Effective medium approximations

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:

${\displaystyle \sum _{i}\,\delta _{i}\,{\frac {\sigma _{i}-\sigma _{e}}{\sigma _{i}+(n-1)\sigma _{e}}}\,=\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)}$

In a system of Euclidean spatial dimension ${\displaystyle n}$ that has an arbitrary number of components,[4] the sum is made over all the constituents. ${\displaystyle \delta _{i}}$ and ${\displaystyle \sigma _{i}}$ are respectively the fraction and the conductivity of each component, and ${\displaystyle \sigma _{e}}$ is the effective conductivity of the medium. (The sum over the ${\displaystyle \delta _{i}}$'s is unity.)

${\displaystyle {\frac {1}{n}}\,\delta \alpha +{\frac {(1-\delta )(\sigma _{m}-\sigma _{e})}{\sigma _{m}+(n-1)\sigma _{e}}}\,=\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2)}$

This is a generalization of Eq. (1) to a biphasic system with ellipsoidal inclusions of conductivity ${\displaystyle \sigma }$ into a matrix of conductivity ${\displaystyle \sigma _{m}}$.[5] The fraction of inclusions is ${\displaystyle \delta }$ and the system is ${\displaystyle n}$ dimensional. For randomly oriented inclusions,

${\displaystyle \alpha \,=\,{\frac {1}{n}}\sum _{j=1}^{n}\,{\frac {\sigma -\sigma _{e}}{\sigma _{e}+L_{j}(\sigma -\sigma _{e})}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(3)}$

where the ${\displaystyle L_{j}}$'s denote the appropriate doublet/triplet of depolarization factors which is governed by the ratios between the axis of the ellipse/ellipsoid. For example: in the case of a circle {${\displaystyle L_{1}=1/2}$, ${\displaystyle L_{2}=1/2}$} and in the case of a sphere {${\displaystyle L_{1}=1/3}$, ${\displaystyle L_{2}=1/3}$, ${\displaystyle L_{3}=1/3}$}. (The sum over the ${\displaystyle L_{j}}$ 's is unity.)

The most general case to which the Bruggeman approach has been applied involves bianisotropic ellipsoidal inclusions.[6]

The figure illustrates a two-component medium.[4] Consider the cross-hatched volume of conductivity ${\displaystyle \sigma _{1}}$, take it as a sphere of volume ${\displaystyle V}$ and assume it is embedded in a uniform medium with an effective conductivity ${\displaystyle \sigma _{e}}$. If the electric field far from the inclusion is ${\displaystyle {\overline {E_{0}}}}$ then elementary considerations lead to a dipole moment associated with the volume

${\displaystyle {\overline {p}}\,\propto \,V\,{\frac {\sigma _{1}-\sigma _{e}}{\sigma _{1}+2\sigma _{e}}}\,{\overline {E_{0}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(4)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,.}$

This polarization produces a deviation from ${\displaystyle {\overline {E_{0}}}}$. If the average deviation is to vanish, the total polarization summed over the two types of inclusion must vanish. Thus

${\displaystyle \delta _{1}{\frac {\sigma _{1}-\sigma _{e}}{\sigma _{1}+2\sigma _{e}}}\,+\,\delta _{2}{\frac {\sigma _{2}-\sigma _{e}}{\sigma _{2}+2\sigma _{e}}}\,=\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(5)}$

where ${\displaystyle \delta _{1}}$ and ${\displaystyle \delta _{2}}$ are respectively the volume fraction of material 1 and 2. This can be easily extended to a system of dimension ${\displaystyle n}$ that has an arbitrary number of components. All cases can be combined to yield Eq. (1).

Eq. (1) can also be obtained by requiring the deviation in current to vanish [7] [8] . 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. (2).

A more general derivation applicable to bianisotropic materials is also available.[6]

The main approximation is that all the domains are located in an equivalent mean field. 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. The threshold values are in general not correctly predicted. It is 33% in the EMA, in three dimensions, far from the 16% expected from percolation theory and observed in experiments. However, in two dimensions, the EMA gives a threshold of 50% and has been proven to model percolation relatively well.[9] [10][11]

The Maxwell Garnett equation reads:[12]

${\displaystyle \left({\frac {\varepsilon _{\mathrm {eff} }-\varepsilon _{m}}{\varepsilon _{\mathrm {eff} }+2\varepsilon _{m}}}\right)=\delta _{i}\left({\frac {\varepsilon _{i}-\varepsilon _{m}}{\varepsilon _{i}+2\varepsilon _{m}}}\right),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(6)}$

where ${\displaystyle \varepsilon _{\mathrm {eff} }}$ is the effective dielectric constant of the medium, ${\displaystyle \varepsilon _{i}}$ of the inclusions, and ${\displaystyle \varepsilon _{m}}$ of the matrix; ${\displaystyle \delta _{i}}$ is the volume fraction of the inclusions.

The Maxwell Garnett equation is solved by:

${\displaystyle \varepsilon _{\mathrm {eff} }\,=\,\varepsilon _{m}\,{\frac {2\delta _{i}(\varepsilon _{i}-\varepsilon _{m})+\varepsilon _{i}+2\varepsilon _{m}}{2\varepsilon _{m}+\varepsilon _{i}-\delta _{i}(\varepsilon _{i}-\varepsilon _{m})}},\,\,\,\,\,\,\,\,(7)}$[13][14]

so long as the denominator does not vanish. A simple MATLAB calculator using this formula is as follows.

% This simple MATLAB calculator computes the effective dielectric
% constant of a mixture of an inclusion material in a base medium
% according to the Maxwell Garnett theory as introduced in:
% http://en.wikipedia.org/wiki/Effective_Medium_Approximations
% INPUTS:
%   eps_base: dielectric constant of base material;
%   eps_incl: dielectric constant of inclusion material;
%   vol_incl: volume portion of inclusion material;
% OUTPUT:
%   eps_mean: effective dielectric constant of the mixture.

function [eps_mean] = MaxwellGarnettFormula(eps_base, eps_incl, vol_incl)

small_number_cutoff = 1e-6;

if vol_incl < 0 || vol_incl > 1
    disp(['WARNING: volume portion of inclusion material is out of range!']);
end
factor_up = 2*(1-vol_incl)*eps_base+(1+2*vol_incl)*eps_incl;
factor_down = (2+vol_incl)*eps_base+(1-vol_incl)*eps_incl;
if abs(factor_down) < small_number_cutoff
    disp(['WARNING: the effective medium is singular!']);
    eps_mean = 0;
else
    eps_mean = eps_base*factor_up/factor_down;
end

For the derivation of the Maxwell Garnett equation we start with an array of polarizable particles. By using the Lorentz local field concept, we obtain the Clausius-Mossotti relation:

${\displaystyle {\frac {\varepsilon -1}{\varepsilon +2}}={\frac {4\pi }{3}}\sum _{j}N_{j}\alpha _{j}}$

Where ${\displaystyle N_{j}}$ is the number of particles per unit volume. By using elementary electrostatics, we get for a spherical inclusion with dielectric constant ${\displaystyle \varepsilon _{i}}$ and a radius ${\displaystyle a}$ a polarisability ${\displaystyle \alpha }$:

${\displaystyle \alpha =\left({\frac {\varepsilon _{i}-1}{\varepsilon _{i}+2}}\right)a^{3}}$

If we combine ${\displaystyle \alpha }$ with the Clausius Mosotti equation, we get:

${\displaystyle \left({\frac {\varepsilon _{\mathrm {eff} }-1}{\varepsilon _{\mathrm {eff} }+2}}\right)=\delta _{i}\left({\frac {\varepsilon _{i}-1}{\varepsilon _{i}+2}}\right)}$

Where ${\displaystyle \varepsilon _{\mathrm {eff} }}$ is the effective dielectric constant of the medium, ${\displaystyle \varepsilon _{i}}$ of the inclusions; ${\displaystyle \delta _{i}}$ is the volume fraction of the inclusions.
As the model of Maxwell Garnett is a composition of a matrix medium with inclusions we enhance the equation:

${\displaystyle \left({\frac {\varepsilon _{\mathrm {eff} }-\varepsilon _{m}}{\varepsilon _{\mathrm {eff} }+2\varepsilon _{m}}}\right)=\delta _{i}\left({\frac {\varepsilon _{i}-\varepsilon _{m}}{\varepsilon _{i}+2\varepsilon _{m}}}\right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(8)}$

In general terms, the Maxwell Garnett EMA is expected to be valid at low volume fractions ${\displaystyle \delta _{i}}$, since it is assumed that the domains are spatially separated and electrostatic interaction between the chosen inclusions and all other neighbouring inclusions is neglected.[15] The Maxwell Garnett formula, in contrast to Bruggeman formula, ceases to be correct when the inclusions become resonant. In the case of plasmon resonance, the Maxwell Garnett formula is correct only at volume fraction of the inclusions ${\displaystyle \delta _{i}<10^{-5}}$.[16] The applicability of effective medium approximation for dielectric multilayers [17] and metal-dielectric multilayers [18] 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.