Out-of-equilibrium modeling of lyotropic liquid crystals: from binary simulations to multi-component theory

TL;DR

This paper develops a thermodynamically consistent GENERIC framework for lyotropic liquid crystals, coupling concentration, momentum, and order via energy and entropy functionals, and extending from binary to multicomponent mixtures. It constructs the Poisson and friction matrices to encode reversible and irreversible dynamics, yielding time-evolution equations that conserve energy and produce entropy. A Julia-based solver demonstrates binary-two-component behavior, reproducing topological defect cores and flow-driven droplet morphologies in agreement with experiments under Couette and Poiseuille-like flows. The work offers a versatile, open platform for simulating multi-component lyotropic LCs and can be extended to multi-interface systems, active materials, and external field applications, bridging theory, computation, and experiment.

Abstract

We present a thermodynamically consistent theoretical framework for lyotropic liquid crystals (LCs) based on the GENERIC (General Equation for the Non-Equilibrium Reversible-Irreversible Coupling) formalism. This formalism ensures conservation of energy and production of entropy, while coupling concentration, momentum balance, and liquid crystalline order. Starting from a binary nematic-isotropic mixture, we derive a theory for these key variables, which is then extended to multi-component systems. The binary equations are solved numerically using a Julia-based solver that relies on an upwind finite-difference scheme, enabling stable and efficient simulations capable of handling multiple time scales while satisfying fundamental mathematical constraints. The results of simulations are consistent with experimental observations of topological core defects in chromonic LCs, as well as flow-driven droplet shape transitions under Couette and Poiseuille flows. This work provides a platform for simulations of multi-component lyotropic LCs that can be extended to systems with multiple interfaces, active materials, and materials subject to external fields.

Figures (5)

• Figure 1: Plots of Eq. \ref{['phi_equilbrium']} (in black) and Eq. \ref{['S_equilbrium']} (in red) using parameters $T=1$, $C_1 = C_2 =1$, $\chi= 3$, $A_0=1$, and $U=20$. $x$-axis denotes the concentration range, while $y$-axis indicates the order parameter. One of the intersection points is indicated with a red dot, denoting $\phi_e$ and $q_e$
• Figure 2: Comparison of topological core defects in a nematic medium. (\ref{['Lav_exp']}) Experimental birefringence map of a DSCG mixture in a thin cell with tangential anchoring (scale bar: 20 $\mu$m), showing optical retardance (color bar) and director orientation (white lines); adapted from zhou2017fine. (\ref{['Lav_Final']}) Final-time-step simulation depicting the director field (white lines, scaled according to the concentration profile), the scalar order parameter (left), and the concentration distribution (right), with their corresponding color bars shown in (\ref{['Lav_Order_colorbar']}) and (\ref{['Lav_conc_colorbar']}).
• Figure 3: Comparison of an axial droplet under semi-Couette flow. Subfigures (\ref{['Couette_IC']})--(\ref{['Couette_Final']}) show simulation snapshots: in each panel, the left side displays the director field (gray lines, scaled by the order parameter) and the scalar order parameter, while the right side depicts the concentration field. (\ref{['Couette_IC']}) displays the initial conditions. Subfigures (\ref{['Couette_colorbar_order']}) and (\ref{['Couette_colorbar_concentration']}) provide the corresponding color bars, while (\ref{['Couette_velocity_profile']}) presents the schematic representation of the velocity profile imposed in the simulation. (\ref{['Couette_exp']}) compares the experimental evolution of a poly(isobutylene) (PIB, Parabol 1300) droplet deformation under shear flow; adapted from Vananroye2006.
• Figure 4: Comparison of an axial droplet under parabolic flow. Subfigures (\ref{['Parabolic_IC_Q']})--(\ref{['Parabolic_Final_phi']}) present simulation snapshots: (\ref{['Parabolic_IC_Q']})--(\ref{['Parabolic_Final_Q']}) depict the director field (gray lines, scaled by the order parameter) and the scalar order parameter, while (\ref{['Parabolic_IC_phi']})--(\ref{['Parabolic_Final_phi']}) show the corresponding concentration fields. Initial conditions are in (\ref{['Parabolic_IC_Q']}) and (\ref{['Parabolic_IC_phi']}), intermediate time steps in (\ref{['Parabolic_Snap1_Q']}), (\ref{['Parabolic_Snap2_Q']}), (\ref{['Parabolic_Snap1_phi']}), and (\ref{['Parabolic_Snap2_phi']}), and the final state in (\ref{['Parabolic_Final_Q']}) and (\ref{['Parabolic_Final_phi']}). Subfigures (\ref{['parabolic_color_order']}) and (\ref{['parabolic_color_phi']}) display the respective color bars. (\ref{['Parabolic_velocity_profile']}) shows the parabolic velocity profile imposed in the simulation. (\ref{['Parabolic_exp']}) presents an experimental still of a 5CB droplet ($\sim$20 $\mu$m) deformed in a pressure-driven microfluidic flow and observed under a polarized microscope; adapted from Tadej2019Sculpting.
• Figure F.1: Basic Integration Steps for the Advection–Diffusion–Relaxation System