Entropic Elasticity of Diluted Central Force Networks

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 $p_c<p<\mathrm{}{p}_r$. We also find that the shear modulus at fixed $T$ vanishes as $\mu \sim ({p-p}_c{)}^f$ with an exponent $f$ that is, within numerical uncertainty, the same as the exponent $t$ that describes the conductivity of randomly diluted resistor networks.