Scaling of Entropic Shear Rigidity

TL;DR

This paper investigates how the shear modulus scales near gelation and vulcanization transitions, revealing different behaviors in dense melts versus phantom networks and predicting crossover phenomena in non-dense systems.

Contribution

It provides heuristic and analytical insights into the scaling laws of shear rigidity, connecting them to percolation theory and network density effects.

Findings

In dense melts, shear modulus scales with sub-linear chain sizes.

In phantom networks, the scaling matches resistor network conductivity.

A crossover in scaling behavior occurs in non-dense systems.

Abstract

The scaling of the shear modulus near the gelation/vulcanization transition is explored heuristically and analytically. It is found that in a dense melt the effective chains of the infinite cluster have sizes that scale sub-linearly with their contour length. Consequently, each contributes k_B T to the rigidity, which leads to a shear modulus exponent d\nu. In contrast, in phantom elastic networks the scaling is linear in the contour length, yielding an exponent identical to that of the random resistor network conductivity, as predicted by de Gennes'. For non-dense systems, the exponent should cross over to d\nu when the percolation length becomes much larger than the density-fluctuation length.