Journal of Statistical Physics

, Volume 113, Issue 1–2, pp 151–176

On the Derivation of a Density Functional Theory for Microphase Separation of Diblock Copolymers

  • Rustum Choksi
  • Xiaofeng Ren

Abstract

We consider here the problem of phase separation in copolymer melts. The Ohta–Kawasaki density functional theory gives rise to a nonlocal Cahn–Hilliard-like functional, promoting the use of ansatz-free mathematical tools for the investigation of minimizers. In this article we re-derive this functional as an offspring of the self-consistent mean field theory, connecting all parameters to the fundamental material parameters and clearly identifying all the approximations used. As a simple example of an ansatz-free investigation, we calculate the surface tension in the strong segregation limit, independent of any phase geometry.

microphase separation diblock copolymers mean field theory density functional theory 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. 1.
    G. Alberti, R. Choksi, and F. Otto, Uniform Energy Distribution for Minimizers of a Nonlocal Functional Describing Microphase Separation of Diblock Copolymers,in preparation.Google Scholar
  2. 2.
    M. Bahiana and Y. Oono, Cell dynamical system approach to block copolymers, Phys. Rev. A 41:6763-6771 (1990).Google Scholar
  3. 3.
    R. Balian, From Microphysics to Macrophysics: Methods and Applications of Statistical Physics, Two Volumes (Springer-Verlag, Berlin, 1991).Google Scholar
  4. 4.
    F. S. Bates and G. H. Fredrickson, Block copolymers-designer soft materials, Physics Today 52:32-38 (Feb, 1999).Google Scholar
  5. 5.
    Y. Bohbot-Raviv and Z.-G. Wang, Discovering new ordered phases of block copolymers, Phys. Rev. Lett. 85:3428-3431 (2000).Google Scholar
  6. 6.
    J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys. 28:258-267 (1958).Google Scholar
  7. 7.
    R. Choksi, Scaling laws in microphase separation of diblock copolymers, J. Nonlinear Sci. 11:223-236 (2001).Google Scholar
  8. 8.
    R. Choksi, R. V. Kohn, and F. Otto, Domains branching in uniaxial ferromagnets: A scaling law for the minimum energy, Comm. Math. Phys. 201:61-79 (1999).Google Scholar
  9. 9.
    P. G. de Gennes, Scaling Concepts in Polymer Physics (Cornell University, Ithaca, NY, 1979).Google Scholar
  10. 10.
    G. Dal Maso, Introduction to Gamma-Convergence, Progress in nonlinear differential equations and their applications, Vol. 8 (Birkhauser, Boston, 1993).Google Scholar
  11. 11.
    F. Drolet and G. H. Fredrickson, Combinatorial screening of complex block copolymer assembly with self-consistent field theory, Phys. Rev. Lett. 83, 4317-4320 (1999).Google Scholar
  12. 12.
    S. F. Edwards, The theory of polymer solutions at intermediate concentration, Proc. Phys. Soc. (London) 88:265-280 (1966).Google Scholar
  13. 13.
    G. H. Fredrickson, V. Ganesan, and F. Drolet, Field-theoretic computer simulation methods for polymer and complex fluids, Macromolecules 16(2002).Google Scholar
  14. 14.
    P. Fife and D. Hilhorst, The Nishiura-Ohnishi free boundary problem in the 1D case, SIAM J. Math. Anal. 33:589-606 (2001).Google Scholar
  15. 15.
    A. Friedman, Stochastic Differential Equations and Applications, Vol. 1 (Academic Press, New York, 1975).Google Scholar
  16. 16.
    N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, Frontiers in Physics (Addison-Wesley, 1992).Google Scholar
  17. 17.
    A. Y. Grosberg and A. R. Khokhlov, Statistical Physics of Macromolecules (American Institute of Physics (AIP) Press, New York, 1994).Google Scholar
  18. 18.
    I. W. Hamley, The Physics of Block Copolymers (Oxford Science Publications, 1998).Google Scholar
  19. 19.
    T. Hashimoto, M. Shibayama, and H. Kawai, Domain-boundary structure of styrene-isoprene block copolymer films cast from solution 4, molecular-weight dependence of lamellar microdomains, Macromolecules 13:1237-1247 (1980).Google Scholar
  20. 20.
    T. Hashimoto, M. Shibayama, and H. Kawai, 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 (1983).Google Scholar
  21. 21.
    T. Hashimoto, M. Fujimura, and H. Kawai, Domain-boundary structure of styrene-isoprene block copolymer films cast from solution 5. Molecular-weight dependence of spherical microdomains, Macromolecules 13:1660-1669 (1980).Google Scholar
  22. 22.
    T. Hashimoto, H. Tannaka, and H. Hasegawa, Molecular Conformation and Dynamics of Macromolecules in Condensed Systems, M. Nagasawa, ed. (Elsevier, Amsterdam, 1998).Google Scholar
  23. 23.
    E. Helfand, Theory of inhomogeneous polymers: Fundamentals of Gaussian random walk model, J. Chem. Phys. 62:999-1005 (1975).Google Scholar
  24. 24.
    E. Helfand and Tagami, Theory of the interface between immiscible polymers II, J. Chem. Phys. 56:3592-3601 (1972).Google Scholar
  25. 25.
    E. Helfand and Z. R. Wasserman, Block copolymer theory 4. Narrow interphase approximations, Macromolecules 9:879-888 (1976).Google Scholar
  26. 26.
    E. Helfand and Z. R. Wasserman, Block copolymer theory 5. Spherical domains, Macromolecules 11:960(1978).Google Scholar
  27. 27.
    E. Helfand and Z. R. Wasserman, Block copolymer theory 6. Cylindrical domains, Macromolecules 13:994-998 (1980).Google Scholar
  28. 28.
    K. M. Hong and J. Noolandi, Theory of inhomogeneous multicomponent polymer systems, Macromolecules 14:727-736 (1981).Google Scholar
  29. 29.
    K. Kawasaki, T. Ohta, and M. Kohrogui, Equilibrium morphology of block copolymer melts 2, Macromolecules 21:2972-2980 (1988).Google Scholar
  30. 30.
    L. Leibler, Theory of microphase separation in block copolymers, Macromolecules 13:1602-1617 (1980).Google Scholar
  31. 31.
    R. L. Lescanec and M. Muthukumar, Density functional theory of phase transitions in diblock copolymer systems, Macromolecules 26:3908-3916 (1993).Google Scholar
  32. 32.
    F. Liu and N. Goldenfeld, Dynamics of phase separation in block copolymer melts, Phys. Rev. A 39:4805(1989).Google Scholar
  33. 33.
    J. Malenkevitz and M. Muthukumar, Density functional theory of lamellar ordering in diblock copolymers, Macromolecules 24:4199-4205 (1991).Google Scholar
  34. 34.
    M. W. Matsen and F. Bates, Unifying weak-and strong-segregation block copolymer theories, Macromolecules 39:1091-1098 (1996).Google Scholar
  35. 35.
    M. W. Matsen and M. Schick, Stable and unstable phases of a diblock copolymer melt, Phys. Rev. Lett. 72:2660-2663 (1994).Google Scholar
  36. 36.
    C. B. Muratov, Theory of domain patterns in systems with long-range interactions of coulomb type, Phys. Rev. E 66:066108(2002).Google Scholar
  37. 37.
    Y. Nishiura and I. Ohnishi, Some mathematical aspects of the micro-phase separation in diblock copolymers, Physica D 84:31-39 (1995).Google Scholar
  38. 38.
    T. Ohta, personal communication.Google Scholar
  39. 39.
    T. Ohta, Y. Enomoto, J. Harden, and M. Doi, Anomalous rheological behavior of ordered phases of block copolymers I, Macromolecules 26:4928-4934 (1993).Google Scholar
  40. 40.
    T. Ohta and K. Kawasaki, Equilibrium morphology of block copolymer melts, Macromolecules 19 2621-2632 (1986).Google Scholar
  41. 41.
    I. Ohnishi, Y. Nishiura, M. Imai, and Y. Matsushita, Analytical solutions describing the phase separation driven by a free energy functional containing a long-range interaction term, CHAOS 9:329-341 (1999).Google Scholar
  42. 42.
    X. Ren and J. Wei, On energy minimizers of the diblock copolymer problem, Interfaces and Free Boundaries 5:193-238 (2003).Google Scholar
  43. 43.
    X. Ren and J. Wei, Concentrically layered energy equilibria of the di-block copolymer problem, Eur. J. Appl. Math 13:479-496 (2002).Google Scholar
  44. 44.
    X. Ren and J. Wei, On the multiplicity of two nonlocal variational problems, SIAM J. Math. Anal. 31:909-924 (2000).Google Scholar
  45. 45.
    X. Ren and J. Wei, On the spectra of 3-D lamellar solutions of the diblock copolymer problem, SIAM J. Math. Anal, to appear.Google Scholar
  46. 46.
    X. Ren and J. Wei, Wriggled lamellar solutions and their stability in the diblock copolymer problem, preprint.Google Scholar
  47. 47.
    A. N. Semenov, Contributions to the theory of microphase layering in block-copolymer melts, Sov. Phys. JETP 61:733-742 (1985).Google Scholar
  48. 48.
    A. N. Semenov, Microphase separation in diblock-copolymer melts: Ordering of micelles, Macromolecules 22:2849-2851 (1989).Google Scholar
  49. 49.
    T. Teramoto and Y. Nishiura, Double gyroid morphology in a gradient system with nonlocal effects, J. Phys. Soc. Japan 71:1611-1614 (2002).Google Scholar

Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • Rustum Choksi
    • 1
  • Xiaofeng Ren
    • 2
  1. 1.Department of MathematicsSimon Fraser UniversityCanada
  2. 2.Department of MathematicsUtah State UniversityUSA

Personalised recommendations