Instability of toroidal nematics

Toroidal nematics are nematic liquid crystals confined within a circular torus and subject to planar degenerate anchoring on the boundary of the torus. They may be droplets floating in an isotropic environment or cavities carved out of a solid substrate. A universal solution of Frank's elastic free energy is an equilibrium configuration for the nematic director field, irrespective of the values of the elastic constants, whose vector lines are the coaxial parallels of the torus. We explore the local stability of this configuration and identify a range of parameters where the main drive towards instability does not come from the surface-like elastic constant $K_{24}$ being large, but from the the ratio $K_2/K_3$ of the twist to bend elastic constants being small, which also makes our study relevant to chromonic liquid crystals.