Miscellaneous

Defects in liquid crystals

My PhD thesis focuses on the defect structures of nematic liquid crystals under various geometric constraints, studied within the Landau–de Gennes theoretical framework through a combination of numerical simulations and analytical insights. One particular motivation is liquid crystal colloids (LCCs).

Leveraging the efficiency of spectral numerical methods, I systematically investigate defect configurations induced by spherical colloidal inclusions with strong homeotropic anchoring. The core algorithm follows a discretize-then-minimize strategy, where the liquid crystal order parameter is approximated using spectral expansions, and the resulting minimization problem is solved in terms of spectral coefficients using optimization techniques such as L-BFGS (see [WZC2017] for details). To accommodate complex geometries, the spectral method is implemented either via coordinate transformations (e.g., bispherical coordinates) or phase-field representations, depending on the setup. A mesh-free analog of this approach, based on neural networks, is known as the deep Ritz method, introduced by E and Yu (2018). 

Here is the  C++ code is designed to simulate defect structures in liquid crystals in a nematic shell or outside two spherical particles(https://github.com/1Dimension/LC_Defects).  Feel free to reach out to me if you have any questions about code.

[WZC2017] Wang, Y., Zhang, P., & Chen, J. Z. (2017). Topological defects in an unconfined nematic fluid induced by single and double spherical colloidal particles. Physical Review E, 96(4), 042702. https://journals.aps.org/pre/abstract/10.1103/PhysRevE.96.042702

[WZC2018] Y. Wang, P. Zhang, and J. Z. Y. Chen. Formation of three-dimensional colloidal crystals in a nematic liquid crystal. Soft Matter, 2018.

[NWLJML2020] Noh, J., Wang, Y., Liang, H. L., Jampani, V. S. R., Majumdar, A., & Lagerwall, J. P. (2020). Dynamic tuning of the director field in liquid crystal shells using block copolymers. Physical Review Research, 2(3), 033160. https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.2.033160