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 $\mathbf{P}$ (and naturally coupled to $\mathbf{P}$ nematic director $\mathbf{n}$). 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 $\mathbf{P}$ 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 ($\beta$). We find that in the easy-plane anisotropy case (when the configuration with $\mathbf{P}$ orthogonal to $\mathbf{n}$ is energetically favorable), the uniform $N_F$ state loses its stability with respect to one-dimensional (1D) or two-dimensional (2D) modulation. If $f\ne 0$, the 2D modulation threshold (${\beta }_{c2}$ value) is always higher than its 1D counterpart value ${\beta }_{c1}$. There is no instability at all if one neglects the flexodipolar coupling ($\beta =0$). In the easy-axis case (when $\mathbf{n}$ prefers to align along $\mathbf{P}$), both instability (1D and 2D) thresholds are the same, and the instability can occur even at $\beta =0$. 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)] for single-splay or double-splay nematics.

• Received 1 December 2020
• Accepted 5 January 2021

DOI:https://doi.org/10.1103/PhysRevE.103.012704