Reentrant disordered phase in a system of repulsive rods on a Bethe-like lattice

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 of fully packed rigid rods on generalized Husimi trees: A route to the square-lattice limit

Although hard rigid rods (k-mers) defined on the square lattice have been widely studied in the literature, their entropy per site, s(k), in the full-packing limit is only known exactly for dimers (k=2) and numerically for trimers (k=3). Here, we investigate this entropy for rods with k≤7, by defining and solving them on Husimi lattices built with diagonal and regular square-lattice clusters of effective lateral size L, where L defines the level of approximation to the square lattice. Due to an L-parity effect, by increasing L we obtain two systematic sequences of values for the entropies s_{L}(k) for each type of cluster, whose extrapolations to L→∞ provide estimates of these entropies for the square lattice. For dimers, our estimates for s(2) differ from the exact result by only 0.03%, while that for s(3) differs from best available estimates by 3%. In this paper, we also obtain a new estimate for s(4). For larger k, we find that the extrapolated results from the Husimi tree calculations do not lie between the lower and upper bounds established in the literature for s(k). In fact, we observe that, to obtain reliable estimates for these entropies, we should deal with levels L that increase with k. However, it is very challenging computationally to advance to solve the problem for large values of L and for large rods. In addition, the exact calculations on the generalized Husimi trees provide strong evidence for the fully packed phase to be disordered for k≥4, in contrast to the results for the Bethe lattice wherein it is nematic.

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.

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.

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.

Relaxation in two-dimensional suspensions of rods as driven by Brownian diffusion

Relaxation in a two-dimensional suspensions containing rods was studied by using dynamic Monte Carlo simulations. An off-lattice model with continuous positional and orientational degrees of freedom was considered. The initial state was produced by using a random sequential adsorption model. During the relaxation, the rods underwent translational and rotational Brownian motions. The simulations were run at different values of number density ρ (the number of rods per unit area) and of the initial orientation order parameter S_{i}. The rods were assumed to have core-shell structures. The evolutions of both the connectivity and the order parameter have revealed different relaxation behavior.

Sedimentation of a suspension of rods: Monte Carlo simulation of a continuous two-dimensional problem

The sedimentation of a two-dimensional suspension containing rods was studied by means of Monte Carlo (MC) simulations. An off-lattice model with continuous positional and orientational degrees of freedom was considered. The initial state before sedimentation was produced using a model of random sequential adsorption. During such sedimentation, the rods undergo translational and rotational Brownian motions. The MC simulations were run at different initial number densities (the numbers of rods per unit area), ρ_{i}, and sedimentation rates, u. For sediment films, the spatial distributions of the rods, the order parameters, and the electrical conductivities were examined. Different types of sedimentation-driven self-assembly and anisotropy of the electrical conductivity were revealed inside the sediment films. This anisotropy can be finely regulated by changes in the values of ρ_{i} and u.

Vertical drying of a suspension of sticks: Monte Carlo simulation for continuous two-dimensional problem

The vertical drying of a two-dimensional colloidal film containing zero-thickness sticks (lines) was studied by means of kinetic Monte Carlo (MC) simulations. The continuous two-dimensional problem for both the positions and orientations was considered. The initial state before drying was produced using a model of random sequential adsorption with isotropic orientations of the sticks. During the evaporation, an upper interface falls with a linear velocity in the vertical direction, and the sticks undergo translational and rotational Brownian motions. The MC simulations were run at different initial number concentrations (the numbers of sticks per unit area), p_{i}, and solvent evaporation rates, u. For completely dried films, the spatial distributions of the sticks, the order parameters, and the electrical conductivities of the films in both the horizontal, x, and vertical, y, directions were examined. Significant evaporation-driven self-assembly and stratification of the sticks in the vertical direction was observed. The extent of stratification increased with increasing values of u. The anisotropy of the electrical conductivity of the film can be finely regulated by changes in the values of p_{i} and u.

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.

Phase transitions in a system of long rods on two-dimensional lattices by means of information theory

The orientational phase transitions that occur in the deposition of longitudinal polymers of length k (in terms of lattice units) are characterized by information theory techniques. We calculate the absolute value of an order parameter δ, which weights the relative orientations of the deposited rods, which varies between 0.0 (random orientation) and 1.0 (fully oriented in either of the two equivalent directions in an L×L square lattice). A Monte Carlo (MC) algorithm is implemented to induce a dynamics allowing for accommodation of the rods for any given density or coverage θ (ratio of the occupied sites over all the sites in the lattice). The files storing δ(t) (with time t measured in MC steps) are then treated by data recognizer wlzip based on data compressor techniques yielding the information content measured by a parameter η(θ). This allows us to recognize two maxima separated by a well-defined minimum for η(θ) provided k≥7. The first maximum is associated with an isotropic-nematic (I-N) phase transition occurring at intermediate density, while the second maximum is associated with some kind of nematic-isotropic transition at high coverage. In the cases of k<7, the curves for η(θ) are almost constant, presenting a very broad maximum which can hardly be associated with a phase transition. The study varies L and k, allowing for a basic scaling of the found critical densities towards the thermodynamic limit. These calculations confirm the tendency obtained by different methods in the case of the intermediate-density I-N phase transition, while this tendency is established here in the case of the high-density phase transition.

Diffusion-driven self-assembly of rodlike particles: Monte Carlo simulation on a square lattice

The diffusion-driven self-assembly of rodlike particles was studied by means of Monte Carlo simulation. The rods were represented as linear k-mers (i.e., particles occupying k adjacent sites). In the initial state, they were deposited onto a two-dimensional square lattice of size L×L up to the jamming concentration using a random sequential adsorption algorithm. The size of the lattice, L, was varied from 128 to 2048, and periodic boundary conditions were applied along both x and y axes, while the length of the k-mers (determining the aspect ratio) was varied from 2 to 12. The k-mers oriented along the x and y directions (k_{x}-mers and k_{y}-mers, respectively) were deposited equiprobably. In the course of the simulation, the numbers of intraspecific and interspecific contacts between the same sort and between different sorts of k-mers, respectively, were calculated. Both the shift ratio of the actual number of shifts along the longitudinal or transverse axes of the k-mers and the electrical conductivity of the system were also examined. For the initial random configuration, quite different self-organization behavior was observed for short and long k-mers. For long k-mers (k≥6), three main stages of diffusion-driven spatial segregation (self-assembly) were identified: the initial stage, reflecting destruction of the jamming state; the intermediate stage, reflecting continuous cluster coarsening and labyrinth pattern formation; and the final stage, reflecting the formation of diagonal stripe domains. Additional examination of two artificially constructed initial configurations showed that this pattern of diagonal stripe domains is an attractor, i.e., any spatial distribution of k-mers tends to transform into diagonal stripes. Nevertheless, the time for relaxation to the steady state essentially increases as the lattice size growth.

Monte Carlo simulation of evaporation-driven self-assembly in suspensions of colloidal rods

The vertical drying of a colloidal film containing rodlike particles was studied by means of kinetic Monte Carlo (MC) simulation. The problem was approached using a two-dimensional square lattice, and the rods were represented as linear k-mers (i.e., particles occupying k adjacent sites). The initial state before drying was produced using a model of random sequential adsorption (RSA) with isotropic orientations of the k-mers (orientation of the k-mers along horizontal x and vertical y directions are equiprobable). In the RSA model, overlapping of the k-mers is forbidden. During the evaporation, an upper interface falls with a linear velocity of u in the vertical direction and the k-mers undergo translation Brownian motion. The MC simulations were run at different initial concentrations, p_{i}, (p_{i}∈[0,p_{j}], where p_{j} is the jamming concentration), lengths of k-mers (k∈[1,12]), and solvent evaporation rates, u. For completely dried films, the spatial distributions of k-mers and their electrical conductivities in both x and y directions were examined. Significant evaporation-driven self-assembly and orientation stratification of the k-mers oriented along the x and y directions were observed. The extent of stratification increased with increasing value of k. The anisotropy of the electrical conductivity of the film can be finely regulated by changes in the values of p_{i}, k, and u.

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.

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

The phase diagram of a system of monodispersed hard rectangles of size m × mk on a square lattice is numerically determined for m = 2,3 and aspect ratio k = 1,2,...,7. We show the existence of a disordered phase, a nematic phase with orientational order, a columnar phase with orientational and partial translational order, and a solidlike phase with sublattice order, but no orientational order. The asymptotic behavior of the phase boundaries for large k is determined using a combination of entropic arguments and a Bethe approximation. This allows us to generalize the phase diagram to larger m and k, showing that for k ≥ 7, the system undergoes three entropy-driven phase transitions with increasing density. The nature of the different phase transitions is established and the critical exponents for the continuous transitions are determined using finite size scaling.