Equation of state of hard-disk fluids under single-file confinement

TL;DR

This paper derives exact and approximate equations of state for single-file hard-disk fluids confined in narrow pores, analyzing low- and high-pressure behaviors and comparing results with simulations.

Contribution

It provides the first exact transfer-matrix solution for the system and introduces simplified approximations for practical use.

Findings

Exact third and fourth virial coefficients obtained.

Asymptotic high-pressure equation has a simple pole at close packing.

Proposed approximations show good agreement with simulations.

Abstract

The exact transfer-matrix solution for the longitudinal equilibrium properties of the single-file hard-disk fluid is used to study the limiting low- and high-pressure behaviors analytically as functions of the pore width. In the low-pressure regime, the exact third and fourth virial coefficients are obtained, which involve single and double integrals, respectively. Moreover, we show that the standard irreducible diagrams do not provide a complete account of the virial coefficients in confined geometries. The asymptotic equation of state in the high-pressure limit is seen to present a simple pole at the close-packing linear density, as in the hard-rod fluid, but, in contrast to the latter case, the residue is $2$. Since, for an arbitrary pressure, the exact transfer-matrix treatment requires the numerical solution of an eigenvalue integral equation, we propose here two simple approximations to the equation of state, with different complexity levels, and carry out an extensive assessment of their validity and practical convenience vs the exact solution and available computer simulations.