2.2 Electromagnetic Properties of Materials
2.2.4 Dielectric Mixing Model
The dielectric behavior of composite materials depends on the compositions. Great attempts on describing bulk dielectric properties of mixtures in terms of their compositions have been proposed through dielectric mixing modeling. In this section, several mixing models will be discussed from different aspects.
Classical mixing model
Figure 2-5 shows spherical inclusions with permittivity εi occupy random positions in the environment of permittivity εe. Let the fraction Fr of the total volume be occupied by the inclusion phase, and the volume fraction 1-Fr left for the host. Thus, a classical mixture rule with spherical inclusions called Maxwell Garnett mixing formula (Maxwell Garnett, 1904) with the polarizability expression gives:
(
i e)
where εeff is the effective permittivity of a mixture, and this formula is in wide use in the very diverse fields of application. However, this analysis treated the media as a plain and pure dielectric, and no charge flow took place when fields were incident on the materials. A bold and straightforward application of the Maxwell Garnett mixing formula gives for the complex effective permittivity of lossy materials:
( ) ( )
in which the inclusions are assumed to be spherical, and complex permittivity, which includes the electrical conductivity, of the environment isεe =εe − jσ/ω =εe′ − jεe′′, and the same with the complex permittivity of the inclusion as εi =εi′− jεi′′. If the volume fraction of the inclusion phase is small, the effective conductivityσeff =ωεeff′′ , calculated from Eq. [2-21], is:
However, the derivation of the Maxwell Garnett mixing formula was based on the algebraic dependence of the internal filed, and it is known that for time-dependent field, losses entail exponential attenuation of the field amplitudes which can be considerable if the extent of the lossy medium is large compared with the penetration depth. Hence the requirements of allowed use of the Maxwell Garnett mixing formula for time-dependent fields is that the inclusion size must not be larger than the skin depth of the wave in the lossy medium 2/
(
ωμiσi)
with μi being the magnetic permeability of the inclusion material.Fig. 2-5 Dielectric spheres are guests in the dielectric background host. (after Sihvola, 1999)
Mixing model for anisotropic mixtures
The original Maxwell Garnett mixing formula basically has an assumption that the inclusion and the background material are isotropic. However, mixtures such as soil and suspensions may have anisotropic inclusions and/or anisotropic background material.
Now we consider inclusions as ellipsoids, as shown in Fig. 2-6, the depolarization factor Nx (the factor in the ax-direction of Cartesian co-ordinate system) is
( ) ( )( )( )
∫
∞+ +
+
= +
0 2 2 2 2
2 x x y z
z y x
x s a s a s a s a
a ds a
N a [2-23]
where s is the integration variable, whose value is between zero and infinity. For other depolarization factor Ny (Nz), interchange ay and ax (az and ax) in the above integral.
Collecting those in a single dyadic (the elementary dyadic analysis could refer to Appendix A), the depolarization dyadic for an ordinary ellipsoid reads:
( )
where vi is unit vectors along the three orthogonal eigendirections, and the symmetric and positive-definite dyadic
If the ellipsoid, which is anisotropic with permittivity εi, is located in an anisotropic material εe =εrε0, and exposed to a uniform external electric field Ee, as shown in Fig. 2-7.
The internal electric field Ei can be shown to be
( )
[
e i e]
e ei L E
E = ε + ′⋅ ε −ε −1⋅ε ⋅ [2-26]
where the transformed depolarization dyadic L′, which is that of real geometry of the ellipsoid after it has been transformed affinely by the anisotropy of the environments, can be calculated from:
anisotropic, it is natural to accept that a mixture composed of reciprocal materials must display reciprocal electromagnetic behavior. Therefore, the Maxwell Garnett equation can be adapted in the following from:
( ) ( )
[
− −1+ 1− −1⋅ ′]
−1+
= e Fr i e Fr e L
eff ε ε ε ε
ε [2-28]
Fig. 2-6 The geometry of an ellipsoid. The semi-axis ax, ax,and ax fix the Cartesian co-ordinate system. (after Sihvola, 1999)
Fig. 2-7 Anisotropic ellipsoid (with permittivity εi) in anisotropic environment (εe =εrε0).
Ei refers to the internal electric field. (after Sihvola, 1999)
Generalized mixing model
The Bruggeman formula is an important mixing rule which is widely used in
electromagnetic literatures, and it is also known by other names: Polder-van Santen formula and de Loor formula. The essence of the Bruggeman mixing rule is the absolute equality between phases in the mixture (Sihvola, 1999), and it for the case when the inclusions are randomly oriented ellipsoids is
(
−)
=∑
+(
−)
where Nj are the depolarization factors as shown in Eq. [2-24]. Dobson et al. (1985) used the de Loor formula for determining the dielectric constant εm of the four-component mixing for a soil mixture, which is composed of dry soil solids, bound water in the Stern layer, bulk water in the Gouy layer, and air. In additional, Dobson et al. (1985) assumed that the ellipsoid depolarization factors lead to Nj = (0,0,1), εeff has a potential range of εs≤ εeff≤ εm , thus the εm to dry soil solids, air, free Gouy layer water, and Stern layer water, respectively. Hallikainen et al. (1985) found a relationship between εs and soil gravity density γs from an empirical fitting of the experimental data as:
Dobson et al. (1985) also adopted the Birchak formula which is also called Power-law model [Birchak et al. 1974] or refractive index model, to provide a four-phase semi-empirical dielectric mixing model for soil mixture,
α
refer to the solid soils, air, free water and bound water, respectively. The exponent α is a constant shape factor, and Mironov et al. (2004) presented a generalized refractive mixing dielectric model based on α = 0.5.
Heimovaara et al. (1994b) and Lin (2003b) formulated the four-component dielectric mixing equation based on the Dobson’s semi-empirical formula, in terms of physical parameters of soil as: