Layered Liquids: A complex tensor theory of simple smectics*

Liquid crystals are ubiquitous in nature, often appearing in biological and industrial systems. Proper hydrodynamic study and simulation of these materials requires an appropriate order parameter capable of describing their complex internal structures. The strength of the Q-tensor for nematic phases arises from the tensor’s ability to simultaneously describe both the local degree of nematic order and orientation. Smectic phases, however, break both translational and rotational symmetry so require a more composite order parameter.

In this work, we take inspiration from the nematic tensor order parameter and derive a novel mathematical description for confined smectic liquid crystals, characterised by a smectic complex tensor order parameter, E. We argue that our theory amounts to a considerable computational simplification to alternative descriptions while bringing the strengths of Q-tensor theory. We express the free energy density in terms of this new tensor and demonstrate how it can be used to describe smectic defects in a variety of confinements through numerical minimisation.