Distortion flexoelectric

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

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 . 

Flexoelectricity - what is it How does it arise in liquid crystals What are its consequences What role does it play in liquid crystal phases, structures, and textures How is it measiu-ed What is its role, both realized and potential, in applications of liquid crystals How was it discovered and what is its history in the context of the development of liquid crystal science and technology in the last 50 years The name flexoelectricity clearly indicates the dual role of curvature distortions and electrical effects in liquid crystals, but just how are these two fundamental sets of concepts related by this phenomenon This book attempts to lay out the answers to these questions, with a combination of broad reviews and focused insights into the role of flexoelectricity in the science and technology of liquid crystals. In this introduction there is first a little informal review of history along with some general comments on the fundamentals and the special challenges presented by this phenomenon, and then there is a brief sketch of the chapters of this book. 

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. 

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 ... 

In this chapter, we will be describing different electro-optic techniques that have been developed specifically for measurement of the flexocoeffi-cients. In these techniques, an electric field is applied to a nematic liquid crystal in a cell with well-defined boundary treatments. The net torque on the medium, which involves dielectric, elastic and flexoelectric components, is set to zero both in the bulk and at the two surfaces in order to And the equilibrium director configuration if a DC field is applied. In AC techniques, the above torques are balanced with the viscous torque. The optical properties of the medium are in turn calculated for the distorted profiles and compared with experimental measurements. The flexoelectric, and in some... 

On the other hand, the flexoelectric energy for bend distortion of the director is ... 

Homeotropic cells offer another way to detect the flexoelectric response via observing the bend director distortions induced by an electric field parallel to the substrates (the Helfrich method ). Takezoe s group applied it to ClPbislOBB and found 63 20 pCm (which is the order of flexocoef-... 

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- ... 

The second application uses the converse flexoelectric effect, i.e. a field-induced splay-bend distortion, to generate a fast, symmetric and thresholdless linear electro-optic effect in a cholesteric liquid crystal. 

G. Durand, Flexoelectricity in nematics Continuous distortions and instabilities, Mol. Cryst. Liq. Cryst. 113, 237-246, (1984). 

The problem of the surface polarization have been raised [12] macroscopically in connection with the bulk flexoelectric distortion [13] discussed in the next chapter. On the microscopic level, we can distinguish between three different mechanisms of Psurf, explained with the help of Fig. 10.10. 

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)...

In Eq. (11.72) there are two terms related, respectively to the splay and bend distortions with corresponding flexoelectric coefficients C and e-. Indeed, the divn... 

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