Elastic deformation and disclination

We consider the possible deformations of a nematic liquid crystal confined between two parallel substrates with tangential anchoring condition (parallel to the substrates but no preferred direction on the plane of the substrate). We use the one elastic constant approximation (A n = K22 = Kss = K), the elastic energy is given by 

Because of the anchoring condition, the liquid crystal director is parallel to the plane of the substrate and depends on the coordinates x and y on the plane. The liquid crystal director is described by 

Using the Euler-Lagrange method (which will be presented in Chapter 5) to minimize the free energy we get 

When we switch from Cartesian coordinates to cylindrical coordinates, the above equation 

We consider the case where 0 only depends on the azimuthal angle p, but not the radius r. The solution to the above equation is 

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