August  2013, 33(8): 3671-3705. doi: 10.3934/dcds.2013.33.3671

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

Received  May 2012 Revised  November 2012 Published  January 2013

Diblock copolymers are a class of materials formed by the reaction of two linear polymers. The different structures taken on by these polymers grant them special properties, which can prove useful in applications such as the development of new adhesives and asphalt additives. We consider a model for the formation of diblock copolymers first proposed by Ohta and Kawasaki [26]. Their model yields a Cahn-Hilliard-like equation, where a nonlocal term is added to the standard Cahn-Hilliard energy. We study the long-term dynamics of this model on one-dimensional domains through a combination of bifurcation theoretic results and numerical simulations. Our results shed light on how the complicated bifurcation behavior of the diblock copolymer model is related to the better known bifurcation structure of the Cahn-Hilliard equation. In addition, we demonstrate that this knowledge can be used to predict the long-term dynamics of solutions originating close to the homogeneous equilibrium. In particular, we show that the periodicity of the long-term limit of such solutions can be predicted by tracking certain secondary bifurcation points in the bifurcation diagram, and that the long-term limit is in general not given by the global energy minimizer.
Citation: Ian Johnson, Evelyn Sander, Thomas Wanner. Branch interactions and long-term dynamics for the diblock copolymer model in one dimension. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3671-3705. doi: 10.3934/dcds.2013.33.3671
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show all references

References:
[1]

Master's thesis, George Mason University, 2011. Google Scholar

[2]

Physical Review A, 41 (1990), 6763-6771. Google Scholar

[3]

Annual Review of Physical Chemistry, 41 (1990), 525-557. Google Scholar

[4]

Physics Today, 52 (1999), 32-38. Google Scholar

[5]

Discrete and Continuous Dynamical Systems, 27 (2010), 25-52. doi: 10.3934/dcds.2010.27.25.  Google Scholar

[6]

in "Surikaisekikenkyusko Kokyuroku," Vol. 1330, RIMS, Kyoto, (2003), 10-17. Google Scholar

[7]

SIAM Journal on Applied Mathematics, 69 (2009), 1712-1738. doi: 10.1137/080728809.  Google Scholar

[8]

Journal of Statistical Physics, 113 (2003), 151-176. doi: 10.1023/A:1025722804873.  Google Scholar

[9]

Nature Materials, 8 (2009), 843-849. Google Scholar

[10]

SIAM Journal on Applied Dynamical Systems, 10 (2011), 707-743. doi: 10.1137/100801378.  Google Scholar

[11]

Discrete and Continuous Dynamical Systems, 15 (2006), 1049-1078. doi: 10.3934/dcds.2006.15.1049.  Google Scholar

[12]

Congressus Numerantium, 30 (1981), 265-284.  Google Scholar

[13]

Physica D, 238 (2009), 1241-1255. doi: 10.1016/j.physd.2009.04.006.  Google Scholar

[14]

Proceedings of the Royal Society of Edinburgh Sect. A, 125 (1995), 351-370. doi: 10.1017/S0308210500028079.  Google Scholar

[15]

Discrete and Continuous Dynamical Systems, 25 (2009), 399-429. doi: 10.3934/dcds.2009.25.399.  Google Scholar

[16]

Physica D, 238 (2009), 645-665. doi: 10.1016/j.physd.2008.12.009.  Google Scholar

[17]

SIAM Journal on Mathematical Analysis, 42 (2010), 1602-1638. doi: 10.1137/090768643.  Google Scholar

[18]

Macromolecular Chemistry and Physics, 204 (2003), 1047-1055. Google Scholar

[19]

International Journal of Bifurcation and Chaos, 17 (2007), 1221-1263. doi: 10.1142/S0218127407017781.  Google Scholar

[20]

Macromolecules, 26 (1993), 5503-5511. Google Scholar

[21]

Translations of Mathematical Monographs, 209, Iwanami Series in Modern Mathematics, American Mathematical Society, Providence, RI, 2002.  Google Scholar

[22]

Y. Nishiura and I. Ohnishi, Rugged landscape with fine structure,, unpublished preprint., ().   Google Scholar

[23]

Physica D, 84 (1995), 31-39. doi: 10.1016/0167-2789(95)00005-O.  Google Scholar

[24]

Y. Nishiura and H. Suzuki, Higher dimensional SLEP equation and applications to morphological stability in polymer problems,, SIAM Journal on Mathematical Analysis, 36 (): 916.  doi: 10.1137/S0036141002420157.  Google Scholar

[25]

Chaos, 9 (1999), 329-341. doi: 10.1063/1.166410.  Google Scholar

[26]

Macromolecules, 19 (1986), 2621-2632. Google Scholar

[27]

Discrete and Continuous Dynamical Systems, 24 (2009), 979-1003. doi: 10.3934/dcds.2009.24.979.  Google Scholar

[28]

Interfaces and Free Boundaries, 5 (2003), 193-238. doi: 10.4171/IFB/78.  Google Scholar

[29]

Physica D, 178 (2003), 103-117. doi: 10.1016/S0167-2789(02)00808-4.  Google Scholar

[30]

Journal of Nonlinear Science, 13 (2003), 175-208. doi: 10.1007/s00332-002-0521-1.  Google Scholar

[31]

Journal of Nonlinear Science, 17 (2007), 471-503. doi: 10.1007/s00332-007-9005-7.  Google Scholar

[32]

Journal of the Physical Society of Japan, 77 (2008), 034802. Google Scholar

[33]

Physical Review E, 69 (2004), 031803. Google Scholar

[34]

Macromolecular Theory and Simulations, 14 (2005), 256-266. Google Scholar

[35]

Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4612-4838-5.  Google Scholar

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