We develop a Landau theory for bend flexoelectricity in liquid crystals of bent-core molecules. In the nematic phase of the model, the bend flexoelectric coefficient increases as we reduce the temperature toward the nematic to polar phase transition. At this critical point, there is a second-order transition from high-temperature uniform nematic phase to low-temperature nonuniform polar phase composed of twist-bend or splay-bend deformations. To test the predictions of Landau theory, we perform Monte Carlo simulations to find the director and polarization configurations as functions of temperature, applied electric field, and interaction parameters.

We report a dynamic light-scattering study of the fluctuation modes in a thermotropic liquid crystalline mixture of monomer and dimer compounds that exhibits the twist-bend nematic (N TB ) phase. The results reveal a spectrum of overdamped fluctuations that includes two nonhydrodynamic modes and one hydrodynamic mode in the N TB phase, and a single nonhydrodynamic mode plus two hydrodynamic modes (the usual nematic optic axis or director fluctuations) in the higher temperature, uniaxial nematic phase. The properties of these fluctuations and the conditions for their observation are comprehensively explained by a Landau-de Gennes expansion of the free-energy density in terms of heliconical director and helical polarization fields that characterize the N TB structure, with the latter serving as the primary order parameter. A "coarse-graining" approximation simplifies the theoretical analysis and enables us to demonstrate quantitative agreement between the calculated and experimentally determined temperature dependence of the mode relaxation rates.

In liquid crystals, if flexoelectric couplings between polar order and director gradients are strong enough, the uniform nematic phase can become unstable to the formation of a modulated polar phase. Previous theories have predicted two types of modulation: twist bend and splay bend; the twist-bend phase has been found in recent experiments. Here, we investigate other types of modulation, using lattice simulations and Landau theory. In addition to twist bend and splay bend, we also find polar blue phases, with 2D or 3D modulations of both the director and the polar order. We compare polar blue phases with chiral blue phases, and discuss opportunities for observing them experimentally.