Zhang Pingwen, Zhang Lei and collaborators published a review paper on "Modeling and Calculation of Liquid Crystals"

On August 4, 2021, Professor Zhang Pingwen of the School of Mathematical Sciences of Peking University and Director of the National Engineering Laboratory of Big Data Analysis and Applied Technology, Associate Professor Zhang Lei of Beijing International Mathematical Research Center and Researcher Wang Wei of the School of Mathematical Sciences of Zhejiang University, published in Acta Numerica A review paper entitled " Modelling and Computation of Liquid Crystals "
.

Liquid crystal is a soft substance between liquid and crystal
.

In the past four decades, the research of liquid crystal has made great progress, which not only greatly promotes basic scientific research, but also has been widely used in industry

The thesis first introduces three types of liquid crystal mathematical models and their connections at different levels, namely molecular model, tensor model and vector model
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These models are either established by mechanics based on the principles of mechanics and constitutive assumptions, or established by physicists based on the molecular structure using the average field theory of statistical physics

Figure 1: Model reduction from micro-molecular theory to macro-continuum theory

The thesis then introduces the calculation methods of liquid crystals, including the energy minimization method and gradient flow method for calculating the stable defects of liquid crystal, and the numerical method of the liquid crystal hydrodynamic model, and lists various types of restrictions in different geometric regions and conditions.
Defect configuration of nematic liquid crystal
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Since liquid crystals usually have multiple ordered phases, how to find out all the phases and the connection relationship between different phases by calculation? The thesis first reviews the calculation methods for calculating transition states and transfer paths, such as the string method for finding the minimum energy path; then introduces the concept of landscape solution proposed in recent years, and the calculation method of using saddle point dynamics to construct the landscape solution.

Figure 2: The solution landscape of nematic liquid crystals confined in a square area

At the end of the thesis, it is concluded that the research of liquid crystal mainly focuses on how the topological and geometric characteristics of molecules affect the mesoscopic structure, which determines the properties of liquid crystal materials
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The scientific value of liquid crystals is not only to raise new mathematical problems, but also to promote new research methods

Acta Numerica is a top review journal in the field of numerical analysis and scientific computing (mathematics journal with the highest impact factor)
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The journal was founded in 1992 and invites well-known experts in the field to write review papers every year.

Introduction to the author of the paper:

Zhang Pingwen is a professor at the School of Mathematical Sciences of Peking University, an academician of the Chinese Academy of Sciences, an academician of the Academy of Sciences of the Developing Countries, and a fellow of the American Society for Industrial and Applied Mathematics
.

He is currently a member of the Standing Committee of the Party Committee of Peking University, the vice president of Peking University, the director of the National Engineering Laboratory of Big Data Analysis and Application Technology, the director of the Peking University Big Data Science Research Center, and the chairman of the Chinese Society of Industrial and Applied Mathematics

Zhang Lei, Associate Professor/Researcher, Director of Beijing International Mathematics Research Center, Peking University, PI, Quantitative Biology Center
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The main research areas of computing and applied mathematics, algorithms and applications including rare event and solution landscape, computing systems biology, computational materials science

Wang Wei, Associate Professor/Researcher, Dean of the School of Mathematical Sciences, Zhejiang University, graduated from the School of Mathematical Sciences, Peking University with a PhD in 2013, and his supervisor is Professor Zhang Pingwen
.
The main research fields are applied analysis and partial differential equations, including liquid crystal mathematical theory, fluid free boundary value problems, etc.
The research results are in Comm.
Pure Appl.
Math.
, Arch.
Ration.
Mech.
Anal, J.
Funct.
Anal.
and other journals Announced, was selected into the "Young Talents Support Project" of the China Association for Science and Technology, and was funded by the Outstanding Youth Science Fund of the National Natural Science Foundation of China
.