Distortions, splay-, bend

The three elastic constants are the Frank elastic constants, called after Frank, who introduced them already in 1958. They originate from the deformation of the director field as shown in Fig. 15.52. A continuous small deformation of an oriented material can be distinguished into three basis distortions splay, twist and bend distortions They are required to describe the resistance offered by the nematic phase to orientational distortions. As an example, values for Miesowicz viscosities and Frank elastic constants are presented in Table 15.10. It should be mentioned that those material constants are not known for many LCs and LCPs. Nevertheless, they have to be substituted in specific rheological constitutive equations in order to describe the rheological peculiarities of LCPs. Accordingly, the viscosity and the dynamic moduli will be functions of the Miesowicz viscosities and/or the Frank elastic constants. Several theories have been presented that are more or less able to explain the rheological peculiarities. Well-known are the Leslie-Ericksen theory and the Larson-Doi theory. It is far beyond the scope of this book to go into detail of these theories. The reader is referred to, e.g. Aciemo and Collyer (General References, 1996). 

Here V n and V x n are the divergence and the curl of n. The three contributions to Wd are associated with the three independent modes of distortion splay, twist, and bend, depicted in Fig, 10-6. Terms of higher order than quadratic in Vn are only required if spatial distortions become severe. The Frank constants K, K2, and are of the order u /a,... 

As before, we shall begin by considering a planar sample in which the director is confined to the xy plane. In such a case, a wedge disclination involves only splay and bend distortions and we need to take into account only the splay-bend anisotropy (kjj + fejj). 

Fig. 3.13.3. A hybrid aligned cell for the determination of the anisotropy of the flexoelectric coefficients. In this geometry, the director has a splay-bend distortion which gives rise to a flexoelectric polarization P. On applying an electric field E, the director is twisted by an angle (j> cc — which can be measured optically.

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. 

Practically aU other methods developed for the measurement of flexo-coefficients are indirect . These exploit the fact that the polarization resulting from the splay-bend distortion couples linearly with an applied electric field E. This contributes to the total free energy of the sample, and hence alters the distortion of the director field compared to that in the absence of flexoelectric polarization. An external electric field of course acts on the dielectric anisotropy (As) of the nematic, which, like the orientational or-... 

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 ... 

Fig. 2.2. Schematic diagram of a hybrid-aligned nematic cell. The field-free director (shown by the continuous curved line) has a splay-bend curvature distortion in the xz plane. A DC field applied along the y axis rotates the polarization and the director (shown by the curved dashed line) acquires a 4>(z) profile. (Reproduced from Dozov et al. with the permission of EDP Sciences, http //publications.edpsciences.org.)...

The second application uses the converse flexoelectric effect, i.e. a field-induced splay-bend distortion, to generate a fast, symmetric and thresholdless linear electro-optic effect in a cholesteric liquid crystal. 

Fig. 7.3. The deviation of the optic axis in a cholesteric (hard-twisted chiral nematic, p <, where p is the cholesteric pitch and A is the wavelength of light) when an electric field E is applied perpendicular to the helical axis. The cholesteric geometry allows a fiexoelectric polarization to be induced in the direction of E. The plane containing the director, which is perpendicular to the page in the middle figure and is shown in the lower figure, illustrates the splay-bend distortion and the corresponding polarization that arises. (After Rudquist, inspired by Meyer and Patel.

Fig. 8.6 Splay, bend and twist distortions in nematics confined between two glasses that align liquid crystal at the surfaces either homogeneously (for splay and twist) or homeotropically (for bend)...

In Fig. 8.10b, we see that the fluctuation mode i(q) is a mixture of the splay and bend distortions, and the component 2(q) is a mixture of twist and bend distortions. This may be clarified as follows the splay-bend (SB) mode on the left side of Fig. 8.10b corresponds to realignment of the molecules within the, z-plane as q evolves and there is no twist here. In contrast, on the right side of the same figure the molecules are deflected from the q z-plane of the figure therefore, the twist and bend are present but the splay is absent (TB mode). 

We discuss the splay-bend distortion induced by an electric voltage applied to a cell similar to that shown in Fig. 11.16 using two transparent electrodes at z = 0 and z = d. The distortion is easy to observe optically for the cell birefringence. The splay-bend cell behaves like a birefringent plate discussed in Section 11.1.1 but now the plate birefringence is controlled by the field. The optical anisotropy... 

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)...

There is a free energy penalty for distortion of a nematic phase, much like the energy required to compress a spring. For a nematic, there are three fundamental types of distortion, splay, twist, and bend, and three elastic constants K, AT, and... 

The only curvature strains of the director field which must be considered correspond to the splay, bend, and twist distortions (Fig. 2.17). Other types of deformation either do not change the elastic energy (e.g., the above mentioned pure shears) or are forbidden due to the symmetry. In nematic liquid crystals the cylindrical symmetry of the structure, as well as the absence of polarity (head to tail symmetry) must be taken into account. 

FIGURE 4.4. Crossed electric and magnetic fields, (a, b) Magnetic field stabilizes director orientation, the first-order Frederiks transitions in an electric field are possible, (c, d, e) Both magnetic and electric fields are destabilizing (c) twist, (d) splay-bend, and (e) twist-splay-bend distortions are possible dependent on the value of the electric and magnetic fields. 

For each in a uniaxial phase there are two normal modes corresponding to a splay-bend distortion n q) and a twist-bend distortion ri2(q) biaxial liquid crystal phases have five normal modes for each value of q. The free energy density can be written in terms of the normal coordinates for torsional displacement in a uniaxial nematic as ... 

Here, Eq is the amplitude of the incident optical field, cOo the frequency of the incident light, V the scattering volume and R the distance between the scattering volume and the detector. The scattered light consists of two modes, splay/bend (a=l) and twist/bend (a=2), as shown in Fig. 2. It can be seen from Fig. 3 (a) that the mode I fluctuations can contain only contributions from the bend and splay distortions. Mode 2 fluctuations are in the perpendicular plane... 

Figure 38. Field-induced tilt (left) and corresponding splay-bend distortion when looking at the Bouli-gand plane along the field direction (right). The same director pattern will be found in any cut made perpendicular to the tilted optic axis, whereas the director is homogeneous in any cut perpendicular to the nontilted axis.

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