Correlation lengths in quasi-one-dimensional systems via transfer matrices

Thermodynamic properties of quasi-one-dimensional fluids

We calculate thermodynamic and structural quantities of a fluid of hard spheres of diameter σ in a quasi-one-dimensional pore with accessible pore width W smaller than σ by applying a perturbative method worked out earlier for a confined fluid in a slit pore [Franosch et al. Phys. Rev. Lett. 109, 240601 (2012)]. In a first step, we prove that the thermodynamic and a certain class of structural quantities of the hard-sphere fluid in the pore can be obtained from a purely one-dimensional fluid of rods of length σ with a central hard core of size σW=σ2-W2 and a soft part at both ends of length (σ - σW)/2. These rods interact via effective k-body potentials veff(k) (k ≥ 2). The two- and the three-body potential will be calculated explicitly. In a second step, the free energy of this effective one-dimensional fluid is calculated up to leading order in (W/σ)2. Explicit results for, e.g., the perpendicular pressure, surface tension, and the density profile as a function of density, temperature, and pore width are presented and partly compared with results from Monte-Carlo simulations and standard virial expansions. Despite the perturbative character of our approach, it encompasses the singularity of the thermodynamic quantities at the jamming transition point.

Ordering properties of anisotropic hard bodies in one-dimensional channels

The phase behavior and structural properties of hard anisotropic particles (prisms and dumbbells) are examined in one-dimensional channels using the Parsons-Lee (PL) theory, and the transfer-matrix and neighbor-distribution methods. The particles are allowed to move freely along the channel, while their orientations are constrained such that one particle can occupy only two or three different lengths along the channel. In this confinement setting, hard prisms behave as an additive mixture, while hard dumbbells behave as a non-additive one. We prove that all methods provide exact results for the phase properties of hard prisms, while only the neighbor-distribution and transfer-matrix methods are exact for hard dumbbells. This shows that non-additive effects are incorrectly included into the PL theory, which is a successful theory of the isotropic-nematic phase transition of rod-like particles in higher dimensions. In the one-dimensional channel, the orientational ordering develops continuously with increasing density, i.e., the system is isotropic only at zero density, while it becomes perfectly ordered at the close-packing density. We show that there is no orientational correlation in the hard prism system, while the hard dumbbells are orientationally correlated with diverging correlation length at close packing. On the other hand, positional correlations are present for all the systems, the associated correlation length diverging at close packing.

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

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.

Anomalous phase behavior of quasi-one-dimensional attractive hard rods

We study a two-state model of attractive hard rods using the transfer matrix method, where the centers of the particles are confined to a straight line, but the orientations of the rods can be parallel or perpendicular to the confining line. The rods are modeled as hard rectangles with length L and width D and decorated with attractive sites at both ends of the rectangles. We find that the particles align parallel to the line and form long chains at low densities, while they turn out of the line and form a Tonks gas at high densities. With increasing the stickiness between the rods, the structural change between parallel and perpendicular states becomes stronger and the pressure vs density curve becomes almost a horizontal line at the transition pressure. We show that such a behavior is reminiscent of the first-order phase transition. This manifests in the validity of the lever rule of the phase transitions for very sticky cases.

Inherent structure landscape of hard spheres confined to narrow cylindrical channels

The inherent structure landscape for a system of hard spheres confined to a hard cylindrical channel, such that spheres can only contact their first and second neighbors, is studied using an analytical model that extends previous results [Phys. Rev. Lett. 115, 025702 (2015)PRLTAO0031-900710.1103/PhysRevLett.115.025702] to provide a comprehensive picture of jammed packings over a range of packing densities. In the model, a packing is described as an arrangement of k helical sections, separated by defects, that have alternating helical twist directions and where all spheres satisfy local jamming constraints. The structure of each helical section is determined by a single helical twist angle, and a jammed packing is obtained by minimizing the length of the channel per particle with respect to the k helical section angles. An analysis of a small system of N=20 spheres shows that the basins on the inherent structure landscape associated with these helical arrangements split into a number of distinct jammed states separated by low barriers giving rise to a degree of hierarchical organization. The model accurately predicts the geometric properties of packings generated using the Lubachevsky and Stillinger compression scheme (N=10^{4}) and provides insight into the nature of the probability distribution of helical section lengths.

Analytical canonical partition function of a quasi-one-dimensional system of hard disks

The exact canonical partition function of a hard disk system in a narrow quasi-one-dimensional pore of given length and width is derived analytically in the thermodynamic limit. As a result, the many body problem is reduced to solving the single transcendental equation. The pressures along and across the pore, distributions of contact distances along the pore, and disks' transverse coordinates are found analytically and presented in the whole density range for three different pore widths. The transition from the solidlike zigzag to the liquidlike state is found to be quite sharp in the density scale but shows no genuine singularity. This transition is quantitatively described by the distribution of zigzag's windows through which disks exchange their positions across the pore. The windowlike defects vanish only in the densely packed zigzag, which is in line with a continuous Kosterlitz-Thouless transition.

Marginally jammed states of hard disks in a one-dimensional channel

We have studied a class of marginally jammed states in a system of hard disks confined in a narrow channel-a quasi-one-dimensional system-whose exponents are not those predicted by theories valid in the infinite dimensional limit. The exponent γ which describes the distribution of small gaps takes the value 1 rather than the infinite dimensional value 0.41269⋯. Our work shows that there exist jammed states not found within the tiling approach of Ashwin and Bowles. The most dense of these marginal states is an unusual state of matter that is asymptotically crystalline.

Highly confined mixtures of parallel hard squares: A density-functional-theory study

Using the fundamental-measure density-functional theory, we study theoretically the phase behavior of extremely confined mixtures of parallel hard squares in slit geometry. The pore width is chosen such that configurations consisting of two consecutive big squares, or three small squares, in the transverse direction, perpendicular to the walls, are forbidden. We analyze two different mixtures with edge lengths of species selected so as to allow or forbid one big plus one small square to fit into the channel. For the first mixture we obtain first-order transitions between symmetric and asymmetric packings of particles: Small and big squares are preferentially adsorbed at different walls. Asymmetric configurations are shown to lead to more efficient packing at finite pressures. We argue that the stability region of the asymmetric phase in the pressure-composition plane is bounded so that the symmetric phase is stable at low and very high pressure. For the second mixture, we observe strong demixing between phases which are rich in different species. Demixing occurs in the lateral direction, i.e., the dividing interface is perpendicular to the walls, and phases exhibit symmetric density profiles. The possible experimental realization of this behavior (which in practical terms is precluded by jamming) in strictly two-dimensional systems is discussed. Finally, the phase behavior of a mixture with periodic boundary conditions is analyzed and the differences and similarities between the latter and the confined system are discussed. We claim that, although exact calculations exclude the existence of true phase transitions in (1+ε)-dimensional systems, density-functional theory is still successful in describing packing properties of large clusters of particles.