Effective medium theories

Simple effective medium calculations can be done here.
EMC -- the much more complex effective media calculator is created by Volker Ossenkopf.
The "available electromagnetic scattering programs" page is maintained by Thomas Wriedt (Bremen).

The different effective medium theories available on the market have all the same origin, i.e. Maxwell's equations for the static limit. The Bergman theorem (D. Bergman, "The dielectric constant of a composite material - a problem in classical physics", Physics Reports 43, 377-407, [1978]) simply results from this fact. The theorem connects the effective dielectric function of a two-phase composite with the (micro)geometry information about the composite (described by the spectral function G[L]) and the dielectric functions of the two components:

,
where f is the volume filling factor of component 1.

In words, the theorem states that

all possible geometric resonances of a two-phase composite occur for real values of the variable t in the interval [0,1].
with the integration over L, one scans all possible resonance positions.
whether a resonance occurs or not is determined by the spectral function G(L) which carries all geometry information.

Therefore, the spectral representation clearly distinguishes between the influence of the geometrical quantities and that of the dielectric properties of the components on the effective behaviour of the system. The spectral representation generally holds as long as the quasistatic approximation is valid. No further restrictions have to be made.

Although the spectral function G(L) is generally unknown for an arbitrary two-phase composite, it's analytically known or can be numerically derived for any existing mixing rule (Stognienko et al. 1995). Because each spectral function has to be non-negative, normalized to unity in the interval [0,1], and obey (for isotropic systems) the first moment equation
,
the derivation of G(L) and the check whether these restrictions are fulfiled or not is useful to valididate any mixing rule. Note, there are mixing rules in the literature which are not correct with respect to the Bergman spectral representation. E.g. the spectral function for the mixing rule by C.J.F. Böttcher (in: "Theory of Electric Polarization", Elsevier, Amsterdam, p. 415 [1952])

which fulfiles only the normalization restriction.

In the following table some mixing rules and their corresponding spectral functions are listed.


Mixing rule				 Spectral function
Bruggeman

(D.A.G. Bruggeman, "Berechnung verschiedener physikalischer Konstanten

von heterogenen Substanzen", Ann. Phys. (Leipzig) 24, 636-679 [1935])




Maxwell Garnett

(J.C.M. Garnett, "Colours in metal glasses and in metallic films",

Phil. Trans. R. Soc. Lond. 203, 385-420 [1904])




Looyenga

(H. Looyenga, "Dielectric constants of heterogeneous mixtures",

Physica 31, 401-406 [1965])




PCA/CCA aggregate equivalent

(Stognienko et al. 1995, Henning and Stognienko 1996)




Monecke

(J. Monecke, "Bergman spectral representation of a simple expression

for the dielectric response of a symmetric two-component composite",

J. Phys.: Cond. Mat. 6, 907-912 [1994])




Hollow sphere equivalent

(see, e.g., C.F. Bohren and D.R. Huffman "Absorption and Scattering

of Light by Small Particles", Wiley, New York, p. 149 [1983])

Mixing rule	Spectral function
Bruggeman (D.A.G. Bruggeman, "Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen", Ann. Phys. (Leipzig) 24, 636-679 [1935])
Maxwell Garnett (J.C.M. Garnett, "Colours in metal glasses and in metallic films", Phil. Trans. R. Soc. Lond. 203, 385-420 [1904])
Looyenga (H. Looyenga, "Dielectric constants of heterogeneous mixtures", Physica 31, 401-406 [1965])
PCA/CCA aggregate equivalent (Stognienko et al. 1995, Henning and Stognienko 1996)
Monecke (J. Monecke, "Bergman spectral representation of a simple expression for the dielectric response of a symmetric two-component composite", J. Phys.: Cond. Mat. 6, 907-912 [1994])
Hollow sphere equivalent (see, e.g., C.F. Bohren and D.R. Huffman "Absorption and Scattering of Light by Small Particles", Wiley, New York, p. 149 [1983])

Back to the general home page of Astronomy and Astrophysics in Jena.

R. Stognienko, Feb 1997