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Spectral Alignment of Mesogens for Three-Dimensional Disclination Localization in Liquid Crystals

This paper introduces a deterministic framework combining Spectral Alignment of Mesogens (SPAM) and Discrete Layer Analysis (DLA) to robustly extract three-dimensional disclination centerlines from unoriented mesogen-scale director fields, overcoming the limitations of traditional thresholding methods in identifying topological defects across various liquid-crystalline media.

Original authors: Saptarshi Saha, Amit Acharya, Gerald J. Wang

Published 2026-09-09

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CC BY 4.0

Original authors: Saptarshi Saha, Amit Acharya, Gerald J. Wang

Original paper licensed under CC BY 4.0 (https://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

Liquid crystals are the materials that make our screens glow, but they are also a unique state of matter found in nature, sitting somewhere between a solid and a liquid. In these materials, tiny rod-shaped molecules, called mesogens, try to line up in the same direction, creating a sense of order. However, this order is not always perfect. Sometimes, the molecules cannot align smoothly everywhere, and they are forced to twist or turn in ways that create unavoidable knots and breaks in the pattern. These breaks are known as topological defects. In three-dimensional liquid crystals, the most important of these defects are thin, line-like structures called disclinations. Understanding exactly where these lines are, how long they are, and how they connect is crucial for predicting how liquid crystal materials will behave, whether they are being used in a display, a soft robot, or a new type of engine.

The challenge for scientists has always been how to find these invisible lines inside a chaotic jumble of molecules. Traditional methods generally fall into two categories. One approach involves using the "SOP-leveraged" method, which looks for areas where the molecules are messy or disordered. These methods measure how much the local alignment has broken down and draw a boundary around that messy zone. The problem is that this approach identifies a fuzzy, three-dimensional blob of disorder rather than the single, one-dimensional line at its center. It is like trying to find the exact path of a river by only looking at the width of the floodplain; the floodplain tells you the river is there, but it does not tell you the precise course of the water. When multiple defects are close together, these fuzzy blobs can merge, making it impossible to tell where one line ends and another begins. Another approach, such as the authors' earlier work known as TADA, focuses on assigning directions to molecules in two-dimensional assemblies, but extending this to complex three-dimensional structures has remained a significant hurdle.

To solve this, researchers at Carnegie Mellon University, including Saptarshi Saha, Amit Acharya, and Gerald J. Wang, developed a new way of thinking about the problem. Instead of looking for disorder, they looked for a specific kind of consistency. In liquid crystals, the direction a molecule points is physically the same as the direction it points if you flip it around; a rod pointing north is identical to a rod pointing south. This creates a mathematical puzzle: can you assign a consistent "up" or "down" direction to every single molecule in a sample so that neighbors agree with each other? In a perfect material, you can. But around a disclination line, it is impossible to make everyone agree without creating a sudden break in the pattern. The researchers realized that these unavoidable breaks form thin, sheet-like surfaces that end exactly at the disclination lines. The core novelty of their work is the formulation of this problem as a global optimization task: finding the best way to assign directions to every molecule so that the fewest number of neighbors disagree. To solve this, they utilized an algorithm called Spectral Alignment of Mesogens (SPAM), which provides a deterministic approximation of the global extremums for this type of binary constrained optimization problem.

Once the directions are assigned, the computer identifies the pairs of neighbors that are forced to disagree. These pairs form a cloud of points that maps out the invisible sheet where the alignment breaks. To turn this cloud of points into a usable line, the researchers used a second step called Discrete Layer Analysis, or DLA. This process identifies the spatial boundaries of the sheet using a two-step criterion. First, it analyzes the density and flatness of the points to identify the "table-like" middle of the sheet. Second, it looks for the specific change in shape and sparsity that occurs at the very edge of the sheet, where the line defect lives. The algorithm detects these edges and connects them together to reconstruct the exact path of the disclination line. This allows them to measure the length of the line and see how different lines connect, something that was difficult with the old methods.

The researchers tested this new approach using computer simulations of liquid crystals. They created samples with known defects, including straight lines running through the material and closed loops that circle back on themselves. In every case, the new method successfully traced the exact path of the defects. They also tested it on a more difficult scenario where a cylinder was placed inside the liquid crystal, forcing the molecules to wrap around it and creating a complex web of multiple defect lines. The old methods struggled here, often producing tangled, folded sheets that were hard to untangle. The new method produced clean, smooth sheets that made it easy to see exactly where each line was and how long it was.

One of the most revealing tests involved removing the molecules right at the center of the defect, simulating a situation where the core of the defect is hidden or missing from the data. The traditional method, which relies on seeing the messy core, failed completely when the center was gone. The new method, however, still found the line. This is because it relies on the global pattern of the molecules surrounding the defect, not just the mess at the center. As long as the molecules around the hole cannot be aligned consistently, the method knows a line must be there.

This work provides a clearer, more precise way to map the hidden architecture of liquid crystals. By shifting the focus from measuring disorder to finding the boundaries of consistency, the researchers have given scientists a tool to see the skeleton of these materials rather than just their skin. This distinction is vital for building better models of how these materials work, especially in complex environments where multiple defects interact. The method works directly on the molecular data, turning a chaotic cloud of points into a precise map of the lines that govern the material's behavior, offering a new level of clarity for the study of soft matter physics.

Technical Summary: Spectral Alignment of Mesogens for Three-Dimensional Disclination Localization in Liquid Crystals

Problem Statement
Topological defects, specifically disclinations, govern the material properties and dynamics of liquid crystals. A fundamental challenge in analyzing molecular data (such as from molecular dynamics simulations) is the efficient identification of three-dimensional disclination centerlines. Traditional diagnostic methods rely on thresholding local scalar order parameters (derived from the Q-tensor) to identify regions of reduced orientational order. However, these methods measure finite-width volumes of disorder rather than the one-dimensional centerlines of the defects themselves. In complex assemblies with multiple defects, these broadened low-order regions often overlap or merge, obscuring the true geometry, connectivity, and arclength of the defect lines. Furthermore, the apparent location and thickness of these regions depend heavily on the chosen coarse-graining length scale and threshold values.

Methodology
The authors propose a deterministic framework comprising two stages: Spectral Alignment of Mesogens (SPAM) and Discrete Layer Analysis (DLA). This approach shifts the focus from local disorder thresholds to the global topological obstruction inherent in nematic line fields.

Spectral Alignment of Mesogens (SPAM):
- Formulation: The problem is cast as assigning a binary sign ( $s_i \in \{-1, +1\}$ ) to every mesogen's unoriented director $\hat{n}_i$ to create a globally consistent oriented vector field $m_i = s_i \hat{n}_i$ . The goal is to maximize the alignment of neighboring vectors such that $s_i s_j (\hat{n}_i \cdot \hat{n}_j) \geq 0$ .
- Optimization: This is formulated as an Ising-like binary optimization problem on a distance-weighted interaction graph derived from the mesogen point set. The energy function is $\theta(A, s) = -s^T A s$ , where $A$ is a weighted director-compatibility matrix.
- Solution: Instead of using propagation-based methods (like the previously developed TADA) which depend on an arbitrary seed and can produce geometrically complex discontinuities, SPAM relaxes the binary constraints to a continuous spectral problem. It solves for the leading eigenvector of the matrix $A$ (maximizing the Rayleigh quotient) and binarizes the result via the sign function. This yields a deterministic global approximation that is invariant to the initial orientation of the input vectors.
Discrete Layer Analysis (DLA):
- Layer Construction: Once signs are assigned, neighboring mesogens with antagonistic assigned vectors ( $v_i \hat{n}_i \cdot v_j \hat{n}_j < 0$ ) define a discrete point set representing a quasi-two-dimensional discontinuity layer.
- Centerline Extraction: In 3D, these discontinuities are surfaces that terminate along disclination lines. DLA treats this point set as a sampled surface. It uses local volumetric point density and PCA to perform a total least squares estimate of an underlying plane to distinguish between points in the smooth interior of the layer and points near the layer's termination.
- Reconstruction: Points identified as terminations are connected via a spatial neighbor graph to reconstruct the one-dimensional disclination centerlines, allowing for the calculation of component connectivity and arclength.

Key Results
The framework was validated on molecular dynamics simulations of calamitic liquid-crystalline assemblies, including:

Prescribed Solitary Defects: Straight half-integer disclination lines ("comet" and "trefoil" configurations) and closed loops ("wedge" and "wedge–twist" configurations). SPAM-DLA successfully reconstructed the centerlines and measured their arclengths, matching the known geometries.
Multiple-Defect Geometry: A cylindrical-exclusion setup inducing a complex 3D defect landscape. Here, SPAM produced discontinuity layers with significantly reduced geometric complexity (fewer folds and concavities) compared to propagation-based TADA assignments. This regularization made the subsequent boundary extraction by DLA more robust and reliable.
Energy Comparison: SPAM consistently achieved alignment energies comparable to or lower than the best TADA assignments (found by sweeping propagation seeds), particularly in complex multi-defect geometries.
Robustness to Missing Data: In scenarios where near-core director data was removed (simulating occlusion or resolution limits), SPAM continued to detect the orientability obstruction and reconstruct the defect lines. In contrast, scalar-order-parameter methods failed to identify the defects because they rely directly on local disorder signals which were absent in the missing data regions.

Significance and Claims
The paper claims to provide a robust, deterministic framework for localizing and quantifying 3D disclination lines directly from unoriented mesogen-scale director fields. The primary significance lies in the conceptual shift from measuring "finite-width regions of reduced order" to extracting the "termination of a global sign-discontinuity surface."

Distinction of Observables: The work clarifies that the thickness of a reduced-order region (measured by scalar order parameters) and the arclength of the topological defect (measured by SPAM-DLA) are distinct observables. Treating them separately avoids conflating core thickness with defect length.
Geometric Regularization: By replacing propagation-based sign assignment with a global spectral optimization, the method produces geometrically coherent discontinuity surfaces, which are essential for reliable boundary extraction in complex 3D geometries.
Applicability: The framework is presented as applicable to a wide range of liquid-crystalline media, including passive nematics, active fluids, and nematic elastomers, and is particularly relevant for calibrating continuum models proposed for line defects.

The authors emphasize that while the method relies on specific parameters (neighbor graph construction, distance weighting, and local geometric neighborhoods for DLA), it offers a deterministic alternative to stochastic or seed-dependent approaches, providing a clear geometric route to defect quantification without annealing schedules.

Technical Summary: Spectral Alignment of Mesogens for Three-Dimensional Disclination Localization in Liquid Crystals

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