Flashing a look at the stability of the uniform ferroelectric nematic phase

Abstract: Recent discovery of the ferroelectric nematic phase N_F resurrects a question about stability of the uniform N_F state with respect to the formation of either standard for solid ferroelectrics domain structure, or often occurring in liquid crystals space modulation of the polarization vector P (and naturally coupled to P nematic director. In this work within Landau mean-field theory we investigate the linear stability of the minimal model admitting the conventional paraelectric nematic N and N_F phases. Our minimal model, (besides the standard terms of the expansion over P and director gradients) includes, also standard for liquid crystals, director flexoelectric coupling term f and often overlooked in the literature (although similar by its symmetry to the director flexoelectric coupling) the flexo-dipolar coupling. We find that in the easy-plane anisotropy case the uniform N_F state loses its stability with respect to one-dimensional or two-dimensional modulation. For non-zero f the 2D modulation threshold is always higher than its 1D counterpart. No any instability at all if one neglects the dipole-flexoelectric coupling. In the easy-axis case the both instability thresholds are the same, and the instability can occur even without flexo-dipolr coupling. We speculate that the phases with 1D or 2D modulations can be identified with discussed in the literature [see M.P.Rosseto, J.V.Selinger, Physical Review E, volume 101, page 052707 (2020)] single splay or double splay nematics.