Wriggled lamellar solutions and their stability in the diblock copolymer problem

SIAM Journal on Mathematical Analysis

Volumn 37, Issue 2, 2006, Pages 455-489

  Ren, Xiaofeng ^a   Wei, Juncheng ^b  

^a UTAH STATE UNIVERSITY   (United States)

^b CHINESE UNIVERSITY OF HONG KONG   (Hong Kong)

Author keywords

Bifurcation; Diblock copolymer; Distortion; Perfect lamellar solution; Singular perturbation; Stability; Wriggled lamellar solution

Indexed keywords

DISTORTION; PERFECT LAMELLAR SOLUTION; SINGULAR PERTURBATION; WRIGGLED LAMELLAR SOLUTION;

BIFURCATION (MATHEMATICS); EIGENVALUES AND EIGENFUNCTIONS; MONOMERS; NUMERICAL METHODS; PERTURBATION TECHNIQUES; PROBLEM SOLVING;

BLOCK COPOLYMERS;

EID: 33644939600     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036141003433589     Document Type: Article

References (27)

1. D. ANDELMAN, F. BROÇHARD, AND J.-F. JOANNY, Phase transitions in Langmuir monolayers of polar molecules, J. Chem. Phys., 86 (1987), pp. 3673-3681.
2. R. CHOKSI, Scaling laws in microphase separation of diblock copolymers, J. Nonlinear Sci., 11 (2001), pp. 223-236.
3. R. CHOKSI AND X. REN, On the derivation of a density functional theory for microphase separation of diblock copolymers, J. Statist. Phys., 113 (2003), pp. 151-176.
4. M. G. CRANDALL AND P. H. RABINOWITZ, Bifurcation, perturbation of simple eigenvalues, and linearized stability, Arch. Rational Mech. Anal., 52 (1973), pp. 161-180.
5. P. C. FIFE AND D. HILHORST, The Nishiura-Ohnishi free boundary problem in the 1D case, SIAM J. Math. Anal., 33 (2001), pp. 589-606.
6. M. HENRY, Singular limit of a fourth order problem arising in the micro-phase separation of diblock copolymers, Adv. Differential Equations, 6 (2001), pp. 1049-1114.
7. C.B. MURATOV, Theory of domain patterns in systems with long-range interactions of Coulomb type, Phys. Rev. E, 66 (2002), 066108.
8. Y. NISHIURA, Coexistence of infinitely many stable solutions to reaction-diffusion systems in the singular limit, in Dynamics Reported: Expositions in Dynamical Systems, Vol. 3, C. R. K. T. Jones, U. Kirchgraber, and H. O. Walther, eds., Springer-Verlag, New York, 1994.
9. Y. NISHIURA AND I. OHNISHI, Some mathematical aspects of the microphase separation in diblock copolymers, Phys. D, 84 (1995), pp. 31-39.
10. 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 (1999), pp. 329-341.
11. T. OHTA AND K. KAWASAKI, Equilibrium morphology of block copolymer melts, Macromolecules, 19 (1986), pp. 2621-2632.
12. X. REN AND J. WEI, On the multiplicity of solutions of two nonlocal variational problems, SIAM J. Math. Anal., 31 (2000), pp. 909-924.
13. X. REN AND J. WEI, Concentrically layered energy equilibria of the di-block copolymer problem, European J. Appl. Math., 13 (2002), pp. 479-496.
14. X. REN AND J. WEI, On energy minimizers of the di-block copolymer problem, Interfaces Free Bound., 5 (2003), pp. 193-238.
15. X. REN AND J. WEI, On the spectra of three-dimensional lamellar solutions of the diblock copolymer problem, SIAM J. Math. Anal., 35 (2003), pp. 1-32.
16. X. REN AND J. WEI, Soliton-stripe patterns in charged Langmuir monolayera, J. Nonlinear Sci., 13 (2003), pp. 603-624.
17. X. REN AND J. WEI, Triblock copolymer theory: Free energy, disordered phase and weak segregation, Phys. D, 178 (2003), pp. 103-117.
18. X. REN AND J. WEI, Triblock copolymer theory: Ordered ABC lamellar phase, J. Nonlinear Sci., 13 (2003), pp. 175-208.
19. X. REN AND J. WEI, The soliton-stripe pattern in the Seul-Andelman membrane, Phys. D, 188 (2004), pp. 277-291.
20. X. REN AND J. WEI, Stability of spot and ring solutions of the diblock copolymer equation, J. Math. Phys., 45 (2004), pp. 4106-4133.
21. X. REN AND J. WEI, Droplet solutions in the diblock copolymer problem with skewed monomer composition, Calc. Var. Partial Differential Equations, to appear.
22. X. REN AND J. WEI, Sphere and Spherical Lamellar Patterns in the Ohta-Kawasaki Theory of Diblock Copolymer Melts, preprint.
23. D. H. SATTINGER, Stability of bifurcating solutions by Leray-Schauder degree, Arch. Rational Mech. Anal., 43 (1971), pp. 154-166.
24. J. V. SELINGER, Z.-G. WANG, R. F. BRUINSMA, AND C. M. KNOBLER, Chiral symmetry breaking in Langmuir monolayers and smectic films, Phys. Rev. Lett., 70 (1993), pp. 1139-1142.
25. M. SEUL AND D. ANDELMAN, Domain shapes and patterns: The phenomenology of modulated phases, Science, 267 (1995), pp. 476-483.
26. J. SMOLLER, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983.
27. Y. TSORI, D. ANDELMAN, AND M. SCHIOK, Defects in lamellar diblock copolymers: Chevron-and Ω-shaped tilt boundaries, Phys. Rev. E, 61 (2000), pp. 2848-2858.