On the orientational ordering of long rods on a lattice

Abstract

Abstract. -We argue that a system of straight rigid rods of length k on square lattice with only hard-core interactions shows two phase transitions as a function of density ρ for k ≥ 7. The system undergoes a phase transition from the low-density disordered phase to a nematic phase as ρ is increased from 0 at ρ = ρc1, and then again undergoes a reentrant phase transition from the nematic phase to a disordered phase at ρ = ρc2 < 1.Introduction. -The study of systems of long rod-like molecules in solution with only excluded volume interaction has a long history. Onsager had noted that such a system would show long-range orientational order at large enough density [1]. Flory studied a lattice model of long rod-like molecules, and based on a mean-field approximation, argued that the lattice model would also show an isotropic-nematic phase transition as a function of density [2]. Zwanzig studied a model of hard rods in the continuum, but with only a finite number of allowed orientations [3]. For the continuum problem, there is general agreement that in three dimensions, the isotropic-nematic transition occurs at large enough density, if the length-towidth ratio of the molecules is large enough [4]. In the case of two-dimensional systems of hard needles in a plane, there is no spontaneous breaking of continuous symmetry, but the high-density phase has orientational correlations that decay as a power-law [5,6].The situation is much less clear in the case of lattice models of straight hard rods of length k. The only analytically soluble case is that of dimers (k = 2), for which it is known that the orientational correlations have an exponential decay for all densities, except the full-coverage limit, which shows power-law correlations in all dimensions [7,8]. Power-law correlations at ρ = 1 were also seen in numerical study on the square lattice for k = 3 [9], and k = 4 [10]. Early Monte Carlo studies of semi-flexible k-mers on a square lattice showed no long-range orientational order for any density or temperature [11]. DeGennes and Prost, in 1995, in their book on liquid crystals, wrote "it is not quite certain that such a lattice model, even for large p (long molecule), will lead to a transition" [12]. While there has been some work on this general problem since then, the question still remains unsettled [13] .In this paper, we present the results of our Monte Carlo study of this model on the square lattice. We find very strong numerical evidence that the system shows nematic order at intermediate densities for k ≥ 7. We are not able to study systems near ρ = 1 in our

Nematic-disordered phase transition in systems of long rigid rods on two-dimensional lattices

“…The amplitude of this power-law term is rather small. This is related to the fact that for the k-mer problem, various perturbation series involve terms like k −k [12], which then leads to large correlation lengths. The HDD phase has power-law correlations at least for lengths up to ξ * .…”

“…[12] that the second phase transition may be viewed as a binding-unbinding transition of k species of vacancies. For studying such a characterization, we break the square lattice into k sublattices.…”

“…Similar behavior is expected in higher dimensions. In two dimensions, numerical studies have shown that k min = 7 [12]. The existence of the nematic phase, and hence the first transition from the isotropic to nematic phase, has been proved rigorously [13].…”

“…In Ref. [12], a variational estimate of the entropy of the nematic and the HDD phases suggests that 1 − ρ * 2 should vary as 1/k 2 for large k. Linares et. al.…”

“…For k > 2, the existence of a phase transition, with only hard-core interactions, remained unsettled for a long time [11]. Ghosh and Dhar recently argued that k-mers on a square lattice, for k ≥ k min , would undergo two phase transitions, and the nematic phase would exist for only an intermediate range of densities ρ * 1 < ρ < ρ * [12]. Similar behavior is expected in higher dimensions.…”