Pretilt at the boundaries

Using the transformation (3.120) again shows that the requirement (3.162) can be rewritten as 

It is then clear from (3.163) that H must tend to zero as because 

Therefore the distorted solution (3.161) first becomes available at the critical field strength 

For H 0 there is in general no constant solution to the equilibrium equation (3.111) satisfying the pretilt boundary conditions (3.159) for small o- Nevertheless, the energy of the distorted solution satisfies 

However, this distorted solution remains relatively small for 0 0o 1 til the field magnitude reaches another critical magnitude Hc2, as we now demonstrate. Consider 

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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. 

The director deformation for the zero director pretilt at the boundaries starts at a certain threshold voltage [78, 125]... 

For a finite director pretilt at the boundaries a sharp threshold disappears however, it is possible to find a voltage at which the director angle at the center of the layer becomes equal to the boundary pretilt angle 6s [116]. This voltage is very close to the optical threshold for the supertwist transition. 

The above discussion of the Fredericks transition can be extended to include (1) pretilt at the boundaries where the boundary conditions (p 0) = (j) L) = 0 are replaced by (p 0) = (p L) = < o (2) the tilted fields where the magnetic field is applied across the sample at a fixed angle (3) weak anchoring where the director is weakly anchored to both boundary plates. A more detailed discussion can be found in Stewart (2004) and Virga (1994). 

The attempts at optimization of the TVC steepness parameter show that the latter increases for the higher elastic ratios Kzz/Kn and increases for lower values of the dielectric anisotropy Ae/e , while the effect of the optical path difference And and the elastic ratio KZ3IK22 is small [86-93]. It was shown, that the maximum steepness parameter is obtained when And 2A, i.e., near the second Mauguin minimum [88, 90] (4.56). If the director pretilt angle at the boundaries is not equal to zero, the TVC steepness parameter decreases [89, 93]. Reference [90] proposed a phenomenological formula for the evaluation of the steepness parameter pso of the TVC based on the experimental data for the binary mixtures of compounds belonging to twelve structurally and physically different liquid crystal classes... 

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. 

Figure 90. Surface-stabilized configuration with less than optimum efficiency, switchable between two symmetric states with low optical contrast. The surface pretilt angle has been chosen equal to the smectic tilt angle 0in this example. For a strong boundary condition with zero pretilt, a different extreme limiting condition with approaching zero at the boundary is also conceivable, without any essential difference in the performance of the cell.

Also, the optical state (transmission, color) is very often practically the same on both sides of a zigzag wall, as in Fig. 93b. Indeed, if the director lies parallel to the surface (pretilt o =0) at the outer boundaries, the chevron looks exactly the same whether the layers fold to the right or to the left. However, if the boundary condition demands a certain pretilt a 5 0, as in Fig. 105, the two chevron structures are no longer identical. The director distribution across the cell now depends on whether the director at the boundary tilts in the same direction relative to the surface as does the cone axis, or whether the tilt is in the opposite direction. In the first case we say that the chevron has a C1 structure, in the second a C2 structure (see also Fig. 106). We may say that the Cl structure is natural in the sense that if the rubbing direction (r) is the same at both surfaces, so that the pretilt a is symmetrically inwards, the smectic layer has a natural tendency (already in the SmA phase) to fold accordingly. However, if less evident at first sight, the C2 structure is certainly possible, as demonstrated in Figs. 105 and 106. 

Figure 3.12 Graphs obtained via equation (3.176) showing the dependency of the maximum distortion angle dm upon the dimensionless magnetic field H when the prescribed pretilt 9o of the director at the boundaries takes the values 0 , 2 and 10° and K = Kz — K. He = classical value of the critical threshold...

As the baseline for our calculations described in the Appendix, we take the typical LCD cell configuration of thickness d=8 ym filled with E7 liquid crystal mixture with a cholesteric additive giving a natural cholesteric pitch P=40 ym. The boundary conditions imposed on the liquid crystal are total twist 0q=9O , pretilt ao=3 and strong surface anchoring. The polarizers are crossed and parallel to the nematic director at the neighboring liquid crystal surfaces. In calculations where the total twist... 

ClTand C2T models are the half splayed states [20, 21], and the boundary surfaces in the C2 state do not have wide regions wherein the molecules can exist in a stable state (Fig. 5.1.14). The c-directors at the surfaces are almost perpendicular to the substrate in the high pretilt angle C2U model. The molecules at the surfaces can move easily in the C2 state if the pretilt angle is low. These states are assumed to switch between two elastically equivalent states that produce the stable memory effect. 

Equation (28) holds only for the zero pretilt angle of the director at the limiting boundaries. Any finite pretilt angle results in a loss of the threshold character of the effect (a continuous distortion instead of the Frederiks transition). 

Figure 107. Polarization switching taking place in the chevron plane at fixed polar boundary conditions. In this case the condition is one of high pretilt a for n with the P vector pointing into the liquid crystal from the boundary.

Figure 3.11 The FVeedericksz transition when the director is strongly anchored on the boundaries with a fixed pretilt of q as shown in (a). There is a FVeedericksz transition at. Tc = 0 but, for small the difference between and the distortion SiXigle 9m remains relatively small until He Hc = demonstrated in...

A theoretical consideration of the case of a pitch that is comparable to the layer thickness for a purely dielectric destabilization of a planar texture in a field 11 has been given both numerically [122] and analytically [123, 124]. In the latter case the perturbation theory was used to search for the structure of the director field just above the threshold of the instability. Two variables, the polar angle 6 and the azimuthal angle 0 were considered, with orientation of the director at opposite walls differing by a twist angle a (pretilt angles at boundaries were also taken into account). It has been shown that two types of instability can be observed depending on the elastic moduli of the material a total twist of the structure between... 

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