Binary Alloy Phase Diagrams: Liquidus, Solidus & Eutectic

Q: What is the eutectic point?

The eutectic point is the unique composition and temperature at which a liquid freezes directly into two solid phases simultaneously, at the lowest melting temperature of the whole system. Gibbs' phase rule (F = C − P + 1) gives zero degrees of freedom there, so both the temperature and all three phase compositions are uniquely fixed.

Q: How does the lever rule work?

The lever rule uses the distances along a horizontal tie line drawn across the two-phase region to calculate the relative amounts of liquid and solid: f_solid = (C₀ − C_L)/(C_S − C_L), where C₀ is the overall composition and C_L, C_S are the liquid and solid compositions at the tie-line ends. It behaves exactly like a balanced physical lever.

Liquidus, solidus, and the lever rule

Every binary diagram plots composition (0–100% of element B) against temperature. The liquidus is the upper curve — above it the alloy is entirely molten. The solidus is the lower curve — below it the alloy is entirely solid. Between them lies a lens-shaped two-phase region where liquid and solid coexist in equilibrium. For a pure metal these two lines meet at a single melting point; for an alloy, the gap between them means solidification occurs over a temperature range, with the first crystals appearing at the liquidus and the last liquid vanishing at the solidus.

Lever rule (fraction solid at a given tie line):
f_solid = (C₀ − C_L) / (C_S − C_L)

C₀ = overall alloy composition
C_L, C_S = liquid and solid compositions at the tie-line ends

The lever rule behaves exactly like a balanced physical lever, and it is a direct consequence of conservation of mass — draw a horizontal tie line across the two-phase region, and the fraction of solid is the opposite arm's length divided by the total tie-line length.

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The eutectic reaction: one liquid, two solids

Many alloy systems show only limited solid solubility, so the liquidus splits into two branches that descend from each pure metal's melting point and meet at a low-temperature minimum: the eutectic point — Greek for "easily melted." There, a single liquid transforms simultaneously into two distinct solid phases (Liquid → α + β) at a fixed temperature and composition. Because three phases coexist at once, Gibbs' phase rule F = C − P + 1 gives zero degrees of freedom for a two-component system, so temperature and all compositions are uniquely fixed — this is why a true eutectic alloy freezes into a fine, lamellar intergrowth of both phases, sharply and at the lowest possible temperature of the whole system.

This sharp freezing is exactly why engineers choose eutectic compositions on purpose: tin-lead and tin-silver-copper solders melt and freeze at one clean low temperature rather than a sluggish pasty range, and aluminium-silicon casting alloys sit close to their eutectic to improve fluidity in the mould, which is why they dominate automotive engine blocks. The same phase-diagram logic, applied to iron-carbon, underlies the hardening and tempering processes used to tune the strength of steels.

Frequently asked questions

What is the difference between the liquidus and the solidus?

The liquidus is the line above which the alloy is fully liquid; the solidus is the line below which it is fully solid. Between them, liquid and solid coexist, so alloys freeze over a temperature range rather than at one point.

What is the eutectic point?

The unique composition and temperature at which a liquid freezes directly into two solid phases at once, at the system's lowest melting temperature. Gibbs' phase rule fixes both the temperature and all phase compositions there.

How does the lever rule work?

It uses the distances along a horizontal tie line to calculate phase fractions: f_solid = (C₀ − C_L)/(C_S − C_L) — behaving exactly like a balanced physical lever.

Try it live

Everything above runs in your browser — open Binary Alloy Phase Diagram and drag the cursor to trace the liquidus, solidus and eutectic point, reading phase composition by the lever rule.

▶ Open Binary Alloy Phase Diagram simulation