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Branch interactions and long-term dynamics for the diblock copolymer model in one dimension
1. | Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, United States, United States |
References:
[1] |
M. Atkins, "Long Term Dynamics of the Diblock Copolymer Model on Higher Dimensional Domains," Master's thesis, George Mason University, 2011. |
[2] |
M. Bahiana and Y. Oono, Cell dynamical system approach to block copolymers, Physical Review A, 41 (1990), 6763-6771. |
[3] |
F. Bates and G. H. Fredrickson, Block copolymer thermodynamics: Theory and experiment, Annual Review of Physical Chemistry, 41 (1990), 525-557. |
[4] |
F. Bates and G. H. Fredrickson, Block copolymers - designer soft materials, Physics Today, 52 (1999), 32-38. |
[5] |
D. Blömker, B. Gawron and T. Wanner, Nucleation in the one-dimensional stochastic Cahn-Hilliard model, Discrete and Continuous Dynamical Systems, 27 (2010), 25-52.
doi: 10.3934/dcds.2010.27.25. |
[6] |
R. Choksi, Mathematical aspects of microphase separation of diblock copolymers, in "Surikaisekikenkyusko Kokyuroku," Vol. 1330, RIMS, Kyoto, (2003), 10-17. |
[7] |
R. Choksi, M. A. Peletier and J. F. Williams, On the phase diagram for microphase separation of diblock copolymers: An approach via a nonlocal Cahn-Hilliard functional, SIAM Journal on Applied Mathematics, 69 (2009), 1712-1738.
doi: 10.1137/080728809. |
[8] |
R. Choksi and X. Ren, On the derivation of a density functional theory for microphase separation of diblock copolymers, Journal of Statistical Physics, 113 (2003), 151-176.
doi: 10.1023/A:1025722804873. |
[9] |
D. A. Christian, A. Tian, W. G. Ellenbroek, I. Levental, K. Rajagopal, P. A. Janmey, A. J. Liu, T. Baumgart and D. E. Discher, Spotted vesicles, striped micelles and Janus assemblies induced by ligand binding, Nature Materials, 8 (2009), 843-849. |
[10] |
J. P. Desi, H. Edrees, J. Price, E. Sander and T. Wanner, The dynamics of nucleation in stochastic Cahn-Morral systems, SIAM Journal on Applied Dynamical Systems, 10 (2011), 707-743.
doi: 10.1137/100801378. |
[11] |
J. P. Desi, E. Sander and T. Wanner, Complex transient patterns on the disk, Discrete and Continuous Dynamical Systems, 15 (2006), 1049-1078.
doi: 10.3934/dcds.2006.15.1049. |
[12] |
E. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems, Congressus Numerantium, 30 (1981), 265-284. |
[13] |
K. Glasner and R. Choksi, Coarsening and self-organization in dilute diblock copolymer melts and mixtures, Physica D, 238 (2009), 1241-1255.
doi: 10.1016/j.physd.2009.04.006. |
[14] |
M. Grinfeld and A. Novick-Cohen, Counting stationary solutions of the Cahn-Hilliard equation by transversality arguments, Proceedings of the Royal Society of Edinburgh Sect. A, 125 (1995), 351-370.
doi: 10.1017/S0308210500028079. |
[15] |
T. Hartley and T. Wanner, A semi-implicit spectral method for stochastic nonlocal phase-field models, Discrete and Continuous Dynamical Systems, 25 (2009), 399-429.
doi: 10.3934/dcds.2009.25.399. |
[16] |
X. Kang and X. Ren, Ring pattern solutions of a free boundary problem in diblock copolymer morphology, Physica D, 238 (2009), 645-665.
doi: 10.1016/j.physd.2008.12.009. |
[17] |
N. Q. Le, On the convergence of the Ohta-Kawasaki equation to motion by nonlocal Mullins-Sekerka law, SIAM Journal on Mathematical Analysis, 42 (2010), 1602-1638.
doi: 10.1137/090768643. |
[18] |
S. Mahajan, S. Renker, P. Simon, J. Gutmann, A. Jain, S. Gruner, L. Fetters, G. Coates and U. Wiesner, Synthesis and characterization of amphiphilic poly(ethylene oxide)-block-poly(hexyl methacrylate) copolymers, Macromolecular Chemistry and Physics, 204 (2003), 1047-1055. |
[19] |
S. Maier-Paape, K. Mischaikow and T. Wanner, Structure of the attractor of the Cahn-Hilliard equation on a square, International Journal of Bifurcation and Chaos, 17 (2007), 1221-1263.
doi: 10.1142/S0218127407017781. |
[20] |
H. Nakazawa and T. Ohta, Microphase separation of ABC-type triblock copolymers, Macromolecules, 26 (1993), 5503-5511. |
[21] |
Y. Nishiura, "Far-from-Equilibrium Dynamics,'' Translations of Mathematical Monographs, 209, Iwanami Series in Modern Mathematics, American Mathematical Society, Providence, RI, 2002. |
[22] |
Y. Nishiura and I. Ohnishi, Rugged landscape with fine structure, unpublished preprint. |
[23] |
Y. Nishiura and I. Ohnishi, Some mathematical aspects of the micro-phase separation in diblock copolymers, Physica D, 84 (1995), 31-39.
doi: 10.1016/0167-2789(95)00005-O. |
[24] |
Y. Nishiura and H. Suzuki, Higher dimensional SLEP equation and applications to morphological stability in polymer problems, SIAM Journal on Mathematical Analysis, 36 (2004/05), 916-966.
doi: 10.1137/S0036141002420157. |
[25] |
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), 329-341.
doi: 10.1063/1.166410. |
[26] |
T. Ohta and K. Kawasaki, Equilibrium morphology of block copolymer melts, Macromolecules, 19 (1986), 2621-2632. |
[27] |
X. Ren, Shell structure as solution to a free boundary problem from block copolymer morphology, Discrete and Continuous Dynamical Systems, 24 (2009), 979-1003.
doi: 10.3934/dcds.2009.24.979. |
[28] |
X. Ren and J. Wei, On energy minimizers of the diblock copolymer problem, Interfaces and Free Boundaries, 5 (2003), 193-238.
doi: 10.4171/IFB/78. |
[29] |
X. Ren and J. Wei, Triblock copolymer theory: Free energy, disordered phase and weak segregation, Physica D, 178 (2003), 103-117.
doi: 10.1016/S0167-2789(02)00808-4. |
[30] |
X. Ren and J. Wei, Triblock copolymer theory: Ordered ABC lamellar phase, Journal of Nonlinear Science, 13 (2003), 175-208.
doi: 10.1007/s00332-002-0521-1. |
[31] |
X. Ren and J. Wei, Single droplet pattern in the cylindrical phase of diblock copolymer morphology, Journal of Nonlinear Science, 17 (2007), 471-503.
doi: 10.1007/s00332-007-9005-7. |
[32] |
R. Tamate, K. Yamada, J. Vinals and T. Ohta, Structural rheology of microphase separated diblock copolymers, Journal of the Physical Society of Japan, 77 (2008), 034802. |
[33] |
P. Tang, F. Qiu, H. Zhang and Y. Yang, Morphology and phase diagram of complex block copolymers: ABC linear triblock copolymers, Physical Review E, 69 (2004), 031803. |
[34] |
R. Wang, J. Hu, Z. Jiang and D. Zhou, Morphology of ABCD tetrablock copolymers predicted by self-consistent field theory, Macromolecular Theory and Simulations, 14 (2005), 256-266. |
[35] |
E. Zeidler, "Nonlinear Functional Analysis and its Applications. I. Fixed-Point Theorems," Springer-Verlag, New York, 1986.
doi: 10.1007/978-1-4612-4838-5. |
show all references
References:
[1] |
M. Atkins, "Long Term Dynamics of the Diblock Copolymer Model on Higher Dimensional Domains," Master's thesis, George Mason University, 2011. |
[2] |
M. Bahiana and Y. Oono, Cell dynamical system approach to block copolymers, Physical Review A, 41 (1990), 6763-6771. |
[3] |
F. Bates and G. H. Fredrickson, Block copolymer thermodynamics: Theory and experiment, Annual Review of Physical Chemistry, 41 (1990), 525-557. |
[4] |
F. Bates and G. H. Fredrickson, Block copolymers - designer soft materials, Physics Today, 52 (1999), 32-38. |
[5] |
D. Blömker, B. Gawron and T. Wanner, Nucleation in the one-dimensional stochastic Cahn-Hilliard model, Discrete and Continuous Dynamical Systems, 27 (2010), 25-52.
doi: 10.3934/dcds.2010.27.25. |
[6] |
R. Choksi, Mathematical aspects of microphase separation of diblock copolymers, in "Surikaisekikenkyusko Kokyuroku," Vol. 1330, RIMS, Kyoto, (2003), 10-17. |
[7] |
R. Choksi, M. A. Peletier and J. F. Williams, On the phase diagram for microphase separation of diblock copolymers: An approach via a nonlocal Cahn-Hilliard functional, SIAM Journal on Applied Mathematics, 69 (2009), 1712-1738.
doi: 10.1137/080728809. |
[8] |
R. Choksi and X. Ren, On the derivation of a density functional theory for microphase separation of diblock copolymers, Journal of Statistical Physics, 113 (2003), 151-176.
doi: 10.1023/A:1025722804873. |
[9] |
D. A. Christian, A. Tian, W. G. Ellenbroek, I. Levental, K. Rajagopal, P. A. Janmey, A. J. Liu, T. Baumgart and D. E. Discher, Spotted vesicles, striped micelles and Janus assemblies induced by ligand binding, Nature Materials, 8 (2009), 843-849. |
[10] |
J. P. Desi, H. Edrees, J. Price, E. Sander and T. Wanner, The dynamics of nucleation in stochastic Cahn-Morral systems, SIAM Journal on Applied Dynamical Systems, 10 (2011), 707-743.
doi: 10.1137/100801378. |
[11] |
J. P. Desi, E. Sander and T. Wanner, Complex transient patterns on the disk, Discrete and Continuous Dynamical Systems, 15 (2006), 1049-1078.
doi: 10.3934/dcds.2006.15.1049. |
[12] |
E. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems, Congressus Numerantium, 30 (1981), 265-284. |
[13] |
K. Glasner and R. Choksi, Coarsening and self-organization in dilute diblock copolymer melts and mixtures, Physica D, 238 (2009), 1241-1255.
doi: 10.1016/j.physd.2009.04.006. |
[14] |
M. Grinfeld and A. Novick-Cohen, Counting stationary solutions of the Cahn-Hilliard equation by transversality arguments, Proceedings of the Royal Society of Edinburgh Sect. A, 125 (1995), 351-370.
doi: 10.1017/S0308210500028079. |
[15] |
T. Hartley and T. Wanner, A semi-implicit spectral method for stochastic nonlocal phase-field models, Discrete and Continuous Dynamical Systems, 25 (2009), 399-429.
doi: 10.3934/dcds.2009.25.399. |
[16] |
X. Kang and X. Ren, Ring pattern solutions of a free boundary problem in diblock copolymer morphology, Physica D, 238 (2009), 645-665.
doi: 10.1016/j.physd.2008.12.009. |
[17] |
N. Q. Le, On the convergence of the Ohta-Kawasaki equation to motion by nonlocal Mullins-Sekerka law, SIAM Journal on Mathematical Analysis, 42 (2010), 1602-1638.
doi: 10.1137/090768643. |
[18] |
S. Mahajan, S. Renker, P. Simon, J. Gutmann, A. Jain, S. Gruner, L. Fetters, G. Coates and U. Wiesner, Synthesis and characterization of amphiphilic poly(ethylene oxide)-block-poly(hexyl methacrylate) copolymers, Macromolecular Chemistry and Physics, 204 (2003), 1047-1055. |
[19] |
S. Maier-Paape, K. Mischaikow and T. Wanner, Structure of the attractor of the Cahn-Hilliard equation on a square, International Journal of Bifurcation and Chaos, 17 (2007), 1221-1263.
doi: 10.1142/S0218127407017781. |
[20] |
H. Nakazawa and T. Ohta, Microphase separation of ABC-type triblock copolymers, Macromolecules, 26 (1993), 5503-5511. |
[21] |
Y. Nishiura, "Far-from-Equilibrium Dynamics,'' Translations of Mathematical Monographs, 209, Iwanami Series in Modern Mathematics, American Mathematical Society, Providence, RI, 2002. |
[22] |
Y. Nishiura and I. Ohnishi, Rugged landscape with fine structure, unpublished preprint. |
[23] |
Y. Nishiura and I. Ohnishi, Some mathematical aspects of the micro-phase separation in diblock copolymers, Physica D, 84 (1995), 31-39.
doi: 10.1016/0167-2789(95)00005-O. |
[24] |
Y. Nishiura and H. Suzuki, Higher dimensional SLEP equation and applications to morphological stability in polymer problems, SIAM Journal on Mathematical Analysis, 36 (2004/05), 916-966.
doi: 10.1137/S0036141002420157. |
[25] |
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), 329-341.
doi: 10.1063/1.166410. |
[26] |
T. Ohta and K. Kawasaki, Equilibrium morphology of block copolymer melts, Macromolecules, 19 (1986), 2621-2632. |
[27] |
X. Ren, Shell structure as solution to a free boundary problem from block copolymer morphology, Discrete and Continuous Dynamical Systems, 24 (2009), 979-1003.
doi: 10.3934/dcds.2009.24.979. |
[28] |
X. Ren and J. Wei, On energy minimizers of the diblock copolymer problem, Interfaces and Free Boundaries, 5 (2003), 193-238.
doi: 10.4171/IFB/78. |
[29] |
X. Ren and J. Wei, Triblock copolymer theory: Free energy, disordered phase and weak segregation, Physica D, 178 (2003), 103-117.
doi: 10.1016/S0167-2789(02)00808-4. |
[30] |
X. Ren and J. Wei, Triblock copolymer theory: Ordered ABC lamellar phase, Journal of Nonlinear Science, 13 (2003), 175-208.
doi: 10.1007/s00332-002-0521-1. |
[31] |
X. Ren and J. Wei, Single droplet pattern in the cylindrical phase of diblock copolymer morphology, Journal of Nonlinear Science, 17 (2007), 471-503.
doi: 10.1007/s00332-007-9005-7. |
[32] |
R. Tamate, K. Yamada, J. Vinals and T. Ohta, Structural rheology of microphase separated diblock copolymers, Journal of the Physical Society of Japan, 77 (2008), 034802. |
[33] |
P. Tang, F. Qiu, H. Zhang and Y. Yang, Morphology and phase diagram of complex block copolymers: ABC linear triblock copolymers, Physical Review E, 69 (2004), 031803. |
[34] |
R. Wang, J. Hu, Z. Jiang and D. Zhou, Morphology of ABCD tetrablock copolymers predicted by self-consistent field theory, Macromolecular Theory and Simulations, 14 (2005), 256-266. |
[35] |
E. Zeidler, "Nonlinear Functional Analysis and its Applications. I. Fixed-Point Theorems," Springer-Verlag, New York, 1986.
doi: 10.1007/978-1-4612-4838-5. |
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