Entropic Elasticity of Diluted Central Force Networks

Abstract

At zero temperature, the elastic constants of diluted central force networks are known to vanish at a concentration ${p}_{r}$ (of either sites or bonds) that is substantially higher than the corresponding geometric percolation concentration ${p}_{c}$. We study such diluted lattices at finite temperatures and show that there is an entropic contribution to the moduli similar to that in cross-linked polymer networks. This entropic elasticity vanishes at ${p}_{c}$ and increases linearly with $T$ for ${p}_{c}<p<{p}_{r}$. We also find that the shear modulus at fixed $T$ vanishes as $\ensuremath{\mu}\ensuremath{\sim}({p\ensuremath{-}p}_{c}{)}^{f}$ with an exponent $f$ that is, within numerical uncertainty, the same as the exponent $t$ that describes the conductivity of randomly diluted resistor networks.