Nonlinear Methods for Shape Optimization Problems in Liquid Crystal Tactoids

TL;DR

This paper addresses predicting equilibrium tactoid shapes and director configurations in nematic liquid crystals by solving a nonlinear shape-order optimization problem within the Landau–de Gennes Q-tensor framework. It introduces an all-at-once finite-element formulation coupled with a quasi-Newton solver and nested iteration to efficiently minimize the free energy with respect to both the mesh coordinates ${\mathbf{X}}$ and the Q-tensor field ${\mathcal{Q}}$, subject to a volume constraint ${\mathcal{C}}({\mathbf{X}})=0$. The method is validated on two- and three-dimensional tactoids, including subproblems with fixed shape or fixed field, and demonstrates substantial reductions in iteration count and runtime while producing physically consistent defect structures and tangential anchoring. The results offer a robust computational tool to explore tactoid morphologies across parameter regimes and motivate extensions to multigrid, continuation strategies, and other LC phases.

Abstract

Anisotropic fluids, such as nematic liquid crystals, can form non-spherical equilibrium shapes known as tactoids. Predicting the shape of these structures as a function of material parameters is challenging and paradigmatic of a broader class of problems that combine shape and order. Here, we consider a discrete shape optimization approach with finite elements to find the configuration of two-dimensional and three-dimensional tactoids using the Landau--de Gennes framework and a Q-tensor representation. Efficient solution of the resulting constrained energy minimization problem is achieved using a quasi-Newton and nested iteration algorithm. Numerical validation is performed with benchmark solutions and compared against experimental data and earlier work. We explore physically motivated subproblems, whereby the shape and order are separately held fixed, respectively, to explore the role of both and examine material parameter dependence of the convergence. Nested iteration significantly improves both the computational cost and convergence of numerical solutions of these highly deformable materials.

Figures (10)

• Figure 1: Applying QN without NI for Subproblem A (Fixed Shape). Results shown on grid $M_9$. The color bar indicates the value of $S$, i.e., the order of the director field in the domain. Left plots, (a), (c), (e), depict the order's distribution, illustrating two defects for higher $\tau$. Right plots, (b), (d), (f), show the directors' anchoring to the boundary as $\tau \rightarrow 100$ and also demonstrate the disorder around the defects.
• Figure 2: Applying QN with NI for Subproblem A (Fixed Shape). Results shown on grid $M_9$. The color bar indicates the value of $S$, i.e., the order of the director field in the domain. Left plots, (a), (c), (e), depict the order's distribution illustrating two defects for higher $\tau$. Right plots, (b), (d), (f), show the directors' anchoring to the boundary as $\tau \rightarrow 100$ and also demonstrate the disorder around the defects.
• Figure 3: Applying QN with NI for Subproblem B (Fixed Field): Grids for each NI level for $\omega=0.01$ (top row) and $\omega=1$ (bottom row).
• Figure 4: Applying QN with NI for Subproblem B (Fixed Field). Results shown on grid $M_9$. The color bar indicates the value of $S$, i.e., the order of the director field in the domain. Left plots, (a), (c), (e), depict the order's distribution. Right plots, (b), (d), (f), show the fixed horizontally-aligned directors. Dramatic shape change is shown as $\omega \rightarrow 1$.
• Figure 5: Applying QN with NI for the full $({\bf{X}},Q)$$2D$ problem: Grids for each NI level for $\omega=0.01$ (top row) and $\omega=1$ (bottom row).

Theorems & Definitions (1)

• Remark