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An Efficient Boundary Integral Scheme for the Threshold Dynamics Method II: Applications to Wetting Dynamics

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Abstract

In this paper, we extend the boundary integral scheme for the threshold dynamics method to treat the case where the material interface is nonsmooth and may undergo topological changes. The scheme is then applied to study the wetting dynamics in both two and three dimensions. Numerical experiments show that the scheme is more efficient as compared with the existing method using uniform grids, making accurate simulation of wetting dynamics on a chemically patterned solid surface in three dimensions within practical reach.

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References

  1. Alberti, G., DeSimone, A.: Wetting of rough surfaces: a homogenization approach. Proc. R. Soc. A 451, 79–97 (2005)

    Article  MathSciNet  Google Scholar 

  2. Bonn, D., Eggers, J., Indekeu, J., Meunier, J., Rolley, E.: Wetting and spreading. Rev. Mod. Phys. 81(2), 739 (2009)

    Article  Google Scholar 

  3. Bonn, D., Ross, D.: Wetting transitions. Rep. Prog. Phys. 64(9), 1085 (2001)

    Article  Google Scholar 

  4. Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28(2), 258–267 (1958)

    Article  Google Scholar 

  5. Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. III. Nucleation in a two-component incompressible fluid. J. Chem. Phys. 31(3), 688–699 (1959)

    Article  Google Scholar 

  6. Chen, H., Leng, H., Wang, D., Wang, X.P.: An efficient threshold dynamics method for topology optimization for fluids. arXiv preprint arXiv:1812.09437 (2018)

  7. Chen, X., Wang, X., Xu, X.: Analysis of the Cahn–Hilliard equation with a relaxation boundary condition modeling the contact angle dynamics. Arch. Ration. Mech. Anal. 213(1), 1–24 (2014)

    Article  MathSciNet  Google Scholar 

  8. Dutt, A., Rokhlin, V.: Fast Fourier transforms for nonequispaced data. SIAM J. Sci. Comput. 14, 1368–1393 (1993)

    Article  MathSciNet  Google Scholar 

  9. Dutt, A., Rokhlin, V.: Fast Fourier transforms for nonequispaced data. II. Appl. Comput. Harmon. Anal. 2, 85–100 (1995)

    Article  MathSciNet  Google Scholar 

  10. Erbil, H.Y.: The debate on the dependence of apparent contact angles on drop contact area or three-phase contact line: a review. Surf. Sci. Rep. 69(4), 325–365 (2014)

    Article  Google Scholar 

  11. Esedoglu, S., Otto, F.: Threshold dynamics for networks with arbitrary surface tensions. Commun. Pur. Appl. Math. 68(5), 808–864 (2015)

    Article  MathSciNet  Google Scholar 

  12. Extrand, C.W.: Model for contact angles and hysteresis on rough and ultraphobic surfaces. Langmuir 18, 7991–7999 (2002)

    Article  Google Scholar 

  13. Gao, M., Wang, X.P.: An efficient scheme for a phase field model for the moving contact line problem with variable density and viscosity. J. Comput. Phys. 272, 704–718 (2014)

    Article  MathSciNet  Google Scholar 

  14. de Gennes, P., Brochard-Wyart, F., Quere, D.: Capillarity and Wetting Phenomena. Springer, Berlin (2003)

    MATH  Google Scholar 

  15. Greengard, L., Lee, J.Y.: Accelerating the nonuniform fast Fourier transform. SIAM Rev. 46(3), 443–454 (2004)

    Article  MathSciNet  Google Scholar 

  16. Greengard, L., Lin, P.: Spectral approximation of the free-space heat kernel. Appl. Comput. Harmon. Anal. 9(1), 83–97 (2000)

    Article  MathSciNet  Google Scholar 

  17. Jiang, S., Wang, D., Wang, X.P.: An efficient boundary integral scheme for the MBO threshold dynamics method via the NUFFT. J. Sci. Comput. 74, 474–490 (2018)

    Article  MathSciNet  Google Scholar 

  18. Leung, S., Zhao, H.: A grid based particle method for moving interface problems. J. Comput. Phys. 228(8), 2993–3024 (2009)

    Article  MathSciNet  Google Scholar 

  19. Müller, D.E.: A method for solving algebraic equations using an automatic computer. Math. Tables Aids Comput. 10, 208–215 (1956)

    Article  MathSciNet  Google Scholar 

  20. Osting, B., Wang, D.: A diffusion generated motion for orthogonal matrix-valued fields. Math. Comp. (2019) (accepted)

  21. Quere, D.: Wetting and roughness. Annu. Rev. Mater. Res. 38, 71–99 (2008)

    Article  Google Scholar 

  22. Randol, B.: On the Fourier transform of the indicator function of a planar set. Trans. Am. Math. Soc. 139, 271–278 (1969)

    MathSciNet  MATH  Google Scholar 

  23. Wang, D.: Threshold dynamics method: theories, algorithms, and applications. Ph.D. thesis, Hong Kong University of Science and Technology (2017)

  24. Wang, D., Li, H., Wei, X., Wang, X.P.: An efficient iterative thresholding method for image segmentation. J. Comput. Phys. 350, 657–667 (2017)

    Article  MathSciNet  Google Scholar 

  25. Wang, D., Osting, B.: A diffusion generated method for computing Dirichlet partitions. J. Comput. Appl. Math. 351, 302–316 (2019)

    Article  MathSciNet  Google Scholar 

  26. Wang, D., Osting, B., Wang, X.P.: Interface dynamics for an Allen–Cahn-type equation governing a matrix-valued field. Multiscale Model. Sim. (2019) (accepted)

  27. Wang, D., Wang, X.P.: The iterative convolution-thresholding method (ICTM) for image segmentation. arXiv preprint arXiv:1904.10917 (2019)

  28. Wang, D., Wang, X.P., Xu, X.: An improved threshold dynamics method for wetting dynamics. J. Comput. Phys. 392, 291–310 (2019)

    Article  MathSciNet  Google Scholar 

  29. Whyman, G., Bormashenko, E., Stein, T.: The rigorous derivative of young, Cassie–Baxter and Wenzel equations and the analysis of the contact angle hysteresis phenomenon. Chem. Phys. Lett. 450, 355–359 (2008)

    Article  Google Scholar 

  30. Womble, D.E.: A front-tracking method for multiphase free boundary problems. SIAM J. Numer. Anal. 26(2), 380–396 (1989)

    Article  MathSciNet  Google Scholar 

  31. Xu, X., Wang, D., Wang, X.P.: An efficient threshold dynamics method for wetting on rough surfaces. J. Comput. Phys. 330, 510–528 (2017)

    Article  MathSciNet  Google Scholar 

  32. Xu, X., Wang, X.P.: The modified Cassie’s equation and contact angle hysteresis. Colloid Polym. Sci. 291, 299–306 (2013)

    Article  Google Scholar 

  33. Young, T.: An essay on the cohesion of fluids. Philos. Trans. R. Soc. Lond. 95, 65–87 (1805)

    Article  Google Scholar 

  34. Zhao, H., Chan, T., Merriman, B., Osher, S.J.: A variational level set approach to multiphase motion. J. Comput. Phys. 127(1), 179–195 (1996)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

S. Jiang was supported by the National Science Foundation under Grant DMS-1720405 and by the Flatiron Institute, a division of the Simons Foundation. The work of X.P. Wang was partially supported by the Hong Kong Research Grants Council (GRF Grants 16302715, 16324416, and 16303318). Part of the work was performed when the authors were participating in the HKUST-ICERM VI-MSS program ‘Integral Equation Methods, Fast Algorithms and Their Applications to Fluid Dynamics and Materials Science’ held in 2017.

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Authors and Affiliations

  1. Department of Mathematics, University of Utah, Salt Lake City, UT, 84112, USA

    Dong Wang

  2. Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ, 07102, USA

    Shidong Jiang

  3. Department of Mathematics, The Hong Kong University of Science and Technology, Kowloon, Hong Kong, China

    Xiao-Ping Wang

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  1. Dong Wang
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  2. Shidong Jiang
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  3. Xiao-Ping Wang
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Correspondence to Dong Wang.

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Wang, D., Jiang, S. & Wang, XP. An Efficient Boundary Integral Scheme for the Threshold Dynamics Method II: Applications to Wetting Dynamics. J Sci Comput 81, 1860–1881 (2019). https://doi.org/10.1007/s10915-019-01067-1

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