  Mesogenic dimers in the twist-bend nematic phase exhibit much higher
flexoelectric polarization than in their uniform nematic phase. In order to
theoretically investigate this data, we extend the symmetry based linear
elastic theory of the twist-bend nematic phase developed in Phys. Rev. E 92,
030501(R) 2015, by including flexoelectricity under the action of an external
electric field perpendicular to the helical axis. We show that at the nematic
towards twist-bend nematic transition, a new flexoelectric mode becomes active.
Consequently, the present model predicts the increase of the effective
flexoelectric coefficient when the system is entering the twist-bend nematic
phase. The influence of the flexoelectric coupling on the equilibrium
wavevector and the spontaneous heliconical tilt angle are investigated. The
electroclinic coefficient is calculated. Finally we argue that the helix could
be unwound giving rise to a splay-bend nematic phase.
