Bianisotropic homogenized composite materials arising from ellipsoidal constituent particles with statistical distributions of orientations, shapes, and sizes

The Bruggeman and Maxwell Garnett formalisms to homogenize particulate composite materials were implemented to accommodate statistical distributions of orientation, shape, and size of ellipsoidal particles. These particles were assumed to be ≥10 times smaller than the smallest relevant electromagnetic wavelength. The corresponding homogenized composite material (HCM) – which belongs to the most general electromagnetic class of linear materials, namely bianisotropic materials – was characterized by a 6×6 constitutive dyadic K__HCM. Numerical studies were conducted to explore the relationship between the constituent parameters of the HCM and the nature of the probability density functions (PDFs) characterizing the statistical distributions. Whereas the Gaussian PDF was used for both the particle orientation and size, the Maxwell–Boltzmann PDF was used for the particle shape. The HCM was found to generally exhibit biaxial bianisotropic symmetry for distributions of orientation involving spheroidal particles. This symmetry becomes uniaxial bianisotropic when the standard deviation of the distribution approaches zero or exceeds 3. For distributions of shapes involving ellipsoidal particles, the HCM exhibits varying degrees of biaxial bianisotropy as the shape parameter is varied. For the distribution of sizes involving spherical particles, the uniaxial bianisotropic HCM exhibits increasing degrees of loss as the upper limit on the size distribution increases. The Bruggeman and Maxwell Garnett formalisms delivered estimates of K__HCM that were generally in close agreement with typical differences being a few percent.