Flexoelectric domains

Figure 6.26 The flexoelectric effect in BaTiO (a) the evolution of the potential energy curve under a homogeneous stress and in a strain gradient (b) domain switching via mechanical stress imposed by a probe (Original data from Lu et al. (2012))...

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... 

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. 

H.P. Hinov and Y. Marinov, Theoretical considerations and experimental illustration of the electro-optic behavior of longitudinal flexoelectric domains under the joint action of DC and AC voltages the case of strong anchoring. Mol. Cryst. Liq. Cryst. 503(1), 45-68, (2009). 

H.P. Hinov, I. Bivas, M.D. Mitov, K. Shoumaxov and Y. Marinov, A further experimental study of parallel surface-induced flexoelectric domains (PSIFED) (flexo-dielectric walls), Liq. Cryst. 30(11), 1293-1317, (2003). doi 10.1080/02678290310001607198... 

H.P. Hinov, On the coexistence of the flexo-dielectric walls-flexoelectric domains for the nematic MBBA - A new estimation of the modulus of the difference between the flexoelectric coefficients of splay and bend eij — esx, Mol. Cryst. Liq. Cryst. 524(1), 26-35, (2010). doi 10.1080/15421400903568161... 

The elemental motile unit of an OHC strikingly resembles the flexoelectric domain structure whose calculated period 7rifn(ei ) also depends linearly on the inverse electric field (Fig. 6.9). Such a repetitive arc and pillar nano-architecture, containing sharp points at the confluence of any two adjacent arcs, is inherently polar and enhances the flexoelectric mechanism (e.g., a sinusoidal-shaped membrane will not enhance flexoelectricity while one of the half-waves is reduced the opposite one will be extended and vice versa). This arc motif is repeated a few thousand times along the... 

This means that to consider the spontaneous flexoelectric effect influence on the substance physical properties one has to rewrite all earlier analytical expressions for long nanorods and nanowires without flexoelectric effect [8, 78] by the substitution g 2 and X for g and Xs in the expressions for the corresponding property. Note that for polydomain (if any) wires the predicted effect of R decrease with /44 increase should lead to the decrease of the intrinsic domain-wall width defined as 2 R. Below we demonstrate the spontaneous flexo-effect influence on the critical parameters (temperature and radius) of size-induced phase transition and correlation radius using the results [8,78] obtained without flexoeffect. 

It is clear from the Fig. 4.28a, b that in ferroelectric phase (i.e. at R>Rcr) the correlation radius monotonically decreases with the increase of the flexoelectric coefficient/44. At the same time, in paraelectric phase correlation radius increases with the increase of the flexoelectric coefficient, since the critical temperature (see Eq. (4.22)) increases with the increase of the flexoelectric coefficient. This opens the possibility to control the phase diagram and polar properties (e.g. via influence on domain wall width) by the choice of the material with necessary flexoelectric coefficient at given temperature or nanoparticle radius. 

The divergences of dielectric permittivity and correlation radius at the critical value of the flexoelectric coefficient (related to the critical radius) give new possibilities to control the physical properties of ferroelectric materials. The effect of the correlation radius renormalization by the flexoelectric effect alters the intrinsic width of domain walls. The predicted effects are useful for design of ferroelectric nanowires with radius up to several nanometers, which have ultra-thin domain walls and reveal polar properties close to those in bulk samples. 

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]. 

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

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 flexoelectric modulated structure in nematic liquid crystals is also known as the variable grating mode [14], since for U > Uth the period of domains w is varied according to the law [5, 8, 10-14] (Fig. 5.1)... 

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]). 

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. 

In Section 5.1.1 we discussed the longitudinal flexoelectric domains in nematics whose period depends on the inducing field, w When chang-... 

FIGURE 6.22. Flexoelectric domains in the zeroth, first, and second Grandjean zones of the wedge-shaped cell at two different polarities (a) and (b) of the applied d.c. field. The director at opposite boundaries is oriented parallel to the wedge edge (vertically in the figure) [87]. 

The other flexoelectric structure can arise in a very thin planar cholesteric layer at a certain set of material parameters. The instabihty occurs in the form of spiral domains [89] whose handedness depends on the sign of an electric field. While the threshold voltage for the hnear domains discussed above slightly depends on cell thickness, Uth 27rRT/ eii - 633 , the threshold for the spiral domains is proportional to thickness... 

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