Conductivity in the two-dimensional-site percolation problem

Abstract

Measurements of the electrical conductivity of a 137 \ifmmode\times\else\texttimes\fi{} 137-site simple-square screen mesh versus concentration of removed sites are reported. The new critical exponent is reported to be $t=1.38\ifmmode\pm\else\textpm\fi{}0.12$ within an estimated critical region $\ensuremath{\Delta}x\ensuremath{\sim}0.1$ about a critical percolation concentration ${x}_{c}=0.413\ifmmode\pm\else\textpm\fi{}0.005$. A new effective medium theory, which includes overlap effects of adjacent removed sites, is derived, which gives $\frac{\ensuremath{\sigma}(x)}{\ensuremath{\sigma}(0)}=1\ensuremath{-}\ensuremath{\pi}x+\frac{\ensuremath{\pi}{x}^{2}}{2}$, in agreement with the experiment, except within the critical region.