The twist-bend nematic phase, $N_{\mathrm{TB}}$, may be viewed as a heliconical molecular arrangement in which the director $\mathbf{n}$ precesses uniformly about an extra director field, $\mathbf{t}$. It corresponds to a nematic ground state exhibiting nanoscale periodic modulation. To demonstrate the stability of this phase from the elastic point of view, a natural extension of the Frank elastic energy density is proposed. The elastic energy density is built in terms of the elements of symmetry of the new phase in which intervene the components of these director fields together with the usual Cartesian tensors. It is shown that the ground state corresponds to a deformed state for which $K_{22}>K_{33}$. In the framework of the model, the phase transition between the usual and the twist-bend nematic phase is of second order with a finite wave vector. The model does not require a negative $K_{33}$ in agreement with recent experimental data that yield $K_{33}>0$. A threshold is predicted for the molecular twist power below which no transition to a twist-bend nematic may occur.

Recent experiments have reported a novel splay nematic phase, which has alternating domains of positive and negative splay. To model this phase, previous studies have considered a one-dimensional (1D) splay modulation of the director field, accompanied by a 1D modulation of polar order. When the flexoelectric coupling between splay and polar order becomes sufficiently strong, the uniform nematic state becomes unstable to the formation of a modulated phase. Here we reexamine this theory in terms of a recent approach to liquid crystal elasticity, which shows that pure splay deformation is double splay rather than planar single splay. Following that reasoning, we propose a structure with a two-dimensional (2D) splay modulation of the director field, accompanied by a 2D modulation of polar order, and show that the 2D structure generally has a lower free energy than the 1D structure.

The recently discovered twist-bend nematic phase, N_tb, is a non-uniform equilibrium nematic phase that presents a spontaneous bend with a precession of the nematic director, n, on a conical helix with a tilt angle θ and helical pitch P. The stability of the N_tb phase has been recently demonstrated from the elastic point of view by extending the Frank elastic energy density of the nematic phase to include the symmetry element of the helical axis, t. In the present article, we investigate the influence of an external bulk field (magnetic or electric) on the N_tb phase. Using symmetry arguments we derive the expression for the flexoelectric polarisation in twist-bend nematic phases. We show that, besides the standard contribution related to the spatial variation of the nematic director, two new contributions connected with the existence of the helical axis appear. In the ground state, where the nematic deformation is a pure heliconical deformation, the new contribution vanishes identically, and the total flexoelectric polarisation is perpendicular to the nematic director. Furthermore, as an example, we study the role of an external magnetic field applied parallel to the helical axis for a material with positive magnetic susceptibility anisotropy. We show that the field modifies the range of values of the coupling parameter between the director and the helical axis, thus shifting the interval of values for which this coupling results in the N_tb phase. GRAPHICAL ABSTRACT

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 $\mathit{\Delta }$ 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 $\mathit{\Delta }$ mode of director deformation. Because of geometric frustration, the $\mathit{\Delta }$ 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.

From the elastic point of view, the twist-bend nematic phase, (N_TB), has a ground state characterised by a spontaneous bend with a precession of the director n along the direction of an axis t. The tilt angle θ formed between the director n and the axis t gives rise to a heliconical structure with a nanometric helical pitch p. Using an elastic continuum theory recently proposed as an extension of the Frank energy for these materials, we investigate the influence of an external magnetic field on the pitch of the N_TB phase. For a material with positive magnetic susceptibility anisotropy, when the magnetic field H is perpendicular to t, it is shown that the pitch of the nematic director n precession increases with H, and at a critical value of it becomes infinite as in a cholesteric liquid crystal, that is, one gets a uniform orientation of n. Our results include the behaviour of a cholesteric under field as a limiting case, and are obtained in the simple case where θ is assumed as uniform throughout the sample.