Spinodal decomposition in the linear Cahn-Hilliard model

10. Stanislaus Maier-Paape, Thomas Wanner:
    Spinodal decomposition in the linear Cahn-Hilliard model
    ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik 78(S3), pp. S1003-S1004, 1998.

Abstract

This short contribution comments on parts of a series of two papers by Maier-Paape, Wanner (1998, 2000) on spinodal decomposition for the Cahn-Hilliard equation $\frac{\partial u}{\partial t} = -\Delta(\epsilon^2 \Delta u + f(u))$ in $\Omega$, subject to $\frac{\partial u}{\partial \nu} = \frac{\partial \Delta u}{\partial \nu} = 0$ on $\partial \Omega$, where $\Omega \subset \mathbb{R}^n$, $n = 1,2,3$, is a bounded domain with sufficiently smooth boundary, $0<\epsilon \ll 1$ is a small parameter, and $f$ is cubic-like, for example $f(u) = u - u^3$. In a small neighborhood of a homogeneous equilibrium $u_0 \equiv \mu$ of the above equation, with $\mu\in\mathbb{R}$ in the spinodal region, we find that a strongly unstable manifold of $u_0$ has dominating effects. Although this is demonstrated here for the linearized equation only, it will be essential for the nonlinear situation as well. Furthermore we comment on the notion of a characteristic wavelength for the strongly unstable directions. These patterns indeed correspond to typically observed spinodally decomposed states which arise for instance in materials sciences where $u$ corresponds to the concentration of one of the components of a binary alloy.

Links

The published version of the paper can be found at https://doi.org/10.1002/zamm.19980781571.

Bibtex

@article{maier:wanner:98b,
   author = {Stanislaus Maier-Paape and Thomas Wanner},
   title = {Spinodal decomposition in the linear {C}ahn-{H}illiard model},
   journal = {ZAMM.\ Zeitschrift f\"ur Angewandte Mathematik und Mechanik},
   year = 1998,
   volume = 78,
   number = {S3},
   pages = {S1003--S1004},
   doi = {10.1002/zamm.19980781571}
   }