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Topological Analysis of the Diblock Copolymer Equation

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Mathematical Challenges in a New Phase of Materials Science

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 166))

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.

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Notes

  1. 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. 2.

    We assume again that the equilibrium is hyperbolic, which is true away from the bifurcation points.

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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).

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  1. Department of Mathematical Sciences, George Mason University, Fairfax, VA, 22030, USA

    Thomas Wanner

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  1. Thomas Wanner
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Correspondence to Thomas Wanner .

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  1. WPI Advanced Institute for Materials Research, Tohoku University, Sendai, Miyagi, Japan

    Yasumasa Nishiura

  2. WPI Advanced Institute for Materials Research, Tohoku University, Sendai, Miyagi, Japan

    Motoko Kotani

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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

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