Bend director distortions, nematics

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 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). 

The spatial and temporal response of a nematic phase to a distorting force, such as an electric (or magnetic) field is determined in part by three elastic constants, kii, k22 and associated with splay, twist and bend deformations, respectively, see Figure 2.9. The elastic constants describe the restoring forces on a molecule within the nematic phase on removal of some external force which had distorted the nematic medium from its equilibrium, i.e. lowest energy conformation. The configuration of the nematic director within an LCD in the absence of an applied field is determined by the interaction of very thin layers of molecules with an orientation layer coating the surface of the substrates above the electrodes. The direction imposed on the director at the surface is then... 

Here u is the position of a layer plane and z is the position coordinate locally parallel to the director n, where n is parallel to the average molecular axis, which is assumed to remain normal to the layer plane, du/dz = e is the compressional (or dilational) strain. Thus, layer bending and layer compression are characterized by a splay (or layer-bend) modulus K and a compression modulus B. Other kinds of distortion present in nematics, such as bend or twisting of the director n, are not compatible with layers that remain nearly parallel, and hence are forbidden. Equation (10-36) is not invariant to rotations of frame, and its validity is limited to weak distortions a rotationally invariant expression has been given by -Grinstein and Pelcovits (1981).---------------------------------------------------------... 

Here K, K2 and iTs are elastic moduli associated with the three elementary types of deformations splay, twist and bend, respectively. Though the three elastic moduli are of the same order of magnitude the ordering K2 < K < K3 holds for most nematics. As a consequence of the orientational elasticity a local restoring torque (later referred to as elastic torque) acts on the distorted director field which tends to reduce the spatial variations. 

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-... 

Any spatial distortion of n leads to elastic restoring torques, which are determined in the standard continuum description of nematics (exclusively used in this review) by three elastic constants Ki (splay), K2 (twist) and Ks (bend). In addition, the electric field E gives rise to an electric torque on the director. The balance of these torques, reflected in the resulting equilibrium director configuration, corresponds to the minimum of the orientational free energy J- n). For positive dielectric anisotropy (ca = — x) the dielectric torque (oc is destabihzing in the pla-... 

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.

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. 

It is well known that nematic liquid crystals are nonpolar. However, for a certain asymmetrical shape of the molecules, splay or bend deformations of the director field lead to an electrical polarization [87]. This feature is known as the flexoelectric effect. Theoretically, the influence of an electric field on CLCs for the case where the helical axis is oriented parallel to the plane of the sample was first considered by Goossens [88]. Experimentally, the flexoelectric electro-optic effect in CLCs can be observed in conventional sandwich cells with transparent electrodes when the helix axis of the CLC lies parallel to the glass surfaces [89]. In the absence of an electric field, the CLC behaves as a uniaxial material with its optic axis perpendicular to the director and parallel to the helix axis. When an electric field is applied normal to the pitch axis, the helix distorts, as shown in Figure 6.6. Thus, the optical axis is reoriented and the medium becomes biaxial. The deviation direction... 

Backflow effects may accompany the transient process of the director reorientation [64,65]. The process is opposite to the flow orientation of the director known from rheological experiments. Disregarding the back-flow, we can use the same equations for the splay (with A",) and bend (K33) small-angle distortions. The backflow effects renormalize the rotational viscosity of a nematic ... 

At present, at least three types of steady-state dielectrically driven pattern are known for nematics. The electric-field-induced periodic bend distortion in the form of parallel stripes has been observed in a homeotropi-cally oriented layer of 5-CB ( a= 13) in the presence of a stabilizing magnetic field [75, 76]. The stripes with a wavevector q were parallel to the electric field E and stationary at low fields. It was shown that a stable periodic pattern of the director minimizes the free energy of the cell when the ela.stic moduli and A 33 are similar to each other. In these experiments the Frederiks transition is of first order, the nonde-formed and deformed areas coexist at a given voltage, and the front between them may propagate along the direction y perpendicular to both fields [77]. 

As discussed in Sec. 2.2.2.1, the foundations of the continuum model were laid by Oseen [61] and Zocher [107] some seventy years ago, and this model was reexamined by Frank [65], who introduced the concept of curvature elasticity to describe the equilibrium free energy. This theory is used, to this day, to determine splay, twist, and bend distortions in nematic materials. The dynamic models or how the director field behaves in changing from one equilibrium state to another have taken much longer to evolve. This is primarily due to the interdependency of the director it (r, t) and v (r, /) fields, which in the case of chiral nematics is made much more complex due to the long-range, spiraling structural correlations. The most widely used dynamic theory for chiral... 

Recognition of the flexoelectric effect is due to Meyer [18], who in 1969 derived an expression equivalent to Eq. (97). He also had a helpful molecular picture of the effect, as illustrated in Fig. 34 a. To the left in this picture, the unstrained nematic structure is shown with a horizontal director in the case of wedge-shaped and crescent-shaped molecules. To the right the same molecules are shown adjusted in their distribution corresponding to a splay and a bend distortion, respectively. In either case the distortion is... 

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