Q-tensor model of twist-bend and splay nematic phases

The twist-bend nematic phase (NTB) is characterized by a conically twisting director and by a dramatic softening of the bend elastic constant before the transition to the NTB phase. In the recently found splay nematic phase (NS) the splay elastic constant tends to zero, resulting in a splay modulation perpendicular to the director. We model both phases with a single Q-tensor free energy including a term that breaks the degeneracy between the splay and bend elastic constant, and a flexoelectric term coupling the divergence of the Q-tensor with polarization. The NTB or NS phase is obtained by a change of sign of one elastic parameter. Measured elastic constants show that the N-NTB transition is mainly driven by the increase of the nematic order, while the NS transition is due to flexoelectric coupling.