Phase Equilibria and Phase Diagrams

Introduction

Materials rarely exist as perfectly uniform substances. During solidification, heat treatment, welding, casting, and service, a material may contain two or more phases—physically distinct regions with different structures and properties. Phase diagrams provide a graphical method for predicting which phases are thermodynamically stable at a specified temperature, pressure, and composition. They also support estimates of phase compositions and amounts.

For mechanical engineers, phase diagrams are essential in selecting alloys, designing heat treatments, controlling casting microstructures, interpreting failure, and understanding why steels, aluminium alloys, brasses, and solders develop particular properties. A phase diagram describes equilibrium possibilities; it does not, by itself, predict how rapidly equilibrium will be reached or the final microstructure under rapid processing.

Learning Objectives

By the end of this chapter, you should be able to:

• distinguish components, phases, constituents, and microstructural features;
• apply the Gibbs phase rule to unary and multicomponent systems;
• interpret unary, binary, and selected ternary diagrams;
• determine phase compositions using tie lines;
• calculate phase fractions using the lever rule;
• distinguish equilibrium microstructures from non-equilibrium products;
• relate invariant reactions and phase fields to processing and properties;
• recognize the limitations of equilibrium diagrams in engineering practice.

1. Fundamental Concepts

Components and phases

A component is a chemically independent constituent used to describe the composition of a system. Pure iron is a one-component (unary) system. An iron–carbon alloy is commonly treated as a two-component (binary) system, even though steels may contain additional alloying elements.

A phase is a homogeneous portion of a system having uniform chemical composition and structure, separated from other portions by an interface. Examples include liquid metal, ferrite (body-centred cubic iron), austenite (face-centred cubic iron), and cementite (iron carbide, Fe₃C). A phase may occur in several grains; grains are crystallographic regions, not necessarily different phases.

A microconstituent is a recognizable structural aggregate that may contain one or more phases. Pearlite, for example, is a lamellar mixture of ferrite and cementite. Calling pearlite a phase is therefore incorrect in the strict thermodynamic sense.

Equilibrium

At equilibrium, temperature, pressure, and chemical potentials are such that no spontaneous macroscopic change can lower the system's Gibbs free energy. Equilibrium phase diagrams identify the stable phase assemblage under specified conditions.

Real processing may produce metastable states. A metastable phase is not the lowest-free-energy state, but it can persist because transformation requires nucleation, diffusion, or interface motion. Martensite in steel is a familiar example.

Thermodynamic driving force

For a closed system at constant temperature and pressure, the stable state minimizes Gibbs free energy, G = H - TS, where G is Gibbs free energy, H enthalpy, T absolute temperature in kelvin, and S entropy. A phase transformation is thermodynamically favoured when it decreases the total G, but kinetics determine whether and how rapidly it occurs.

2. Gibbs Phase Rule

The general Gibbs phase rule is

where F is the number of degrees of freedom, C is the number of components, and P is the number of phases in equilibrium. A degree of freedom is an independently variable intensive property, such as temperature, pressure, or composition, that can be changed without changing the number of phases.

For condensed systems at essentially constant pressure, the reduced phase rule is

This form is normally used for metallic phase diagrams. Pressure is treated as fixed, usually near atmospheric pressure.

For a one-component system at constant pressure:

• one phase gives F = 1 - 1 + 1 = 1, so temperature may vary;
• two phases give F = 1 - 2 + 1 = 0, so coexistence occurs at a fixed temperature;
• three phases give F = 1 - 3 + 1 = 0, an invariant three-phase point.

For a binary system at constant pressure, a single-phase field has F = 2, meaning temperature and composition can vary independently. A two-phase field has F = 1, so once temperature is selected, the equilibrium compositions of the two phases are fixed. A three-phase invariant reaction has F = 0.

The phase rule counts thermodynamic degrees of freedom, not the number of phases, grains, or processing variables such as cooling rate.

3. Unary Phase Diagrams

A unary diagram represents a one-component system, with pressure and temperature commonly plotted as the axes. It shows the stability regions of different crystal structures and liquid or vapour.

For water, solid, liquid, and vapour fields meet at the triple point. For iron, important solid allotropes include α-ferrite (body-centred cubic), γ-austenite (face-centred cubic), and δ-ferrite (body-centred cubic at high temperature). The transitions between these forms are polymorphic transformations.

A boundary line represents two phases in equilibrium. Crossing a boundary changes the stable phase assemblage. At an ordinary point on a boundary, temperature and pressure are related; only one independent intensive variable remains. At a triple point, three phases coexist and the system is invariant under the full phase rule.

The slope of a phase boundary can be related to the Clapeyron equation:

Here P is pressure, T is absolute temperature, ΔS and ΔV are the entropy and volume changes per mole between phases, and ΔH is the molar enthalpy change. The equation applies to a reversible phase boundary at equilibrium. It explains why boundary slopes depend on the signs and magnitudes of latent heat and volume change. It should not be applied casually to non-equilibrium transformations.

4. Binary Phase Diagrams

A binary diagram usually plots temperature vertically and composition horizontally. The horizontal axis may be expressed in mass percent or mole percent; these are not interchangeable. The composition scale must always be identified.

Important features include:

• liquidus: above this line, the alloy is completely liquid;
• solidus: below this line, the alloy is completely solid, or solid phases occupy the field;
• solvus: separates a single solid solution from a two-solid-phase field;
• phase field: region in which a specified phase or phase assemblage is stable;
• tie line: an isothermal line across a two-phase field;
• invariant point: a point at which three phases coexist in a constant-pressure binary system.

Common binary system types

In an isomorphous system, the two components have complete solid solubility. The copper–nickel system is a classic approximation. The diagram contains a liquid field, a solid-solution field, and a two-phase liquid-plus-solid field.

In a eutectic system, two components have limited solid solubility and a liquid transforms into two solid phases at the eutectic temperature:

The eutectic reaction is invariant at constant pressure. A hypoeutectic alloy contains less solute than the eutectic composition; a hypereutectic alloy contains more.

In a peritectic reaction, liquid and one solid phase react to form another solid phase:

In a eutectoid reaction, one solid phase transforms into two different solid phases:

The iron–carbon eutectoid transformation, in the idealized Fe–Fe₃C diagram, is a key example: austenite transforms to ferrite plus cementite, forming pearlite.

Reading a phase field

To interpret a binary diagram:

1. Locate the alloy's overall composition on the horizontal axis.
2. Locate the temperature on the vertical axis.
3. Identify the field containing the intersection.
4. If the point lies in a two-phase field, draw a horizontal tie line.
5. Read the compositions at the tie-line intersections with the phase boundaries.
6. Calculate phase amounts with the lever rule if the axis is a mass-fraction scale.

The overall composition is not generally equal to the composition of either phase in a two-phase field. It is the weighted average of the phase compositions.

5. Tie Lines and the Lever Rule

Suppose an alloy of overall composition C₀ lies in an α + β field. At the selected temperature, the tie line meets the α boundary at Cα and the β boundary at Cβ. Conservation of mass gives

with

where Wα and Wβ are the mass fractions of α and β. Solving gives the lever-rule relations:

These equations assume a binary system, a uniform overall composition, equilibrium, and a composition axis expressed consistently with the desired fraction basis. If compositions are in mass percent, the results are mass fractions. Mole fractions require a mole-balance interpretation.

The name lever rule follows from the fact that each phase fraction is proportional to the length of the tie-line segment opposite that phase. The phase fraction of α uses the distance from C₀ to Cβ, not the distance from C₀ to Cα.

Worked example: eutectic alloy

An alloy contains 40 wt% B. At a particular temperature in the α + β field, the tie line gives Cα = 10 wt% B and Cβ = 70 wt% B. The phase fractions are

Thus the equilibrium mixture contains 50 wt% α and 50 wt% β. A check gives 0.50 + 0.50 = 1.00 and 0.50×10 + 0.50×70 = 40 wt% B.

If the same alloy lies in a liquid-plus-α field where C_L = 30 wt% B and Cα = 5 wt% B, the same equations apply with L replacing β. The liquid fraction is

provided C₀ lies between Cα and C_L.

6. Microstructure Development During Solidification

A phase diagram predicts the equilibrium phase assemblage, but microstructure develops through nucleation, growth, diffusion, and heat extraction. In a binary isomorphous alloy, solidification generally proceeds through a liquid-plus-solid field. The first solid may have a composition different from the final bulk composition. If diffusion in the solid is slow, compositional segregation can remain.

For a eutectic alloy at the eutectic composition, the liquid transforms at one temperature into a fine mixture of α and β. The eutectic microconstituent is not necessarily a single phase. For a hypoeutectic alloy, primary α forms before the eutectic temperature; the remaining liquid then transforms into the eutectic mixture. The final structure consists of primary α plus eutectic α + β. A hypereutectic alloy similarly contains primary β plus eutectic material.

The amount of eutectic microconstituent can also be estimated by a lever rule immediately above the eutectic temperature, using the liquid composition and the composition of the primary solid. This is distinct from the fractions of α and β within the eutectic mixture itself.

Cooling rate affects scale and morphology. Faster cooling commonly produces finer spacing and can suppress diffusion-controlled transformations. However, equilibrium phase fractions at a specified temperature are determined by the diagram, not by grain size alone.

7. The Iron–Carbon System as an Engineering Model

The Fe–Fe₃C diagram is widely used for steels and cast irons. It is technically a metastable diagram because cementite is not the most stable carbon-containing phase under all conditions; nevertheless, it is highly useful for ordinary heat treatment and many casting applications.

Important phases include:

• ferrite (α): low carbon solubility, body-centred cubic, relatively soft and ductile;
• austenite (γ): face-centred cubic, substantially greater carbon solubility;
• δ-ferrite: high-temperature body-centred cubic iron;
• cementite (Fe₃C): hard and brittle intermetallic compound;
• liquid: molten alloy.

At the eutectoid composition, austenite transforms to pearlite. Hypoeutectoid steel forms proeutectoid ferrite plus pearlite. Hypereutectoid steel forms proeutectoid cementite plus pearlite. These descriptions apply to sufficiently slow cooling for near-equilibrium transformations.

In industrial heat treatment, cooling may be rapid enough to form bainite or martensite, neither of which is obtained by simply reading equilibrium phase fields. Time–temperature–transformation and continuous-cooling-transformation diagrams are required for kinetic predictions.

8. Ternary Phase Diagrams

A ternary diagram represents three components at fixed pressure, usually at a selected temperature. Composition is plotted in an equilateral triangle. Each vertex represents 100% of one component and 0% of the others; every point represents compositions whose percentages sum to 100%.

For a point with component fractions A, B, and C:

or, in percentage form, A% + B% + C% = 100%. Lines parallel to a triangle side represent constant concentration of the component at the opposite vertex. A ternary diagram must be read with care because neither axis is a conventional Cartesian axis.

At constant temperature and pressure, the phase rule becomes

when temperature and pressure are fixed. Thus a one-phase region has two compositional degrees of freedom, a two-phase region has one, and a three-phase equilibrium is compositionally invariant within the fixed-temperature section.

In a ternary two-phase field, a tie line connects the compositions of the coexisting phases. In a three-phase triangle, an overall composition inside the triangle decomposes into the three corner-phase compositions. Phase fractions can be found by a generalized lever rule based on area ratios, although the calculation is more demanding than in a binary diagram.

Ternary diagrams are useful for ceramics, solder alloys, glass compositions, superalloys, and cementitious systems. Their limitations include visual complexity, dependence on temperature sections, and the need to distinguish composition triangles from liquidus projection diagrams.

9. Applied Case: Selecting a Heat Treatment for a Hypoeutectoid Steel

Background. An engineering firm produces a medium-carbon steel component requiring improved strength while retaining useful toughness. The nominal composition is below the eutectoid carbon content.

Problem. The production team must choose a treatment that produces a uniform austenitic starting structure before controlled cooling. Direct quenching from an uncertain temperature may leave ferrite or produce nonuniform properties.

Analysis. The Fe–Fe₃C diagram indicates that the steel must be heated above the upper critical boundary, commonly designated A₃ for hypoeutectoid steel, so that the equilibrium structure becomes austenite. Heating only into the α + γ region would leave proeutectoid ferrite. The diagram therefore supplies the required phase condition, while furnace practice supplies the time needed for temperature equalization and carbon diffusion.

Approach. The component is heated uniformly into the single-phase austenite field, held sufficiently for thermal and compositional homogenization, and then cooled according to the desired transformation path. Slow cooling produces ferrite plus pearlite; faster controlled cooling may produce finer pearlite or, with suitable rates, bainite. Quenching can produce martensite, but its prediction requires kinetic diagrams and a subsequent tempering decision.

Outcome. The phase diagram prevents an inappropriate austenitizing temperature, while hardness and toughness testing verifies the actual result. Metallographic examination checks whether proeutectoid ferrite, pearlite, bainite, or martensite is present.

Learning points. Equilibrium diagrams identify stable phase fields and critical temperatures. They do not specify holding times, cooling rates, transformation kinetics, residual stresses, or final properties. Sound heat-treatment design combines phase-equilibrium information with kinetic data, process control, and inspection.

10. Limitations and Best Practice

Phase diagrams are models with defined assumptions. Common limitations include:

• equilibrium may not be reached during industrial cooling;
• diagrams may omit minor alloying elements that alter phase boundaries;
• metastable diagrams, such as Fe–Fe₃C, may be more practical than fully stable diagrams but require explicit interpretation;
• composition gradients, oxidation, porosity, and segregation can invalidate a uniform-composition assumption;
• diagrams do not directly give hardness, toughness, strength, grain size, or residual stress;
• measured boundaries depend on experimental uncertainty and thermodynamic databases.

Best practice is to identify the composition basis, pressure assumption, temperature, phase-field meaning, and equilibrium status before calculating. Use tie lines only within the relevant two-phase field, check that fractions sum to unity, and distinguish phase fractions from microconstituent fractions. For real process design, combine phase diagrams with diffusion data, TTT/CCT diagrams, thermodynamic software, microscopy, and mechanical testing. Computational thermodynamics can evaluate multicomponent systems, but the engineer must still validate databases and interpret results physically.

Chapter Summary

Phase diagrams map equilibrium phase stability as a function of temperature, pressure, and composition. Components are chemically independent constituents, whereas phases are homogeneous regions with distinct structures and compositions. The Gibbs phase rule, F = C - P + 2, and its constant-pressure form, F = C - P + 1, quantify thermodynamic degrees of freedom.

Unary diagrams describe polymorphism and phase boundaries in one-component systems. Binary diagrams contain liquidus, solidus, solvus, single-phase fields, and two-phase fields. Tie lines provide the compositions of coexisting phases. The lever rule, derived from mass balance, gives their fractions. Eutectic, peritectic, and eutectoid reactions are invariant transformations involving three phases.

Ternary diagrams extend composition analysis to three components using a triangular representation. They are especially useful for ceramics, solders, glasses, and multicomponent alloys. Equilibrium diagrams predict stable assemblages, but kinetics, cooling rate, diffusion, and metastability determine practical microstructures. Effective engineering use therefore combines phase diagrams with processing knowledge, kinetic diagrams, characterization, and testing.

Key Terms

Component: Chemically independent constituent used to describe a system.

Phase: Homogeneous region with uniform structure and composition.

Microconstituent: Recognizable structural aggregate that may contain multiple phases.

Equilibrium: State of minimum Gibbs free energy under specified conditions.

Metastable phase: Non-equilibrium phase that persists because transformation is kinetically hindered.

Gibbs phase rule: Relation between components, phases, and degrees of freedom.

Degree of freedom: Independent intensive variable that may change without altering phase count.

Unary system: System containing one component.

Binary system: System containing two components.

Ternary system: System containing three components.

Liquidus: Boundary above which an alloy is completely liquid.

Solidus: Boundary below which the alloy is completely solid or solid-phase assemblages are stable.

Solvus: Boundary separating one solid solution from a two-solid-phase field.

Tie line: Isothermal line connecting equilibrium phase compositions in a two-phase field.

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

Eutectic reaction: Liquid transforming into two solid phases.

Peritectic reaction: Liquid and one solid reacting to form another solid.

Eutectoid reaction: One solid phase transforming into two solid phases.

Polymorphism: Ability of a material to exist in more than one crystal structure.

Pearlite: Ferrite–cementite microconstituent formed by eutectoid transformation in steel.

Review Questions

Short Answer Questions

1. Define a phase and distinguish it from a grain.
2. What is meant by a degree of freedom in the Gibbs phase rule?
3. State the constant-pressure Gibbs phase rule for condensed systems.
4. What information does a tie line provide in a binary two-phase field?
5. Why is pearlite classified as a microconstituent rather than a phase?

Long Answer Questions

1. Explain the construction and interpretation of a binary eutectic phase diagram, including liquidus, solidus, solvus, and the eutectic reaction.
2. Derive the lever rule from conservation of mass for an α + β binary alloy.
3. Compare equilibrium, metastable, and non-equilibrium states using steel as an example.
4. Explain how unary phase diagrams and the Clapeyron equation describe phase boundaries.
5. Describe how a ternary composition triangle represents alloy composition and phase equilibria.

Analytical / Application Questions

1. An alloy of overall composition 35 wt% B lies in an α + β field where Cα = 5 wt% B and Cβ = 65 wt% B. Calculate the mass fractions of α and β and verify the composition balance.
2. A hypoeutectoid steel is heated into the α + γ field rather than above A₃. Predict the phases remaining during heating and explain the likely consequence for a subsequent quench.
3. Explain why a phase diagram alone cannot predict whether a quenched steel will contain pearlite, bainite, or martensite. Identify the additional information required.

Multiple Choice Questions with Answers

Question 1

For a binary condensed system at constant pressure containing two phases, the number of degrees of freedom is:

A. 0

B. 1

C. 2

D. 3

Correct Answer: B. 1

Question 2

In a binary α + β field, the overall composition is 40 wt% B, Cα = 10 wt% B, and Cβ = 70 wt% B. What is the mass fraction of α?

A. 0.25

B. 0.40

C. 0.50

D. 0.75

Correct Answer: C. 0.50

Question 3

Which reaction is eutectic?

A. L → α + β

B. L + α → β

C. γ → α

D. α → L + β

Correct Answer: A. L → α + β

Question 4

A solvus line separates:

A. Liquid from vapour

B. A liquid field from a solid field

C. A single solid solution from a two-solid-phase field

D. Two liquid phases

Correct Answer: C. A single solid solution from a two-solid-phase field

Question 5

In an isomorphous binary system, the solid phase has:

A. No solubility for either component

B. Complete solid solubility over the composition range

C. Only intermetallic compounds

D. Three coexisting phases everywhere

Correct Answer: B. Complete solid solubility over the composition range

Question 6

At constant temperature and pressure, a ternary one-phase field has how many compositional degrees of freedom?

A. 0

B. 1

C. 2

D. 3

Correct Answer: C. 2

Question 7

Which statement about the lever rule is correct?

A. Each phase fraction is proportional to the adjacent tie-line segment

B. Each phase fraction is proportional to the opposite tie-line segment

C. It applies only to unary systems

D. It determines cooling rate

Correct Answer: B. Each phase fraction is proportional to the opposite tie-line segment

Question 8

The eutectoid transformation in the Fe–Fe₃C model is approximately:

A. L → α + β

B. L + α → γ

C. γ → α + Fe₃C

D. α → γ + L

Correct Answer: C. γ → α + Fe₃C

Question 9

A phase diagram primarily predicts:

A. Equilibrium phase stability

B. Exact grain size

C. Impact toughness directly

D. Cooling time directly

Correct Answer: A. Equilibrium phase stability

Question 10

Why can a quenched steel not be interpreted using only an equilibrium phase diagram?

A. Composition has no effect on steel

B. Equilibrium diagrams do not describe transformation kinetics

C. Steel contains no phases

D. Temperature is irrelevant during quenching

Correct Answer: B. Equilibrium diagrams do not describe transformation kinetics