Frank Elastic Energy of A Flattened Nematic Droplet With Boojums in Bipolar Coordinates

The article shows an analytical calculation of the Frank–Oseen elastic energy for a flattened droplet of nematic liquid crystal containing two-point defects (boojums) at its poles. To solve the problem, a modified coordinate system structurally similar to bipolar coordinates was proposed and employed, adapted to the geometry of a droplet with two distinct poles. It is shown that, under the assumption of a tangential director distribution, the deformation energy within the droplet reduces to an integral of the squared divergence of the director field, while the contribution from the director’s curl is zero. By evaluating this integral, an exact expression for the elastic energy is obtained, proportional to the characteristic size of the droplet and a parameter defining its flattening. The result has demonstrated the effectiveness of using curvilinear coordinates for calculating the energetics of topological defects in confined liquid-crystalline systems.