Singularity tracking methods for a unified description of topology, geometry, and the dynamics of disclinations in 3D nematic liquid crystals

Disclinations are pervasive in both passive and active nematic liquid crystals. In the former, they can be used to align and propel colloidal particles, actuate surfaces, and transport biomaterials; while in the latter they spontaneously nucleate and recombine, facilitate density gradients, and promote extrusion of nematogens and layered growth of the material. In three dimensions, characterizing the structure and understanding the dynamic behavior of line defects poses theoretical challenges. We introduce a disclination density tensor, a function of the tensor order parameter, that characterizes disclinations in 3D nematics. The evolution of the disclination density tensor obeys conservation of topological charge and can be used to obtain the velocity of disclination lines in both passive and active nematics. We also report numerical results in 3D to validate the analytical predictions of disclination motion for a variety of configurations.