Phase Equilibria and Phase Diagrams

Introduction

Materials used in machine technology rarely consist of a single perfectly uniform phase. Their properties depend on the phases present, their compositions, proportions, morphology, and thermal history. A phase is a physically and chemically homogeneous portion of a material with a definite structure and composition. Ice and liquid water are different phases of the same chemical substance; ferrite, austenite, and cementite are different phases or constituents encountered in steels.

Phase diagrams provide a graphical representation of the equilibrium relationships among phases in an alloy system. They enable engineers to predict which phases form at a specified temperature and composition, estimate the composition and amount of each phase, and select suitable heat-treatment, casting, welding, and forming conditions. This chapter develops the phase rule, binary phase diagrams, the lever rule, and the technologically essential iron–carbon system.

Learning Objectives

After studying this chapter, the reader should be able to:

1. Define phase, component, equilibrium, solubility, and degrees of freedom.
2. Apply the Gibbs phase rule to simple systems.
3. Interpret liquidus, solidus, solvus, eutectic, eutectoid, peritectic, and related boundaries.
4. Determine phase compositions and phase fractions using the lever rule.
5. Distinguish equilibrium phases from nonequilibrium microstructural constituents.
6. Interpret the iron–iron carbide diagram and relate it to steel and cast-iron processing.
7. Use phase-diagram information to make practical materials-processing decisions.

1. Phases, Components, and Equilibrium

A component is a chemically independent constituent required to describe the composition of a system. Pure copper is a one-component system. A copper–nickel alloy is commonly treated as a two-component system, or binary system. In the iron–iron carbide diagram, the components are conventionally taken as iron and cementite, Fe₃C, although the diagram represents the technologically important iron–carbon alloys.

A system is at phase equilibrium when no phase changes occur with time at constant temperature, pressure, and overall composition. Equilibrium requires sufficient time for diffusion and does not imply that all atoms are stationary. At equilibrium, atoms continue to move microscopically, but the average phase amounts and compositions remain constant.

A phase has:

• a uniform crystal structure or physical state;
• a definite range of composition;
• distinct physical and chemical properties; and
• an interface separating it from another phase.

A phase should not be confused with a microstructural constituent. Pearlite, for example, is a two-phase lamellar mixture of ferrite and cementite, not a single thermodynamic phase.

2. Gibbs Phase Rule

The Gibbs phase rule relates the number of phases, components, and independent variables that can be changed without changing the number of phases:

where:

• F is the number of degrees of freedom or variance;
• C is the number of components;
• P is the number of phases; and
• the number 2 represents temperature and pressure.

The rule applies to systems at equilibrium and assumes that the phases are macroscopic, pressure–volume effects are treated conventionally, and external fields or significant surface effects are absent.

For condensed alloy systems at approximately constant pressure, pressure is usually fixed. The condensed phase rule is therefore:

For a one-component system with one phase, F = 1 − 1 + 1 = 1. Temperature may vary while the phase remains stable. For a one-component system with two phases, F = 1 − 2 + 1 = 0. The temperature is fixed; this is the condition at which two phases coexist in equilibrium, such as ice and water at the melting point at a specified pressure.

For a binary system, C = 2. In a single-phase field, F = 2, so both temperature and composition can vary. In a two-phase field, F = 1, so at a selected temperature the compositions of the two equilibrium phases are fixed by the tie line, although their relative amounts may vary. At an invariant three-phase reaction, F = 0; temperature and phase compositions are fixed.

3. Reading a Binary Phase Diagram

A binary phase diagram normally plots temperature vertically and composition horizontally. The horizontal axis may be expressed as mass percent or mole percent of one component. The choice must be checked before calculations because mass fraction and mole fraction are not interchangeable.

Important boundaries include:

• Liquidus: above this boundary, the alloy is completely liquid. On cooling through it, the first solid forms.
• Solidus: below this boundary, the alloy is completely solid. Between liquidus and solidus, liquid and solid coexist.
• Solvus: separates a single solid solution from a two-solid-phase field.
• Tie line: a horizontal line at a specified temperature connecting the equilibrium compositions of phases in a two-phase region.
• Phase field: a region in which a specified phase or combination of phases is stable.
• Terminal solid solution: a solid solution occurring near one end of a binary diagram, such as α near copper or nickel in a limited-solubility system.

The liquidus and solidus do not usually give the composition of the original alloy. They give the composition of the phases that coexist at the selected temperature. The overall alloy composition is located independently on the composition axis.

Common invariant reactions

At constant pressure, invariant reactions in binary condensed systems have three phases and zero degrees of freedom. Their names describe the transformation:

The arrows indicate equilibrium reversibility. In practice, cooling may produce undercooling, segregation, or metastable products, so the actual transformation temperature and structure may differ from the equilibrium prediction.

Figure: Schematic binary eutectic diagram showing liquid, two-phase fields, and the eutectic point.

4. The Lever Rule

The lever rule determines the mass fraction of each phase in a binary two-phase region. It follows from conservation of mass. Consider an alloy with overall composition C₀ in a two-phase field containing phases α and β. At the selected temperature, the tie-line compositions are C_α and C_β. If W_α and W_β are the mass fractions of α and β, then:

and the component balance is:

Solving gives:

All compositions must be expressed on the same basis, normally mass percent of component B. The fractions are dimensionless and should sum to 1, or 100% when expressed as percentages. The rule applies only when the phase compositions are read from an equilibrium tie line and the system contains exactly two phases.

Worked Example 1: α–β phase field

At a particular temperature, a binary alloy contains 40 mass% B. The tie-line endpoints are C_α = 20 mass% B and C_β = 70 mass% B. Determine the phase fractions.

Thus the alloy contains 60% α and 40% β by mass. The result also illustrates the inverse-distance idea: the amount of α is proportional to the tie-line segment opposite α.

Lever rule for liquid–solid regions

The same equations apply in an L + α region. If the overall composition lies closer to the liquidus composition, the solid fraction is relatively large or small according to the opposite-segment relationship. The result is a phase fraction, not necessarily a microstructural fraction visible in a polished specimen, because subsequent cooling may transform one of the phases.

5. Solidification and Microstructural Interpretation

When a liquid alloy cools through a liquidus boundary, crystals of the primary solid phase nucleate and grow. In an equilibrium process, diffusion in both liquid and solid is sufficiently rapid to maintain uniform compositions. Industrial solidification is often non-equilibrium: diffusion in the solid is slow, producing coring, segregation, and a composition gradient within grains.

The equilibrium phase diagram predicts the final phases after adequate homogenization, whereas a cooling curve and kinetic analysis are needed to predict the actual transformation path. Cooling rate affects grain size, dendrite spacing, segregation, and the possibility of metastable phases.

For a hypoeutectic alloy, primary α forms before the eutectic temperature is reached; the remaining liquid then transforms into the eutectic mixture. For a hypereutectic alloy, primary β forms first. The eutectic constituent is therefore a mixture, not a single phase.

6. Iron–Carbon and Iron–Iron Carbide Diagram

The iron–carbon diagram is central to steels and cast irons. The technologically useful diagram is usually the metastable iron–iron carbide diagram, written Fe–Fe₃C. Cementite, Fe₃C, is metastable relative to graphite, but its formation is favored in many steels by practical cooling rates and alloying conditions.

Important phases and constituents include:

• Ferrite, α: body-centred cubic iron, with very low carbon solubility and relatively low strength but good ductility.
• Austenite, γ: face-centred cubic iron, capable of dissolving much more carbon than ferrite; stable above the eutectoid temperature within the appropriate composition range.
• δ-ferrite: body-centred cubic high-temperature iron phase.
• Cementite, Fe₃C: hard and brittle iron carbide containing approximately 6.67 mass% carbon.
• Pearlite: a lamellar mixture of ferrite and cementite formed by eutectoid transformation.
• Ledeburite: the eutectic constituent formed from liquid in the Fe–Fe₃C system; after further cooling it contains transformed austenite plus cementite.

The major invariant reactions are approximately:

Values may vary slightly among references because of experimental methods, pressure assumptions, and whether compositions are reported as mass or atomic percentages. The eutectoid composition is especially important: steels below it are called hypoeutectoid, those near it eutectoid, and those above it up to the cast-iron range hypereutectoid.

Hypoeutectoid steel

For a steel containing less than approximately 0.76 mass% carbon, austenite first produces proeutectoid ferrite on cooling. At 727 °C, the remaining austenite transforms into pearlite. The final equilibrium structure is therefore proeutectoid ferrite plus pearlite.

Eutectoid steel

Near 0.76 mass% carbon, austenite transforms almost entirely into pearlite at the eutectoid temperature. Pearlite strength increases as the interlamellar spacing decreases, which is influenced by cooling rate and transformation temperature.

Hypereutectoid steel

For carbon contents above approximately 0.76 mass% and below the cementite limit, proeutectoid cementite forms first. The remaining austenite transforms into pearlite. The final structure is proeutectoid cementite plus pearlite.

Cast irons

Alloys above approximately 2.11 mass% carbon are conventionally classified as cast irons, though the exact boundary is a classification convention. The Fe–Fe₃C diagram helps interpret white cast iron, in which carbon is largely associated with cementite, while the stable Fe–graphite system is more relevant to grey, ductile, and malleable cast irons. Silicon, cooling rate, inoculation, and section thickness strongly influence whether graphite or cementite forms.

7. Worked Example 2: Phase Fractions in Hypoeutectoid Steel

Consider an equilibrium 0.40 mass% C steel just below 727 °C. Approximate the ferrite composition as 0.022 mass% C and cementite composition as 6.67 mass% C. The phases in equilibrium are ferrite and cementite, although the microstructural constituents are proeutectoid ferrite and pearlite.

The total ferrite fraction is:

The total cementite fraction is:

Thus the equilibrium phase fractions are approximately 94.3% ferrite and 5.7% cementite by mass.

To estimate the amounts of proeutectoid ferrite and pearlite, use the eutectoid tie line. The pearlite fraction is approximately the fraction of austenite present immediately above 727 °C:

The proeutectoid ferrite fraction is therefore approximately:

The distinction is important: approximately 51.2% of the microstructure is pearlite, but pearlite itself contains ferrite and cementite. Phase fractions and constituent fractions answer different questions.

8. Processing Applications

Heat treatment

Phase diagrams establish the temperatures at which austenite forms, ferrite or cementite precipitates, and eutectoid transformation occurs. They provide the equilibrium basis for annealing, normalizing, and selecting austenitizing temperatures. However, phase diagrams do not predict transformation time or the formation of martensite. Time–temperature–transformation and continuous-cooling-transformation diagrams are required for those kinetic decisions.

Casting

Liquidus and solidus temperatures determine the solidification interval. A wide freezing range increases the possibility of mushy-zone feeding problems and segregation. Eutectic compositions often have good fluidity because solidification occurs over a relatively narrow temperature interval, but shrinkage, gas porosity, and hot tearing still depend on mold design and process conditions.

Welding

The weld pool and heat-affected zone experience rapid, nonuniform thermal cycles. The equilibrium diagram identifies possible phases, but cooling rate, local composition, and restraint determine the actual microstructure and properties. In steels, excessive hardenability and rapid cooling can produce hard, brittle martensite and hydrogen-assisted cracking; preheating and controlled post-weld cooling may be required.

Alloy design

A useful alloy should be evaluated for phase stability, processing window, cost, toxicity, recyclability, and service environment. Alloying additions can shift phase boundaries, form intermetallic compounds, refine grains, or alter diffusion rates. A binary diagram is a foundation, not a complete representation of a commercial multicomponent alloy.

9. Illustrative Case: Selecting a Heat Treatment for a Medium-Carbon Steel

Background. A manufacturer produces a 0.40 mass% carbon steel shaft requiring a combination of strength, toughness, and dimensional stability.

Problem. The production team must choose between full annealing and normalizing before machining and final service.

Analysis. The Fe–Fe₃C diagram indicates that 0.40% carbon steel is hypoeutectoid. On slow equilibrium cooling, proeutectoid ferrite forms before the remaining austenite transforms to pearlite. Heating above the upper critical boundary produces austenite; cooling in a furnace gives relatively coarse ferrite–pearlite, whereas air cooling produces finer constituents.

Approach. Full annealing uses slow furnace cooling and is selected when maximum softness and machinability are important. Normalizing uses air cooling from the austenite region and generally produces finer grains and higher strength. The selected austenitizing temperature must be sufficiently above the relevant critical boundary for complete transformation, but excessive overheating should be avoided because it promotes grain growth and oxidation.

Outcome. For a shaft needing improved strength and a refined structure before machining, normalizing is the better preliminary treatment. If machining force and softness dominate, annealing may be preferred. Final selection must be verified by hardness, tensile, impact, dimensional, and metallographic tests.

Learning points. The phase diagram identifies the equilibrium transformation path and critical temperature range. It does not by itself determine the optimum heating time, cooling rate, residual stress, distortion, or final toughness. Processing decisions must combine thermodynamics, kinetics, geometry, equipment capability, and quality requirements.

10. Limitations, Safety, and Contemporary Practice

Phase diagrams are equilibrium maps. Industrial processes are often too rapid for equilibrium to be reached, and diffusion may be incomplete. Important limitations include metastability, hysteresis, segregation, uncertainty in boundary data, multicomponent interactions, pressure effects, and the difference between mass-based and atom-based composition scales.

Engineers should use validated databases and critically assess the source of thermodynamic data. Computational thermodynamics, including CALPHAD-based software, can estimate multicomponent phase equilibria and guide alloy development, but predictions require appropriate databases and experimental validation. Machine learning can assist phase-boundary prediction and microstructure classification, yet it does not replace conservation laws, thermodynamic consistency, or responsible engineering judgment.

Safe practice requires controlled furnace operation, calibrated temperature measurement, appropriate personal protective equipment, safe handling of hot specimens, and documented process limits. Environmental decisions should consider energy consumption, alloying-element toxicity, emissions, and recyclability. In regulated applications such as pressure vessels, aerospace components, and medical devices, phase predictions must be supported by qualification tests and traceable quality systems.

Best Practices for Phase-Diagram Analysis

1. Identify the diagram type and composition basis before reading values.
2. Locate the overall alloy composition independently from the phase boundaries.
3. Draw a horizontal tie line at the specified temperature.
4. Read the phase compositions at the tie-line endpoints.
5. Apply the lever rule only within a two-phase field.
6. Check that all phase fractions sum to 1 or 100%.
7. Distinguish equilibrium phases from microstructural constituents.
8. Consider kinetics, cooling rate, segregation, and alloying additions.
9. Use experimentally validated data for production decisions.
10. Report approximate values with appropriate significant figures.

Chapter Summary

Phase equilibria describe the stable relationships among phases at specified temperature, pressure, and composition. The Gibbs phase rule, F = C − P + 1 for condensed systems at fixed pressure, establishes the degrees of freedom. Binary diagrams use liquidus, solidus, and solvus boundaries to identify single- and two-phase regions. Tie lines give the compositions of coexisting phases, while the lever rule gives their mass fractions through conservation of mass.

The Fe–Fe₃C diagram is fundamental for understanding steels and cast irons. Ferrite, austenite, cementite, pearlite, and ledeburite have different meanings and properties. Hypoeutectoid steels form proeutectoid ferrite plus pearlite, eutectoid steels form pearlite, and hypereutectoid steels form proeutectoid cementite plus pearlite under equilibrium cooling. Phase diagrams guide casting, heat treatment, welding, and alloy design, but they must be supplemented by kinetic diagrams, processing data, testing, and safety controls.

Key Terms

Phase: Homogeneous region with distinct structure and composition.

Component: Chemically independent constituent used to describe a system.

Equilibrium: Condition in which phase conditions remain constant with time.

Phase rule: Relationship among components, phases, and degrees of freedom.

Degree of freedom: Independent intensive variable that may be changed without changing phase number.

Binary system: System containing two components.

Liquidus: Boundary above which an alloy is completely liquid.

Solidus: Boundary below which an alloy is completely solid.

Solvus: Boundary separating one solid solution from two solid phases.

Tie line: Isothermal line connecting compositions of coexisting phases.

Lever rule: Mass-balance method for calculating phase fractions.

Eutectic: Liquid transforming into two solid phases.

Eutectoid: Solid transforming into two different solids.

Peritectic: Liquid and solid reacting to form a new solid.

Ferrite: Low-carbon body-centred cubic iron phase.

Austenite: Face-centred cubic iron phase with relatively high carbon solubility.

Cementite: Iron carbide, Fe₃C.

Pearlite: Ferrite–cementite microstructural constituent.

Proeutectoid phase: Phase forming before the eutectoid reaction.

CALPHAD: Computational approach for calculating phase equilibria from thermodynamic databases.

Review Questions

Short Answer Questions

1. Define a phase and distinguish it from a microstructural constituent.
2. State the condensed Gibbs phase rule and define each symbol.
3. What information is obtained from the endpoints of a tie line?
4. Why is pearlite not considered a single thermodynamic phase?
5. State the approximate composition and temperature of the eutectoid reaction in the Fe–Fe₃C system.

Long Answer Questions

1. Explain the construction and interpretation of a binary eutectic phase diagram.
2. Derive the lever rule from the component mass balance and state its assumptions.
3. Describe the phases and constituents important in the Fe–Fe₃C diagram.
4. Compare equilibrium cooling, annealing, normalizing, and rapid cooling in terms of phase-diagram and kinetic information.
5. Discuss the limitations of using binary equilibrium diagrams to design industrial multicomponent alloys.

Analytical / Application Questions

1. A binary alloy contains 35 mass% B at a temperature where α contains 15 mass% B and β contains 65 mass% B. Calculate the mass fraction of each phase and verify the result.
2. Explain the expected equilibrium microstructure of a 0.40 mass% carbon steel cooled slowly from the austenite region. Distinguish phase fractions from constituent fractions.
3. A casting alloy lies in a wide liquid–solid freezing range. Analyze two likely processing problems and recommend measurements or process controls to reduce them.

Multiple Choice Questions with Answers

Question 1

For a condensed binary system at fixed pressure containing two phases, the Gibbs phase rule gives how many degrees of freedom?

A. 0

B. 1

C. 2

D. 3

Correct Answer: B. 1

Question 2

Which boundary separates a single solid solution from a two-solid-phase region?

A. Liquidus

B. Solidus

C. Solvus

D. Eutectic isotherm

Correct Answer: C. Solvus

Question 3

An alloy has C₀ = 40 mass% B, Cα = 20 mass% B, and Cβ = 70 mass% B. What is the mass fraction of α?

A. 0.20

B. 0.40

C. 0.60

D. 0.80

Correct Answer: C. 0.60

Question 4

Which reaction is the eutectoid reaction in the iron–iron carbide system?

A. L ⇌ α + β

B. γ ⇌ α + Fe₃C

C. L + δ ⇌ γ

D. α + Fe₃C ⇌ L

Correct Answer: B. γ ⇌ α + Fe₃C

Question 5

Pearlite is best described as:

A. A single iron carbide phase

B. A liquid–solid mixture

C. A ferrite–cementite microstructural constituent

D. A pure austenitic structure

Correct Answer: C. A ferrite–cementite microstructural constituent

Question 6

In a hypoeutectoid steel cooled slowly through the eutectoid temperature, the proeutectoid phase is generally:

A. Cementite

B. Austenite

C. Ferrite

D. Martensite

Correct Answer: C. Ferrite

Question 7

If an overall alloy composition lies closer to Cα than to Cβ in an α + β field, which phase has the larger mass fraction?

A. α

B. β

C. The phases are always equal

D. The liquid phase

Correct Answer: A. α

Question 8

Which statement about a phase diagram is correct?

A. It directly predicts transformation time

B. It is primarily an equilibrium map

C. It applies only to pure metals

D. It eliminates the need for experiments

Correct Answer: B. It is primarily an equilibrium map

Question 9

The approximate eutectic composition in the Fe–Fe₃C diagram is:

A. 0.022 mass% C

B. 0.76 mass% C

C. 2.11 mass% C

D. 4.3 mass% C

Correct Answer: D. 4.3 mass% C

Question 10

For a 0.40 mass% carbon steel under equilibrium cooling, the final constituent structure is approximately:

A. Pearlite plus proeutectoid ferrite

B. Pearlite plus proeutectoid cementite

C. Austenite plus martensite

D. Cementite only

Correct Answer: A. Pearlite plus proeutectoid ferrite