Cholesteric liquid crystals helical axis

Fig. 9.8 Three different geometries of a shear of a cholesteric liquid crystal helical axis hllx (I), hllz (II) and hlly (III)...

A sheet of cholesteric liquid crystal (pitch is / q) is sandwiched between two glass plates with the gap d being ten microns. The helical axis of cholesteric liquid crystals in the absence of a magnetic field is defined along the Z axis, shown in Figure 6.4, and 6 is the deformation angle between... 

Fig. 4.5.4. Helfrich s model of permeation in a cholesteric liquid crystal. At low shear rates flow takes place along the helical axis without the helical structure itself...

Fig. 4.6.3. Deformation of a planar structure due to a magnetic field acting along the helical axis of cholesteric liquid crystal composed of molecules of positive diamagnetic anisotropy. A similar deformation superposed in an orthogonal direction results in the square-grid pattern (see fig. 4.6.4). (Helfrich. )...

Though this type of periodic structure with multiple arches of the director is difficult to generate in a nematic, it is already present in a cholesteric liquid crystal when viewed in a plane whose normal makes an oblique angle with the helical axis. The flexoelectric effect changes the periodicity of this structure under a DC field applied normal to the helical axis, effectively rotating the latter. This can be used in tmn to measure (ei — 63). ... 

Kent Display is a pioneer of cholesteric liquid crystal displays (ChLCDs) in which the director of the liquid crystal twists around a helical axis [3]. The remarkable property is that the cholesteric material reflects light of certain wavelengths depending on the pitch over which the director rotates. When an electric held is applied. 

Cholesteric liquid crystal (ChLC)-driven electronic paper has been investigated mainly by Kent Display [2], Since ChLCs are chiral molecules, the particular color of light depends on the pitch denoted as P, where the pitch is the distance along the helical axis for ChLC to twist 360 and is determined by the amotmt and type of chiral additive within the liquid crystal mixture. ChLCDs are driven by switching the different textures of the ChLC electrically [1, 3], as shown in Fig. 4. 

It should be noted that cholesteric liquid crystals (chiral nematics) having point group symmetry Dqo are also periodic with flie pitch considerably exceeding a molecular size. The preferable direction of the local molecular orientatiOTi, i.e. the director oriented along the Coo axis, rotates additionally through subsequent infinitesimal angles in the direction perpendicular to that axis. Hence a helical structure forms with a screw axis and continuous translation group. 

Our task is to find the spectrum of eigenmodes propagating along the helical axis of a cholesteric liquid crystal and discuss some consequences of that. It is very rare and even unique case when, despite chirality and anisotropy of a medium, there is an analytical solution found many years ago by De Vries [6]. Here, we follow a rather simple and very elegant analytic solution of this problem given by Kats [7]. 

Let both the helical axis and the electric field are parallel to the normal z of a cholesteric liquid crystal layer of thickness d and >0. In the case of a very weak field the elastic forces tend to preserve the original stack-like arrangement of the cholesteric quasi-layers as shown in Fig. 12.15a. On the contrary, in a very strong field, the dielectric torque causes the local directors to be parallel to the cell normal, as shown in Fig. 12.15c. At intermediate fields, due to competition of the elastic and electric forces an undulation pattern appears pictured in Fig. 12.15b. Such a structure has two wavevectors, one along the z-axis (nld) and the other along the arbitrary direction x within the xy-plane. The periodicity of the director pattern results in periodicity in the distribution of the refractive index. Hence, a diffraction grating forms. Let us find a threshold field for this instability. 

We showed in last section that in a uniform anisotropic medium, for each propagation direction, there are two eigenmodes which are linearly polarized. The polarization state of the eigenmodes is invariant in space. In this section, we discuss the propagation of light in a special case of a non-uniform anisotropic medium a cholesteric liquid crystal which locally is optically uniaxial, but the optic axis twists uniformly in space [6,7]. Choose the z axis of the lab frame to be parallel to the helical axis of the cholesteric liquid crystal. The pitch P of the liquid crystal is the distance over which the liquid crystal director twists In. The components of the liquid crystal director of a right-handed cholesteric liquid crystal q > 0) are given by... 

Optical ProportiGS. All of the optical applications of LCTs have considered the cholesteric phase. The cholesteric phase is interesting because thin films of a cholesteric liquid crystal oriented such that the helical axis is perpendicular to the plane of the film will selectively reflect circularly polarized light with a wavelength determined by the pitch of the helix. For low molar mass liquid crystals. 

The above solutions describe the characteristics of linear wave propagation in a cholesteric liquid crystal along the helical axis. In particular, if X - 11 < la I, then m. is purely imaginary, and the corresponding wave should be totally reflected. We realize that x 1 or X = 2 nc/(j)t =p (la 1 1 usually) is just the condition for Bragg reflection from the helical structure. If we transform Eq. (5) back into the lab frame, we have... 

Knowing the characteristics of linear wave propagation, we can then discuss the nonlinear optical effects in a cholesteric liquid crystal. In Sec. Ill, we shall consider the problem of third-harmonic generation along the helical axis in such a medium. Emphasis is on the derivation of collinear phase-matching conditions. 

We also attempted to observe second-harmonic generation in cholesteric liquid crystals. Hie negative results indicate that the molecular arrangement of the liquid crystals in a plane perpendicular to the helical axis has an over-all inversion symmetry. 

When the anisotropy of the electrical conductivity is positive, Aa > 0, the planar texture of a cholesteric liquid crystal in a field parallel to the helical axis (Fig. 6.8) is unstable for any sign of Aer [67, 68]. 

The EHD behavior of cholesteric liquid crystals is very similar to that of nematics. When the anisotropy of the electrical conductivity is positive (cTa>0), the planar texture of a cholesteric liquid crystal in a field parallel to the helical axis is unstable for any sign of [263, 264]. The instability is caused by the torque induced by the electrical conductivity acting against the elastic torque of the cholesteric and, although the cause is different from the purely dielectric case (see Sec. 9.3.2.2 of this Chapter), the result obtained is the same that is, the appearance of a two-dimensional periodic pattern for the distribution of the director. 

A property that may be admitted by noncen-trosymmetry may very well be ruled out by one of the other symmetry operations of the medium. As an example we will finally consider whether some of the properties discussed so far would be allowed in the cholesteric liquid crystals, which lack a center of inversion. A cholesteric is simply a chiral version of a nematic, abbreviated N, characterized by the same local order but with a helical superstructure, which automatically appears if the molecules are chiral or if a chiral dopant is added (see Fig. 30). Could such a cholesteric phase be spontaneously polarized If there were a polarization P, it would have to be perpendicular to n, because of the condition of Eq. (6), and thus along the helical axis direction m. However, the helical N phase has an infinity of twofold rotation axes perpendicular to m and the symmetry operation represented by any of these would invert P. Hence P=0. A weaker requirement would be to ask for piezoelectricity. (Due to the helical configuration, the liquid has in fact some small... 

The flow properties of cholesteric liquid crystals are surprisingly different from those of the nematics. The most important difference is that, in some directions (along the helical axis), the viscosity measured in Poiseuille flow geometries (see Appendix B) is about six orders of magnitude larger than in the isotropic phase, or in the cholesteric phase when the flow direction is perpendicular to the helix axis. In this latter case, the viscosity is similar to that of nematics, although the behavior is somewhat non-Newtonian above a pitch-dependent threshold shear rate. It was found that the shear rate above which the fluid becomes non-Newtonian is inversely proportional to the square of the pitch. The apparent viscosities as the function of shear rate of materials with different pitch values are shown in Figure 4.6. 

Note that the tilt does not necessarily require a layer structure, and the same concept can be employed to explain the piezoelectricity of the cholesteric liquid crystal. In this case, the shear along the helical axis leads to a tilt of the director toward the helical axis and a polarization normal to the shear plane. This "shear electricity" was predicted by Prosh and experimentally was verified in case of cholesteric elastomers, where the molecules are weakly cross-linked and thus can sustain elastic strains. ... 

At this point we make a slight generalization so that the theory can be applied to cholesteric liquid crystals as well. The basic difference between cholesteric and nematic liquid crystals lies in the fact that the equilibrium state of cholesterics is characterized by a nonvanishing twist in the director field. If we denote the cholesteric helical axis as the X -axis, the equilibrium state is characterized by... 

Consider a sample of cholesteric liquid crystal placed in a magnetic field H, with the field direction perpendicular to the cholesteric helical axis as shown in Fig. 10. From Eqs. [5], and [6], the free energy of the system over one period (or pitch) of the helix, X, can be written as... 

---