture.
For the derivation of the Maxwell Garnett equation we start with an array of polarizable particles.
In the case of plasmon resonance, the Maxwell Garnett formula is correct only at volume fraction of the inclusions
[23] The applicability of effective medium approximation for dielectric multilayers [24] and metal-dielectric multilayers [25] have been studied, showing that there are certain cases where the effective medium approximation does not hold and one needs to be cautious in application of the theory.
Maxwell Garnett Equation describes optical properties of nanocomposites which consist in a collection of perfectly spherical nanoparticles.
However, due to confinement effect, the optical properties can be influenced by the nanoparticles size distribution.
As shown by Battie et al.,[26] the Maxwell Garnett equation can be generalized to take into account this distribution.
The Maxwell Garnett equation only describes the optical properties of a collection of perfectly spherical nanoparticles.
To overcome this limit, Y. Battie et al.[27] have developed the shape distributed effective medium theory (SDEMT).
The SDEMT theory was used to extract the shape distribution of nanoparticles from absorption [28] or ellipsometric spectra.
In this paper Bruggeman's approach was used, but electromagnetic field for electric-dipole oscillation mode inside the picked particle was computed without applying quasi-static approximation.
this formula has a simple form [18] For a network consisting of a high density of random resistors, an exact solution for each individual element may be impractical or impossible.