A static elastic theory for usual, chiral, and twist-bend nematic liquid crystals orderings

To describe the elastic properties of the twist-bend nematic ($\rm N_{\rm TB}$) phase, a continuum description is proposed to tackle the orientational properties of the nematic, cholesteric, and twist-bend nematic phase. The elastic energy density is an extension of the usual Frank elastic energy density, by including an extra element of symmetry represented by the axis $ \textbf {t}$, allowing periodic distortions. This general energy density indicates the stability of at least three phases allowed by the elements of symmetry and can be faced as a framework to study the static distortion in the nematic ordering. The study of order transitions reveals a periodically modulated structure which appears as a ground state, exhibiting a twist-bend molecular organization. Similar arguments demonstrate that the nematic twist-bend $\rm N_{TB}$ phase is indeed a heliconical structure. It is possible to show analytically that the pitch of this structure is in the nanometric range, in agreement with experimental observations.