Mathematical and computational aspects of solidification of pure substances.

*(English)*Zbl 0990.80006The problem of solidification of pure substances, especially with respect to a body with some microscopic geometry is presented in this article. On the microscopic level, a small neighborhood of an arbitrary point in the liquid phase is chosen and the transient short-time process of the solid growth is investigated. Crystal growth from a pure liquid is important for the semiconductor processing whereas solidification of mixtures creating alloys is important for mechanical engineering. A brief summarizing of physical background is included together with the classical model of Stefan problem and the model with Gibbs-Thompson surface tension condition is presented. Phase field theory and the formulation of the phase field model for solidification of a pure substance is explained and some variants of a model are depicted. A weak solution for the phase field equation is defined and theoretical results concerning uniqueness and some properties of this solution are presented. Some examples of qualitative computational studies by the phase field model are involved, too. A wide literature and references for problems and results concerning this topic conclude this article.

Reviewer: Angela Handlovičová (Bratislava)

##### MSC:

80A22 | Stefan problems, phase changes, etc. |