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A rigorous derivation of mean-field models for diblock copolymer melts

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Abstract

We study the free boundary problem describing the micro phase separation of diblock copolymer melts in the regime that one component has small volume fraction such that micro phase separation results in an ensemble of small balls of one component. Mean-field models for the evolution of a large ensemble of such spheres have been formally derived in Glasner and Choksi (Physica D, 238:1241–1255, 2009), Helmers et al. (Netw Heterog Media, 3(3):615–632, 2008). It turns out that on a time scale of the order of the average volume of the spheres, the evolution is dominated by coarsening and subsequent stabilization of the radii of the spheres, whereas migration becomes only relevant on a larger time scale. Starting from the free boundary problem restricted to balls we rigorously derive the mean-field equations in the early time regime. Our analysis is based on passing to the homogenization limit in the variational framework of a gradient flow.

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References

  1. Alberti G., Choksi R., Otto F.: Uniform energy distribution for an isoperimetric problem with long-range interactions. J. Am. Math. Soc. 22(2), 569–605 (2009)

    Article  MathSciNet  Google Scholar 

  2. Bates F.S., Fredrickson G.H.: Block copolymers—designer soft materials. Phys. Today 52(2), 32–38 (1999)

    Article  Google Scholar 

  3. Chen X., Oshita Y.: Periodicity and uniqueness of global minimizers of an energy functional containing a long-range interaction. SIAM J. Math. Anal. 37, 1299–1332 (2005)

    Article  MathSciNet  Google Scholar 

  4. Chen X., Oshita Y.: An application of the modular function in nonlocal variational problems. Arch. Ration. Mech. Anal. 186(1), 109–132 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  5. Choksi, R., Peletier, M.A.: Small volume fraction limit of the diblock copolymer problem I: sharp interface functional, Preprint (2009)

  6. Escher J., Nishiura Y.: Smooth unique solutions for a modified Mullins-Sekerka model arising in diblock copolymer melts. Hokkaido Math. J. 31(1), 137–149 (2002)

    MATH  MathSciNet  Google Scholar 

  7. Fife P., Hilhorst D.: The Nishiura-Ohnishi free boundary problem in the 1D case. SIAM J. Math. Anal. 33, 589–606 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  8. Glasner K., Choksi R.: Coarsening and self-organization in dilute diblock copolymer melts and mixtures. Phys. D 238, 1241–1255 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  9. Hamley I.W.: The Physics of Block Copolymers. Oxford Science Publications, Oxford (1998)

    Google Scholar 

  10. Helmers M., Niethammer B., Ren X.: Evolution in off-critical diblock-copolymer melts. Netw. Heterog. Media 3(3), 615–632 (2008)

    MATH  MathSciNet  Google Scholar 

  11. Henry M., Hilhorst D., Nishiura Y.: Singular limit of a second order nonlocal parabolic equation of conservative type arising in the micro-phase separation of diblock copolymers. Hokkaido Math. J. 32, 561–622 (2003)

    MATH  MathSciNet  Google Scholar 

  12. Le, N.Q.: On the convergence of the Ohta-Kawasaki evolution equation to motion by Nishiura-Ohnishi law, Preprint (2009)

  13. Modica L.: The gradient theory of phase transitions and the minimal interface criterion. Arch. Ration. Mech. Anal. 98, 357–383 (1987)

    Article  MathSciNet  Google Scholar 

  14. Niethammer B.: The LSW model for Ostwald ripening with kinetic undercooling. Proc. Roy. Soc. Edinb. 130A, 1337–1361 (2000)

    Article  MathSciNet  Google Scholar 

  15. Niethammer B., Otto F.: Ostwald ripening: the screening length revisited. Calc. Var. Partial Differ. Equ. 13(1), 33–68 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  16. Niethammer B., Velázquez J.J.L.: Well-posedness for an inhomogeneous LSW-model in unbounded domains. Math. Annalen 328(3), 481–501 (2004)

    Article  MATH  Google Scholar 

  17. Nishiura Y., Ohnishi I.: Some aspects of the micro-phase separation in diblock copolymers. Physica D 84, 31–39 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  18. Nishiura, Y., Suzuki, H.: Higher dimensional SLEP equation and applications to morphological stability in polymer problems. SIAM J. Math. Anal. 36(3), 916–966 (2004/05)

    Google Scholar 

  19. Ohnishi I., Nishiura Y., Imai M., Matsushita Y.: Analytical solutions describing the phase separation driven by a free energy functional containing a long-range interaction term. CHAOS 9(2), 329–341 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  20. Ohta T., Kawasaki K.: Equilibrium morphology of block copolymer melts. Macromolecules 19, 2621–2632 (1986)

    Article  Google Scholar 

  21. Ren X., Wei J.: On the multiplicity of solutions of two nonlocal variational problems. SIAM J. Math. Anal. 31, 909–924 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  22. Ren X., Wei J.J.: Concentrically layered energy equilibria of the di-block copolymer problem. Eur. J. Appl. Math. 13, 479–496 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  23. Ren X., Wei J.J.: On energy minimizers of the diblock copolymer problem. Interfaces Free Bound. 5, 193–238 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  24. Ren X., Wei J.J.: On the spectra of three-dimensional lamellar solutions of the diblock copolymer problem. SIAM J. Math. Anal. 35, 1–32 (2003)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Oxford Centre for Nonlinear PDE, Mathematical Institute, University of Oxford, 24-29 St Giles’, Oxford, OX1 3LB, UK

    Barbara Niethammer

  2. Department of Mathematics, Okayama University, Tsushimanaka 3-1-1, Okayama, 700-8530, Japan

    Yoshihito Oshita

  3. PRESTO, JST, 4-1-8 Honcho Kawaguchi, Saitama, Japan

    Yoshihito Oshita

Authors
  1. Barbara Niethammer
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  2. Yoshihito Oshita
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Correspondence to Yoshihito Oshita.

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Communicated by J. Ball.

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Niethammer, B., Oshita, Y. A rigorous derivation of mean-field models for diblock copolymer melts. Calc. Var. 39, 273–305 (2010). https://doi.org/10.1007/s00526-010-0310-x

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  • DOI: https://doi.org/10.1007/s00526-010-0310-x

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