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

Abstract

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 Öttinger. 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.

Figures (11)

• Figure 1: Schematic of the simulation model. Small black circles with an arrow represent N-MPCD particles/nematogens. They are confined in the light-blue area whereas the extended region contains virtual particles (small empty circles with an arrow). The virtual particles are used to enforce tangent boundary conditions and avoid collision in partially empty cells.
• Figure 2: Comparison between the second and fourth scalar order parameters, computed from Eqs.(\ref{['scalar_order_003a']}) and (\ref{['scalar_order_003b']}), and the closure approximation from Eq. (\ref{['scalar_order_005']}). The closure expression accurately captures the behavior of $S_{4}(U)$ near small values of $U-U_{\text{c}}$, while exhibiting a slight overestimation for large values of $U$.
• Figure 3: The emergence of nematic order near $U=U_c$ with different mean-field potential models, i.e., Maier–Saupe (MS), Marrucci–Greco (MG), and Ilg–Karlin–Öttinger (IKÖ) mean-field potentials. The scalar order parameter, $S(U)$, is obtained from N-MPCD simulations (symbols) and analytical predictions based on \ref{['scalar_order_006']} and \ref{['scalar_order_007']}-\ref{['scalar_order_008']} (continuous curves). The agreement across all models confirms the validity of the closure scheme and its ability to reproduce phase transitions and ordered profiles.
• Figure 4: Normalized length scale $\lambda = l/\xi_{\text{N}}$ estimated from the closure-based derivation of LdG and elastic coefficients, as a function of system size $l$ and interaction strength $U$, in simulation units. The nematic coherence length $\xi_{\text{N}}$ is computed using Eqs.(\ref{['ldg_coefficients_013']}) and (\ref{['elastic_coeff_001']}) under the equal elastic constant approximation for the LdG elastic energy density, assuming equal rotational and translational diffusion coefficients for simplicity. The closure function $S_{4}(S) =S\left[1-\left(1-S\right)^{\nu}\right]$ is taken with $\nu=1$, and stationary values of the order parameter are used to ensure consistency with equilibrium conditions.
• Figure 5: Average equilibrium configurations in square domains of different sizes for N-MPCD fluids, simulated through MS interactions of different strength, $U$. The color bar defines $S^{\text{c}}$, as dictated by the bar at the top. The small interior lines label the local director, $\hat{\mathbf{n}}^{\text{c}}$. Relative system sizes are not realistic but chosen to ease readability and interpretability of the figures.