The dielectric properties of heterogeneous materials for various condensed-matter
systems are important for several technologies, e.g. impregnated polymers for
high-density capacitors, polymer carbon black mixtures for automotive tires and current
limiters in circuit protection. These multiscale systems lead to challenging problems
of connecting microstructural features (shape, spatial arrangement and size
distribution of inclusions) to macroscopic materials response (permittivity,
conductivity). In this paper, we briefly discuss an ab initio computational
electrostatics approach, based either on the use of the field calculation package
FLUX3D (or FLUX2D) and a conventional finite elements method, or the use of the field
calculation package PHI3D and the resolution of boundary integral equations, for
calculating the effective permittivity of two-component dielectric heterostructures.
Numerical results concerning inclusions of permittivity ε1 with various
geometrical shapes periodically arranged in a host matrix of permittivity
ε2 are provided. Next we discuss these results in terms of
phenomenological mixing laws, analytical theory and connectedness. During the pursuit
of these activities, several interesting phenomena were discovered that will stimulate
further investigation.