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.
Contents
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
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
In a system of Euclidean spatial dimension
Elliptical and ellipsoidal inclusions
This is a generalization of Eq. (1) to a biphasic system with ellipsoidal inclusions of conductivity
where the
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
This polarization produces a deviation from
where
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
Formula
The Maxwell Garnett equation reads:
where
The Maxwell Garnett equation is solved by:
so long as the denominator does not vanish. A simple MATLAB calculator using this formula is as follows.
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:
Where
If we combine
Where
As the model of Maxwell Garnett is a composition of a matrix medium with inclusions we enhance the equation:
Validity
In general terms, the Maxwell Garnett EMA is expected to be valid at low volume fractions