Flexoelectric effects external fields

Since the flexoelectric effect is associated with curvature distortions of the director field it seems natural to expect that the splay and bend elastic constants themselves may have contributions from flexoelectricity. The shape polarity of the molecules invoked by Meyer will have a direct mechanical influence independently of flexoelectricity and can be expected to lower the relevant elastic constants.The flexoelectric polarization will generate an electrostatic self-energy and hence make an independent contribution to the elastic constants. In the absence of any external field, the electric displacement D = 0 and the flexoelectric polarization generates an internal field E = —P/eo, where eq is the vacuum dielectric constant. Considering only a director deformation confined to a plane, and described by a polar angle 9 z), and in the absence of ionic screening, the energy density due to a splay-bend deformation reads as ... 

In liquid crystals with the capability of flexoelectric effect, in the absence of external electric fields, the state with uniform director configuration, which has no induced polarization, is the ground state and is stable. When an electric field is applied to the liquid crystal, the uniform orientation becomes unstable, because any small orientation deformation produced by thermal fluctuation or boundary condition will induce a polarization which will interact with the electric field and results in a lower free energy. The torque on the molecules due to the applied field and... 

We now turn to the changes that occur in the macroscopic structure of a liquid crystal due to a destabilization and reorientation of the director under direct action of an electric or magnetic field. The external field might be coupled either to the dielectric (diamagnetic) anisotropy (magnetically or electrically driven uniform Frederiks transition and periodic pattern formation) or to the macroscopic polarization (flexoelectric effect and ferroelectric switching) of the substance. The fluid is considered to be nonconductive. 

As discussed for nematics, flexoelectric effects are caused by the linear coupling of an external electric field with the flexoelectric polarization. In cholesterics, these effects are manifested in the three specific phenomena described below. 

As has been shown, the splay and bend distortions of a nematic create electric polarization. There is also a converse effect the external electric field causes a distortion due to the flexoelectric mechanism. For example, if the banana-shape molecules with transverse dipoles are placed in the electric field, the dipoles are partially aligned along the field and their banana shape induces some bend. This effect takes place even in nematics with zero dielectric anisotropy. 

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