# Phase boundary

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In thermal equilibrium, each phase (i.e. liquid, solid etc.) of physical matter 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 research field, called interface science, in physics and mathematics 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, 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 [1] 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 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. [2]

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 ${\displaystyle d=3}$ was solved by Shoshana Kamin.

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The enthalpy of vaporization, also known as the (latent) heat of vaporization or heat of evaporation, is the amount of energy (enthalpy) that must be added to a liquid substance to transform a quantity of that substance into a gas. The enthalpy of vaporization is a function of the pressure at which that transformation takes place.

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Josef Stefan was an ethnic Carinthian Slovene physicist, mathematician, and poet of the Austrian Empire.

A phase diagram in physical chemistry, engineering, mineralogy, and materials science is a type of chart used to show conditions at which thermodynamically distinct phases occur and coexist at equilibrium.

Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension is what allows objects with a higher density than water such as razor blades and insects to float on a water surface without becoming even partly submerged.

Freezing, also known as solidification, is a phase transition where a liquid turns into a solid when its temperature is lowered below its freezing point. In accordance with the internationally established definition, freezing means the solidification phase change of a liquid or the liquid content of a substance, usually due to cooling.

Benoît Paul Émile Clapeyron was a French engineer and physicist, one of the founders of thermodynamics.

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In mathematics and its applications, particularly to phase transitions in matter, a Stefan problem is a particular kind of boundary value problem for a system of partial differential equations (PDE), in which the boundary between the phases can move with time. The classical Stefan problem aims to describe the evolution of the boundary between two phases of a material undergoing a phase change, for example the melting of a solid, such as ice to water. This is accomplished by solving heat equations in both regions, subject to given boundary and initial conditions. At the interface between the phases the temperature is set to the phase change temperature. To close the mathematical system a further equation, the Stefan condition, is required. This is an energy balance which defines the position of the moving interface. Note that this evolving boundary is an unknown (hyper-)surface; hence, Stefan problems are examples of free boundary problems.

Premelting refers to a quasi-liquid film that can occur on the surface of a solid even below melting point. The thickness of the film is temperature dependent. This effect is common for all crystalline materials. Premelting shows its effects in frost heave, the growth of snowflakes and, taking grain boundary interfaces into account, maybe even in the movement of glaciers.

A dispersion is a system in which distributed particles of one material are dispersed in a continuous phase of another material. The two phases may be in the same or different states of matter.

James Thomson FRS FRSE LLD was a British engineer and physicist, born in Belfast, and older brother of William Thomson.

In chemistry, phase-boundary catalysis (PBC) is a type of heterogeneous catalytic system which facilitates the chemical reaction of a particular chemical component in an immiscible phase to react on a catalytic active site located at a phase boundary. The chemical component is soluble in one phase but insoluble in the other. The catalyst for PBC has been designed in which the external part of the zeolite is hydrophobic, internally it is usually hydrophilic, notwithstanding to polar nature of some reactants. In this sense, the medium environment in this system is close to that of an enzyme. The major difference between this system and enzyme is lattice flexibility. The lattice of zeolite is rigid, whereas the enzyme is flexible.

Gunduz Caginalp was a mathematician whose research has also contributed over 100 papers to physics, materials science and economics/finance journals, including two with Michael Fisher and nine with Nobel Laureate Vernon Smith. He started Cornell University in 1970 and received an AB in 1973 "Cum Laude with Honors in All Subjects" and Phi Beta Kappa, Master's in 1976 and PhD in 1978. He has held positions at The Rockefeller University, Carnegie-Mellon University and the University of Pittsburgh, where he was a Professor of Mathematics until his death on December 7th, 2021. He was born in Turkey, and spent his first seven years and ages 13–16 there, and the middle years in New York City.

Miscibility is the property of two substances to mix in all proportions, forming a homogeneous mixture. The term is most often applied to liquids but also applies to solids and gases. For example, water and ethanol are miscible because they mix in all proportions.

A phase-field model is a mathematical model for solving interfacial problems. It has mainly been applied to solidification dynamics, but it has also been applied to other situations such as viscous fingering, fracture mechanics, hydrogen embrittlement, and vesicle dynamics.

The Gibbs–Thomson effect, in common physics usage, refers to variations in vapor pressure or chemical potential across a curved surface or interface. The existence of a positive interfacial energy will increase the energy required to form small particles with high curvature, and these particles will exhibit an increased vapor pressure. See Ostwald–Freundlich equation. More specifically, the Gibbs–Thomson effect refers to the observation that small crystals are in equilibrium with their liquid melt at a lower temperature than large crystals. In cases of confined geometry, such as liquids contained within porous media, this leads to a depression in the freezing point / melting point that is inversely proportional to the pore size, as given by the Gibbs–Thomson equation.

In mathematics, a free boundary problem is a partial differential equation to be solved for both an unknown function and an unknown domain . The segment of the boundary of which is not known at the outset of the problem is the free boundary.

## References

1. G. Lamé, B. P. Clapeyron, Memoire sur la solidification par refroiddissement d'un globe solide, Ann. Chem. Physics, 47, 250–256 (1831).
2. J. Stefan, Über einige Probleme der Theorie der Warmeleitung, S.-B Wien Akad. Mat. Natur, 98, 173–484, (1889).