Besides the microscopic and macroscopic param- eters discussed above, in materials research and engineering, some other macroscopic properties are often used to describe materials.
1.3.6.1 Linear and nonlinear materials
Linear materials respond linearly with externally applied electric and magnetic fields. In weak field ranges, most of the materials show linear responses to applied fields. In the characterization of materials’ electromagnetic properties, usually weak fields are used, and we assume that the materials under study are linear and that the applied electric and magnetic fields do not affect the properties of the materials under test.
However, some materials easily show nonlinear properties. One typical type of nonlinear material is ferrite. As discussed earlier, owing to the nonlinear relationship betweenBandH, if different strength of magnetic field H is applied, different value of permeability can be obtained. High-temperature superconducting thin films also easily show non- linear properties. In the characterization of HTS thin films and the development microwave devices using HTS thin films, it should be kept in mind that the surface impedance of HTS thin films are dependent on the microwave power.
1.3.6.2 Isotropic and anisotropic materials The macroscopic properties of an isotropic mate- rial are the same in all orientations, so they can be represented by scalars or complex numbers. How- ever, the macroscopic properties of an anisotropic
material have orientation dependency, and they are usually represented by tensors or matrixes. Some crystals are anisotropic because of their crystalline structures. More discussion on anisotropic materi- als can be found in Chapter 8, and further discus- sion on this topic can be found in (Kong 1990).
1.3.6.3 Monolithic and composite materials According to the number of constituents, materi- als can be classified into monolithic or composite materials. A monolithic material has a single con- stituent. While a composite material has several constituents, and usually one of the constituents is calledhost medium, the others are calledinclusions or fillers. The properties of a composite material are related to the properties and fractions of the constituents, so the electromagnetic properties of composites can be tailored by varying the proper- ties and fractions of the constituents. The study of the electromagnetic properties of composite mate- rials has attracted much attention, with the aim of developing composites with expected electromag- netic properties.
The prediction of the properties of a composite from those of the constituents of the composite is a long-standing problem for theoretical and experimental physics. The mixing laws relating the macroscopic electromagnetic properties of composite materials to those of their individual constituents have been a subject of enquiry since the end of the nineteenth century. The ability to treat a composite with single effective permittivity and effective permeability is essential to work in many fields, for example, remote sensing, industrial and medical applications of microwaves, materials science, and electrical engineering.
The mean-field method and effective-medium method are two traditional approaches in predict- ing the properties of composite materials (Banhegyi 1994). In the mean-field method, we calculate the upper and lower limits of properties represent- ing the parallel and perpendicular arrangements of the constituents. A practical method is to approxi- mate the composite structure by elements of ellip- soidal shape, and various techniques are avail- able to calculate the composite permittivity. For isotropic composites, closer limits can be calculated and, depending on morphological knowledge, more
sophisticated limits are possible. In an effective- medium method, we assume the presence of an imaginary effective medium, whose properties are calculated using general physical principles, such as average fields, potential continuity, average polariz- ability, and so on. Detailed discussion on effective- medium theory can be found in (Choy 1999).
To achieve more accurate prediction, numerical methods are often used in predicting the properties of composite materials. Numerical computation of the effective dielectric constant of discrete random media is important for practical applications such as geophysical exploration, artificial dielectrics, and so on. In such dielectrics, a propagating electromagnetic wave undergoes dispersion and absorption. Some materials are naturally absorptive owing to viscosity, whereas inhomogeneous media exhibit absorption due to geometric dispersion or multiple scattering. The scattering characteristics of the individual particles (or the inclusions) in the composite could be described by a transition or T-matrix and the frequency-dependent dielectric properties of the composite are calculated using multiple scattering theory and appropriate correla- tion functions between the particles (Varadan and Varadan 1979; Bringi et al. 1983; Varadan et al.
1984; Varadan and Varadan 1985).
More discussions on monolithic and compos- ite materials can be found in (Sihvola 1999; Nee- lakanta 1995; Priou 1992; Van Beek 1967). In the following, we concentrate on the dielec- tric permittivity of composite materials, and we mainly discuss dielectric–dielectric composites and dielectric-conductor composites.
Dielectric–dielectric composites
The host media of composite materials are usually dielectric materials, and if the inclusions are also dielectric materials, such composites are called dielectric–dielectric composites. The shapes and structures of the inclusions affect the overall properties of the composites.
A composite with spherical inclusions is the sim- plest and a very important case. Consider a mixture with a host medium of permittivity ε0 containing n inclusions in unit volume, with each of the inclusions having polarizabilityα. The permittivity ε0 of the host medium can take any value,
including complex ones. The effective permittivity εeff of a composite is defined as the ratio between the average electric displacement D and the average electric field E: D =εeffE. The electric displacement D depends on the polarization P in the material, D =ε0E +P, and the polarization can be calculated from the dipole momentspof the ninclusions,P =np. This treatment assumes that the dipole moments are the same for all inclusions.
If the inclusions are of different polarizabilities, the polarization has to be summed by weighting each dipole moment with its number density, and the overall polarization thus consists of a sum or an integral over all the individual inclusions.
The dipole moment p depends on the polariz- ability and the exciting field Ee: p=αEe. For spherical inclusions, the exciting fieldEe is:Ee= E +P/(3ε0). From the above equations, the effec- tive permittivity can be calculated as a function of the dipole moment densitynα:
εeff=ε0+3ε0 nα
3ε0−nα (1.94) Equation (1.94) can also be written in the form of the Clausius–Mossotti formulas
εeff−ε0
εeff+2ε0 = nα
3ε0 (1.95) If the composite contains inclusions with different polarization, for example N types of spheres with different permittivities, Eq. (1.95) should be modified into (Sihvola 1989a; Sihvola and Lindell 1989b):
εeff−ε0
εeff+2ε0 =
N
i=1
niαi
3ε0 (1.96) Let the permittivity of the background medium be ε0, that of the inclusions beε1, and the volume fraction of the inclusions be f1. The polarizability of this kind of inclusions depends on the ratio between the inside and the outside fields when the inclusions are in a static field. According to Sihvola(1989a) and Sihvola and Lindell (1989b), the polarizability of a spherical inclusion with radius a1 is
α=4π ε0a13 ε1−ε0
ε1+2ε0 (1.97)
So the effective permittivity of this mixture is εeff−ε0
εeff+2ε0 =f1
ε1−ε0
ε1+2ε0 (1.98) This formula is known as the Rayleigh’s formula.
The success of a mixture formula for a compos- ite relies on the accuracy in the modeling of its real microstructure details. Besides the Rayleigh’s formula, several other formulas have been derived using different approximations of the microstruc- tural details of the composite. Several other pop- ular formulas for the effective permittivity εeff of two-phase nonpolar dielectric mixtures with host medium of permittivityε0 and spherical inclusion of permittivityε1with volume fractionf1are listed in the following:
Looyenga’s formula:
ε
1 eff3 =f1ε
1
13 +(1−f1)ε
1
03 (1.99) Beer’s formula:
ε
1 eff2 =f1ε
1
12 +(1−f1)ε
1
02 (1.100) Lichtenecher’s formula:
lnεeff=f1lnε1+(1−f1)lnε0 (1.101) In the above formulas (1.98–1.101), the interpar- ticle actions between the inclusions are neglected.
The above formulas can be extended to multiphase composites, but they are not applicable to compos- ites with layered inclusions, because they ignore the interactions between the different layers in an inclusion. The properties of composites with lay- ered spherical inclusions are discussed in (Sihvola and Lindell 1989b).
Dielectric-conductor composites
In a dielectric-conductor composite, the host me- dium is a dielectric material, while the inclusions are conductors. Such composite materials have exten- sive electrical and electromagnetic applications, such as antistatic materials, electromagnetic shields, and radar absorbers. Here we do not consider the frequency dependence of the electromagnetic prop- erties of dielectric-conductor composites, and only consider the static limit (ω→0). Discussions on the frequency dependence of the properties of such
Volume concentration (a)
(b) (c)
Vp
e′ e′′
Permittivity
Figure 1.30 Percolation in a dielectric-conductor composite. (a) Change of static permittivity near the percolation threshold, (b) the case when the volume concentration of the fillers is less than the percolation threshold (Vp), and (c) the case when the volume concentration of the fillers is close to the percolation threshold (Vp). In (b) and (c), circle denotes inclusions, otherwise the host medium
composites can be found in (Potschkeet al. 2003) and the references given therein.
For a dielectric-conductor composite, there exists a phenomenon calledpercolation. When the volume concentration of the conductive inclusions approaches the percolation threshold, the dielectric composite becomes conductive. One can observe a significant change in permittivity of the com- posites filled with conductive inclusions when it percolates. As shown in Figure 1.30(a), near the percolation threshold, the real part of permittiv- ity of the composite increases quickly along with the increase of the volume concentration of the conductive inclusions and reaches its maximum value at the percolation threshold; while the imag- inary part of permittivity monotonically increases with the increase of the volume concentration of the conductive inclusions. The origin of percola- tion phenomenon is the connection of the con- ductive inclusions. Figures 1.30(b) and (c) show distributions of the conductive inclusions in the
host medium when the volume concentration of the inclusions is less than and close to the percolation threshold respectively.
The location of the percolation threshold and the concentration dependence of permittivity and conductivity around the threshold depend on the properties of the host medium, conductive inclusions, and the morphology of the composite.
Because of the rich physics phenomena near the percolation threshold, in percolation research, constructing models of permittivity or conductivity near the percolation threshold is of great theoretical importance and application meaning.
It should be emphasized that the geometry of the inclusions plays an important role in deter- mining the percolation threshold and the electro- magnetic properties of a dielectric-conductor com- posite. The general geometry of an inclusion is elliptic sphere, which, at special conditions, can be disk, sphere, and needle. In recent years, com- posites with fiber inclusions have attracted great
attentions (Lagarkov et al. 1998). Fiber can be taken as a very thin and long needle. The mechan- ical and electrical performance of polymer mate- rials may be greatly improved by adding carbon or metal fibers, and the resulted fiber-reinforced composites have a wide range of practical applica- tions due to their unique mechanical, chemical, and physical properties. Fiber-filled composites present more possibilities of tailoring the dielectric prop- erties. For example, high values of dielectric con- stant can be obtained at a low concentration of fiber inclusions, and composites filled with metal fibers possess pronounced microwave dielectric dispersion, which are very important for the devel- opment microwave absorbing materials.
Finally, it should be indicated that percolation phenomena also exists in many other systems, for example, the superconductivity of metal- superconductor composites and leakage of fluids through porous media.