Flexoelectric instability

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. 

N.V. Madhusudana, J.F. Palierne, Ph. Martinot-Lagarde and G. Durand, Twist instability of a flexoelectric nematic domain in an external field, Phys. Rev. A 30, 2153R-2154R, (1984). doi 10.1103/PhysRevA.30.2153... 

V.A. Raghunathan and N.V. Madhusudana, A new threshold flexoelectric instability in nematic liquid crystals. Mol. Cryst. Liq. Cryst. Lett. 6, 103-111, (1989). 

Flexocoefficients have also been obtained from the threshold voltages and wave numbers of electric field-induced flexoelectric instabilities (which are discussed in more detaii in Chapter 4 by Buka et al. ) for ClPbislOBB and for the molecule 4-((3-(4-(4-(decyloxy) benzoyloxy)benzoyloxy) phenylimino) methyl)-3-hydroxyphenyl 4-(6-(4 -cyanobiphenyl-4-yloxy)hexyloxy) benzoate (BCCB), which is a dimer composed of a calamitic and a BC molecule. In both cases the flexocoefficients were found to have the same order as that of calamities. 

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- ... 

S.P. Palto, N.J. Mottram and M.A. Osipov, Flexoelectric instability and spontaneous chiral-symmetry breaking in a nematic liquid crystal with asymmetric boundary conditions, Phys. Rev. E 75(6), 061707/1-8, (2007). 

Fig. 11.30 Flexoelectric instability. Photos of flexoelectric domains with a period variable by electric field (nematic cell thickness 12 pm)...

Fig. 11.31 Flexoelectric instability. Periodic structure of the field induced director distribution along the y-axis represented by projections and...

In both the cases considered, an optical contrast of the patterns observed in isotropic liquids is very small. Certainly, the anisotropy of Uquid crystals brings new features in. For instance, the anisotropy of (helectric or diamagnetic susceptibility causes the Fredericks transition in nematics and wave like instabilities in cholesterics (see next Section), and the flexoelectric polarizaticm results in the field-controllable domain patterns. In turn, the anisotropy of electric conductivity is responsible for instability in the form of rolls to be discussed below. All these instabilities are not observed in the isotropic liquids and have an electric field threshold controlled by the corresponding parameters of anisotropy. In addition, due to the optical anisotropy, the contrast of the patterns that are driven by isotropic mechanisms , i.e. only indirectly dependent on anisotropy parameters, increases dramatically. Thanks to this, one can easily study specific features and mechanisms of different instability modes, both isotropic and anisotropic. The characteristic pattern formation is a special branch of physics dealing with a nonlinear response of dissipative media to external fields, and liquid crystals are suitable model objects for investigation of the relevant phenomena [39]. 

The flexoelectric domain instability takes place due to the linear coupling between the flexoelectric polarization P and the external electric field E. The corresponding term in the nematic free energy F gives... 

The first paper [87] which considered the role of flexoelectric polarization (3.19) in nematic liquid crystals predicted the possibility of an oscillatory instability, the wave vector of rolls being parallel to the initial director Lq. Later Madhusudana et al. [88] and Thom et al. [89] pointed to the important role of a flexoelectric contribution to the equations of nematodynamics in an electric field. 

Let us now briefly describe electrohydrodynamic instabilities in polymer nematics. The first observation of the Kapustin-Williams domains in nematic polymers were reported in [117, 118]. The qualitative picture of the phenomenon is, in fact, the same as that for the conventional nematics (domains perpendicular to the initial director orientation in a planar cell, typical divergence of the threshold voltage at a certain, critical frequency, etc.). The only principal difference is a very slow dynamics of the process of the domain formation (hours for high-molecular mass compounds [117]). The same authors have observed longitudinal domains in very thin samples which may be referred to as the flexoelectric domains [5-14] discussed in Section 5.1.1. 

Several cases of dielectric, hydrodynamic, and flexoelectric instabilities and domain structures have been observed and extensively studied in CLCs. Their appearance depends on the initial orientation of molecules, the physical parameters of the material, and the applied electric field. In CLCs with positive dielectric anisotropy Ae > 0, an electric field applied along the helix axis of a planar (Grandjean) texture can induce a two-dimensional spatially periodic deformation which has the form of a square grid [96], The period and threshold voltage of this field-induced instability depend on the elastie constants, the dieleetric anisotropy, and the sample thickness [97],... 

In Section 9.3.1.3 we discussed the longitudinal flexoelectric domains in nematics, the period of which depends on the inducing field, wo=E. When changing the sign of the field, the phases of the distortion angles G(y) and 0(y) are changed for the opposite ones, but the direction of the wavevector of the instability (perpendicular to the initial... 

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