Ordering in Confined Two-Dimensional Nematic Systems: Mesoscopic Simulations Based on Different Mean-Field Potentials

We use nematic Multi-particle Collision Dynamics (N-MPCD) simulations to study confined nematic liquid crystals in square domains, with three distinct mean-field potentials: the classical Maier-Saupe and Marrucci-Greco models, and a more recent model due to Ilg, Karlin, and \"Ottinger. These potentials incorporate diverse physical features, including spatial gradients and nonlinear dependencies on the order parameter, to describe nematic ordering at mesoscopic scales. We derive coarse-grained equations from a Fokker-Planck description with tensorial closures, and analyse the emergence of order as a function of interaction strength, $U$, in two dimensions. The critical interaction strength depends on the choice of the mean-field potential. We also analytically estimate the nematic coherence length in three dimensions, to establish a rigorous correspondence between the N-MPCD parameters (the system size $R$ and $U$) and the continuum Landau-de Gennes theoretical parameters. We systematically study equilibrium and metastable configurations, including relaxation pathways to stable equilibria, on square domains, for all three mean-field potentials. Our results confirm universal equilibrium and metastable configurations for all three mean-field potentials. Our results also suggest that the N-MPCD predictions are consistent with the continuum Landau-de Gennes predictions, regardless of the choice of the underlying mean-field potential and approximations, for large $R$ and $U$. There are differences for small $R$ and for $U$ near the critical interaction strength, that need to be further explored and quantified for new-age multiscale and multiphysics theories.