Computer study of the percolation threshold in a two-dimensional anisotropic system of conducting sticks

Abstract

We report here a Monte Carlo study of the percolation threshold in two-dimensional systems of conducting sticks. This is an extension of the work of Pike and Seager, who have considered only the isotropic sample of randomly-oriented---, equal-length---sticks system. Our study is concerned with the dependence of the percolation threshold on the macroscopic anisotropy of systems in which there is a preferred orientation of the sticks ensemble, as well as on the distribution of the sticks' lengths. In particular, we studied systems in which the orientation is determined by random alignments within a given interval or in which the alignments are normally distributed around a given direction. Similarly, for the sticks' lengths we have studied systems of equal lengths, of normally distributed lengths, and of log-normally distributed lengths. The results have shown that the percolation threshold always increases with the macroscopic anisotropy. Extrapolation of the results, from those of the finite sticks ensembles used to the infinite ensemble case, has indicated that in the infinite ensemble the percolation threshold is isotropic. It is found that the broader the stick-length distribution, the lower the mean of the distribution needed for the onset of percolation. Application of the present results for the evaluation of the conductivity indicates that the anisotropy dependence of the conductivity in systems of conducting fibers is determined by both the anisotropy dependence of the percolation threshold and the anisotropy dependence of the critical exponent. If (as found experimentally) a practically infinite two-dimensional system has a conductivity anisotropy, it must be attributed to the anisotropy in the latter parameter.