Director distribution

The surface flexoelectric energy, which is found from (4.2) and (4.3). Attaining the minimum of the nematic free energy, (4.5) or (4.6), it is possible to derive the equilibrium director distribution in a static case. To find the response times, we have to solve the equations of nematodynamics in the electric field. The corresponding analysis shows that the director reorientation is always accompanied by the macroscopic flow, the so-called backflow [5]. (The only exclusion is the pure twist rotation of the director [1].) Backflow considerably affects the characteristic times of the electrooptical effects in uniform structures, especially in the case of strong deformations of the initial director orientation [3, 5]. 

Based on the director distribution we can derive the electrooptical response of a nematic liquid crystal cell (such as birefringence), rotation of the polarization plane of the incident light, total internal reflection, absorption, or some other important characteristics of the cell. In this chapter we will consider in detail these particular features of the electrooptical phenomena in uniform structures. Special attention will be paid to their possible applications. Electrooptics of the isotropic phase and polymer nematics, including Polymer Dispersed Liquid Crystals (PDLC), are also discussed. 

The most important geometries of electrically controlled birefringence (ECB) are shown in Fig. 4.1. A compromise between dielectric and elastic torques results in director reorientation from the initial alignment 0(z) with the maximum deviation 0m at the center of the layer (the Prederiks transition). The effect occurs when the electric field exceeds a certain threshold value 

FIGURE 4.1. S(a) and B(b) Prederiks transitions in (a) homogeneous and (b) homeotropic nematic cells with (a) positive and (b) negative values of dielectric anisotropy. The transitions take place when the applied voltage U exceeds certain threshold values Us or Ub- 

Let us consider first the splay Frederiks transition or the S-effect. (All the expressions are also valid for the B-effect if the following exchange of parameters are made Ku ATss, n .) In this case, the 

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Fig. 6. The two generic shapes of molecules which exhibit flexoelectric polarisation under distortion of the equilibrium director distribution...

Interestingly, the macroorder of LC side chain polymers can be frozen in at the glass transition. No electric or magnetic field is required to maintain the director distribution... 

Transmission electron microscopy (TEM) studies conducted by Thomas et al. [62,63] have permitted direct visualization of the molecular director distribution in flow-oriented thin films prepared from semiflex-ible thermotropic LCPs of the following structure ... 

In the distorted nematic the director distribution is inhomogeneous, and hence n = n(r) in Eq. (1.46). Taking into account that the length scale of the director variation is much larger than the molecular size we can apply the gradient expansion of the director n(r) with respect to some point Ro inside the molecule ... 

Fig. 8.9a, we are interested in the coordinate dependence of the azimuthal director angle cp(z), which is rigidly fixed at the two boundaries, tp(0) = 0, equilibrium director distribution to be found corresponds to the minimum of the elastic free energy for the cell as a whole. First, we should write an expression for the density of Frank elastic energy. The director at any point z is given by n = cos 4> z) + sin( (z)j. There is no z-compcment, = 0, even no pretilt at the boundaries. 

Consider a disclination with its ends fixed at the opposite plates of a planar nematic cell. Such a disclination connects the two glass plates as in Fig. 8.13a. If we are looking at it from the top along the z-direction we can see the director distribution n (x, y) in the Ay-plane around the disclination. In a polarization microscope, in the same cell, we can see different n(jc, y) patterns corresponding to disclinations shown in Fig. 8.14. A point in the middle of each sketch shows the disclination under discussion that has its own strength s. 

F. 8.15 Geometry for calcnlatimi of the director distribution around a disclination L. P is the azimuthal angle for an arbitrary point r in the x, y plane of the nematic layer (p is the director angle in point r... 

Fig. 11.31 Flexoelectric instability. Periodic structure of the field induced director distribution along the y-axis represented by projections and...

The appearance of the high harmonics in the director distribution results in considerably faster electro-optical switching. The dynamics of the cholesteric helix in the electric field is described by the balance of viscous, elastic and electric torques in the infinitely thick sample is given by... 

With the aim of quantitatively predicting the orientational order of rigid solutes of small dimensions dissolved in the nematic liquid crystal solvent, 4-n-pentyl-4 cyanobiphenyl (5CB), an atomistic molecular dynamics (MD) computer simulation has been applied. It is found that for the cases examined the alignment mechanism is dominated by steric and van der Waals dispersive forces. A computer simulation of the deuterium NMR spectra of molecules in a thin nematic cell has been carried out and the director distribution in the cell has been studied. An experiment for the direct estimation of an element of the order matrix from H NMR spectra of strongly dipolar coupled spins that is based on the multiple quantum spin state selected detection of single quantum transitions has been proposed. The experiment also enables obtaining nearly accurate starting dipolar... 

Figure 8.5 Device structure, simulated on-state LC director distribution, and corresponding light transmittance of an IPS cell. Electrode width VF=4 pm and electrode gap L = 8 pm.

Figure 8.22 shows the calculated iso-contrast contour of the four-domain MVA-LCD without film compensation. In the contrast ratio calculation, we first use continuum theory [52] to calculate the LC director distribution at Von = 5 Vj ,s and Vojf= 0, respectively, and then use the... 

Figure 8.21 (a) Schematic top-view of the four-domain LC director distribution in the voltage-on state,... 

To optimize the lens design, we need to know the LC director profiles of the above foiu device configurations. Several commercial software packages are available and can be used to calculate the LC directors distribution. The parameters used in the simulations are // = 14.9, e = 3.3, = 20.3 pN, = 33.8 pN, rig = 1.9653, and = 1-5253. First, we need... 

Liquid crystalline phases formed by chiral molecules (i.e. molecules differing from their mirror image) show unique macroscopic properties. The best-known example is the cholesteric phase which is termodynamically equivalent to the nematic phase. In the later phase the free-energy of the system corresponds to a uniform director distribution in the whole sample. On the other hand in cholesterics the molecules tend to form a helical structure the helical axis being perpendicular to the director. A similar helical structure develops in the smectic C phase when the molecules are chiral. In this case the helical axis is parallel to the layer normal the tilt angle is constant while the azimuthal angle is rotating in space. The pitch of the helix in these systems is typically in the order of a micron. 

The problem of light propagation becomes much more complicated in spatially inhomogeneous liquid crystal layers. There is no general method to solve the Maxwell equations for an arbitrary director distribution or, more generally speaking, for an arbitrary spatial dependence of the dielectric tensor. On the other hand in some important special cases exact solutions were found and useful approximations were worked out for other conditions. In this section we survey these results. 

Hyperhybrid cells. We take the unperturbed director distribution to be... 

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. 

Let the structure of the nematic liquid crystal be called uniform if the director distribution in this structure, and, accordingly, its optical and electrooptical properties are uniform, i.e., do not depend on the coordinates in the plane parallel to the substrates. The first observations of the optical and electrooptical properties of uniform nematic structures were carried out by Mauguin, Tsvetkov, and Prederiks over 50 years ago [1-4]. Since the beginning of the 1970s these structures have attracted growing interest, as they are the most promising ones for practical applications in Liquid Crystal Displays (LCDs). 

Let the plane TM wave be incident at an angle ie onto a nematic layer with the initially homeotropic director distribution (Fig. 4.34)... 

Let us note that 9 is not the director angle at the boundary. This is so only for special types of director distributions, e.g., in the flexoelectric effect (Fig. 4.30(a, b)) or in the quasi-homeotropic reverse pretilt configuration (Fig. 4.3(b)). In the B effect, on the contrary, the director angle is maximum in the middle of the layer (Fig. 4.1(b)), and, consequently, we cannot analyze the boundary region with the TIR method. 

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