🔬 condensed matter

Landau-de Gennes numerical simulation of nematic liquid crystals utilizing radial basis functions

This paper presents a novel Landau-de Gennes numerical simulation for nematic liquid crystals using radial basis functions, which offers enhanced geometric flexibility and computational efficiency over traditional grid methods while effectively capturing topological features and resolving singular defects through an adaptive node refinement scheme.

Original authors: Jin-Sheng Wu, Ivan I. Smalyukh

Published 2026-03-17

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CC BY 4.0

Original authors: Jin-Sheng Wu, Ivan I. Smalyukh

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

Imagine you are trying to predict how a crowd of people will stand in a room. If the room is a perfect, empty square, it's easy to guess where everyone will stand. But what if the room is filled with weirdly shaped furniture, like a giant donut, a pretzel, or a complex sculpture? Suddenly, the crowd has to twist and turn in complicated ways to avoid bumping into things.

This is exactly the challenge scientists face when studying liquid crystals (the stuff inside your LCD screens). These molecules want to line up in neat rows, but when they encounter complex shapes like tiny colloidal particles or droplets, they get confused, creating "traffic jams" called defects.

Here is a simple breakdown of what Jin-Sheng Wu and Ivan Smalyukh did in their new paper:

1. The Old Way: The "Grid" Problem

Traditionally, scientists used a cubic grid (like a 3D checkerboard) to simulate these molecules.

The Analogy: Imagine trying to map a winding river using only square tiles. You can't fit the curves perfectly. You have to use thousands of tiny squares to approximate the curve, or you end up with a jagged, blocky mess.
The Problem: If you want to see the fine details of a "traffic jam" (a defect) near a complex shape, you need to make the squares incredibly small. This requires a massive computer, takes forever to run, and wastes a lot of memory on empty space where nothing is happening.

2. The New Way: The "Radial Basis Function" (RBF) Magic

The authors introduced a new method using Radial Basis Functions (RBFs).

The Analogy: Instead of a rigid checkerboard, imagine a fishing net made of elastic rubber bands. You can place the knots (nodes) of the net exactly where you need them.
- Where the molecules are calm and straight? The knots are far apart (sparse).
- Where the molecules are twisting wildly around a defect? The knots bunch up tightly (dense).
The Benefit: This method is mesh-free. You aren't forced to use a grid. You can drop "nodes" anywhere in 3D space, even inside a pretzel-shaped particle, without worrying about the geometry.

3. The "Smart" Refinement: Adaptive Nodes

The paper introduces a clever trick called adaptive node refinement.

The Analogy: Think of a security camera system.
- In a quiet hallway, the camera only checks in once an hour.
- But if someone starts running or fighting (a defect), the camera instantly zooms in and starts recording every second.
How it works: The computer automatically detects where the liquid crystal molecules are getting stressed (high energy) and adds more calculation points there. It removes points where everything is calm. This saves huge amounts of computing power while giving a super-clear picture of the messy parts.

4. What They Tested

They didn't just do math; they tested this on some wild shapes:

Handlebodies: They simulated liquid crystals around particles shaped like pretzels or donuts with multiple holes (genus-5 shapes).
The Result: Their new method predicted exactly where the "traffic jams" (defects) would form. They even matched these predictions with real-life experiments using microscopes, proving their math works.

5. Why This Matters

Speed: Because they don't waste time calculating empty space, their simulations run about 50% faster than the old grid methods.
Flexibility: They can model shapes that were previously too hard or impossible to simulate accurately.
Future Tech: This helps scientists design better materials, like self-assembling nano-structures or more efficient liquid crystal displays, by understanding exactly how these molecules behave around complex obstacles.

In a nutshell:
The authors swapped a rigid, blocky grid for a flexible, smart net. This allows them to see the intricate dance of liquid crystal molecules around complex shapes with high precision, using less computer power and time. It's like upgrading from a pixelated, low-res map to a high-definition, zoomable satellite view.

1. Problem Statement

Traditional numerical simulations of nematic liquid crystals (LCs) using the Landau-de Gennes (LdG) theory typically rely on cubic grid-based finite difference methods. While effective for simple geometries, these methods face significant limitations when modeling complex systems:

Geometric Rigidity: Cubic grids struggle to conform to arbitrary or complex boundary shapes (e.g., colloidal particles with non-trivial topologies like handlebodies) without introducing artificial symmetries or requiring complex mesh generation.
Computational Inefficiency: Resolving fine-scale features, such as singular defects (disclinations), often requires uniform high-resolution grids across the entire domain, leading to excessive computational costs and memory usage.
Limited Application: While Radial Basis Functions (RBFs) have been successfully applied in fields like seismology and fluid dynamics, their application to tensorial LdG modeling of LCs (specifically for Q-tensor systems) had not been demonstrated prior to this work.

2. Methodology

The authors developed a mesh-free numerical framework based on the Radial Basis Function-generated Finite Difference (RBFFD) method to solve the LdG equations.

Governing Equations: The system minimizes the total free energy functional ( $F_{total}$ ), which includes:
- Bulk Free Energy: Landau-de Gennes expansion (thermotropic terms $A, B, C$ ) and elastic energy (one-constant approximation $L$ ).
- Surface Energy: Conically degenerate surface anchoring conditions at colloidal interfaces.
- Minimization: Equilibrium structures are found via gradient descent ( $\frac{dQ}{dt} \propto -\frac{\delta F}{\delta Q}$ ) using a combination of Adaptive Runge-Kutta (ARK23) and Fast Inertia Relaxation Engine (FIRE) algorithms.
RBFFD Implementation:
- Basis Functions: The Q-tensor components are approximated as a linear combination of radial basis functions (specifically Gaussian functions $\phi(r) = e^{-\epsilon|r-r_i|^2}$ ).
- Derivative Approximation: Finite difference coefficients ( $d_j$ ) are calculated by solving a linear system where the operator is applied to the basis functions. To ensure stability and high-order accuracy (4th to 8th order), the method incorporates supplementary polynomials up to the second degree.
- Mesh-Free Nature: Nodes are placed freely in space, unconstrained by grid geometry, allowing for arbitrary domain shapes.
Adaptive Node Refinement:
- A crucial innovation is an adaptive scheme where node density is dynamically adjusted based on the local elastic distortion.
- Nodes are treated as particles with "charges" proportional to the ideal separation distance ( $r_0 \propto (\partial_i \partial_j Q_{ab})^{-1/2}$ ).
- Electrostatic-like repulsive forces move nodes toward regions of high gradient (defect cores), achieving resolutions of 1.5–7 nm near defects while maintaining sparser spacing elsewhere.
Experimental Validation:
- The simulations were validated against experimental data using 3-photon excitation fluorescence polarizing microscopy (3PEF-PM) and conventional Polarizing Optical Microscopy (POM).
- Samples included LCs confined in handlebody-shaped colloids (genus 1 and 5) and droplets within complex polymer matrices.

3. Key Contributions

First Tensorial LdG-RBF Implementation: This work presents the first successful application of RBFFD to the tensorial Landau-de Gennes modeling of nematic LCs, moving beyond previous vectorial Frank-Oseen models.
Adaptive Mesh-Free Refinement: The introduction of an electrostatic-based adaptive node refinement scheme allows for the automatic resolution of singular defects without manual mesh generation or multigrid methods.
Geometric Flexibility: The method successfully handles complex topologies (e.g., genus-5 handlebodies) and arbitrary boundary conditions that are difficult to model with cubic grids.
Computational Efficiency: The approach demonstrates significant speed improvements and reduced memory overhead for specific geometries compared to traditional grid methods.

4. Key Results

Defect Resolution: The simulations accurately captured the topological constraints of complex geometries. For example:
- Genus-1 (Torus): Correctly predicted two $+1$ and two $-1$ surface defects (total winding number 0).
- Genus-5 Handlebody: Predicted a total winding number of $-8$ (3 positive and 11 negative defects), consistent with the generalized hairy ball theorem ( $\chi = 2-2g$ ).
- The adaptive scheme successfully resolved the splitting of integer defects into pairs of half-integer defects near the surface.
Elastic Multipoles: The method accurately reproduced known elastic multipole patterns around spherical colloids with different anchoring conditions:
- Homeotropic ( $\theta_e = 0^\circ$ ) & Planar ( $\theta_e = 90^\circ$ ): Quadrupolar distortions ( $l=2$ ).
- Conical ( $\theta_e = 45^\circ$ ): Hexadecapolar distortions ( $l=4$ ).
Benchmarking: Comparisons with regular grid simulations showed:
- Accuracy: Excellent agreement in director field distortions, defect positions, and multipole moments ( $b_l$ ).
- Efficiency: RBFFD required roughly 50% less computation time for the same number of iterations when using a fixed node set.
- Node Economy: RBFFD achieved the same geometric representation with significantly fewer nodes (e.g., 800k nodes vs. >1.7M nodes for a $120^3$ cubic grid) by eliminating inactive nodes inside solid particles.
Parallelization: The method is highly amenable to parallelization (GPU/CPU), as the calculation of differential coefficients for each node is independent.

5. Significance

This study establishes RBFFD as a powerful, versatile tool for simulating soft matter systems with complex geometries.

Scalability: By reducing the number of required nodes and eliminating the need for complex mesh generation, the method enables the simulation of larger LC systems (colloidal dispersions, emulsions) that were previously computationally prohibitive.
Physical Insight: The ability to resolve fine defect structures in arbitrary topologies allows for a deeper understanding of topological solitons, knotted structures, and defect interactions in confined LCs.
Future Potential: The framework opens the door for combining vectorial and tensorial approaches adaptively within the same simulation and for studying highly anisotropic geometries, potentially revolutionizing the design of LC-based metamaterials and photonic devices.

In conclusion, Wu and Smalyukh demonstrate that the RBFFD method offers a superior balance of geometric flexibility, computational efficiency, and high-order accuracy for Landau-de Gennes simulations, particularly for systems involving singular defects and complex boundaries.