Escape of liquid crystal director in disclinations

For a disclination with the strength S, as one approaches the center of the disclination, the elastic energy diverges, as shown by Equation (1.124). In reality this will not occur. The liquid crystal will transform either into isotropic phase at the center of the disclination or a different deformation where there is no singularity. Here we only discuss the cases of cylindrical confinements (two-dimensional confinement) where it is possible to obtain analytical solutions. The mechanism of liquid crystal director escape in spherical confinement (three-dimensional confinement) is similar to that of two-dimensional. 

The total free energy per unit length of the configuration with the isotropic core is 

The total free energy increases with the radius of the cylinder. Note that we neglect the surface energy of the interface between the nematic liquid crystal and the isotropic core. 

Using Euler-Lagrange method to minimize the elastic energy, we obtain 

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