The European Physical Journal E

, Volume 20, Issue 4, pp 449–457 | Cite as

Self-assembled pattern formation of block copolymers on the surface of the sphere using self-consistent field theory

  • J. F. Li
  • J. Fan
  • H. D. Zhang
  • F. Qiu
  • P. Tang
  • Y. L. Yang
Regular Article


The spherical surface is spatially discretized with triangular lattices to numerically calculate the Laplace-Beltrami operator contained in the self-consistent field theory (SCFT) equations using a finite volume method. Based on this method we have developed a spherical alternating-direction implicit (ADI) scheme for the first time to help extend real-space implementation of SCFT in 2D flat space to the surface of the sphere. By using this method, we simulate the equilibrium microphase separation morphology of block copolymers including AB diblocks, ABC linear triblocks and ABC star triblock copolymers occurred on the spherical surface. In general, two classes of microphase separation morphologies such as striped patterns for compositionally symmetric block copolymers and spotted patterns for asymmetric compositions have been found. In contrast to microphase separation morphology in 2D flat space, the geometrical characteristics of a sphere has a large influence on the self-assembled morphology. For striped patterns, several of spiral-form and ring-form patterns are found by changing the ratio of the radius of a sphere to the averaging width of the stripes. The specific pattern such as the striped and spotted pattern with intrinsic dislocations or defects stems from formed periodic patterns due to microphase separation of block copolymers arranged on the curved surface.


83.80.Uv Block copolymers 36.20.-r Macromolecules and polymer molecules 68.08.De Structure: measurements and simulations 


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  1. 1.
    A.M. Jackson, J.W. Myerson, F. Stellacci, Nature Mater. 3, 330 (2004).CrossRefGoogle Scholar
  2. 2.
    Y.Y. Wu, G.S. Cheng, K. Katsov, S.W. Sides, J.F. Wang, J. Tang, G.H. Fredrickson, M. Moskovits, G.D. Stucky, Nature Mater. 3, 816 (2004).CrossRefGoogle Scholar
  3. 3.
    H. Yabu, T. Higuchi, M. Shimomura, Adv. Mater. 17, 2062 (2005).CrossRefGoogle Scholar
  4. 4.
    G.J.A. Sevink, A.V. Zvelindovsky, J.G.E.M. Fraaije, H.P. Huinink, J. Chem. Phys. 115, 8226 (2001).CrossRefADSGoogle Scholar
  5. 5.
    H.P. Huinink, J.C.M. Brokken-Zijp, M.A. van Dijk, G.J.A. Sevink, J. Chem. Phys. 112, 2452 (2000).CrossRefADSGoogle Scholar
  6. 6.
    M.A. van Dijk, R. van den Berg, Macromolecules 28, 6773 (1995).CrossRefGoogle Scholar
  7. 7.
    C. Varea, J.L. Aragon, R.A. Barrio, Phys. Rev. E 60, 4588 (1999).CrossRefADSGoogle Scholar
  8. 8.
    Y. Jiang, T. Lookman, A. Saxena, Phys Rev. E 61, R57 (2000).Google Scholar
  9. 9.
    Y. Jiang, T. Lookman, A. Saxena, Biophys. J. 78, 182A (2000).Google Scholar
  10. 10.
    P. Tang, F. Qiu, H.D. Zhang, Y.L. Yang, Phys. Rev. E 72, 016710 (2005).CrossRefADSGoogle Scholar
  11. 11.
    F. Drolet, G.H. Fredrickson, Phys. Rev. Lett. 83, 4317 (1999).CrossRefADSGoogle Scholar
  12. 12.
    F. Drolet, G.H. Fredrickson, Macromolecules 34, 5317 (2001).CrossRefGoogle Scholar
  13. 13.
    P. Tang, F. Qiu, H.D. Zhang, Y.L. Yang, Phys. Rev. E 69, 031803 (2004).CrossRefADSGoogle Scholar
  14. 14.
    P. Tang, F. Qiu, H.D. Zhang, Y.L. Yang, J. Phys. Chem. B 108, 8434 (2004).CrossRefGoogle Scholar
  15. 15.
    W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes (Cambridge University Press, Cambridge, England, 1989).Google Scholar
  16. 16.
    D.R. Nelson, T. Piran, S. Weinberg, Statistical Mechanics of Membranes and Surfaces (World Scientific, Singapore, 1989).Google Scholar
  17. 17.
    E.J. Helfand, J. Chem. Phys 62, 999 (1975).CrossRefADSGoogle Scholar
  18. 18.
    S.F. Edwards, Proc. Phys. Soc. London 85, 613 (1965).MATHCrossRefGoogle Scholar
  19. 19.
    M.W. Matsen, M.Schick, Phys. Rev. Lett. 72, 2660 (1994).CrossRefADSGoogle Scholar
  20. 20.
    J.R. Baumgardner, P.O. Frederickson, SIAM J. Numer. Anal. 22, (1985) 1107.Google Scholar
  21. 21.
    A.R. Bausch, M.J. Bowick, A. Cacciuto, A.D. Dinsmore, M.F. Hsu, D.R. Nelson, M.G. Nikolaides, A. Travesset, D.A. Weitz, Science 299, 1716 (2003).CrossRefADSGoogle Scholar
  22. 22.
    T. Kohyama, D.M. Kroll, G. Gompper, Phys. Rev. E 68, 061905 (2003).CrossRefADSGoogle Scholar
  23. 23.
    M. Meyer, VisMath Proceedings, Berlin, Germany, 2002, diffGeorOps.pdf. Google Scholar
  24. 24.
    M.W. Matsen, F.S. Bates, Macromolecules 29, 1091 (1996).CrossRefGoogle Scholar
  25. 25.
    J. Gomatam, F. Amdjadi, Phys. Rev. E 56, 3913 (1997).MathSciNetCrossRefADSGoogle Scholar
  26. 26.
    M. Bowick, A. Cacciuto, D.R. Nelson, A. Travesset, Phys. Rev. Lett. 89, 185502 (2002).CrossRefADSGoogle Scholar
  27. 27.
    J.J. Thomson, Philos Mag. 7, 237 (1904).MATHMathSciNetGoogle Scholar
  28. 28.
    D.R. Nelson, Nano Lett. 2, 1125 (2002).CrossRefGoogle Scholar
  29. 29.
    T. Stehle, S.J. Gamblin, Y.W. Yan, S.C. Harrison, Structure 4, 165 (1996).CrossRefGoogle Scholar
  30. 30.
    M. Fialkowski, A. Bitner, B.A. Grzykowski, Nature Materials 4, 93 (2005).CrossRefGoogle Scholar

Copyright information

© EDP Sciences, Società Italiana di Fisica and Springer-Verlag 2006

Authors and Affiliations

  • J. F. Li
    • 1
    • 2
  • J. Fan
    • 1
    • 2
  • H. D. Zhang
    • 1
    • 2
  • F. Qiu
    • 1
    • 2
  • P. Tang
    • 1
    • 2
  • Y. L. Yang
    • 1
    • 2
  1. 1.The Key Laboratory of Molecular Engineering of PolymersMinistry of Education of ChinaShanghaiPRC
  2. 2.Department of Macromolecular ScienceFudan UniversityShanghaiPRC

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