Flexoelectricity direct

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 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]. 

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. 

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. 

General expressions for the flexocoefiicients of nematic liquid crystals have been obtained in terms of the direct correlation function using the powerful density functional approach. These expressions have been used to obtain some interesting numerical results using the Perkus-Yevic approximation for the pair correlation function. The results from the density functional theory have also been used in computer simulations of flexoelectricity using model bent-core molecules interacting via the Gay-Berne potential. Alternative general expressions for the flexocoefiicients have... 

This chapter is arranged as follows. In Section 1.2 we consider in more detail the dipolar and quadrupolar mechanisms of flexoelectricity, and in Section 1.3 we derive the general expressions for the flexocoefficients in terms of the direct pair correlation function. These results are used in Section 1.4 to obtain approximate expressions for the flexocoefficients in the molecular-field approximation taking into account both intermolecular repulsion and attraction. In that section we also consider the dependence of the flexocoefficients on the absolute value of the molecular dipole and on the orientation of the electric dipole with respect to the molecular long axes and the steric dipole. In Section 1.5 the effect of dipole-dipole correlations is analysed and in Section 1.6 we discuss the mean-field theory of flexoelectricity, which allows us to account for the real molecular shape. 

A molecular-statistical theory of the flexoelectric effect in the nematic phase can be derived in a general way using the density-functional approach to the theory of liquid crystals. In this approach, the free energy of a liquid crystal, F, is a functional of the density po(a ) = Po/(w) where /(w) is the orientational distribution fimction. The general structure of the functional F p) is not known, but the functional derivatives are known and are related to the direct correlation functions of the nematic phase. 

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. 

The general expressions for the flexoelectric coefficients obtained using the density functional approach can be used to estimate the contributions from different types of intermolecular correlations, including short-range dipole-dipole correlations. We use the following approximation for the direct pair correlation function ... 

The linear terms in the expansion Eq. (1.39) do not contribute to the flexoelectric coefficients because the dipole-dipole interaction potential is odd both in di and d2 and hence the corresponding contributions vanish after averaging over the orientation of the molecular axes. Thus it is necessary to take into account the quadratic terms in the expansion of the direct correlation function. Then the contribution from the dipole-dipole correlations to the flexocoefficients can be written in the form ... 

The molecular-statistical theory of flexoelectricity, presented in the previous sections, does not allow us to establish a direct relation between the flexocoefficients and the details of a particular molecular structure (except for permanent electric and steric dipoles) because the theory is based on simple model interaction potentials. A different version of the mean-field theory, which takes into consideration the real molecular shape, has recently been proposed by Ferrarini et This approach is based on the... 

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. 

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. 2.4. A transverse electric field, indicated by the arrow pointed towards the right at the top of the figiu-e, tilts the apolar director as shown by double headed arrows in a specific direction due to the flexoelectric effect on a 90° twisted nematic cell. The tilting direction reverses if the field direction is reversed. The transmitted intensity mesisured with a polarized light beam traversing the cell vertically as indicated by the dashed line will be identical in the two cases. On the other hand, with an oblique beam, the transmitted intensities for the two tilted director structures will be different, and can be used to me siu-e the flexocoefficient (adapted from Kischfai et cU. ).

A direct flexing method for measuring flexoelectric coefficients. 70... 

In the following sections of this chapter we will summarize the direct, as well as the converse, flexoelectric measurements in fluid and elastomeric (dry or swollen) bent-core nematic liquid crystals, and try to explain these observations using the structural model outlined above. 

Although most of the scientists working on liquid crystals may think that flexoelectricity is a special property of liquid crystals, it was actually first discussed in 1964 for crystals as a response to strain (stress) gradients. The direct and converse flexoelectric coupling constants Cijki were described with a fourth-rank tensor, as... 

The direct method introduced in the previous section was first employed for studying the flexoelectric response of a bent-core nematic liquid crystal. 

The 63 values obtained from the measm-ements on ClPbislOBB using Eq. (3.27) are plotted in Fig. 3.12. It can be seen that 63 deduced from these measm-ements is in satisfactory agreement with that obtained from the direct flexoelectric effect based on polarization current detection imder an apphed cm-vatm-e strain (also plotted in Fig. 3.12). 

Summarizing, experimental observations suggest that the giant (direct or converse) flexoelectricity of bent-core nematics is related to the polar smectic clusters occurring in them. In order to explore the exact mechanism for how clusters contribute to the flexoelectric response, further experimental and theoretical studies are needed. 

MBBA.50 The wave number gd of the patterns is not much influenced by the inclusion of flexoelectricity in contrast to considerable changes with respect to the direction of the wave vector. 

In this chapter we consider several important aspects of the flexoelectric effect for chiral polar smectic liquid crystals and for the variety of phases. First, we discuss the reason for indirect interlayer interactions, which extend to more distant layers, and the lock-in to multi-layer structures. Second, although it was believed for a long time that polarization in tilted chiral smectics is always perpendicular to the tilt with the smectic layer normal, a component in the direction of the tilt may exist. And third, in multi-layer structures, the flexoelectricaUy induced polarization can be extremely large but is difficult to measure. 

The flexoelectric contribution to the free energy has to take into account both the non-uniformity of director and the polarization. As in the SmC phase the non-uniformity appears due to the changes of the tilt direction, which can be expressed in the derivatives of the tilt. In the SmC phase, where the modulation periods are long and the variation of the tilt direction from layer to layer is small, we can consider the contri-... 

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