title: The sharp-interface limit of the Cahn-Hilliard system with elasticity
Creators: name: Kwak, David Jung Chul
Publisher: name: Universität Regensburg
Year: 2008
Subjects: subject: Nichtlineare partielle Differentialgleichung , Elastizität , Phasenseparation , freies Randwert-Problem , phase separation , free boundary problem
subject: 510 Mathematik
subject: 49Q15
subject: 41A99
subject: 76M45
subject: 74B10
subject: 74N25

description: The aim of this work is to analyse the morphological changes in alloys with elastic misfit. The phase separation of a binary alloy takes place in three stages. t a first stage when a uniform mixture is brought into an unstable state many small nuclei of one component of the alloy are created, which grow in an intermediate state rapidly up to a certain order of size. The evolution on this short time scale is called spinodal decomposition. Now the two phases are separated but still form a finely mixed microstructure which gives rise to high interfacial energy. In the last stage of the phase separation, which takes place on a much slower time scale, atoms diffuse from smaller particles towards larger particles to decrease the surface area. Consequently, large particles grow and smaller ones shrink, a process which is known as Ostwald ripening or aging. In alloys with different kind of atoms the free energy which drives the process does not only depend on the surface area of the particles. If the difference in size of the atoms is not too large, they can fit on a common lattice but at the cost of a distortion of the lattice. The energy arising from this distortion is proportional to the volume of the particles and is thus unimportant for very small particles. As the coarsening process advances, the typical particle size becomes larger and the elastic energy becomes comparable to the surface energy. The effects of elastic misfit or inhomogeneous elastic properties can change the coarsening process drastically. The shape of the particles changes from spherical to cuboidal or plate shape, particles can align or even split, the coarsening process can slow down or might even stop. Since the elastic energy not necessarily favours large particles there is some hope that it is possible to use elastic effects to stabilise the coarsening process (inverse coarsening''). For the phenomenon of phase separation of binary alloys two models are widely accepted and used: a phase-field model (Cahn-Hilliard) and a sharp interface model (Mullins-Sekerka). We extend these models by elastic effects. Then, the two models are mathematically rigorously put into relation: we look at the asymptotic behaviour of solutions of the phase-field model. Here techniques from partial differential equations, functional analysis and geometric measure theory are applied. The mail result is that solutions of the phase-field model converge along a subsequence to a varifold-type solution of the sharp-interface model. The varifold notation enables us to generalise the notion of surface and its curvature in a way that the desired asymptotic limit. Furthermore, we have identified some situations where inverse coarsening'' is expected. In the non-elastic case the surface energy yields to the coarsening behaviour. So, in this sense the elastic effect surmounts the surface energy.
description: Bei binären Legierungen tritt bei Zimmertemperatur das Phänomen der Phasenseparation auf: die Legierung trennt sich wieder in ihre Ursprungsmaterialien auf. Bei metallenen Legierungen besteht die Hoffnung, dass die Elastizität dazu führt, dass das Gemisch doch stabil bleibt. Für die mathematische Beschreibung dieses Phänomens haben sich zwei Modelle etabliert: ein sogenanntes Phasenfeld-Modell und ein Modell mit freiem Rand. Ziel der vorliegenden Arbeit war es diese beiden Modelle mathematisch rigoros zueinander in Beziehung zu setzen. Dabei wurden Techniken aus der Partiellen Differentialgleichung, Funktionalanalysis und der Geometrischen Maßtheorie benutzt. Das Ergebnis kann folgendermaßen verkürzt dargestellt werden: Lösungen des Phasenfeldmodells konvergieren entlang einer Teilfolge zu einer Varigfaltigkeits-Lösung des freien Randwert-Problems. Varigfaltigkeiten dienen dazu, den Begriff der Fläche und ihrer Krümmung zu verallgemeinern. Weiterhin wurden Fälle identifiziert, in denen das sogenannte "inverse Vergröberung", also das Gegenteil der Phasenseparation, erwartet werden können.