Electrical conductivity of a two-dimensional model for a structurally anisotropic composite
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SORDER, AND PHASE TRANSITION IN CONDENSED SYSTEM
Electrical Conductivity of a TwoDimensional Model for a Structurally Anisotropic Composite B. Ya. Balagurov Emanuel Institute of Biochemical Physics, Russian Academy of Sciences, ul. Kosygina 4, Moscow, 119334 Russia
email: [email protected], [email protected] Received September 9, 2009
Abstract—The electrical conductivity of a twodimensional structurally anisotropic model for a composite is considered. The model represents an isotropic matrix with a system of nonconducting inclusions in the form of infinitely thin straight line segments (scratches). The scratches make an angle θ or –θ with a preferred axis (for definiteness, axis y) at the same probability, and their centers are chaotically distributed. An approximate ˆ e of this effective medium method is used to obtain a general expression for the effective conductivity tensor σ ˆ e are model that is valid over a wide concentration range. In this approximation, both components of tensor σ shown to vanish at the same percolation threshold, which is expressed explicitly. The conductivity of the model in a critical region is considered in terms of the similarity hypothesis. DOI: 10.1134/S106377611002010X
1. INTRODUCTION In the standard percolation theory [1–3], promi nence is given to macroscopically isotropic heteroge neous systems. However, it is also interesting (from both theoretical and practical standpoints) to study the electrophysical properties of less well understood anisotropic composite materials. Among the works dealing with the electrical conductivity of this class of composites, we note the following works. In [4–9], various methods were used to study the conductivity of composites with natural anisotropy, i.e., layered graphitelike and filamentary TCNQtype composites. Bernasconi [4] performed a numerical experiment on the cubic lattice (problem of associa tions). The authors of [5–7] considered conductivity in the vicinity of a percolation threshold, and the author of [8, 9] analyzed conductivity over the entire concentration range. Those works revealed a number of specific features characteristic of anisotropic com posites. In particular, it was found that an initially sharply anisotropic composite becomes almost isotro pic when the metal–insulator phase transition point is approached. In this case, the corresponding critical indices coincide with their isotropic values. The author of [8, 9] also investigated an intermediate con centration range, where the properties of anisotropic composites differ radically from those of isotropic ones. The authors of [10–14] studied structurally aniso tropic composites whose anisotropy was induced arti ficially, namely, by the introduction of identically ori ented inclusions (e.g., elongated inclusions) in an iso tropic matrix. In [10, 11], computer methods were
used to determine percolation thresholds (critical concentrations) for twodimensional models with a system of scratches having a preferred orientation. According to the resul
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