Eutectic system

Eutectic reaction

The eutectic reaction is defined as follows:

This type of reaction is an invariant reaction, because it is in thermal equilibrium; another way to define this is the Gibbs free energy equals zero. Tangibly, this means the liquid and two solid solutions all coexist at the same time and are in chemical equilibrium. There is also a thermal arrest for the duration of the change of phase during which the temperature of the system does not change.

The resulting solid macrostructure from a eutectic reaction depends on a few factors. The most important factor is how the two solid solutions nucleate and grow. The most common structure is a lamellar structure, but other possible structures include rodlike, globular, and acicular.

Non-eutectic compositions

Compositions of eutectic systems that are not at the eutectic composition can be classified as hypoeutectic or hypereutectic. Hypoeutectic compositions are those with a smaller percent composition of species β and a greater composition of species α than the eutectic composition (E) while hypereutectic solutions are characterized as those with a higher composition of species β and a lower composition of species α than the eutectic composition. As the temperature of a non-eutectic composition is lowered the liquid mixture will precipitate one component of the mixture before the other. In a hypereutectic solution, there will be a proeutectoid phase of species β whereas a hypoeutectic solution will have a proeutectic α phase.

Alloys

Eutectic alloys have two or more materials and have a eutectic composition. When a non-eutectic alloy solidifies, its components solidify at different temperatures, exhibiting a plastic melting range. Conversely, when a well-mixed, eutectic alloy melts, it does so at a single, sharp temperature. The various phase transformations that occur during the solidification of a particular alloy composition can be understood by drawing a vertical line from the liquid phase to the solid phase on the phase diagram for that alloy.

Some uses include:

Others

Eutectoid

When the solution above the transformation point is solid, rather than liquid, an analogous eutectoid transformation can occur. For instance, in the iron-carbon system, the austenite phase can undergo a eutectoid transformation to produce ferrite and cementite, often in lamellar structures such as pearlite and bainite. This eutectoid point occurs at 727 °C (1,341 °F) and about 0.76% carbon.

Peritectoid

A peritectoid transformation is a type of isothermal reversible reaction that has two solid phases reacting with each other upon cooling of a binary, ternary, ..., n ary alloy to create a completely different and single solid phase. The reaction plays a key role in the order and decomposition of quasicrystalline phases in several alloy types.

Peritectic

Peritectic transformations are also similar to eutectic reactions. Here, a liquid and solid phase of fixed proportions react at a fixed temperature to yield a single solid phase. Since the solid product forms at the interface between the two reactants, it can form a diffusion barrier and generally causes such reactions to proceed much more slowly than eutectic or eutectoid transformations. Because of this, when a peritectic composition solidifies it does not show the lamellar structure that is found with eutectic solidification.

Such a transformation exists in the iron-carbon system, as seen near the upper-left corner of the figure. It resembles an inverted eutectic, with the δ phase combining with the liquid to produce pure austenite at 1,495 °C (2,723 °F) and 0.17% carbon.

At the peritectic decomposition temperature the compound, rather than melting, decomposes into another solid compound and a liquid. The proportion of each is determined by the lever rule. In the Al-Au phase diagram, for example, it can be seen that only two of the phases melt congruently, AuAl₂ and Au₂Al , while the rest peritectically decompose.

Eutectic calculation

The composition and temperature of a eutectic can be calculated from enthalpy and entropy of fusion of each components.

The Gibbs free energy, G, depends on its own differential

G = H − T S ⇒ { H = G + T S ( ∂ G ∂ T ) P = − S ⇒ H = G − T ( ∂ G ∂ T ) P .

Thus, the G/T derivative at constant pressure is calculated by the following equation

( ∂ G / T ∂ T ) P = 1 T ( ∂ G ∂ T ) P − 1 T 2 G = − 1 T 2 ( G − T ( ∂ G ∂ T ) P ) = − H T 2

The chemical potential μ i is calculated if we assume the activity is equal to the concentration.

μ i = μ i ∘ + R T ln ⁡ a i a ≈ μ i ∘ + R T ln ⁡ x i

At the equilibrium, μ i = 0 , thus μ i ∘ is obtained by:

μ i = μ i ∘ + R T ln ⁡ x i = 0 ⇒ μ i ∘ = − R T ln ⁡ x i .

Using and integrating gives

( ∂ μ i / T ∂ T ) P = ∂ ∂ T ( R ln ⁡ x i ) ⇒ R ln ⁡ x i = − H i ∘ T + K

The integration constant K may be determined for a pure component with a melting temperature T ∘ and an enthalpy of fusion H ∘ Eq.

x i = 1 ⇒ T = T i ∘ ⇒ K = H i ∘ T i ∘

We obtain a relation that determines the molar fraction as a function of the temperature for each component.

R ln ⁡ x i = − H i ∘ T + H i ∘ T i ∘

The mixture of n components is described by the system

{ ln ⁡ x i + H i ∘ R T − H i ∘ R T i ∘ = 0 ∑ i = 1 n x i = 1 { ∀ i < n ⇒ ln ⁡ x i + H i ∘ R T − H i ∘ R T i ∘ = 0 ln ⁡ ( 1 − ∑ i = 1 n − 1 x i ) + H n ∘ R T − H n ∘ R T n ∘ = 0

that can be solved by

[ Δ x 1 Δ x 2 Δ x 3 ⋮ Δ x n − 1 Δ T ] = [ 1 / x 1 0 0 0 0 − H 1 ∘ R T 2 0 1 / x 2 0 0 0 − H 2 ∘ R T 2 0 0 1 / x 3 0 0 − H 3 ∘ R T 2 ⋮ ⋱ ⋱ ⋱ ⋱ ⋮ 0 0 0 0 1 / x n − 1 − H n − 1 ∘ R T 2 − 1 1 − ∑ i = 1 n − 1 x i − 1 1 − ∑ i = 1 n − 1 x i − 1 1 − ∑ i = 1 n − 1 x i − 1 1 − ∑ i = 1 n − 1 x i − 1 1 − ∑ i = 1 n − 1 x i − H n ∘ R T 2 ] − 1 . [ ln ⁡ x 1 + H 1 ∘ R T − H 1 ∘ R T 1 ∘ ln ⁡ x 2 + H 2 ∘ R T − H 2 ∘ R T 2 ∘ ln ⁡ x 3 + H 3 ∘ R T − H 3 ∘ R T 3 ∘ ⋮ ln ⁡ x n − 1 + H n − 1 ∘ R T − H n − 1 ∘ R T n − 1 ∘ ln ⁡ ( 1 − ∑ i = 1 n − 1 x i ) + H n ∘ R T − H n ∘ R T n ∘ ]