Phase boundary
Encyclopedia
The behavior of phase boundaries has been a developing subject of interest and an active research field in physics
and mathematics
for almost two centuries. One reason behind this is that phase
boundaries naturally arise in many physical processes due to immiscibility of two or more substances with different physical properties. Hence, various phenomena such as capillarity effect, growth of grain boundaries
, physics of binary alloys, formation of snow flakes fall under the category of interface science.
One of the oldest problems in the area dates back to Lame 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
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
.
The proof of existence and uniqueness of a solution to the Stefan problem
was done in many stages. Proving the general existence of the solutions turned out to be a difficult problem for
(see e.g. \cite{3}) that was finally solved by Enverbek Meirmenov.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
and mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
for almost two centuries. One reason behind this is that phase
Phase (matter)
In the physical sciences, a phase is a region of space , throughout which all physical properties of a material are essentially uniform. Examples of physical properties include density, index of refraction, and chemical composition...
boundaries naturally arise in many physical processes due to immiscibility of two or more substances with different physical properties. Hence, various phenomena such as capillarity effect, growth of grain boundaries
Grain boundary
A grain boundary is the interface between two grains, or crystallites, in a polycrystalline material. Grain boundaries are defects in the crystal structure, and tend to decrease the electrical and thermal conductivity of the material...
, physics of binary alloys, formation of snow flakes fall under the category of interface science.
One of the oldest problems in the area dates back to Lame 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
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
filling the half-space
Half-space
In geometry, a half-space is either of the two parts into which a plane divides the three-dimensional euclidean space. More generally, a half-space is either of the two parts into which a hyperplane divides an affine 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
Stefan problem
In mathematics and its applications, particularly to phase transitions in matter, a Stefan problem is a particular kind of boundary value problem for a partial differential equation , adapted to the case in which a phase boundary can move with time...
.
The proof of existence and uniqueness of a solution to the Stefan problem
Stefan problem
In mathematics and its applications, particularly to phase transitions in matter, a Stefan problem is a particular kind of boundary value problem for a partial differential equation , adapted to the case in which a phase boundary can move with time...
was done in many stages. Proving the general existence of the solutions turned out to be a difficult problem for
(see e.g. \cite{3}) that was finally solved by Enverbek Meirmenov.