Thermodynamic and structural behavior of one-dimensional divalent patchy hard rods: Wertheim's first-order thermodynamic perturbation theory versus exact results

TL;DR

This study examines the thermodynamic and structural properties of one-dimensional divalent patchy hard rods, comparing exact theoretical predictions with results from Wertheim's perturbation theory, highlighting differences between finite-range and sticky interactions.

Contribution

The paper introduces an exact one-dimensional model for divalent patchy rods, improving upon Wertheim's theory by incorporating an exact relation between density and unbonded sites.

Findings

Finite-range SW sites lead to richer structural behavior than sticky sites.

The correlation length exhibits a maximum (Widom line) in the monotonic regime.

High-pressure correlation length scales as p^2 for finite-range, p^3 in sticky limit.

Abstract

We investigate the thermodynamic and structural properties of divalent patchy hard rods confined to a one-dimensional channel by modeling the bonding sites as attractive square-well (SW) patches located at the rod tips. The zero-range sticky limit is recovered by letting the well width vanish while keeping the stickiness parameter finite. While Wertheim's first-order thermodynamic perturbation theory (TPT1) becomes exact in this sticky limit, it fails for finite-range site-site interactions. We show that the theory can be made exact in one dimension by replacing the standard law of mass action with an exact relation between the density and the fraction of unbonded sites, together with an exact bonding free-energy contribution. Finite-range SW sites produce a richer structural behavior than sticky sites, including monotonic and oscillatory asymptotic decay of the pair correlation function, separated by the Fisher--Widom line. In the monotonic regime, the correlation length exhibits an absolute maximum defining the Widom line, while in the oscillatory regime it may display a local maximum and minimum, whose locus defines the ``Extrema of the Correlation length under Oscillatory decay'' (ECO) line. These features disappear in the sticky limit, where the system remains entirely in the oscillatory regime. We also show that the high-pressure behavior of the correlation length changes from $\xi\sim p^2$ for finite-range SW sites to $\xi\sim p^3$ in the sticky limit.