Effective medium approximations

Effective medium approximations or effective medium theory (sometimes abbreviated as EMA or EMT) pertains 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

They can be discrete models such as applied to resistor networks or continuum theories as applied to elasticity or viscosity but 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.

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.

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. Let us 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

The Maxwell Garnett equation reads:

$\left( \frac{\varepsilon_\mathrm{eff}-\varepsilon_m}{\varepsilon_\mathrm{eff}+2\varepsilon_m} \right) =\delta_i \left( \frac{\varepsilon_i-\varepsilon_m}{\varepsilon_i+2\varepsilon_m}\right),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(6)$

where $\varepsilon_\mathrm{eff}$ is the effective dielectric constant of the medium, $\varepsilon_i$ is the one of the inclusions and $\varepsilon_m$ is the one of the matrix; $\delta_i$ is the volume fraction of the inclusions.

The Maxwell Garnett equation is solved by:

$\varepsilon_\mathrm{eff}\,=\,\varepsilon_m\,\frac{2\delta_i(\varepsilon_i - \varepsilon_m) + \varepsilon_i + 2\varepsilon_m}{2\varepsilon_m + \varepsilon_i + \delta_i(\varepsilon_m-\varepsilon_i)},\,\,\,\,\,\,\,\,(7)$

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, $\varepsilon_i$ is the one 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 .

See also

• Constitutive equation
• Percolation threshold