Topological defects are ubiquitous across various scientific disciplines, from crystallography and liquid crystals to cosmology and biology. These defects represent mismatches in order and often exhibit characteristic behaviors, including reconnections, where line-shaped defects approach, collide, and exchange endpoints. Nematic liquid crystals, characterized by a unidirectional director vector (\hat{n}) and its singularities (disclinations), provide an ideal model system to study such phenomena due to their optical properties and experimental controllability. Previous research has extensively studied nematic disclinations, covering topics such as defect generation and ordering, defect interactions with microparticles and light, molecular manipulation by defects, and defect control through surface alignment. However, most observations have been limited to two-dimensional (2D) information obtained via transmitted light microscopy. Recent work has demonstrated the ability to observe 3D disclination line dynamics, including reconnections and loop shrinkage, using dye localization techniques. However, this analysis primarily focused on nearly parallel disclinations within a single plane. The current study aims to expand on these findings by investigating the more fundamentally three-dimensional (3D) scenario of intersecting reconnections, where disclinations in separate planes approach at a finite angle. This detailed examination, using confocal microscopy, will provide valuable insight into the 3D dynamics of these topological defects and allow for a more comprehensive comparison with theoretical models.

The study builds upon previous research on topological defects and their dynamics, particularly in nematic liquid crystals. The authors cite works detailing the fundamental properties of liquid crystals and topological defects, including the role of elastic constants (K1, K2, K3) in determining the system's behavior. Several papers are referenced that explore the 2D and 3D dynamics of disclinations, including the generation, interaction, and annihilation of these defects. The authors' previous work on 3D disclination dynamics, specifically reconnections of nearly parallel disclinations, serves as a basis for this study. Existing theoretical approaches for understanding 3D disclination interactions and velocities are also discussed, forming the backdrop against which the current research's findings are compared and evaluated. The literature review highlights the need for a more detailed investigation of intersecting reconnections, a more complex and fundamentally 3D configuration than those previously studied extensively.

The experiment employed a nematogen MLC-2037 (Merck), chosen for its low optical anisotropy (Δn = 0.0649) and negative dielectric anisotropy (Δε = -3.1). A fluorescent dye (coumarin 545T) was added to label the disclinations, allowing for 3D visualization. The sample, contained within a 130 µm-thick cell with planar alignment, was subjected to an AC voltage to induce a turbulent state, generating a high density of disclinations. Switching off the voltage initiated a relaxation process where the disclinations interacted and reconnected. Three-dimensional dynamics were captured using a Leica SP8 laser-scanning confocal microscope with resonant and piezo objective scanners. Twelve intersecting reconnection events were observed and analyzed. Image analysis utilized the snake method to extract the 3D coordinates of the disclinations from the confocal images. The time evolution of the minimum distance (δ(t)) between the two disclinations and the angle (ψ(t)) between their tangent vectors were determined. The experimental results were then compared with theoretical predictions, primarily those derived from a model that assumes equal nematic elastic constants. To address quantitative discrepancies between experimental and theoretical results, the equations of motion were derived for a more generalized case involving unequal elastic constants, specifically focusing on the reduced twist constant (K2) observed in the material MLC-2037. Furthermore, 3D simulations of Q-tensor evolution were performed to probe the influence of system size and disclination length on the observed mobility ratio.

Experimental observations revealed a square-root law for the time dependence of the distance between reconnecting disclinations (δ(t) ≈ C|t - t₀|^(1/2)), consistent with previous findings for in-plane reconnections. The inter-disclination angle (ψ) was found to decrease over time, indicating that the disclinations tend towards parallel alignment as they reconnect. Comparing the experimental data with a theoretical model (Ref. [35]), which assumes equal nematic elastic constants, revealed qualitative agreement regarding the scaling of distance and angle but significant quantitative discrepancies in the relative mobilities (μ₁ and μ₂). The experimental data showed a mobility ratio (μ₂/μ₁) significantly less than 1, indicating a relatively slower change in the angle compared to the distance. A modified theoretical derivation that incorporates unequal elastic constants (specifically a reduced twist constant K2) failed to explain this discrepancy. The analysis of the experimental data using integrated forms of the equations of motion (Eqs. 8 and 9) confirmed the validity of the modified equations but highlighted the persistent discrepancy in the mobility ratio (μ₂/μ₁ = 0.26 ± 0.08). Further investigation revealed a slight negative correlation between the mobility ratio and the initial angle between the disclinations. 3D simulations using a reduced twist constant showed that the mobility ratio (μ₂/μ₁) approaches the theoretical value of 1 only for small system sizes where disclination deformations are minimal. In larger systems, the mobility ratio decreases with increasing initial angle, mirroring the experimental trend. This suggests that disclination deformations play a crucial role in the observed quantitative discrepancies.

The findings highlight the importance of considering disclination deformations in theoretical models of reconnection dynamics. The quantitative disagreement between experimental results and theoretical predictions based on simplified models (assuming equal elastic constants and straight disclinations) suggests that the simplified assumptions are insufficient to capture the complex 3D dynamics of reconnecting disclinations. The observed dependence of the mobility ratio on the initial angle and system size further strengthens this conclusion. The reduced twist constant, while considered theoretically, does not fully account for the observed difference. The study thus underscores the need for a more comprehensive theoretical framework that explicitly incorporates disclination deformations, possibly involving advanced numerical techniques to model the director field and resulting elastic energy contributions accurately. The experimental results, supported by numerical simulations, offer valuable insights into the 3D dynamics of topological defects and challenge the assumptions of previous theoretical models.

This study provided experimental and numerical evidence for a square-root law governing the distance between reconnecting disclinations and a decreasing inter-disclination angle. The significant deviation of the experimentally measured mobility ratio from theoretical predictions highlights the importance of considering disclination deformations in theoretical models. Future research should focus on developing theoretical frameworks that incorporate disclination deformations, extend the analysis to longer sections of the disclinations, consider the impact of surface alignment, and explore experiments with liquid crystals having different ratios of elastic constants. Understanding the 3D dynamics of topological defects is crucial not only for fundamental science but also for applications in areas such as microparticle manipulation and light control.

The study's limitations include the assumption of infinitely long disclinations in the theoretical model, which may not perfectly represent the experimental scenario. The analysis focuses on the vicinity of the closest points of the disclinations, potentially overlooking influences from the remainder of the defect lines. The experimental system's surface alignment might influence the disclination dynamics, although efforts were made to minimize this influence by focusing on events occurring away from the surface. The simulations were performed for a specific range of parameters and system sizes, limiting the generalizability of the conclusions to other conditions.