Nanometric pitch in modulated structures of twist-bend nematic liquid crystals

The extended Frank elastic energy density is used to investigate the existence of a stable periodically modulated structure that appears as a ground state exhibiting a twist-bend molecular arrangement. For an unbounded sample, we show that the twist-bend nematic phase N-TB is characterized by a heliconical structure with a pitch in the nanometric range, in agreement with experimental results. For a sample of finite thickness, we show that the wavevector of the stable periodic structure depends not only on the elastic parameters, but also on the anchoring energy, easy axis direction, and the thickness of the sample.