Interpretation of saddle-splay and the Oseen-Frank free energy

In this talk, we re-examine the oldest question in liquid-crystal physics: What are the elastic modes of a nematic liquid crystal? For this analysis, we use a recent mathematical construction [1], which breaks the director gradient tensor into four distinct types of mathematical objects. Based on this construction, we suggest an interpretation of saddle-splay as bulk rather than surface elasticity. This interpretation leads to an alternative way to think about several previous results in liquid-crystal physics, including: (1) free energy balance between cholesteric and blue phases, (2) director deformations in hybrid-aligned-nematic cells, (3) spontaneous twist of achiral liquid crystals confined in a torus or a cylinder, and (4) curvature of smectic layers.
[1] T. Machon and G. P. Alexander, Phys. Rev. X 6, 011033 (2016).