Defects, Dislocations and Disclinations

To understand the relationship to the director it is helpful to consider a planar sample in which the director orientation is parallel to the glass surfaces and not a function of the thickness. This assumption is of course not valid close to the singularities. We will firstly consider nematic liquid crystals with a curved structure, splay, twist and bend, as indicated below  

The elastic free energy per unit volume is yc d4 /dx), where f = nx is the tilt of the director and the splay constant. The free energy density may be written for splay as = k dnx/dxf, for twist F = niduxldyf and for bend as = k-i-i dnx/dzy. In the more general form it may be written 

In the case of the disclinations, only two elastic constants are involved, kxx and 22, that to a first approximation are taken as being equal to k. If we consider cylindrical coordinates (r, a) and now seek solutions for eqn (3.30) in which the director orientation [J/ is independent of r, then the elastic free energy is 

Minimization of eqn (3.31) gives J/ a constant which describes the uniformly orientated nematic sample, or iJ/ = ia+c, where c is a constant. In the nematic case, the orientational order is taken to be polar, and hence S= 1/2, 1, 3/ 2. with 0 C 7t. The angle between two successive dark brushes is therefore Aa = Ail//S = n/2S, and thus the number of dark brushes per singularity is 2n/Aa = 4 S. A few examples of the types of molecular orientation in 

The curves represent the projection of the director field in the xy plane. For S I, a change in C merely causes a rotation of the figure by C/(l — S), while for 5 = 1 the pattern itself is changed. The disclinations are characterized by their strengths , which are defined as the number of multiples of 2n that the director rotates in a complete circuit around the diselination core. A value of+1 indieates that the director is rotated through 2n. The micrograph of a liquid crystal material, shown below, illustrates the various disclinations described above. 

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At the late stage of lamella orientation, classical topological defects (dislocations and disclinations) dominate [40, 41] (Fig. 8h and Fig. 9), and their movement and annihilation can be followed in Fig. 8h-i and Fig. 9. The latter presents an example of the apparent topological defect interactions and their transformations. Displayed are two dislocations of PMMA, which have an attractive interaction due to their opposite core sign. Therefore, in the next annealing step the dislocation is shifted... 

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