Point disclinations

Assume a point disclination located in a nematic droplet of radius R. The point disclination can be classified according to their Poincare characteristic angle a as a knot point (a = 0), focus point (0 a 7t/2), center (a = 7t/2), saddle-focus point (tt/2 a tt) or saddle point (a = 7t/2). For a knot point, one has a spherically symmetrical radial configuration and then 

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. 

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. 

If the inequality of the three elastic constants is taken into account, the disclination configurations are slightly different. Even though the energy for same m is different. The integration of defects produces the textures of liquid crystals. Owing to the thermal fluctuation of the molecules and 

The disclinations may split or annihilate due to interactions. The discli-nations of the same sign tend to repel each other while those of opposite signs attract each other and may eventually annihilate. 

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Thus a nematic phase is indicated by a mixture of two and four point discli-nations, while a smectic phase exhibits only four point disclinations. 

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 there is translational order, since they are line defects in this lattice order. Unlike crystals, there is a type of line defect unique to liquid crystals, termed disclination. A disclination is a discontinuity of orientation of the director field. Point defects such as point disclinations are observed at free surfaces. 

R.B. Meyer, Point Disclinations at a Nematic-Isotropic Liquid Interface, Mol. Cryst. Liq. Cryst., 16, 355-369 (1972). 

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. 

In order to discuss the hexatic phase it is necessary to introduce the idea of a disclination. Imagine a two-dimensional close packed hexagonal lattice drawn on a deformable sheet. If one chooses a particular lattice site as the centre of coordinates, the lattice will consist of six 60° sectors centred on this point. One now has two alternatives. 

Figure 13 depicts a coalescence event that took place in 7 sec. Like other coalescence events in which new disclinations are produced, a 2tt disclination appeared at the position of the former boundary between the mesophase spherules. Frenkel (27) has pointed out that the time constant t for such coalescence phenomena is given approximately by... 

The spatial geometry of disclination reactions in bulk mesophase has recently been presented by Zimmer and Weitz (29) Working with coarse-structured mesophase prepared by lengthy pyrolysis of A240 petroleum pitch at 400°C, they defined the disclination arrays on a succession of polished sections spaced at about 7 ym. In this way a +tt disclination was traced through a branching point (i.e., a reaction point) to form a -tt and a +2tt disclination. Thus a reaction... 

Figurc 18. Voronoi construction for two nearly square arrangements of points, showing how a small displacement of the points can create disclination quadrupole (right) from a configuration with no disclinations (left).

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. 

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... 

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. 

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