In this book, a particular emphasis is given to the electrical properties of mixtures. Although the reader will find by the course of the present text that plain volume-average homogenisation along the lines of mass density is used in certain corners of the electromagnetics community, in general the averaging needs to be done in a manner where the laws of electromagnetics are respected and give their own perspective to the averaging process.

Dielectric materials-or, as they are also called, dielectrics-are such media that do not conduct electricity. In the present text a definition so narrow is not strictly followed but certainly the most important property of dielectric materials in our discussion is their ability to store, not conduct, electrical energy. A measure for this property is the permittivity or dielectric constant of the material. In fact, permittivity is only a higher-level invention to calculate approximatively the electric response of matter. Underneath it a great amount of detailed physics is hidden. Let us try to start the discussion of dielectric materials with a look at the various polarisation mechanisms.

We analyse the simplest model for a dielectric mixture. Isotropic dielectric spheres are embedded as inclusions in an isotropic dielectric environment. The two material components of the mixture are known by various names: the inclusion phase is often called guest, and the environment host, or matrix. If we look at this geometry from such a distant position that only averages matter, it may feel natural to associate a macroscopic permittivity with the mixture, which can be calculated if the permittivities of the two components are known. The basic mixing rule, the so-called Maxwell Garnett formula, will be derived. The discussion will be extended to more complicated mixing principles which involve interaction of inclusions in dense materials.

A whole chapter has now been devoted to the introduction of the basic Maxwell Garnett mixing formula and various analytical exercises to illustrate the theory. The mixing analysis was in many respects idealised. The mixture consisted of two phases and the inclusions were assumed to be spherical in shape. It is perhaps time to relax these assumptions and try to see whether it is possible to say more about mixtures. In the present section, let us generalise the Maxwell Garnett theory to allow variation in the structure and materials of the mixture.

Chapter 5 discusses anisotropic mixtures and dielectric anisotropy.

This chapter discusses a very special type of materials: such materials which become electrically excited by magnetic field, and vice versa. These materials are called bi-anisotropic media. A special case of bi-anisotropic media is the class of bi-isotropic media, such materials which are magnetoelectric but not sensitive to field direction. In general, bi-anisotropy could be understood to mean that the electric polarisation is caused by any other type of excitation, in addition to the ordinary electrically caused polarisation component, but commonly adopted use of the term “bi-anisotropy” restricts the cross-polarisation to magnetoelectric effects.

This chapter discusses the regime of nonlinear materials. When this is done, the characterisation of the polarisation of materials becomes much more difficult because of very many new possible connections between the various components of the fields and polarisations. Therefore the discussion of the mixing and homogenisation principles that follows cannot be as exhaustive as it has been for isotropic and anisotropic media.

The chapter is an attempt to dispel impressions that conceptual difficulties regarding the randomness are not given attention. Let us try to articulate the weaknesses and problems of the classical approach to mixing rules. Also the question is addressed between which limits the effective properties of heterogeneous media can vary if the case is really so that we cannot predict an exact macroscopic permittivity for a mixture.

The reader may have grown exhausted by this time by the endless emphasis on the Maxwell Garnett-type mixtures. Indeed, the discussion has heretofore concentrated heavily on a single approach to the homogenisation. One of the components is treated as environment, and the inclusion phase is considered as a perturbation against this background. And in particular, the local field that excites a single scatterer in the mixture was calculated by replacing all neighbours by a uniform polarisation density in which a hole was carved. This treatment is clearly approximate, as was pointed out in Chapter 8 where the limitations of Maxwell Garnett philosophy were discussed. But on the other hand, no exact solution exists for the electrostatic problem in a random heterogeneous geometry. Multiple opinions can flourish when nobody knows for certain. It is evident that the Maxwell Garnett model cannot remain at the scene as the only ruling formula. Many rival mixing rules are being used in the modelling of heterogeneous materials. In the present chapter, some of the most common competitor models are presented.

In this chapter possibilities are sought how the effective-medium models could be hardened; in other words, to generalise them so that their domain of applicability would extend to frequencies for which the inclusions and inhomogeneities can no longer be considered very small. Certainly, the scattering of single bodies in free space has been studied thoroughly for various canonical shapes. See the extensive collection by Bowman et al. for these results. However, the aim here is not to look for full solutions of Maxwell equations in random media but rather, how the most important correction terms could be added to mixing rules if we wish to try to describe the dynamic response of the dielectric inhomogeneity and randomness of the material.

This chapter discusses how the mixing process affects the temporally dispersive properties of materials. The time variation of the electromagnetic phenomena under consideration is very simple. It is sinusoidal, and the frequency of the wave variation is a measure for the time derivatives that are needed in Maxwell equations. Use of complex vectors helps to eliminate totally the time-dependence in the field quantities, and admittedly the remaining equations are easier to solve than the original time-varying system.

The evident message of mixing formulas is that a mixture has properties that are dependent and determined by those of its constituents but different from them-just like the phenotype of a child may differ in some respects from parents' appearance. Although the dielectric properties of a mixture are a certain average of the component permittivities we have observed in previous chapters that sometimes the whole character of the dielectric behaviour of the mixture is changed by the mixing process. The effective medium can possess properties that are totally absent from the inclusions and the environment. Let us concentrate on description and analysis of such effects in this chapter.

In this chapter, let us study how well the proposed mixing theories agree with the real world. After all, an effective description of matter is ultimately a project that grows from practical needs. As we have noticed in the earlier chapters, in none of the mixing models could the exact effective permittivity be derived with full rigour. Approximations had to be made in the analysis to be able to take into account the effect of the randomness of the structure on the dielectric interaction. Were those assumptions justified? The approximations earn their legitimation from the success of the mixing models in predicting macroscopic properties of heterogeneous media. Because dielectric data exist for many types of natural and human-made materials, it is time to ask what kind of averages of the components are the dielectric properties of these materials and how do they match against the models.

Our path through various attempts to explain and understand dielectric properties of heterogeneous materials brought us finally to the comparison of the models with the properties of real-life materials. In the last chapter on applications, we saw that very often a gap remains between our theory-based mixing models and the true properties of actual existent materials. Engineers are painfully aware of this, and they-instead of using a classical mixing rule-often resort to empirical relations between the nonelectrical and dielectric properties of media of their interest. Such formulas may be weighted averages of the properties of the constituents, for example the refractive model or the Looyenga formula, or in the extreme case such a relation can be a regression curve upon experimental data.

This appendix discusses a collection of dyadic relations.

In this appendix, some of the important mixing rules and other formulas are repeated. The effective permittivity ε_eff is for a mixture where the environment has permittivity ε_e and the inclusions, which occupy a volume fraction f are of permittivity ε_i.