Archive for Rational Mechanics and Analysis

, Volume 186, Issue 1, pp 109–132 | Cite as

An Application of the Modular Function in Nonlocal Variational Problems

Article

Abstract

Using the modular function and its invariance under the action of a modular group and an heuristic reduction of a mathematical model, we present a mathematical account of a hexagonal pattern selection observed in di-block copolymer melts.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alberti G. and Müller S. (2001). A new approach to variational problems with multiple scales. Comm. Pure Appl. Math. 54: 761–825 MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bahiana M. and Oono Y. (1990). Cell dynamical system approach to block copolymers. Phys. Rev. 41: 6763–6771 CrossRefADSGoogle Scholar
  3. 3.
    Bates F.S. and Fredrickson G.H. (1999). Block Copolymers-Designer Soft Materials. Physics Today 32: 32–38 Google Scholar
  4. 4.
    Chen X. and Oshita Y. (2005). Periodicity and uniqueness of global minimizers of an energy functional containing a long-range interaction. SIAM J. Math. Anal. 37: 1299–1332 CrossRefMathSciNetGoogle Scholar
  5. 5.
    Choksi R. and Ren X.F. (2003). On the derivation of a density functional theory for microphase separation of diblock copolymers. J. Stat. Phys. 113: 151–176 MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Fife P. and Hilhorst D. (2001). The Nishiura-Ohnishi Free Boundary Problem in the 1D case. SIAM J. Math. Anal. 33: 589–606 MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Hamley, I.W.: The Physics of Block Copolymers, Oxford University Press, Oxford, 1999Google Scholar
  8. 8.
    Hashimoto T., Shibayama M. and Kawai H. (1983). Ordered Structure in Block Polymer Solutions. 4. Scaling Rules on Size of Fluctuations with Block Molecular Weight, Concentration and Temperature in Segregation and Homogeneous Regimes. Macromolecules 16: 1093–1101 CrossRefADSGoogle Scholar
  9. 9.
    Hashimoto T., Shibayama M. and Kawai H. (1980). Domain-Boundary Structure of Styrene-Isoprene Block Copolymer Films Cast from Solution. 4. Molecular-Weight Dependence of Lamellar Microdomains. Macromolecules 13: 1237–1247 CrossRefADSGoogle Scholar
  10. 10.
    Henry M., Hilhorst D. and Nishiura Y. (2003). Singular limit of a second order nonlocal parabolic equation of censervative type arising in the micro-phase separation of diblock copolymers. Hokkaido Math. J. 32: 561–622 MATHMathSciNetGoogle Scholar
  11. 11.
    Lang, S.: Introduction to Arakelov theory. Springer-Verlag, 1988Google Scholar
  12. 12.
    Leibler L. (1980). Theory of Microphase Separation in Block Copolymers. Macromolecules 13: 1602–1617 CrossRefADSGoogle Scholar
  13. 13.
    Lehner, J.: Discontinuous groups and automorphic functions American mathematical society, 1964Google Scholar
  14. 14.
    Modica L. (1987). The gradient theory of phase transitions and the minimal interface criterion. Arch. Ration. Mech. Anal. 98: 123–142 MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Müller S. (1993). Singular perturbations as a selection criterion for periodic minimizing sequences. Calc. Var. Partial Differential Equations 1: 169–204 MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Nishiura Y. and Ohnishi I. (1995). Some mathematical aspects of the micro-phase separation in diblock copolymers. Phys. D 84: 31–39 CrossRefMathSciNetGoogle Scholar
  17. 17.
    Ohta T. and Kawasaki K. (1986). Equilibrium morphology of block copolymer melts. Macromolecules 19: 2621–2632 CrossRefADSGoogle Scholar
  18. 18.
    Ohnishi I., Nishiura Y., Imai M. and Matsushita Y. (1999). Analytical solution describing the phase separation driven by a free energy functional containing a long-range interaction term. Chaos 9: 329–341 MATHCrossRefADSMathSciNetGoogle Scholar
  19. 19.
    Oshita Y. (2003). On stable stationary solutions and mesoscopic patterns for FitzHugh–Nagumo equations in higher dimensions. J. Differential Equations 188: 110–134 MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Oshita Y. (2004). Stable stationary patterns and interfaces arising in reaction-diffusion systems. SIAM J. Math. Anal. 36: 479–497 MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Ren X. and Wei J. (2002). Concentrically layered energy equilibria of the di-block copolymer problem. European J. Appl. Math. 13: 479–496 MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Ren X. and Wei J. (2003). On energy minimizers of the diblock copolymer problem. Interfaces Free Bound. 5: 193–238 MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Ren X. and Wei J. (2003). On the spectra of three-dimensional lamellar solutions of the Diblock copolymer problem. SIAM J. Math. Anal. 35: 1–32 MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Theil F. (2006). A proof of crystallization in two dimensions. Comm. Math. Phys. 262: 209–236 MATHCrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PittsburghPittsburghUSA
  2. 2.Graduate School of Mathematical SciencesUniversity of TokyoTokyoJapan
  3. 3.Graduate School of Natural Science and TechnologyOkayama UniversityOkayamaJapan

Personalised recommendations