Equivalent local force conditions minimizing the Frank free energy for topological defect equilibria in nematic liquid crystals

Recently, Miyoshi et al . (Miyoshi et al . 2024 Proc. R. Soc. A 480 , 20240405 (doi: 10.1098/rspa.2024.0405 )) derived analytical formulas for the Frank free energy associated with multiple topological defects in nematic liquid crystals confined to an arbitrary simply connected domain. The energy formulas, derived using an analogy with the so-called Kirchhoff–Routh path function in point vortex dynamics, required the evaluation of contour integrals involving analytical formulas associated with the crystal alignment field. Equilibria for topological defects were then obtained by finding local extrema of this Frank free energy. By contrast, in vortex dynamics, it is rare to find point vortex equilibria by minimizing the Kirchhoff–Routh path function that is a regularized kinetic energy; more commonly, a local ‘non-self-induction rule’ is used that is equivalent to each point vortex being free of net force. It is shown here that an analogous equivalent set of local force conditions can also be used to find equilibria for topological defects in nematic liquid crystals in confined domains, and without any need to evaluate the Frank free energy. This observation is significant theoretically and also because, at a technical level, the new local force conditions are easier to formulate and solve.