Effective medium approximations

Introduction

These analytical models describe the macroscopic properties of a medium based on the properties and the relative fractions of its components. They are continuous theories and do not relate directly to percolating systems. Indeed, among the numerous effective medium approximations (EMA or EMT), only Bruggeman’s symmetrical theory is able to predict a threshold.

There are many different effective medium approximations¹, each of them being more or less accurate in distinct conditions. Nevertheless, they all assume that the macroscopic system is homogeneous and generally fail to predict the properties of a multiphasic medium close to the percolation threshold where it is highly asymmetrical.

The properties under consideration are usually the conductivity $\sigma$ or the dielectric constant $\epsilon$ of the medium. These values are interchangeable in the formulas.

Bruggeman's Model

Formulas

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 dimension $n$ that has an arbitrary number of components². The sum is made over all the constituents. $\delta _{i}$ and $\sigma _{i}$ are respectively the fraction and the conductivity of each component, $\sigma _{e}$ is the 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 $\sigma _{m}$³. The fraction of inclusions is $\delta$ and the system is $n$ dimensional. For randomly oriented inclusions;

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

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)

Derivation

The figure illustrates a two-component medium². 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

${\overline {p}}\,\propto \,V\,{\frac {\sigma _{1}-\sigma _{e}}{\sigma _{1}+2\sigma _{e}}}\,{\overline {E_{0}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(4)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,.$File:Bruggeman Effective Medium.jpg

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^4,5. 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).

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 highly anisotropic. Also, at this stage the shape of the clusters themselves are far from spherical and the effective medium approximation is not expected to model the system well in that region. Indeed, the threshold predicted by the EMA in three dimensions is 33%, 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^6,7,8.

Maxwell-Garnett's Equation

Formula

$\epsilon _{e}\,=\,\epsilon \,{\frac {\epsilon _{m}(1+2\delta )-\epsilon (2\delta -2)}{\epsilon (2+\delta )+\epsilon _{m}(1-\delta )}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(6)$

where $\epsilon _{e}$ is the effective dielectric constant of the medium, $\epsilon _{i}$ is the one of the inclusions and $\epsilon _{m}$ is the one of the matrix; $\delta$ is the volume fraction of the embedded material.

Validity

In general terms, the Maxwell-Garnett EMA is expected to be valid at low volume fractions since it is assumed that the domains are spatially separated⁹.

Page started by John Rozen aka Goldenvu 04:37, 24 February 2007 (UTC)