Distortion in liquid crystals

Similar to the situation for quadrupole-induced relaxation, the quadrupole splitting in liquid crystals is zero if the molecular symmetry is tetrahedral or higher. Electric field gradients are zero for such symmetries so there can be no quadrupolar interaction. However, one expects to see small splittings from tetrahedral or octahedral derivatives because of structural distortions. These predominately arise from specific interactions with extraneous materials such as lipophilic headgroups in surfactant systems, as seen, for instance, in both cationic and anionic octahedral cobalt(III) species [23], Much larger splittings will be expected from other structure... 

The liquid crystals can be deformed by applying external fields. Even a small electric or magnetic field, shear force, surface anchoring, etc., is able to make significant distortion or deformation to liquid crystals. Thus, n is actually a function of position r. According to the symmetry of liquid crystals there exist three kinds of deformations in liquid crystals splay, twist and bend deformations, shown in Figure 1.17. The short bars in the figure represent the projections of the local directors. 

The strain increases the energy of the solid as a stress is applied. The distortion of the director in liquid crystals causes an additional energy in a similar way. The energy is proportional to the square of the deformations and the correspondent coefficients are defined as the splay elastic constant, K, twisted elastic constant K22 and bend elastic constant Kx, i.e., the respective energies are the half of... 

Flexoelectricity - what is it How does it arise in liquid crystals What are its consequences What role does it play in liquid crystal phases, structures, and textures How is it measiu-ed What is its role, both realized and potential, in applications of liquid crystals How was it discovered and what is its history in the context of the development of liquid crystal science and technology in the last 50 years The name flexoelectricity clearly indicates the dual role of curvature distortions and electrical effects in liquid crystals, but just how are these two fundamental sets of concepts related by this phenomenon This book attempts to lay out the answers to these questions, with a combination of broad reviews and focused insights into the role of flexoelectricity in the science and technology of liquid crystals. In this introduction there is first a little informal review of history along with some general comments on the fundamentals and the special challenges presented by this phenomenon, and then there is a brief sketch of the chapters of this book. 

Let us now assume that the orientational deformation is small, which corresponds to the linear flexoeffect considered in this chapter. Then the one-particle distribution function of the distorted nematic liquid crystal is given by Eq. (1.6) ... 

Preedericksz transition in planar geometry is uniform in the plane of the layer and varies only in the z direction. However, in some exceptional cases, when the splay elastic constant Ki is much larger than the twist elastic constant K2 (e.g., in liquid crystal polymers), a spatially periodic out-of-plane director distortion becomes energetically favourable. The resulting splay-twist (ST) Freedericksz state is manifested in experiments in the form of a longitudinal stripe pattern running parallel to the initial director alignment no x. 

The basic difference between deformations in a liquid crystal and in a solid is that in liquid crystals there is no translational displacement of the molecules on distortion of a sample. This is due to slippage between liquid layers. A purely shear deformation of a liquid crystal conserves elastic energy. The elasticity of an isotropic liquid is related to changes in density. In liquid crystals, variations in density can also be characterized by a suitable modulus, but the elasticity which is related to the local variation in the orientation of the director is their principal characteristic. 

In this section we discuss electrohydrodynamic (EHD) instabilities, that is electric-field-induced phenomena that are caused by the flow of a liquid crystal (see also [8,219]. The reason for the flow is electrical conductivity, which has been disregarded in previous sections. The flow may arise either independently of the anisotropic properties of substance, as in isotropic liquids (isotropic modes of the electrohydrodynamic instability), or may be driven by the conductivity anisotropy, as in liquid crystals (anisotropic modes). The threshold for EHD instabilities depends on many parameters, such as the electrical and viscoelastic properties of substance, the temperature, and the applied field frequency. Due to flow distortion of the director alignment, the instability is usually accompanied by a characteristic optical... 

The other two instabilities shown in Fig. 28 may be observed only in liquid crystals (nematic, cholesteric, and smectic C). The first is the Carr-Helfrich instability, which is caused by a low-frequency electric field and occurs in the form of elongated vortices with their axis perpendicular to the original director alignment. The vortices cause a distortion of the director orientation, which is observed optically as a one-dimensional periodic pattern (Kapustin-Williams domains). The other anisotropic mode is observed only in highly conductive liquid crystals. For its interpretation the inertial term dvidt for the fluid velocity must be taken into account, which is why this mode may be called inertial mode. 

Closely related phenomenon to the piezoelectricity in liquid crystals is the flexoelectricity introduced by R.B. Meyer. Flexoelectricity means a linear coupling between the distortion of the director and the electric polarization. The constituent molecules of the nematic liquid crystals are rotating around their axes, and in absence of electric fields they are nonpolar. However, polar axes can arise in a liquid crystal made up from polar pear- or banana-shape molecules when they are subjected to splay or bend deformations, respectively. In these cases, the polar structures correspond to more efficient packing of the molecules (see Figure 8.17). 

F. M. Leslie, Distortion of Twisted Orientation Patterns in Liquid Crystals by Magnetic Fields, MoL Cryst and Liq. Cryst, 12, p. 57 (1970). 

In general, the distortions on the electronic wave function of liquid crystal molecules caused by an applied field do not cause appreciable change to its contribution to the refractive indices (see Chapter 10). However, the orientation of the molecules can be dramatically altered by the apphed field. This process alters the overall optical properties of the medium and is the principal mechanism used in liquid-crystal-based electro-optical devices. As noted in Section 6.2.2, the electrically induced orientational refractive index changes could be Pockel or Kerr effect. In this and the next sections, we shall focus on nematic liquid crystals in which the director axis reorientation is a Kerr-like effect that is, the process is quadratic in the applied field. 

A splay or bending distortion may create a polarisation in liquid crystals. This phenomenon is called the flexoelectric effect and was first studied theoretically by Meyer [197] in the context of nematic liquid crystals convenient summaries can be found in Chandrasekhar [38, pp.205-212] or de Gennes and Frost [110, pp.l35-139]. The flexoelectric effect for SmC liquid crystals has been investigated by Dahl and Lagerwall [62] and a brief development can be found in [110, pp.347-349]. Flexoelectric effects in smectics have also been discussed by Lagerwall [158]. 

AH distortions of the nematic phase may be decomposed into three basic curvatures of the director, as depicted in Figure 6. Liquid crystals are unusual fluids in that such elastic curvatures may be sustained. Molecules of a tme Hquid would immediately reorient to flow out of an imposed mechanical shear. The force constants characterizing these distortions are very weak, making the material exceedingly sensitive and easy to perturb. 

In order to compensate for the distortions in the wavefront due to the atmosphere we must introduce a phase correction device into the optical beam. These phase correction devices operate by producing an optical path difference in the beam by varying either the refractive index of the phase corrector (refractive devices) or by introducing a variable geometrical path difference (reflective devices, i.e. deformable mirrors). Almost all AO systems use deformable mirrors, although there has been considerable research about liquid crystal devices in which the refractive index is electrically controlled. 

Currently, a wide variety of methods exists for calculating the molecular structure of large liquid crystal molecules which make use of pre-determined functional forms for the interactions in a molecule and semi-empirical information to parametrise the potentials. In general the interaction terms represent the energy cost of distorting bonds and bond angles from equilibrium. These can be expressed as... 

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