Renormalization-Group Approach to the Vulcanization Transition

TL;DR

This paper applies a renormalization group approach to analyze the vulcanization transition, revealing universal critical behavior and connecting it to percolation theory, with implications for understanding amorphous solid formation.

Contribution

It introduces a minimal model incorporating thermal motion, quenched constraints, and particle repulsion, and computes critical exponents near the upper critical dimension of six.

Findings

Derived a Ginzburg criterion for the critical region width.

Calculated universal critical exponents to lowest order.

Established a connection between vulcanization and percolation beyond mean-field theory.

Abstract

The vulcanization transition - the crosslink-density-controlled equilibrium phase transition from the liquid to the amorphous solid state - is explored analytically from a renormalization group perspective. The analysis centers on a minimal model that accounts for both the thermal motion of the constituents and the quenched random constraints imposed on their motion by the crosslinks, as well as particle-particle repulsion which suppresses density fluctuations. A correlation function involving fluctuations of the amorphous solid order parameter, the behavior of which signals the vulcanization transition, is examined, its physical meaning is elucidated, and the associated susceptibility is constructed and analyzed. A Ginzburg criterion for the width (in crosslink density) of the critical region is derived and is found to be consistent with a prediction due to de Gennes. Certain universal critical exponents characterizing the vulcanization transition are computed, to lowest nontrivial order, within the framework of an expansion around the upper critical dimension of six. This expansion shows that the connection between vulcanization and percolation extends beyond mean-field theory, at least to first order in the departure from the upper critical dimension. The relationship between the present approach to vulcanized matter and other approaches is explored in the light of this connection. To conclude, some expectations for how the vulcanization transition is realized in two dimensions, developed with H. E. Castillo, are discussed.