Pattern formation in nonlinear dynamics of nematic liquid crystals above the flexoelectric instability threshold

For many decades, researchers have been studying various types of electrohydrodynamic instabilities in liquid crystals. A significant amount of experimental data has been collected; however, the theoretical interpretations of the results typically rely on linear analysis. In response to this limitation, we investigate the nonlinear stage of the flexoelectric instability in nematics, focusing on liquid crystals with negative dielectric permittivity and electrical conductivity anisotropies. We base our analysis on a comprehensive set of nonlinear electrohydrodynamic equations for nematics subjected to an external alternating electric field. The equations predict an instability that is driven by the flexoelectric effect. To examine the specific features of this phenomenon, we use a model that was proposed in our previous publications {Pikina et al., “Nonlinear electrohydrodynamics of liquid crystals,” Zh. Eksp. Teor. Fiz. 164, 129 (2023) [JETP 137, 114 (2023)] and Pikina et al., “Dynamic exoelectric instabilities in nematic liquid crystals,” Phys. Rev. E 110, 024701 (2024)}, which allows us to perform numerical simulations of the nonlinear dynamics. We examine patterns that are formed above the instability threshold. Through numerical simulations, we identify static and dynamic patterns that occur on time scales much longer than the period of the external electric field. The static patterns are one-dimensional structures, whereas the dynamic patterns are standing or traveling one-dimensional waves. The type of realized pattern depends on the material and experimentally controlled parameters. We found that the standing waves are stable with respect to small transverse perturbations, whereas the propagating waves are unstable. We present a Ginzburg–Landau-like phenomenology that applies near the instability threshold. This approach allows us to rationalize our numerical findings with a few parameters.