Effective medium approximations

Effective medium approximations or effective medium theory (sometimes abbreviated as EMA or EMT) are physical models that describe the macroscopic properties of a medium based on the properties and the relative fractions of its components. 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^[1], 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 σ 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 celebrated 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^[2], the sum is made over all the constituents. δ_i and σ_i are respectively the fraction and the conductivity of each component, and σ_e is the effective conductivity of the medium. (The sum over the δ_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 σ into a matrix of conductivity σ_m^[3]. The fraction of inclusions is δ 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 {L₁ = 1 / 2, L₂ = 1 / 2} and in the case of a sphere {L₁ = 1 / 3, L₂ = 1 / 3, L₃ = 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.^[4]

Derivation

The figure illustrates a two-component medium^[2]. Let us consider the cross-hatched volume of conductivity σ₁, take it as a sphere of volume V and assume it is embedded in a uniform medium with an effective conductivity σ_e. If the electric field far from the inclusion is $\overline{E_0}$ then elementary considerations lead to a dipole moment^{[disambiguation needed ]} associated with the volume

$\overline{p}\, \propto \,V\,\frac{\sigma_1 - \sigma_e}{\sigma_1 + 2\sigma_e}\,\overline{E_0}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(4)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,.$

This polarization^{[disambiguation needed ]} 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 δ₁ and δ₂ 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 ^{[5] [6]} . 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.^[7]

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 ^{[8] [9] [10]} .

Maxwell Garnett's Equation

In the Maxwell Garnett Approximation the effective medium consists of a matrix medium with ε_m and inclusions with ε_i.

Derivation

For the derivation of the Maxwell-Garnett equation we start with an array of polarizable particles. Only by using the Lorentz local field concept, it is straightforward to get the Clausius Mosotti equation.

$\frac{\varepsilon-1}{\varepsilon+2}= \frac{4\pi}{3} \sum_j N_j \alpha_j$

By using elementary electrostatic, we get for a spherical inclusion with dielectric constant ε_i and a radius a a polarisability α:

$\alpha = \left( \frac{\varepsilon-1}{\varepsilon+2} \right) a^{3}$

If we combine α with the Clausius Mosotti eqation, we get:

$\left( \frac{\varepsilon_{eff}-1}{\varepsilon_{eff}+2} \right) = \delta_i \left( \frac{\varepsilon_i-1}{\varepsilon_i+2} \right)$

Where $\varepsilon_{eff}$ is the effective dielectric constant of the medium, ε_i is the one of the inclusions; δ_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:

Formula

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

This is the Maxwell Garnett equation.^[11]
Where $\varepsilon_{eff}$ is the effective dielectric constant of the medium, ε_i is the one of the inclusions and ε_m is the one of the matrix; δ_i is the volume fraction of the inclusions.
Only if $\varepsilon \in \mathbb{R}$ we can simplify the Maxwell Garnett equation to:

$\varepsilon_{eff}\,=\,\varepsilon_m\,\frac{\varepsilon_i(1 + 2\delta_i) - \varepsilon_m(2\delta_i - 2)}{\varepsilon_m(2 + \delta_i) + \varepsilon_i(1 - \delta)}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(7)$

where $\varepsilon_{eff}$ is the effective dielectric constant of the medium, ε_i is the one of the inclusions and ε_m is the one of the matrix; δ 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 δ_i since it is assumed that the domains are spatially separated ^[12].

See also

• Percolation threshold

References

1. ^ Tinga, W. R.; Voss, W. A. G.; Blossey, D. F. (1973). "Generalized approach to multiphase dielectric mixture theory". J. Appl. Phys. 44 (9): 3897. doi:10.1063/1.1662868. http://link.aip.org/link/?JAPIAU/44/3897/1. 
2. ^ ^{a b} Landauer, Rolf (April 1978). "Electrical conductivity in inhomogeneous media". AIP Conference Proceedings. 40. American Institute of Physics. pp. 2–45. doi:10.1063/1.31150. http://link.aip.org/link/?APCPCS/40/2/1. Retrieved 2010-02-07. 
3. ^ Granqvist, C. G.; Hunderi, O. (1978). "Conductivity of inhomogeneous materials: Effective-medium theory with dipole-dipole interaction". Phys. Rev. B 18 (4): 1554–1561. doi:10.1103/PhysRevB.18.1554. http://link.aps.org/doi/10.1103/PhysRevB.18.1554. 
4. ^ Weiglhofer, W. S.; Lakhtakia, A.; Michel, B. (1998). "Maxwell Garnett and Bruggeman formalisms for a particulate composite with bianisotropic host medium". Microw. Opt. Technol. Lett. 15 (4): 263–266. doi:10.1002/(SICI)1098-2760(199707)15:4<263::AID-MOP19>3.0.CO;2-8. http://www3.interscience.wiley.com/journal/53983/abstract?CRETRY=1&SRETRY=0. 
5. ^ Stroud, D. (1975). "Generalized effective-medium approach to the conductivity of an inhomogeneous material". Phys. Rev. B 12 (8): 3368–3373. doi:10.1103/PhysRevB.12.3368. http://link.aps.org/doi/10.1103/PhysRevB.12.3368. 
6. ^ Davidson, A.; Tinkham, M. (1976). "Phenomenological equations for the electrical conductivity of microscopically inhomogeneous materials". Phys. Rev. B 13 (8): 3261–3267. doi:10.1103/PhysRevB.13.3261. http://link.aps.org/doi/10.1103/PhysRevB.13.3261. 
7. ^ Weiglhofer, W. S.; Lakhtakia, A.; Michel, B. (1998). "Maxwell Garnett and Bruggeman formalisms for a particulate composite with bianisotropic host medium". Microw. Opt. Technol. Lett. 15 (4): 263–266. doi:10.1002/(SICI)1098-2760(199707)15:4<263::AID-MOP19>3.0.CO;2-8. http://www3.interscience.wiley.com/journal/53983/abstract?CRETRY=1&SRETRY=0. 
8. ^ Kirkpatrick, Scott (1973). "Percolation and conduction". Rev. Mod. Phys. 45 (4): 574–588. doi:10.1103/RevModPhys.45.574. http://link.aps.org/doi/10.1103/RevModPhys.45.574. 
9. ^ Zallen, Richard (1998). The Physics of Amorphous Solids. Wiley-Interscience. ISBN 978-0471299417. 
10. ^ Rozen, John; Lopez, René; Haglund, Richard F. Jr.; Feldman, Leonard C. (2006). "Two-dimensional current percolation in nanocrystalline vanadium dioxide films". Appl. Phys. Lett. 88 (8): 081902. doi:10.1063/1.2175490. http://link.aip.org/link/?APPLAB/88/081902/1. 
11. ^ Choy, Tuck C. (1999). Effective Medium Theory. Oxford: Clarendon Press. ISBN 978-0-19-851892-1. 
12. ^ Jepsen, Peter Uhd; Fischer, Bernd M.; Thoman, Andreas; Helm, Hanspeter; Suh, J. Y.; Lopez, René; Haglund, R. F. Jr. (2006). "Metal-insulator phase transition in a VO₂ thin film observed with terahertz spectroscopy". Phys. Rev. B 74 (20): 205103. doi:10.1103/PhysRevB.74.205103. http://link.aps.org/doi/10.1103/PhysRevB.74.205103. 

Further reading

• Lakhtakia (Ed.), A. (1996). Selected Papers on Linear Optical Composite Materials [Milestone Vol. 120]. Bellingham, WA, USA: SPIE Press. ISBN 0-8194-2152-9. 

• Tuck, Choy (1999). Effective Medium Theory (1st ed.). Oxford: Oxford University Press. ISBN 978-0-19-851892-1. 

• Lakhtakia (Ed.), A. (2000). Electromagnetic Fields in Unconventional Materials and Structures. New York: Wiley-Interscience. ISBN 0-471-36356-1. 

• Weiglhofer (Ed.); Lakhtakia (Ed.), A. (2003). Introduction to Complex Mediums for Optics and Electromagnetics. Bellingham, WA, USA: SPIE Press. ISBN 0-819-44947-4. 

• Mackay, T. G.; Lakhtakia, A. (2010). Electromagnetic Anisotropy and Bianisotropy: A Field Guide (1st ed.). Singapore: World Scientific. ISBN 978-981-4289-61-0.