Flexoelectric moduli

In this section we will consider the orientational deformations of a nematic director which are flexoelectric in nature, i.e., induced due to the interaction of the flexoelectric polarization (see (4.2)) with the external electric field. Our consideration will be limited to spatially uniform fields E the case when E depends on coordinates is discussed in Chapter 5. We also discuss semiphenomenological approaches for the determination of nematic flexoelectric moduli ei and 63. Different types of electrooptical phenomena, where flexoelectric distortion plays a dominant role will be considered. Some of them are promising for potential applications. 

The flexoelectric distortion in Hybrid Aligned Nematic (HAN) configuration was studied [190, 191]. In HAN cells the director is homeotropic on one of the substrates and homogeneous on the other. The measme-ments of transmitted intensity in HAN samples allow us to evaluate for MBBA both a nematic anchoring energy W for the homeotropic orientation VF 6 X 10 erg/cm and the sum of the flexoelectric moduli 11 + 33 —4.5 X 10 dyn / [191]- In an electric field normal to the initial director plane, the flexoelectric effect results in the appearance of twist deformation, observed by the rotation of linearly polarized light [190]. 

Nevertheless, the method was realized with radial and tangential director orientation on the cylinder surfaces with acoustical modulation of the polarization signal [195]. The flexoelectric moduli were deduced firom the relations... 

The temperature dependence of flexoelectric moduli in MBBA was studied [202]. The coefficient 633 was shown to depend on temperatures even weaker than the order parameter S(T)... 

Taking into account flexoelectricity, it is possible to explain the appearance of a certain angle a, which the Kapustin-Williams domains form in some cases with the y-axis (the usual domain strips are parallel to the 2/-axis, Fig. 5.5). This oblique roll motion was observed in [90] and cannot be explained within the framework of the usual three-dimensional Carr-Helfrich model with strong anchoring at the boundaries [91]. The angle of the domain pattern a was shown [88, 89] to depend on the flexoelectric moduli eii, 633, the dielectric Ae, and the conductive Aa anisotropy. In certain intervals of the en — 633 and 611/633 values the angle A = 0 (the usual Kapustin-Williams domains) or A = 7t/2 (the longitudinal domains, also seen in experiment near the nematic-smectic A transition [91]). 

TABLE 7.1. Relative values of the flexoelectric moduli of DOBAMBC [29]. 

H.P. Hinov, On the coexistence of the flexo-dielectric walls-flexoelectric domains for the nematic MBBA - A new estimation of the modulus of the difference between the flexoelectric coefficients of splay and bend eij — esx, Mol. Cryst. Liq. Cryst. 524(1), 26-35, (2010). doi 10.1080/15421400903568161... 

Furthermore, by taking the length elastic modulus as C22 = 1 x 10 m/N ° we calculated for the apparent piezocoefficient a value of C12 = 8 x f0 C/N = 8 pm/V. This is also close to the experimental value of 20 pm/V, showing that even a very weak membrane flexoelectricity at the nanolevel combined with a thousandfold repetition of the elemental motile unit, can explain the huge apparent piezoelectricity of OHC at the microlevel. 

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). ... 

The first of them is the flexoelectric polarization, introduced in [31] and dependent on the curvature of the director field (div L and curl L) at constant modulus of the order parameter S. For uniform director orientation (L = const) Pf = 0. The second term is the so-called order polarization. It depends on the gradient of the nematic (quadrupolar) order parameter... 

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