# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 2013, 33 (8) : 3671-3705. doi: 10.3934/dcds.2013.33.3671
##### 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 (): 916.  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 (): 916.  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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