Effective medium theory

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.

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 ${\displaystyle \sigma }$ or the dielectric constant ${\displaystyle \epsilon }$ 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.

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