Flexoelectric polarization

Today the electrooptical properties of liquid crystals form well-developed branches both in the physics and technology of liquid crystals. In addition, electrooptical measurements are the basis of a number of precise methods for determining the physical parameters of a material, such as its elastic and viscosity coefficients, optical anisotropy, spontaneous polarization, flexoelectric coefficients, anchoring energies at interfaces, etc. 

We now turn to the changes that occur in the macroscopic structure of a liquid crystal due to a destabilization and reorientation of the director under direct action of an electric or magnetic field. The external field might be coupled either to the dielectric (diamagnetic) anisotropy (magnetically or electrically driven uniform Frederiks transition and periodic pattern formation) or to the macroscopic polarization (flexoelectric effect and ferroelectric switching) of the substance. The fluid is considered to be nonconductive. 

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. 

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]... 

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.

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. 

The aim of these studies is to understand and control the electrical properties of surfaces on a micrometric scale, that might be extremely useful for the application of these materials in the field of liquid crystal display (LCD) technology. In fact, it is well known that the LC anchoring properties depend not only on the substrate morphology but also on its electrical properties [68]. The electric polarization in nematic and other non-polar liquid crystals has essentially three origins flexoelectricity [65], orderelectricity [66,67] (related to the gradient in the order parameter) and the polarization of the substrate... 

It should be noted, however, that the flexoelectric effect is not necessarily related to the ordering of molecular dipoles. Frost and Marcerou proposed another microscopic mechanism of the flexoelectric effect, which requires neither the asymmetry of the molecular shape nor the permanent molecular dipole. The macroscopic polarization may simply appear in the direction of the gradient of average density of the molecular quadrupole moments. The quadrupole mechanism of flexoelectricity is more general because, in principle, it should manifest itself in any anisotropic material with a non-zero quadrupole density including solid crystals d and elastomers. 

During recent decades the molecular theory of flexoelectricity in nematic liquid crystals was developed further by various authors. " In particular, explicit expressions for the flexocoefiicients were obtained using the molecular-field approximation taking into account both steric repulsion and attraction between the molecules of polar shape. The influence of dipole-dipole correlations and molecular flexibility was later considered. Recently flexoelectric coefficients have been calculated numerically using the mean-field theory based on a simple surface intermolecular interaction model. This approach allows us to take into consideration the real molecular shape and to evaluate the flexocoefiicients for mesogenic molecules of different structures including dimers with flexible spacers. 

The results of Straley can be obtained by neglecting the pair attraction interaction potential V(xi, X2, ri2) in the equation for the direct correlation function. Indeed, the Straley theory of flexoelectricity was developed for the system of hard polar rods, while for thermotropic liquid crystals both the molecular shape and the intermolecular attraction are important. 

As mentioned above, the dipolar flexoelectric coefficients are determined by the polar molecular shape, which can be characterized by the molecular steric dipole. For a molecule having the shape of a truncated cone, as shown in Fig. 1.2, the steric dipole is in the direction of the long molecular axis a and is proportional to the cone angle 7, while for a bent-rod molecule the steric dipole is parallel to the short axis b and is proportional to the bend angle 7x- The relation between the flexocoefficients and the molecular shape is determined by the distance of closest approach 12 = i2(xi,X2,ri2), which reflects the polarity of the shape. 

In the present section we consider only the flexoelectric effect relating to the polar molecular shape, and we therefore assume that the pair attraction interaction potential V(xi,X2,ri2) is even in ai,a2,bi,b2. In this case the first term in Eq. (1.30) does not contribute to the flexoelectric coefficients, which are determined mainly by steric dipoles. 

It follows from Eqs (1.31) and (1.32) that the predominant contribution to the flexoelectric coefficients is determined by the isotropic intermolecular attraction modulated by the polar molecular shape. Indeed, in the general case the maximum attraction interaction energy V(R) kT where R is the equilibrium distance between the two molecules. It follows then that A ... 

The expressions for the flexoelectric coefficients presented in this section are derived using the molecular-field approximation. Therefore, care should be taken in the description of nematic liquid crystals composed of strongly polar molecules. In such liquid crystal materials (for example, cyanobiphenyls) the flexoelectric coefficients may be strongly affected by the short-range dipole-dipole correlations, which are considered in the following section. 

It follows from Eq. (1.45) that the contribution from the dipole-dipole correlations strongly depends on the value of the molecular dipoles, i.e. A dj (i if d > d . Therefore the correlation contribution can be substantial in liquid crystal materials composed of strongly polar molecules. The effect of dipole-dipole correlations on the flexoelectric coefficients has been discussed in experimental papers. 

It should be noted that dipole-dipole correlations may contribute to the flexoelectric coefficients only when mesogenic molecules have both a longitudinal and a sufficiently large transverse dipole. This may explain why the correlation contribution seems to be very important for oxycyanobiphenyls (and not for cyanobiphenyls, which do not possess any transverse dipoles). For weakly polar molecules d dx 1 D, and in this case the contribution from the dipole-dipole correlations is two orders of magnitude smaller than for 80CB and can be neglected. 

A.K. Tagantsev, Pyroelectric, piezoelectric, flexoelectric, and thermal polarization effects in ionic crystals, Sov. Phys. Uspekhi, 30(7), 588-603, (1987). doi 10.1070/PU1987v030n07 ABEH002926... 

The orientationally ordered nematic has a non-zero quadrupole density, irrespective of the shape or polar nature of the constituent molecules. Prost and Marcerou realized that, as a spatial gradient in the quadrupole density is equivalent to polarization, all nematics have non-zero flexocoefficients. In other words, flexoelectricity is a universal property of all nematic liquid crystals. As the nematic is a prime example of soft condensed matter, the magnitude of the flexocoefficient can be simply estimated as... 

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 ... 

The torque acting on the director is given by n x h, where h is called the molecular field, which can be derived from the Euler-Lagrange equation. The energy density corresponding to the flexoelectric polarization is given by — Pfl E and the molecular field can be expressed in the form ... 

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