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.
Similar content being viewed by others
REFERENCES
G. Alberti, R. Choksi, and F. Otto, Uniform Energy Distribution for Minimizers of a Nonlocal Functional Describing Microphase Separation of Diblock Copolymers,in preparation.
M. Bahiana and Y. Oono, Cell dynamical system approach to block copolymers, Phys. Rev. A 41:6763-6771 (1990).
R. Balian, From Microphysics to Macrophysics: Methods and Applications of Statistical Physics, Two Volumes (Springer-Verlag, Berlin, 1991).
F. S. Bates and G. H. Fredrickson, Block copolymers-designer soft materials, Physics Today 52:32-38 (Feb, 1999).
Y. Bohbot-Raviv and Z.-G. Wang, Discovering new ordered phases of block copolymers, Phys. Rev. Lett. 85:3428-3431 (2000).
J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys. 28:258-267 (1958).
R. Choksi, Scaling laws in microphase separation of diblock copolymers, J. Nonlinear Sci. 11:223-236 (2001).
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).
P. G. de Gennes, Scaling Concepts in Polymer Physics (Cornell University, Ithaca, NY, 1979).
G. Dal Maso, Introduction to Gamma-Convergence, Progress in nonlinear differential equations and their applications, Vol. 8 (Birkhauser, Boston, 1993).
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).
S. F. Edwards, The theory of polymer solutions at intermediate concentration, Proc. Phys. Soc. (London) 88:265-280 (1966).
G. H. Fredrickson, V. Ganesan, and F. Drolet, Field-theoretic computer simulation methods for polymer and complex fluids, Macromolecules 16(2002).
P. Fife and D. Hilhorst, The Nishiura-Ohnishi free boundary problem in the 1D case, SIAM J. Math. Anal. 33:589-606 (2001).
A. Friedman, Stochastic Differential Equations and Applications, Vol. 1 (Academic Press, New York, 1975).
N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, Frontiers in Physics (Addison-Wesley, 1992).
A. Y. Grosberg and A. R. Khokhlov, Statistical Physics of Macromolecules (American Institute of Physics (AIP) Press, New York, 1994).
I. W. Hamley, The Physics of Block Copolymers (Oxford Science Publications, 1998).
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).
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).
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).
T. Hashimoto, H. Tannaka, and H. Hasegawa, Molecular Conformation and Dynamics of Macromolecules in Condensed Systems, M. Nagasawa, ed. (Elsevier, Amsterdam, 1998).
E. Helfand, Theory of inhomogeneous polymers: Fundamentals of Gaussian random walk model, J. Chem. Phys. 62:999-1005 (1975).
E. Helfand and Tagami, Theory of the interface between immiscible polymers II, J. Chem. Phys. 56:3592-3601 (1972).
E. Helfand and Z. R. Wasserman, Block copolymer theory 4. Narrow interphase approximations, Macromolecules 9:879-888 (1976).
E. Helfand and Z. R. Wasserman, Block copolymer theory 5. Spherical domains, Macromolecules 11:960(1978).
E. Helfand and Z. R. Wasserman, Block copolymer theory 6. Cylindrical domains, Macromolecules 13:994-998 (1980).
K. M. Hong and J. Noolandi, Theory of inhomogeneous multicomponent polymer systems, Macromolecules 14:727-736 (1981).
K. Kawasaki, T. Ohta, and M. Kohrogui, Equilibrium morphology of block copolymer melts 2, Macromolecules 21:2972-2980 (1988).
L. Leibler, Theory of microphase separation in block copolymers, Macromolecules 13:1602-1617 (1980).
R. L. Lescanec and M. Muthukumar, Density functional theory of phase transitions in diblock copolymer systems, Macromolecules 26:3908-3916 (1993).
F. Liu and N. Goldenfeld, Dynamics of phase separation in block copolymer melts, Phys. Rev. A 39:4805(1989).
J. Malenkevitz and M. Muthukumar, Density functional theory of lamellar ordering in diblock copolymers, Macromolecules 24:4199-4205 (1991).
M. W. Matsen and F. Bates, Unifying weak-and strong-segregation block copolymer theories, Macromolecules 39:1091-1098 (1996).
M. W. Matsen and M. Schick, Stable and unstable phases of a diblock copolymer melt, Phys. Rev. Lett. 72:2660-2663 (1994).
C. B. Muratov, Theory of domain patterns in systems with long-range interactions of coulomb type, Phys. Rev. E 66:066108(2002).
Y. Nishiura and I. Ohnishi, Some mathematical aspects of the micro-phase separation in diblock copolymers, Physica D 84:31-39 (1995).
T. Ohta, personal communication.
T. Ohta, Y. Enomoto, J. Harden, and M. Doi, Anomalous rheological behavior of ordered phases of block copolymers I, Macromolecules 26:4928-4934 (1993).
T. Ohta and K. Kawasaki, Equilibrium morphology of block copolymer melts, Macromolecules 19 2621-2632 (1986).
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).
X. Ren and J. Wei, On energy minimizers of the diblock copolymer problem, Interfaces and Free Boundaries 5:193-238 (2003).
X. Ren and J. Wei, Concentrically layered energy equilibria of the di-block copolymer problem, Eur. J. Appl. Math 13:479-496 (2002).
X. Ren and J. Wei, On the multiplicity of two nonlocal variational problems, SIAM J. Math. Anal. 31:909-924 (2000).
X. Ren and J. Wei, On the spectra of 3-D lamellar solutions of the diblock copolymer problem, SIAM J. Math. Anal, to appear.
X. Ren and J. Wei, Wriggled lamellar solutions and their stability in the diblock copolymer problem, preprint.
A. N. Semenov, Contributions to the theory of microphase layering in block-copolymer melts, Sov. Phys. JETP 61:733-742 (1985).
A. N. Semenov, Microphase separation in diblock-copolymer melts: Ordering of micelles, Macromolecules 22:2849-2851 (1989).
T. Teramoto and Y. Nishiura, Double gyroid morphology in a gradient system with nonlocal effects, J. Phys. Soc. Japan 71:1611-1614 (2002).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Choksi, R., Ren, X. On the Derivation of a Density Functional Theory for Microphase Separation of Diblock Copolymers. Journal of Statistical Physics 113, 151–176 (2003). https://doi.org/10.1023/A:1025722804873
Issue Date:
DOI: https://doi.org/10.1023/A:1025722804873