Axial Line Disclinations

In order to simplify the presentation the one-constant approximation will be assumed for the nematic energy density given by equation (2.67). To analyse axial disclinations we consider configurations in which the director referred to Cartesian axes takes the form 

It will be instructive to solve this equation in the cylindrical coordinates introduced in Fig. 3.19 by setting 

The equilibrium equation (3.336) then becomes Laplace s equation in cylindrical coordinates, which is, in the obvious notation (cf. equation (C.7) in Appendix C), 

For axial disclinations, 6 is expected to be independent of r and so in this case equation (3.338) collapses to 

The integer n is called the Prank index of the disclination. It is worth noting that in some articles, for example Bouligand [22] or Chandrasekhar [38], the order or strength of a disclination has been defined by s =.  

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Figure 3.20 Examples of flux lines which are tangential to the orientation of the director n around an axial line disclination located perpendicular to the page and passing through the point indicated by a black dot. Various cases of Rrank index n are given together with the associated solution B provided by equation (3.341). The strength of such disclinations is often defined by s = f. The bold lines represent the singular radial lines obtained from equation (3.347). The constant 0o has been set to zero except for the examples of Prank index n = 2.

When a nematic-smectic A transition occurs in a capillary tube (Fig. 8 a), smectic layers nucleate at disclination points and a singular line forms along the axis, with cylindrical layers (as shown in Fig. 8 b), but the presence of beads along the axial defect shows that the situation is less schematic [42]. 

We axe now in a position to picture the flux lines following the orientation of n around an axial disclination for various values of the FVank index and constant o-To find the flux lines in particular cases we solve the differential equation (3.342) after substituting for 0 in the solution (3.341) for fixed values of the arbitrary constant < o- In all cases, except the solution forn = 2 in equation (3.348), changing the constant o merely rotates the flux lines shown in Fig. 3.20 below, and so we set ( 0 = 0 except for two of the instances when n = 2. It suffices to demonstrate the technique for a few cases the others are in a similar style. For n = —2 and 00 = 0, equation (3.342) can be solved to find that the flux lines away from the disclination are given by... 

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