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Finite element analysis of a nematic liquid crystal Landau-de Gennes model with quartic elastic terms

This paper presents an energy-stable numerical scheme for a Landau-de Gennes model of nematic liquid crystals with quartic elastic terms, rigorously proving its convergence and $\Gamma$ -convergence while successfully simulating isotropic-to-nematic phase transitions.

Original authors: Jacob Elafandi, Franziska Weber

Published 2026-08-20

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CC BY 4.0

Original authors: Jacob Elafandi, Franziska Weber

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Between the fluid state of water and the rigid structure of a diamond, there exists a fascinating middle ground known as a liquid crystal. These materials flow like liquids but possess an internal order more like a solid. Imagine a crowd of people walking through a busy station; if they are all facing the same direction while moving, they exhibit a partial order similar to the nematic phase of liquid crystals. This unique state of matter is the foundation of the screens on our phones and televisions, yet the physics governing how these molecules arrange themselves, shift, and form patterns is incredibly complex. Scientists have long used mathematical models to predict how these materials behave, but the most common models often simplify the reality too much, missing the intricate details of how the molecules interact when they are under stress or changing phases.

Researchers Jacob Elafandi and Franziska Weber have tackled this complexity by developing a new, more detailed mathematical framework to describe these materials. They focused on a specific type of liquid crystal called a nematic, where the rod-like molecules tend to align in a general direction. While previous models could describe simple, uniform alignments, they struggled to capture the messy, dynamic reality of what happens when the material transitions from a disordered state to an ordered one, or when defects—places where the alignment breaks down—appear and move. The authors introduced a model that includes higher-order terms in the energy calculations, essentially adding more layers of detail to the physics equations. This allows the model to account for situations where the material constants, which dictate how stiff or flexible the molecular alignment is, vary significantly. By doing so, they created a tool that can simulate the formation of complex shapes and defects that simpler models miss.

To test their new model, the team did not just rely on theory; they built a powerful computer simulation to watch how these virtual liquid crystals evolve over time. They designed a numerical scheme, a step-by-step computational method, that guarantees the energy of the system decreases steadily, mimicking the natural tendency of physical systems to settle into a stable state. This stability is crucial because it ensures the computer simulations do not produce wild, unrealistic results as they run. The researchers proved mathematically that their method works correctly and that as they made the computer grid finer, the results would converge to the true physical solution. They then ran these simulations to watch the material change from a disordered, isotropic state into an ordered, nematic state, a process known as a phase transition.

The simulations revealed a rich variety of behaviors that depended heavily on the specific properties of the material being modeled. In one set of experiments, the researchers created a droplet of the ordered material, known as a tactoid, surrounded by a disordered fluid. When the material constants were set to values that favored one type of deformation over others, the droplet did not simply shrink in a perfect circle. Instead, it developed sharp corners and kinks, eventually splitting into smaller, rotating defects. These defects, which look like tiny vortices, drifted apart or annihilated each other in complex patterns. In contrast, when they used a simpler, older model that ignored these higher-order details, the droplet shrank in a perfectly round, symmetrical fashion before splitting, missing the jagged, irregular shapes that are actually observed in real-world experiments.

The study also explored what happens when a bubble of ordered material is trapped inside a disordered fluid. In these scenarios, the bubble expanded outward until it hit the container walls. Depending on the material properties, the bubble would either form new defects at its edges or absorb them, creating a dynamic interplay between the ordered and disordered regions. The researchers found that their new model could reproduce the specific, non-spherical shapes and the formation of these defects with a level of detail that the standard models could not. They observed that the inclusion of the extra terms in their equations allowed the simulation to capture the "kinks" and irregular boundaries that physical experiments have shown to exist, confirming that the added complexity was necessary to describe the true behavior of the material.

Ultimately, this work provides a more accurate lens through which to view the microscopic world of liquid crystals. By proving that their new, more complex model is mathematically sound and by demonstrating through simulations that it captures richer, more realistic dynamics, the authors have offered a better tool for understanding these materials. Their findings suggest that to truly understand how liquid crystals form defects and transition between states, one cannot rely on simplified approximations. The extra mathematical detail they introduced is essential for predicting the formation of the complex, non-spherical structures that define the behavior of these materials in the real world, paving the way for more precise control in future applications.

Technical Summary: Finite Element Analysis of a Nematic Liquid Crystal Landau-de Gennes Model with Quartic Elastic Terms

Problem Statement
The paper addresses the numerical analysis of a generalized Landau-de Gennes model for nematic liquid crystals that incorporates quartic elastic energy terms. While the standard Landau-de Gennes model utilizes quadratic elastic energy density, recent work by Golovaty et al. [15] proposed a higher-order generalization to better capture phase transitions and connect rigorously to the Oseen-Frank director field model, particularly in regimes with highly disparate elastic constants ( $K_1, K_2, K_3$ ). The authors investigate a specific energy functional $F(Q)$ that includes these quartic terms, specifically adding a term proportional to $|Q|^2|\nabla Q|^2$ (denoted with coefficient $L_5$ ) to the model proposed by Golovaty et al. The primary challenges addressed are the lack of existing provably stable numerical schemes for such models and the rigorous proof of convergence for discrete minimizers as the mesh size vanishes.

Methodology
The authors develop a fully discrete finite element scheme for the gradient flow of the energy functional $F(Q)$ .

Model Formulation: The energy $F(Q)$ consists of five elastic terms involving $div Q$ and $curl Q$ coupled with tensor coefficients $S_1 = \frac{s_0}{3}I + Q$ and $S_2 = \frac{2s_0}{3}I - Q$ , plus the bulk energy $W(Q)$ and the additional quartic term $L_5|Q|^2|\nabla Q|^2$ .
Numerical Scheme: A nonlinearly implicit finite element scheme (Scheme 3.1) is constructed using a semi-implicit time-stepping approach. The scheme is designed to satisfy a discrete energy dissipation law, ensuring unconditional stability independent of the mesh size $h$ and time step $\Delta t$ .
Existence and Uniqueness: The solvability of the nonlinear system at each time step is established using the Leray-Schauder fixed point theorem. Uniqueness is proven under a constraint on the time step size ( $\Delta t \leq Ch^{d+2}$ ).
Convergence Analysis: The paper rigorously proves the $\Gamma$ -convergence of the discrete energies $F_h$ to the continuous energy $F$ as $h \to 0$ . A critical component of this proof is the construction of a recovery sequence, which relies on the coercivity of the energy provided by the $L_5 > 0$ term. The authors utilize truncation operators and mollification to handle the potential unboundedness and lack of smoothness of the $Q$ -tensor field.

Key Contributions

Energy-Stable Scheme: The paper presents the first provably energy-stable and convergent numerical scheme for the gradient flow of a Landau-de Gennes model with quartic elastic terms.
$\Gamma$ -Convergence Proof: The authors provide a rigorous proof that global discrete minimizers converge to global minimizers of the continuous energy. They demonstrate that the inclusion of the $L_5$ term is mathematically necessary to construct the recovery sequence required for this proof, although numerical experiments suggest the dynamics remain similar when $L_5$ is small or zero.
Extension of Theory: The work extends the theoretical framework established by Golovaty et al. [15], showing that their results regarding the connection to the Oseen-Frank model and the recovery of uniaxial states in the limit of vanishing correlation length hold for this modified model with simple adaptations.

Numerical Results
The authors perform numerical experiments in two dimensions to validate the scheme and explore the physical implications of the quartic model:

Convergence Tests: The scheme exhibits second-order convergence with respect to both spatial element width and time step size.
Phase Transitions: Simulations of isotropic-to-nematic phase transitions successfully reproduce the formation of defects and tactoids (nematic droplets) with non-spherically symmetric boundaries, consistent with physical observations [14].
Elastic Constant Disparity: By varying the elastic constants (specifically $L_1$ ), the simulations capture different defect dynamics. For instance, in degree 1 and degree -1 tactoid simulations, the quartic model produces defect structures (such as V-shaped tails or specific vortex constellations) that differ significantly from the standard quadratic Landau-de Gennes model.
Comparison with Quadratic Models: The quartic model is shown to capture richer dynamics than the standard quadratic one-constant approximation. Specifically, the quartic model allows for the formation of tactoids with kinked boundaries, a feature observed in physical experiments but not captured by the standard quadratic model.

Significance and Claims
The paper claims that the proposed numerical method is robust and capable of simulating complex liquid crystal behaviors that are inaccessible to standard quadratic models. The authors emphasize that while the $L_5$ term is crucial for the mathematical proof of $\Gamma$ -convergence, the numerical results indicate that the model behaves similarly for small or zero $L_5$ , suggesting the physical relevance of the quartic terms derived from the Oseen-Frank energy. The work provides a foundation for the rigorous numerical analysis of higher-order Landau-de Gennes models, offering a tool to study materials with highly disparate elastic constants and complex defect dynamics. The authors note that the techniques developed for $\Gamma$ -convergence may be applicable to other related models with quartic energy terms.

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