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| '''Effective medium approximations''' or '''effective medium theory''' (sometimes abbreviated as EMA or EMT) pertains to [[computer modeling|analytical]] or [[scientific theory|theoretical]] modeling that describes the [[macroscopic]] properties of [[Advanced composite materials (engineering)|composite material]]s. 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 [[homogeneous|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.<ref name=Cai-book>
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| {{Cite book
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| | last1 = Wenshan | first1 = Cai
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| | first2 = Vladimir | last2 = Shalaev
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| | authorlink = Vladimir Shalaev
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| | title =Optical Metamaterials: Fundamentals and Applications
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| | publisher =Springer
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| | date =November 2009
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| | pages =Chapter 2.4
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| | url =http://books.google.com/?id=q8gDF2pbKXsC&pg=PA59&dq=artificial+dielectrics#v=onepage&q=artificial%20dielectrics&f=false
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| | isbn =978-1-4419-1150-6}}</ref><ref name= wang-pan>
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| {{cite journal
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| | doi=10.1016/j.mser.2008.07.001
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| | url= http://ningpan.net/publications/151-200/156.pdf
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| | format=Free PDF download
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| | title=Predictions of effective physical properties of complex multiphase materials
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| | year=2008
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| | last1=Wang
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| | first1=M
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| | last2=Pan
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| | first2=N
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| | journal=Materials Science and Engineering: R: Reports
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| | volume=63
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| | pages=1}}</ref>
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| ==Applications==
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| 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]].
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| There are many different effective medium approximations,<ref>{{cite journal |last1=Tinga |first1=W. R. |last2=Voss |first2=W. A. G.|last3=Blossey|first3=D. F. |year=1973 |title=Generalized approach to multiphase dielectric mixture theory |journal=J. Appl. Phys. |volume=44 |issue= 9|pages=3897 |url=http://link.aip.org/link/?JAPIAU/44/3897/1 |doi=10.1063/1.1662868|bibcode = 1973JAP....44.3897T }}</ref> 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.
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| The properties under consideration are usually the [[electrical conductivity|conductivity]] <math>\sigma</math> or the [[dielectric constant]] <math>\epsilon</math> 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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| == Bruggeman's Model ==
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| === Formulas ===
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| 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:
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| ==== Circular and spherical inclusions ====
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| <math>\sum_i\,\delta_i\,\frac{\sigma_i - \sigma_e}{\sigma_i + (n-1) \sigma_e}\,=\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)</math>
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| In a system of Euclidean spatial dimension <math> n </math> that has an arbitrary number of components,<ref name=landauer>{{cite conference |url=http://link.aip.org/link/?APCPCS/40/2/1 |title=Electrical conductivity in inhomogeneous media |last1=Landauer |first1=Rolf |date=April 1978 |publisher=American Institute of Physics |accessdate=2010-02-07 |booktitle=AIP Conference Proceedings |volume=40 |pages=2–45 |location=|doi=10.1063/1.31150}}</ref> the sum is made over all the constituents. <math>\delta_i</math> and <math>\sigma_i</math> are respectively the fraction and the conductivity of each component, and <math>\sigma_e</math> is the effective conductivity of the medium. (The sum over the <math>\delta_i</math>'s is unity.)
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| ==== Elliptical and ellipsoidal inclusions ====
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| <math>\frac{1}{n}\,\delta\alpha+\frac{(1-\delta)(\sigma_m - \sigma_e)}{\sigma_m + (n-1)\sigma_e}\,=\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2)</math>
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| This is a generalization of Eq. (1) to a biphasic system with ellipsoidal inclusions of conductivity <math>\sigma</math> into a matrix of conductivity <math>\sigma_m</math>.<ref>{{cite journal|last1=Granqvist|first1=C. G. |last2=Hunderi |first2=O. |year=1978 |title=Conductivity of inhomogeneous materials: Effective-medium theory with dipole-dipole interaction |journal=Phys. Rev. B |volume=18 |issue=4 |pages=1554–1561 |url=http://link.aps.org/doi/10.1103/PhysRevB.18.1554 |doi=10.1103/PhysRevB.18.1554|bibcode = 1978PhRvB..18.1554G }}</ref> The fraction of inclusions is <math>\delta</math> and the system is <math>n</math> dimensional. For randomly oriented inclusions,
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| <math>\alpha\,=\,\sum_{j=1}^{n}\,\frac{\sigma - \sigma_e}{\sigma_e + L_j(\sigma - \sigma_e)}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(3)</math>
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| where the <math>L_j</math>'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 {<math>L_1=1/2</math>, <math>L_2=1/2</math>} and in the case of a sphere {<math>L_1=1/3</math>, <math>L_2=1/3</math>, <math>L_3=1/3</math>}. (The sum over the <math>L_j</math> 's is unity.)
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| The most general case to which the Bruggeman approach has been applied involves bianisotropic ellipsoidal inclusions.<ref name="www3.interscience.wiley">{{cite journal|last1=Weiglhofer|first1=W. S. |last2=Lakhtakia |first2=A. |last3=Michel |first3=B. |year=1998 |title=Maxwell Garnett and Bruggeman formalisms for a particulate composite with bianisotropic host medium|journal=Microw. Opt. Technol. Lett. |volume=15 |issue=4 |pages=263–266 |url=http://www3.interscience.wiley.com/journal/53983/abstract?CRETRY=1&SRETRY=0 |doi=10.1002/(SICI)1098-2760(199707)15:4<263::AID-MOP19>3.0.CO;2-8}}</ref>
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| === Derivation ===
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| The figure illustrates a two-component medium.<ref name=landauer/> Let us consider the cross-hatched volume of conductivity <math>\sigma_1</math>, take it as a sphere of volume <math>V</math> and assume it is embedded in a uniform medium with an effective conductivity <math>\sigma_e</math>. If the [[electric field]] far from the inclusion is <math>\overline{E_0}</math> then elementary considerations lead to a [[Electric dipole moment|dipole moment]] associated with the volume
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| <math>\overline{p}\, \propto \,V\,\frac{\sigma_1 - \sigma_e}{\sigma_1 + 2\sigma_e}\,\overline{E_0}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(4)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,.</math><!-- Deleted image removed: [[File:Bruggeman_Effective_Medium.jpg]] -->
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| This [[polarization density|polarization]] produces a deviation from <math>\overline{E_0}</math>. If the average deviation is to vanish, the total polarization summed over the two types of inclusion must vanish. Thus
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| <math>\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)</math>
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| where <math>\delta_1</math> and <math>\delta_2</math> are respectively the volume fraction of material 1 and 2. This can be easily extended to a system of dimension <math>n</math> that has an arbitrary number of components. All cases
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| can be combined to yield Eq. (1).
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| Eq. (1) can also be obtained by requiring the deviation in current to vanish
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| <ref>{{cite journal |last1=Stroud |first1=D. |year=1975 |title=Generalized effective-medium approach to the conductivity of an inhomogeneous material |journal=Phys. Rev. B |volume=12 |issue=8 |pages=3368–3373 |url=http://link.aps.org/doi/10.1103/PhysRevB.12.3368 |doi=10.1103/PhysRevB.12.3368 |bibcode = 1975PhRvB..12.3368S }}</ref>
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| <ref>{{cite journal |last1=Davidson |first1=A. |last2=Tinkham |first2=M. |year=1976 |title=Phenomenological equations for the electrical conductivity of microscopically inhomogeneous materials |journal=Phys. Rev. B |volume=13 |issue=8 |pages=3261–3267 |url=http://link.aps.org/doi/10.1103/PhysRevB.13.3261 |doi=10.1103/PhysRevB.13.3261 |bibcode = 1976PhRvB..13.3261D }}</ref>
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| . 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).
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| A more general derivation applicable to bianisotropic materials is also available.<ref name="www3.interscience.wiley" />
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| === Modeling of percolating systems ===
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| The main approximation is that all the domains are located in an equivalent mean field.
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| 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.
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| The threshold values are in general not correctly predicted. It is 33% in the EMA, in three dimensions, far
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| from the 16% expected from percolation theory and observed in experiments. However, in
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| two dimensions, the EMA gives a threshold of 50% and has been proven to model percolation
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| relatively well
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| <ref>{{cite journal |last1=Kirkpatrick |first1=Scott |year=1973 |title=Percolation and conduction |journal=Rev. Mod. Phys. |volume=45 |issue=4 |pages=574–588 |url=http://link.aps.org/doi/10.1103/RevModPhys.45.574 |doi=10.1103/RevModPhys.45.574 |bibcode = 1973RvMP...45..574K }}</ref>
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| <ref>{{cite book |title=The Physics of Amorphous Solids |last=Zallen |first=Richard |authorlink= |year=1998 |publisher=Wiley-Interscience |isbn= 978-0-471-29941-7 |page= |pages= }}</ref>
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| <ref>{{cite journal |last1=Rozen |first1=John |last2=Lopez |first2=René |last3=Haglund |first3=Richard F. Jr. |last4=Feldman |first4=Leonard C. |year=2006 |title=Two-dimensional current percolation in nanocrystalline vanadium dioxide films |journal=Appl. Phys. Lett. |volume=88 |issue=8 |pages=081902 |url=http://link.aip.org/link/?APPLAB/88/081902/1 |doi=10.1063/1.2175490 |bibcode = 2006ApPhL..88h1902R }}</ref>
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| .
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| == Maxwell-Garnett Equation ==
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| In the Maxwell Garnett Approximation the effective medium consists of a matrix medium with <math>\varepsilon_m</math> and inclusions with <math>\varepsilon_i</math>.
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| === Formula ===
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| The Maxwell-Garnett equation reads:<ref name=TuckChoy>{{cite book|last=Choy|first=Tuck C.|title=Effective Medium Theory|year=1999|publisher=Clarendon Press|location=Oxford|isbn=978-0-19-851892-1}}</ref>
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| :<math>\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)</math>
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| where <math>\varepsilon_{eff}</math> is the effective dielectric constant of the medium, <math>\varepsilon_i</math> is the one of the inclusions and <math>\varepsilon_m</math> is the one of the matrix; <math>\delta_i</math> is the volume fraction of the inclusions.
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| The Maxwell-Garnett equation is solved by:
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| :<math>\varepsilon_{eff}\,=\,\varepsilon_m\,\frac{2(1 - \delta_i)\varepsilon_m + (1 + 2\delta_i)\varepsilon_i}{(2 + \delta_i)\varepsilon_m + (1 - \delta_i)\varepsilon_i},\,\,\,\,\,\,\,\,(7)</math>
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| so long as the denominator does not vanish. A simple MATLAB calculator using this formula is as follows.
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| <source lang="matlab">
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| % This simple MATLAB calculator computes the effective dielectric
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| % constant of a mixture of an inclusion material in a base medium
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| % according to the Maxwell-Garnett theory as introduced in:
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| % http://en.wikipedia.org/wiki/Effective_Medium_Approximations
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| % INPUTS:
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| % eps_base: dielectric constant of base material;
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| % eps_incl: dielectric constant of inclusion material;
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| % vol_incl: volume portion of inclusion material;
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| % OUTPUT:
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| % eps_mean: effective dielectric constant of the mixture.
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| function [eps_mean] = MaxwellGarnettFormula(eps_base, eps_incl, vol_incl)
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| small_number_cutoff = 1e-6;
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| if vol_incl < 0 || vol_incl > 1
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| disp(['WARNING: volume portion of inclusion material is out of range!']);
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| end
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| factor_up = 2*(1-vol_incl)*eps_base+(1+2*vol_incl)*eps_incl;
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| factor_down = (2+vol_incl)*eps_base+(1-vol_incl)*eps_incl;
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| if abs(factor_down) < small_number_cutoff
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| disp(['WARNING: the effective medium is singular!']);
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| eps_mean = 0;
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| else
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| eps_mean = eps_base*factor_up/factor_down;
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| end
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| </source>
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| === Derivation ===
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| 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]]:
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| :<math>\frac{\varepsilon-1}{\varepsilon+2}= \frac{4\pi}{3} \sum_j N_j \alpha_j</math>
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| By using elementary electrostatics, we get for a spherical inclusion with dielectric constant <math>\varepsilon_i</math> and a radius <math>a</math> a polarisability <math>\alpha</math>:
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| :<math> \alpha = \left( \frac{\varepsilon_i-1}{\varepsilon_i+2} \right) a^{3}</math>
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| If we combine <math>\alpha</math> with the Clausius Mosotti equation, we get:
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| :<math> \left( \frac{\varepsilon_{eff}-1}{\varepsilon_{eff}+2} \right) = \delta_i \left( \frac{\varepsilon_i-1}{\varepsilon_i+2} \right)</math>
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| Where <math>\varepsilon_{eff}</math> is the effective dielectric constant of the medium, <math>\varepsilon_i</math> is the one of the inclusions; <math>\delta_i</math> is the volume fraction of the inclusions.<br />
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| As the model of Maxwell Garnett is a composition of a matrix medium with inclusions we enhance the equation:
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| :<math>\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)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(8)</math>
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| === Validity ===
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| In general terms, the Maxwell Garnett EMA is expected to be valid at low volume fractions <math>\delta_i </math> since it is assumed that the domains are spatially separated
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| .<ref>{{cite journal |last1=Jepsen |first1=Peter Uhd |last2=Fischer |first2=Bernd M. |last3=Thoman|first3=Andreas |last4=Helm|first4=Hanspeter |last5=Suh |first5=J. Y. |last6=Lopez |first6=René | last7= Haglund |first7=R. F. Jr. |year=2006 |title=Metal-insulator phase transition in a VO<sub>2</sub> thin film observed with terahertz spectroscopy |journal=Phys. Rev. B |volume=74 |issue=20 |pages=205103 |url=http://link.aps.org/doi/10.1103/PhysRevB.74.205103 |doi=10.1103/PhysRevB.74.205103 |bibcode = 2006PhRvB..74t5103J }}</ref>
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| ==See also==
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| * [[Constitutive equation]]
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| * [[Percolation threshold]]
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| ==References==
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| {{reflist}}
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| ==Further reading==
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| * {{cite book |title=Selected Papers on Linear Optical Composite Materials [Milestone Vol. 120]|last=Lakhtakia (Ed.) |first=A. |year=1996 |publisher=SPIE Press |location=Bellingham, WA, USA|isbn=0-8194-2152-9 }}
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| * {{cite book |title=Effective Medium Theory|last=Tuck |first=Choy |edition=1st |year=1999 |publisher=Oxford University Press |location=Oxford|isbn=978-0-19-851892-1 }}
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| * {{cite book |title=Electromagnetic Fields in Unconventional Materials and Structures|last=Lakhtakia (Ed.) |first=A. |authorlink=Akhlesh Lakhtakia |year=2000 |publisher=Wiley-Interscience|location=New York|isbn=0-471-36356-1 }}
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| * {{cite book |title=Introduction to Complex Mediums for Optics and Electromagnetics |last1=Weiglhofer (Ed.) |first2=A.|last2=Lakhtakia (Ed.) |authorlink=Akhlesh Lakhtakia |year=2003 |publisher=SPIE Press |location=Bellingham, WA, USA|isbn=0-8194-4947-4 }}
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| * {{cite book |title=Electromagnetic Anisotropy and Bianisotropy: A Field Guide|last1=Mackay |first1=T. G. |last2=Lakhtakia |first2=A. |authorlink=Akhlesh Lakhtakia|edition=1st |year=2010 |publisher=World Scientific |location=Singapore|isbn=978-981-4289-61-0 }}
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| [[Category:Condensed matter physics]]
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| [[Category:Physical chemistry]]
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