Complementary tensorial measures of disclination line winding character

Whereas disclinations in two-dimensional nematics are point defects with quantized, half-integer winding numbers, the disclination lines of three-dimensional nematics are much more complicated: The nematic director winds about a rotation vector that can vary continuously in time and along the defect’s contour, resulting in continuously variable winding character (wedge, twist, and their intermediates). This poses challenges for characterizing disclination lines in nematic orientation fields from experiment or simulation. Recently, a second-rank tensorial calculation was proposed that allows the rotation vector to be determined from the orientation field in the vicinity of a disclination (Schimming and Viñals, Soft Matter 18: 2234 (2022); Schimming and Viñals, arXiv:2308.04496 (2023)). Here, we demonstrate that the rotation vector can also be calculated from an alternative measure, based on a second-rank pseudotensor defined on either the director field or the Q-tensor field. Furthermore, the two tensorial descriptions are complementary, in that each fails for a different type of disclination winding. We also explore the importance of saddle-splay distortions as a scalar measure of disclination winding character.