Elasticity near the vulcanization transition

TL;DR

This paper presents a semi-microscopic theory of the shear modulus near the vulcanization transition, showing it grows continuously with classical critical behavior and that external stresses do not break particle localization symmetry.

Contribution

It introduces a new statistical-mechanical model that accounts for thermal fluctuations and quenched disorder in vulcanization, revealing the nature of elasticity near the transition.

Findings

Shear modulus grows continuously with a third-power law at the transition.

External stresses do not break the spherical symmetry of particle localization.

The theory aligns with classical critical exponents for the transition.

Abstract

Signatures of the vulcanization transition--amorphous solidification induced by the random crosslinking of macromolecules--include the random localization of a fraction of the particles and the emergence of a nonzero static shear modulus. A semi-microscopic statistical-mechanical theory is presented of the latter signature that accounts for both thermal fluctuations and quenched disorder. 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 cross-link 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.