Pattern flexoelectric

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]. 

A. Buka, T. Toth-Katona, N. Eber, A. Krekhov and W. Pesch, Chapter 4. The role of flexoelectricity in pattern formation. In eds. A. Buka and N. Eber, Elexoelectricity in Liquid Crystals. Theory, Experiments and Applications, Imperial College Press, London, 2012. pp. 101-135. 

In this chapter the influence of flexoelectricity on pattern formation induced by an electric Held in nematics will be summarized. Two types of patterns will be discussed in the linear regime, the equilibrium structure of flexoelectric domains and the dissipative electroconvection (EC) rolls. In a separate section, recent experimental and theoretical results on the competition and crossover between the flexoelectric domains and EC patterns will be described. 

Patterns, i.e. regular spatiotemporal structures, can easily be generated in liquid crystals via a large variety of external stresses, e.g., by mechanical shear, temperature or pressure gradients, electric or magnetic fields, etc. representative examples can be found in Buka and Kramer. Here we concentrate on patterns induced by electric fields in nematics and in particular on the implications of flexoelectricity. 

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 chapter is organized as follows Section 4.2 describes flexodomains in the planar geometry. Particular emphasis is placed on the most recent theoretical results, where for the first time arbitrary ratios of the elastic constants ifi, if2 are considered as well as driving by an AC electric field. Section 4.3 deals with the effects of flexoelectricity on dissipative EC patterns. The focus is on qualitatively new phenomena that are not covered by the standard model of EC. In Section 4.4 we analyse the competition between flexodomains and EC patterns at low AC driving frequencies. The chapter ends with a discussion and some concluding remarks in Section 4.5. 

Flexoelectric patterns also exist for nematic layers with asjrmmetric boundary conditions, i.e. with homeotropic anchoring on one surface and planar anchoring on the other one hybrid-aligned nematics).The critical voltage and the critical wave number obtained with the one-elastic-constant approximation are in a good agreement with experimental re-sults. ... 

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. 

V.A. Delev, A.P. Krekhov and L. Kramer, Crossover between flexoelectric stripe patterns and electroconvection in hybrid ahgned nematics. Mol. Cryst. Liq. Cryst. 366(1), 849-856, (2001). doi 10.1080/10587250108024026... 

There is a very interesting example of the flexoelectric torque acting on the director in the bulk. In a typical planar nematic cell the director is strongly anchored at both interfaces, n = (1,0,0) and the electric field is directed along z. The conductivity is low and the dielectric anisotropy is either zero or small negative, such that the dielectric torque may only weakly stabilize the initial planar structure. Upon the dc field application, a pattern in the form of stripes parallel to the initial director orientation in the bulk nollx is observed in the polarization microscope. The most interesting feature of these domains is substantial field dependence of their spatial period as shown in Fig. 11.30 [34]. 

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]. 

FIGURE 4.31. Distribution of the director for (a) a deformation quadratic in the field (b) a deformation linear in the field and (c) diffraction patterns from phase lattices caused by the quadratic and linear flexoelectric effect [188]. 

Taking into account flexoelectricity, it is possible to explain the appearance of a certain angle a, which the Kapustin-Williams domains form in some cases with the y-axis (the usual domain strips are parallel to the 2/-axis, Fig. 5.5). This oblique roll motion was observed in [90] and cannot be explained within the framework of the usual three-dimensional Carr-Helfrich model with strong anchoring at the boundaries [91]. The angle of the domain pattern a was shown [88, 89] to depend on the flexoelectric moduli eii, 633, the dielectric Ae, and the conductive Aa anisotropy. In certain intervals of the en — 633 and 611/633 values the angle A = 0 (the usual Kapustin-Williams domains) or A = 7t/2 (the longitudinal domains, also seen in experiment near the nematic-smectic A transition [91]). 

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