Symmetry of the Fréedericksz transition in nonchiral nematic liquid crystals.

Abstract

The Fr\'eedericksz transition is a nonuniaxial, nonchiral nematic liquid crystal (LC) in the presence of the electric E field is considered using the Curie symmetry principle. All possible symmetries of (i) the LC point symmetry group ${\mathit{G}}_{\mathit{N}}$, (ii) the strong boundary LC orientation at the infinite plane-parallel plates of slab, and (iii) the direction of the E field are analyzed. The free energy polynomial J(${\mathit{c}}_{\mathit{i}}$) is expressed in terms of the invariant polynomials of components of the three-dimensional axial vector c. Possible primary and secondary bifurcations are determined for all classes of nonchiral nematic LC's (biaxial, tetrahedral, cubic, and icosahedral). It is shown that different kinds of such LC's, subjected to the E fields of different orientations, can be described by the same polynomial J(${\mathit{c}}_{\mathit{i}}$), invariant with respect to the action of a symmetry group ${\mathit{G}}_{\mathrm{Fr}}$ of the Fr\'eedericksz transition. In the framework of the symmetry approach, the influence of the thermal fluctuations of the nematic directors on the Fr\'eedericksz transition is studied and mean squares of these fluctuations are found.