Stability of the uniform ferroelectric nematic phase

The recent discovery of the ferroelectric nematic phase N_{F} resurrects a question about the stability of the uniform N_{F} state with respect to the formation of either a standard for the solid ferroelectric domain structure or for the often occurring liquid crystal space modulation of the polarization vector P (and naturally coupled to P nematic director n). 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 (in addition to the standard terms of the expansion over the P and director gradients) includes the standard for liquid crystals, the director flexoelectric coupling term (f), and, often overlooked in the literature (although similar by its symmetry to the director flexoelectric coupling), the flexodipolar coupling (β). We find that in the easy-plane anisotropy case (when the configuration with P orthogonal to n is energetically favorable), the uniform N_{F} state loses its stability with respect to one-dimensional (1D) or two-dimensional (2D) modulation. If f≠0, the 2D modulation threshold (β_{c2} value) is always higher than its 1D counterpart value β_{c1}. There is no instability at all if one neglects the flexodipolar coupling (β=0). In the easy-axis case (when n prefers to align along P), both instability (1D and 2D) thresholds are the same, and the instability can occur even at β=0. We speculate that the phases with 1D or 2D modulations can be identified as discussed in the literature [see M. P. Rosseto and J. V. Selinger, Phys. Rev. E 101, 052707 (2020)2470-004510.1103/PhysRevE.101.052707] for single-splay or double-splay nematics.