Interaction between electric field and liquid crystals with spontaneous polarization: Derivation of suitable free energy expression for a cell with an applied voltage

The interaction between electric fields and liquid crystals is reconsidered, taking into account spatially varying anisotropic dielectric constant and ferro- resp. flexoelectricity. The general static case as well as planar samples with applied constant voltage is treated, and explicit formulas for the free energy are given. Some results: The consequences of the simple fact that the curl of the electric field is zero have been overlooked in many articles. The Landau free energy expansion for the chiral smectic C phase have been developed by R.B. Meyer1 and Pikin-Indenbom2. In the Pikin-Indenbom theory the P2-term, controlling the size of the spontaneous polarization, has been given the coefficient ½ χ, where χ is the dielectric susceptibility. However, there are several independent sources to the dielectric susceptibility, and not all of them do contribute to the formation of spontaneous polarization. This indicates that the size of the spontaneous polarization and the size of the induced polarization should be controlled by two separate quadratic terms, with unequal coefficients. This would imply a general reformulation of the theory. The self-energy of the ferroelectric polarization is dependent on the conformation of the liquid crystal, and acts as an extra elastic force, stiffening the liquid crystal. The resultant effective elastic force is proportional to the spontaneous polarization. This effect is mostly not screened by ions.