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

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 becomes exact in this sticky limit, it fails for finite-range site-site interactions. Because the present model is mathematically equivalent to an exactly solvable one-dimensional nearest-neighbor fluid, we use the exact solution to reformulate the thermodynamics in terms of association-theory variables, including the fraction of unbonded sites and a generalized law of mass action. 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" 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 ξ ∼ p2 for finite-range SW sites to ξ ∼ p3 in the sticky limit.