Stability of spot and ring solutions of the diblock copolymer equation

Journal of Mathematical Physics

Volumn 45, Issue 11, 2004, Pages 4106-4133

  Ren, Xiaofeng ^a   Wei, Juncheng ^b  

^a UTAH STATE UNIVERSITY   (United States)

^b CHINESE UNIVERSITY OF HONG KONG   (Hong Kong)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 22544461938     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1782280     Document Type: Article

References (38)

1. Alikakos, N., Bronsard, L., and Fusco, G., " Slow motion in the gradient theory of phase transitions via energy and spectrum, " Calculus Var. Partial Differ. Equ. 6, 39-66 (1998).
2. Alikakos, N., and Fusco, G., " Slow dynamics for the Cahn-Hilliard equation in higher space dimension, part I: Spectral estimates " Commun. Partial Differ. Equ. 19, 1397-1447 (1994).
3. Alikakos, N., and Fusco, G., " Slow dynamics for the Cahn-Hilliard equation in higher space dimension: The motion of bubbles, " Arch. Ration. Mech. Anal. 141, 1-61 (1998).
4. Bates, F. S., and Fredrickson, G. H., " Block copolymers: Designer soft materials, " Phys. Today 52, 32-38 (1999).
5. Cahn, J. W., and Hilliard, J. E., " Free energy of a nonuniform system. I. Interfacial free energy, " J. Chem. Phys. 28, 258-267 (1958).
6. Chen, X., and Taniguchi, M., " Instability of spherical interfaces in a nonlinear free boundary problem, " Adv. Differ. Equ. 5, 747-772 (2000).
7. Choksi, R., " Scaling laws in microphase separation of diblock copolymers, " J. Nonlinear Sci. 11, 223-236 (2001).
8. Choksi, R., and Ren, X., " On the derivation of a density functional theory for microphase separation of diblock copolymers, " J. Stat. Phys. 113, 151-176 (2003).
9. De Giorgi, E., " Sulla convergenza di alcune successioni di integrali del tipo della'area, " Rendiconti di Matematica 8, 277-294 (1975).
10. Fife, P. C., and Hilhorst, D., " The Nishiura-Ohnishi free boundary problem in the 1D case, " SIAM J. Math. Anal. 33, 589-606 (2001).
11. Hamley, I.W., The Physics of Block Copolymers (Oxford Science, New York, 1998).
12. Helffer, B., and Sjöstrand, J., " Multiple wells in the semiclassical limit. I, " Commun. Partial Differ. Equ. 9, 337-408 (1984).
13. Henry, M., " Singular limit of a fourth order problem arising in the microphase separation of diblock copolymers, " Adv. Differ. Equ. 6, 1049-1114 (2001).
14. Kohn, R., and Sternberg, P., " Local minimizers and singular perturbations, " Proc. - R. Soc. Edinburgh, Sect. A: Math. 111, 69-84 (1989).
15. Modica, L., " The gradient theory of phase transitions and the minimal interface criterion, " Arch. Ration. Mech. Anal. 98, 123-142 (1987).
16. Müller, S., " Singular perturbations as a selection criterion for periodic minimizing sequences, " Calculus Var. Partial Differ. Equ. 1, 169-204 (1993).
17. Muratov, C. B., " Theory of domain patterns in systems with long-range interactions of Coulomb type, " Phys. Rev. E 66, 066108 (2002).
18. Nakazawa, H., and Ohta, T., " Microphase separation of ABC-type triblock copolymers, " Macromolecules 26, 5503-5511 (1993).
19. Nishiura, Y., and Ohnishi, I., " Some mathematical aspects of the microphase separation in diblock copolymers, " Physica D 84, 31-39 (1995).
20. Ohnishi, I., Nishiura, Y., Imai, M., and Matsushita, Y., " Analytical solutions describing the phase separation driven by a free energy functional containing a long-range interaction term, " Chaos 9, 329-341 (1999).
21. Ohta, T., and Kawasaki, K., " Equilibrium morphology of block copolymer melts, " Macromolecules 19, 2621-2632 (1986).
22. Pohozaev, S., " Eigenfunctions of the equation Δu+λf(u)=0, " Sov. Math. Dokl. 6, 1408-1411 (1965).
23. Ren, X., and Truskinovsky, L., " Finite scale microstructures in nonlocal elasticity, " in Recognition of the Sixtieth Birthday of Roger L. Fosdick (Blacksburg, VA, 1999);
24. J. Elast. 59, 319-355 (2000).
25. Ren, X., and Wei, J., " On the multiplicity of solutions of two nonlocal variational problems, " SIAM J. Math. Anal. 31, 909-924 (2000).
26. Ren, X., and Wei, J., " Concentrically layered energy equilibria of the diblock copolymer problem, " Eur. J. Appl. Math. 13, 479-496 (2002).
27. Ren, X., and Wei, J., " On energy minimizers of the diblock copolymer problem, " Interfaces Free Boundaries 5, 193-238 (2003).
28. Ren, X., and Wei, J., " On the spectra of 3D lamellar solutions of the diblock copolymer problem, " SIAM J. Math. Anal. 35, 1-32 (2003).
29. Ren, X., and Wei, J., " Soliton-stripe patterns in charged Langmuir monolayers, " J. Nonlinear Sci. 13, 603-624 (2003).
30. Ren, X., and Wei, J., " Triblock copolymer theory: Free energy, disordered phase and weak segregation, " Physica D 178, 103-117 (2003).
31. Ren, X., and Wei, J., " Triblock copolymer theory, Ordered ABC lamellar phase, " J. Nonlinear Sci. 13, 175-208 (2003).
32. Ren, X., and Wei, J., " Chiral symmetry breaking and the soliton-stripe pattern in Langmuir monoloayers and smectic films, " Nonlinearity 17, 617-632 (2004).
33. Ren, X., and Wei, J., " The soliton-stripe pattern in the Seul-Andelman membrane, " Physica D 188, 277-291 (2004).
34. Ren, X., and Wei, J., " Wriggled lamellar solutions and their stability in the diblock copolymer problem " (preprint).
35. Taniguchi, M., " Multiple existence and linear stability of equilibrium balls in a nonlinear free boundary problem, " Q. Appl. Math. 58, 283-302 (2000).
36. Teramoto, T., and Nishiura, Y., " Double gyroid morphology in a gradient system with nonlocal effects, " J. Phys. Soc. Jpn. 71, 1611-1614 (2002).
37. Tsori, Y., Andelman, D., and Schick, M., " Defects in lamellar diblock copolymers: Chevron- and O-shaped boundaries, " Phys. Rev. E 61, 2848-2858 (2000).
38. Ward, M., " Metastable bubble solutions for the Allen-Cahn equation with mass conservation, " SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 56, 1247-1279 (1996).