In materials science, effective medium approximations (EMA) or effective medium theory (EMT) pertain to analytical or theoretical modeling that describes the macroscopic properties of composite materials. 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 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 including the effective permittivity and permeability of the materials 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, and effective medium theory.T.C. Choy, "Effective Medium Theory", Oxford University Press, (2016) 241 p. There are two widely used formulae.M. Scheller, C. Jansen, M. Koch, "Applications of Effective Medium Theories in the Terahertz Regime" in Recent Optical and Photonic Technologies, ed. by K.Y. Kim, Intech, Croatia, Vukovar (2010), p. 231.
Effective permittivity and permeability are averaged dielectric and magnetic characteristics of a microinhomogeneous medium. They both were derived in quasi-static approximation when the electric field inside a mixture particle may be considered as homogeneous. So, these formulae can not describe the particle size effect. Many attempts were undertaken to improve these formulae.
Applications There are many different effective medium approximations, 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 or the dielectric constant 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.
EMAs can be discrete models, such as applied to resistor networks, or continuum theories as applied to elasticity or viscosity. However, 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.
Bruggeman's model For a mixture of two materials with permittivities and with corresponding volume fractions and , D.A.G. Bruggeman proposed a formula in 1935 of the following form:
Here the positive sign before the square root must be altered to a negative sign in some cases in order to get the correct imaginary part of effective complex permittivity which is related with electromagnetic wave attenuation. The formula is symmetric with respect to swapping the 'd' and 'm' roles. This formula is based on the equality
where is the jump of electric displacement flux all over the integration surface, is the component of microscopic electric field normal to the integration surface, is the local relative complex permittivity which takes the value inside the picked metal particle, the value inside the picked dielectric particle and the value outside the picked particle, is the normal component of the macroscopic electric field. Formula (4) comes out of Maxwell's equality . Thus only one picked particle is considered in Bruggeman's approach. The interaction with all the other particles is taken into account only in a mean field approximation described by . Formula (3) gives a reasonable resonant curve for plasmon excitations in metal nanoparticles if their size is 10 nm or smaller. However, it is unable to describe the size dependence for the resonant frequency of plasmon excitations that are observed in experiments
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 Bruggeman formula takes the form:
Circular and spherical inclusions
In a system of Euclidean spatial dimension that has an arbitrary number of components, the sum is made over all the constituents. and are respectively the fraction and the conductivity of each component, and is the effective conductivity of the medium. (The sum over the 's is unity.)
Elliptical and ellipsoidal inclusions
This is a generalization of Eq. (1) to a biphasic system with ellipsoidal inclusions of conductivity into a matrix of conductivity . The fraction of inclusions is and the system is dimensional. For randomly oriented inclusions,
where the '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 (, ) and in the case of a sphere (, , ). (The sum over the 's is unity.)
The most general case to which the Bruggeman approach has been applied involves bianisotropic ellipsoidal inclusions.
Derivation The figure illustrates a two-component medium. Consider the cross-hatched volume of conductivity , take it as a sphere of volume and assume it is embedded in a uniform medium with an effective conductivity . If the electric field far from the inclusion is then elementary considerations lead to a dipole moment associated with the volume
This polarization produces a deviation from . If the average deviation is to vanish, the total polarization summed over the two types of inclusion must vanish. Thus
where and are respectively the volume fraction of material 1 and 2. This can be easily extended to a system of dimension 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. 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.
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.
Maxwell Garnett equation In the Maxwell Garnett approximation, the effective medium consists of a matrix medium with and inclusions with . Maxwell Garnett was the son of physicist William Garnett, and was named after Garnett's friend, James Clerk Maxwell. He proposed his formula to explain coloured pictures that are observed in glasses doped with metal nanoparticles. His formula has a form
where is effective relative complex permittivity of the mixture, is relative complex permittivity of the background medium containing small spherical inclusions of relative permittivity with volume fraction of . This formula is based on the equality
where is the absolute permittivity of free space and is electric dipole moment of a single inclusion induced by the external electric field . However this equality is good only for homogeneous medium and . Moreover, the formula (1) ignores the interaction between single inclusions. Because of these circumstances, formula (1) gives too narrow and too high resonant curve for plasmon excitations in metal nanoparticles of the mixture.
Formula The Maxwell Garnett equation reads:
where is the effective dielectric constant of the medium, of the inclusions, and of the matrix; is the volume fraction of the inclusions.
The Maxwell Garnett equation is solved by:Levy, O., & Stroud, D. (1997). Maxwell Garnett theory for mixtures of anisotropic inclusions: Application to conducting polymers. Physical Review B, 56(13), 8035.Liu, Tong, et al. "Microporous Co@ CoO nanoparticles with superior microwave absorption properties." Nanoscale 6.4 (2014): 2447-2454.
so long as the denominator does not vanish. A simple MATLAB calculator using this formula is as follows. % This simple MATLAB calculator computes the effective dielectric % constant of a mixture of an inclusion material in a base medium % according to the Maxwell-Garnett theory. % INPUTS: % eps_base: dielectric constant of base material; % eps_incl: dielectric constant of inclusion material; % vol_incl: volume portion of inclusion material; % OUTPUT: % eps_mean: effective dielectric constant of the mixture.
function eps_mean = MaxwellGarnettFormula(eps_base, eps_incl, vol_incl)
small_number_cutoff = 1e-6;
if vol_incl < 0 || vol_incl > 1 disp('WARNING: volume portion of inclusion material is out of range!'); end factor_numer = 2 * (1 - vol_incl) * eps_base + (1 + 2 * vol_incl) * eps_incl; factor_denom = (2 + vol_incl) * eps_base + (1 - vol_incl) * eps_incl; if abs(factor_denom) < small_number_cutoff disp('WARNING: the effective medium is singular!'); eps_mean = 0; else eps_mean = eps_base * factor_numer / factor_denom; end end
Derivation 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:
Where is the number of particles per unit volume. By using elementary electrostatics, we get for a spherical inclusion with dielectric constant and a radius a polarisability :
If we combine with the Clausius Mosotti equation, we get:
Where is the effective dielectric constant of the medium, of the inclusions; 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:
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 separated and electrostatic interaction between the chosen inclusions and all other neighbouring inclusions is neglected. The Maxwell Garnett formula, in contrast to Bruggeman formula, ceases to be correct when the inclusions become resonant. In the case of plasmon resonance, the Maxwell Garnett formula is correct only at volume fraction of the inclusions . The applicability of effective medium approximation for dielectric multilayers and metal-dielectric multilayers have been studied, showing that there are certain cases where the effective medium approximation does not hold and one needs to be cautious in application of the theory.
Generalization of the Maxwell Garnett equation to describe the nanoparticle size distribution