Quadrupolar Flexoelectricity

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... 

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. 

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. 

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. ... 

Kumar et al. proposed that a chain formed by intercalated bent-core molecules with alternating dipoles may have a considerable quadrupolar moment and they assumed that this idea is transferable to antiferroelectric layers. They estimated an increment in the quadrupole moment proportional to the number of molecules (m) in such clusters. As the typical cluster size indicated by X-ray measurements is about 4-5 layers of 20-30 nm, m may be of the order of 1,000. Therefore, on the one hand, the presence of clusters with an antiferroelectric SmCP structure may cause a huge increase in the quadrupolar contribution to the flexoelectric response compared to that of individual molecules. 

Besides the elastic and the electric torques the so-called flexoelectric (or flexo) torques on the director play an important role as well. Their effect on pattern-forming instabilities in nematics is the main issue of this chapter. Flexotorques originate from the fact that typically (in some loose analogy to piezoelectricity) any director distortion is accompanied by an electric flexopolarization Pa (characterized by the two ffexocoefScients ei, 63). From a microscopic point of view, finite ei and 03 naturally arise when the nematic molecules have a permanent dipole moment. But also for molecules with a quadrupolar moment, finite ei and 63 are possible (see also Chapter 1 in this book ). Flexopolarization has to be incorporated into the free energy P n) for finite E. It is not surprising that this leads to quantitative modifications of phenomena, which exist also for ci = 63 = 0. Though, for example, the Freedericksz threshold field Ep is not modified, the presence of flexoelectricity leads to considerable modifications of the Freedericksz distorted state for E > Ep- ... 

L.M. Bhnov, M.l. Barnik, H. Ohoka, M. Ozaki and K. Yoshino, Separate measurements of the flexoelectric and surface polarization in a model nematic liquid crystal p-methoxybenzyhdene-p -butylanihne Validity of the quadrupolar approach, Phys. Rev. E 64(3), 031707/1-7, (2001). 

Blinov, L.M., Bamik, M.I., Ohoka, H., Ozaki, M., Yoshino, K. Separate Measurements of the Flexoelectric and Surface Polarization in a Model Nematic Liquid Crystal MBBA on Validity of the Quadrupolar Approach. Phys. Rev. E 64, 031707-031713 (2001)... 

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]. 

Fig. 11.25 Quadrupolar flexoelectric polarization. Undistorted nematic phase consisted solely of molecular quadrupoles (a) and appearance of a polar axis and flexoelectric polarization due to splay distortion (b). Note that in the lower part of (b) the density of positive charges is larger than in the upper part whereas in sketch (a) these densities are equal...

The second turn of the discussion around the nature of a SHG in nematic liquid crystals arised when the SHG was observed in oriented layers of 4-methoxybenzylidene-4 -butylaniline (MBBA)/ The phenomenon has been explained by the lack of the symmetry center in the nematic phase. The zero-field SHG in MBBA was also investigated but the nature of the effect was connected with the flexoelectric polarization of surface layers. Such a polarization has to remove the inversion center in surface liquid crystalline layers and to allow the SHG to be detectable. Another explanation of the zero-field SHG in terms of the electric quadrupolar interaction was suggested in. ... 

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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