Flexoelectricity dipolar

In this book the flexoelectric effect is mainly considered from the phenomenological point of view. At the same time it is very interesting and important to reveal the molecular origin of flexoelectricity and, in particular, to consider different types of intermolecular interactions that may be responsible for the dipolar ordering in a deformed liquid crystal, and to study the effects of intermolecular correlations and the molecular structure. This problem can only be solved using a molecular-statistical theory, which eventually allows us to express the flexoelectric coefficients in terms of molecular model parameters using various approximations. 

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. 

At the same time experimental facts indicate that the difference between the flexocoefficients is non-zero and even rather large for a number of nematic materials, and it strongly depends on the absolute value and the orientation of the permanent dipole within the molecular structure.Moreover, the difference between the flexocoefficients determines the flexoelectro-optic effect, which has been extensively studied experimentally. There exist also some other experimental data which, in principle, allowed us to distinguish between dipolar and quadrupolar flexoelectricity. This can be considered as an argument in favour of the dipolar interpretation of the flexoelectric effect. On the other hand, the actual ratio of the dipolar and quadrupolar contributions to the flexocoefficients of particular nematic materials remains unknown. It is only possible to speculate... 

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. 

As mentioned above, the first expressions for the flexoelectric coefficients were obtained by Helfrich and Petrov and Derzhanski while a systematic molecular-statistical theory was developed later by Straley. The results of these two approaches were compared by Marcerou and Prost who concluded that the theories of Helfrich and Petrov and Derzhanski and of Straley describe different mechanisms for the dipolar flexoelectric effect because Straley s theory 5nelds values for the flexocoefficients that are two orders of magnitude smaller than the experimental ones, and which therefore can be neglected. 

Let us now discuss the approximate expressions for the flexoelectric coefficients, Eq. (1.31), in more detail. Firstly, note that the expressions for both coefficients and es contain terms proportional to both S and S. It has been assumed in the literature that the dipolar contribution to the flexoelectric coefficients is always proportional to while the quadrupole contribution is proportional to S, and even the method of separation between the dipolar and quadrupolar flexoelectric effect has been proposed based on these preliminary results. The results of the consistent molecular theory presented in this section allow us to conclude that the relation e S for the dipolar contribution is due to the shortcomings of the semi-phenomenological approach. The results of this section also cast some doubt on the quantitative ratio of the dipolar and quadrupole contributions based on a comparison of the two terms in the expression e = eoS + C2S. At the same time, the absence of the linear term in S in the dependence e S) for a number of nematic materials stUl points to the predominant role of the quadrupole flexoeffect for those materials. 

Secondly, it follows from Eqs (1.31) and (1.32) that the longitudinal molecular dipole d provides a much smaller contribution to the flexocoef-ficients than the transverse dipole d , since A/k 10. Thus we conclude that the dipole flexoeffect is expected to be important only for molecules with large transverse dipoles. Note that the significant dipole flexoeffect has indeed been determined for nematics composed of molecules with large transverse dipoles. For cyanobiphenyl liquid crystals Marcerou and Frost did not find any dipolar flexoelectric effect, which may be determined not only by the tendency to form dimers with antiparallel dipoles but also by a relatively small contribution from transverse molecular dipoles to the flexoelectric coefficients. 

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. 

In addition to the pear-shaped molecules, bent-shaped molecules were used to illustrate the dipolar origin of the flexoelectric effects in nematic liquid crystals. It was assumed that the constituent molecules of the nematic liquid crystals are free to rotate around their axes, and in the absence of electric fields, their dipole moments average out so the net polarization of the material is zero. However, when liquid crystals made from polar pear- or banana-shaped molecules are subjected to splay or bend deformations, respectively, they can become macroscopically polar, because the polar structures correspond to a more efficient packing of the molecules. It follows from symmetry considerations that the deformation-induced fiexo-electric polarization Pa can be written as ... 

The magnitude of the effect is characterized by two flexoelectric coefficients, ei and 63, for splay and bend, respectively. As far as the microscopic origin of these phenomenological flexocoefficients is concerned, two different mechanisms - dipolar and quadrupolar - have been identified they are discussed in detail in Chapter 1. ... 

Phenomenologically, there is a close analogy between flexoelectricity of thermotropics and lyotropics. Equations (6.1) and (6.2) are in correspondence since a bend deformation of the director is not allowed either in a two-dimensional bilayer or in a lamellar lyotropic phase. From dimensional arguments we have concluded that the three-dimensional to two-dimensional correspondence leads to / = e d, where d is the bilayer thickness. However, the molecular mechanisms of the two are very different, e.g., molecular shape asymmetry is not a precondition for dipolar membrane flexoelectricity (see Petrov and below). 

A. Todorov, A.G. Petrov and J.H. Fendler, Flexoelectricity of charged and dipolar bilayer Hpid membranes studied by stroboscopic interferometry, Langmuir 10(7), 2344 2350, (1994). doi 10.1021/la00019a053... 

Fig. 11.24 Dipolar flexoelectric polarization. Pear-shape and banana-shape molecules in undistorted nematic liquid crystals without any polar axes (a) and appearance of polar axes and flexoelectric polarization along the z-direction in the same nematics due, correspondingly, splay and bend distortion (b)...

We already discussed this case in relation to the surface polarizafimi (Section 10.1.3). Generally both dipolar and quadrupolar mechanisms contribute to Py but the temperature dependence of the corresponding coefficients is different, oc S(T) for the quadrupolar mechanism, but Cd oc S (T) for the dipolar one. The flexoelectric coefficients have the dimension of (CGSQ/cm or C/m) and the order of magnitude, e 10 CGS units (or 3 pC/m). The flexoelectricity is also observed in the SmA phase [27]. 

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