Abstract
We demonstrate how topological methods can be used to study pattern formation and pattern evolution in phase-field models of materials science. In the context of the diblock copolymer model for microphase separation, we will present new quantitative results on the microstructure topology during the initial phase separation from a homogeneous state, both for a deterministic and a stochastic version of the model. We also describe the long-term dynamics of the model and associated questions of multistability, which can be addressed using rigorous topological methods aimed at determining the structure of the global attractor of the system.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
We assume for the purposes of this survey that equilibrium solutions are generally hyperbolic, i.e., the linearization of the evolution equation at the stationary solutions has no zero eigenvalue.
- 2.
We assume again that the equilibrium is hyperbolic, which is true away from the bifurcation points.
References
Bahiana, M., Oono, Y.: Cell dynamical system approach to block copolymers. Phys. Rev. A 41, 6763–6771 (1990)
Bates, P.W., Fife, P.C.: The dynamics of nucleation for the Cahn-Hilliard equation. SIAM J. Appl. Math. 53 (4), 990–1008 (1993)
Blömker, D., Gawron, B., Wanner, T.: Nucleation in the one-dimensional stochastic Cahn-Hilliard model. Discret. Contin. Dyn. Syst. Ser. A 27 (1), 25–52 (2010)
Blömker, D., Maier-Paape, S., Wanner, T.: Phase separation in stochastic Cahn-Hilliard models. In: Miranville, A. (ed.) Mathematical Methods and Models in Phase Transitions, pp. 1–41. Nova Science Publishers, New York (2005)
Blömker, D., Maier-Paape, S., Wanner, T.: Second phase spinodal decomposition for the Cahn-Hilliard-Cook equation. Trans. Am. Math. Soc. 360 (1), 449–489 (2008)
Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system I. Interfacial free energy. J. Chem. Phys. 28, 258–267 (1958)
Choksi, R., Peletier, M.A., Williams, J.F.: On the phase diagram for microphase separation of diblock copolymers: an approach via a nonlocal Cahn-Hilliard functional. SIAM J. Appl. Math. 69 (6), 1712–1738 (2009)
Choksi, R., Ren, X.: On the derivation of a density functional theory for microphase separation of diblock copolymers. J. Stat. Phys. 113, 151–76 (2003)
CHomP: Computational Homology Project. http://chomp.rutgers.edu (2015)
Cochran, G.S., Wanner, T., Dłotko, P.: A randomized subdivision algorithm for determining the topology of nodal sets. SIAM J. Sci. Comput. 35 (5), B1034–B1054 (2013)
Cook, H.: Brownian motion in spinodal decomposition. Acta Metall. 18, 297–306 (1970)
Day, S., Kalies, W.D., Mischaikow, K., Wanner, T.: Probabilistic and numerical validation of homology computations for nodal domains. Electron. Res. Announcements Am. Math. Soc. 13, 60–73 (2007)
Day, S., Kalies, W.D., Wanner, T.: Verified homology computations for nodal domains. SIAM J. Multiscale Model. Simul. 7 (4), 1695–1726 (2009)
Desi, J.P., Edrees, H., Price, J., Sander, E., Wanner, T.: The dynamics of nucleation in stochastic Cahn-Morral systems. SIAM J. Appl. Dyn. Syst. 10 (2), 707–743 (2011)
Desi, J.P., Sander, E., Wanner, T.: Complex transient patterns on the disk. Discret. Contin. Dyn. Syst. Ser. A 15 (4), 1049–1078 (2006)
Dłotko, P., Kaczynski, T., Mrozek, M., Wanner, T.: Coreduction homology algorithm for regular CW-complexes. Discret. Comput. Geom. 46 (2), 361–388 (2011)
Doedel, E.: AUTO: a program for the automatic bifurcation analysis of autonomous systems. In: Proceedings of the Tenth Manitoba Conference on Numerical Mathematics and Computing, Vol. I (Winnipeg, Man., 1980), vol. 30, pp. 265–284 (1981)
Edelman, A., Kostlan, E.: How many zeros of a random polynomial are real? Bull. Am. Math. Soc. 32 (1), 1–37 (1995)
Edelsbrunner, H., Harer, J.L.: Computational Topology. American Mathematical Society, Providence (2010)
Farahmand, K.: Topics in Random Polynomials. Pitman Research Notes in Mathematics, vol. 393. Longman, Harlow (1998)
Gameiro, M., Mischaikow, K., Kalies, W.: Topological characterization of spatial-temporal chaos. Phys. Rev. E 70 (3), 035,203, 4 (2004)
Gameiro, M., Mischaikow, K., Wanner, T.: Evolution of pattern complexity in the Cahn-Hilliard theory of phase separation. Acta Materialia 53 (3), 693–704 (2005)
Grinfeld, M., Novick-Cohen, A.: Counting stationary solutions of the Cahn-Hilliard equation by transversality arguments. Proc. R. Soc. Edinb. 125A, 351–370 (1995)
Harker, S., Mischaikow, K., Mrozek, M., Nanda, V.: Discrete Morse theoretic algorithms for computing homology of complexes and maps. Found. Comput. Math. 14 (1), 151–184 (2014)
Hyde, J.M., Miller, M.K., Hetherington, M.G., Cerezo, A., Smith, G.D.W., Elliott, C.M.: Spinodal decomposition in Fe-Cr alloys: experimental study at the atomic level and comparison with computer models — II. Development of domain size and composition amplitude. Acta Metallurgica et Materialia 43, 3403–3413 (1995)
Hyde, J.M., Miller, M.K., Hetherington, M.G., Cerezo, A., Smith, G.D.W., Elliott, C.M.: Spinodal decomposition in Fe-Cr alloys: experimental study at the atomic level and comparison with computer models — III. Development of morphology. Acta Metallurgica et Materialia 43, 3415–3426 (1995)
Johnson, I., Sander, E., Wanner, T.: Branch interactions and long-term dynamics for the diblock copolymer model in one dimension. Discret. Contin. Dyn. Syst. Ser. A 33 (8), 3671–3705 (2013)
Kaczynski, T., Mischaikow, K., Mrozek, M.: Computational Homology. Applied Mathematical Sciences, vol. 157. Springer-Verlag, New York (2004)
Kramár, M., Goullet, A., Kondic, L., Mischaikow, K.: Quantifying force networks in particulate systems. Physica D 283, 37–55 (2014)
Krishan, K., Gameiro, M., Mischaikow, K., Schatz, M., Kurtuldu, H., Madruga, S.: Homology and symmetry breaking in Rayleigh-Benard convection: experiments and simulations. Phys. Fluids 19, 117,105 (2007)
Maier-Paape, S., Miller, U., Mischaikow, K., Wanner, T.: Rigorous numerics for the Cahn-Hilliard equation on the unit square. Revista Matematica Complutense 21 (2), 351–426 (2008)
Maier-Paape, S., Mischaikow, K., Wanner, T.: Structure of the attractor of the Cahn-Hilliard equation on a square. Int. J. Bifurc. Chaos 17 (4), 1221–1263 (2007)
Maier-Paape, S., Wanner, T.: Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions. Part I: probability and wavelength estimate. Commun. Math. Phys. 195 (2), 435–464 (1998)
Maier-Paape, S., Wanner, T.: Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions: nonlinear dynamics. Arch. Ration. Mech. Anal. 151 (3), 187–219 (2000)
Miller, M.K., Hyde, J.M., Hetherington, M.G., Cerezo, A., Smith, G.D.W., Elliott, C.M.: Spinodal decomposition in Fe-Cr alloys: experimental study at the atomic level and comparison with computer models — I. Introduction and methodology. Acta Metallurgica et Materialia 43, 3385–3401 (1995)
Mischaikow, K., Wanner, T.: Probabilistic validation of homology computations for nodal domains. Ann. Appl. Probab. 17 (3), 980–1018 (2007)
Mischaikow, K., Wanner, T.: Topology-guided sampling of nonhomogeneous random processes. Ann. Appl. Probab. 20 (3), 1068–1097 (2010)
Mrozek, M., Batko, B.: Coreduction homology algorithm. Discret. Comput. Geom. 41 (1), 96–118 (2009)
Mrozek, M., Wanner, T.: Coreduction homology algorithm for inclusions and persistent homology. Comput. Math. Appl. 60 (10), 2812–2833 (2010)
Munkres, J.R.: Elements of Algebraic Topology. Addison-Wesley, Menlo Park (1984)
Nanda, V.: PERSEUS. http://www.sas.upenn.edu/~vnanda/perseus (2015)
Nishiura, Y., Ohnishi, I.: Some mathematical aspects of the micro-phase separation in diblock copolymers. Physica D 84 (1–2), 31–39 (1995)
Ohta, T., Kawasaki, K.: Equilibrium morphology of block copolymer melts. Macromolecules 19, 2621–2632 (1986)
Plum, M.: Computer-assisted proofs for semilinear elliptic boundary value problems. Jpn. J. Ind. Appl. Math. 26 (2–3), 419–442 (2009)
RedHom: Simplicial and cubical homology. http://capd.sourceforge.net/capdRedHom (2015)
Ren, X., Wei, J.: On energy minimizers of the diblock copolymer problem. Interfaces Free Bound. 5 (2), 193–238 (2003)
Rohrer, G.S., Miller, H.M.: Topological characteristics of plane sections of polycrystals. Acta Materialia 58 (10), 3805–3814 (2010)
Rump, S.M.: INTLAB – INTerval LABoratory. In: Csendes, T. (ed.) Developments in Reliable Computing, pp. 77–104. Kluwer Academic Publishers, Dordrecht (1999). http://www.ti3.tuhh.de/rump/
Sander, E., Wanner, T.: Monte Carlo simulations for spinodal decomposition. J. Stat. Phys. 95 (5–6), 925–948 (1999)
Sander, E., Wanner, T.: Unexpectedly linear behavior for the Cahn-Hilliard equation. SIAM J. Appl. Math. 60 (6), 2182–2202 (2000)
Stephens, T., Wanner, T.: Rigorous validation of isolating blocks for flows and their Conley indices. SIAM J. Appl. Dyn. Syst. 13 (4), 1847–1878 (2014)
Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer-Verlag, New York/Berlin/Heidelberg (1988)
Teramoto, T., Nishiura, Y.: Morphological characterization of the diblock copolymer problem with topological computation. Jpn. J. Ind. Appl. Math. 27 (2), 175–190 (2010)
Wanner, T.: Maximum norms of random sums and transient pattern formation. Trans. Am. Math. Soc. 356 (6), 2251–2279 (2004)
Wanner, T.: Computer-assisted equilibrium validation for the diblock copolymer model. Discret. Contin. Dyn. Syst. Ser. A (2016, to appear)
Wanner, T., Fuller Jr., E.R., Saylor, D.M.: Homological characterization of microstructure response fields in polycrystals. Acta Materialia 58 (1), 102–110 (2010)
Acknowledgements
The author would like to acknowledge the hospitality of the Research Institute for Mathematical Sciences (RIMS) at Kyoto University, where part of this work was done within the international research project Toward a New Fusion Research of Mathematics and Materials Science. The author was partially supported by National Science Foundation grants DMS-0907818, DMS-1114923, and DMS-1407087. The numerical simulations for this work were run on ARGO, a research computing cluster provided by the Office of Research Computing at George Mason University, VA (URL: http://orc.gmu.edu).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer Japan
About this paper
Cite this paper
Wanner, T. (2016). Topological Analysis of the Diblock Copolymer Equation. In: Nishiura, Y., Kotani, M. (eds) Mathematical Challenges in a New Phase of Materials Science. Springer Proceedings in Mathematics & Statistics, vol 166. Springer, Tokyo. https://doi.org/10.1007/978-4-431-56104-0_2
Download citation
DOI: https://doi.org/10.1007/978-4-431-56104-0_2
Published:
Publisher Name: Springer, Tokyo
Print ISBN: 978-4-431-56102-6
Online ISBN: 978-4-431-56104-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)