Ordering of hard rectangles in strong confinement

Phases of the hard-plate lattice gas on a three-dimensional cubic lattice

We study the phase diagram of a lattice gas of 2×2×1 hard plates on the three-dimensional cubic lattice. Each plate covers an elementary plaquette of the cubic lattice, with the constraint that a site can belong to utmost one plate. We focus on the isotropic system, with equal fugacities for the three orientations of plates. We show, using grand canonical Monte Carlo simulations, that the system undergoes two phase transitions when the density of plates is increased: the first from a disordered fluid phase to a layered phase, and the second from the layered phase to a sublattice-ordered phase. In the layered phase, the system breaks up into disjoint slabs of thickness two along one spontaneously chosen Cartesian direction, corresponding to a twofold (Z_{2}) symmetry breaking of translation symmetry along the layering direction. Plates with normals perpendicular to this layering direction are preferentially contained entirely within these slabs, while plates straddling two adjacent slabs have a lower density, thus breaking the symmetry between the three types of plates. We show that the slabs exhibit two-dimensional power-law columnar order even in the presence of a nonzero density of vacancies. In contrast, interslab correlations of the two-dimensional columnar order parameter decay exponentially with the separation between the slabs. In the sublattice-ordered phase, there is twofold symmetry breaking of lattice translation symmetry along all three Cartesian directions. We present numerical evidence that the disordered to layered transition is continuous and consistent with universality class of the three-dimensional O(3) model with cubic anisotropy, while the layered to sublattice transition is first-order in nature.

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.

Entropy-driven phases at high coverage adsorption of straight rigid rods on two-dimensional square lattices

Polymers are frequently deposited on different surfaces, which has attracted the attention of scientists from different viewpoints. In the present approach polymers are represented by rigid rods of length k (k-mers), and the substrate takes the form of an L×L square lattice whose lattice constant matches exactly the interspacing between consecutive elements of the k-mer chain. We briefly review the classical description of the nematic transition presented by this system for k≥7 observing that the high-coverage (θ) transition deserves a more careful analysis from the entropy point of view. We present a possible viewpoint for this analysis that justifies the phase transitions. Moreover, we perform Monte Carlo (MC) simulations in the grand canonical ensemble, supplemented by thermodynamic integration, to first calculate the configurational entropy of the adsorbed phase as a function of the coverage, and then to explore the different phases (and orientational transitions) that appear on the surface with increasing the density of adsorbed k-mers. In the limit of θ→1 (full coverage) the configurational entropy is obtained for values of k ranging between 2 and 10. MC data are discussed in comparison with recent analytical results [D. Dhar and R. Rajesh, Phys. Rev. E 103, 042130 (2021)2470-004510.1103/PhysRevE.103.042130]. The comparative study allows us to establish the applicability range of the theoretical predictions. Finally, the structure of the high-coverage phase is characterized in terms of the statistics of k×l domains (domains of l parallel k-mers adsorbed on the surface). A distribution of finite values of l (l≪L) is found with a predominance of k×1 (single k-mers) and k×k domains. The distribution is the same in each lattice direction, confirming that at high density the adsorbed phase goes to a state with mixed orientations and no orientational preference. An order parameter measuring the number of k×k domains in the adsorbed layer is introduced.

Confinement effects on the random sequential adsorption packings of elongated particles in a slit

The behavior of a system of two-dimensional elongated particles (discorectangles) packed in a slit between the two parallel walls was analyzed using a simulation approach. The packings were produced using the random sequential adsorption model with continuous positional and orientational degrees of freedom. The aspect ratio (length-to-width ratio, ɛ=l/d) of the particles was varied within the range ɛ∈[1;32] while the distance between the walls was varied within the range h/d∈[1;80]. The properties of deposits in jammed state [the coverage, the order parameter, and the long-range (percolation) connectivity between particles] were studied numerically. The values of ɛ and h significantly affected the structure of the packings and the percolation connectivity. Particularly, the observed nontrivial dependencies of the jamming coverage φ(ɛ) or φ(h) were explained by the interplay of the different geometrical factors related to confinement, particle orientation degrees of freedom and excluded volume effects.

Hard core lattice gas with third next-nearest neighbor exclusion on triangular lattice: One or two phase transitions?

We obtain the phase diagram of the hard core lattice gas with third nearest neighbor exclusion on the triangular lattice using Monte Carlo simulations that are based on a rejection-free flat histogram algorithm. In a recent paper [Darjani et al., J. Chem. Phys. 151, 104702 (2019)], it was claimed that the lattice gas with third nearest neighbor exclusion undergoes two phase transitions with increasing density with the phase at intermediate densities exhibiting hexatic order with continuously varying exponents. Although a hexatic phase is expected when the exclusion range is large, it has not been seen earlier in hard core lattice gases with short range exclusion. In this paper, by numerically determining the entropies for all densities, we show that there is only a single phase transition in the system between a low-density fluid phase and a high density ordered sublattice phase and that a hexatic phase is absent. The transition is shown to be first order in nature, and the critical parameters are determined accurately.

Re-entrant transition as a bridge of broken ergodicity in confined monolayers of hexagonal prisms and cylinders

The entropy-driven monolayer assembly of hexagonal prisms and cylinders was studied under hard slit confinement. At the conditions investigated, the particles have two distinct and dynamically disconnected rotational states: unflipped and flipped, depending on whether their circular/hexagonal face is parallel or perpendicular to the wall plane. Importantly, these two rotational states cast distinct projection areas over the wall plane that favor either hexagonal or tetragonal packing. Monte Carlo simulations revealed a re-entrant melting transition where an intervening disordered Flipped-Unflipped (FUN) phase is sandwiched between a fourfold tetratic phase at high concentrations and a sixfold triangular solid at intermediate concentrations. The FUN phase contains a mixture of flipped and unflipped particles and is translationally and orientationally disordered. Complementary experiments were conducted with photolithographically fabricated cylindrical microparticles confined in a wedge cell. Both simulations and experiments show the formation of phases with comparable fraction of flipped particles and structure, i.e., the FUN phase, triangular solid, and tetratic phase, indicating that both approaches sample analogous basins of particle-orientation phase-space. The phase behavior of hexagonal prisms in a soft-repulsive wall model was also investigated to exemplify how tunable particle-wall interactions can provide an experimentally viable strategy to dynamically bridge the flipped and unflipped states.

Husimi-lattice solutions and the coherent-anomaly-method analysis for hard-square lattice gases

Although lattice gases composed of particles preventing up to their kth nearest neighbors from being occupied (the kNN models) have been widely investigated in the literature, the location and the universality class of the fluid-columnar transition in the 2NN model on the square lattice are still a topic of debate. Here, we present grand-canonical solutions of this model on Husimi lattices built with diagonal square lattices, with 2L(L+1) sites, for L⩽7. The systematic sequence of mean-field solutions confirms the existence of a continuous transition in this system, and extrapolations of the critical chemical potential μ_{2,c}(L) and particle density ρ_{2,c}(L) to L→∞ yield estimates of these quantities in close agreement with previous results for the 2NN model on the square lattice. To confirm the reliability of this approach, we employ it also for the 1NN model, where very accurate estimates for the critical parameters μ_{1,c} and ρ_{1,c}-for the fluid-solid transition in this model on the square lattice-are found from extrapolations of data for L⩽6. The nonclassical critical exponents for these transitions are investigated through the coherent anomaly method (CAM), which in the 1NN case yields β and ν differing by at most 6% from the expected Ising exponents. For the 2NN model, the CAM analysis is somewhat inconclusive, because the exponents sensibly depend on the value of μ_{2,c} used to calculate them. Notwithstanding, our results suggest that β and ν are considerably larger than the Ashkin-Teller exponents reported in numerical studies of the 2NN system.

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.

Alternative characterization of the nematic transition in deposition of rods on two-dimensional lattices

We revisit the problem of excluded volume deposition of rigid rods of length k unit cells over square lattices. Two new features are introduced: (a) two new short-distance complementary order parameters, called Π and Σ, are defined, calculated, and discussed to deal with the phases present as coverage increases; (b) the interpretation is now done beginning at the high-coverage ordered phase which allows us to interpret the low-coverage nematic phase as an ergodicity breakdown present only when k≥7. In addition the data analysis invokes both mutability (dynamical information theory method) and Shannon entropy (static distribution analysis) to further characterize the phases of the system. Moreover, mutability and Shannon entropy are compared, and we report the advantages and disadvantages they present for their use in this problem.

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.

Thermodynamic behavior of binary mixtures of hard spheres: Semianalytical solutions on a Husimi lattice built with cubes

We study binary mixtures of hard particles, which exclude up to their kth nearest neighbors (kNN) on the simple cubic lattice and have activities z_{k}. In the first model analyzed, point particles (0NN) are mixed with 1NN ones. The grand-canonical solution of this model on a Husimi lattice built with cubes unveils a phase diagram with a fluid and a solid phase separated by a continuous and a discontinuous transition line which meet at a tricritical point. A density anomaly, characterized by minima in isobaric curves of the total density of particles against z_{0} (or z_{1}), is also observed in this system. Overall, this scenario is identical to the one previously found for this model when defined on the square lattice. The second model investigated consists of the mixture of 1NN particles with 2NN ones. In this case, a very rich phase behavior is found in its Husimi lattice solution, with two solid phases-one associated with the ordering of 1NN particles (S1) and the other with the ordering of 2NN ones (S2)-beyond the fluid (F) phase. While the transitions between F-S2 and S1-S2 phases are always discontinuous, the F-S1 transition is continuous (discontinuous) for small (large) z_{2}. The critical and coexistence F-S1 lines meet at a tricritical point. Moreover, the coexistence F-S1,F-S2, and S1-S2 lines meet at a triple point. Density anomalies are absent in this case.

Phase transitions in a system of hard Y-shaped particles on the triangular lattice

We study the different phases and the phase transitions in a system of Y-shaped particles, examples of which include immunoglobulin-G and trinaphthylene molecules, on a triangular lattice interacting exclusively through excluded volume interactions. Each particle consists of a central site and three of its six nearest neighbors chosen alternately, such that there are two types of particles which are mirror images of each other. We study the equilibrium properties of the system using grand canonical Monte Carlo simulations that implement an algorithm with cluster moves that is able to equilibrate the system at densities close to full packing. We show that, with increasing density, the system undergoes two entropy-driven phase transitions with two broken-symmetry phases. At low densities, the system is in a disordered phase. As intermediate phases, there is a solidlike sublattice phase in which one type of particle is preferred over the other and the particles preferentially occupy one of four sublattices, thus breaking both particle symmetry as well as translational invariance. At even higher densities, the phase is a columnar phase, where the particle symmetry is restored, and the particles preferentially occupy even or odd rows along one of the three directions. This phase has translational order in only one direction, and breaks rotational invariance. From finite-size scaling, we demonstrate that both the transitions are first order in nature. We also show that the simpler system with only one type of particle undergoes a single discontinuous phase transition from a disordered phase to a solidlike sublattice phase with an increasing density of particles.

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<H<3D) planar order with two layers, homeotropic order, and even combined bilayer structures (one layer is homeotropic, while the other is planar) can be stabilized at high densities. Moreover, first-order phase transitions can be seen between different structures. One of them emerges between a monolayer and a bilayer with planar orders at relatively low packing fractions.

Columnar-disorder phase boundary in a mixture of hard squares and dimers

A mixture of hard squares, dimers, and vacancies on a square lattice is known to undergo a transition from a low-density disordered phase to a high-density columnar ordered phase. Along the fully packed square-dimer line, the system undergoes a Kosterliz-Thouless-type transition to a phase with power law correlations. We estimate the phase boundary separating the ordered and disordered phases by calculating the interfacial tension between two differently ordered phases within two different approximation schemes. The analytically obtained phase boundary is in good agreement with Monte Carlo simulations.