Phase diagrams and the lever rule

Why it matters

A phase diagram is the map that tells you, for any alloy composition and temperature, which phases are present, what each one contains and how much of each there is. Casting, soldering, brazing, welding and every heat treatment are planned from it, and the iron–carbon diagram used for steels is simply a more detailed example of the rules learned here.

Key ideas

Basic terms.

• Component: a pure element or compound that makes up the alloy (e.g. Cu and Ni).
• Phase: a physically distinct, homogeneous region with its own structure and composition (liquid, α solid solution, an intermetallic compound).
• Solid solution: a phase in which solute atoms sit in the solvent lattice, substitutionally or interstitially. Hume-Rothery rules for complete substitutional solubility: atomic radii within about 15 %, same crystal structure, similar electronegativity and same valence. Cu–Ni meets all four.
• Equilibrium phase diagram: shows the phases present under very slow cooling (equilibrium) at constant pressure (1 atm), so pressure is not a variable.

Gibbs phase rule. At constant pressure, F = C − P + 1, where F is the number of variables (temperature, phase compositions) that can be changed independently without changing the number of phases. In a binary system: a single-phase field has F = 2, a two-phase field F = 1 (fix T and the phase compositions are fixed), and a three-phase point F = 0 (an invariant reaction at fixed T and fixed compositions).

Isomorphous systems (complete solubility, e.g. Cu–Ni). Two lines: the liquidus (above it all liquid) and the solidus (below it all solid), with a lens-shaped L + α field between them. An alloy solidifies over a temperature range, not at one temperature.

Reading a two-phase field.

1. Phases present: from the field the point lies in.
2. Compositions: draw a horizontal tie line through the point at that temperature; where it meets the two boundaries gives each phase's composition.
3. Amounts: use the lever rule — the fraction of a phase equals the length of the tie-line segment on the opposite side of the alloy point divided by the total tie-line length. The rule is a mass balance and works with weight fractions when compositions are in wt %.

Binary eutectic systems (e.g. Pb–Sn, Al–Si). Limited solid solubility gives two terminal solid solutions α and β, three two-phase fields (L + α, L + β, α + β) and a horizontal eutectic line. At the eutectic point the liquid transforms at one temperature into two solids: L → α + β. It is the lowest temperature at which liquid can exist in the system, which is why eutectic solders (Pb–61.9 Sn at 183 °C) melt sharply at a low temperature.

• A eutectic alloy solidifies entirely at the eutectic temperature to a fine lamellar α + β structure.
• A hypoeutectic alloy (less solute than eutectic) first forms primary (proeutectic) α in the L + α field; the remaining liquid reaches the eutectic composition and solidifies as eutectic. A hypereutectic alloy forms primary β first.
• The solvus line gives the limit of solid solubility, which falls on cooling; crossing it precipitates the second phase (the basis of precipitation hardening).

Invariant reactions (all at fixed T, three phases).

• Eutectic: L → α + β.
• Eutectoid: γ → α + β (one solid to two solids; e.g. austenite → pearlite).
• Peritectic: L + α → β.
• Peritectoid: α + β → γ.
• Monotectic: L1 → L2 + α.

Non-equilibrium effects. Faster cooling in castings means the solid cannot re-homogenise by diffusion, so isomorphous alloys show coring (composition varies from the centre of a dendrite to its edge). Homogenising anneals remove it. Very fast cooling can suppress equilibrium phases altogether.

Cooling curves. A pure metal or eutectic alloy shows a flat arrest at a single temperature; an alloy in an isomorphous system shows a change of slope at the liquidus and another at the solidus. Such curves are how phase diagrams are measured.

Formulas

F = C − P + 1 Condensed Gibbs phase rule at constant pressure; F = degrees of freedom, C = number of components, P = number of phases in equilibrium.

W_L = (Cα − C0) / (Cα − CL) and Wα = (C0 − CL) / (Cα − CL) Lever rule; C0 = overall alloy composition, CL and Cα = compositions of the two phases at the ends of the tie line (all in the same wt % units); W = weight fraction. Valid only inside a two-phase field. WL + Wα = 1.

Vα = (Wα/ρα) / (Wα/ρα + Wβ/ρβ) Converts weight fraction to volume fraction using the phase densities ρ (kg/m³).

Worked examples

Example 1 (standard): isomorphous Cu–Ni. Given: a Cu–35 wt % Ni alloy at 1250 °C lies in the L + α field. The tie line meets the liquidus at CL = 31.5 wt % Ni and the solidus at Cα = 42.5 wt % Ni (read from the diagram).

1. Phases: liquid and α.
2. Compositions: liquid 31.5 wt % Ni, α 42.5 wt % Ni.
3. W_L = (Cα − C0)/(Cα − CL) = (42.5 − 35)/(42.5 − 31.5) = 7.5/11.0 = 0.68.
4. Wα = (35 − 31.5)/11.0 = 0.32. Check: 0.68 + 0.32 = 1.

Example 2 (GATE level): hypoeutectic Pb–Sn. Given (from the diagram): eutectic at 183 °C and 61.9 wt % Sn; at 183 °C the α boundary is 18.3 wt % Sn and the β boundary 97.8 wt % Sn. Alloy: Pb–40 wt % Sn.

1. Just above 183 °C (L + α): primary α = (61.9 − 40)/(61.9 − 18.3) = 21.9/43.6 = 0.502; liquid of eutectic composition = 0.498. This liquid becomes the eutectic microconstituent.
2. Just below 183 °C (α + β): total α = (97.8 − 40)/(97.8 − 18.3) = 57.8/79.5 = 0.727; total β = 0.273.
3. Eutectic α (the α inside the eutectic lamellae) = total α − primary α = 0.727 − 0.502 = 0.225.

Example 3 (phase rule). At the Pb–Sn eutectic point, P = 3 (L, α, β) and C = 2, so F = 2 − 3 + 1 = 0: temperature and all three compositions are fixed.

Common mistakes

• Using the near-side segment of the tie line in the lever rule (the "inverse" lever is the right one).
• Using the overall alloy composition as the composition of a phase; phase compositions come from the tie-line ends.
• Applying the lever rule in a single-phase field (the only phase is 100 %).
• Mixing wt % and at % in one calculation.
• Forgetting that the eutectic microconstituent contains both α and β, so "total α" and "primary α" differ.
• Confusing eutectic (liquid → two solids) with eutectoid (solid → two solids) and peritectic (liquid + solid → new solid).

For GATE PI

Expect lever-rule numericals (fractions of phases or of primary and eutectic constituents), phase-rule counting at invariant points, identification of reactions from their equations, and one-mark questions on liquidus, solidus, solvus and the meaning of the eutectic point. Practise drawing the tie line and labelling both arms before substituting numbers.

Quick check

1. How many degrees of freedom exist in a two-phase field of a binary system at constant pressure?
2. An alloy of 25 wt % B lies between α (20 wt % B) and β (30 wt % B). What fraction is α?
3. Write the peritectic reaction.
4. What does the solvus line show?
5. Why does a eutectic alloy show a single flat arrest on its cooling curve?

Answers: 1. One; 2. 0.50; 3. L + α → β on cooling; 4. The limit of solid solubility versus temperature; 5. It solidifies at one fixed temperature (F = 0 at the eutectic point).