Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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A NONLINEAR CONVEX SPLITTING FOURIER SPECTRAL SCHEME FOR THE CAHN-HILLIARD EQUATION WITH A LOGARITHMIC FREE ENERGY

  • Received : 2018.03.16
  • Accepted : 2018.10.11
  • Published : 2019.01.31
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Abstract

For a simple implementation, a linear convex splitting scheme was coupled with the Fourier spectral method for the Cahn-Hilliard equation with a logarithmic free energy. However, an inappropriate value of the splitting parameter of the linear scheme may lead to incorrect morphologies in the phase separation process. In order to overcome this problem, we present a nonlinear convex splitting Fourier spectral scheme for the Cahn-Hilliard equation with a logarithmic free energy, which is an appropriate extension of Eyre's idea of convex-concave decomposition of the energy functional. Using the nonlinear scheme, we derive a useful formula for the relation between the gradient energy coefficient and the thickness of the interfacial layer. And we present numerical simulations showing the different evolution of the solution using the linear and nonlinear schemes. The numerical results demonstrate that the nonlinear scheme is more accurate than the linear one.

Keywords

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FIGURE 1. Spinodal decomposition kinetics of polycarbonateand polystyrene 50 : 50 at various times: (a) 50, (b) 100, (c)300, (d) 500, (e) 700, and (f) 1000s. Reprinted with permissionfrom [1]. Copyright (2001) American Chemical Society.

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FIGURE 2. φeq for various θ with θc = 1.

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FIGURE 3. Interfacial layer at the equilibrium state for various θ with (a) ϵ = 0.5h and (b) ϵ = h. Here, θc = 1 is used.

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FIGURE 4. Surface fitting: ϵ(θ,m) = 0.1h(4.835θ2−3.752θm+0.005m2 − 4.01θ + 4.078m+ 0.693).

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FIGURE 5. Evolution of φ(x, t) with ϵ(θ, 8) for various θ. Ineach subfigure, horizontal dashed lines represent the rangefrom φ = −0.9φeq to φ = 0.9φeq.

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FIGURE 6. C(θ, φeq) for various θ with θc = 1.

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FIGURE 7. Evolution of φ(x, y, t) using (a) the linear scheme with s = 1.0548, (b) the linear scheme with s = 115.0843, and (c) the nonlinear scheme. Here, θ = 0.8, θc = 1, and ϵ = ϵ(θ, 8) are used. The times are t = 100, 400, 700, and 1000 (from left to right).

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FIGURE 8. Evolution of φ(x, y, t) with φ= −0.3. Here, θ =0.8, θc = 1, and ϵ = ϵ(θ, 8) are used. The times are shown below each subfigure.

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FIGURE 9. Evolution of φ(x, y, t) with φ= 0. Here, θ = 0.8, θc = 1, and ϵ = ϵ(θ, 8) are used. The times are shown below each subfigure.

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FIGURE 10. Evolution of the energy with φ= −0.3 and 0.

TABLE 1. Numbers of grid points in the interfacial layer (fromφ = −0.9φeq to φ = 0.9φeq).

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TABLE 2. Values of s for various θ with θc = 1.

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References

  1. J. T. Cabral, J. S. Higgins, T. C. B. McLeish, S. Strausser, and S. N. Magonov, Bulk spinodal decomposition studied by atomic force microscopy and light scattering, Macromolecules 34 (2001), 3748-3756. https://doi.org/10.1021/ma0017743
  2. J. W. Cahn, On spinodal decomposition, Acta Metall. 9 (1961), 795-801. https://doi.org/10.1016/0001-6160(61)90182-1
  3. 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
  4. 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
  5. M. Cheng and A. D. Rutenberg, Maximally fast coarsening algorithms, Phys. Rev. E 72 (2005), 055701(R). https://doi.org/10.1103/PhysRevE.72.055701
  6. J.-W. Choi, H. G. Lee, D. Jeong, and J. Kim, An unconditionally gradient stable numerical method for solving the Allen-Cahn equation, Phys. A 388 (2009), no. 9, 1791-1803. https://doi.org/10.1016/j.physa.2009.01.026
  7. M. I. M. Copetti and C. M. Elliott, Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy, Numer. Math. 63 (1992), no. 1, 39-65. https://doi.org/10.1007/BF01385847
  8. D. J. Eyre, Unconditionally gradient stable time marching the Cahn-Hilliard equation, in Computational and mathematical models of microstructural evolution (San Francisco, CA, 1998), 39-46, Mater. Res. Soc. Sympos. Proc., 529, 1998.
  9. 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
  10. D. Jeong and J. Kim, A practical numerical scheme for the ternary Cahn-Hilliard system with a logarithmic free energy, Phys. A 442 (2016), 510-522. https://doi.org/10.1016/j.physa.2015.09.038
  11. D. Jeong and J. Kim, Practical estimation of a splitting parameter for a spectral method for the ternary Cahn-Hilliard system with a logarithmic free energy, Math. Methods Appl. Sci. 40 (2017), no. 5, 1734-1745. https://doi.org/10.1002/mma.4093
  12. J. 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
  13. D. Lee, J.-Y. Huh, D. Jeong, J. Shin, A. Yun, and J. Kim, Physical, mathematical, and numerical derivations of the Cahn-Hilliard equation, Comp. Mater. Sci. 81 (2014), 216-225. https://doi.org/10.1016/j.commatsci.2013.08.027
  14. The MathWorks, Inc., Natick, MA, USA.
  15. J. Shin, D. Jeong, and J. Kim, A conservative numerical method for the Cahn-Hilliard equation in complex domains, J. Comput. Phys. 230 (2011), no. 19, 7441-7455. https://doi.org/10.1016/j.jcp.2011.06.009
  16. J. Shin, S. Kim, D. Lee, and J. Kim, A parallel multigrid method of the Cahn-Hilliard equation, Comp. Mater. Sci. 71 (2013), 89-96. https://doi.org/10.1016/j.commatsci.2013.01.008
  17. J. Shin, H. G. Lee, and J.-Y. Lee, Convex Splitting Runge-Kutta methods for phase-field models, Comput. Math. Appl. 73 (2017), no. 11, 2388-2403. https://doi.org/10.1016/j.camwa.2017.04.004
  18. H. Song, Energy SSP-IMEX Runge-Kutta methods for the Cahn-Hilliard equation, J. Comput. Appl. Math. 292 (2016), 576-590. https://doi.org/10.1016/j.cam.2015.07.030
  19. S. Zhai, Z. Weng, and X. Feng, Fast explicit operator splitting method and time-step adaptivity for fractional non-local Allen-Cahn model, Appl. Math. Model. 40 (2016), no. 2, 1315-1324. https://doi.org/10.1016/j.apm.2015.07.021