On the theory of conductivity of anisotropic composites: Lattice model
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DISORDER, AND PHASE TRANSITION IN CONDENSED SYSTEM
On the Theory of Conductivity of Anisotropic Composites: Lattice Model B. Ya. Balagurov Emanuel Institute of Biochemical Physics, Russian Academy of Sciences, Moscow, 119334 Russia email: [email protected]; [email protected] Received May 12, 2014
Abstract—The electrical conductivity in a disordered anisotropic lattice model is considered using analytic methods. The effective conductivity of a weakly heterogeneous lattice in an approximation quadratic in the ˆ (r) from its mean value 〈 σ ˆ 〉 is determined. In the case of a low deviation of the local conductivity tensor σ concentration (c 1) of “defective” bonds, the conductivity in the binary lattice model is calculated in an approximation linear in c. The equations of the effective medium method are derived for an anisotropic lat tice. The results are compared with the relevant results for the continuum model. DOI: 10.1134/S1063776114090106
1. INTRODUCTION Analysis of the conductivity of anisotropic com posites encounters the same fundamental difficulties as in the case of isotropic heterogeneous media, which are associated with disorder in such systems. In addi tion, the existence of supplementary parameters of the problem associated with allowance for anisotropy complicates the problem still further. For this reason, theoretical works devoted to analyzing the conductiv ity of anisotropic composites [1–3] are mostly qualita tive by nature. Nevertheless, these publications revealed some peculiar features (as compared to the isotropic case) in the behavior of the effective conduc tivity of such systems. In particular, it was shown that with increasing concentrations of dielectric (or per fectly conducting) inclusions, the isotropization of the properties of composites takes place, even with a strongly anisotropic matrix. As such a composite approaches the percolation threshold, it becomes almost isotropic with “isotropic” critical indices. The quantitative approach to calculating the elec trophysical properties of anisotropic composites was proposed earlier [4, 5]. In [4], a weakly heterogeneous medium was considered; to calculate the conductivity of this medium, a consistent perturbation theory was developed (the expansion in the small deviation of ˆ (r) from its mean value 〈 σ ˆ 〉). conductivity tensor σ The explicit expressions for the components of the ˆ e were obtained in an effective conductivity tensor σ approximation quadratic in this deviation. In [5], a general approach to calculating the conductivity ten sor for anisotropic composites with a low concentra tion of inclusions of arbitrary shape was proposed. ˆ e was expressed in an approximation linear in Tensor σ concentration in terms of the dipole polarizability of
the inclusion, which was determined in a certain transformed system with the inclusion surrounded by an isotropic matrix. In the general case, diversified information on the properties of heterogeneous anisotropic media can be obtained by analyzing lattice models. N
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