Unwinding of the twist-bend nematic structure under an external field

The recently discovered twist-bend nematic phase, $\rm N_{tb}$, is a non-uniform equilibrium nematic phase that presents a spontaneous bend with a precession of the nematic director, $\textbf{n}$, on a conical helix with a tilt angle $\theta$ and helical pitch $P$. The stability of the $\rm N_{tb}$ 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 helical axis, $\bf t$. In the present communication, we investigate the influence of an external bulk field (magnetic or electric) on the pitch of the $\rm N_{tb}$ phase. For a material with positive magnetic susceptibility anisotropy, when the magnetic field $\bf{H}$ is parallel to the helical axis, $\bf{t}$, the field modifies the coupling parameter between the director and the helical axis, thus shifting the interval of values for which this coupling results in the $\rm N_{tb}$ phase. If $\bf{H}$ is perpendicular to $\bf{t}$, it is shown that the pitch of the nematic director $\bf n$ precession increases with $\bf{H}$, and at a critical value of $H_c$ 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.