In metallurgy

Metallurgy

Metallurgy is a domain of materials science that studies the physical and chemical behavior of metallic elements, their intermetallic compounds, and their mixtures, which are called alloys. It is also the technology of metals: the way in which science is applied to their practical use...

, the Scheil-Gulliver equation (or Scheil equation) describes solute

Solution

In chemistry, a solution is a homogeneous mixture composed of only one phase. In such a mixture, a solute is dissolved in another substance, known as a solvent. The solvent does the dissolving.- Types of solutions :...

 redistribution during solidification of an alloy

Alloy

An alloy is a mixture or metallic solid solution composed of two or more elements. Complete solid solution alloys give single solid phase microstructure, while partial solutions give two or more phases that may or may not be homogeneous in distribution, depending on thermal history...

. This approach approximates non-equilibrium solidification by assuming a local equilibrium

Thermodynamic equilibrium

In thermodynamics, a thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, radiative equilibrium, and chemical equilibrium. The word equilibrium means a state of balance...

 of the advancing solidification front at the solid-liquid interface. This allows the use of equilibrium phase diagram

Phase diagram

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 can occur at equilibrium...

s in solidification analysis.

Unlike equilibrium solidification, solute does not diffuse back into the solid and is rejected completely into the liquid

Liquid

Liquid is one of the three classical states of matter . Like a gas, a liquid is able to flow and take the shape of a container. Some liquids resist compression, while others can be compressed. Unlike a gas, a liquid does not disperse to fill every space of a container, and maintains a fairly...

. Complete mixing of solute in the liquid is also assumed as a result of convection

Convection

Convection is the movement of molecules within fluids and rheids. It cannot take place in solids, since neither bulk current flows nor significant diffusion can take place in solids....

 and/or stirring.

Derivation

The following figure shows the solute redistribution for non-equilibrium solidification where there is no diffusion

Diffusion

Molecular diffusion, often called simply diffusion, is the thermal motion of all particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size of the particles...

 of solute in the solid and complete mixing of solute in the liquid. and .
Equilibrium is assumed at the interface which allows the use of an equilibrium phase diagram.



The hatched areas in the figure represent the amount of solute in the solid and liquid. Considering that the total amount of solute in the system must be conserved, the areas are set equal as follows:
.

Since the distribution coefficient

Partition coefficient

In chemistry and the pharmaceutical sciences, a partition- or distribution coefficient is the ratio of concentrations of a compound in the two phases of a mixture of two immiscible solvents at equilibrium. The terms "gas/liquid partition coefficient" and "air/water partition coefficient" are...

 is
(determined from the phase diagram)

and mass must be conserved

Conservation of mass

The law of conservation of mass, also known as the principle of mass/matter conservation, states that the mass of an isolated system will remain constant over time...

the mass balance may be rewritten as
.

Using the boundary condition
at

the following integration may be performed:
.

Integrating results in the Scheil-Gulliver equation for composition of the liquid during solidification:


or for the composition of the solid:
.