Tilt director

More complex configurations with a tilted director orientation at the surface have also been reported by Hobdell Windel (1995) and Madhusudana Sumathy (1983). It is illustrated that the knot defect has the highest energy while the saddle defect gives the lowest. 

Fig. 2.4. A transverse electric field, indicated by the arrow pointed towards the right at the top of the figiu-e, tilts the apolar director as shown by double headed arrows in a specific direction due to the flexoelectric effect on a 90° twisted nematic cell. The tilting direction reverses if the field direction is reversed. The transmitted intensity mesisured with a polarized light beam traversing the cell vertically as indicated by the dashed line will be identical in the two cases. On the other hand, with an oblique beam, the transmitted intensities for the two tilted director structures will be different, and can be used to me siu-e the flexocoefficient (adapted from Kischfai et cU. ).

In a real nematic cell, however, it is very difficult to satisfy the conditions (4.15). Moreover, to avoid degenerate solutions, when both clockwise and counterclockwise director rotations are possible (Fig. 4.1), the boundaries are specially prepared with tilted director orientation... 

FIGURE 5.9. Electrohydrodynamic instability with tilted director orientations at the boundaries [63]. 

A new type of electrohydrodynamic instability in liquid crystals with tilted directors at the boundaries was revealed by Pikin et al. [63]. This instability is characterized by a domain pattern parallel to the projection of the initial director orientation. It is supposed that the director angle with respect to the substrate remains unchanged (x>2 -plane) while the periodic two-dimensional vortex motion appears in the yz-plane, where the 2 -axis is perpendicular to the substrates (Fig. 5.9). These domain structures seem to be observed in [64, 65] with the oblique director orientation achieved by means of oblique SiO evaporation. When the voltage exceeds its critical value the second domain pattern appears along the y-axis. 

Berreman and Heffner [59] considered the cholesteric Grandjean texture with tilted director orientation on the boundaries, Fig. 6.16. In the absence of the tilt, the free energy g is minimum at the following thicknesses d of the Cano wedge ... 

Figure 12.5. Cross-sections of director fields corresponding to the X and x lines X with ) n — X and (c) = — x with [h) n — X and (d) = — i. Points indicate the director is perpendicular to the page, whereas marked ends represent tilted directors. The line defect is indicated by a heavy dot (courtesy of J. Bajc).

Fig. 5.18. The chiral smectic C phase (S ) has the molecular axes on average tilted with respect to the layer normal of the smectic, and the in-plane component of the local director the so-called tilt director, traces out a helicoidal path in the mesophase.

We may also apply the Curie principle to this phenomenon. The SmA phase has (or oo/m) symmetry, with one C axis along the director (optic axis), infinitely many C2 axes perpendicular to this axis, and in addition one horizontal and infinitely many vertical mirror planes. The mirror planes are absent in the SmA phase (D or o°22). If we apply an electric field E (of symmetry perpendicular to the C axis, the only common symmetry element left is one C2 axis along E, which permits a tilt around C2. In Fig. 46 we also see that, in particular, the plane E, n) is a mirror plane in the nonchi-ral SmA phase, and consequently neither of the two tilting directors shown in the figure are allowed. In the SmA phase E, n) is not a mirror plane, hence one of these tilt directions will be preferred. (Which one cannot be predicted as this is a material property.)... 

Illustration of the difference in the conoscopic images for homeotropic and tilted director structures. The images are taken on a free-standing film of a liquid crystal mixture ZKS 2504 in the SmA and SmC phases, respectively. ... 

Figure 73 Schematic reperaentatioo of the geometry of the glass substrate containing the gold electrode patterns, (a) Onas-sectioiiBl view of the LB layets deposited between a pair of gold electrodes, (b) Oblique view of a section of the surface i h paiteroed electrodes. is the field directioo parallel bs the layers, Z is the smectic layer normal, n is the tilt director, and is the projection of n onto the sufastralc. (From Ref. 101.)...

As witli tlie nematic phase, a chiral version of tlie smectic C phase has been observed and is denoted SniC. In tliis phase, tlie director rotates around tlie cone generated by tlie tilt angle [9,32]. This phase is helielectric, i.e. tlie spontaneous polarization induced by dipolar ordering (transverse to tlie molecular long axis) rotates around a helix. However, if tlie helix is unwound by external forces such as surface interactions, or electric fields or by compensating tlie pitch in a mixture, so tliat it becomes infinite, tlie phase becomes ferroelectric. This is tlie basis of ferroelectric liquid crystal displays (section C2.2.4.4). If tliere is an alternation in polarization direction between layers tlie phase can be ferrielectric or antiferroelectric. A smectic A phase foniied by chiral molecules is sometimes denoted SiiiA, altliough, due to the untilted symmetry of tlie phase, it is not itself chiral. This notation is strictly incorrect because tlie asterisk should be used to indicate the chirality of tlie phase and not tliat of tlie constituent molecules. 

This transition is usually second order [18,19 and 20]. The SmC phase differs from the SmA phase by a tilt of the director with respect to the layers. Thus, an appropriate order parameter contains the polar (0) and azimuthal ((]i) angles of the director ... 

Chiral Smectic. In much the same way as a chiral compound forms the chiral nematic phase instead of the nematic phase, a compound with a chiral center forms a chiral smectic C phase rather than a smectic C phase. In a chiral smectic CHquid crystal, the angle the director is tilted away from the normal to the layers is constant, but the direction of the tilt rotates around the layer normal in going from one layer to the next. This is shown in Figure 10. The distance over which the director rotates completely around the layer normal is called the pitch, and can be as small as 250 nm and as large as desired. If the molecule contains a permanent dipole moment transverse to the long molecular axis, then the chiral smectic phase is ferroelectric. Therefore a device utilizing this phase can be intrinsically bistable, paving the way for important appHcations. 

Fig. 10. The chiral smectic Chquid crystal stmcture. The director, n, rotates about the normal to the smectic layers, keeping the tilt angle. A, constant.

The alignment of the director at the nematic free surface of real systems is not found to exhibit universal behaviour. Depending on the mesogen, homeotro-pic, tilted and planar anchoring have been observed. Clearly, to study this interface in a simulation a potential which exhibits a nematic phase in co-... 

Smectic A and C phases are characterized by a translational order in one dimension and a liquid-like positional order in two others. In the smectic A phase the molecules are oriented on average in the direction perpendicular to the layers, whereas in the smectic C phase the director is tilted with respect to the layer normal. A simple model of the smectic A phase has been proposed by McMillan [8] and Kobayashi [9] by extending the Maier-Saupe approach for the case of one-dimensional density modulation. The corresponding mean field, single particle potential can be expanded in a Fourier series retaining only the leading term ... 

In the smectic mesophases the molecules are oriented, as in a nematic mesophase, with their principal axis roughly parallel to the director, but they are also defining layers. These layers can be perpendicular to the director, as in the smectic A mesophase (SmA), or tilted, as in the smectic C (SmC). The SmA and SmC mesophases are the less ordered and more common smectic mesophases. Other less common types of smectic mesophases are known, which differ in the degree or kind of molecular ordering within and between the layers [2]. 

A very different model of tubules with tilt variations was developed by Selinger et al.132,186 Instead of thermal fluctuations, these authors consider the possibility of systematic modulations in the molecular tilt direction. The concept of systematic modulations in tubules is motivated by modulated structures in chiral liquid crystals. Bulk chiral liquid crystals form cholesteric phases, with a helical twist in the molecular director, and thin films of chiral smectic-C liquid crystals form striped phases, with periodic arrays of defect lines.176 To determine whether tubules can form analogous structures, these authors generalize the free-energy of Eq. (5) to consider the expression... 

In a bulk SmC material the tilt angle 0 is fixed, but the director is free to take up any azimuthal orientation, defining a tilt cone of degenerate azimuthal... 

Since P must remain normal to z and n, the polarization vector forms a helix, where P is everywhere normal to the helix axis. While locally a macroscopic dipole is present, globally this polarization averages to zero due to the presence of the SmC helix. Such a structure is sometimes termed a helical antiferroelectric. But, even with a helix of infinite pitch (i.e., no helix), which can happen in the SmC phase, bulk samples of SmC material still are not ferroelectric. A ferroelectric material must possess at least two degenerate states, or orientations of the polarization, which exist in distinct free-energy wells, and which can be interconverted by application of an electric field. In the case of a bulk SmC material with infinite pitch, all orientations of the director on the tilt cone are degenerate. In this case the polarization would simply line up parallel to an applied field oriented along any axis in the smectic layer plane, with no wells or barriers (and no hysteresis) associated with the reorientation of the polarization. While interesting, such behavior is not that of a true ferroelectric. 

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