Uniform-director configuration

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... 

The equilibrium ordering of a hybrid nematic film can exhibit either a distorted (hybridly bent) or undistorted (hybrid biaxial and uniform) director structure. Which of the two possible configurations will actually occur depends on the temperature and film thickness. The existence of either of the two undistorted structures depends on the strength of the surface coupling. 

We have considered several examples which illustrate the diversity of the interesting effects associated with lightwave-liquid-crystal interaction in cells with a nonuniform initial director distribution. Many such cells can be constructed. We have already seen that their qualitative properties depend not only on the boundary conditions but also on the specific physical properties of the LC material. This fact should be stressed, since in cells with a uniform director orientation the differences in the Franck constants, say, from one LC to another lead merely to quantitative differences. In many cases, even the description of the equilibrium structure for nonuniform cells encounters serious mathematical difficulties. Conservation laws may provide a powerful technique for solving problems of this type. For instance, a theorem of E. Noether was used in Ref 23 to derive analytic expressions for the equilibrium structure of complex configurations such as homeotropic-planar oriented cholesterics and cholesterics in magnetic fields with a homeotropic orientation at the walls. 

Fig. 1 Field-driven director configurations in (a) a nematic liquid (conventional Fredericks effect) and (b) a nematic elastomer slab floating in liquid between rigid electrodes. They initially have a uniform planar orientation before imposing field E in the direction shown by the arrow. In (a), the director at the surfaces are anchored, and the rotation angle of director has a finite distribution along the field axis, with the maximum at the middle layer of the cell. The recovery force for the director originates from the Frank elasticity. In (b), the director is capable of uniform rotation under electric fields, and the Frank elasticity plays no role in the recovery force...

In a uniaxial nematic, the particle-like soliton amounts to a director configuration distorted in a region of finite size, outside of which the director field is uniform. As a rule, such solitons are unstable with respect to a decrease in size and subsequent disappearance on scales smaller than the... 

In this section we analyze how light interacts with an LC with a twisted initial director orientation and find configurations for which the effects of director non uniformity are particularly pronounced. 

In smectic A phases where the smectic layers are perpendicular to the molecules, the orientation of the whole structure is, in principle, fixed once the orientation of the molecules is defined by the interface. The surface orientation of achiral smectic A phases is then the same as that of the nematic phase [88, 89]. However, since splay deformations of smectic layers (director bend deformations) are forbidden and layer bend deformations (director splay deformations) require a lot of energy, smectic phases tend to adopt uniform configurations, even between two walls inducing two different orientations. In the latter case, the surface orientation of the smectic phase differs from that of the nematic phase, and depends on the layer configuration in the bulk [90, 91]. 

This surface bistability is at the basis of chiral smectic C surface stabilized ferroelectric liquid crystal (SSFLC) devices [92]. As their name indicates, these devices are made of thin cells in which the walls, imposing the orientation of the molecules at the surfaces, unwind the spontaneous smectic C helix and stabilize two uniform configurations of the director in the cell. Switching between these two states can be done by applying an electric field. 

NMR, and particularly deuterium NMR, are the best experimental techniques for the determination of the director field and the molecular dynamics in submicron nematic droplets. The theoretical static proton NMR spectra of the nematic micro-droplets for a bipolar and a radial configuration are shown in Fig. 17 a. The corresponding motionally averaged spectra are shown in Fig. 17 b. Here the motional averaging is produced by translational diffusion which induces slow molecular rotations due to the non-uniform orientational ordering in the droplet [206]. 

In spherical geometry the situation is more complex as in planar cells since the equilibrium configuration of the director is not uniform. In radial spherical droplets there is a singularity in the centre of the droplet and therefore the lowest eigenvalue of the fluctuations is of the order ofl/R [24], The bulk elasticity thus always dominates the relaxation rate, regardless of the anchoring strength. 

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