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Unconditionally Energy Stable and Bound-Preserving Schemes for Phase-Field Surfactant Model with Moving Contact Lines

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Abstract

Phase-field surfactant model with moving contact lines (PFS-MCL) has been extensively investigated in the study of droplet dynamics on solid surfaces in the presence of surfactants. This model consists of two Cahn–Hilliard type equations, governing the dynamics of interface and surfactant concentration, with the dynamical boundary condition for moving contact lines. Moreover, the total free energy has logarithmic singularity due to Flory–Huggins potential. Based on the convexity of Flory–Huggins potential and the convex splitting technique, we propose a set of unconditionally energy stable and bound-preserving schemes for PFS-MCL model. The proposed schemes are decoupled, uniquely solvable, and mass conservative. These properties are rigorously proved for the first-order fully discrete scheme. In addition, we prove that the second-order fully discrete scheme satisfies all these properties except for the energy stability. We numerically validate the desired properties for both first-order and second-order schemes. We also present numerical results to systematically study the influence of surfactants on contact line dynamics and its parameter dependence. Furthermore, droplet spreading and retracting on chemically patterned surfaces are numerically investigated. It is observed that surfactants can affect contact angle hysteresis by changing advancing/receding contact angles.

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Acknowledgements

The authors gratefully acknowledge many helpful discussions with Jie Shen and Jie Xu during the preparation of the paper. The authors would also like to dedicate this paper to Hui Zhang for his departure. The work of Zhen Zhang was partially supported by the NSFC Grant (No. 11731006) and (No. 12071207), NSFC Tianyuan-Pazhou grant (No. 12126602), the Guangdong Basic and Applied Basic Research Foundation (2021A1515010359) and the Guangdong Provincial Key Laboratory of Computational Science and Material Design (No. 2019B030301001).

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  1. Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, People’s Republic of China

    Chenxi Wang

  2. Department of Mathematics, Southern University of Science and Technology (SUSTech), Shenzhen, 518055, People’s Republic of China

    Chenxi Wang & Zhen Zhang

  3. Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA, 24060, USA

    Yichen Guo

  4. Guangdong Provincial Key Laboratory of Computational Science and Material Design, International Center for Mathematics, National Center for Applied Mathematics (Shenzhen), Southern University of Science and Technology (SUSTech), Shenzhen, 518055, People’s Republic of China

    Zhen Zhang

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Wang, C., Guo, Y. & Zhang, Z. Unconditionally Energy Stable and Bound-Preserving Schemes for Phase-Field Surfactant Model with Moving Contact Lines. J Sci Comput 92, 20 (2022). https://doi.org/10.1007/s10915-022-01863-2

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