Computer simulation study of two-dimensional nematogenic lattice models based on dispersion interactions

Abstract

We have considered a nematogenic lattice model, consisting of three-component unit vectors, associated with a two-dimensional lattice, and interacting via London–Heller–de Boer dispersion potential restricted to nearest neighbours, already studied by simulation in three dimensions (Mol. Phys. 42 (1981) 1205; Int. J. Mod. Phys. B 13 (1999) 3879; Phys. Rev. E 65 (2002) 041706). The model is defined by Δ jk=ε[(γ 2−γ)S jk+γ 2(− 3 2 h jk+1)] , where h jk=(3a ja k−τ jk) 2, S jk=P 2(a j)+P 2(a k) , r = x j− x k, s = r /| r |, a j= u j· s , a k= u k· s , τ jk= u j· u k and α ̄ = 1 3 (α ||+2α ⊥), γ= α ||−α ⊥ 3 α ̄ . Here the two-component vectors x j∈Z 2 define centre-of-mass coordinates of the particles, and u k are three-component unit vectors defining their orientations; α ||, α ⊥ are the eigenvalues of the molecular polarizability tensor, γ denotes its relative anisotropy, and ε is a positive quantity setting energy and temperature scales (i.e., T ∗=k BT/ε ). In two dimensions, and in contrast to the Lebwohl–Lasher lattice model, the potential's anisotropic character does not prevent existence of orientational order at finite temperature. Monte Carlo calculations were carried out using the two values γ=± 1 2 , and comparisons are reported with a mean field ( MF) treatment as well. In both cases, some orientational order was found to survive up to temperatures well above the disordering transition of the three-dimensional counterpart, possibly at all finite temperatures. When γ=− 1 2 , the model produces homeotropic anchoring, and MF gives results in good agreement with simulation. On the other hand, when γ=+ 1 2 , the model produces a ground state where particles are aligned along a lattice axis; both MF and simulation results for the second-rank ordering tensor show a low-temperature region where the system becomes biaxial, with the main director aligned along a lattice axis; at higher temperature there is a transition to uniaxial order with negative order parameter, and director orthogonal to the lattice plane. MF predictions now agree qualitatively with simulation, but, in quantitative terms, the transition temperature is overestimated by some 50%.