Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content

Advertisement

Springer Nature Link
Log in

Thermodynamically Consistent Algorithms for Models of Diblock Copolymer Solutions Interacting with Electric and Magnetic Fields

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

We derive thermodynamically consistent models for diblock copolymer solutions coupled with the electric and magnetic field, respectively. These models satisfy the second law of thermodynamics and are therefore thermodynamically consistent. We then design a set of 2nd order, linear, semi-discrete schemes for the models using the energy quadratization method and the supplementary variable method, respectively, which preserve energy dissipation rates of the models. The spatial discretization is carried out subsequently using 2nd order finite difference methods, leading to fully discrete, energy-dissipation-rate preserving algorithms that are thermodynamically consistent. Convergence rates are numerically confirmed through mesh refinement tests and several numerical examples are given to demonstrate the role of mobility in pattern formation, defect removing effect of both electric and magnetic fields as well as the hysteresis effect with respect to applied external fields in copolymer solutions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21

Similar content being viewed by others

References

  1. Badia, S., Guillén-González, F., Gutiérrez-Santacreu, J.V.: Finite element approximation of nematic liquid crystal flows using a saddle-point structure. J. Comput. Phys. 230(41), 1686–1706 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Böker, A., Elbs, H., Hänsel, H., Knoll, A., Ludwigs, S., Zettl, H., Zvelindovsky, A.V., Sevink, G.J.A., Urban, V., Abetz, V., Müller, A.H.E., Krausch, G.: Electric field induced alignment of concentrated block copolymer solutions. Macromolecules 36(21), 8078–8087 (2003)

    Article  Google Scholar 

  3. Chen, C., Li, X., Zhang, J., Yang, X.: Efficient linear, decoupled, and unconditionally stable scheme for a ternary Cahn-Hilliard type Nakazawa-Ohta phase-field model for tri-block copolymers. Appl. Math. Comput. 388(1), 125463 (2020)

    MathSciNet  MATH  Google Scholar 

  4. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Cheng, Q., Yang, X., Shen, J.: Efficient and accurate numerical schemes for a hydro-dynamically coupled phase field diblock copolymer model. J. Comput. Phys. 341, 44–60 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  7. Choksi, R., Maras, M., Williams, J.F.: 2D phase diagram for minimizers of a Cahn-Hilliard functional with long-range interactions. Siam J. Appl. Dyn. Syst. 10(4), 1344–1362 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Choksi, R., Peletier, M.A., Williams, J.F.: On the phase diagram for microphase separation of diblock copolymers: an approach via a nonlocal Cahn-Hilliard functional. Siam J. Appl. Math. 69(6), 1712–1738 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Choksi, R., Ren, X.: On the derivation of a density functional theory for microphase separation of diblock copolymers. J. Stat. Phys. 113(1–2), 151–176 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Choksi, R., Ren, X.: Diblock copolymer/homopolymer blends: derivation of a density functional theory. Physica D 203(1–2), 100–119 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Copetti, M.I.M., Elliott, C.M.: Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy. Numer. Math. 63(1), 39–65 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  12. Faghihi, N., Mkhonta, S., Elder, K.R., Grant, M.: Magnetic islands modelled by a phase-field-crystal approach. Eur. Phys. J. B 91, 55 (2018)

    Article  MathSciNet  Google Scholar 

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

  14. Guillén-González, F., Tierra, G.: On linear schemes for a Cahn-Hilliard diffuse interface model. J. Comput. Phys. 234, 140–171 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Guillén-González, F., Tierra, G.: Second order schemes and time-step adaptivity for Allen-Cahn and Cahn-Hilliard models. Comput. Math. Appl. 68(8), 821–846 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration Structure-preserving Algorithms for Ordinary Differential Equations. Springer, Berlin (2006)

    MATH  Google Scholar 

  17. Hamley, I.W.: The Physics of Block Copolymers. Oxford University Press, Oxford (1998)

    Google Scholar 

  18. Hamley, I.W.: Developments in Block Copolymer Science and Technology. Wiley, New York (2004)

    Book  Google Scholar 

  19. Hamley, I.W.: Block copolymers in solution: fundamentals and applications. Wiley, New York (2005)

    Book  Google Scholar 

  20. Hong, Q., Li, J., Wang, Q.: Supplementary variable method for structure-preserving approximations to partial differential equations with deduced equations. Appl. Math. Lett. 110, 106576 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  21. Landau, L.D., Mikhaílovich Lifshitz, E.: Electrodynamics of Continuous Media. Elsevier, Amsterdam (1984)

    Google Scholar 

  22. Ly, D.Q., Makatsoris, C.: Effects of the homopolymer molecular weight on a diblock copolymer in a 3D spherical confinement. BMC Chem. 13(1), 24 (2019)

    Article  Google Scholar 

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

    Article  Google Scholar 

  24. Ohta, T., Nonomura, M.: Elastic property of bilayer membrane in copolymer-homopolymer mixtures. Eur. Phys. J. B 2(1), 57–68 (1998)

    Article  Google Scholar 

  25. Onsager, L.: Reciprocal relations in irreversible processes. I. Phys. Rev. 37, 405–426 (1931)

    Article  MATH  Google Scholar 

  26. Onsager, L.: Reciprocal relations in irreversible processes. II. Phys. Rev. 38, 2265–2279 (1931)

    Article  MATH  Google Scholar 

  27. Onsager, L., Machlup, S.: Fluctuations and irreversible processes. Phys. Rev. 91, 1505–1512 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  28. Orizaga, S., Glasner, K.: Instability and reorientation of block copolymer microstructure by imposed electric fields. Phys. Rev. E. 93(5), 052504 (2016)

    Article  Google Scholar 

  29. Pestera, C.W., Liedel, C., Ruppel, M., Böker, A.: Block copolymers in electric fields. Prog. Polym. Sci. 64, 182–214 (2017)

    Article  Google Scholar 

  30. Pinna, M., Schreier, L., Zvelindovsky, A.V.: Mechanisms of electric-field-induced alignment of block copolymer lamellae. Soft Matter 5, 970–973 (2009)

    Article  Google Scholar 

  31. Seymour, M., Sanches, F., Elde, K., Provatas, N.: Phase-field crystal approach for modeling the role of microstructure in multiferroic composite materials. Phys. Rev. E 92, 184109 (2015)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  33. Shen, J., Jie, X., 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 

  34. Shirokoff, D., Choksi, R., Nave, J.-C.: Sufficient conditions for global minimality of metastable states in a class of non-convex functionals: a simple approach via quadratic lower bounds. J. Nonlinear Sci. 25(3), 539–582 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  35. van den Berg, J.B., Williams, J.F.: Validation of the bifurcation diagram in the 2D Ohta-Kawasaki problem. Nonlinearty 30(4), 1584–1638 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  36. van den Berg, B., Jan, Williams J.F: Rigorously computing symmetric stationary states of the Ohta-Kawasaki problem in three dimensions. Siam J. Math. Aanl. 51(1), 131–158 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  37. Wang, Q.: Generalized onsager principle and its applications, Chapter 3. In: Xiangyang, L. (ed.) Frontiers and Progress of Current Soft Matter Research. Springer, Singapore (2021)

    Google Scholar 

  38. Wu, J., Wang, Z., Yin, Y., Jiang, R., Li, B., Shi, A.C.: A simulation study of phase behavior of double-hydrophilic block copolymers in aqueous solutions. Macromolecules 48(24), 8897–8906 (2014)

    Article  Google Scholar 

  39. Xiangfa, W., Dzenis, Y.A.: Phase-field modeling of the formation of lamellar nanostructures in diblock copolymer thin films under inplanar electric fields. Phys. Rev. E. 77(511), 031807 (2008)

    Google Scholar 

  40. Xu, T., Zvelindovsky, A.V., Sevink, G.J., Gang, O., Ocko, B., Zhu, Y., Gido, S.P., Russell, T.P.: Electric field induced sphere-to-cylinder transition in diblock copolymer thin films. Macromolecules 37(18), 6980–6984 (2004)

    Article  Google Scholar 

  41. Xu, T., Zvelindovsky, A.V., Sevink, G.J., Gang, O., Ocko, B., Zhu, Y., Gido, S.P., Russell, T.P.: Electric field alignment of asymmetric diblock copolymer thin films. Macromolecules 38(26), 10788–10798 (2005)

    Article  Google Scholar 

  42. Yamada, K., Kawabata, E.Y.Y., Kato, T.O.T.: Mesoscopic simulation of phase behaviors and structures in an amphiphile-solvent system. Phys. Rev. E 89(6), 062310 (2014)

    Article  Google Scholar 

  43. Yang, X., Zhao, J.: On linear and unconditionally energy stable algorithms for variable mobility Cahn-Hilliard equation with logarithmic Flory-Huggins potential. Comm. Comp. Phys. 25, 703–728 (2019)

    MathSciNet  Google Scholar 

  44. Yang, X., Li, J., Gregory Forest, M., Wang, Q.: Hydrodynamic theories for flows of active liquid crystals and the Generalized Onsager principle. Entropy 18(6), 202 (2016)

    Article  MathSciNet  Google Scholar 

  45. Yang, Z., Dong, S.: A roadmap for discretely energy-stable schemes for dissipative systems based on a generalized auxiliary variable with guaranteed positivity. J. Comput. Phys. 404, 109121 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  46. Yin, J., Wang, Y., Chen, J.Z., Zhang, P., Zhang, L.: Construction of a pathway map on a complicated energy landscape. Phys. Rev. Lett. 124(9), 090601 (2020)

    Article  MathSciNet  Google Scholar 

  47. Yu, B., Zhang, L.: Global optimization-based dimer method for finding saddle points. Discrete Cont. Dyn.-B 26(1), 741–753 (2021)

    MathSciNet  MATH  Google Scholar 

  48. Zhang, J., Chen, C., Yang, X.: Efficient and energy stable method for the Cahn-Hilliard phase-field model for diblock copolymers. Appl. Numer. Math. 151, 263–281 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  49. Zhang, J., Chen, C., Yang, X., Pan, K.: Efficient numerical scheme for a penalized Allen-Cahn type Ohta-Kawasaki phase-field model for diblock copolymers. J. Comput. Appl. Math. 378, 112905 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  50. Zhang, J., Yang, X.: A new magnetic-coupled Cahn-Hilliard phase-field model for diblock copolymers and its numerical approximations. Appl. Math. Lett. 107, 106412 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  51. Zhao, J., Yang, X., Gong, Y., Zhao, X., Yang, X., Li, J., Wang, Q.: A general strategy for numerical approximations of non-equilibrium models-part i: thermodynamical systems. Int. J. Numer. Anal. Mod. 15(6), 884–918 (2018)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Research at CSRC is partially supported by the National Natural Science Foundation of China (award 11971051 and NSAF-U1930402). Qi Wang’s research is partially supported by a DOE grant (DE-SC0020272), National Science Foundation grants (award DMS-1815921 and OIA-1655740) and a GEAR award from SC EPSCoR/IDeA Program.

Author information

Authors and Affiliations

  1. Beijing Computational Science Research Center, Beijing, 100193, China

    Xiaowen Shen

  2. Department of Mathematics, University of South Carolina, Columbia, SC, 29208, USA

    Qi Wang

Authors
  1. Xiaowen Shen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qi Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qi Wang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shen, X., Wang, Q. Thermodynamically Consistent Algorithms for Models of Diblock Copolymer Solutions Interacting with Electric and Magnetic Fields. J Sci Comput 88, 43 (2021). https://doi.org/10.1007/s10915-021-01470-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-021-01470-7

Keywords

Search

Navigation