Hard-needle elastomer in one spatial dimension

TL;DR

This paper provides exact statistical mechanics analysis of a one-dimensional liquid crystal elastomer model with hard needles, revealing critical behavior and universal scaling laws near zero tension.

Contribution

It introduces an exact transfer-matrix solution for a 1D liquid crystal elastomer model with elastic and steric interactions, identifying critical scaling behavior.

Findings

Nematic order parameter decays linearly with tension without elastic interactions.

Universal scaling form of nematic order with elastic energy constant.

Asymptotic behavior of order parameter at zero tension independent of spring equilibrium distance.

Abstract

We perform exact Statistical Mechanics calculations for a system of elongated objects (hard needles) that are restricted to translate along a line and rotate within a plane, and that interact via both excluded-volume steric repulsion and harmonic elastic forces between neighbors. This system represents a one-dimensional model of a liquid crystal elastomer, and has a zero-tension critical point that we describe using the transfer-matrix method. In the absence of elastic interactions, we build on previous results by Kantor and Kardar, and find that the nematic order parameter $Q$ decays linearly with tension $\sigma$. In the presence of elastic interactions, the system exhibits a standard universal scaling form, with $Q / |\sigma|$ being a function of the rescaled elastic energy constant $k / |\sigma|^\Delta$, where $\Delta$ is a critical exponent equal to $2$ for this model. At zero tension, simple scaling arguments lead to the asymptotic behavior $Q \sim k^{1/\Delta}$, which does not depend on the equilibrium distance of the springs in this model.