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.

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.

Extended law of corresponding states: square-well oblates.

The vapour-liquid coexistence collapse in the reduced temperature,Tr=T/Tc, reduced density,ρr=ρ/ρc, plane is known as a principle of corresponding states, and Noro and Frenkel have extended it for pair potentials of variable range. Here, we provide a theoretical basis supporting this extension, and show that it can also be applied to short-range pair potentials where both repulsive and attractive parts can be anisotropic. We observe that the binodals of oblate hard ellipsoids for a given aspect ratio (κ= 1/3) with varying short-range square-well interactions collapse into a single master curve in theΔB2*-ρrplane, whereΔB2*=(B2(T)-B2(Tc))/v0,B2is the second virial coefficient, andv0is the volume of the hard body. This finding is confirmed by both REMC simulation and second virial perturbation theory for varying square-well shells, mimicking uniform, equator, and pole attractions. Our simulation results reveal that the extended law of corresponding states is not related to the local structure of the fluid.

Demixing and tetratic ordering in some binary mixtures of hard superellipses.

We examine the fluid phase behavior of binary mixtures of hard superellipses using the scaled particle theory. The superellipse is a general two-dimensional convex object that can be tuned between the elliptical and rectangular shapes continuously at a given aspect ratio. We find that the shape of the particle affects strongly the stability of isotropic, nematic, and tetratic phases in the mixture even if the side lengths of both species are fixed. While the isotropic-isotropic demixing transition can be ruled out using the scaled particle theory, the first order isotropic-nematic and the nematic-nematic demixing transition can be stabilized with strong fractionation between the components. It is observed that the demixing tendency is strongest in small rectangle-large ellipse mixtures. Interestingly, it is possible to stabilize the tetratic order at lower densities in the mixture of hard squares and rectangles where the long rectangles form a nematic phase, while the squares stay in the tetratic order.

Three-step melting of hard superdisks in two dimensions.

We explore the link between the melting scenarios of two-dimensional systems of hard disks and squares through replica-exchange Monte Carlo simulations of hard superdisks. The well-known melting scenarios are observed in the disk and square limits, while we observe an unusual three-step scenario for dual shapes. We find that two mesophases mediate the melting: a hexatic phase and another fluid phase with a D_{2} local symmetry, we call it rhombatic, where both bond and particle orientational orders are quasi-long-range. Our results show that not only can the melting process of liquid-crystal forming molecules be complicated, where elongated shapes stabilize several mesophases, but also that of anisotropic quasispherical molecules.

Phase diagram of hard squares in slit confinement.

This work shows a complete phase diagram of hard squares of side length σ in slit confinement for H < 4.5, H being the wall to wall distance measured in σ units, including the maximal packing fraction limit. The phase diagram exhibits a transition between a single-row parallel 1-[Formula: see text] and a zigzag 2-[Formula: see text] structures for H c (2) = (2[Formula: see text] - 1) < H < 2, and also another one involving the 1-[Formula: see text] and 2-[Formula: see text] structures (two parallel rows) for 2 < H < H c (3) (H c (n) = n - 1 + [Formula: see text]/n is the critical wall-to-wall distance for a (n - 1)-[Formula: see text] to n-[Formula: see text] transition and where n-[Formula: see text] represents a structure formed by tilted rectangles, each one clustering n stacked squares), and a triple point for H t [Formula: see text] 2.005. In this triple point there coexists the 1-[Formula: see text], 2-[Formula: see text], and 2-[Formula: see text] structures. For regions H c (3) < H < H c (4) and H c (4) < H < H c (5), very similar pictures arise. There is a (n - 1)-[Formula: see text] to a n-[Formula: see text] strong transition for H c (n) < H < n, followed by a softer (n - 1)-[Formula: see text] to n-[Formula: see text] transition for n < H < H c (n + 1). Again, at H [Formula: see text] n there appears a triple point, involving the (n - 1)-[Formula: see text], n-[Formula: see text], and n-[Formula: see text] structures. The similarities found for n = 2, 3 and 4 lead us to propose a tentative phase diagram for H c (n) < H < H c (n + 1) (n ∈ [Formula: see text], n > 2), where structures (n - 1)-[Formula: see text], n-[Formula: see text], and n-[Formula: see text] fill the phase diagram. Simulation and Onsager theory results are qualitatively consistent.

Positional ordering of hard adsorbate particles in tubular nanopores.

The phase behavior and structural properties of a monolayer of hard particles is examined in such a confinement where the adsorbed particles are constrained to the surface of a narrow hard cylindrical pore. The diameter of the pore is chosen such that only first- and second-neighbor interactions occur between the hard particles. The transfer operator method of [Percus and Zhang, Mol. Phys. 69, 347 (1990)MOPHAM0026-897610.1080/00268979000100241] is reformulated to obtain information about the structure of the monolayer. We have found that a true phase transition is not possible in the examined range of pore diameters. The monolayer of hard spheres undergoes a structural change from fluidlike order to a zigzaglike solid one with increasing surface density. The case of hard cylinders is different in the sense that a layering takes place continuously between a low-density one-row and a high-density two-row monolayer. Our results reveal a clear discrepancy with classical density functional theories, which do not distinguish smecticlike ordering in bulk from that in narrow periodic pores.

Ordering transitions of weakly anisotropic hard rods in narrow slitlike pores.

The effect of strong confinement on the positional and orientational ordering is examined in a system of hard rectangular rods with length L and diameter D (L>D) using the Parsons-Lee modification of the second virial density-functional theory. The rods are nonmesogenic (L/D<3) and confined between two parallel hard walls, where the width of the pore (H) is chosen in such a way that both planar (particle's long axis parallel to the walls) and homeotropic (particle's long axis perpendicular to the walls) orderings are possible and a maximum of two layers is allowed to form in the pore. In the extreme confinement limit of H≤2D, where only one-layer structures appear, we observe a structural transition from a planar to a homeotropic fluid layer with increasing density, which becomes sharper as L→H. In wider pores (2D

Critical behavior of hard squares in strong confinement.

We examine the phase behavior of a quasi-one-dimensional system of hard squares with side-length σ, where the particles are confined between two parallel walls and only nearest-neighbor interactions occur. As in our previous work [Gurin, Varga, and Odriozola, Phys. Rev. E 94, 050603 (2016)]2470-004510.1103/PhysRevE.94.050603, the transfer operator method is used, but here we impose a restricted orientation and position approximation to yield an analytic description of the physical properties. This allows us to study the parallel fluid-like to zigzag solid-like structural transition, where the compressibility and heat capacity peaks sharpen and get higher as H→H_{c}=2sqrt[2]-1≈1.8284 and p→p_{c}=∞. Here H is the width of the channel measured in σ units and p is the pressure. We have found that this structural change becomes critical at the (p_{c},H_{c}) point. The obtained critical exponents belong to the universality class of the one-dimensional Ising model. We believe this behavior holds for the unrestricted orientational and positional case.

Ordering of hard rectangles in strong confinement.

Using transfer operator and fundamental measure theories, we examine the structural and thermodynamic properties of hard rectangles confined between two parallel hard walls. The side lengths of the rectangle (L and D, L>D) and the pore width (H) are chosen such that a maximum of two layers is allowed to form when the long sides of the rectangles are parallel to the wall, while only one layer is possible in case the rectangles are perpendicular to the wall. We observe three different structures: (i) at low density, the rectangles align mainly parallel to the wall, (ii) at intermediate or high density, two fluid layers form in which the rectangles are parallel to the wall, and (iii) a dense single fluid layer with rectangles aligned mainly perpendicular to the wall. The transition between these structures is smooth without any non-analytic behaviour in the thermodynamic quantities; however, the fraction of particles perpendicular (or parallel) to the wall can exhibit a relatively sudden change if L is close to H. In this case, interestingly, even three different structures can be observed with increasing density.

Anomalous structural transition of confined hard squares.

Structural transitions are examined in quasi-one-dimensional systems of freely rotating hard squares, which are confined between two parallel walls. We find two competing phases: one is a fluid where the squares have two sides parallel to the walls, while the second one is a solidlike structure with a zigzag arrangement of the squares. Using transfer matrix method we show that the configuration space consists of subspaces of fluidlike and solidlike phases, which are connected with low probability microstates of mixed structures. The existence of these connecting states makes the thermodynamic quantities continuous and precludes the possibility of a true phase transition. However, thermodynamic functions indicate strong tendency for the phase transition and our replica exchange Monte Carlo simulation study detects several important markers of the first order phase transition. The distinction of a phase transition from a structural change is practically impossible with simulations and experiments in such systems like the confined hard squares.

Phase behaviour and correlations of parallel hard squares: from highly confined to bulk systems.

We study a fluid of two-dimensional parallel hard squares in bulk and under confinement in channels, with the aim of evaluating the performance of fundamental-measure theory (FMT). To this purpose, we first analyse the phase behaviour of the bulk system using FMT and Percus-Yevick (PY) theory, and compare the results with molecular dynamics and Monte Carlo simulations. In a second step, we study the confined system and check the results against those obtained from the transfer matrix method and from our own Monte Carlo simulations. Squares are confined to channels with parallel walls at angles of 0° or 45° relative to the diagonals of the parallel hard squares, respectively, which allows for an assessment of the effect of the external-potential symmetry on the fluid structural properties. In general FMT overestimates bulk correlations, predicting the existence of a columnar phase (absent in simulations) prior to crystallization. The equation of state predicted by FMT compares well with simulations, although the PY approach with the virial route is better in some range of packing fractions. The FMT is highly accurate for the structure and correlations of the confined fluid due to the dimensional crossover property fulfilled by the theory. Both density profiles and equations of state of the confined system are accurately predicted by the theory. The highly non-uniform pair correlations inside the channel are also very well described by FMT.

Beyond the single-file fluid limit using transfer matrix method: Exact results for confined parallel hard squares.

We extend the transfer matrix method of one-dimensional hard core fluids placed between confining walls for that case where the particles can pass each other and at most two layers can form. We derive an eigenvalue equation for a quasi-one-dimensional system of hard squares confined between two parallel walls, where the pore width is between σ and 3σ (σ is the side length of the square). The exact equation of state and the nearest neighbor distribution functions show three different structures: a fluid phase with one layer, a fluid phase with two layers, and a solid-like structure where the fluid layers are strongly correlated. The structural transition between differently ordered fluids develops continuously with increasing density, i.e., no thermodynamic phase transition occurs. The high density structure of the system consists of clusters with two layers which are broken with particles staying in the middle of the pore.

Pair correlation functions of two- and three-dimensional hard-core fluids confined into narrow pores: exact results from transfer-matrix method.

The effect of confinement is studied on the local structure of two- and three-dimensional hard-core fluids. The hard disks are confined between two parallel lines, while the hard spheres are in a cylindrical hard pore. In both cases only nearest neighbour interactions are allowed between the particles. The vertical and longitudinal pair correlation functions are determined by means of the exact transfer-matrix method. The vertical pair correlation function indicates that the wall induced packing constraint gives rise to a zigzag (up-down sequence) shaped close packing structure in both two- and three-dimensional systems. The longitudinal pair correlation function shows that both systems transform continuously from a one-dimensional gas-like behaviour to a zigzag solid-like structure with increasing density.

Towards understanding the ordering behavior of hard needles: analytical solutions in one dimension.

We re-examine the ordering behavior of a one-dimensional fluid of freely rotating hard needles, where the centers of mass of the particles are restricted to a line. Analytical equations are obtained for the equation of state, order parameter, and orientational correlation functions using the transfer-matrix method if some simplifying assumptions are applied for either the orientational freedom or the contact distance between two needles. The two-state Zwanzig model accounts for the orientational ordering, but it produces unphysical pressure at high densities and there is no orientational correlation. The four-state Zwanzig model gives reasonable results for orientational correlation function, but the pressure is still poorly represented at high densities. In the continuum limit, apart from the orientational correlation length it is managed to reproduce all relevant bulk properties of the hard needles using an approximate formula for the contact distance. The results show that the orientational correlation length diverges at zero and infinite pressures. The high-density behavior of the fluid of needles is not resolved.

Orientational ordering of hard zigzag needles in one dimension.

The orientational ordering and the tilt angle behavior of a one-dimensional fluid of hard zigzag needles are examined by means of transfer matrix method and Onsager theory. The centers of mass of the particles are restricted to a line, while the orientational unit vectors are allowed to rotate freely in two dimensions. It is shown that zigzag needles do not undergo an isotropic-nematic phase transition, but the system is always in an orientationally ordered phase where the order parameter increases with the density. For hard needles and any other kinds of particles with an axis of symmetry the orientational distribution function is symmetric around its maximum value and the nematic director is perpendicular to the layer. For zigzag needles, which have nonconvex shape without an axis of symmetry, the orientational order is anisotropic around its maximum value and the nematic director is density dependent even at very high densities, i.e., the structure of one-dimensional fluid is always tilted. It is found that the density dependence of the tilted structure depends strongly on the shape of the zigzags. Surprisingly, the Onsager theory produces quite accurate results for the order parameters and tilt angles even in very dense systems.

Nematic and smectic ordering in a system of two-dimensional hard zigzag particles.

The orientational and positional ordering of the two-dimensional system of hard zigzag particles has been investigated by means of Onsager theory. Analytical results are obtained for the transition densities of the isotropic-nematic and the nematic-smectic phase transitions. It is shown that the stability of the nematic and smectic phases is very sensitive to the molecular shape. In the hard needle limit, only the isotropic-nematic phase transition takes place, while increasing the tail length and the bent angle between the central core and the tails destabilizes the nematic phase. On the other hand the stability of the smectic phase is due to the increasing excluded area cost with bent angle and the tail length. The zigzag particles pack in a layered structure such that they are tilted and form semi-ideal gas in the layers to push the high cost excluded area regions into the interstitial regions. The predictions of Onsager theory are in good agreement with MC simulation data.