Random networks of cross-linked directed polymers

TL;DR

This paper investigates how random permanent cross-links affect the structure and elasticity of directed polymer networks, revealing a continuous gelation transition and universal shear modulus independent of microscopic details.

Contribution

It introduces a semimicroscopic replica field theory to analyze the gelation transition and elastic properties of cross-linked directed polymers, highlighting the universality of the shear modulus.

Findings

Identification of a continuous gelation transition with increasing cross-link density.

Derivation of a universal in-plane shear modulus at the gelation point.

Demonstration that negligible extent cross-links do not alter the tilt modulus.

Abstract

We explore the effect of random permanent cross-links on a system of directed polymers confined between two planes with their end-points free to slide on them. We treat the cross-links as quenched disorder and we use a semimicroscopic replica field theory to study the structure and elasticity of this system. Upon increasing the cross-link density, we get a continuous gelation transition signaled by the emergence of a finite in-plane localization length. The distribution of localization length turns out to depend on the height along the preferred direction of the directed polymers. The gelation transition also gives rise to a finite in-plane shear modulus which we calculate and turns out to be universal, i.e., independent of the energy and length scales of the polymers and the cross-links. Using a symmetry argument, we show that cross-links of negligible extent along the preferred axis of the directed polymers do not cause any renormalization to the tilt modulus of the uncross-linked system.