Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

AN UNCONDITIONALLY GRADIENT STABLE NUMERICAL METHOD FOR THE OHTA-KAWASAKI MODEL

  • Kim, Junseok (Department of Mathematics Korea University) ;
  • Shin, Jaemin (Institute of Mathematical Sciences Ewha W. University)
  • Received : 2015.11.27
  • Published : 2017.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

We present a finite difference method for solving the Ohta-Kawasaki model, representing a model of mesoscopic phase separation for the block copolymer. The numerical methods for solving the Ohta-Kawasaki model need to inherit the mass conservation and energy dissipation properties. We prove these characteristic properties and solvability and unconditionally gradient stability of the scheme by using Hessian matrices of a discrete functional. We present numerical results that validate the mass conservation, and energy dissipation, and unconditional stability of the method.

Keywords

References

  1. M. Bahiana and Y. Oono, Cell dynamical system approach to block copolymers, Phys. Rev. A 41 (1990), 6763-6771. https://doi.org/10.1103/PhysRevA.41.6763
  2. J. W. Barrett, J. F. Blowey, and H. Garcke, Finite Element Approximation of the Cahn-Hilliard Equation with Degenerate Mobility, SIAM J. Numer. Anal. 37 (1999), no. 1, 286-318. https://doi.org/10.1137/S0036142997331669
  3. F. S. Bates and G. H. Fredrickson, Block copolymers-designer soft materials, Phys. Today. 37 (1999), 32-38.
  4. A. Brandt, Multi-level adaptive solutions to boundary-value problems, Math. Comp. 31 (1977), no. 138, 333-390. https://doi.org/10.1090/S0025-5718-1977-0431719-X
  5. J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys. 28 (1958), 258-267. https://doi.org/10.1063/1.1744102
  6. L. Q. Chen and J. Shen, Applications of semi-implicit Fourier-spectral method to phase field equations, Comput. Phys. Commun. 108 (1998), 147-158. https://doi.org/10.1016/S0010-4655(97)00115-X
  7. R. Choksi, M. Maras, and J. F. Williams, 2D phase diagram for minimizers of a Cahn-Hilliard functional with long-range interactions, SIAM J. Appl. Dyn. Syst. 10 (2011), no. 4, 1344-1362. https://doi.org/10.1137/100784497
  8. R. Choksi, M. A. Peletier, and J. F. Williams, On the phase diagram for microphase separation of diblock copolymers: an approach via a nonlocal Cahn-Hilliard functional, SIAM J. Appl. Math. 69 (2009), no. 6, 1712-1738. https://doi.org/10.1137/080728809
  9. F. Drolet and G. H. Fredrickson, Combinatorial screening of complex block copolymer assembly with self-consistent field theory, Phys. Rev. Lett. 83 (1999), 4317-4320. https://doi.org/10.1103/PhysRevLett.83.4317
  10. Q. Du and R. A. Nicolaides, Numerical analysis of a continuum model of phase transition, SIAM J. Numer. Anal. 28 (1991), no. 5, 1310-1322. https://doi.org/10.1137/0728069
  11. D. J. Eyre, An unconditionally stable one-step scheme for gradient systems, Unpublished article, http://www.math.utah.edu/-eyre/research/methods/stable.ps (1998).
  12. D. J. Eyre, Unconditionally gradient stable time marching the Cahn-Hilliard equation, Computational and mathematical models of microstructural evolution (San Francisco, CA, 1998), 39-46, Mater. Res. Soc. Sympos. Proc., 529, MRS, Warrendale, PA, 1998.
  13. D. Furihata, A stable and conservative finite difference scheme for the Cahn-Hilliard Equation, Numer. Math. 87 (2001), no. 4, 675-699. https://doi.org/10.1007/PL00005429
  14. H. Gomez, V. M. Calo, Y. Bazilevs, and T. J. Hughes, Isogeometric analysis of the Cahn-Hilliard phase-field model, Comput. Methods Appl. Mech. Engrg. 197 (2008), no. 49-50, 4333-4352. https://doi.org/10.1016/j.cma.2008.05.003
  15. Z. Guan, C.Wang, and S. M.Wise, A convergent convex splitting scheme for the periodic nonlocal Cahn-Hilliard equation, Numer. Math. 128 (2014), no. 2, 377-406. https://doi.org/10.1007/s00211-014-0608-2
  16. D. Jeong, J. Shin, Y. Li, Y. Choi, J. H. Jung, S. Lee, and J. S. Kim, Numerical analysis of energy-minimizing wavelengths of equilibrium states for diblock copolymers, Curr. Appl. Phys. 14 (2014), 1263-1272. https://doi.org/10.1016/j.cap.2014.06.016
  17. J. S. Kim, A numerical method for the Cahn-Hilliard equation with a variable mobility, Commun. Nonlinear Sci. Numer. Simul. 12 (2007), no. 8, 1560-1571. https://doi.org/10.1016/j.cnsns.2006.02.010
  18. J. S. Kim, Phase-field models for multi-component fluid flows, Commun. Comput. Phys. 12 (2012), no. 3, 613-661. https://doi.org/10.4208/cicp.301110.040811a
  19. J. S. Kim and H. O. Bae, An unconditionally gradient stable adaptive mesh refinement for the Cahn-Hilliard equation, J. Korean Phys. Soc. 53 (2008), 672-679. https://doi.org/10.3938/jkps.53.672
  20. J. S. Kim, K. Kang, and J. S. Lowengrub, Conservative multigrid methods for Cahn-Hilliard fluids, J. Comput. Phys. 193 (2004), no. 2, 511-543. https://doi.org/10.1016/j.jcp.2003.07.035
  21. Y. Nishiura and I. Ohnishi, Some mathematical aspects of the micro-phase separation in diblock copolymers, Phys. D 84 (1995), no. 1-2, 31-39. https://doi.org/10.1016/0167-2789(95)00005-O
  22. T. Ohta and K. Kawasaki, Equilibrium morphology of block copolymer melts, Macromol. 19 (1986), 2621-2632. https://doi.org/10.1021/ma00164a028
  23. M. Pinna and A. V. Zvelindovsky, Large scale simulation of block copolymers with cell dynamics, Eur. Phys. J. B 85 (2012), 1-18. https://doi.org/10.1140/epjb/e2011-20818-1
  24. J. Qin, G. S. Khaira, Y. Su, G. P. Garner, M. Miskin, H. M. Jaeger, and J. J. de Pablo, Evolutionary pattern design for copolymer directed self-assembly, Soft Matter 9 (2013), 11467-11472. https://doi.org/10.1039/c3sm51971f
  25. J. Shin, Y. Choi, and J. S. Kim, An unconditionally stable numerical method for the viscous Cahn-Hilliard equation, Discret. Contin. Dyn. Syst. Ser. B 19 (2014), no. 6, 1737-1747. https://doi.org/10.3934/dcdsb.2014.19.1737
  26. U. Trottenberg, C. Oosterlee, and A. Schuller, Multigrid, Academic press, London, 2001.