Defects lattice disclinations

Carrying this idea over to the helical-isotropic transition, there are two differences. First, we must use disclinations topological line singularities in the director field of the liquid crystal—rather than crystal defects. The second difference is that the helical phase, which has no defects, melts to the blue phase, which is characterized by a stable defect lattice of line disclinations rather than by a random collection of defects. Indeed, there is more than one way to create such a lattice thus BPI and BPII. The helical phase therefore melts to BPI, BPI melts to BPn, and, with a final onset of randomly positioned defects, BPII melts to the isotropic phase. 

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. 

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.)...

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. 

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). 

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. 

One can think of the blue phase as a lattice of double-twist tubes (which necessitates a lattice of disclinations) or a lattice of disclinations (which necessitates a lattice of double-twist tubes) [20]. Thus, a theory involving a lattice of double-twist tubes becomes implicitly a theory for a lattice of defects. 

Calculating the free energy for a lattice of defects is a difficult task. Meiboom et al. [12] start by writing the free energy per unit length F isc of a single disclination line, which consists of four terms ... 

The distribution of defects in mesophases is often regular, owing to their fluidity, and this introduces pattern repeats. For instance, square polygonal fields are frequent in smectics and cholesteric liquids. Such repeats occur on different scales - at the level of structural units or even at the molecular level. Several types of amphiphilic mesophase can be considered as made of defects . In many examples the defect enters the architecture of a unit cell in a three-dimensional array and the mesophase forms a crystal of defects [119]. Such a situation is found in certain cubic phases in water-lipid systems [120] and in blue phases [121] (see Chap. XII of Vol. 2 of this Handbook). Several blue phases have been modeled as being cubic centred lattices of disclinations in a cholesteric matrix . Mobius disclinations are assumed to join in groups of 4x4 or 8x8, but in nematics or in large-pitch cholesterics such junctions between thin threads are unstable and correspond to brief steps in recombinations. An isotropic droplet or a Ginsburg decrease to zero of the order parameter probably stabilizes these junctions in blue phases. 

Figure 8.14a, c shows double twist cylinders, and the bold black lines in Fig. 8.14b, d show disclinations (defect Unes). In each double twist cylinder, the molecules are radially twisted towards each other through 90°. The molecules are parallel to the cylinder axis at the cylinder center and are tilted by 45° at the outer radial periphery. In other words, the molecules twist from —45° to -t45° through the cylinder, which corresponds to a quarter pitch. The diameter of a double twist cylinder is typically about 100 nm, and a simple calculation shows that approximately 200 molecules with a diameter of 0.5 nm mildly twist against each other. The lattice constant for blue phase I corresponds to a one helical pitch, and the lattice constant for blue phase II corresponds to one half helical pitch. We generally see a very small mismatch in pitch length with that of the lower-temperature chiral nematic phase. Peculiar to soft matter, a complex hierarchical structure is formed in... 

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