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<p_r. We also find that the shear modulus at fixed T vanishes as μ∼(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.