Phase transitions in a system of hard rectangles on the square lattice

Exclusion statistics for structured particles on topologically correlated states. II. Multicomponent lattice gases

Statistical thermodynamics of particles having a spectrum of topological correlated states and observing statistical exclusion is developed to describe mixtures of species of arbitrary size and shape. A generalized statistical distribution is obtained through a configuration space ansatz recently introduced for single species accounting for the multiple exclusion statistical phenomena, where spatially correlated particle states can be simultaneously excluded by more than one particle. Statistical exclusion on a correlated states spectrum is characterized by exclusion statistical parameters β_{cij} which are self-consistently determined within the multiple exclusion from thermodynamic boundary conditions. Self-exclusion and cross-exclusion frequency functions e_{ij}(n) and average cumulative exclusion functions G_{ij}(n) are introduced to characterize the state exclusion spectrum as density varies. Haldane's statistics and Wu's distribution for statistically independent excluding species are recovered in the limit of uncorrelated states for single species as well as for mixtures of self- and cross-excluding species with constant mutual statistical exclusion. The multiple exclusion statistics formalism is applied to the k-mer problem on a square lattice rationalized as a mixture of two differently oriented self-excluding and cross-excluding pseudospecies. An isotropic-nematic and a high-density nematic-isotropic (disordered) phase transitions is predicted only for k≥7. The isotropic-nematic transition is continuum as expected, but the high-density transition results in a first-order one. The formalism provides phase coexistence lines and the chemical potential dependence of the low- and high-density branches in the nematic regime. The theoretical approach to lattice gases presented in this work offers a unique general framework applicable to mixtures of entropy-complex lattice gases. From this framework, k-mer phase transitions can be reproduced, and significant configuration features can be derived from the state exclusion spectrum functions.

Exclusion statistics for structured particles on topologically correlated states. I. Single species lattice gases

A statistical thermodynamics description of particles having a set of spatially correlated states with statistical exclusion is developed. A general approximation for the density of states is presented from a state-counting ansatz recently introduced accounting for the multiple state exclusion statistical phenomena as a consequence of state spatial correlations. The multiple exclusion statistics is characterized by an exclusion correlation constant g_{c} which is consistently determined within the formalism from proper thermodynamic limits. The analytical form of g_{c} is given in terms of the Lambert function from the particle-lattice geometry. A generalized statistical distribution is obtained reducing to Haldane's statistics and Wu's distribution in the limiting case of particles on a set of spatially uncorrelated states. The problem of hard rods (k-mers) on a square lattice is studied with this formalism. From the entropy density dependence of the isotropic (I) and fully oriented nematic (N) phases, the approximation predicts two transitions, I→N and high-coverage N→I (disordered), only for k≥7 with the entropy at saturation matching to the known value from a Monte Carlo (MC) simulation. Critical coverage of both transitions is given for k=7 to k=20 in the first and second orders of approximations, in qualitative and quantitative agreement with results from MC simulations. State exclusion frequency e(n) and exclusion average G(n) functions are introduced and given in terms of the chemical potential to obtain a thermodynamic characterization of the state exclusion evolution on density. Results of chemical potential and state exclusion are shown for ideal lattice gases of k-mers, squares, and rectangles on a square lattice. Analytical results are compared with fast-relaxation grand canonical MC simulations.

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.

Freezing phase transition in hard-core lattice gases on the triangular lattice with exclusion up to seventh next-nearest neighbor

Hard-core lattice-gas models are minimal models to study entropy-driven phase transitions. In the k-nearest-neighbor lattice gas, a particle excludes all sites up to the kth next-nearest neighbors from being occupied by another particle. As k increases from one, it extrapolates from nearest-neighbor exclusion to the hard-sphere gas. In this paper we study the model on the triangular lattice for k≤7 using a flat histogram algorithm that includes cluster moves. Earlier studies focused on k≤3. We show that for 4≤k≤7, the system undergoes a single phase transition from a low-density fluid phase to a high-density sublattice-ordered phase. Using partition function zeros and nonconvexity properties of the entropy, we show that the transitions are discontinuous. The critical chemical potential, coexistence densities, and critical pressure are determined accurately.

Triangles on a triangular lattice: Insights into self-assembly in two dimensions driven by shape complementarity

A series of models for reversible filling of a triangular lattice with equilateral triangles has been developed and investigated. There are eight distinct models that vary in the set of prohibitions. In zeroth approximation, these models allow one to estimate the influence of the particles' shape and complementarity of their pair configurations on the self-assembly of dense monolayers formed by reversible filling. The most symmetrical models were found to be equivalent to hard-disk models on the hexagonal lattice. When any contact of hard triangles by vertices is prohibited, the dense monolayers are disordered, and their entropy tends to the constant. If only one pair configuration is prohibited, the close-packed layer appears through the continuous phase transition. In other cases, the weak first-order transition resulting in the self-assembly of close-packed layers is observed.

Phase transition from nematic to high-density disordered phase in a system of hard rods on a lattice

A system of hard rigid rods of length k on hypercubic lattices is known to undergo two phase transitions when chemical potential is increased: from a low density isotropic phase to an intermediate density nematic phase, and on further increase to a high-density phase with no orientational order. In this paper, we argue that, for large k, the second phase transition is a first-order transition with a discontinuity in density in all dimensions greater than 1. We show that the chemical potential at the transition is ≈kln[k/lnk] for large k, and that the density of uncovered sites drops from a value ≈(lnk)/k^{2} to a value of order exp(-ak), where a is some constant, across the transition. We conjecture that these results are asymptotically exact, in all dimensions d≥2. We also present evidence of coexistence of nematic and disordered phases from Monte Carlo simulations for rods of length 9 on the square lattice.

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.

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.

Rejection-free cluster Wang-Landau algorithm for hard-core lattice gases

We introduce a rejection-free, flat histogram, cluster algorithm to determine the density of states of hard-core lattice gases. We show that the algorithm is able to efficiently sample low entropy states that are usually difficult to access, even when the excluded volume per particle is large. The algorithm is based on simultaneously evaporating all the particles in a strip and reoccupying these sites with a new appropriately chosen configuration. We implement the algorithm for the particular case of the hard-core lattice gas in which the first k next-nearest neighbors of a particle are excluded from being occupied. It is shown that the algorithm is able to reproduce the known results for k=1,2,3 both on the square and cubic lattices. We also show that, in comparison, the corresponding flat histogram algorithms with either local moves or unbiased cluster moves are less accurate and do not converge as the system size increases.

Entropy of fully packed hard rigid rods on d-dimensional hypercubic lattices

We determine the asymptotic behavior of the entropy of full coverings of a L×M square lattice by rods of size k×1 and 1×k, in the limit of large k. We show that full coverage is possible only if at least one of L and M is a multiple of k, and that all allowed configurations can be reached from a standard configuration of all rods being parallel, using only basic flip moves that replace a k×k square of parallel horizontal rods by vertical rods, and vice versa. In the limit of large k, we show that the entropy per site S_{2}(k) tends to Ak^{-2}lnk, with A=1. We conjecture, based on a perturbative series expansion, that this large-k behavior of entropy per site is superuniversal and continues to hold on all d-dimensional hypercubic lattices, with d≥2.

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.

Glassy dynamics and equilibrium state on the honeycomb lattice: Role of surface diffusion and desorption on surface crowding

The phase behavior and adsorption kinetics of hard-core particles on a honeycomb lattice are studied by means of random sequential adsorption with surface diffusion. We concentrate on reversible adsorption by introducing a desorption process into our previous model and varying the equilibrium rate constant as a control parameter. We find that an exact prediction of the temporal evolution of fractional surface coverage and the surface pressure dynamics of reversible adsorption can be achieved by use of the blocking function of a system with irreversible adsorption of highly mobile particles. For systems out of equilibrium we observe several features of glassy dynamics, such as slow relaxation dynamics, the memory effect, and aging. In particular, the analysis of our system in the limit of small desorption probability shows simple aging behavior with a power-law decay. A detailed discussion of Gibbs adsorption isotherm for nonequilibrium adsorption is given, which exhibits a hysteresis between this system and its equilibrium counterpart.

Phase transitions in hard-core lattice gases on the honeycomb lattice

We study lattice gas systems on the honeycomb lattice where particles exclude neighboring sites up to order k (k=1,...,5) from being occupied by another particle. Monte Carlo simulations were used to obtain phase diagrams and characterize phase transitions as the system orders at high packing fractions. For systems with first-neighbors exclusion (1NN), we confirm previous results suggesting a continuous transition in the two-dimensional Ising universality class. Exclusion up to second neighbors (2NN) lead the system to a two-step melting process where, first, a high-density columnar phase undergoes a first-order phase transition with nonstandard scaling to a solidlike phase with short-range ordered domains and, then, to fluidlike configurations with no sign of a second phase transition. 3NN exclusion, surprisingly, shows no phase transition to an ordered phase as density is increased, staying disordered even to packing fractions up to 0.98. The 4NN model undergoes a continuous phase transition with critical exponents close to the three-state Potts model. The 5NN system undergoes two first-order phase transitions, both with nonstandard scaling. We, also, propose a conjecture concerning the possibility of more than one phase transition for systems with exclusion regions further than 5NN based on geometrical aspects of symmetries.

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.

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.

Multiple Exclusion Statistics

A new distribution for systems of particles in equilibrium obeying the exclusion of correlated states is presented following Haldane's state counting. It relies upon an ansatz to deal with the multiple exclusion that takes place when the states accessible to single particles are spatially correlated and it can be simultaneously excluded by more than one particle. Haldane's statistics and Wu's distribution are recovered in the limit of noncorrelated states of the multiple exclusion statistics. In addition, an exclusion spectrum function G(n) is introduced to account for the dependence of the state exclusion on the occupation number n. The results of thermodynamics and state occupation are shown for ideal lattice gases of linear particles of size k (k-mers) where the multiple exclusion occurs. Remarkable agreement is found with grand-canonical Monte Carlo simulations from k=2 to 10 where the multiple exclusion dominates as k increases.

Nematic and gas-liquid transitions for sticky rods on square and cubic lattices

Using grand-canonical Monte Carlo simulations, we investigate the phase diagram of hard rods of length L with additional contact (sticky) attractions on square and cubic lattices. The phase diagram shows a competition between gas-liquid and ordering transitions (which are of demixing type on the square lattice for L≥7 and of nematic type on the cubic lattice for L≥5). On the square lattice, increasing attractions initially lead to a stabilization of the isotropic phase. On the cubic lattice, the nematic transition remains of weak first order upon increasing the attractions. In the vicinity of the gas-liquid transition, the coexistence gap of the nematic transition quickly widens. These features are different from nematic transitions in the continuum.

Phase diagram of a system of hard cubes on the cubic lattice

We study the phase diagram of a system of 2×2×2 hard cubes on a three-dimensional cubic lattice. Using Monte Carlo simulations, we show that the system exhibits four different phases as the density of cubes is increased: disordered, layered, sublattice ordered, and columnar ordered. In the layered phase, the system spontaneously breaks up into parallel slabs of size 2×L×L where only a very small fraction cubes do not lie wholly within a slab. Within each slab, the cubes are disordered; translation symmetry is thus broken along exactly one principal axis. In the solidlike sublattice-ordered phase, the hard cubes preferentially occupy one of eight sublattices of the cubic lattice, breaking translational symmetry along all three principal directions. In the columnar phase, the system spontaneously breaks up into weakly interacting parallel columns of size 2×2×L, where only a very small fraction cubes do not lie wholly within a column. Within each column, the system is disordered, and thus translational symmetry is broken only along two principal directions. Using finite-size scaling, we show that the disordered-layered phase transition is continuous, while the layered-sublattice and sublattice-columnar transitions are discontinuous. We construct a Landau theory written in terms of the layering and columnar order parameters which is able to describe the different phases that are observed in the simulations and the order of the transitions. Additionally, our results near the disordered-layered transition are consistent with the O(3) universality class perturbed by cubic anisotropy as predicted by the Landau theory.

Controlled Symmetry Breaking in Colloidal Crystal Engineering with DNA

The programmed crystallization of particles into low-symmetry lattices represents a major synthetic challenge in the field of colloidal crystal engineering. Herein, we report an approach to realizing such structures that relies on a library of low-symmetry Au nanoparticles, with synthetically adjustable dimensions and tunable aspect ratios. When modified with DNA ligands and used as building blocks for colloidal crystal engineering, these structures enable one to expand the types of accessible lattices and to answer mechanistic questions about phase transitions that break crystal symmetry. Indeed, crystals formed from a library of elongated rhombic dodecahedra yield a rich phase space, including low-symmetry lattices (body-centered tetragonal and hexagonal planar). Molecular dynamics simulations corroborate and provide insight into the origin of these phase transitions. In particular, we identify an unexpected asymmetry in the DNA shell, distinct from both the particle and lattice symmetries, which enables directional, nonclose-packed interactions.

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.

Diffusion dynamics and steady states of systems of hard rods on a square lattice

It is known from grand canonical simulations of a system of hard rods on two-dimensional lattices that an orientationally ordered nematic phase exists only when the length of the rods is at least seven. However, a recent microcanonical simulation with diffusion kinetics, conserving both total density and zero nematic order, reported the existence of a nematically phase-segregated steady state with interfaces in the diagonal direction for rods of length six [Phys. Rev. E 95, 052130 (2017)2470-004510.1103/PhysRevE.95.052130], violating the equivalence of different ensembles for systems in equilibrium. We resolve this inconsistency by demonstrating that the kinetics violate detailed balance condition and drives the system to a nonequilibrium steady state. By implementing diffusion kinetics that drive the system to equilibrium, even within this constrained ensemble, we recover earlier results showing phase segregation only for rods of length greater than or equal to seven. Furthermore, in contrast to the nonequilibrium steady state, the interface has no preferred orientational direction. In addition, by implementing different nonequilibrium kinetics, we show that the interface between the phase segregated states can lie in different directions depending on the choice of kinetics.

Isotropic-nematic transition for hard rods on a three-dimensional cubic lattice

Using grand-canonical Monte Carlo (GCMC) simulations, we investigate the isotropic-nematic phase transition for hard rods of size L×1×1 on a three-dimensional cubic lattice. We observe such a transition for L≥6. For L=6, the nematic state has a negative order parameter, reflecting the co-occurrence of two dominating orientations. For L≥7, the nematic state has a positive order parameter, corresponding to the dominance of one orientation. We investigate rod lengths up to L=25 and find evidence for a very weakly first-order isotropic-nematic transition, while we cannot completely rule out a second-order transition. It was not possible to detect a density jump at the transition, despite using large systems containing several 10^{5} particles. The probability density distributions P(Q) from the GCMC simulations near the transition are very broad, pointing to strong fluctuations. Our results complement earlier results on the demixing (pseudonematic) transition for an equivalent system in two dimensions, which is presumably of Ising type and occurs for L≥7. We compare our results to lattice fundamental measure theory (FMT) and find that FMT strongly overestimates nematic order and consequently predicts a strong first-order transition. The rod packing fraction of the nematic coexisting states, however, agree reasonably well between FMT and GCMC.

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.

Random sequential adsorption of polydisperse mixtures on lattices

Random sequential adsorption of linear and square particles with excluded volume interaction is studied numerically on planar lattices considering Gaussian distributions of lateral sizes of the incident particles, with several values of the average μ and of the width-to-average ratio w. When the coverage θ is plotted as function of the logarithm of time t, the maximum slope is attained at a time t_{M} of the same order of the time τ of incidence of one monolayer, which is related to the molecular flux and/or sticking coefficients. For various μ and w, we obtain 1.5τ<t_{M}<5τ for linear particles and 0.3τ<t_{M}<τ for square particles. At t_{M}, the coverages with linear and square particles are near 0.3 and 0.2, respectively. Extrapolations show that coverages may vary with μ up to 20% and 2% for linear and square particles, respectively, for μ≥64, fixed time, and constant w. All θ vs logt plots have approximately the same shape, but other quantities measured at times of order t_{M} help to distinguish narrow and broad incident distributions. The adsorbed particle-size distributions are close to the incident ones up to long times for small w, but appreciably change in time for larger w, acquiring a monotonically decreasing shape for w=1/2 at times of order 100τ. At t_{M}, incident and adsorbed distributions are approximately the same for w≤1/8 and show significant differences for w≥1/2; this result may be used as a consistency test in applications of the model. The pair correlation function g(r,t) for w≤1/8 has a well defined oscillatory structure at 10t_{M}, with a minimum at r≈μ and maximum at r≈1.5μ, but this structure is not observed for w≥1/4.

Percolation of heteronuclear dimers irreversibly deposited on square lattices

The percolation problem of irreversibly deposited heteronuclear dimers on square lattices is studied. A dimer is composed of two segments, and it occupies two adjacent adsorption sites. Each segment can be either a conductive segment (segment type A) or a nonconductive segment (segment type B). Three types of dimers are considered: AA, BB, and AB. The connectivity analysis is carried out by accounting only for the conductive segments (segments type A). The model offers a simplified representation of the problem of percolation of defective (nonideal) particles, where the presence of defects in the system is simulated by introducing a mixture of conductive and nonconductive segments. Different cases were investigated, according to the sequence of deposition of the particles, the types of dimers involved in the process, and the degree of alignment of the deposited objects. By means of numerical simulations and finite-size scaling analysis, the complete phase diagram separating a percolating from a nonpercolating region was determined for each case. Finally, the consistency of our results was examined by comparing with previous data in the literature for linear k-mers (particles occupying k adjacent sites) with defects.

Columnar order and Ashkin-Teller criticality in mixtures of hard squares and dimers

We show that critical exponents of the transition to columnar order in a mixture of 2×1 dimers and 2×2 hard squares on the square lattice depends on the composition of the mixture in exactly the manner predicted by the theory of Ashkin-Teller criticality, including in the hard-square limit. This result settles the question regarding the nature of the transition in the hard-square lattice gas. It also provides the first example of a polydisperse system whose critical properties depend on composition. Our ideas also lead to some interesting predictions for a class of frustrated quantum magnets that exhibit columnar ordering of the bond energies at low temperature.

Polydispersed rods on the square lattice

We study the grand-canonical solution of a system of hard polydispersed rods placed on the square lattice using transfer matrix and finite-size scaling calculations. We determine the critical line separating an isotropic from a nematic phase. No second transition to a disordered phase is found at high density, contrary to what is observed in the monodispersed case. The estimates of critical exponents and the central charge on the critical line are consistent with the Ising universality class.

Asymptotic behavior of the isotropic-nematic and nematic-columnar phase boundaries for the system of hard rectangles on a square lattice

A system of hard rectangles of size m×mk on a square lattice undergoes three entropy-driven phase transitions with increasing density for large-enough aspect ratio k: first from a low-density isotropic to an intermediate-density nematic phase, second from the nematic to a columnar phase, and third from the columnar to a high-density sublattice phase. In this paper we show, from extensive Monte Carlo simulations of systems with m=1,2, and 3, that the transition density for the isotropic-nematic transition is ≈A(1)/k when k≫1, where A(1) is independent of m. We estimate A(1)=4.80±0.05. Within a Bethe approximation and virial expansion truncated at the second virial coefficient, we obtain A(1)=2. The critical density for the nematic-columnar transition when m=2 is numerically shown to tend to a value less than the full packing density as k(-1) when k→∞. We find that the critical Binder cumulant for this transition is nonuniversal and decreases as k(-1) for k≫1. However, the transition is shown to be in the Ising universality class.

Multiple phase transitions in extended hard-core lattice gas models in two dimensions

We study the k-NN hard-core lattice gas model in which the first k next-nearest-neighbor sites of a particle are excluded from occupation by other particles on a two-dimensional square lattice. This model is the lattice version of the hard-disk system with increasing k corresponding to decreasing lattice spacing. While the hard-disk system is known to undergo a two-step freezing process with increasing density, the lattice model has been known to show only one transition. Here, based on Monte Carlo simulations and high-density expansions of the free energy and density, we argue that for k = 4,10,11,14,⋯, the lattice model undergoes multiple transitions with increasing density. Using Monte Carlo simulations, we confirm the same for k = 4,...,11. This, in turn, resolves an existing puzzle as to why the 4-NN model has a continuous transition against the expectation of a first-order transition.