Lattice disclinations

Both groups claimed that the mechanism of nanoparticle stabilization of blue phases is similar to that reported for polymer stabilized blue phases [412], with the nanoparticles accumulating or being trapped in the lattice disclinations, which finds support from numerical modeling of colloidal particles in blue phases recently described by Zumer and co-workers [429]. 

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). 

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. 

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 4. Isolated topological defects in a triangular lattice, (a) Isolated -1 and +1 disclinations. A vector aligned along a local lattice direction is rotated by 60° upon parallel transport around a unit strength disclination. (6) An isolated dislocation. The heavy line represents a Burgers circuit around the dislocation, and the Burgers vector of the dislocation is the amount by which the circuit fails to close. The core of the dislocation is a tightly bound pair of +1 and -1 disclinations (Reproduced from [78] by permission of Oxford University Press.)...

W. Pantleon, in Local Lattice Rotations and Disclinations in Microstructures of Distorted Crystalline Materials, Eds. P. Klimanek, A.E. Romanov, M. Seefeld, (SEITEC Publications Ltd.)... 

Fig. 4.8.3. Unit celk of BP disclination lattices. O is simple cubic, O , 0 + and O — are body-centred cubic. The tubes represent disclination lines whose cores are supposed to be isotropic (liquid) material. (From Berreman. )...

Topologically, it turns out that the helical structure of the cholesteric cannot be deformed continuously to produce a cubic lattice without creating defects. Thus BP I and BP II are unique examples in nature of a regular three-dimensional lattice composed of disclination lines. Possible unit cells of such a disclination network, arrived at by minimizing the Oseen-Frank free energy, are shown in fig. 4.8.3. The tubes in the diagram represent disclination lines, whose cores are supposed to consist of isotropic (liquid) material. Precisely which of these configurations represents the true situation is a matter for further study. 

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. 

Bourrat X, Roche EJ, Lavin JG, Lattice imaging of disclinations in carbon fibers. Carbon, 28, 236, 1990. 

The concept of defects came about from crystallography. Defects are dismptions of ideal crystal lattice such as vacancies (point defects) or dislocations (linear defects). In numerous liquid crystalline phases, there is variety of defects and many of them are not observed in the solid crystals. A study of defects in liquid crystals is very important from both the academic and practical points of view [7,8]. Defects in liquid crystals are very useful for (i) identification of different phases by microscopic observation of the characteristic defects (ii) study of the elastic properties by observation of defect interactions (iii) understanding of the three-dimensional periodic structures (e.g., the blue phase in cholesterics) using a new concept of lattices of defects (iv) modelling of fundamental physical phenomena such as magnetic monopoles, interaction of quarks, etc. In the optical technology, defects usually play the detrimental role examples are defect walls in the twist nematic cells, shock instability in ferroelectric smectics, Grandjean disclinations in cholesteric cells used in dye microlasers, etc. However, more recently, defect structures find their applications in three-dimensional photonic crystals (e.g. blue phases), the bistable displays and smart memory cards. 

The defects that can occur in BCP nanopatterns can take several forms and it is beyond the scope of this chapter to detail these in full, however, it is worth providing a general overview. They take the form of many structural defects in other systems and can be broadly described as dislocations and disclinations and a good review is provided elsewhere (Krohner and Antony, 1975). In the simplest explanation, a dislocation is a defect that affects the positional order of atoms in a lattice and the displacement of atoms from their ideal positions is a symmetry of the medium Screw and edge dislocations representing insertion of planes or lines of atoms are typical of dislocations. For a discUnation the defects (lines, planes or 3D shapes) the rotational symmetry is altered through displacements that do not comply with the symmetry of the environment. Kleman and Friedel give an excellent review of the application of these topics to modern materials science (Kleman and Friedel, 2008). 

Figure 14.1 Blue phase LC structure at the microscopic level (a) double-twist alignment of LC molecules, (b) double-twist cylinder, (c) lattice cubic formed by double-twist cylinders, and (d) disclination lines.

As the temperature increases, up to three types of blue phases BPI, BPII, and BPIII may exist [14]. BPIII is believed to possess amorphous stmcture. BPI (Figure 14.2(a)) and BPII (Figure 14.2(c)) are composed of double-twist cylinders arranged in cubic lattices. Inside each cylinder, the LC director rotates spatially about any radius of the cylinder. These double-twist cylinders are then fitted into a three-dimensional stmcture. However, they cannot fill the full space without defects. Therefore, blue phase is a coexistence of double-twist cylinders and disclinations. Defects occur at the points where the cylinders are in contact (Figures. 14.2(b) and 14.2(d)). BPI is known to have body-center cubic stmcture and BPII simple cubic stmcture. 

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