Wedge disclination

In this theory, yielding is produced by local molecular kinks at the nanoscale level. The formation of a kink pair is modeled by a wedge disclination. Permanent yielding is achieved when the surrounding molecules perform a similar process, which can relieve the local stored elastic energy of the initial kink pair. 

Figure 8. Wedge disclinations, with rotational strength -it, in a cubic crystal and in a liquid crystal. Note that the molecular scale of the liquid crystal is much larger than the atomic scale of the cubic crystal.

Fine deformed microstructures with strong preferred orientations could be produced by a single stroke of the wire probe (25), as illustrated by Figure 17. A vertical section made on a specimen quenched immediately after deformation confirmed that the underlying structure was fibrous with tt and 2tt wedge disclinations (30). The relaxation or coarsening after deformation indicated that the mesophase was sufficiently fluid for disclination motion... 

Figure 20 presents mesophase rods produced by extrusion alone and by a light draw after extrusion. As Jenkins and Jenkins observed (34), the strong preferred orientation induced by extrusion was easily disturbed by pyrolysis bubbles or even by small flow irregularities. However, modest draws (e.g., draw ratio = 2) after extrusion produced fibrous morphologies with good uniformity. At these draw levels, the nodes and crosses characteristic of wedge disclinations could be resolved on transverse sections. 

The carbonization experiments emphasize the need to fix or stabilize an oriented mesophase microstructure that has been imposed by mechanical deformation. The array of wedge disclinations imposed by drawing during cooldown is not thermally stable, and disclination reactions will proceed to coarsen the microstructure and reduce the preferred orientation if the mesophase softens sufficiently during carbonization. 

QuasicrystaUine phases form at compositions close to the related crystalline phases. When solidified, the resultant strucmre has icosahedra threaded by a network of wedge disclinations, having resisted reconstruction into crystalline units with three-dimensional translational periodicity. The most well-known examples of quasicrystals are inorganic phases from the ternary intermetallic systems Al-Li-Cu, Al-Pd-Mn, Zn-Mg-Ln, Al-Ni-Co, Al-Cu-Co, and Al-Mn-Pd. In 2007, certain blends of polyisoprene, polystyrene, and poly(2-vinylpyridine) were found to form star-shaped copolymers that assemble into the first known organic quasicrystals (Hayashida et al., 2007). 

The configuration of Equation (1.32) is associated with a twist disclination. Both twist and wedge disclinations are shown in Figure 1.24, where the disclination line is normal to this sheet of paper. 

The helical structure of the c-director in the smectic C phase makes the defects different from those in the smectic C phase. As the Volterra process produces a screw dislocation, for example, along the z axis and the Burger vector b = d, it must be accompanied by a parallel wedge disclination in the c-director, in the form... 

From the nature of the director patterns it is clear that dark brushes of the schlieren type will not be seen under the polarizing microscope for light propagating normal to the film (see 4.1.1). Twist disclinations may therefore be expected to be less conspicuous than wedge disclinations, and few observations have been reported of their existence in ordinary nematics. They do, however, reveal themselves under favourable circumstances in twisted nematics, often as loops separating regions of different twist. ... 

A stability analysis has shown that twist disclinations are less favourable than wedge disclinations in elastically anisotropic media. This may explain why the former are so rarely seen in ordinary nematics. 

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. 

As before, we shall begin by considering a planar sample in which the director is confined to the xy plane. In such a case, a wedge disclination involves only splay and bend distortions and we need to take into account only the splay-bend anisotropy (kjj + fejj). 

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... 

Fig. 5.8.5. Disclinations in the c-director field of smectic C. (a) 5 = 1 wedge disclination with a radial configuration (i) sink, (ii) source and (iii) meridian section of (i). (b)s= 1 wedge disclination with a circular configuration (i) vortex, (ii) antivortex and (iii) meridian section of (i) the nails signifying that the director is tilted with respect to the plane of the paper, (c) (i) = — 1 wedge disclination and (ii) s = 1 twist disclination.

Fig. 6.4.4. Disclinations in the columnar phase (a) and (b) —n/i and ff/3 longitudinal wedge disclinations about the sixfold axis L (c) and (d) n transverse wedge disclinations about the binary axes and 0 respectively (e) -n transverse wedge disclination leading to the formation of walls (/) two n disclinations at right angles to each other, one about 7 and the other about 0. (Bouligand. ")...

Two transverse wedge disclinations may occur in association as shown in fig. 6.4.4 (/). The angle between the rotation axis may be 90°, 60° or 30°. Such defects have been observed experimentally. Fig. 6.4.5 presents a photograph which may be interpreted as arising from two 71 disclinations... 

The symmetry of the columnar phase also permits the occurrence of twist disclinations in the hexagonal lattice and of hybrids consisting of a twist disclination in the hexagonal lattice and a wedge disclination in the director field. According to Bouligand these defects are not likely to exist. 

We have so far assumed Kxx=K22=K =K. Real nematics are of course, elastically anisotropic. From the values of the energies of the disclinations calculated for the case where Ki =K22 K22 it follows that the wedge disclinations are more stable than the twist disclinations if A 22>A =Af i=A 33, and vice versa. Anisimov and Dzyaloshin-skii [38] have shown that lines of half-integral strength may be stable against three-dimensional perturbations if the twist elastic constant (22 i/2(A, j-i-A"3 3). More precisely,... 

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... 

FIGURE 12.52 The combination of basic structural units into microdomains within a carbon fiber (a) skin region (b) core region (c) a hairpin defect and (d) a wedge disclination. (From Bennet, S.C. and Johnson, D.J. in Society of Chemical Industry, London, p. 377, 1978.)... 

Fig. 2.23 Disclinations, among aligned molecules in a chain-folded crystal lamella (a) a twist disclination that can be a part of a dispiration loop (b) a fold-over of two neighboring molecules and (c) a wedge disclination (from Li and Gilman (1970) courtesy of the American Institute of Physics).

According to Li and Gilman (1970) the energies of the twist-disclination and wedge-disclination loops are, respectively,... 

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