Deformations flexoelectric smectic phases

The term piezoelectric was borrowed from the physics of solids by analogy to the piezoelectric effect in crystals without center of symmetry. As a rule, the piezoelectric polarization manifests itself as a charge on the surfaces of a crystal due to a translational deformation, e.g. compression or extension. Piezo-effects are also characteristic of polar liquid crystalline phases, e.g., of the chiral smectic C phase. The polarization, we are interested now, is caused by the mechanical curvature (or flexion) of the director field, and, following De Gennes, we call it flexoelectric. 

This effect is observed in a geometry where the cholesteric axis h is homogeneously oriented in the plane of the cell (along x) and an electric field is applied to the electrodes of a sandwich cell along the z axis [137,138]. In this case, the helical structure, even the ideal one, is incompatible with the planar boundary conditions, and splayed and bended regions form near the boundaries. Thus, according to Eq. (38), the flexoelectric polarization arises in those regions which can interact with the electric field. The distortion is very similar to that observed in the ferroelectric smectic C phase (see Fig. 24) for a so-called deformed helix ferroelectric effect [139]. 

Flexoelectric phenomena in smectic C phases are considerably more complicated than in smectic A phases, even in the case of preserved layer thickness. Thus under the assumption of incompressible layers, there are not less than nine independent flexoelectric contributions. This was first shown in [ 127]. If we add such deformations which do not preserve the layer spacing, there are a total of 14 flexocoefficients [128]. If we add chirality, as in smectic C, we thus have, in principle, 15 different sources of polarization to take into account. 

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