Determination of the symmetry classes of orientational ordering tensors

Abstract

The orientational order of nematic liquid crystals is traditionally studied by means of the secondrank ordering tensor . 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 , 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 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 2h 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.

Identification of low-symmetry phases in nematic liquid crystals

“…Surely, it is not an easy task the interpretation of the physical meaning of the j = 2 order parameters, when no particular symmetry is imposed [20][21][22]. Accordingly, in [5] we considered the second-rank order parameters, i.e. j = 2, and here we stick to that choice.…”

“…However, it is convenient for the sake of the presentation to distinguish between the linear space V M 2 , generated by the orthonormal frame (m 1 , m 2 , m 3 ) which describes the orientation of a generic molecule and V L 2 , with basis obtained from the laboratory frame of reference (ℓ 1 , ℓ 2 , ℓ 3 ). We follow [5,[25][26][27], and associate an orthonormal basis to a generic molecule; such a basis spans the five-dimensional linear space of traceless symmetric…”

“…In a previous work [5] we have proposed a method to perform the analysis of the data leading to the identification of the optimal orientation for the laboratory reference frame, and therefore to provide a systematic procedure to determine the symmetry class (the "phase" of the system), the symmetry axes (the "directors"), and the scalar order parameters of a liquid crystal compound whose second-rank ordering tensor is obtained through experiments or simulations.…”

“…In this paper we review the theory developed in [5], and provide a more systematic and practical discussion of the related algorithm.…”

“…In all sections we stick to the notation adopted in [5]; in particular, for point symmetry groups, i.e. subgroups of O(3), we comply with the standard Schönflies notation [6][7][8].…”