Conductivity fluctuations in polymer's networks

TL;DR

This paper models polymer networks as anisotropic fractals near one dimension, analyzing their percolation and conductivity fluctuations with a focus on critical exponents and universal distribution functions.

Contribution

It introduces a novel percolation model on anisotropic fractals with fractional dimensions and derives nonanalytic critical exponents and universal conductivity distribution functions.

Findings

Critical exponents are strongly nonanalytic functions of psilon.

The conductivity distribution at threshold is universal in a specific fluctuating variable.

Finite size conductivity moments decrease exponentially with psilon.

Abstract

Polymer's network is treated as an anisotropic fractal with fractional dimensionality D = 1 + \epsilon close to one. Percolation model on such a fractal is studied. Using the real space renormalization group approach of Migdal and Kadanoff we find threshold value and all the critical exponents to be strongly nonanalytic functions of \epsilon, e.g. the critical exponent of the conductivity was obtained to be \epsilon^{-2}\exp(-1-1/\epsilon). The main part of the finite size conductivities distribution function at the threshold was found to be universal if expressed in terms of the fluctuating variable, which is proportional to the large power of the conductivity, but with dimensionally-dependent low-conductivity cut-off. Its reduced central momenta are of the order of \exp(-1/\epsilon) up to the very high order.