Modulated phases of nematic liquid crystals induced by tetrahedral order

Recent theoretical research has developed a general framework to understand director deformations and modulated phases in nematic liquid crystals. In this framework, there are four fundamental director deformation modes: twist, bend, splay, and a fourth mode Delta related to saddle-splay. The first three of these modes are known to induce modulated phases. Here, we consider modulated phases induced by the fourth mode. We develop a theory for tetrahedral order in liquid crystals, and show that it couples to the Delta mode of director deformation. Because of geometric frustration, the Delta mode cannot fill space by itself, but rather must be accompanied by twist or splay. Hence, it may induce a spontaneous cholesteric phase, with either handedness, or a splay nematic phase.