Disclinations splitting

The textures in homeotropic lamellar phases of lecithin are studied in lecithin-water phases by polarizing microscopy and in dried phases by electron microscopy. In the former, we observe the La phase (the chains are liquid, the polar heads disordered)—the texture displays classical FriedeVs oily streaks, which we interpret as clusters of parallel dislocations whose core is split in two disclinations of opposite sign, with a transversal instability of the confocal domain type. In the latter case, the nature of the lamellar phase is less understood. However, the elementary defects (negative staining) are quenched from the La phase they are dislocations or Grandjean terraces, where the same transversal instability can occur. We also observed dislocations with an extended core these defects seem typical of the phase in the electron microscope. 

The disclinations may split or annihilate due to interactions. The discli-nations of the same sign tend to repel each other while those of opposite signs attract each other and may eventually annihilate. 

For example, LC-organized split ring resonators is presented in Fig. 4.17 [113], where GNPs were trapped in disclination lines surroimding micron sized silica beads that were organized into an array by the nematic continuum phase. There were other developed interesting types of self-assembled nanostructures and it has been claimed that the next frontier toward commercialization was bringing the fabrication cost down with nanochemistry-based bottom-up self-assembly approaches [114]. 

Table 5.1. Multiplication rules of five classes of elements of the group Q that control the merger and splitting of disclinations in a cholesteric phase.

The multiplication rules (5.20) are specific of the classes of elements, rather than the elements themselves. The results are given in Table 5.1, that can be used to predict the result of merger or splitting of disclinations. 

The topological classification prohibits transformations of disclinations from one class to another. For example, x n X n + i), and x(n + ) cannot be continuously transformed one into another, despite the apparent similarity in the value of the strength. On the other hand, different lines can be transformed by splitting. For example, according to Table 5.1 and... 

An important property of x dislocations is their ability to split into combinations of A and t disclinations. Of course, these transformations must obey the multiplication rules (5.23). For example, a x line from the class can split into a pair of A and r lines (classes Cx and Q, respectively). An example is shown in Figure 5.11(b), (c) the core splits into a A and r+... 

Figure 5.11. Equivalence in the presentation of / lines (a) wedge /" disclination = screw dislocation (b) /-twist disclination = edge dislocation and (c) splitting of the core of a dislocation into a pair of disclinations.

Splitting of / disclination lines has been observed in the so-called Cano wedges one of the disclinations is always a X, i.e., it does not carry any material singularity. The first dislocations near the center of the wedge have a small Burgers vector b = p/2), while b increases for dislocations far from the center. 

When the cholesteric layers are tilted with respect to the bounding plate, the disclinations might occur to match the twisted structure in the bulk with an (usually unidirectional or conical) orientational field of anchoring forces [47]-[49]. Here again, the / dislocations split into X and r pairs. The X lines... 

Figure 5.21. (a) Splitting of the core of a large Burgers vector dislocation into two disclinations and (b) an oily streak with a semicircular end in a system of parallel and equidistant layers. 

Another type of interpretation of the textures was proposed by Nas-tishin et al., who suggested that the By phase is a smectic and columnar phase at the same time. The geometry of the helical filaments (ribbons) is that one of the central region of a screw disclination with a giant Burgers vector split into two disclination lines of strength V2/ which boimd the ribbon. 

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