Boundary director fields

The thermodynamical equilibrium of nematics would correspond to a spatially uniform (constant n(r)) director orientation. External influences, like boundaries or external fields, often lead to spatial distortions of the director field. This results in an elastic increment, fd, of the volume/ree energy density which is quadratic in the director gradients [2, 3] ... 

The equilibrium configuration in a liquid crystal sample is strongly influenced by the sample boundaries. The confining surfaces can induce order, disorder, or can align liquid crystal molecules in a given direction. Surface interactions not only have influence on the static properties of a confined liquid crystal, but can also have a strong effect on the director dynamics. By studying temporal fluctuations of the director field in confined samples, information about the surface-liquid crystal interaction can be obtained. 

According to Eq. (7.1) P is zero for the two cases of uniform director fields and pure twist. Hence both cases can serve as a zero state as far as flexoelectric excitations are concerned. It is important to note that a twist is not associated with a polarization (i.e. C2 is identically zero, cf. Fig. 7.2). An imstrained nematic has a centre of symmetry (centre of inversion). On the other hand, none of the elementary deformations - splay, twist or bend have a centre of symmetry. According to Curie s principle they could then be associated with the separation of charges analogous to the piezoeffect in solids. This is true for splay and bend but not for twist because of an additional symmetry in that case if we twist the adjacent directors in a nematic on either side of a reference point, there is always a two-fold symmetry axis along the director of the reference point. In fact, any axis perpendicular to the twist axis is such an axis. Due to this symmetry no vectorial property can exist perpendicular to the director. In other words, a twist does not lead to the separation of charges. This is the reason why twist states appear naturally in liquid crystals and are extremely common. It also means that an electric field cannot induce a twist just by itself in the bulk of a nematic. If anything it reduces the twist. A twist can only be induced in a situation where a field turns the director out of a direction that has previously been fixed by boundary conditions (which, for instance, happens in the pixels of an IPS display). 

Unoriented poly (p-hydroxybenzoic acid-co-2,6-hydroxynaphthoic acid) exhibited smoothly wandering director fields in three dimensions. Alignment with a 1.1 T magnetic field for 30 min at 300 C transformed this structure to domains with an anisotropic shape within which the polymer was highly oriented, and the global order parameter amounted to 0.85 [110]. Boundaries were of the splay-bend type and involved a 180 director rotation. At lower field strengths, the domains were less... 

The study of defects in liquid crystal systems is rooted in the understanding of defects in the solid state. For instance, crystals are rarely perfect and usually contain a variety of defects, e.g., point defects, line defects, or dislocations, and planar defects such as grain boundaries. In addition to these typical imperfections of the solid state, liquid crystals can also exhibit defects known as disclinations. These defects are not usually found in solids and result from the fact that mesophases have liquid-like structures that can give rise to continuous but sharp changes in the orientations of the molecules, i.e., sharp changes in orientation occur in the director field. 

Let us assume that a liquid is incompressible, B oo, and discuss orientational (or torsimial) elasticity of a nematic. In a solid, the stress is caused by a change in the distance between neighbor points in a nematic the stress is caused by the curvature of the director field. Now a curvature tensor dnjdxj plays the role of the strain tensor ,y. Here, indices i,j = 1, 2, 3 and Xj correspond to the Cartesian frame axes. The linear relationship between the curvature and the torsional stress (i.e., Hooke s law) is assumed to be valid. The stress can be caused by boundary conditions, electric or magnetic field, shear, mechanical shot, etc. We are going to write the key expression for the distortion fi-ee energy density gji, related to the director field curvature . To discuss a more general case, we assume that gji t depends not only on quadratic combinations of derivatives dnjdxj, but also on their linear combinations ... 

But how to force the system relax to a particular state selected by an experimentalist Berreman and Heffner [20] suggested to exploit the backflow ejfect discussed in Section. 11.2.6. We know that, upon relaxation of the director from the field-ON quasi-homeotropic state (barrier state B) to a field-OFF state, a flow appears within the cell. The direction of the flow depends on the curvature of the director field, which is more pronounced near the electrodes. Moreover it has the opposite sign at the top and bottom electrodes, see the molecules distribution in state B in Fig. 12.17. Due to this, the close-to-electrode flows create a strong torque exerted on the director mostly in the middle of the cell that holds the director to be more or less parallel to the boundaries in favour of the n = 2) initial state in Fig. 12.17. 

As an example let us consider nematics. We mentioned that in nematics the internal free-energy minimum corresponds to a constant director field within the entire sample. The actual director configuration in the layer is, however, influenced by the boundary conditions and, if present, by external fields. By now there are well-developed techniques to align nematics parallel or perpendicularly (or in any direction) relative to glass substrates. In this way it is possible to prepare single-crystals such as those shown in Fig. 2a and 2b, or films deformed in a controlled manner (Fig. 2c and 2d). 

A helical director field also occurs in the chiral smectic-C phase and those smectic phases where the director is tilted with respect to the layer normal (Figure 1.13(c)). In these cases, the pitch axis is parallel to the layer normal and the director inclined with respect to the pitch axis. Very complicated defect structures can occur in the temperature range between the cholesteric (or isotropic) phase and a smectic phase. The incompatibility between a cholesteric-like helical director field (with the director perpendicular to the pitch axis) and a smectic layer structure (with the layer normal parallel or almost parallel to the director) leads to the appearance of grain boundaries which in turn consist of a regular lattice of screw dislocations. The resulting structures of twist grain boundary phases are currently extensively studied. The state of the art in this topical field is summarized in Chapter 10. 

After several minutes of annealing, necessary for the elimination of disclinations and relaxation of the director field, the display is ready for operation. The equilibrium configuration of the director field, satisfying the boundary conditions on the limit glass plates, is the helical one... 

Figure 5.15. Cholesteric textures in spherical droplets with tangential director anchoring at the boundary. Top A monopole configuration with a point defect N =l in the field / of normals to the cholesteric layers and an attached nonsingular line k — 1, stable when R/p 1 (microphotograph in crossed polarizers). Bottom A boojum configuration with an isolated k — 2 surface point defect at R/p 1 (no crossed polarizers). The insert shows the director field at the surface of the droplet.

Twist grain boundary (TGB) phases [l]-[4] usually appear in the temperature range between a cholesteric N phase with short pitch and a smectic phase, typically SmA or SmC. In particular, they are expected to appear close to a N /SmA/SmC triple point [5]. One of their remarkable properties is the selective reflection of circularly polarized light [2], [3]. This feature shows that the director field has a helical structure similar to the cholesteric phase. On the other hand. X-ray investigations of TGB phases indicate a layer structure as occurring in smectic phases [6]. Chirality of the system is an essential precondition for the occurrence of TGB phases. In mixtures of... 

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... 

The introduction of is not as straightforward as that of K24, and causes serious mathematical problems. The free energy contribution corresponding to the 13 term is not bound from below, and the simple application of the variational principle with a nonzero Ki coefficient may lead to discontinuities in the director field at the boundaries. This problem is known as the Oldano-Bar-bero paradox [213, 325]. Consider the simple one-dimensional splay geometry having the director field n=[sin d(z), 0, cos 0(z)] in one constant approximation Kn=K =K. The free energy density is... 

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