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In thermal equilibrium, each phase (i.e.
liquid Liquid is a state of matter with a definite volume but no fixed shape. Liquids adapt to the shape of their container and are nearly incompressible, maintaining their volume even under pressure. The density of a liquid is usually close to th ...
,
solid Solid is a state of matter where molecules are closely packed and can not slide past each other. Solids resist compression, expansion, or external forces that would alter its shape, with the degree to which they are resisted dependent upon the ...
etc.) of physical
matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic pa ...
comes to an end at a transitional point, or spatial interface, called a phase boundary, due to the immiscibility of the matter with the matter on the other side of the boundary. This immiscibility is due to at least one difference between the two substances' corresponding physical properties. The behavior of phase boundaries has been a developing subject of interest and an active interdisciplinary research field, called interface science, for almost two centuries, due partly to phase boundaries naturally arising in many physical processes, such as the capillarity effect, the growth of
grain boundaries In materials science, a grain boundary is the interface between two grains, or crystallites, in a polycrystalline material. Grain boundaries are two-dimensional crystallographic defect, defects in the crystal structure, and tend to decrease the ...
, the physics of binary alloys, and the formation of snow flakes. One of the oldest problems in the area dates back to Lamé and Clapeyron who studied the freezing of the ground. Their goal was to determine the thickness of solid crust generated by the cooling of a liquid at constant
temperature Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...
filling the half-space. In 1889, Stefan, while working on the freezing of the ground developed these ideas further and formulated the two-phase model which came to be known as the Stefan Problem. J. Stefan, Über einige Probleme der Theorie der Warmeleitung, S.-B Wien Akad. Mat. Natur, 98, 173–484, (1889). The proof for the existence and uniqueness of a solution to the Stefan problem was developed in many stages. Proving the general existence and uniqueness of the solutions in the case of d=3 was solved by Shoshana Kamin.


References

Phase transitions Applied mathematics {{applied-math-stub