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Three decoupled, second-order accurate, and energy stable schemes for the conserved Allen–Cahn-type block copolymer (BCP) model

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Abstract

In this paper, the numerical approximations of a new, Allen–Cahn-type block copolymer (BCP) model describing the phase transition of the block copolymer and homopolymer mixtures are considered. We first derive a new Allen–Cahn-type coupled phase-field model by using the L2-gradient flow and add two nonlocal Lagrange multipliers to the system to conserve the mass for each component. Then, we develop a series of efficient, unconditionally energy stable, non-iterative schemes based on the SAV, 3S-SAV, and new Lagrange multiplier approaches. At each time level, the developed numerical schemes are reduced to decoupled linear equations with constant coefficients, and their unconditional energy stabilities are strictly proved. Numerical examples are provided to validate the accuracy and energy stability of the schemes, and ample simulations are conducted to show the various pattern morphologies.

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References

  1. Avalos, E., Higuchi, T., Teramoto, T., Yabu, H., Nishiura, Y.: Frustrated phases under three-dimensional confinement simulated by a set of coupled Cahn–Hilliard equations. Soft Matter 12, 5905–5914 (2016)

    Article  Google Scholar 

  2. Mai, Y., Eisenberg, A.: Self-assembly of block copolymers. Chem. Soc. Rev. 41(18), 5969–5985 (2012)

    Article  Google Scholar 

  3. Glasner, K.: Evolution and competition of block copolymer nanoparticles. SIAM J. Appl. Math. 79(1), 28–54 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  4. Winey, K.I., Thomas, E.L., Fetters, L.J.: Isothermal morphology diagrams for binary blends of diblock copolymer and homopolymer. Macromolecules 25(10), 2645–2650 (1992)

    Article  Google Scholar 

  5. Hashimoto, T., Koizumi, S., Hasegawa, H.: Ordered structure in blends of block copolymers.2.self-assembly for immiscible lamella-forming copolymers. Macromolecules 27(6), 1562–1570 (1994)

    Article  Google Scholar 

  6. Jin, Z., Fan, H.: Self-assembly of nanostructured block copolymer nanoparticles. Soft Matter 10(46), 9212–9219 (2014)

    Article  Google Scholar 

  7. Zhang, K., Gao, L., Chen, Y., Yang, Z.: Onionlike spherical polymer composites with controlled dispersion of gold nanoclusters. Chem. Mater. 20(1), 23–25 (2008)

    Article  Google Scholar 

  8. Rahikkala, A., Soininen, A.J., Ruokolainen, J., Mezzenga, R., Raula, J., Kauppinen, E.I.: Self-assembly of PS-b-P4VP block copolymers of varying architectures in aerosol nanospheres. Soft Matter 9(5), 1492–1499 (2013)

    Article  Google Scholar 

  9. Deng, R., Liu, S., Li, J., Liao, Y., Tao, J., Zhu, J.: Mesoporous block copolymer nanoparticles with tailored structures by hydrogen-bonding-assisted self-assembly. Adv. Mater. 24(14), 1889–1893 (2012)

    Article  Google Scholar 

  10. Avalos, E., Teramoto, T., Komiyama, H., Yabu, H., Nishiura, Y.: Transformation of block copolymer nanoparticles from ellipsoids with striped lamellae into onionlike spheres and dynamical control via coupled Cahn–Hilliard equations. ACS Omega 3(1), 1304–1314 (2018)

    Article  Google Scholar 

  11. Li, Q., Mei, L.: Efficient, decoupled, and second-order unconditionally energy stable numerical schemes for the coupled Cahn-Hilliard system in copolymer/homopolymer mixtures. Comput. Phys. Commun. 260, 107290 (2021)

    Article  MathSciNet  Google Scholar 

  12. Martini, M., Sodini, G.E.: Numerical methods for a system of coupled Cahn-Hilliard equations. Commun. Appl. Ind. Math. 12(1), 1–12 (2021)

    MathSciNet  MATH  Google Scholar 

  13. Cheng, Q., Liu, C., Shen, J.: A new Lagrange multiplier approach for gradient flows. Comput. Methods Appl. Mech. Engrg. 367, 113070 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  14. Li, Y., Zhang, L., Xia, Q., Yu, Q., Kim, J.: An unconditionally energy-stable second-order time-accurate numerical scheme for the coupled Cahn–Hilliard system in copolymer/homopolymer mixtures. Comput. Mater. Sci. 200, 110809 (2021)

    Article  Google Scholar 

  15. Rubinstein, J., Sternberg, P.: Nonlocal reaction-diffusion equations and nucleation. IMA J. Appl. Math. 48(3), 249–264 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  16. Zhang, J., Yang, X.: Numerical approximations for a new L2-gradient flow based phase field crystal model with precise nonlocal mass conservation. Comput. Phys. Commun. 243, 51–67 (2019)

    Article  MathSciNet  Google Scholar 

  17. Lee, H.G.: A new conservative Swift-Hohenberg equation and its mass conservative method. J. Comput. Appl. Math. 375, 112815 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  18. Li, Q., Mei, L.: Numerical approximation of the two-component PFC models for binary colloidal crystals: efficient, decoupled, and second-order unconditionally energy stable schemes. J. Sci. Comput. 88(3), 60 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  19. Chen, C., Zhang, J., Yang, X.: Efficient numerical scheme for a new hydrodynamically-coupled conserved Allen-Cahn type Ohta-Kawaski phase-field model for diblock copolymer melt. Comput. Phys. Commun. 256, 107418 (2020)

    Article  MathSciNet  Google Scholar 

  20. Zhang, J., Yang, X.: Unconditionally energy stable large time stepping method for the L2-gradient flow based ternary phase-field model with precise nonlocal volume conservation. Comput. Methods Appl. Mech. Engrg. 361, 112743 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  21. Zhang, J., Chen, C., Yang, X., Chu, Y., Xia, Z.: Efficient, non-iterative, and second-order accurate numerical algorithms for the anisotropic Allen–Cahn Equation with precise nonlocal mass conservation. J. Comput. Appl. Math. 363, 444–463 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  22. Li, X., Li, T., Tu, R., Pan, K., Chen, C., Yang, X.: Efficient energy stable scheme for volume-conserved phase-field elastic bending energy model of lipid vesicles. J. Comput. Appl. Math. 385, 113177 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  23. Yang, X.: A novel fully-decoupled, second-order and energy stable numerical scheme of the conserved Allen-Cahn type flow-coupled binary surfactant model. Comput. Methods Appl. Mech. Engrg. 373, 113502 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  24. Jeong, D., Kim, J.: Conservative Allen-Cahn-Navier-Stokes system for incompressible two-phase fluid flows. Comput. Fluids 156, 239–246 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  25. Chai, Z., Sun, D., Wang, H., Shi, B.: A comparative study of local and nonlocal Allen-Cahn equations with mass conservation. Int. J. Heat Mass Transf. 122, 631–642 (2018)

    Article  Google Scholar 

  26. Shen, J., Yang, X.: Numerical approximations of Allen–Cahn and Cahn–Hilliard equations. Discrete Contin. Dyn. Syst. 28(4), 1669–1691 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Li, D., Qiao, Z.: On second order semi-implicit Fourier spectral methods for 2D Cahn–Hilliard equations. J. Sci. Comput. 70(1), 301–341 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  28. Eyre, D.J.: Unconditionally gradient stable time marching the Cahn–Hilliard equation. MRS Proceedings. vol. 529 (1998)

  29. Diegel, A.E., Wang, C., Wise, S.M.: Stability and convergence of a second-order mixed finite element method for the Cahn-Hilliard equation. IMA J. Numer. Anal. 36(4), 1867–1897 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  30. Yang, X., Zhao, J., Wang, Q.: Numerical approximations for the molecular beam epitaxial growth model based on the invariant energy quadratization method. J. Comput. Phys. 333, 104–127 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  31. Yang, X., Zhao, J.: Efficient linear schemes for the nonlocal Cahn-Hilliard equation of phase field models. Comput. Phys. Commun. 235, 234–245 (2019)

    Article  MathSciNet  Google Scholar 

  32. Li, Q., Mei, L., Yang, X., Li, Y.: Efficient numerical schemes with unconditional energy stabilities for the modified phase field crystal equation. Adv. Comput. Math. 45(3), 1551–1580 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  33. Shen, J., Xu, J., Yang, J.: The scalar auxiliary variable (SAV) approach for gradient flows. J. Comput. Phys. 353, 407–416 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  34. Shen, J., Xu, J., Yang, J.: A new class of efficient and robust energy stable schemes for gradient flows. SIAM Rev. 61(3), 474–506 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  35. Li, Q., Mei, L., Li, Y.: Efficient second-order unconditionally stable numerical schemes for the modified phase field crystal model with long-range interaction. J. Comput. Appl. Math. 389, 113335 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  36. Liu, Z., Li, X.: Step-by-step solving schemes based on scalar auxiliary variable and invariant energy quadratization approaches for gradient flows. Numer. Algorithms 89(1), 65–86 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  37. Cheng, Q., Liu, C., Shen, J.: Generalized SAV approaches for gradient systems. J. Comput. Appl. Math. 394, 113532 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  38. Yang, J., Kim, J.: The stabilized-trigonometric scalar auxiliary variable approach for gradient flows and its efficient schemes. J. Engrg. Math. 129, 18 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  39. Liu, Z., Li, X.: The exponential scalar auxiliary variable (E-SAV) approach for phase field models and its explicit computing. SIAM J. Sci. Comput. 42(3), 630–655 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  40. Ohta, T., Kawasaki, K.: Equilibrium morphology of block copolymer melt. Macromolecules 19(10), 2621–2632 (1986)

    Article  Google Scholar 

  41. 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  MATH  Google Scholar 

  42. Zhao, J.: A revisit of the energy quadratization method with a relaxation technique. Appl. Math. Lett. 120, 107331 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  43. Jiang, M., Zhang, Z., Zhao, J.: Improving the accuracy and consistency of the scalar auxiliary variable (SAV) method with relaxation. J. Comput. Phys. 456, 110954 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  44. Gong, Y., Hong, Q., Wang, Q.: Supplementary variable method for thermodynamically consistent partial differential equations. Comput. Methods Appl. Mech. Engrg. 381, 113746–19 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  45. Shen, J., Xu, J.: Convergence and error analysis for the scalar auxiliary variable (SAV) schemes to gradient flows. SIAM J. Numer. Anal. 56(5), 2895–2912 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  46. Li, X., Shen, J., Rui, H.: Energy stability and convergence of SAV block-centered finite difference method for gradient flows. Math. Comp. 88(319), 2047–2068 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  47. Wang, M., Huang, Q., Wang, C.: A second order accurate scalar auxiliary variable (SAV) numerical method for the square phase field crystal equation. J. Sci. Comput. 88, 33 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  48. Cheng, Q., Wang, C.: Error estimate of a second order accurate scalar auxiliary variable (SAV) numerical method for the epitaxial thin film equation. Adv. Appl. Math. Mech. 13(6), 1318–1354 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  49. Guo, J., Wang, C., Wise, S.M., Yue, X.: An H2 convergence of a second-order convex-splitting, finite difference scheme for the three-dimensional Cahn-Hilliard equation. Commun. Math. Sci. 14(2), 489–515 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  50. Cheng, K., Wang, C., Wise, S.M., Yue, X.: A second-order, weakly energy-stable pseudo-spectral scheme for the Cahn-Hilliard equation and its solution by the homogeneous linear iteration method. J. Sci. Comput. 69(3), 1083–1114 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  51. Yan, Y., Chen, W., Wang, C., Wise, S.M.: A second-order energy stable BDF numerical scheme for the Cahn-Hilliard equation. Commun. Comput. Phys. 23(2), 572–602 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  52. Li, Q., Mei, L., You, B.: A second-order, uniquely solvable, energy stable BDF numerical scheme for the phase field crystal model. Appl. Numer. Math. 134, 46–65 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  53. Cheng, K., Feng, W., Wang, C., Wise, S.M.: An energy stable fourth order finite difference scheme for the Cahn-Hilliard equation. J. Comput. Appl. Math. 362, 574–595 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  54. Guo, J., Wang, C., Wise, S.M., Yue, X.: An improved error analysis for a second-order numerical scheme for the Cahn-Hilliard equation. J. Comput. Appl. Math. 388, 113300 (2021)

    Article  MathSciNet  MATH  Google Scholar 

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Funding

Qi Li’s work is supported by the Fundamental Research Funds for the Central Universities, CHD (No. 300102121302), and the National Natural Science Foundation of China (No. 12101073). Supei Zheng’s work is supported by the National Natural Science Foundation of China (No. 11971075).

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Authors and Affiliations

  1. School of Science, Chang’an University, Xi’an, Shaanxi, 710064, People’s Republic of China

    Qi Li & Supei Zheng

  2. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi, 710049, People’s Republic of China

    Liquan Mei

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  1. Qi Li
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Correspondence to Supei Zheng.

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Li, Q., Zheng, S. & Mei, L. Three decoupled, second-order accurate, and energy stable schemes for the conserved Allen–Cahn-type block copolymer (BCP) model. Numer Algor 92, 1233–1259 (2023). https://doi.org/10.1007/s11075-022-01338-3

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