Theoretical model for the discrete flexoelectric effect and a description for the sequence of intermediate smectic phases with increasing periodicity

A general phenomenological description and a simple molecular model is proposed for the “discrete” flexoelectric effect in tilted smectic liquid crystal phases. This effect defines a polarization in a smectic layer induced by a difference of director orientations in the two smectic layers adjacent to it. It is shown that the “discrete” flexoelectric effect is determined by electrostatic dipole-quadrupole interaction between positionally correlated molecules located in adjacent smectic layers, while the corresponding dipole-dipole interaction is responsible for a coupling between polarization vectors in neighboring layers. It is shown that a simple phenomenological model of a ferrielectric smectic liquid crystal, which has recently been proposed in the literature, can be used to describe the whole sequence of intermediate chiral smectic $C*$ phases with increasing periods, and to determine the nonplanar structure of each phase without additional assumptions. In this sequence the phases with three- and four-layer periodicities have the same structure, as observed in the experiment. The theory predicts also the structure of intermediate phases with longer periods that have not been studied experimentally so far. The structures of intermediate phases with periodicities of up to nine layers are presented together with the phase diagrams, and a relationship between molecular chirality and the three-dimensional structure of intermediate phases is discussed. It is considered also how the coupling between the spontaneous polarization determined by molecular chirality and the induced polarization determined by the discrete flexoelectric effect stabilizes the nonplanar structure of intermediate phases.