# Effective medium approximations

Effective medium approximations (abbreviated as 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. At the constituent level, the values of the materials vary and are inhomogeneous. Precise calculation of the many constituent values is nearly impossible. However, theories have been developed that can produce acceptable approximations which in turn describe useful parameters and properties of the composite material as a whole. In this sense, effective medium approximations are descriptions of a medium (composite material) based on the properties and the relative fractions of its components and are derived from calculations.

## Applications

There are many different effective medium approximations, each of them being more or less accurate in distinct conditions. 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.

The properties under consideration are usually the conductivity $\sigma$ or the dielectric constant $\epsilon$ of the medium. 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.

EMAs can be discrete models, such as applied to resistor networks, or continuum theories as applied to elasticity or viscosity. However, most of the current theories have difficulty in describing percolating systems. Indeed, among the numerous effective medium approximations, only Bruggeman's symmetrical theory is able to predict a threshold. This characteristic feature of the latter theory puts it in the same category as other mean field theories of critical phenomena.

## Bruggeman's model

### Formulas

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:

#### Circular and spherical inclusions

$\sum _{i}\,\delta _{i}\,{\frac {\sigma _{i}-\sigma _{e}}{\sigma _{i}+(n-1)\sigma _{e}}}\,=\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)$ In a system of Euclidean spatial dimension $n$ that has an arbitrary number of components, the sum is made over all the constituents. $\delta _{i}$ and $\sigma _{i}$ are respectively the fraction and the conductivity of each component, and $\sigma _{e}$ is the effective conductivity of the medium. (The sum over the $\delta _{i}$ 's is unity.)

#### Elliptical and ellipsoidal inclusions

${\frac {1}{n}}\,\delta \alpha +{\frac {(1-\delta )(\sigma _{m}-\sigma _{e})}{\sigma _{m}+(n-1)\sigma _{e}}}\,=\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2)$ This is a generalization of Eq. (1) to a biphasic system with ellipsoidal inclusions of conductivity $\sigma$ into a matrix of conductivity $\sigma _{m}$ . The fraction of inclusions is $\delta$ and the system is $n$ dimensional. For randomly oriented inclusions,

$\alpha \,=\,{\frac {1}{n}}\sum _{j=1}^{n}\,{\frac {\sigma -\sigma _{e}}{\sigma _{e}+L_{j}(\sigma -\sigma _{e})}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(3)$ where the $L_{j}$ '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 {$L_{1}=1/2$ , $L_{2}=1/2$ } and in the case of a sphere {$L_{1}=1/3$ , $L_{2}=1/3$ , $L_{3}=1/3$ }. (The sum over the $L_{j}$ 's is unity.)

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

### Derivation

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

${\overline {p}}\,\propto \,V\,{\frac {\sigma _{1}-\sigma _{e}}{\sigma _{1}+2\sigma _{e}}}\,{\overline {E_{0}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(4)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,.$ This polarization produces a deviation from ${\overline {E_{0}}}$ . If the average deviation is to vanish, the total polarization summed over the two types of inclusion must vanish. Thus

$\delta _{1}{\frac {\sigma _{1}-\sigma _{e}}{\sigma _{1}+2\sigma _{e}}}\,+\,\delta _{2}{\frac {\sigma _{2}-\sigma _{e}}{\sigma _{2}+2\sigma _{e}}}\,=\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(5)$ where $\delta _{1}$ and $\delta _{2}$ are respectively the volume fraction of material 1 and 2. This can be easily extended to a system of dimension $n$ 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   . 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.

### Modeling of percolating systems

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. 

## Maxwell Garnett equation

In the Maxwell Garnett approximation, the effective medium consists of a matrix medium with $\varepsilon _{m}$ and inclusions with $\varepsilon _{i}$ .

### Formula

$\left({\frac {\varepsilon _{\mathrm {eff} }-\varepsilon _{m}}{\varepsilon _{\mathrm {eff} }+2\varepsilon _{m}}}\right)=\delta _{i}\left({\frac {\varepsilon _{i}-\varepsilon _{m}}{\varepsilon _{i}+2\varepsilon _{m}}}\right),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(6)$ where $\varepsilon _{\mathrm {eff} }$ is the effective dielectric constant of the medium, $\varepsilon _{i}$ of the inclusions, and $\varepsilon _{m}$ of the matrix; $\delta _{i}$ is the volume fraction of the inclusions.

The Maxwell Garnett equation is solved by:

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


### Derivation

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:

${\frac {\varepsilon -1}{\varepsilon +2}}={\frac {4\pi }{3}}\sum _{j}N_{j}\alpha _{j}$ Where $N_{j}$ is the number of particles per unit volume. By using elementary electrostatics, we get for a spherical inclusion with dielectric constant $\varepsilon _{i}$ and a radius $a$ a polarisability $\alpha$ :

$\alpha =\left({\frac {\varepsilon _{i}-1}{\varepsilon _{i}+2}}\right)a^{3}$ If we combine $\alpha$ with the Clausius Mosotti equation, we get:

$\left({\frac {\varepsilon _{\mathrm {eff} }-1}{\varepsilon _{\mathrm {eff} }+2}}\right)=\delta _{i}\left({\frac {\varepsilon _{i}-1}{\varepsilon _{i}+2}}\right)$ Where $\varepsilon _{\mathrm {eff} }$ is the effective dielectric constant of the medium, $\varepsilon _{i}$ of the inclusions; $\delta _{i}$ 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:

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

In general terms, the Maxwell Garnett EMA is expected to be valid at low volume fractions $\delta _{i}$ , since it is assumed that the domains are spatially separated and electrostatic interaction between the chosen inclusions and all other neighbouring inclusions is neglected. 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 $\delta _{i}<10^{-5}$ .

## Effective medium theory for resistor networks

For a network consisting of a high density of random resistors, an exact solution for each individual element may be impractical or impossible. In such case, a random resistor network can be considered as a two-dimensional graph and the effective resistance can be modelled in terms of graph measures and geometrical properties of networks. Assuming, edge length << electrode spacing and edges to be uniformly distributed, the potential can be considered to drop uniformly from one electrode to another. Sheet resistance of such a random network ($R_{sn}$ ) can be written in terms of edge (wire) density ($N_{E}$ ), resistivity ($\rho$ ), width ($w$ ) and thickness ($t$ ) of edges (wires) as:

$R_{sn}\,=\,{\frac {\pi }{2}}{\frac {\rho }{w\,t\,{\sqrt {N_{E}}}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(9)$ 