Torque flexoelectric

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... 

The torque acting on the director is given by n x h, where h is called the molecular field, which can be derived from the Euler-Lagrange equation. The energy density corresponding to the flexoelectric polarization is given by — Pfl E and the molecular field can be expressed in the form ... 

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- ... 

For the parameter combination o < 0 and <7a < 0, which can be found in some nematic compounds, electroconvection is definitely excluded within the standard model. Nevertheless, EC has surprisingly been observed in this case (for recent examples see, e.g. Kochowska et alP and Toth-Katona et al ). The theoretical analysis has proved that flexoelectricity is crucial for understanding this non-standard EC because in Eq. (4.7) the contribution V Pfi to Pel is dominant. It is interesting that the flexotorque on the director is determined by the difference (ei — 63) of the flexocoefScients while the sum (ei - - 63) governs the flexocharge and thus its contribution to the viscous torque. Further details will be sketched in Section 4.3.2. 

This means that the flexoelectric torque F will contain two contributions ... 

Note that, in the free energy density expansion (11.77), the flexoelectric term is proportional to the first derivative 39/3z. Therefore, upon integration over the cell thickness, it gives only surface energy terms W( di2). Correspondingly, the torque balance (11.78) shows the absence of the flexoelectric torque in the bulk of a cell for the uniform field... 

Equation (11.82) is valid for the flexoelectric distortion b) as well, but it does not have a sine-form profile and we cannot expect a simple relaxation process. Moreover, this distortion is controlled by boundary conditions that generally include the Rapini-type surface torque, elastic and flexoelectric torques ... 

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]. 

The period of the stripes and the threshold voltage for their appearance have been found [35] by minimising the free energy of the nematic in an electric field, taking into account the flexoelectric (P/E) and dielectric Ea(En) /47i terms. The solution of the torque balance equations for angles

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In liquid crystals with the capability of flexoelectric effect, in the absence of external electric fields, the state with uniform director configuration, which has no induced polarization, is the ground state and is stable. When an electric field is applied to the liquid crystal, the uniform orientation becomes unstable, because any small orientation deformation produced by thermal fluctuation or boundary condition will induce a polarization which will interact with the electric field and results in a lower free energy. The torque on the molecules due to the applied field and... 

FIGURE 4.30. Geometry of an investigation of the flexoelectric effect [26]. The expressions for surface torques are given under the drawings. 

It is shown in [26] that the (a) and (c) deformations in Fig. 4.30 do not have a threshold, i.e., they can occur with any small qxternal voltage, and the stabilizing dielectric torque (because of Ae < 0) partially depresses the flexoelectric deformation. For (b) and (d) the deformation has a threshold and, with the conditions Wsi Ws2, the threshold depends on the polarity of the field. When the flexoelectric and dielectric torques are simultaneously destabilizing, the flexoelectric effect reduces the threshold of the Frederiks transition. In the opposite case, the dielectric torque raises the threshold of the flexoelectric effect. 

According to (5.84) the flexoelectric torque dominates over the dielectric torque if the amplitude of the potential tp, denoted as vq (Table 5.2), is sufficiently small... 

In experiment, a deviation of the optical axis in the plane of a cell is observed. The sign and the magnitude of the deviation angle depend on the polarity and strength of the applied field, respectively. In the field-off state the helix is undistorted and the cell behaves like a uniaxial optical plate with the optical axis coincident with h. When the field is applied molecules leave the x, 2 -plane due to the flexoelectric deformation coming from the surface regions where the flexoelectric torque M = CfE (cf = d = 63) is developed. Now the optical axis does not coincide with the initial orientation of the helix h but forms an angle with respect to it, linearly dependent on E. The field-induced distortion of the helix is shown in Fig. 6.21 [81]. 

The dynamics of the flexoelectric effect have been studied [97, 98]. For a nematic layer with a geometry similar to that shown in Fig. 12c and finite dielectric anisotropy, the torque belance equation for the bulk of a sample, written in the linearized form (sin0=0, cos0 l)... 

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