Binary Phase Diagram: Alloy Composition & the Lever Rule

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Two-phase region: lever rule yields ~60% solid

At 30 wt% B and 800°C in a Cu-Ni-like system, the alloy sits in the two-phase (L+S) region. The lever rule determines the fraction of each phase from the tie-line endpoints.

Formula

f_L = (C_S - C₀) / (C_S - C_L) (lever rule, liquid fraction)
f_S = (C₀ - C_L) / (C_S - C_L) (lever rule, solid fraction)
T_liquidus ≈ Tm_A + (Tm_B - Tm_A) × (wt%B / 100) for ideal isomorphous system

Mapping the Phases of Alloys

A binary phase diagram is the roadmap of alloy metallurgy. For any combination of two elements, it tells you exactly which phases are stable at a given temperature and composition. The diagram is constructed from cooling curve experiments and thermodynamic calculations, and it underpins every decision in alloy design, casting, and heat treatment.

The Isomorphous System

The simplest binary system is isomorphous — the two components are completely soluble in each other in both liquid and solid states (like Cu-Ni). The diagram has just two curves: the liquidus (above which everything is liquid) and the solidus (below which everything is solid). Between them, liquid and solid solutions coexist. The compositions of these coexisting phases are read from the endpoints of a horizontal tie line.

The Lever Rule

When you're in a two-phase region, the lever rule tells you how much of each phase is present. Think of the tie line as a seesaw balanced at the overall composition. The fraction of each phase is proportional to the opposite arm of the lever. This simple geometric construction is one of the most powerful tools in metallurgical engineering — it connects the phase diagram directly to microstructure.

Engineering Applications

Phase diagrams guide real engineering decisions. They tell foundry engineers the pouring temperature for a casting. They tell metallurgists which heat treatment will produce the desired microstructure. They predict whether an alloy will have a narrow or wide freezing range (important for avoiding casting defects). The Fe-C phase diagram alone is the foundation of the entire steel industry.

FAQ

What is a binary phase diagram?

A binary phase diagram maps the stable phases of a two-component alloy as a function of composition (x-axis) and temperature (y-axis). It shows which phases — liquid, solid solution, or two-phase mixtures — exist under equilibrium conditions.

How does the lever rule work?

In a two-phase region, draw a horizontal tie line at the given temperature. The fraction of each phase is determined by the 'lever' distances: fraction of phase α = (C₀ - Cβ)/(Cα - Cβ), where C₀ is overall composition and Cα, Cβ are the phase boundary compositions.

What is the difference between liquidus and solidus lines?

The liquidus line is the upper boundary — above it, the alloy is fully liquid. The solidus line is the lower boundary — below it, the alloy is fully solid. Between them is the two-phase region where liquid and solid coexist.

Why do alloys melt over a range of temperatures?

Unlike pure metals which melt at a single temperature, alloys have different compositions in their liquid and solid phases. This means melting occurs progressively over a range between the solidus and liquidus temperatures.

Sources

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