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Spinodal Decomposition in the Presence of Elastic Interactions

In: Geometric Analysis and Nonlinear Partial Differential Equations

Author

Listed:
  • Harald Garcke

    (Universität Regensburg, Naturwissenschaftliche Fakultät I — Mathematik)

  • Stanislaus Maier-Paape

    (RWTH Aachen, Institut für Mathematik)

  • Ulrich Weikard

    (Universität Duisburg, Institut für Mathematik)

Abstract

Summary Spinodal decomposition, i.e., the separation of a homogeneous mixture into different phases, can be modeled by the Cahn-Hilliard equation — a fourth order semilinear parabolic equation. If elastic stresses due to a lattice misfit become important, the Cahn-Hilliard equation has to be coupled to an elasticity system to take this into account. It is the goal of this paper to understand how elastic effects influence the formation of patterns during spinodal decomposition and to analyze what kind of morphologies one has to expect. It is shown that with a probability close to one, the dynamics of randomly chosen initial data in the neighborhood of a uniform mixture will be dominated by an invariant manifold which is tangential to the most unstable eigenfunctions of the linearized operator. For example in the case of cubic anisotropy it is shown that the most unstable eigenfunctions reflect the cubic anisotropy and the anisotropy will influence the dynamics quite drastically.

Suggested Citation

  • Harald Garcke & Stanislaus Maier-Paape & Ulrich Weikard, 2003. "Spinodal Decomposition in the Presence of Elastic Interactions," Springer Books, in: Stefan Hildebrandt & Hermann Karcher (ed.), Geometric Analysis and Nonlinear Partial Differential Equations, pages 603-635, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-55627-2_32
    DOI: 10.1007/978-3-642-55627-2_32
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