Modelling of mesoscopically heterogeneous systems (effective medium)


For decades, understanding of mesoscopically heterogeneous systems is of a great interest from fundamental as well as from practical perspectives, mainly because great deal of the functional materials for applications are currently produced in the form of ceramics and composites. The word ‘mesoscopic’ points out to the size of inhomogeneities, which are all much larger than a typical interatomic distance. Consequently they can be often treated as composed of regions with homogeneous material (macroscopic) properties separated by thin boundaries. Depending on the concrete system the inhomogeneities are called grains (ceramics), clusters (composites), or microdomains (ferroics).  The overall properties are not simple average of components’ properties, but are crucially impinged by electric charges or mechanical stresses induced at the boundaries. This can lead to a deterioration or also an improvement in the properties, depending on the target requirements.  The micro-geometry can crucially tune the properties of the samples, e.g. the dielectric matrix with metal nearly percolated inclusions yields high-permittivity systems, highly appreciated in applications. For practice, the main goal is to achieve the ability to construct the composite of the required properties.

Realistic three-dimensional model of BaTiO3 ceramics
a) Realistic three-dimensional model of BaTiO3 ceramics, whith constructed mesh for calculations using finite element method. Clearly visible are pores between ferroelectric grains.
b) For comparison, the experimental micrograph of the same material.

Therefore the main theoretical task concerns the relationship between the overall properties (dielectric, elastic, etc.), and the mesoscopic (regular or random) geometry and properties of the constituents. If the wavelength of the probing field is large in comparison with the dimension of inhomogeneities (the long-wavelength limit), the effective medium approaches are widely used, and leads to various mixing formulas describing the effective response (for example, a ‘classical’ Maxwell-Garnett relation). For two-component systems the exact analytic dependence of  properties of the effective permittivity is studied using integral representation allowing separation of the micro-geometry and material properties, which turns important for interpretation of experimental data and construction of new models. Description of more complicated systems, such as those with multiple components is much more complicated and awaits for theoretical progress. The construction and analysis of the appropriate finite element models is a complementary method in study of the effective properties.

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