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

The twist-bend nematic phase (N_{TB}) is characterized by a conically twisting director and by a dramatic softening of the bend elastic constant before the transition to the N_{TB} phase. In the recently found splay nematic phase (N_{S}) 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 N_{TB} or N_{S} phase is obtained by a change of sign of one elastic parameter. Measured elastic constants show that the N-N_{TB} transition is mainly driven by the increase of the nematic order, while the N_{S} transition is due to flexoelectric coupling.