Acta Numerica (2022), pp. 1{89 © Cambridge University Press, 2022 doi:10.1017/S09624929XXXXXX Printed in the United Kingdom Modeling and Computation of Liquid Crystals Wei Wang Department of Mathematics, Zhejiang University, Hangzhou 310027, China
[email protected] Lei Zhang Beijing International Center for Mathematical Research, Center for Quantitative Biology, Peking University, Beijing 100871, China
[email protected] Pingwen Zhang School of Mathematical Sciences, Peking University, Beijing 100871, China
[email protected] Liquid crystal is a typical kind of soft matter that is intermediate between crystalline solids and isotropic fluids. The study of liquid crystals has made tremendous progress over the last four decades, which is of great importance on both fundamental scientific researches and widespread applications in in- dustry. In this paper, we review the mathematical models and their connec- tions of liquid crystals, and survey the developments of numerical methods for finding the rich configurations of liquid crystals. CONTENTS 1 Introduction 2 2 Mathematical models of liquid crystals 6 arXiv:2104.02250v1 [math.NA] 6 Apr 2021 3 Mathematical analysis for different liquid crystal models 26 4 Numerical methods for computing stable defects of liquid crystals 38 5 Numerical methods for computing liquid crystal hydrodynamics 51 6 Numerical methods for computing transition pathways and solution landscape of liquid crystals 58 7 Conclusion and future directions 69 8 Appendix 74 References 76 2 Acta Numerica 1. Introduction Liquid crystals (LCs) are classical examples of partially ordered materi- als that translate freely as liquid and exhibit some long-range order above a critical concentration or below a critical temperature.