Defects in Nematics and Cholesterics

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. 

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. 

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... 

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 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]) ... 

The characteristic feature by which cholesterics differ from the nematics is the spontaneous formation of twisted structures, reflecting the existence of a preferred screw sense. For this reason, the defect lines no longer merge and cancel each other as in nematics instead, complicated stable networks of disclination lines may form. The "streaks" in planar cholesteric films that often form a crackle consist of bimdles of thin individual lines (Figure 6.18). A single line itself may show a number of complicated features. ... 

The polarizing light microscopy is the simplest method available to identify LC phases. This optical method has been used since the discovery of liquid crystals and has led to nematic, cholesteric and smectic classifications. The appearance of a specific texture of the melt is usually a function of the types of LC phase, and it is often possible to directly identify the type of LC phase present in a polymer melt by this optical method. The textures of various LC phases are caused by the existence of different types of defect present in the LC phases. It should be noted that microscopic observations are sometimes misleading because the development of specific textures in an LC phase can occur with great difficulty. This problem arises owing to their multiphase nature (the coexistence of polycrystalline and amorphous phases), polydispersity and/or higher viscosities of LCPs melts compared with those of LMLCs. In most cases, LCPs must be annealed for hours or days at suitable temperatures to develop specific textures. 

When L/p I, the cholesteric does not differ much from the nematic phase. No wonder therefore that optical observations for weakly twisted cholesterics reveal thick (nonsingular) and thin (singular) line defects —disclinations similar to that in the nematic phase. Moreover, in droplets of the so-called compensated cholesteric mixtures with extremely small Ljp one can observe point defects [6] which, from the topological point of view, are allowed only in a nematic phase. 

The example above shows that the isolated point defects are not hkely to occur in the bulk of cholesteric phase when L/p 1. This is indeed a general statement, valid for any ordered medium, such as superfluid 3He-A, smectic C, or biaxial nematic with a trihedron of vectors as the order parameter the second homotopy group for the OP space of these media is trivial. However, point defects at the boundary of the cholesteric volumes and all the media listed above are formally allowed by the homotopy theory. 

This range has been called the "distal" region. Although the layer compression modulus is about two orders of magnitude larger than in tire short-pitch cholesterics (see (4.31)), permeation-type flow similar to that observed in cholesterics was observed near the nematic phase, ° shovdng that the apparent viscosity maybe smaller than in the cholesteric, probably due to defects that cause plastic behavior. This and measurements imder periodic deformations indicate the importance of layer defects, which are hard to regulate. 

A useful structural concept introduced by Kleman and FriedeF postulates a quasi-layered structure and explicitly takes into account the natural twist of the system. Concerning defects, we may think of cholesteric liquid crystals as a smectic with an in-plane nematic behavior, similar to the smectic C phase. Instead of using tire concept of a layered structure to account for the twist, we may also consider tire field of twist axis t in addition to the director field n. The two concepts are essentially equivalent, with the twist field being identical with the layer normal. The twist field accordingly suffices the condition t curti = 0, which means that in this twist field no fwist deformation is allowed. The concept of "layers" or twist-field is an approximation, which may not be valid in the core of the defects. We assume that the core structures of cholesterics (especially those with weak chirality) are similar to that of nematics. 

---