Flexoelectricity

We know that the quadratic-in-field coupling of an electric field to the dielectric tensor craitributes to the free energy density with the term g = —EaE / n. When liquid crystals possess macroscopic electric polarization P (spontaneous or induced by some external, other than electric field factors), then an additional, linear-in-field term gE = PE is added to the free energy density. One of such a source of the macroscopic electric polarization is orientational distortion of a liquid crystal. 

Hy favours rolls with axes normal to y, but in the field-free case the rolls degenerate into a square pattern that may be regarded as a linear superposition of crossed convection rolls. When AT is increased well beyond A7 a complex hexagonal structure is found with a nematic-isotropic interface if the temperature of the upper plate is large enough. 

The studies outlined here are the most important ones that established the fundamental of principles of thermal instability in nematics. A number of theoretical and experimental investigations on these and other geometries have since been reported. A particularly interesting study is that of Lekkerkerker who predicted that a homeotropic nematic heated from below (which, it will be recalled, is stable against stationary convection) should become unstable with respect to oscillatory convection. The phenomenon was demonstrated experimentally by Guyon et 

It was shown that, for weak deformations corresponding to the continuum limit, the induced flexoelectric polarization P/ is proportional to first-order space derivatives of n. Higher-order derivatives are negligible in case of small a/l (a molecular dimension, 1 periodicity of deformation). Taking into account 

Illustration of the flexoelectricity assuming polar noncentrosymmetric molecules. Upper row pear-shape molecules Bottom row banana-shape molecules. (After Meyer. 

The molecular statistical approach to calculate the flexoelectric coefficients was developed independently by Helfrich and Derzhanski and Petrov. The calculation is based on the requirement to ensure maximum packing condition. The excess number (AN = N+ - N-) of the molecules with dipole moment n determines the electric polarization P= AN/i. Dividing this by the distortion we get the flexoelectric coefficient. 

Without going into details, we just give the results of Helfrich s calculation. Accordingly  

Molecular models for (a) pear-shape molecules (b) banana-shape molecules, used for calculation of flexoelectric coefficients. 

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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... 

Fig. 6. The two generic shapes of molecules which exhibit flexoelectric polarisation under distortion of the equilibrium director distribution...

Binary molecular co-crystals of 2,5-bis(3-pyridyl)-l,3,4-oxadiazole and 2,5-bis-(4-pyridyl)-l,3,4-oxadiazole with benzene-1,3,5-tricarboxylic and benzene-1,2,4,5-tetracarboxylic acids were studied by X-ray and thermogravimetric analysis of mass loss <2005MI1247>. Dipole moments were used to study the flexoelectric effect in guest-host mixtures of 2,5-(4-pentylbenzene)-l,3,4-oxadiazole with commercial liquid crystal hosts <2005CM6354>. The luminescence properties of many other copolymers were also investigated (see Section 5.06.12.3). 

A series of model nematic liquid crystals (among them oxadiazole derivatives) with transverse dipole moments were used to study the flexoelectric effect in guest-host mixtures with a commercial liquid crystal host <2005CM6354>. 

BLM, the type and concentrations) of the electrolytes bathing the BLM, the ions adsorbed on the BLM surface, and the extent to and frequency with which the BLM is bent [419]. These experimental observations have led to a phenomenological definition of the flexoelectric coefficient, f, as the ratio between the bending-induced transmembrane potential, Uf, and the change of curvature, c, that accompanies the bending of the membrane ... 

Flexoelectricity involves two degrees of freedom of the BLM electrical and mechanical. The system is amenable to simultaneous electrical and optical investigation of mechanical-to-electrical-energy conversion mediated by a bi-molecularly thin membrane. BLMs themselves are unlikely to be used as device components. They offer, however, an eminently suitable means for conducting the fundamental studies which are necessary for the full potential of the membrane-mimetic approach to advanced materials to be realized. 

Converse flexoelectric effects (i.e. voltage-generated curving) have been demonstrated in uranyl-acetate-stabilized phosphatidylserine BLMs by real-time stroboscopic interferometric measurements the obtained satisfactory agreement between the converse and the direct (i.e. curvature-generated voltage) flexoelectric coefficients have been in accord with the Maxwell relationship [8]. 

The enhanced chirality by doping SmC with BSMs can be explained qualitatively in the same way as in the N phase. However, the situation is more complicated in SmC because of spontaneous polarization and flexoelectric effect, and (3) must be replaced by an equation including such effects. Actually, the contribution of flexoelectric effect has been discussed by Gorecka et al. [4]. The other important effect is caused by the fact that the BSMs are in the tilted smectic phase. As mentioned above, the tilt of BSMs induces chirality as observed in the B2 phase. 

The third electro-optical effect using calamitic nematic liquid crystals makes use of a flexoelectric effect manifested by a curved asymmetrical nematic medium. This corresponds to piezoelectricity in crystals. The existence of flexoelectricity in a nematic phase under certain boundary conditions was predicted in the late 1960s and then confirmed experimentally several years later. However, LCDs using this effect, such as bistable nematic displays are only in the development stage and as such they will not be discussed in this monograph. 

Petrov, A.G. (1984) Flexoelectricity of lyotropics and biomembranes. Nuovo Cimento della Societa Italiana di Fisica D, 3 (1), 174-192. 

As noted earlier, the incorporation of chiral groups in the liquid crystal moieties can have the effect of inducing non-linear properties, which include thermochromism, ferroelectricity, antiferroelectricity, electrostriction, and flexoelectricity. In a now classical study, Hult [82] demonstrated that it was possible for supermolecular material 34 to exhibit two-state ferroelectric switching. The remarkable material he investigated, shown in Fig. 30, was found to exhibit two hitherto unclassified mesophases between the smectic... 

Though nematics are non-polar substances, a polarization may emerge in the presence of director gradients, even in the absence of an electric field. This flexoelectric polarization [2, 3]... 

The transition was found to be mediated by nucleation and traveling of sharp fronts (Fig. 11a) that indicates a backward bifurcation, although hysteresis has not been identified directly. Rather, a sharp jump in the contrast (pattern amplitude) with increasing voltage has been detected, with some indications that a low contrast pattern already arises at voltages before the jump occurs in Fig. 1 lb. A preliminary, weakly non-linear analysis has exhibited a bifurcation, which is in fact weakly supercritical at low frequencies. If small changes of the parameters and/or additional effects are included (e.g. flexoelectricity and weak-electrolyte effects) the bifurcation could become a more expressed subcritical one [32, 33]. 

As possible explanations, several ideas have been proposed a hand-waving argument based on destabilization of twist fluctuations" [52], a possibility of an isotropic mechanism based on the non-uniform space charge distribution along the field [53] and the flexoelectric effect [55-57]. 

Fig. 3.13.2. Interpretation of the origin of flexoelectricity in an assembly of quadrupoles (a) in the undeformed state the symmetry is such that there is no bulk polarization, (6) a splay deformation causes the positive charges to approach the upper plane and to be pushed away from the lower one. This dissymmetry gives rise...

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.

Fig. 3.13.5. A periodic electrostatic potential applied by means of interdigitated electrodes coated on one of the plates gives rise to (a) a flexoelectric distortion having a periodicity Id, where d is the spacing between the electrodes and (6) a dielectric distortion having a periodicity d. (Frost and Pershan.

Meyer s idea of flexoelectricity has been generalized to include a contribution due to the gradient of the orientational order parameter. The polarization in this case arises not from the curvature distortion of the director but from the spatial variation of the degree of orientational order of the molecules. In a simple first order theory, one may take P oc V5, where s is the order parameter as defined in 2.3.1. This effect has been termed as order electricity . 

Thus in S. each layer is spontaneously polarized. Since the structure has a twist about the layer normal, the tilt and the polarization direction rotate from one layer to the next (fig. 5.10.1(a)). This implies that there is a constant bend around the helical axis, which gives rise to a flexoelectric contribution to the polarization. 

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