Boundary Line Disclinations

It is instructive to examine boundary disclination lines at a nematic surface or interface by introducing some additional modelling and approximations that incorporate surface tension and the effect due to gravity. Many of these aspects introduced here in this Section are common throughout the literature on liquid crystals and will also form a basis for the discussion on point defects at a free surface of nematic discussed in the next Section. The results presented below are based on those derived by de Gennes [107] and have been further elucidated in physical terms by de Gennes and Prost [110]. 

Consider a horizontal magnetic field H applied at the surface of a nematic liquid crystal when it is assumed that the director n exhibits conical anchoring and therefore makes a fixed angle with the tangent plane to the surface. It is assumed that initially the surface of the nematic is located in the plane z = 0 and that the sample occupies the semi-infinite region 2 0 with n parallel to H as 2 —00, the 

It should be noted here that, for example, 6 in Fig. 3.23(a) could be replaced by the condition 6 = + 6, which is also compatible with the same conical anchoring condition (see de Gennes and Prost [110, p.l31]). However, this means that for 0 C 1 we have 6 6 and so the energy, which must be related to the magnitude of the distortion could be greater for as a boundary condition rather than Hence = — -he is expected to be an energetically more favourable 

To first order in e this energy, for a finite domain in x, is 

Similar arguments apply for the case pictured in Fig. 3.23(b) for 6 = — e except 

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Figure 3.24 A periodic array of boundary line disclinations oriented perpendicular to the page, each separated by the distance L and located at the peaks and troughs of the nematic surface (in bold). The fixed conical anchoring angle is ip and c is the angle of the slope at a peak (or trough) measured relative to the x-axis. The director is parallel to the other curved lines.

To allow a comparison with the earlier work on boundary line disclinations, we can take the special case for = (corresponding to the director being fixed perpendicularly at the surface) and approximate the physical constants by (in cgs units)... 

The study of defects in liquid crystal systems is rooted in the understanding of defects in the solid state. For instance, crystals are rarely perfect and usually contain a variety of defects, e.g., point defects, line defects, or dislocations, and planar defects such as grain boundaries. In addition to these typical imperfections of the solid state, liquid crystals can also exhibit defects known as disclinations. These defects are not usually found in solids and result from the fact that mesophases have liquid-like structures that can give rise to continuous but sharp changes in the orientations of the molecules, i.e., sharp changes in orientation occur in the director field. 

The polarizing optical texture and the corresponding director structures in a capillary with homeotropic boundary conditions are illustrated in Figure 6.4. Note that the disclination line in the middle along the capillary axis is due to the escape to the third direction. The vertical lines indicate regions where the different escape directions meet. And the crosses are point defects. 

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