Japan Journal of Industrial and Applied Mathematics

, Volume 27, Issue 2, pp 175–190 | Cite as

Morphological characterization of the diblock copolymer problem with topological computation

Original Paper Area 1

Abstract

Our subject is the diblock copolymer problem in a three-dimensional space. Using numerical simulations, the double gyroid and orthorhombic morphologies are obtained as energy minimizers. By investigating the geometric properties of these bicontinuous morphologies, we demonstrate the underlying mechanism affecting the triply periodic energy minimizers in terms of a balanced scaling law. We also apply computational homology to their characterization during the dynamics of morphology transition. Our topological approaches detect the morphology of transient perforated layers as they transition from layers to cylinders, and the t−1 law of the Betti number in the phase ordering process.

Keywords

Double gyroid Microphase separation The Betti number 

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References

  1. 1.
    A Collection Papers of T. Hashimoto: ordered structures, order–disorder transition, and physical properties of block copolymers , Dept. of Polym. Chem., Kyoto University (1995)Google Scholar
  2. 2.
    Aksimentiev A., Fiałkowski M., Hołyst R.: Morphology of surfaces in polymer, surfactant, electron and reaction-diffusion systems: methods, measurements and simulations. Adv. Chem. Phys 121, 143–239 (2002)Google Scholar
  3. 3.
    Bahiana M., Oono Y.: Cell dynamical system approach to block copolymers. Phys. Rev. A 41, 6763–6771 (1990)CrossRefGoogle Scholar
  4. 4.
    Bailey T.S., Hardy C.M., Epps T.H. III, Bates F.S.: A noncubic triply periodic network morphology in poly(isoprene-b-styrene-b-ethylene oxide) triblock copolymers. Macromolecules 35, 7007–7017 (2002)CrossRefGoogle Scholar
  5. 5.
    Bates F.S., Fredrickson G.H.: Block copolymers—designer soft materials. Phys Today 52(2), 32–38 (1999)CrossRefGoogle Scholar
  6. 6.
    Braides A.: Γ-Convergence for Beginners. Oxford University Press, Oxford (2002)MATHCrossRefGoogle Scholar
  7. 7.
    Chen X., Oshita Y.: Applications of modular functions to interfacial dynamics. Arch. Rat. Mech. Anal. 186, 109–132 (2007)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Choksi R., Ren X.: On the derivation of a density functional theory for microphase separation of diblock copolymers. J. Stat. Phys. 113, 151–176 (2003)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Choksi R., Sternberg P.: Periodic phase separation: the periodic Cahn-Hilliard and isoperimetric problem. Interfaces Free Boundaries 8, 371–392 (2006)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Eyre, D.J.: Unconditionally gradient stable time marching the Cahn-Hilliard equation in computational and mathematical models of microstructual evolution. In: Bullard, J.W., Kalita, R., Stoneham, M., Chen, L.-Q. (eds.) (MRS, 1998)Google Scholar
  11. 11.
    Gameiro M., Mischaikow K., Wanner T.: Evolution of pattern complexity in the Cahn-Hilliard theory of phase separation. Acta Materialia 53, 693–704 (2005)CrossRefGoogle Scholar
  12. 12.
    Goźd́ź W.T., Hołyst R.: Triply periodic surfaces and multiply continuous structures from the Landau model of microemulsions. Phys. Rev. E 54, 5012–5027 (1996)CrossRefGoogle Scholar
  13. 13.
    Grosse-Brauckmann K.: On gyroid interfaces. J. Colloid Interface Sci. 187, 418–428 (1997)CrossRefGoogle Scholar
  14. 14.
    Hagita K., Teramoto T.: Topological validation of morphology modeling by extended reverse Monte Carlo analysis. Phys. Rev. E 77, 056704 (2008)CrossRefGoogle Scholar
  15. 15.
    Hajduk D.A., Harper P.E., Gruner S.M., Honeker C.C., Kim G., Thomas E.L.: The gyroid: a new equilibrium morphology in weakly segregated diblock copolymers. Macromolecules 27, 4063–4075 (1994)CrossRefGoogle Scholar
  16. 16.
    Ishimura, N., Ishiwata, T., Skajo, T., Sakurai, T., Nagayama, M., Nara, T., Hayami, K., Furihata, D., Matsuo, T.: Computational homology and its applications, Hokkaido University Technical Report Series in Mathematics, No. 124 (in Japanese) (2007)Google Scholar
  17. 17.
    Kaczynski T., Mischaikow K., Mrozek M.: Computational Homology. Springer, New York (2004)MATHGoogle Scholar
  18. 18.
    Kang X., Ren X.: Ring pattern solutions of a free boundary problem in diblock copolymer morphology. Physica D 238, 645–665 (2009)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Langer, J.S.: An introduction to the kinetics of first-order phase transitions, solids far from equilibrium. In: Godreche, G. (ed.). Cambridge University Press, Cambridge (1992)Google Scholar
  20. 20.
    Nishiura, Y.: Far-from-equilibrium dynamics, Translations of Mathematical Monographs Vol. 209, AMS (2002)Google Scholar
  21. 21.
    Nishiura Y., Ohnishi I.: Some mathematical aspects of the micro-phase separation in diblock copolymers. Physica D 84, 31–39 (1995)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Nonomura M., Yamada K., Ohta T.: Formation and stability of double gyroid in microphase-separated diblock copolymers. J. Phys. Condens. Matter 15, L423–L430 (2003)CrossRefGoogle Scholar
  23. 23.
    Ohta T., Kawasaki T.: Equilibrium morphology of block copolymer melts. Macromolecules 19, 2621–2632 (1986)CrossRefGoogle Scholar
  24. 24.
    Ren X., Wei J.: On the multiplicity of two nonlocal variational problems. SIAM J. Math. Anal. 31, 909–924 (2000)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Spivak M.: A Comprehensive Introduction to Differential Geometry. Publish or Perish, Berkley (1979)Google Scholar
  26. 26.
    Teramoto T.: Nosé thermostat for the pattern formation dynamics. Mol. Sim. 33, 71–75 (2007)CrossRefGoogle Scholar
  27. 27.
    Teramoto T., Nishiura Y.: Double gyroid morphology in a gradient system with nonlocal effects. J. Phys. Soc. Jpn. 71, 1611–1614 (2002)CrossRefGoogle Scholar
  28. 28.
    Thomas E.L., Anderson D.M., Henkee C.S., Hoffman D.: Periodic area-minimizing surfaces in block copolymers. Nature 334, 598–601 (1988)CrossRefGoogle Scholar

Copyright information

© The JJIAM Publishing Committee and Springer 2010

Authors and Affiliations

  1. 1.Chitose Institute of Science and TechnologyChitoseJapan
  2. 2.Research Institute of Electronic ScienceHokkaido UniversitySapporoJapan

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