Towards understanding the ordering behavior of hard needles: New analytical solutions in one dimension

TL;DR

This paper develops analytical solutions for the ordering behavior of one-dimensional hard needles, exploring different models and their accuracy in predicting bulk properties and correlations.

Contribution

It introduces new analytical equations for the equation of state and correlation functions of hard needles using the transfer-matrix method with simplifying assumptions.

Findings

Orientational correlation length diverges at zero and infinite pressures.

Four-state Zwanzig model provides reasonable orientational correlations.

Approximate formula reproduces bulk properties in the continuum limit.

Abstract

We re-examine the ordering behavior of a one-dimensional fluid of freely rotating hard needles, where the centers of mass of the particles are restricted to a line. Analytical equations are obtained for the equation of state, order parameter and orientational correlation functions using the transfer-matrix method if some simplifying assumptions are applied for either the orientational freedom or the contact distance between two needles. The two-state Zwanzig model accounts for the orientational ordering, but it produces unphysical pressure at high densities and there is no orientational correlation. The four-state Zwanzig model gives reasonable results for orientational correlation function, but the pressure is still poorly represented at high densities. In the continuum limit, apart from the orientational correlation length it is managed to reproduce all relevant bulk properties of the hard needles using an approximate formula for the contact distance. The results show that the orientational correlation length diverges at zero and infinite pressures. The high density behavior of needles is not resolved.