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This study examines the thermodynamic and structural properties of one-dimensional divalent patchy hard rods, where bonding sites are modeled as attractive square-well patches at the rod tips. The authors demonstrate that Wertheim's first-order thermodynamic perturbation theory, while exact in the sticky (zero-range) limit, fails for finite-range interactions, and they derive an exact formulation for one dimension by replacing the standard law of mass action with precise relations. The finite-range interactions produce richer structural behavior than the sticky limit, including monotonic and oscillatory decay regimes of the pair correlation function separated by the Fisher-Widom line, along with a newly identified "ECO line" describing extrema of the correlation length under oscillatory decay.
Why it matters
Understanding the exact thermodynamic and structural behavior of simplified one-dimensional models provides critical benchmarks for validating and improving perturbation theories used in colloidal science and soft matter physics. These findings have potential implications for designing patchy particle systems with tailored self-assembly properties relevant to materials science and nanotechnology.
arXiv:2605.21003v1 Announce Type: cross
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 $xisim p^2$ for finite-range SW sites to $xisim p^3$ in the sticky limit.