Disclination points/lines

As in crystals, defects in liquid crystals can be classified as point, line or wall defects. Dislocations are a feature of liquid crystal phases where tliere is translational order, since tliese are line defects in tliis lattice order. Unlike crystals, tliere is a type of line defect unique to liquid crystals tenned disclination [39]. A disclination is a discontinuity of orientation of tire director field. 

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 disclination line of m = 1/2 and disclination point of m = 1 are frequently observed in liquid crystals. In some cases high strength discli-nations may be observed, such as in a thin layer of small molecular mass liquid crystals, polymer liquid crystals, lyotropic liquid crystals, binary thermotropic liquid crystals, etc. 

For a three-dimensional nematic liquid crystal for example, the r = 0 case corresponds for example to a defect with d = 2, which means a discli-nation wall for r = 1, d = 1 corresponds to a disclination line for r = 2, d = 0 corresponds to a disclination point. It is known that the order vector space of three dimensional nematic liquid crystals is the projection plane P2 Its homotopy group of the zero rank (r = 0) is... 

When a nematic-smectic A transition occurs in a capillary tube (Fig. 8 a), smectic layers nucleate at disclination points and a singular line forms along the axis, with cylindrical layers (as shown in Fig. 8 b), but the presence of beads along the axial defect shows that the situation is less schematic [42]. 

Inelastic deformation of any solid material is heterogeneous. That is, it always involves the propagation of localized (inhomogeneous) shear. The elements of this localized shear do not occur at random places but are correlated in a solid. This means that the shears are associated with lines rather than points. The lines may delineate linear shear (dislocation lines), or they may delineate rotational shear (disclination lines). The existence of correlation means that when shear occurs between a pair of atoms, the probability is high that an additional shear event will occur adjacent to the initial pair because stress concentrations will lie adjacent to it. This is not the case in a liquid where the two shear events are likely to be uncorrelated. 

Note 1 Diselinations are responsible for some optical textures seen with a polarizing microscope, such as the schlieren texture formed by disclination lines in nearly vertical orientations, whose projections are seen as dark points with two or four emerging dark stripes or brushes (see Definition 4.9.2). 

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. 

Fig. 1. Schematic diagrams of the director field distortions black lines) around particles in an aligned nematic liquid crystal. For a normal anchoring of the liquid crystal molecules at the surface of the particles, there are two possible configurations, a Dipole configuration with a companion point defect (indicated by an arrow) located in the immediate vicinity of the particle, b Quadrupolar Saturn-ring configuration with a disclination ring surrounding the particle at the equator...

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. 

We now consider a twist disclination loop in a twisted nematic. The nematic is supposed to have a planar structure with the director parallel to the xy plane and an imposed twist of q per unit length about the z axis, and the disclination loop of radius R is supposed to be in the xy plane. The director distortions are planar, = cos = sin = 0. On going once round the disclination line at any point on the loop, the director orientation changes by 2tis, the sign of which may be either the same as that of q or opposite. 

Point defects are another class of defects in nematics. When a line, such as the wedge disclination of Fig. 3 a, with 5=1, escapes... 

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]. 

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. 

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... 

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