DOI QR코드

DOI QR Code

DIRECT COMPARISON STUDY OF THE CAHN-HILLIARD EQUATION WITH REAL EXPERIMENTAL DATA

  • DARAE, JEONG (DEPARTMENT OF MATHEMATICS, KANGWON NATIONAL UNIVERSITY) ;
  • SEOKJUN, HAM (DEPARTMENT OF MATHEMATICS, KOREA UNIVERSITY) ;
  • JUNSEOK, KIM (DEPARTMENT OF MATHEMATICS, KOREA UNIVERSITY)
  • Received : 2022.10.18
  • Accepted : 2022.12.20
  • Published : 2022.12.25

Abstract

In this paper, we perform a direct comparison study of real experimental data for domain rearrangement and the Cahn-Hilliard (CH) equation on the dynamics of morphological evolution. To validate a mathematical model for physical phenomena, we take initial conditions from experimental images by using an image segmentation technique. The image segmentation algorithm is based on the Mumford-Shah functional and the Allen-Cahn (AC) equation. The segmented phase-field profile is similar to the solution of the CH equation, that is, it has hyperbolic tangent profile across interfacial transition region. We use unconditionally stable schemes to solve the governing equations. As a test problem, we take domain rearrangement of lipid bilayers. Numerical results demonstrate that comparison of the evolutions with experimental data is a good benchmark test for validating a mathematical model.

Keywords

Acknowledgement

The first author (D. Jeong) was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (NRF-2020R1F1A1A01075937). The corresponding author (J.S. Kim) was supported by the National Research Foundation(NRF), Korea, under project BK21 FOUR. The authors appreciate the reviewers for the valuable comments on the revision of this article.

References

  1. E. D. Siggia, Late stages of spinodal decomposition in binary mixtures, Physical review A, 20(2) (1979), 595.  https://doi.org/10.1103/PhysRevA.20.595
  2. J. T. Cabral, J. S. Higgins, T. C. B. McLeish, S. Strausser, S. N. Magonov Bulk, spinodal decomposition studied by atomic force microscopy and light scattering, Macromolecules, 34(11) (2001), 3748-3756.  https://doi.org/10.1021/ma0017743
  3. H. J. Chung, R. J. Composto, Breakdown of dynamic scaling in thin film binary liquids undergoing phase separation, Physical review letters, 92(18) (2004), 185704.  https://doi.org/10.1103/PhysRevLett.92.185704
  4. B. P. Lee, J. F. Douglas, S. C. Glotzer, Filler-induced composition waves in phase-separating polymer blends, Physical Review E, 60(5) (1999), 5812.  https://doi.org/10.1103/PhysRevE.60.5812
  5. F. A. Castro, C. F. Graeff, J. Heier, R. Hany, Interface morphology snapshots of vertically segregated thin films of semiconducting polymer/polystyrene blends, Polymer, 48(8) (2007), 2380-02386.  https://doi.org/10.1016/j.polymer.2007.02.059
  6. P. Sphingomyelin, Lipid Rafts: Phase Separation in Lipid Bilayers studied with Atomic Force Microscopy, https://analyticalscience.wiley.com/do/10.1002/micro.162/full/ 
  7. R. Blossey, Thin film rupture and polymer flow, Physical Chemistry Chemical Physics, 10(34) (2008), 5177-5183.  https://doi.org/10.1039/b807728m
  8. J. L. Masson, R. Limary, P. F. Green, Pattern formation and evolution in diblock copolymer thin films above the order-disorder transition, The Journal of Chemical Physics, 114(24) (2001), 10963-10967.  https://doi.org/10.1063/1.1370565
  9. A. Novick-Cohen, L. A. Segel, Nonlinear aspects of the Cahn-Hilliard equation, Physica D: Nonlinear Phenomena, 10(3) (1984), 277-298.  https://doi.org/10.1016/0167-2789(84)90180-5
  10. C. M. Elliott, Z. Songmu, On the cahn-hilliard equation, Archive for Rational Mechanics and Analysis, 96(4) (1986), 339-357.  https://doi.org/10.1007/BF00251803
  11. L. A. Caffarelli, N. E. Muler, An L bound for solutions of the Cahn-Hilliard equation, Archive for rational mechanics and analysis, 133(2) (1995), 129-144.  https://doi.org/10.1007/BF00376814
  12. Y. Jingxue, On the existence of nonnegative continuous solutions of the Cahn-Hilliard equation, Journal of differential equations, 97(2) (1992), 310-327.  https://doi.org/10.1016/0022-0396(92)90075-X
  13. C. M. Elliott, H. Garcke, On the Cahn-Hilliard equation with degenerate mobility, Siam journal on mathematical analysis, 27(2) (1996), 404-423.  https://doi.org/10.1137/S0036141094267662
  14. P. C. Fife, Dynamical aspects of the Cahn-Hilliard equations, University of Tennessee; Knoxville; TN; Barret Lectures, 1991. 
  15. P. C. Fife, Models for phase separation and their mathematics, Electron. J. Differ. Eq. Conf., 48 (2000), 1-26. 
  16. M. Grinfeld, A. Novick-Cohen, Counting stationary solutions of the Cahn-Hilliard equation by transversality arguments, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 125(2) (1995), 351-370.  https://doi.org/10.1017/S0308210500028079
  17. P. Rybka, K. H. Hoffnlann, Convergence of solutions to Cahn-Hilliard equation, Communications in partial differential equations, 24(5-6) (1999), 1055-1077.  https://doi.org/10.1080/03605309908821458
  18. D. Furihata, A stable and conservative finite difference scheme for the Cahn-Hilliard equation, Numerische Mathematik, 87(4) (2001), 675-699.  https://doi.org/10.1007/PL00005429
  19. Y. He, Y. Liu, T. Tang, On large time-stepping methods for the Cahn-Hilliard equation, Applied Numerical Mathematics, 57(5-7) (2007), 616-628.  https://doi.org/10.1016/j.apnum.2006.07.026
  20. J. Zhu, L. Q. Chen, J. Shen, V. Tikare, Coarsening kinetics from a variable-mobility Cahn-Hilliard equation: Application of a semi-implicit Fourier spectral method, Physical Review E, 60(4) (1999), 3564.  https://doi.org/10.1103/PhysRevE.60.3564
  21. E. V. L. De Mello, O. T. da Silveira Filho, Numerical study of the Cahn-Hilliard equation in one, two and three dimensions, Physica A: Statistical Mechanics and its Applications, 347 (2005), 429-443.  https://doi.org/10.1016/j.physa.2004.08.076
  22. M. I. M. Copetti, C. M. Elliott, Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy, Numerische Mathematik, 63Z(1) (1992), 39-65. 
  23. T. M. Rogers, R. C. Desai, Numerical study of late-stage coarsening for off-critical quenches in the CahnHilliard equation of phase separation, Physical Review B, 39(16) (1989), 11956.  https://doi.org/10.1103/physrevb.39.11956
  24. C. M. Elliott, D. A. French, Numerical studies of the Cahn-Hilliard equation for phase separation, IMA Journal of Applied Mathematics, 38(2) (1987), 97-128.  https://doi.org/10.1093/imamat/38.2.97
  25. L. Ju, J. Zhang, Q. Du, Fast and accurate algorithms for simulating coarsening dynamics of Cahn-Hilliard equations, Computational Materials Science, 108(2015), 272-282.  https://doi.org/10.1016/j.commatsci.2015.04.046
  26. Y. Zhao, P. Stein, B. X. Xu, Isogeometric analysis of mechanically coupled Cahn-Hilliard phase segregation in hyperelastic electrodes of Li-ion batteries, Computer Methods in Applied Mechanics and Engineering, 297, 325-347.  https://doi.org/10.1016/j.cma.2015.09.008
  27. Y. Shang, L. Fang, M. Wei, C. Barry, J. Mead, D. Kazmer, Verification of numerical simulation of the self-assembly of polymer-polymer-solvent ternary blends on a heterogeneously functionalized substrate, Polymer, 52(6) (2011), 1447-1457.  https://doi.org/10.1016/j.polymer.2011.01.038
  28. H. Mantz, K. Jacobs, K. Mecke, Utilizing Minkowski functionals for image analysis: a marching square algorithm, Journal of Statistical Mechanics: Theory and Experiment, 2008(12) (2008), 12015. 
  29. S. Burger, T. Fraunholz, C. Leirer, R. H. Hoppe, A. Wixforth, M. A. Peter, T. Franke, Comparative study of the dynamics of lipid membrane phase decomposition in experiment and simulation, Langmuir, 29(25) (2013), 7565-7570.  https://doi.org/10.1021/la401145t
  30. J. W. Cahn, On spinodal decomposition, Acta metallurgica, 9(9) (1961), 795-801.  https://doi.org/10.1016/0001-6160(61)90182-1
  31. D. Lee, J. Y. Huh, D. Jeong, J. Shin, A. Yun, J. Kim, Physical, mathematical, and numerical derivations of the Cahn-Hilliard equation, Computational Materials Science, 81 (2014), 216-225.  https://doi.org/10.1016/j.commatsci.2013.08.027
  32. Y. Li, J. Kim, An unconditionally stable hybrid method for image segmentation, Applied Numerical Mathematics, 82 (2014), 32-43.  https://doi.org/10.1016/j.apnum.2013.12.010
  33. W. L. Briggs, A Multigrid Tutorial, SIAM, Philadelphia, 1987. 
  34. U. Trottenberg, C. Oosterlee, A. Schuller, Multigrid, Academic Press, London, 2001. 
  35. M. Donatelli, C. Garoni, C. Manni, S. Serra-Capizzano, H. Speleers, Robust and optimal multi-iterative techniques for IgA collocation linear systems, Computer Methods in Applied Mechanics and Engineering, 284 (2015), 1120-1146.  https://doi.org/10.1016/j.cma.2014.11.036
  36. H. K. Kodali, B. Ganapathysubramanian, A computational framework to investigate charge transport in heterogeneous organic photovoltaic devices, Computer Methods in Applied Mechanics and Engineering, 247 (2012), 113-129.  https://doi.org/10.1016/j.cma.2012.08.012
  37. Y. G. Smirnova, M. Fuhrmans, I. A. B. Vidal, M. Muller, Free-energy calculation methods for collective phenomena in membranes, Journal of Physics D: Applied Physics, 48(34) (2015), 343001. 
  38. J. Fan, T. Han, M. Haataja, Hydrodynamic effects on spinodal decomposition kinetics in planar lipid bilayer membranes, The Journal of Chemical Physics, 133(23) (2010), 235101. 
  39. J. W. Choi, H. G. Lee, D. Jeong, J. Kim, An unconditionally gradient stable numerical method for solving the Allen-Cahn equation, Physica A, 388(9) (2009), 1791-1803. https://doi.org/10.1016/j.physa.2009.01.026