Solid–Liquid Diagrams: Simple Eutectics

Introduction

When two substances mix as liquids but have limited solid solubility, their melting diagram may contain a eutectic. At the eutectic composition, liquid freezes at one lowest temperature into two solid phases together. On either side, one solid generally appears first and the remaining liquid moves toward eutectic composition during cooling. This diagram explains characteristic cooling-curve arrests and why certain alloy mixtures melt or freeze in a narrow interval.

Core explanation

Consider components A and B that are completely miscible in the liquid and, for the simplest model, essentially insoluble in one another as solids. A temperature–composition diagram at fixed pressure has two liquidus branches descending from the pure-component melting points and meeting at a minimum. Above both branches the sample is liquid. Between the A-side liquidus and the eutectic temperature, solid A and liquid coexist; on the B side, solid B and liquid coexist. At the meeting point, liquid of one special composition coexists with solid A and solid B.

For the binary system at fixed pressure, the reduced phase rule is F'=C−P+1. At the eutectic, C=2 and P=3, so F'=0. The eutectic temperature and the compositions of its three equilibrium phases are fixed at that pressure. Heat removed during the eutectic transformation changes amounts while temperature remains at the arrest until one of the three phases disappears. This is analogous to a pure substance's melting arrest, but here the invariant reaction involves one liquid and two solids.

For an A-rich overall composition, cooling from the liquid region crosses the A-side liquidus first. Solid A begins to crystallise. Because this removes A preferentially, the remaining liquid becomes richer in B and follows the liquidus toward the eutectic composition. At the eutectic temperature, the residual liquid transforms to a mixture of solid A and solid B. On the B-rich side, primary solid B appears first and residual liquid moves in the opposite direction toward the same eutectic. A cooling curve may show a change in slope at first crystallisation and a more pronounced temperature arrest at eutectic solidification under near-equilibrium conditions.

The lever rule applies within a two-phase field if the endpoints are known. In an ideal simple eutectic's A-solid plus liquid region, the A-rich solid endpoint may be nearly pure A, while the liquid endpoint lies on the liquidus at the chosen temperature. The overall composition is their mass- or mole-weighted mean on a consistent basis. At a three-phase invariant point, a single scalar composition balance and total balance may not by themselves specify all three independent phase amounts; the thermal history and additional balance information matter.

Real binary solids may have finite solid solubility, intermediate compounds or polymorphs, so the branches and endpoints may be more complicated. A eutectic temperature is the lowest liquidus temperature for a given simple binary equilibrium diagram, not a statement that the mixture is a new pure chemical compound. The resulting fine microstructure can depend on cooling rate: rapid freezing may suppress diffusion and yield structures different from the ideal equilibrium assemblage.

Historically, solders were often chosen for accessible melting ranges and wetting behaviour. A eutectic alloy can transform at one temperature rather than over a broad freezing interval, which can be convenient in joining. Practical solder selection also depends on toxicity rules, mechanical reliability, corrosion and process temperature; a phase diagram alone does not determine suitability.

Step-by-step reasoning

Identify the overall composition and locate it relative to the eutectic composition. Follow a vertical constant-composition cooling path from the liquid region. Record the first liquidus crossed and its primary solid. Track the liquid endpoint along the liquidus toward the eutectic point, then identify the two solids produced from residual liquid at the invariant temperature.

Visual explanation

Sketch a V-shaped pair of liquidus curves meeting at a low point. Label liquid above, A solid plus liquid in the left wedge, B solid plus liquid in the right wedge, and two solids below the eutectic isotherm. A vertical cooling arrow left of the minimum meets the left branch before reaching the horizontal eutectic line.

Real-world analogy

Imagine a mixed queue in which one type leaves first. The people remaining become enriched in the other type until a final group leaves together. This resembles primary crystallisation enriching the residual liquid toward eutectic composition. The analogy does not explain why the equilibrium liquidus has its specific temperature, which comes from chemical potentials.

Real-world example

In alloy processing, a composition near a eutectic can solidify with a fine two-phase microstructure. An A-rich casting instead first forms primary A-rich crystals, with eutectic material developing from the last remaining liquid. Metallographers compare observed structures with equilibrium diagrams while accounting for non-equilibrium cooling effects.

Why?

Why does the residual liquid move toward eutectic composition? When primary A solid removes disproportionately more A than B, the remaining liquid's B fraction rises. Its equilibrium composition must stay on the liquidus at the falling temperature, leading toward the intersection where both solids can coexist with liquid.

Common misconception

The eutectic point does not mean every mixture of A and B freezes entirely at one temperature. Off-eutectic mixtures begin primary crystallisation at their own liquidus temperature and finish at the eutectic. A eutectic composition is also not automatically a stoichiometric compound; the two resulting solids can remain separate phases.

Worked example

At a temperature above the eutectic, an A-rich binary alloy contains solid A with x B≈0 and liquid with x B=0.40. Its overall B fraction is z B=0.10. The liquid fraction is (0.10−0)/(0.40−0)=0.25, leaving 0.75 solid A by moles under the stated basis. The remaining liquid is richer in B than the overall alloy, consistent with primary A crystallisation.

Quick check

1. What phases coexist at a simple binary eutectic at fixed pressure, and what is F'? Answer: Liquid plus A-rich and B-rich solid phases coexist; C=2 and P=3 give F'=2−3+1=0.

Exam focus

Do not confuse overall composition with the evolving residual-liquid composition. Label liquidus and eutectic isotherm, and state whether the overall sample is left, right or exactly at the eutectic. For lever calculations, use endpoints of the specific two-phase field at the selected temperature.

Advanced insight

The eutectic transformation is an invariant reaction at fixed pressure because equality of each component's chemical potential across three phases fully constrains temperature and endpoint compositions. A kinetic cooling curve may show undercooling before nucleation, then recalescence as latent heat is released. The equilibrium eutectic temperature should not be inferred blindly from a strongly nonequilibrium arrest.

Summary

A simple binary eutectic has two liquidus branches meeting at a lowest-temperature invariant point. Off-eutectic liquid first forms a primary solid, then its residual liquid reaches the eutectic composition and freezes into two solids. Cooling paths, tie lines and material balances together determine the phase sequence and proportions.

Practice questions

1. What crystallises first from an A-rich liquid in a simple eutectic diagram? Answer: Primary A solid appears when the cooling path reaches the A-side liquidus, before eutectic solidification of residual liquid. 2. Why does a eutectic cooling curve show a thermal arrest under near-equilibrium conditions? Answer: Latent heat is released while the invariant liquid-plus-two-solids transformation proceeds at fixed temperature and pressure. 3. Does an off-eutectic mixture start freezing at the eutectic temperature? Answer: No. It begins at its own liquidus temperature, which is usually higher, and residual liquid later reaches the eutectic. 4. If liquid is x B=0.50 and solid A is x B=0, with z B=0.20, what is the liquid mole fraction? Answer: (0.20−0)/(0.50−0)=0.40 by the lever rule on the stated mole basis.