Splay-bend

Splay ( bending or curvature ) defined by a splay constant K, or by a curvature elastic modulus = K h [76],... 

Fig. 24. Three principal types of orientational effects induced by electric (E) and magnetic (H) fields in nematic low molecular liquid crystals. At the top of the figure the initial geometries of molecules are shown. Below the different variants of the Frederiks transition — splay-, bend- and twist-effects are represented...

Sathyanarayana P, Mathew M, Sastry VSS, Kundu B, Le KV, Takezoe H, Dhara S (2010) Splay bend elasticity of a bent-core nematic liquid crystal. Phys Rev E (Rapid) 81 010702(R)-... 

In this equation, 8 is the dielectric anisotropy, K - (K +K j)/ is an average splay-bend elastic modulus, and E is the applied electric field. In this presentation, the curve must be linear when the Rapini approximation for the... 

Fig. 3.5.15. Helfrich walls (a) a twist wall parallel to the field, (b) a bend-splay wall parallel to the field, and (c) a splay-bend wall perpendicular to the field.

As before, we shall begin by considering a planar sample in which the director is confined to the xy plane. In such a case, a wedge disclination involves only splay and bend distortions and we need to take into account only the splay-bend anisotropy (kjj + fejj). 

Fig. 3.13.3. A hybrid aligned cell for the determination of the anisotropy of the flexoelectric coefficients. In this geometry, the director has a splay-bend distortion which gives rise to a flexoelectric polarization P. On applying an electric field E, the director is twisted by an angle (j> cc — which can be measured optically.

Here, a = 1,2 denotes the splay-bend and twist-bend mode, respectively, i i,2,3 are the Prank elastic constants, 771 2 are the rotational viscosities, is the component of the fluctuation wave vector parallel to the director and q the component perpendicular to it. 

This chapter is concerned with experimental measurements of flexo-electric coefficients. After a brief introduction to flexoelectricity in nematic liquid crystaJs, some applications exploiting the flexoelectric effect and the influence of this effect on electrohydrodynamic instabilities are pointed out. Flexoelectricity axises in samples with a splay-bend distortion in the director field and as such its measurement is not as direct as for dielectric constants. The theoretical background needed to analyse electro-optic experiments and extract the flexocoefficients is outlined in Section 2.2. Various experimental techniques that have been developed are described in Section 2.3. These involve cells in which the alignment of the nematic director is homeotropic, or planar or hybrid. In the first case, the interdigitated electrode technique is particularly noteworthy, as it has been used to establish several features of flexoelectricity (1) the effect can arise purely from the quadrupolar nature of the medium, and (2) the dipolar contribution relaxes at a relatively low frequency. 

Practically aU other methods developed for the measurement of flexo-coefficients are indirect . These exploit the fact that the polarization resulting from the splay-bend distortion couples linearly with an applied electric field E. This contributes to the total free energy of the sample, and hence alters the distortion of the director field compared to that in the absence of flexoelectric polarization. An external electric field of course acts on the dielectric anisotropy (As) of the nematic, which, like the orientational or-... 

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 ... 

Fig. 2.2. Schematic diagram of a hybrid-aligned nematic cell. The field-free director (shown by the continuous curved line) has a splay-bend curvature distortion in the xz plane. A DC field applied along the y axis rotates the polarization and the director (shown by the curved dashed line) acquires a 4>(z) profile. (Reproduced from Dozov et al. with the permission of EDP Sciences, http //publications.edpsciences.org.)...

Fig. 7.1. The field-induced twist in a hybrid-aligned device, based on the torque between E and the fiexoelectric polarization P permanently induced in the splay-bend due to the hybrid alignment. The arrows indicate the crossed polarizers.

The second application uses the converse flexoelectric effect, i.e. a field-induced splay-bend distortion, to generate a fast, symmetric and thresholdless linear electro-optic effect in a cholesteric liquid crystal. 

Fig. 7.3. The deviation of the optic axis in a cholesteric (hard-twisted chiral nematic, p <, where p is the cholesteric pitch and A is the wavelength of light) when an electric field E is applied perpendicular to the helical axis. The cholesteric geometry allows a fiexoelectric polarization to be induced in the direction of E. The plane containing the director, which is perpendicular to the page in the middle figure and is shown in the lower figure, illustrates the splay-bend distortion and the corresponding polarization that arises. (After Rudquist, inspired by Meyer and Patel.

The fiexoelectric coupling is not chiral, so what is the role of chirality in this case The answer is that the helically twisted state is the only one that is fiexoelectrically neutral (there is no local polarization related to twist) and therefore the only state from where a splay-bend deformation can increase continuously from zero in a symmetric fashion independent of the direction of E, while allowing for a homogeneously space-filling splay-bend. How it increases is illustrated in Fig. 7.5. 

Fig. 7.5. Increasing curvature in response to an increasing electric field the thresholdless field-induced periodic splay-bend deformation. The pattern in the director field is shown to the left (in any oblique cut perpendicular to the optic axis) and the optic axis deflection with increasing field is shown to the right. (From Rudquist et reproduced with kind permission of Taylor Francis, http //www.tandfonline.com.)...

---