Disclinations without gradients: A nonlocal model for topological defects in liquid crystals

Nematic liquid crystals composed of rod-like molecules have an orientational elasticity that accounts for the energetics of the molecular orientation. This elasticity can be described by a unit vector field; the unit vector constraint interacts with even fairly simple boundary conditions to cause disclination defects. Disclinations are entirely a topological consequence of the kinematic constraint, and occur irrespective of the particular energetic model. Because disclinations are topological defects, they cannot be regularized by adding higher gradients, as in phase-field models of interface defects. On the contrary, the higher gradient terms would cause even greater singularities in the energy. In this paper, we formulate an integral-based nonlocal regularized energy for nematic liquid crystals. Our model penalizes disclination cores and thereby enforces a finite width, while the integral regularization ensures that the defect core energy is bounded and finite. The regularization at the same time tends to the standard gradient-based energies away from the disclination, as well as building in the head-tail symmetry. We characterize the formulation in its ability to describe disclinations of various strengths, and then apply it to examine: (1) the stability and decomposition of various disclinations, and the competition between bend and splay energies in determining the relative stability of integer and half-integer disclinations (2) the coalescence of a +12 and−12 disclination pair; we find the disclinations do not move at the same velocities towards each other, suggesting that the asymmetry of the director field plays a dominant role despite the equal-and-opposite topological strengths of the disclinations.