Anisotropic generalization of Stinchcombe's solution for the conductivity of random resistor networks on a Bethe lattice

Abstract

Our study is based on the work of Stinchcombe [1974 J. Phys. C 7 179] and is devoted to the calculations of average conductivity of random resistor networks placed on an anisotropic Bethe lattice. The structure of the Bethe lattice is assumed to represent the normal directions of the regular lattice. We calculate the anisotropic conductivity as an expansion in powers of inverse coordination number of the Bethe lattice. The expansion terms retained deliver an accurate approximation of the conductivity at resistor concentrations above the percolation threshold. We make a comparison of our analytical results with those of Bernasconi [1974 Phys. Rev. B 9 4575] for the regular lattice.

Properties Prediction of Carbon Nanotube Polymer Composites during Melt Processing

“…Other conductivity models were also tested, e.g. the classic percolation model for concentrations below and above the percolation concentration [11], the Fournier model [13] and a model for a Bethe lattice [14].…”

Superposition approach for description of electrical conductivity in sheared MWNT/polycarbonate melts

“…There are two main assumptions of our approach: 1) the percolation probability depends on the history of shear application, 2) the particles in the percolation phase are oriented randomly. If for some other system it will be find out that the percolating clusters are noticeably deformed by the flow before their breakage, one can try to account for this effect using the theory of anisotropic electrical conductivity proposed recently by Semeriyanov et al [29].…”

“…There even exists an opinion that percolation threshold can become anisotropic under application of the shear forces, the possibility considered in the frame of anisotropic percolation theory, in which the occupation probability is different in two perpendicular directions [55]. Theory of anisotropic electrical conductivity developed by Semeriyanov et al on a Bethe lattice [29] does not presume an anisotropic percolation threshold but an anisotropic conductivity of electrical resistors which may differ in directions parallel and perpendicular to the shear direction. The percolation threshold stays unaffected by the shearing and is solely defined by a functionality of the Bethe lattice.…”

“…As percolation models are static ones, they are only applicable to homogeneous systems with statistically distributed filler in the equilibrium state. An anisotropic generalization of the Stinchcombe's approach [28], proposed recently by Semeriyanov et al [29], aims at the description of percolation networks with anisotropic local conductivity represented by resistors with direction-dependent electrical conductivity. Such anisotropy may arise for example from strong shearing of the filler network built from elongated conductive particles.…”

Bethe-Peierls approximation for linear monodisperse polymers re-examined

“…However, it is not completely clear how the dynamic arrest is caused by the local anisotropy. To address this problem we mention the result in [43], where it was shown that the absence of lattice loops may amplify the effect of local anisotropy on averaged transport properties, which is shown for percolating resistors randomly distributed on the lattice. The breakage of the local anisotropy at the glass transition for the spherical beads can be explained in the framework of the theory of glass transition of Edwards and Vilgis [45], where the dynamic arrest is caused by closed path configurations.…”