Phase Diagrams & the Iron–Carbon System

Maps of what forms at each composition and temperature. Learn phases, components and equilibrium, the Gibbs phase rule, unary diagrams, binary isomorphous systems (Cu–Ni) with the tie line and lever rule, eutectic, eutectoid and peritectic reactions, microstructure development on cooling, and the iron–iron carbide diagram (ferrite, austenite, cementite, pearlite, hypo- and hypereutectoid steels, cast irons) — with Python applying the lever rule to find phase and microconstituent fractions in a 0.40% C steel.

Phase diagrams show which phases are stable for any alloy composition and temperature. They are the roadmap for casting, welding, heat treatment and alloy design.

The iron–carbon diagram is the most important in engineering: it explains why steels with slightly different carbon contents behave so differently.

An analogy

A phase diagram is like a weather map for alloys: at each 'location' (composition) and 'altitude' (temperature) it tells you whether you'll find liquid, one solid, or a mixture of two — and the lever rule tells you how much of each.

Phases and equilibrium

A phase is a region of uniform structure and composition (liquid, solid solution, compound). Components are the chemical species (Fe and C). Equilibrium phase diagrams assume slow cooling. Reading a diagram: locate the point (composition, temperature); if in a two-phase field, draw a horizontal tie line — its ends give phase compositions, and the lever rule gives phase amounts.

Binary diagram types

Common features:

• Isomorphous (Cu–Ni) — complete solubility in solid and liquid; liquidus and solidus lines
• Eutectic (Pb–Sn solder, Al–Si) — liquid → two solids at the lowest melting point
• Eutectoid — one solid → two solids (γ → α + Fe₃C)
• Peritectic — liquid + solid → new solid
• Intermetallic compounds — fixed stoichiometry (Fe₃C, CuAl₂)

The Fe–Fe₃C diagram

Phases: α-ferrite (BCC, ≤ 0.022% C, soft), γ-austenite (FCC, ≤ 2.14% C, only stable at high temperature in plain steels), Fe₃C cementite (6.70% C, hard and brittle), δ-ferrite at very high temperature. Steels < 2.14% C; cast irons > 2.14%. Hypoeutectoid steels (< 0.76% C): proeutectoid ferrite + pearlite; eutectoid (0.76%): 100% pearlite; hypereutectoid (> 0.76%): proeutectoid cementite on grain boundaries + pearlite.

Beyond equilibrium

Real processes are rarely at equilibrium: coring in castings, non-equilibrium eutectic, and — most importantly — rapid cooling of steel that bypasses the diagram entirely to form bainite or martensite (next lesson). Alloying elements shift the eutectoid composition and temperature (Ni, Mn lower A1; Cr, Mo, Si raise it). Ternary and multicomponent diagrams (stainless steels: Schaeffler diagram) and computational thermodynamics (CALPHAD, Thermo-Calc) are used for modern alloy design.

Formulas

• Gibbs phase rule: F = C − P + 2 (F = C − P + 1 at constant pressure)
• Lever rule: W_α = (C_β − C₀)/(C_β − C_α); W_β = (C₀ − C_α)/(C_β − C_α)
• Eutectoid (steel): γ (0.76% C) → α (0.022% C) + Fe₃C (6.70% C) at 727 °C
• Eutectic (cast iron): L (4.30% C) → γ (2.14% C) + Fe₃C at 1147 °C
• Pearlite fraction (hypoeutectoid): W_p = (C₀ − 0.022)/(0.76 − 0.022)

Worked example: Degrees of freedom

Given: Binary alloy at the eutectic point (L + α + β), constant P

Find: F.

• F = 2 − 3 + 1

Answer: 0 — invariant point

Worked example: Pearlite in 0.20% C steel

Given: C₀ = 0.20

Find: W_p.

• (0.20 − 0.022)/0.738

Answer: 0.24 (24% pearlite)

Worked example: Hypereutectoid steel

Given: 1.0% C just below 727 °C

Find: Proeutectoid cementite fraction.

• (1.0 − 0.76)/(6.70 − 0.76)

Answer: 0.040 (4%)

Lever rule in a 0.40% C steel

# Fe-Fe3C lever rule: phases and microconstituents in a 0.40% C steel just below 727 C
C0 = 0.40
Ca, Cp, Cfe3c, Ce = 0.022, 0.76, 6.70, 0.76   # ferrite, pearlite (eutectoid), cementite, eutectoid
W_ferrite_total = (Cfe3c - C0) / (Cfe3c - Ca)
W_cementite = 1 - W_ferrite_total
W_pearlite = (C0 - Ca) / (Ce - Ca)
W_proeutectoid = 1 - W_pearlite
print(f"phases: ferrite {W_ferrite_total:.3f}, cementite {W_cementite:.3f}")
print(f"microconstituents: proeutectoid ferrite {W_proeutectoid:.3f}, pearlite {W_pearlite:.3f}")
# Estimated hardness by rule of mixtures (ferrite ~80 HB, pearlite ~240 HB)
print(f"estimated hardness ~ {W_proeutectoid*80 + W_pearlite*240:.0f} HB")
# phases: ferrite 0.943, cementite 0.057
# microconstituents: proeutectoid ferrite 0.488, pearlite 0.512
# estimated hardness ~ 162 HB

A 0.40% C steel (like EN8/C40) is 94% ferrite and 6% cementite as phases, but under the microscope you see about half proeutectoid ferrite grains and half pearlite colonies (fine layers of ferrite and cementite). More carbon → more pearlite → stronger and harder but less ductile; the rough hardness estimate matches typical normalised C40 (~160–200 HB).

Common mistakes

• Wrong: Using the lever rule backwards. — Right: Fraction of α uses the distance to the β side.
• Wrong: Confusing phases with microconstituents. — Right: Pearlite is a microconstituent of ferrite + cementite.
• Wrong: Applying equilibrium diagrams to quenched steel. — Right: Use TTT/CCT diagrams for fast cooling.

Applications

• Steel selection: Carbon content vs strength and weldability.
• Soldering: Eutectic Sn–Pb/Sn–Ag–Cu melts sharply.
• Aluminium casting alloys: Near-eutectic Al–Si for fluidity.
• Welding metallurgy: Heat-affected zone phase changes.

Key takeaways

• F = C − P + 1 at constant pressure.
• Tie line → compositions; lever rule → amounts.
• Fe–C: ferrite, austenite, cementite, pearlite at 727 °C/0.76% C.
• Carbon content controls steel microstructure.

Going further

• CALPHAD: Computational phase diagrams.
• Schaeffler diagram: Stainless steel weld microstructures.
• Cast irons: Grey, ductile, white and malleable.