Orientational ordering and correlations in a quasi-one-dimensional hard-dumbbell fluid

TL;DR

This work addresses how geometry and entropy govern orientational ordering in a quasi-one-dimensional fluid of hard dumbbells with a continuous orientation. It develops an exact transfer-matrix framework to derive closed-form expressions for the equation of state, orientational distribution function, RDFs, and correlation lengths from the spectrum of the transfer operator. A key finding is the continuous crossover from unimodal to bimodal orientational statistics around $\rho_b\simeq 0.93$, with the high-density limit yielding $P/P_T\to 2$ and a geometry-driven isotropic–nematic-like ordering, all while maintaining an exact, benchmarkable description. The results quantify how orientational and positional fluctuations contribute to pressure and structure in confinement, offering precise insights for confined anisotropic fluids and providing stringent tests for approximate theories and simulations.

Abstract

We study a quasi-one-dimensional fluid of hard dumbbells with continuous orientational degrees of freedom using an exact transfer-matrix formulation. The model allows for a complete analytical characterization of thermodynamic properties, orientational ordering, and correlation functions in terms of the spectral properties of an integral operator. We derive exact expressions for the equation of state, the orientational distribution function, and both partial and total radial distribution functions. Their asymptotic behavior is governed by the complex poles of the Laplace-transformed correlation functions, which determine the positional and orientational correlation lengths. As density increases, the system exhibits a continuous crossover from a weakly ordered regime with a unimodal orientational distribution to a strongly constrained regime characterized by bimodal orientational ordering. This crossover is accompanied by a nonmonotonic behavior of the pressure relative to the Tonks gas and by a qualitative change in the decay of correlation functions from oscillatory to monotonic. In the high-pressure limit, we show that orientational and positional fluctuations contribute equally to the pressure, leading to a universal ratio of twice the Tonks pressure. The theoretical predictions are supported by numerical solutions of the discretized transfer operator and by scaling arguments that elucidate the high-pressure behavior of ordering and correlation lengths.

Figures (10)

• Figure 1: Quasi-one-dimensional system of hard dumbbells with centers constrained to move along the $x$ axis and free rotation in the $xy$ plane.
• Figure 2: Contour plot of the reduced contact distance $\sigma(\varphi,\varphi')/a$ for two dumbbells. The locus $-\pi/6 \le \varphi=\varphi' \le \pi/6$, along which the contact distance attains its minimum value $\sigma=a$, is indicated by the black line. The behavior of $\sigma$ in the immediate vicinity of this line, highlighted in red, governs the high-pressure properties of the system. The maximum value, $\sigma=2a$, is reached at the corners $\varphi,\varphi'=\pm\pi/2$.
• Figure 3: The solid line shows the slope of the contact distance along the valley, $\tau \equiv \lim_{\varphi'\rightarrow\varphi}|\nabla \sigma(\varphi,\varphi')/a|=(\cos\varphi)/\sqrt{2}$, for $-\pi/6<\varphi=\varphi'<\pi/6$. Since $\nabla\sigma$ along the valley is orthogonal to the segment $-\pi/6<\varphi=\varphi'<\pi/6$, $\tau$ coincides with the directional derivative of $\sigma$ along that direction. The dashed line shows the corresponding quantity for needles.
• Figure 4: Pressure $P/P_{\rm T}$ relative to that of the Tonks gas of spherical particles with diameter $a = 1$. The solid red curve corresponds to the dumbbell system, while the dashed blue curve shows the Tonks gas with effective diameter $\langle \sigma \rangle_{\rm iso} \simeq 1.5345$. In the latter, the pressure diverges as $\rho \to 1/\langle \sigma \rangle_{\rm iso} \simeq 0.652$ (vertical dotted blue line). The vertical dashed black line indicates the density $\rho_b \simeq 0.93$ at which the ODF changes from unimodal to bimodal. The inset shows a magnified view of $P/P_{\rm T}$ in the range $0.92 < \rho < 1$.
• Figure 5: Orientational distribution function (ODF), $f(\varphi)$, at densities $\rho = 0.40$, $0.91$, $0.93$, $0.95$, $0.97$, and $0.98$. The two vertical dashed lines indicate the angles $\varphi = \pm \pi/6$.