Maxwell-Garnett mixing rule in the presence of multiple scattering: Derivation and accuracy

We give a rigorous and original derivation of the Maxwell-Garnett mixing rule in the dynamical regime for a composite dielectric random medium with small spherical inclusions. For certain configurations of scatterers, we show that contrarily to the common belief, the Maxwell-Garnett formula can remain very accurate at a high concentration of scatterers and incorporate multiple-scattering effects as well as attenuation of the mean field. We provide a realistic numerical example for which this is the case.

Figure 1

Principle of the homogenization in a spherical test volume. We compare the coherent and incoherent scattering cross sections of the random medium with the scattering, extinction, and absorption cross sections of the homogenized volume.

Figure 2

Two random composite media with same inclusion density but different correlation structure. In the top figure (medium 1), the probabilities of occupation of two neighbor sites on the lattice are independent. In the bottom figure (medium 2), the probabilities of occupation of neighbor sites are strongly correlated.

Figure 3

First experiment. Differential coherent scattering cross section for random medium 1 (MC1, thick line) and comparison with the differential cross section of the homogenized medium with the same volume (Mie solution) and MG effective permittivity (squares). There is good agreement between the random and homogenized media. The single-scattering solution (MC1, thin line) and the Born approximation for the homogeneous sphere are also given to show the importance of multiple scattering.

Figure 4

Second experiment. Differential coherent scattering cross section for the weakly (MC1) and strongly (MC2) correlated random media with same density and comparison with the differential cross section of the homogenized test volume with MG effective permittivity (Mie). Only the weakly correlated medium MC1 is correctly homogenized. The Born approximation for the homogeneous medium with MG effective permittivity is also given. It coincides with the single-scattering approximation of both MC1 and MC2 (which are not shown here) and is therefore unable to discriminate these different media.

Figure 5

Third experiment. Incoherent scattering cross section at increasing density for the weakly (MC1) and strongly (MC2) correlated random media and comparison with the absorption cross section of the homogeneous sphere with MG effective permittivity.

Figure 6

Influence of the size of scatterers on the extinction cross section. A comparison is given between the classical theories (MG and Bruggeman) and the extended theories (EMG and EB), which bring a size correction to the effective permittivity. The configuration is that of the first experiment.

Figure 7

Same as Fig. 6 for the absorption.