Semi-microscopic theory of elasticity near the vulcanization transition

TL;DR

This paper develops a semi-microscopic statistical mechanical theory to describe the elastic properties of randomly crosslinked macromolecules near the vulcanization transition, capturing how the shear modulus emerges and behaves.

Contribution

It introduces a new semi-microscopic theory that accounts for disorder and thermal fluctuations, providing a clear calculation of the shear modulus near the transition.

Findings

Shear modulus grows continuously from zero with a third power law.

External stresses do not break the spherical symmetry of localization clouds near the transition.

The theory predicts classical critical behavior at the transition.

Abstract

Randomly crosslinked macromolecules undergo a liquid-to-amorphous solid phase transition at a critical crosslink concentration. This transition has two main signatures: the random localization of a fraction of the monomers and the emergence of a nonzero static shear modulus. In this article, a semi-microscopic statistical mechanical theory of the elastic properties of the amorphous solid state is developed. This theory takes into account both quenched disorder and thermal fluctuations, and allows for the direct computation of the free energy change of the sample due to a given macroscopic shear strain. This leads to an unambiguous determination of the static shear modulus. At the level of mean field theory, it is found (i) that the shear modulus grows continuously from zero at the transition, and does so with the classical exponent, i.e., with the third power of the excess crosslink density and, quite surprisingly, (ii) that near the transition the external stresses do not spoil the spherical symmetry of the localization clouds of the particles.