Fulvio Bisi

The orientational order of nematic liquid crystals is traditionally studied by means of the second-rank ordering tensor S . When this is calculated through experiments or simulations, the symmetry group of the phase is not known a priori , but needs to be deduced from the numerical realisation of S , which is affected by numerical errors. There is no generally accepted procedure to perform this analysis. Here, we provide a new algorithm suited to identifying the symmetry group of the phase. As a by product, we prove that there are only five phase-symmetry classes of the second-rank ordering tensor and give a canonical representation of S for each class. The nearest tensor of the assigned symmetry is determined by group-projection. In order to test our procedure, we generate uniaxial and biaxial phases in a system of interacting particles, endowed with D ∞ h or D 2 h symmetry, which mimic the outcome of Monte–Carlo simulations. The actual symmetry of the phases is correctly identified, along with the optimal choice of laboratory frame.

DOI: http://dx.doi.org/10.1103/PhysRevE.88.039903 ©2013 American Physical Society

Recent measurements in a nematic liquid crystal formed by polyester compounds have given evidence of spontaneous macroscopic polar ordering. A mean-field theory for liquid crystals is built combining short-range, repulsive, steric forces with the average electric dipolar energy exchanged between molecules sharing the same excluded region. Such model is capable of identifying both uniaxial and biaxial polar phases. Under the assumption of a spheroidal shape for the molecules, and through a numerical bifurcation analysis, we assess the stability of phases upon two interaction parameters: the degree of intrinsic biaxiality, and the relative orientation of the permanent electric dipoles.

The excluded volume of a pair of molecules is proportional to the second virial coefficient in hard-core models that represent, for instance, the reference model in perturbation approaches to statistical theories of fluids [see, e.g. Chap. 5 of Kalikmanov, V. (2001), Statistical Physics of Fluids. Texts and Monograph in Physics, Springer, Berlin]. In three space dimensions, there exist exact results for convex molecules and in fact lack of convexity has been a major obstacle in applying the mathematical techniques employed in the convex case. In this paper, we illustrate how a mixed—analytical and numerical—method can be used to obtain exact expressions of the excluded volume for a pair of non-convex molecules conceived as aggregates of hard spheres; these can model van der Waals regions associated to the atoms forming each molecule. To compute the excluded volume for molecules endowed with C_2v symmetry, modelled as chains of tangent hard spheres, we adapt a numerical code available to the scientific community. Because the result is a rather cumbersome expression in term of the relative orientation of the interacting molecules, we expand it over a suitable set of symmetry adapted Wigner functions to build up approximate, but faithful expressions, and we also prove analytical results announced elsewhere [Bisi, F., Durand, G., Rosso, R. & Virga, E. (2008), Polar steric interactions for v-shaped molecules. Phys. Rev. E, 78, 011705].

Basic methods from bifurcation theory are applied to derive a criterion that predicts when a symmetric tricritical point may occur in a transition between condensed ordered phases described by any finite number of scalar order parameters. At such a point, a change of order takes place in the phase transition, which passes from first to second order, or vice versa.

The existence of uniaxial liquid crystals comprising polar molecules, with all the dipoles aligned in a parallel pattern, is classically ruled out. Generally, there are two different avenues to a mean-field theory for liquid crystals: one is based on short-range, repulsive, steric forces, and the other is based on long-range, globally attractive, dispersion forces. Purely polar steric interactions have been shown to have the potential of inducing unexpected orientationally ordered states. In real molecules, anisotropies both in shape and in polarizability coexist; it has been shown that dispersion forces interaction can be combined with hard-core repulsion in a formal theory, based on a steric tensor. Starting from this, we build an interaction Hamiltonian featuring the average electric dipolar energy exchanged between molecules with the same excluded region. Under the assumption that the molecular shape is spheroidal, we propose a mean-field model for polar nematic liquid crystals which can exhibit both uniaxial and biaxial polar phases. By means of a numerical bifurcation analysis, we discuss the stability of the equilibrium against the choice of two model parameters, one describing the degree of molecular shape biaxiality and the other describing the relative orientation of the electric dipole within each molecule. We find only uniaxial stable phases, which are effectively characterized by a single scalar order parameter.

Tetrahedral symmetry in nematic liquid crystals has been studied for several years, since the seminal paper by Fel [ 1 Fel , L. G. ( 1995 ). Phys. Rev. E , 52 , 702 – 717 . [Crossref], [PubMed], [Web of Science ®] [Google Scholar] ]. Amongst statistical theories of nematic liquid crystals, a molecular mean-field model has proven to be quite effective in predicting phase sequences for thermotropic biaxial nematic molecules endowed with D_2h symmetry, as a function of the parameters entering their interaction potential [ 2 Bisi , F. , Virga , E. G. , Gartland, Jr. , E. C. , De Matteis , G. , Sonnet , A. M. , & Durand , G. E. ( 2006 ). Phys. Rev. E , 73 , 051709 . [Google Scholar] , 3 Bisi , F. , Luckhurst , G. R. , & Virga , E. G. ( 2008 ). Phys. Rev. E , 78 , 021710 . [Google Scholar] ]. This model has highlighted the role of a partially repulsive quadrupolar potential of mean torque in promoting condensed phases. It has been shown that the quadrupolar approximation to the excluded-volume interaction between hard spherocuboids can be written precisely as the superposition of two London interactions: one repulsive and one attractive [ 4 Rosso , R. & Virga , E. G. ( 2006 ). Phys. Rev. E , 74 , 021712 . [Google Scholar] ]; furthermore, polar steric interactions have been shown to be capable of inducing orientationally ordered states possibly unexpected [ 5 Bisi , F. , Rosso , R. , Virga , E. G. , & Durand , G. E. ( 2008 ). Phys. Rev. E , 78 , 011705 . [Google Scholar] ]. By adapting a numerical code available to the scientific community, we evaluate the analytical excluded volume function for non-convex tetrahedral molecules, modelled as chains of tangent hard spheres. Since this function is overly complicated, we expand it over a suitable set of Symmetry Adapted Wigner Functions (SAWFs).