We study the phase transition from a nematic phase to a high-density disordered phase in systems of long rigid rods of length k on the square and triangular lattices. We use an efficient Monte Carlo scheme that partly overcomes the problem of very large relaxation times of nearly jammed configurations. The existence of a continuous transition is observed on both lattices for k = 7. We study correlations in the high-density disordered phase, and we find evidence of a crossover length scale ξ * 1400, on the square lattice. For distances smaller than ξ * , correlations appear to decay algebraically. Our best estimates of the critical exponents differ from those of the Ising model, but we cannot rule out a crossover to Ising universality class at length scales ≫ ξ * . On the triangular lattice, the critical exponents are consistent with those of the two dimensional three-state Potts universality class.

PACS. 61.20.Qg Structure of associated liquids: electrolytes, molten salts, etc. - 61.25.Hq Macromolecular and polymer solutions; polymer melts; swelling,

We solve a model of polymers represented by self-avoiding walks on a lattice which may visit the same site up to three times in the grand-canonical formalism on the Bethe lattice. This may be a model for the collapse transition of polymers where only interactions between monomers at the same site are considered. The phase diagram of the model is very rich, displaying coexistence and critical surfaces, critical, critical endpoint and tricritical lines, as well as a multicritical point.From the grand-canonical results, we present an argument to obtain the properties of the model in the canonical ensemble, and compare our results with simulations in the literature. We do actually find extended and collapsed phases, but the transition between them, composed by a line of critical endpoints and a line of tricritical points, separated by the multicritical point, is always continuous.This result is at variance with the simulations for the model, which suggest that part of the line should be a discontinuous transition. Finally, we discuss the connection of the present model with the standard model for the collapse of polymers (self-avoiding self-attracting walks), where the transition between the extended and collapsed phases is a tricritical point.

We study a system of long rigid rods of fixed length k with only excluded volume interaction. We show that, contrary to the general expectation, the self-consistent field equations of the Bethe approximation do not give the exact solution of the problem on the Bethe lattice in this case. We construct a new lattice, called the random locally treelike layered lattice, which allows a dense packing of rods, and we show that the Bethe self-consistent equations are exact for this lattice. For a four-coordinated lattice, k-mers with k ≥ 4 undergo a continuous isotropic-nematic phase transition. For even coordination number q ≥ 6, the transition exists only for k ≥ k(min)(q), and is discontinuous.

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We solve a model of self-avoiding walks with up to two monomers per site on the Bethe lattice. This model, inspired in the Domb-Joyce model, was recently proposed to describe the collapse transition observed in interacting polymers [J. Krawczyk, Phys. Rev. Lett. 96, 240603 (2006)]. When immediate self-reversals are allowed (reversion-allowed model), the solution displays a phase diagram with a polymerized phase and a nonpolymerized phase, separated by a phase transition which is of first order for a nonvanishing statistical weight of doubly occupied sites. If the configurations are restricted forbidding immediate self-reversals (reversion-forbidden model), a richer phase diagram with two distinct polymerized phases is found, displaying a tricritical point and a critical end point.

Articles you may be interested inFast off-lattice Monte Carlo simulations of soft-core spherocylinders: Isotropic-nematic transition and comparisons with virial expansion J. Chem. Phys. 137, 134904 (2012); 10.1063/1.4755959 Critical behavior of long straight rigid rods on two-dimensional lattices: Theory and Monte Carlo simulations Comment on "Monte Carlo simulations of smectic phase transitions in flexible-rigid-flexible molecules" [J. Chem.We present a Monte Carlo algorithm for studying the equilibrium properties of hard rod fluids having only excluded volume interactions. The algorithm does not suffer from slow-down due to jamming even at densities close to the maximum possible. Implementing this algorithm on a two dimensional square lattice, we show the existence of a transition from an ordered nematic phase to a disordered phase at a critical density 0.910, for rods of length 7.

Athermal lattice gases of particles with first neighbor exclusion have been studied for a long time as simple models exhibiting a fluid-solid transition. At low concentration the particles occupy randomly both sublattices, but as the concentration is increased one of the sublattices is occupied preferentially. Here, we study a mixed lattice gas with excluded volume interactions only in the grand-canonical formalism with two kinds of particles: small ones, which occupy a single lattice site and large ones, which, when placed on a site, do not allow other particles to occupy its first neighbors also. We solve the model on a Bethe lattice of arbitrary coordination number q. In the parameter space defined by the activities of both particles, at low values of the activity of small particles (z(1)) we find a continuous transition from the fluid to the solid phase as the activity of large particles (z(2)) is increased. At higher values of z(1) the transition becomes discontinuous, both regimes are separated by a tricritical point. The critical line has a negative slope at z(1) = 0 and displays a minimum before reaching the tricritical point, so that a re-entrant behavior is observed for constant values of z(2) in the region of low density of small particles. The isobaric curves of the total density of particles as a function of the density or the activity of small particles show a minimum in the fluid phase.