The Flexoelectric Polarization

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 ... 

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 ... 

Fig. 5.7. Each layer in the four-layer structure of the SmCpj2 phase has a piezoelectric and flexoelectric polarization. The flexoelectric polarization also has a longitudinal component. The angle <5 marks the angle formed by the polarization direction and the direction perpendicular to the tilt.

Another structure, which possesses longitudinal polarization, is the three-layer structme of the SmC and the SmC j phases. One layer has a different environment (two neighbom-ing layers form the phase difference a) than the other two (one phase difference a and another (3) and therefore the tilt magnitude could be slightly different. The flexoelectric polarization is proportional to the difference of tUt orders above and below the considered layer, which is related to splay, and also to the difference in the... 

From symmetry reasons, two layers out of three have longitudinal components of polarizations, while the flexoelectric polarization in the third layer is strictly perpendicular to the tilt in the layer (Fig. 5.8). 

This chapter considers polarization in polar smectics in general and the flexoelectric polarization in polar smectics in particular. The existence of flexoelectric polarization has been known for a long time. The effects related to the phenomenon are more pronounced in systems formed of polar molecules including chiral polar smectics. Therefore it is not surprising that several important consequences of the flexoelectric phenomenon are present... 

For a hybrid cell the flexoelectric polarization can easily be calculated. In Fig. 11.26a, the profile of the director is n(z) with boundary conditions 9(0) = ti/2 and d(d) = 0. These angles are rigidly fixed. The components of the director are = sinS, Hy = 0, n = cos9. To calculate the polarization we have to find distribution P/(z) using Meyer s equation (11.72), and after integrating over z, to obtain total polarization of the cell. In the considered geometry ... 

Fig. 11.26 A scheme of a hybrid cell that supports the splay-bend distortion and manifests the flexoelectric polarization (a) and an experimental temperature dependence of the sum of flexoelectric coefficients in the nematic phase of liquid crystal 5CB (b)...

We measure pyroelectric coefficient y = dP/dT, using heating the hybrid cell by short ( 10 ns) laser pulses, as shown in Fig. 10.13. The only difference from the surface polarization measurements is using a hybrid cell instead of uniform (planar or homeotropic) cells [28]. The laser pulse produces a temperature increment about AT 0.05 K and the flexoelectric polarization changes. To compensate this change, a charge passes through the external circuit and the current i = dqldt is measured by an oscilloscope. From the identity (A is cell area)... 

The second turn of the discussion around the nature of a SHG in nematic liquid crystals arised when the SHG was observed in oriented layers of 4-methoxybenzylidene-4 -butylaniline (MBBA)/ The phenomenon has been explained by the lack of the symmetry center in the nematic phase. The zero-field SHG in MBBA was also investigated but the nature of the effect was connected with the flexoelectric polarization of surface layers. Such a polarization has to remove the inversion center in surface liquid crystalline layers and to allow the SHG to be detectable. Another explanation of the zero-field SHG in terms of the electric quadrupolar interaction was suggested in. ... 

The first of them is the flexoelectric polarization, introduced in [31] and dependent on the curvature of the director field (div L and curl L) at constant modulus of the order parameter S. For uniform director orientation (L = const) Pf = 0. The second term is the so-called order polarization. It depends on the gradient of the nematic (quadrupolar) order parameter... 

In this section we will consider the orientational deformations of a nematic director which are flexoelectric in nature, i.e., induced due to the interaction of the flexoelectric polarization (see (4.2)) with the external electric field. Our consideration will be limited to spatially uniform fields E the case when E depends on coordinates is discussed in Chapter 5. We also discuss semiphenomenological approaches for the determination of nematic flexoelectric moduli ei and 63. Different types of electrooptical phenomena, where flexoelectric distortion plays a dominant role will be considered. Some of them are promising for potential applications. 

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 effects, which were discussed for nematics in Section 4.5, are caused by the linear coupling of an external fleld with the flexoelectric polarization. In cholesterics, they manifest themselves in two specific phenomena to be considered below. 

This effect is observed in the geometry shown in Fig. 6.7(a) when the cholesteric axis h is homogeneously oriented in the plane of the cell (along x), and an electric field is applied to the electrodes of a sandwich cell along the 2 -axis [37, 81, 82]. In this case, the helical structure, even an ideal one, is incompatible with the planar boundary conditions and splayed and bended regions form near the boundaries. Thus, according to (3.19) the flexoelectric polarization arises in those regions which can interact with the electric field. 

FIGURE 7.4. (a) Flexoelectric polarization of an FLC. (b) The dependence is given of the reduced polarization P)/Ps averaged over a helix period on the relative field EfEu, Eu is the field of helix imwinding. The total polarization (curve 1) is a sum of the spontaneous polarization (curve 2) and the flexoelectric polarization (curve 3). (o) is the experiment and the solid curves are the calculations. 

Fflexoeiectric the Contribution to the free energy density due to the coupling of the flexoelectric polarization Fflex with the applied electric field E, such that... 

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