Nanocomposites and nanomaterials

Effective electrical parameters of systems of fine anisotropic particles 
Sushko M.Ya.
Department of Theoretical Physics and Astronomy, Odesa I.I.Mechnikov National University. 2 Dvoryans’ka St., Odesa 65026, Ukraine. 
E-mail: mrs@onu.edu.ua

     We propose a homogenization theory for the effective dielectric constant (electrical conductivity) of macroscopically homogeneous and isotropic systems of anisotropic particles imbedded in a uniform matrix. The theory is based on our compact group approach [1] and the Hashin-Shtrikman variational theorem [2]. The combination of both makes it possible [3] to close the homogenization procedure and efficiently evaluate many-particle contributions to the desired effective parameters from all terms in the related iterative series. The calculations involved become rigorous in the quasistatic limit. 
     The efficiency of the theory is demonstrated by applying it to systems of (a) spherical and (b) nonsphrerical particles with anisotropic dielectric properties. In particular, in the limit of low concentrations of the filler, we recover the Levy-Stroud result [4] for the effective permittivity of a mixture of randomly oriented spherical crystallites, characterized by a uniaxial dielectric tensor. In the opposite limit, our result lies within the well-known Hashin-Shtrikman bounds [2] on the effective conductivity of polycrystals in terms of the principal conductivities of the constituting crystals. 
     We also discuss the features of our solutions for intermediate values of the filler concentration and present a generalization of the theory to systems of inhomogeneous anisotropic particles. 

1.	Sushko M. Ya. Effective permittivity of mixtures of anisotropic particles // J. Phys. D: Appl. Phys.-2009.- 42. -P. 155410-1–9.
2.	Hashin Z., Shtrikman S. Conductivity of polycrystals // Phys. Rev.-1963.-130.-P. 129–133.
3.	Sushko M. Ya. Effective dielectric response of dispersions of graded particles // Phys. Rev. E.-2017.-96.-P. 062121-1–8.
4.	Levy O., Stroud D. Maxwell Garnett theory for mixtures of anisotropic inclusions: Application to conducting polymers // Phys. Rev. B.-1991.-56.-P. 8035–8046.