Effective Medium Theory

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An effective medium theory ( English effective medium theory , EMT) describes analytical or theoretical models for describing the macroscopic properties of mixed bodies . The properties of these materials consisting of a mixture of different substances are developed from the (weighted) averaging of the respective values ​​of the components. This also applies if the values ​​of the materials vary and are inhomogeneous at the individual component level, which makes an exact calculation of the many component values ​​almost impossible. The EM theories developed, however, provide acceptable approximations which in turn describe useful parameters and properties of the composite material as a whole.

Applications

There are many different effective medium theories, each of which is more or less accurate under certain conditions. The approximations can be discrete models as applied to resistance networks, or continuum theories as applied to elasticity or viscosity . However, they all have in common that they assume a homogeneous macroscopic system and, typical for all averaging field theories, cannot predict the properties of a multiphase medium near the percolation threshold, since in theory there are no correlations over large distances or critical fluctuations.

The properties considered are usually the electrical conductivity or the dielectric constant of the medium. These parameters are interchangeable in the formulas of a number of models because of the broad applicability of Laplace's equation . The problems outside this class are mainly in the area of ​​elasticity and hydrodynamics , since the constants of the effective medium have a higher order tensor character. ${\displaystyle \sigma }$ ${\displaystyle \epsilon }$

Both in practice most commonly used models are the effective medium approximation ( English approximation effective medium , EMA) of Bruggeman and the Maxwell-Garnett theory (MGT). Both are based on the Clausius-Mossotti relationship , which provides the connection between macroscopic and microscopic parameters of a medium. The Bergman-Milton representation is regarded as the most general approach for an effective medium , in which the geometric properties of complex plasmonic composite materials are even defined as geometric functions which are correlated via a spectral density function .

Bruggeman's effective medium approximation

In 1935, Dirk Anton George Bruggeman developed formulas for calculating the dielectric , magnetic and optical properties of heterogeneous materials. Without loss of generality, we will consider the investigation of the effective conductivity (for direct or alternating current) of a system that consists of spherical multicomponent inclusions with different arbitrary conductivities. In this case the Bruggeman's formula takes the following forms.

Circular and spherical inclusions

${\displaystyle \sum _{i}\,\delta _{i}\,{\frac {\sigma _{i}-\sigma _{e}}{\sigma _{i}+(n-1)\sigma _{e}}}\,=\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)}$

In a system of Euclidean spatial dimensions with any number of components, the sum of all components is formed. and are here in each case the proportion and the conductivity of each component, and is the effective conductivity of the medium. (The sum over all is 1.) ${\displaystyle n}$ ${\displaystyle \delta _{i}}$ ${\displaystyle \sigma _{i}}$ ${\displaystyle \sigma _{e}}$ ${\displaystyle \delta _{i}}$

Elliptical and ellipsoidal inclusions

${\displaystyle {\frac {1}{n}}\,\delta \alpha +{\frac {(1-\delta )(\sigma _{m}-\sigma _{e})}{\sigma _{m}+(n-1)\sigma _{e}}}\,=\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2)}$

This is a generalization of equation (1) to a two-phase system with ellipsoidal inclusions of conductivity in a matrix of conductivity . The proportion of inclusions is and the system is -dimensional. For randomly oriented inclusions, ${\displaystyle \sigma }$ ${\displaystyle \sigma _{m}}$ ${\displaystyle \delta }$ ${\displaystyle n}$

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

where denotes the appropriate double / triplet of the depolarization factors, which is determined by the relationships between the axis of the ellipse / ellipsoid. For example: in a circle { , } and {a ball , , }. (The sum over all is 1.) ${\displaystyle L_{j}}$ ${\displaystyle L_{1}=1/2}$ ${\displaystyle L_{2}=1/2}$ ${\displaystyle L_{1}=1/3}$ ${\displaystyle L_{2}=1/3}$ ${\displaystyle L_{3}=1/3}$ ${\displaystyle L_{j}}$

The most general case to which the Bruggeman approach has been applied concerns bianisotropic ellipsoidal inclusions.

Derivation

The illustration shows a two-component medium. The hatched volume of conductivity is viewed as a sphere of the volume and it is assumed that it is embedded in a uniform medium with an effective conductivity . If the electric field is in the distance , then elementary considerations lead to a dipole moment associated with the volume. ${\displaystyle \sigma _{1}}$ ${\displaystyle V}$ ${\displaystyle \sigma _{e}}$ ${\displaystyle {\overline {E_{0}}}}$

${\displaystyle {\overline {p}}\,\propto \,V\,{\frac {\sigma _{1}-\sigma _{e}}{\sigma _{1}+2\sigma _{e}}}\,{\overline {E_{0}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(4)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}$

This polarization creates a deviation from . If the mean deviation is to disappear, the entire polarization that is summed over the two types of inclusions must disappear. So ${\displaystyle {\overline {E_{0}}}}$

${\displaystyle \delta _{1}{\frac {\sigma _{1}-\sigma _{e}}{\sigma _{1}+2\sigma _{e}}}\,+\,\delta _{2}{\frac {\sigma _{2}-\sigma _{e}}{\sigma _{2}+2\sigma _{e}}}\,=\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(5)}$

where and are the volume fraction of material 1 and 2, respectively. ${\displaystyle \delta _{1}}$ ${\displaystyle \delta _{2}}$

This equation can easily be extended to a system of dimension that has any number of components. All cases can be combined to get equation (1). ${\displaystyle n}$

Equation (1) can also be obtained in that the current deviation must vanish. It was derived here from the assumption that the inclusions are spherical and can be modified for shapes with other depolarization factors; which equation (2) leads.

Modeling of percolation systems

The main assumption of the approximation is that all domains are in an equivalent middle field. Unfortunately for a system close to the percolation threshold this is not the case. Here the system will be determined by the largest (fractal) group of leaders and by far-reaching correlations, which, however, do not exist in Bruggeman's simple formula. The thresholds are generally not correctly predicted. A three-dimensional model results in 33%, which is a far cry from the 16% expected from percolation theory and observed in experiments. In two dimensions, however, there is a threshold value of 50% and it has been shown that this models the percolation relatively well.

Maxwell-Garnett theory

In the Maxwell-Garnett equation , the effective medium consists of a matrix medium and inclusions, it reads:

${\displaystyle \left({\frac {\varepsilon _{\mathrm {eff} }-\varepsilon _{m}}{\varepsilon _{\mathrm {eff} }+2\varepsilon _{m}}}\right)=\delta _{i}\left({\frac {\varepsilon _{i}-\varepsilon _{m}}{\varepsilon _{i}+2\varepsilon _{m}}}\right),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(6)}$

where are the effective dielectric constant of the medium, inclusions and matrix. is this the volume fraction of the inclusions. ${\displaystyle \varepsilon _{\mathrm {eff} }}$ ${\displaystyle \varepsilon _{i}}$ ${\displaystyle \varepsilon _{m}}$ ${\displaystyle \delta _{i}}$

The Maxwell-Garnett equation is solved by:

${\displaystyle \varepsilon _{\mathrm {eff} }\,=\,\varepsilon _{m}\,{\frac {2\delta _{i}(\varepsilon _{i}-\varepsilon _{m})+\varepsilon _{i}+2\varepsilon _{m}}{2\varepsilon _{m}+\varepsilon _{i}-\delta _{i}(\varepsilon _{i}-\varepsilon _{m})}},\,\,\,\,\,\,\,\,(7)}$

as long as the denominator does not disappear.

Derivation

An arrangement of polarizable particles is assumed for the derivation of the Maxwell-Garnett equation . Using the Lorentz local field concept one obtains the Clausius-Mossotti relationship :

${\displaystyle {\frac {\varepsilon -1}{\varepsilon +2}}={\frac {4\pi }{3}}\sum _{j}N_{j}\alpha _{j}}$

where is the number of particles per unit volume. ${\displaystyle N_{j}}$

By using elementary electrostatics, we get a polarizability for a spherical inclusion with the dielectric constant and a radius : ${\displaystyle \varepsilon _{i}}$ ${\displaystyle a}$ ${\displaystyle \alpha }$

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

When combined with the Clausius-Mosotti equation, one obtains: ${\displaystyle \alpha }$

${\displaystyle \left({\frac {\varepsilon _{\mathrm {eff} }-1}{\varepsilon _{\mathrm {eff} }+2}}\right)=\delta _{i}\left({\frac {\varepsilon _{i}-1}{\varepsilon _{i}+2}}\right)}$

Here, the effective dielectric constant of the medium, the inclusions. is the volume fraction of the inclusions. ${\displaystyle \varepsilon _{\mathrm {eff} }}$ ${\displaystyle \varepsilon _{i}}$ ${\displaystyle \delta _{i}}$

Since the Maxwell and Garnett model is a composite of a matrix medium with inclusions, the equation can be transformed:

${\displaystyle \left({\frac {\varepsilon _{\mathrm {eff} }-\varepsilon _{m}}{\varepsilon _{\mathrm {eff} }+2\varepsilon _{m}}}\right)=\delta _{i}\left({\frac {\varepsilon _{i}-\varepsilon _{m}}{\varepsilon _{i}+2\varepsilon _{m}}}\right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(8)}$

validity

It is generally assumed that the Maxwell-Garnett approximation is valid at low volume fractions , since it is assumed that the domains are spatially separated and the electrostatic interaction between the selected inclusions and all other neighboring inclusions is neglected. In contrast to the Bruggeman equation, the Maxwell-Garnett equation ceases to be correct when the inclusions become resonant. In the case of plasmon resonance, the Maxwell-Garnett equation is only valid for a volume fraction of the inclusions . ${\displaystyle \delta _{i}}$ ${\displaystyle \delta _{i}<10^{-5}}$

Resistor Networks

For a network made up of a large number of random resistors, an exact solution for each individual element can be impractical or impossible. In such a case, such a resistor network can be viewed as a two-dimensional graph and the effective resistance can be modeled in the form of graph dimensions and geometric properties of networks. Assuming that the edge length is much smaller than the electrode spacing and the edges are evenly distributed, the potential from one electrode to the other can be assumed to fall evenly. The sheet resistance of the network ( ) can be written in terms of the edge (wire) number ( ), the resistivity ( ), the width ( ) and the thickness ( ) of the edges as follows: ${\displaystyle R_{sn}}$ ${\displaystyle N_{E}}$ ${\displaystyle \rho }$ ${\displaystyle w}$ ${\displaystyle t}$

${\displaystyle R_{sn}\,=\,{\frac {\pi }{2}}{\frac {\rho }{w\,t\,{\sqrt {N_{E}}}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(9)}$

literature

• Akhlesh Lakhtakia (Ed.): Selected Papers on Linear Optical Composite Materials (=  SPIE milestone series . Vol. 120). SPIE Press, Bellingham, WA, USA 1996, ISBN 0-8194-2152-9 .
• Choy Tuck: Effective Medium Theory . Oxford University Press, Oxford 1999, ISBN 0-19-851892-7 .
• Akhlesh Lakhtakia (Ed.): Electromagnetic Fields in Unconventional Materials and Structures . Wiley-Interscience, New York 2000, ISBN 0-471-36356-1 .
• A. Weiglhofer, Akhlesh Lakhtakia (Ed.): Introduction to Complex Mediums for Optics and Electromagnetics . SPIE Press, Bellingham, WA, USA 2003, ISBN 0-8194-4947-4 .
• TG Mackay , A. Lakhtakia: Electromagnetic Anisotropy and Bianisotropy: A Field Guide , 1st Edition, World Scientific, Singapore 2010, ISBN 978-981-4289-61-0 .

Individual evidence

1. ^ Cai Wenshan, Vladimir Shalaev: Optical Metamaterials: Fundamentals and Applications . Springer, 2009, ISBN 978-1-4419-1150-6 , Chapter 2.4 Electric Metamaterials ( limited preview in Google book search).
2. ^ M. Wang, N. Pan: Predictions of effective physical properties of complex multiphase materials . In: Materials Science and Engineering: R: Reports . tape 63 , no. 1 , December 20, 2008, p. 1–30 , doi : 10.1016 / j.mser.2008.07.001 ( ningpan.net [PDF]).
3. ↑ A. Piegari: Optical thin films and coatings - From materials to applications (=  Woodhead Publishing Series in Electronic and Optical Materials . Band 49 ). Woodhead Publ., London 2013, ISBN 978-0-85709-594-7 , pp. 154 .
4. ^ AH Sihvola: Electromagnetic Mixing Formulas and Applications . The Institution of Electrical Engineers, London 1999, ISBN 0-85296-772-1 .
5. ↑ DAG Bruggeman: Calculation of various physical constants of heterogeneous substances. I. Dielectric constants and conductivities of mixed bodies made of isotropic substances . In: Annals of Physics, 5th episode . tape 24 , no. 7 December 1935, p. 636--664 , doi : 10.1002 / andp.19354160705 ( digitized on Gallica ).
6. ↑ DAG Bruggeman: Calculation of various physical constants of heterogeneous substances. I. Dielectric constants and conductivities of mixed bodies made of isotropic substances (conclusion) . In: Annals of Physics, 5th episode . tape 24 , no. 8 December 1935, p. 665--679 , doi : 10.1002 / andp.19354160705 ( digitized on Gallica ).
7. ^ ^{A b} Rolf Landauer: Electrical conductivity in inhomogeneous media . In: AIP Conference Proceedings . tape 40 . American Institute of Physics, April 1978, pp. 2-45 , doi : 10.1063 / 1.31150 .
8. ↑ CG Granqvist, O. Hunderi: Conductivity of inhomogeneous materials: Effective-medium theory with dipole-dipole interaction . In: Physical Review B . tape 18 , no. 4 , August 1978, p. 1554–1561 , doi : 10.1103 / PhysRevB.18.1554 .
9. ↑ Werner S. Weiglhofer, Akhlesh Lakhtakia, Bernhard Michel: Maxwell Garnett and Bruggeman formalisms for a particulate composite with bianisotropic host medium . In: Microwave and Optical Technology Letters . tape 15 , no. 4 , July 1997, p. 263-266 , doi : 10.1002 / (SICI) 1098-2760 (199707) 15: 4 <263 :: AID-MOP19> 3.0.CO; 2-8 .
10. ↑ D. Stroud: Generalized effective-medium approach to the conductivity of an inhomogeneous material . In: Physical Review B . tape 12 , no. 8 , October 1975, p. 3368-3373 , doi : 10.1103 / PhysRevB.12.3368 .
11. ^ A. Davidson, M. Tinkham: Phenomenological equations for the electrical conductivity of microscopically inhomogeneous materials . In: Physical Review B . tape 13 , no. 8 April 1976, p. 3261-3267 , doi : 10.1103 / PhysRevB.13.3261 .
12. ^ Scott Kirkpatrick: Percolation and Conduction . In: Reviews of Modern Physics . tape 45 , no. 4 , October 1973, p. 574-588 , doi : 10.1103 / RevModPhys.45.574 .
13. ↑ Richard Zallen: The Physics of Amorphous Solids . Wiley-Interscience, 1998, ISBN 0-471-29941-3 .
14. ^ John Rozen, René Lopez, Richard F. Haglund, Leonard C. Feldman: Two-dimensional current percolation in nanocrystalline vanadiumdioxide films . In: Applied Physics Letters . tape 88 , no. 8 , February 20, 2006, p. 081902 , doi : 10.1063 / 1.2175490 .
15. ^ Tuck C. Choy: Effective Medium Theory . Clarendon Press, Oxford 1999, ISBN 978-0-19-851892-1 .
16. ^ Ohad Levy, David Stroud: Maxwell Garnett theory for mixtures of anisotropic inclusions: Application to conducting polymers . In: Physical Review B . tape 56 , no. October 13 , 1997, p. 8035 , doi : 10.1103 / PhysRevB.56.8035 .
17. ^ Tong Liu, Y. u. Pang, M. et al. Zhu, Satoru Kobayashi: Microporous Co @ CoO nanoparticles with superior microwave absorption properties . In: Nanoscale . tape 6 , no. 4 , 2014, p. 2447--2454 , doi : 10.1039 / C3NR05238A .
18. ↑ Peter Uhd Jepsen, Bernd M. Fischer, Andreas Thoman, Hanspeter Helm, JY Suh, René Lopez, RF Haglund: Metal-insulator phase transition in a thin film observed with terahertz spectroscopy . In: Physical Review B . tape 74 , no. November 20 , 2006, pp. 205103 , doi : 10.1103 / PhysRevB.74.205103 ( dtu.dk ).
19. ↑ BA Belyaev, VV Tyurnev: Electrodynamic Calculation of Effective Electromagnetic Parameters of a Dielectric Medium with Metallic Nanoparticles of a Given Size . In: Journal of Experimental and Theoretical Physics . tape 127 , no. 4 , October 2018, p. 608-619 , doi : 10.1134 / S1063776118100114 .
20. ↑ Ankush Kumar, NS Vidhyadhiraja, Giridhar U. Kulkarni: Current distribution in conducting nanowire networks . In: Journal of Applied Physics . tape 122 , no. 4 , July 28, 2017, p. 045101 , doi : 10.1063 / 1.4985792 .