# Eutectic system

(Redirected from Eutectic point)
A phase diagram for a fictitious binary chemical mixture (with the two components denoted by A and B) used to depict the eutectic composition, temperature, and point. (L denotes the liquid state.)

A eutectic system is a mixture of chemical compounds or elements that have a single chemical composition that solidifies at a lower temperature than any other composition made up of the same ingredients. This composition is known as the eutectic composition and the temperature at which it solidifies is known as the eutectic temperature.

On a phase diagram the intersection of the eutectic temperature and the eutectic composition gives the eutectic point.[1] Non-eutectic mixtures will display solidification of one component of the mixture before the other. Not all binary alloys have a eutectic point; for example, in the silver-gold system the melt temperature (liquidus) and freeze temperature (solidus) both increase monotonically as the mix changes from pure silver to pure gold.[2]

## Eutectic reaction

Four eutectic structures: A) lamellar B) rod-like C) globular D) acicular.

The eutectic reaction is defined as follows:[3]

$\text{Liquid} \xrightarrow[\text{cooling}]{\text{eutectic temperature}} \alpha \,\, \text{solid solution} + \beta \,\, \text{solid solution}$

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.[3]

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.[4]

## Non-eutectic compositions

Compositions of eutectic systems that are not at the eutectic composition are called either hypoeutectic or hypereutectic. Hypoeutectic compositions are those to the left of the eutectic composition while hypereutectic compositions are those to the right. As the temperature of a non-eutectic composition is lowered the liquid mixture will precipitate one component of the mixture before the other.[3]

## Types

### 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:

## Other critical points

### Eutectoid

Iron-carbon phase diagram, showing the eutectoid transformation between austenite (γ) and pearlite.

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.[10]

### Peritectoid

A peritectoid transformation is a type of isothermal reversible reaction that have two solid phases reacting with each other upon cooling of a binary, ternary, ..., $n\!$ alloy to create a completely different and single solid phase.[11] The reaction plays a key role in the order and decomposition of quasicrystalline phases in several alloy types.[12]

### 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.

Gold-aluminium phase diagram (German). Top axis title reads "Weight-percent Gold", lower axis title reads "Atomic-percent Gold"

Peritectic decomposition. Up to this point in the discussion transformations have been addressed from the point of view of cooling. They also can be discussed noting the changes that occur to some solid chemical compounds as they are heated. Rather than melting, at the peritectic decomposition temperature, the compound decomposes into another solid compound and a liquid. The proportion of each is determined by the lever rule. The vocabulary changes slightly. Just as the cooling of water, which leads to ice, is termed freezing, the warming of ice leads to melting. In the Al-Au phase diagram, for example, it can be seen that only two of the phases melt congruently, AuAl2 and Au2Al. 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.[13]

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

$G = H - TS \Rightarrow {\left\{ \begin{array}{l} H = G + TS \\ \\ {\left( {\frac{\partial G}{\partial T}} \right)_P = - S} \end{array} \right.} \Rightarrow H = G - T\left( {\frac{\partial G}{\partial T}} \right)_P .$

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

$\left( {\frac{\partial G / T}{\partial T}} \right)_P = \frac{1}{T}\left( {\frac{\partial G}{\partial T}} \right)_P - \frac{1}{T^{2}}G = - \frac{1}{T^{2}}\left( {G - T\left({\frac{\partial G}{\partial T}} \right)_P } \right) = - \frac{H}{T^{2}}$

The chemical potential $\mu _{i}$ is calculated if we assume the activity is equal to the concentration.

$\mu _i = \mu _i^\circ + RT\ln \frac{a_i}{a} \approx \mu _i^\circ + RT\ln x_i$

At the equilibrium, $\mu_i =0$, thus $\mu_i^\circ$ is obtained by:

$\mu _i = \mu _i^\circ + RT\ln x_i = 0 \Rightarrow \mu _i^\circ = - RT\ln x_i.$

Using and integrating gives

$\begin{array}{l} \left( {\frac{\partial \mu _i / T}{\partial T}} \right)_P = \frac{\partial }{\partial T}\left( {R\ln x_i } \right) \Rightarrow R\ln x_i = - \frac{H_i ^\circ }{T} + K \\ \\ \end{array}$

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

$x_i = 1 \Rightarrow T = T_i^\circ \Rightarrow K = \frac{H_i^\circ }{T_i^\circ }$

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

$R\ln x_i = - \frac{H_i ^\circ }{T} + \frac{H_i^\circ }{T_i^\circ }$

The mixture of n components is described by the system

$\begin{array}{l} \left\{ {{\begin{array}{*{20}c} {\ln x_i + \frac{H_i ^\circ }{RT} - \frac{H_i^\circ }{RT_i^\circ } = 0} \\ {\sum\limits_{i = 1}^n {x_i = 1} } \\ \end{array} }} \right. \\ \\ \end{array}$ $\begin{array}{l} \left\{ {{\begin{array}{*{20}c} {\forall i < n \Rightarrow \ln x_i + \frac{H_i ^\circ }{RT} - \frac{H_i^\circ }{RT_i^\circ } = 0} \\ {\ln \left( {1 - \sum\limits_{i = 1}^{n - 1} {x_i } } \right) + \frac{H_n ^\circ }{RT} - \frac{H_n^\circ }{RT_n^\circ } = 0} \\ \end{array} }} \right. \\ \\ \end{array}$

that can be solved by

$\begin{array}{c} \left[ {{\begin{array}{*{20}c} {\Delta x_1 } \\ {\Delta x_2 } \\ {\Delta x_3 } \\ \vdots \\ {\Delta x_{n - 1} } \\ {\Delta T} \\ \end{array} }} \right] = \left[ {{\begin{array}{*{20}c} {1 / x_1 } & 0 & 0 & 0 & 0 & { - \frac{H_1^\circ }{RT^{2}}} \\ 0 & {1 / x_2 } & 0 & 0 & 0 & { - \frac{H_2^\circ }{RT^{2}}} \\ 0 & 0 & {1 / x_3 } & 0 & 0 & { - \frac{H_3^\circ }{RT^{2}}} \\ 0 & 0 & 0 & \ddots & 0 & { - \frac{H_4^\circ }{RT^{2}}} \\ 0 & 0 & 0 & 0 & {1 / x_{n - 1} } & { - \frac{H_{n - 1}^\circ }{RT^{2}}} \\ {\frac{ - 1}{1 - \sum\limits_{1 = 1}^{n - 1} {x_i } }} & {\frac{ - 1}{1 - \sum\limits_{1 = 1}^{n - 1} {x_i } }} & {\frac{ - 1}{1 - \sum\limits_{1 = 1}^{n - 1} {x_i } }} & {\frac{ - 1}{1 - \sum\limits_{1 = 1}^{n - 1} {x_i } }} & {\frac{ - 1}{1 - \sum\limits_{1 = 1}^{n - 1} {x_i } }} & { - \frac{H_n^\circ }{RT^{2}}} \\ \end{array} }} \right]^{ - 1} .\left[ {{\begin{array}{*{20}c} {\ln x_1 + \frac{H_1 ^\circ }{RT} - \frac{H_1^\circ }{RT_1^\circ }} \\ {\ln x_2 + \frac{H_2 ^\circ }{RT} - \frac{H_2^\circ }{RT_2^\circ }} \\ {\ln x_3 + \frac{H_3 ^\circ }{RT} - \frac{H_3^\circ }{RT_3^\circ }} \\ \vdots \\ {\ln x_{n - 1} + \frac{H_{n - 1} ^\circ }{RT} - \frac{H_{n - 1}^\circ }{RT_{n - 1i}^\circ }} \\ {\ln \left( {1 - \sum\limits_{i = 1}^{n - 1} {x_i } } \right) + \frac{H_n ^\circ }{RT} - \frac{H_n^\circ }{RT_n^\circ }} \\ \end{array} }} \right] \end{array}$

## References

1. ^ Smith & Hashemi 2006, pp. 326–327.
2. ^ http://www.crct.polymtl.ca/fact/phase_diagram.php?file=Ag-Au.jpg&dir=SGTE
3. ^ a b c Smith & Hashemi 2006, p. 327.
4. ^ Smith & Hashemi 2006, pp. 332–333.
5. ^ Muldrew, Ken; Locksley E. McGann (1997). "Phase Diagrams". Cryobiology—A Short Course. University of Calgary. Retrieved 2006-04-29.
6. ^ Senese, Fred (1999). "Does salt water expand as much as fresh water does when it freezes?". Solutions: Frequently asked questions. Department of Chemistry, Frostburg State University. Retrieved 2006-04-29.
7. ^ "Molten salts properties". Archimede Solar Plant Specs.
8. ^ Fichter, Lynn S. (2000). "Igneous Phase Diagrams". Igneous Rocks. James Madison University. Retrieved 2006-04-29.
9. ^ Davies, Nicholas A.; Beatrice M. Nicholas (1992). "Eutectic compositions for hot melt jet inks". US Patent & Trademark Office, Patent Full Text and Image Database. United States Patent and Trademark Office. Retrieved 2006-04-29.
10. ^ Iron-Iron Carbide Phase Diagram Example
11. ^ IUPAC Compendium of Chemical Terminology, Electronic version. "Peritectoid Reaction" Retrieved May 22, 2007.
12. ^ Numerical Model of Peritectoid Transformation. Peritectoid Transformation Retrieved May 22, 2007.
13. ^ International Journal of Modern Physics C, Vol. 15, No. 5. (2004), pp. 675-687

### Bibliography

• Smith, William F.; Hashemi, Javad (2006), Foundations of Materials Science and Engineering (4th ed.), McGraw-Hill, ISBN 0-07-295358-6.