Binary Phase Diagrams: Reading Guide with Lever Rule and Worked Examples

Binary phase diagrams are the foundational tool through which metallurgists determine which phases coexist in any two-component alloy at a given temperature, and in precisely what proportions. This guide develops your ability to read any binary diagram from first principles, derives the lever rule rigorously from mass conservation, and applies it through three fully worked numerical examples spanning isomorphous (Cu-Ni), eutectic (Pb-Sn), and eutectoid (Fe-C) systems.

What Is a Binary Phase Diagram?

A binary phase diagram is an equilibrium map for a two-component system (binary = two components, typically labelled A and B). For every combination of temperature and overall composition, the diagram tells you which phase or phases are thermodynamically stable, and the composition of each phase present. The horizontal axis is composition, expressed in weight percent (wt%) or atomic percent (at%) of one component; the vertical axis is temperature. Constant pressure (1 atm) is assumed throughout.

The word equilibrium is critical. A phase diagram describes the state a system would reach given unlimited time for diffusion and rearrangement. Real alloys cooled at practical rates may pass through non-equilibrium states, producing microstructures that deviate from diagram predictions. Nevertheless, the diagram remains the essential reference frame from which all kinetic deviations are measured.

Phase diagrams are constructed from a combination of calorimetric experiments, diffusion couple analysis, CALPHAD thermodynamic modelling, and direct microstructural observation. For the most common engineering systems, compiled atlases such as the ASM Binary Alloy Phase Diagrams are the primary reference.

Understanding the Axes

The composition axis spans from pure component A on the left to pure component B on the right. A point at 40 wt% B means the alloy contains 40 g of B per 100 g of total alloy. When converting between wt% and at%, the appropriate atomic masses must be used: at% B = (wt% B / M_B) / (wt% A / M_A + wt% B / M_B) × 100.

The temperature axis spans from below the lowest relevant solidus (or solvus) to above the highest liquidus. The melting points of the pure components appear at the left and right extremes of the liquidus, anchoring the diagram. For the iron-carbon system, coverage typically extends from room temperature to above the δ-ferrite liquidus near 1538°C; for solder alloys, it may only span 0 to 300°C.

Phase Regions and the Gibbs Phase Rule

The diagram is divided into distinct phase fields (single-phase regions) and two-phase regions separated by phase boundaries. The number of degrees of freedom available in each region is governed by the Gibbs phase rule:

Applying this to the three possible situations in a binary diagram:

• Single-phase region (P = 1): F = 2 − 1 + 1 = 2. Both temperature and composition can vary independently. The alloy remains single-phase over a range of conditions.
• Two-phase region (P = 2): F = 2 − 2 + 1 = 1. Only one variable is free. At a fixed temperature, the compositions of both phases are uniquely determined. This is why drawing a horizontal tie line at any temperature in a two-phase region immediately gives you the equilibrium phase compositions.
• Invariant point (P = 3): F = 2 − 3 + 1 = 0. Three phases coexist at a single, fixed temperature with three fixed compositions. The reaction proceeds isothermally until one phase is consumed.

Types of Binary Phase Diagrams

Isomorphous Systems

An isomorphous system forms when two components are completely miscible in both the liquid and the solid state at all compositions. The result is a diagram with only two phase boundaries: the liquidus (above which the system is entirely liquid) and the solidus (below which it is entirely solid). The Cu–Ni and Au–Ag systems are the classical metallurgical examples.

Complete solid solubility requires that the components satisfy the Hume-Rothery rules: (1) atomic radii differ by less than 15% (Cu: 0.128 nm, Ni: 0.125 nm — a 2.4% difference); (2) identical crystal structure (both FCC); (3) similar electronegativity; and (4) the same or adjacent valence. Violation of any of these conditions typically produces limited solid solubility and a more complex diagram.

Between the liquidus and solidus lies the two-phase (L + α) region. As an alloy cools through this region, the solid that forms is enriched in the higher-melting component (Ni in Cu–Ni), while the residual liquid is depleted. At any temperature, the lever rule gives the exact weight fractions of liquid and solid present. The iron-carbon phase diagram also contains an isomorphous-type δ L + δ region at very high temperatures.

Eutectic Systems

Most binary alloy systems show only limited mutual solid solubility. In a eutectic system, the liquidus lines from each component converge at a single invariant point, the eutectic point, where three phases coexist: liquid of eutectic composition, and two solid phases (α and β). The eutectic reaction is:

Below the eutectic temperature, the phase diagram contains three regions separated by solvus lines: a single-phase α field (usually the A-rich solid solution), a two-phase α + β field, and a single-phase β field (the B-rich solid solution). The lever rule applies within every two-phase field. The Pb–Sn system has its eutectic at 61.9 wt% Sn and 183°C; Al–Si has a eutectic at 12.6 wt% Si and 577°C. These systems are foundational to solder and casting alloy technology respectively.

Peritectic Systems

A peritectic reaction occurs when a liquid reacts with an existing solid on cooling to form a different solid:

This reaction is notorious for incompleteness during practical cooling rates: diffusion in the solid α phase cannot keep pace with the advancing interface, leaving cored α particles wrapped in a β shell. Fe–C (at 0.17 wt% C and 1493°C), Cu–Zn, and Cu–Sn systems all include peritectic reactions. The lever rule applies in the two-phase regions flanking the peritectic, but the invariant line itself is handled the same way as a eutectic: three phases, zero degrees of freedom.

Eutectoid and Other Invariant Reactions

A eutectoid reaction is the solid-state analogue of the eutectic: one solid transforms to two different solids on cooling:

The most technologically important eutectoid in engineering metallurgy is the eutectoid reaction in the Fe–C system at 0.77 wt% C and 727°C, where austenite (γ) transforms to pearlite (α + Fe₃C) on slow cooling. The rate and mechanism of this transformation determine the final microstructure and mechanical properties of virtually all plain-carbon and low-alloy steels.

Other invariant reaction types include the peritectoid (α + β ↔ γ), the monotectic (L₁ ↔ L₂ + α), and the syntectic (L₁ + L₂ ↔ α). Each appears in specific engineering alloy systems and each features P = 3, F = 0.

Tie Lines: Reading Phase Compositions in Two-Phase Regions

A tie line is a horizontal isotherm drawn across a two-phase region at the temperature of interest. It is horizontal because both coexisting phases must be at the same temperature (thermal equilibrium). Its two endpoints lie on the left and right phase boundaries of the two-phase region, giving the equilibrium compositions of each phase at that temperature.

The procedure is straightforward:

1. Locate the alloy composition C₀ on the composition axis and draw a vertical line upward to the temperature of interest.
2. Confirm the point lies within a two-phase region, not a single-phase field.
3. Draw a horizontal line at that temperature until it intersects both phase boundaries.
4. Read the composition at the left intersection: this is C_α, the composition of the left phase (α).
5. Read the composition at the right intersection: this is C_β, the composition of the right phase (β).
6. Apply the lever rule using C_α, C₀, and C_β to find the weight fraction of each phase.

A common error is to confuse the tie line endpoints (which are the phase compositions) with the alloy composition C₀ (which lies between them). The alloy composition tells you where in the two-phase field the alloy sits, but it is not the composition of either phase present. On the microstructural level, these phase composition differences are the thermodynamic driving force for the composition gradients that form across grain and phase boundaries during solidification or solid-state transformation.

The Lever Rule: Derivation and Application

Derivation from Mass Balance

Consider a binary alloy of total mass M at temperature T in a two-phase (α + β) region. Define:

• C₀ = overall alloy composition (wt% B)
• C_α = equilibrium composition of α phase (left tie line endpoint)
• C_β = equilibrium composition of β phase (right tie line endpoint), C_β > C_α
• M_α, M_β = mass of each phase (M_α + M_β = M)
• W_α = M_α/M, W_β = M_β/M (weight fractions, W_α + W_β = 1)

Writing a mass balance on component B — the total amount of B in the alloy must equal the amount of B in each phase:

C0 × M  =  Cα × Mα  +  Cβ × Mβ

Dividing through by M and substituting Wα = Mα/M, Wβ = Mβ/M:

C0  =  Cα × Wα  +  Cβ × Wβ  ... (1)
Wα + Wβ  =  1                      ... (2)

Substituting Wα = 1 − Wβ into (1):

C0  =  Cα(1 − Wβ) + Cβ × Wβ

C0 − Cα  =  Wβ(Cβ − Cα)

Therefore:

       C0 − Cα                 Cβ − C0
Wβ = ————————     Wα = ————————
       Cβ − Cα                 Cβ − Cα

The denominator (C_β − C_α) is the total tie line length. The numerator for W_β is the left lever arm (distance from C_α to C₀), and the numerator for W_α is the right lever arm (distance from C₀ to C_β).

The mechanical analogy that gives the rule its name: imagine the tie line as a rigid lever balanced at a fulcrum placed at C₀. The α phase acts as a mass concentrated at C_α, and the β phase as a mass at C_β. For the lever to balance, the product of weight and arm length must be equal on each side: W_α(C₀ − C_α) = W_β(C_β − C₀). This is exactly the lever rule. The critical counter-intuitive consequence: the fraction of phase β equals the left arm divided by total length, not the right arm. The phase whose composition is closest to C₀ has the larger fraction.

Worked Example 1: Cu–Ni Isomorphous System at 1250°C

An alloy of C₀ = 58 wt% Ni is held at T = 1250°C, which places it in the two-phase (L + α) region of the Cu–Ni diagram. Reading the tie line intersections from the diagram:

• Liquidus intersection: C_liq ≈ 55 wt% Ni (composition of liquid phase)
• Solidus intersection: C_sol ≈ 63 wt% Ni (composition of solid α phase)

Treating C_liq as C_α and C_sol as C_β:

Tie line length:  Csol − Cliq = 63 − 55 = 8 wt% Ni

Wsolid = (C0 − Cliq) / (Csol − Cliq) = (58 − 55) / 8 = 3/8 = 0.375  (37.5% solid)
Wliquid = (Csol − C0) / (Csol − Cliq) = (63 − 58) / 8 = 5/8 = 0.625  (62.5% liquid)

Verification: 0.375 + 0.625 = 1.000 ✓
Mass balance: 55 × 0.625 + 63 × 0.375 = 34.375 + 23.625 = 58.000 = C0 ✓

Interpretation: At 1250°C, the 58 wt% Ni alloy is 62.5% liquid (enriched in Cu, at only 55 wt% Ni) and 37.5% solid (enriched in Ni, at 63 wt% Ni). As the alloy continues cooling toward the solidus, the solid fraction increases monotonically while both the liquid and solid phases become progressively richer in Ni. The process of non-equilibrium solidification in this system produces coring — a Ni-depleted core (solidified first, at lower Ni content) surrounded by Ni-enriched outer regions (solidified last, at higher Ni) — which can be homogenised by annealing above the solidus.

Worked Example 2: Pb–Sn Eutectic System at 150°C

A Pb–Sn solder alloy of composition C₀ = 40 wt% Sn is held at 150°C, which lies in the two-phase (α + β) region below the eutectic temperature. From the Pb–Sn diagram at 150°C:

• α phase boundary (Pb-rich solid solution limit): C_α ≈ 10 wt% Sn
• β phase boundary (Sn-rich solid solution limit): C_β ≈ 98 wt% Sn

Tie line length:  Cβ − Cα = 98 − 10 = 88 wt% Sn

Wβ = (C0 − Cα) / (Cβ − Cα) = (40 − 10) / 88 = 30/88 = 0.341  (34.1% β phase)
Wα = (Cβ − C0) / (Cβ − Cα) = (98 − 40) / 88 = 58/88 = 0.659  (65.9% α phase)

Mass balance check:  10 × 0.659 + 98 × 0.341 = 6.59 + 33.42 = 40.01 ≈ C0 = 40 ✓

Interpretation: The microstructure at 150°C consists of 65.9% Pb-rich α solid solution (at only 10 wt% Sn) and 34.1% Sn-rich β solid solution (at 98 wt% Sn). Note that C₀ = 40 wt% Sn is closer to the β boundary (98%) than to the α boundary (10%) — wait, 40 is closer to 10 than to 98 — so the α fraction is larger. The alloy is predominantly Pb-rich α, consistent with this hypoeutectic composition.

Worked Example 3: Fe–C Hypoeutectoid Steel at 750°C

A plain-carbon steel of C₀ = 0.40 wt% C (a typical engineering structural steel) is held at 750°C, above the eutectoid temperature of 727°C. This places the alloy in the two-phase (α + γ) field of the Fe–C phase diagram. From the diagram at 750°C:

• Ferrite (α) boundary: C_α ≈ 0.018 wt% C
• Austenite (γ) boundary: C_γ ≈ 0.680 wt% C

Tie line length:  Cγ − Cα = 0.680 − 0.018 = 0.662 wt% C

Wγ = (C0 − Cα) / (Cγ − Cα) = (0.400 − 0.018) / 0.662 = 0.382 / 0.662 = 0.577  (57.7% austenite)
Wα = (Cγ − C0) / (Cγ − Cα) = (0.680 − 0.400) / 0.662 = 0.280 / 0.662 = 0.423  (42.3% ferrite)

Mass balance: 0.018 × 0.423 + 0.680 × 0.577 = 0.0076 + 0.3924 = 0.4000 = C0 ✓

Interpretation: At 750°C this steel is 57.7% austenite and 42.3% ferrite. The result looks counterintuitive: C₀ = 0.40 wt% C appears to lie closer to the γ boundary (0.680) than to the α boundary (0.018)… wait, 0.40 − 0.018 = 0.382, while 0.680 − 0.40 = 0.280. So C₀ is actually closer to the γ boundary, meaning the left arm is longer, which means more γ. This confirms the calculation: W_γ > W_α.

On slow cooling to room temperature, the austenite transforms via the eutectoid reaction to pearlite. Applying the lever rule at room temperature between ferrite (0.022 wt% C) and cementite Fe₃C (6.70 wt% C):

WFe3C = (0.400 − 0.022) / (6.70 − 0.022) = 0.378 / 6.678 = 0.0566  (5.7% cementite)
Wα  = (6.700 − 0.400) / (6.70 − 0.022) = 6.300 / 6.678 = 0.943   (94.3% ferrite)

This predicts a microstructure that is approximately 94% proeutectoid and eutectoid ferrite with 6% cementite — consistent with the mechanical properties (moderate strength, good toughness) expected for a 0.4 wt% C structural steel. Understanding how to apply the lever rule across these phase boundaries is the first step toward predicting the microstructures produced by quenching and tempering or other heat treatments.

Invariant Reactions in Binary Phase Diagrams

Invariant reactions appear as horizontal isotherms on the phase diagram, because F = 0 means temperature cannot change during the reaction. Three phases coexist along the isotherm at three fixed compositions. The lever rule does not apply at the invariant isotherm itself; it applies in the two-phase regions immediately above and below it. The five principal invariant reaction types are summarised in the table below.

The Fe–C system contains three invariant reactions: the peritectic at 1493°C (0.17 wt% C), the eutectic at 1148°C (4.30 wt% C, producing ledeburite), and the eutectoid at 727°C (0.77 wt% C, producing pearlite). The eutectoid is the basis of the heat treatment of steels: different cooling rates through 727°C produce progressively different microstructures — from pearlite (slow cool) to bainite (intermediate) to martensite (fast quench) — all exploitable by quenching and tempering.

Limitations: Equilibrium Assumptions and Real Alloy Behaviour

The lever rule delivers exact results only under equilibrium conditions, where diffusion has been allowed to reach completion and every region of the alloy has the composition dictated by the phase diagram. In practice, several non-equilibrium phenomena cause deviations:

Coring (solidification segregation). During solidification of a solid-solution alloy (e.g., Cu–Ni), the first solid to form is enriched in the high-melting component. As the solid grows, interdiffusion in the solid phase is too slow to maintain the uniform composition predicted by the lever rule. The result is a gradient from a high-Ni core to a lower-Ni rim. The actual solidus temperature of the alloy is effectively depressed, and the final solid composition differs from the equilibrium prediction. Homogenisation annealing restores the equilibrium distribution.

Eutectic microstructure discrepancy. In rapidly solidified alloys, a hypoeutectic alloy may appear to contain a higher proportion of eutectic than the lever rule predicts, because the liquid ahead of the growing α phase becomes enriched in component B faster than diffusion can equilibrate the bulk liquid.

Metastable phases. Certain phases appear kinetically but are not on the equilibrium diagram. Martensite in steels, the metastable GP zones and θ′ precipitates in Al–Cu alloys, and the ω phase in Ti alloys are important examples. The lever rule cannot be applied to metastable phases without constructing or referencing the appropriate metastable phase diagram.

Pressure effects. The standard phase rule adds an extra degree of freedom for pressure: F = C − P + 2. Phase boundaries shift under high pressure (relevant to geophysical applications and high-pressure processing). The diagrams used in most engineering contexts assume 1 atm throughout.

Industrial Significance

The ability to read binary phase diagrams and apply the lever rule is not merely an academic exercise. It underpins the rational design of alloys and heat treatment cycles across virtually every branch of materials engineering:

Alloy composition selection. A phase diagram immediately reveals the composition range over which a fully single-phase microstructure can be achieved (the α or β solvus bounds), the eutectic or peritectic compositions that control casting behaviour, and the solubility limits relevant to precipitation hardening. The lever rule quantifies exactly how much second phase will form at service temperature for a given alloy composition.

Solder and brazing alloys. The Pb–Sn eutectic at 61.9 wt% Sn/183°C was the standard electronics solder for decades; its eutectic composition minimises liquid–solid coexistence time during assembly. Lead-free replacements (Sn–Ag–Cu) are selected from ternary diagram sections using the same tie line logic extended to three-component systems.

Steel heat treatment. The lever rule applied to the Fe–C diagram at the austenitising temperature predicts the volume fractions of austenite and undissolved carbides entering the furnace. After quenching, the relative fractions of martensite, retained austenite, and undissolved carbides can be estimated. These fraction estimates directly predict bulk hardness and toughness before any physical testing is performed.

Casting and solidification engineering. The fraction solid as a function of temperature during solidification (the solidification path) is computed from lever rule calculations at each temperature step. This is the starting point for solidification simulation software (e.g., CALPHAD-based tools such as Thermo-Calc and PANDAT) that predict casting porosity, segregation, and hot tearing susceptibility.

Corrosion-resistant alloys. The composition of the α/β boundary in duplex stainless steels (Fe–Cr–Ni pseudo-binary sections) is controlled to achieve target α:β ratios for optimum strength-corrosion combinations. An understanding of corrosion mechanisms depends on knowing the equilibrium phase assemblage at service temperature.