Defects in cholesterics

Earlier in Section 4.8 we discussed the blue phases observed in cholesterics close to the transition to the isotropic phase. The whole appearance of the blue phase is owed to the defects, which form a three dimensional lattice. 

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The same result can be obtained for biaxial nematics [42] from a topological point of view, the classifications of defects in cholesterics and biaxial nematics are identical. Calculation of the fundamental group for iR = SO 2i)/D2 requires knowledge beyond the scope of this chapter. We simply present the result (for details, see [2], [37], [42]) ... 

One should bear in mind that the topological classification of defects in cholesteric and other layered media such as smectics and ordinary crystals is limited by the condition of the layers equidistance. As a result, some transformations between defects that belong to the same class require very high energy barriers comparable to the energy barriers between different classes. Transformation X X within the class Cx represents such an example. 

Because of the additional translational order, the dislocations can exist in the cholesteric and smectic liquid crystals, which makes the texture of these liquid crystals even more complicated. Each liquid crystal phase shows characteristic textures and thus the optical texture becomes an important means to differentiate the phase of the liquid crystals. Liquid crystalline polymers have the same topologically stable defects as small molecular mass liquid crystals do, but the textures may be different due to the difference in the energetic stability of the same topological defects in both low molecular mass and polymeric liquid crystals (Kleman, 1991). In Chapter 3 we will discuss the textures in detail. 

The singularities in the liquid crystals cause the deformation of the director field of liquid crystals and thus affect the symmetry of liquid crystals. This idea provides an approach to analyze the characteristics of the defects. The order vectors (or scalars, or tensors) of various liquid crystals are not the same. The director n is the order vector of the nematic liquid crystals, but the order for the cholesteric liquid crystals is a symmetric matrix, i.e., a tensor. Because the order vector space is thus a topological one, any configuration of the director field of liquid crystals is thus represented by a point in the order vector space. The order vector space (designated by M) is associated with the symmetry of liquid crystals. The topologically equivalent defects in liquid crystals constitutes the homotopy class. The complete set of homotopy classes constitutes a homotopy group, denoted Hr(M). r is the dimension of the sub-space surrounding a defect, which is related to the dimension of the defect (point, line or wall) d, and the dimension of the liquid crystal sample d by... 

Because of the layered structure, defects in the cholesteric can be likened in many respects to those in smectic A. Both of them exhibit focal conic textures and both allow for the existence of screw and edge dislocations. To discuss these similarities we employ a coarse-grained approximation in which the cholesteric distortions are considered to be small and to vary slowly over a pitch. In this approximation the free energy of distortion may be expressed in terms of layer displacement u parallel to the twist axis ... 

Cholesteric liquid crystals are compounds that go through a transition phase in which they flow like a liquid, yet retain much of the molecular order of a crystalline solid. Liquid crystals are able to reflect iridescent colors, depending on the temperature of their environment. Because of this property they may be applied to the surfaces of bonded assemblies and used to project a visual color picture of minute thermal gradients associated with bond discontinuities. Cholesteric crystals are potentially a simple, reliable, and economical method for evaluating bond defects in metallic composite structures.f Materials with poor heat-transfer properties are difficult to test by this method. The joint must also be accessible from both sides. ... 

We can imagine a cholesteric as a smck of nematic quasi-layers of molecular thickness a with the director slightly turned by ( ) from one layer to the next one. In fact it is Oseen model [18]. Such a structure is, to some extent, similar to lamellar phase. Indeed, the quasi-nematic layers behave like smectic layers in formation of defects, in flow experiments, etc. Then, according to the Landau-Peierls theorem, the fluctuations of molecular positions in the direction of the helical axis blur the one-dimensional, long-range, positional (smectic A phase like) helical order but in reality the corresponding scale for this effect is astronomic. 

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. 

In addition to band-edge lasing, defect mode lasing is possible [97]. A defect in the photonic structure creates fine bands of allowed transmissions inside the photonic band gap. A defect may result from an isotropic or anisotropic layer between adjacent cholesteric layers, but also from phase shifts [106], particle stabilized defects [107], deformation of the cholesteric helix [108] or local polymerization [109]. The threshold of such defect modes are low and can be as low as a few nanojoules per pulse [110]. 

Zapotocky M, Ramos L, Poulin P, Lubensky TC, Weitz DA (1999) Particle-stabilized defect gel in cholesteric liquid crystals. Science 283 209... 

S. V. Shiyanovskii, 1.1. Smalyukh, and O. D. Lavrentovich, Computer simulations and fluorescence con-focal polarizing microscopy of structures in cholesteric liquid crystals, p. 229, in Defects in liquid crystals computer simulations, theory and experiments (Kluwer Academic Publishers, Netherland, 2001). 

Nematic and cholesteric liquid crystals can be used for the nondestructive study of electrical defects in transistors and integrated circuits [81, 82], for the detection defects in film capacitors prepared by vacuum deposition [83], for the visualization of electrically active defects or rapidly diffusing dopants, as well as for quality control at various stages of integrating circuits production [84-86]. The most suitable effect for this purpose would appear to be the B effect [85] and the fiexoelectric effect in spatially nonuniform field [84, 86], which permits the distribution of the electrical potential in operating the integrated circuits to be visualized. 

Schmidtke J, Stifle W (2003) Photonic defect modes in cholesteric liquid crystal films. Eur Phys J E Soft Matter 12 553-564... 

The point defect at a surface of an ordered medium can represent either the end of a line that is topologically stable in the bulk or a true surface point defect with no bulk singularity attached [61]. In cholesteric liquid crystals, all points with A = i 1 are the ends of bulk disclinations. Only when k = 2 An rotations of the director field), the point defect might be an isolated surface singularity. However, even in this case one should take care of the requirement of the layers equidistance. For example, the classical boojum configuration cannot be observed in a cholesteric vessel when 1. 

As already discussed, there are no isolated point defects in the cholesteric phase, 7r2(9 = SO 3)/D2) = 0. However, singular point defects can serve as the ends of linear solitons, as in the case of the monopole structure, in which the nonsingular disclinations can be considered as linear solitons. 

In the regime L/p 1, the elastic theory considers the cholesteric medium as a system of equidistant (and thus parallel) layers and that the curvatrue distortions are predominant, (5.8). The description of defects such as edge dislocations, oily streaks, and focal conic domains in cholesterics is often based on the results obtained for simpler layered medium, namely, the smectic A phase. 

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