Inversion symmetry, flexoelectricity

We need to note that mathematically the splay and bending deformations are vectors, i.e., the flexoelectric coupling constant is a second-rank tensor. Accordingly, the flexoelectricity should not be confused with piezoelectricity, which is described by a third-rank tensor coupling constant. Piezoelectricity requires lack of inversion s)mtimetry of the phase, whereas flexoelectricity can exist in materials with inversion symmetry, such as in nematics with macroscopic symmetry. 

According to Eq. (7.1) P is zero for the two cases of uniform director fields and pure twist. Hence both cases can serve as a zero state as far as flexoelectric excitations are concerned. It is important to note that a twist is not associated with a polarization (i.e. C2 is identically zero, cf. Fig. 7.2). An imstrained nematic has a centre of symmetry (centre of inversion). On the other hand, none of the elementary deformations - splay, twist or bend have a centre of symmetry. According to Curie s principle they could then be associated with the separation of charges analogous to the piezoeffect in solids. This is true for splay and bend but not for twist because of an additional symmetry in that case if we twist the adjacent directors in a nematic on either side of a reference point, there is always a two-fold symmetry axis along the director of the reference point. In fact, any axis perpendicular to the twist axis is such an axis. Due to this symmetry no vectorial property can exist perpendicular to the director. In other words, a twist does not lead to the separation of charges. This is the reason why twist states appear naturally in liquid crystals and are extremely common. It also means that an electric field cannot induce a twist just by itself in the bulk of a nematic. If anything it reduces the twist. A twist can only be induced in a situation where a field turns the director out of a direction that has previously been fixed by boundary conditions (which, for instance, happens in the pixels of an IPS display). 

The second turn of the discussion around the nature of a SHG in nematic liquid crystals arised when the SHG was observed in oriented layers of 4-methoxybenzylidene-4 -butylaniline (MBBA)/ The phenomenon has been explained by the lack of the symmetry center in the nematic phase. The zero-field SHG in MBBA was also investigated but the nature of the effect was connected with the flexoelectric polarization of surface layers. Such a polarization has to remove the inversion center in surface liquid crystalline layers and to allow the SHG to be detectable. Another explanation of the zero-field SHG in terms of the electric quadrupolar interaction was suggested in. ... 

The flexoelectric effect is the liquid crystal analogy to the piezoelectric effect in solids. To see this we only have to make the connection between the translational variable in solids and the angular variable in liquid crystals. For both effects there is a corresponding inverse effect. However, the differences are notable. The piezoeffect is related to an asymmetry of the medium (no inversion center). The flexoelectric effect is due to the asymmetry of the molecules, regardless of the symmetry of the medium. We have seen how the elementary deformations in the director field destroy the center of symmetry in the liquid crystal. Therefore the liquid may itself possess a center of symmetry. In other words, the situation is just the opposite to the one in solids ... 

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