Linear Singularities in Nematics

Consider a disclination with its ends fixed at the opposite plates of a planar nematic cell. Such a disclination connects the two glass plates as in Fig. 8.13a. If we are looking at it from the top along the z-direction we can see the director distribution n (x, y) in the Ay-plane around the disclination. In a polarization microscope, in the same cell, we can see different n(jc, y) patterns corresponding to disclinations shown in Fig. 8.14. A point in the middle of each sketch shows the disclination under discussion that has its own strength s. 

The strength of a disclination is defined as follows. We traverse the disclination line along the closed contour counterclockwise as shown in sketch (b) and count the angle A( ) the director acquires as a result of the traverse. It is evident that after the full turn A( ) = mn where m = 0,1,2... and, by convention, the strength s = ml2. In fact, we deal with a solution of the Laplace equation, see the next paragraph. Let us count A( ) from the horizontal axis in Fig. 8.14. Then, upon the traverse in the counter-clockwise direction, for disclinations of strength s = 1/2 and s = 1, the 

The problem is to find the distribution of the director around a disclination [14]. To solve it we can use the elasticity theory discussed in Section 8.3. Let a liquid crystal layer is situated in the y plane of drawing, and singularity L is parallel to the 

15 Geometry for calcnlatimi of the director distribution around a disclination L. P is the azimuthal angle for an arbitrary point r in the x, y plane of the nematic layer (p is the director angle in point r 

In the one-constant approximation, the distortion free energy per unit volume is 

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