Fourier-space Monte Carlo simulations of two-dimensional nematic liquid crystals

Thermal fluctuations are ubiquitous in mesoscopic and microscopic systems. Take nematic liquid crystals (LCs) as an example; their director fluctuations can strongly scatter light and give rise to random motions and rotations of topological defects and solid inclusions. These stochastic processes contain important information about the material properties of the LC and dictate the transport of the immersed colloidal particles. However, modeling thermal fluctuations of the nematic field remains challenging. Here, we introduce a new Monte Carlo simulation method, namely the Fourier-space Monte Carlo (FSMC) method, which is based on the Oseen-Frank elastic distortion energy model. This method accurately models the thermal fluctuations of a nematic LC's director field. In contrast to the traditional real-space MC method, which perturbs the director locally, the FSMC method samples different eigenmodes of the director distortions in the Fourier space, aligning with the equipartition theorem. We apply FSMC to study defect fluctuations and trajectories in a two-dimensional nematic LC confined to various geometries. Our results show that FSMC can effectively sample degenerate defect configurations and reproduce long-range elastic interactions between defects. In addition, we conduct three-dimensional molecular dynamics simulations using a coarse-grained Gay-Berne potential, which corroborates the findings from FSMC. Taken together, we have developed a new Monte Carlo method to accurately model thermal fluctuations in nematic LCs, which can be useful for searching global free-energy minimum states in nematic, smectic, and other LC mesophases and can also be helpful in modeling the thermal motions of defects and inclusions in LCs.