Flexoelectric structure, elasticity

The factors Kn are elastic constants for the nematic phase and Icb is the Boltzmann constant. Therefore a combination of molecular electronic structure, orientational order and continuum elasticity are all involved in determining the flexoelectric polarisation. Polarisation can also be produced in the presence of an average gradient in the density of quadrupoles. This is... 

All consequences of the fiexoelectric effect considered so far are indirect and are not directly observable. Fiexoelectric interactions change the elastic constant (Eq. 5.16) in the SmC phase, which influences the layer polarization (Eq. 5.42). The layer polarization influences intralayer interactions and interactions to nearest layers and is consequently a source of interactions to more distant layers. The fiexoelectric interactions stabilize different structures with longer periodicities. But these are all indirect effects, which often cannot be separated from other effects. As the fiexoelectric effect is of achiral origin, there is no simple way to isolate the influence of flexoelectricity on the structure or the macroscopic properties. 

In the most simple chiral polar tilted smectics, ferroelectric liquid crystals, the flexoelectric phenomenon influences the structure of the SmC phase only quantitatively. It affects the elastic and chiral couplings and consequently slightly changes the transition temperature to the tilted phase and the pitch of the helicoidal modulation. 

For weak anchoring and 0 by analogy with a nematic (see Eq. 10.77), the free energy of the distortion includes the elastic term due to the bend-distortion (we assume K = Kn = K s) and the flexoelectric term with an average coefficient e. The second elastic term is due to the cholesteric helical structure (modulus K22). ... 

Due to the above-mentioned results [188, 189] the flexoelectric response is considerable in a narrow frequency region limited by the frequency of the structural relaxation fs / d where K is the average elastic co-... 

It is also possible to envisage minimum energy structures with non-zero splay or bend strain, when other terms linear in strain contribute to the free energy. However, a uniformly splayed or bent structure will no longer have the reversal symmetry+n=-n and so will be associated with permanently polarized structures a uniformly bent structure would have to be biaxial with a polar structure perpendicular to the major axis. These structures, illustrated in Fig. 5, are associated with the phenomenon known as flexoelectricity, where polarization is coupled to elastic strain. 

As we have repeatedly stressed, flexoelec-tricity is a phenomenon that is a priori independent of chirality. But we have also seen that some flexoelectric deformations do have a tendency to occur spontaneously in a chiral medium. All except the helical C state are, however, suppressed, because they are not space-filling. A flexoelectric deformation may of course also occur spontaneously in the nonchiral case, namely, under exactly the same conditions where the deformation is space-filling and does not give rise to defect structures. In other words, in creating the twist-bend structure which is characteristic of a helielectric. Imagine, for instance, that we have mesogens which have a pronounced bow shape and, in addition, some lateral dipole. Sterically they would prefer a helicoidal structure, as depicted in Fig. 52, which would minimize the elastic... 

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