Spontaneous polarization in chiral biaxial liquid crystals

A phenomenological theory of polar structures in chiral biaxial liquid crystals is constructed exploiting the properties of a symmetric and traceless tensor order parameter field ${\mathit{Q}}_{\mathrm{\alpha }\mathrm{\beta }}$(r) and of a polar field ${\mathit{P}}_{\mathrm{\alpha }}$(r). Full advantage is taken of the symmetry of the order parameters by systematic use of the method of integrity bases, which allows us to establish an expansion of the most general SO(3)-invariant free-energy density to arbitrary powers in the components ${\mathit{Q}}_{\mathrm{\alpha }\mathrm{\beta }}$ and ${\mathit{P}}_{\mathrm{\alpha }}$. A coordinate-independent parametrization of the invariants is introduced that yields a classification of local polar structures and some predictions about possible topologies of phase diagrams without the necessity of performing numerical calculations. As one prominent result, the theory predicts a polar, chiral biaxial state that exists due to a piezoelectric coupling of a chiral biaxial tensor field and the polarization field and which disappears if tensor is uniaxial. We then provide a general theory of flexopolarization in biaxial systems. A general biaxial system is described by 12 fundamental flexopolarization modes. Special cases, obtained by imposing symmetry restrictions to the tensor field Q, reduce the number of modes. Finally, the theory is applied to chiral phases. Simple polar chiral structures including cholesteric and smectic-${\mathit{C}}^{\mathrm{*}}$ liquid crystals are analyzed. In particular, it is shown that if the smectic-${\mathit{C}}^{\mathrm{*}}$ phase is stabilized due to the piezoelectric coupling between Q, P, and a density wave, then it must be described as a biaxial uniform spiral with at least two nonvanishing commensurate harmonics. The minimization of the quadratic part of the Landau–de Gennes energy supplemented by (flexo)polarization terms may give rise to incommensurate two- or three-dimensional polar structures that can be stabilized by cubic terms.