Disclinations singular

The most favourable relative configuration of identical chiral molecules is that where all neighbouring molecules are twisted relative to each other. This is achieved by a "double-twist" stacking, illustrated in Fig. 4.32. In three-dimensional euclidean space, this double-twist caimot be realised throughout space some "disclination" singularities must occur [61]. How then can this double twist be most closely approached A simple model, involving nothing more than potatoes, and oven and matches, is useful. The lower-... 

Fig. 22. (a) Identification of the angles and 6 used to describe a disclination. (b) Director arrangement of an 5 = I/2 singularity line. The end of the line attached to the sample surface appears as the point s = + V2 (points P). The director alignment or field does not change along the z direction. The director field has been drawn in the upper and the lower surfaces only. 

The local translational and orientational order of atoms or molecules in a sample may be destroyed by singular points, lines or walls. The discontinuities associated with the translational order are the dislocations while the defects associated with the orientational order are the disclinations. Another kind of defect, dispirations, are related to the singularities of the chiral symmetry of a medium. The dislocations were observed long after the research on them began. The dislocations in crystals have been extensively studied because of the requirement in industry for high strength materials. On the contrary, the first disclination in liquid crystals was observed as early as when the liquid crystal was discovered in 1888, but the theoretical treatment on disclinations was quite a recent endeavor. 

The projections of the director on the smectic C layer construct a c-vector (or c-director) field. The distortion of the c-director and the displacement of the layer are two parts of the defects in the smectic C phase. Because the c-director is apolar, there are only the integer disclinations. In addition, there is no escape in the third dimension because the c-director is confined in layers. Neither topological stable singular walls nor points exist in the smectic C sample according to the homotopy argument. 

In a nematic phase, the strength of disclination may have values that are half-numbered or whole-numbered. However because the disclination energy is proportional to s2 (Nehring and Saupe, 1972), it is rare to observe singularities with s > 1. In studies of a series of nematic polymers with two-dimensional mesogenic units Zhou and coworkers (1993) have been... 

Disclination in smectic phases may also show up in the form of schlieren textures. It was believed that only the whole-numbered singularities are present in these phases. However more recent work has shown that certain smectic Ca phases can also give schlieren textures with half-numbered singularities (Watanabe et al, 1989 1992 Niori et al., 1995). Other methods such as X-ray scattering may be needed for an unambiguous characterization of the phase. 

As remarked in chapter 1, the nematic state is named for the threads that can be seen within the fluid under a microscope (fig. 1.1.6(a)). In thin films sandwiched between glass plates these threads can be seen end on. A typical example of the texture in a plane film of thickness about 10 /tm between crossed polarizers - the structures a noyaux or schlieren textures - is given in fig. 1.1.6(6). The black brushes originating from the points are due to line singularities perpendicular to the layer. In analogy with dislocations in crystals, Frank proposed the term disinclinations , which has since been modified to disclinations in current usage. 

Fig. 3.5.11 gives the director configurations for some typical cases. The sections through the (x,y), and the (x, z) or (y, z) planes are identical with the patterns for the +1 and — 1 wedge disclinations in two dimensions. Any pattern on the left-hand side of fig. 3.5.11 may be combined with any one on the right to give a possible point singularity. 

Such disclinations are closely analogous to nematic wedge disclinations ( 3.5.1). The singular line is along the z axis (parallel to the twist axis) and the director pattern is given by... 

In this case the singular line is perpendicular to the twist axis. On going round this line, one gains or loses an integral number of half-pitches. The director pattern around the -edge disclination was first worked out by de Gennes who proposed a nematic twist disclination type of solution ... 

The cholesteric pitch is altered around the singular line where N is an integer. The pattern for i = j is shown in fig. 4.2.4. Again, the energies and interactions in the one-constant approximation are the same as for nematic twist disclinations. A somewhat more elaborate treatment of this model has been presented by Scheffer and the effect of elastic anisotropy has been investigated by Caroli and Dubois-Violette. ... 

Fig. 5.4.2. A rare example of a pair of singular lines, one almost straight and the other almost circular in a toric domain in the (a) smectic A and (h) smectic C phases. Additional disclination lines develop near the centre in the smectic C phase for reasons discussed in 5.8.3. (From A. Perez, M. Brunet and O. Parodi, J. de Physique Lettres, 39, 353 (1978)).

So can have lattice disclinations as well, but they are perfect and energetically favoured only along the twofold axis (which is normal to the plane containing the layer normal and the c-director). Focal conic textures are also seen, though they are always accompanied by additional singular lines of disclination, which arise because of the molecular tilt (fig. 5.8.6). 

Fig. 3 b shows the director escape at the center of a disclination of strength 5= 1 in a thin capillary of radius R. The arrangement is continuous with no singular line. The deformation involves splay and bend, but no... 

In addition to the above-discussed discli-nations, which are referred to as wedge dis-clinations, there are twist disclinations. The director is always parallel to the xy plane, but the axis of rotation (z-axis) is normal to the singular line (y-axis). Figure 4 shows the director patterns for (a) 5=1/2, 6q=0 and (b) 5=1, 6[)=0. is a linear function of the angle 0=tan (z/x). 

As a rule, thin lines of strength 1/2 or singular lines of strength 1 are seen in the threaded textures of nematic thermotropic MCPs [25,57,58]. Rare cases of thick lines have been observed. As we have already pointed out, the reason for this is that the elastic anisotropy is large in these systems. The disclinations with 5= l/2 were also reported to be the most abundant in the Schlieren textures of nematic copolyesters [59-65]. 

Figure 3 Singularities in the schlieren textures of nematic phases (A) a point singularity (the rod-like molecules are shown as short lines), and (B) a nematic thread or n disclination joining two singularities on the surfaces of the glass (the orientation of the director field is shown by the black curved lines).

It is also possible to have points with 5 = +1/2 joined by line singularities in the nematic phase these 7t disclinations, which are commonly known as threads, pass through the preparation almost perpendicularly with the ends attached to the glass surfaces. Figure 3B shows the topology about s = + 1/2 singularity line the end appears as a point on the... 

In Fig. 8.11 an example is given of a Schlieren texture in the nematic phase observed under a polarisation microscope. The polariser and analyser are always crossed and their positions with respect to photos (a) and (b) differ by 45° as shown by small crosses. On both photos characteristic brushes (threads) are seen originated and terminated at some points. The points are linear singularities (disinclinations or just disclinations) to be discussed below. Note the difference between a number of brushes originated or terminated in different points only two brushes in points 1 and 5 and four brushes in points 2, 3 and 4. It is evident that the pictures discussed are related to the local orientation of the director, i.e. to the structure of... 

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