CHAPTER 9Phase Diagrams—

EquilibriumMicrostructural

Development

The microstructure of a slowly cooled “eutec-tic” soft solder (≈ 38 wt % Pb − wt % Sn)consists of a lamellar structure of tin-rich solidsolution (white) and lead-rich solid solution(dark), 375X. (From ASM Handbook, Vol.3: Alloy Phase Diagrams, ASM International,Materials Park, Ohio, 1992.)

Figure 9-1 Single-phase microstructure of commerciallypure molybdenum, 200×. Although there are manygrains in this microstructure, each grain has the same,uniform composition. (From Metals Handbook, 8thed., Vol. 7: Atlas of Microstructures, American Societyfor Metals, Metals Park, Ohio, 1972.)

Figure 9-2 Two-phase microstructure of pearlite found ina steel with 0.8 wt % C, 500×. This carbon content isan average of the carbon content in each of the alter-nating layers of ferrite (with <0.02 wt % C) and ce-mentite (a compound, Fe3C, which contains 6.7 wt %C). The narrower layers are the cementite phase. (FromMetals Handbook, 9th ed., Vol. 9: Metallography andMicrostructures, American Society for Metals, MetalsPark, Ohio, 1985.)

TemperatureGas

Liquid

Solid

1 atmPressure (log scale)

(a) (b)

Water

Ice

Steam100

0

T(°C)

Figure 9-3 (a) Schematic representation of the one-component phase diagram for H2O.(b) A projection of the phase diagram information at 1 atm generates a temperaturescale labeled with the familiar transformation temperatures for H2O (melting at 0◦Cand boiling at 100◦C).

Temperature

Gas Liquid

1 atmPressure (log scale)

(a) (b)

Liquid

T(˚C)

15381394

910

Figure 9-4 (a) Schematic representation of the one-component phase diagram for pureiron. (b) A projection of the phase diagram information at 1 atm generates a tempera-ture scale labeled with important transformation temperatures for iron. This projectionwill become one end of important binary diagrams such as Figure 9–19.

Tem

pera

ture

Melting pointof A

Melting pointof B

A B0 20 40 60 80 100 ← wt % B

100 80 60 40 20 0 ←wt % A

Composition (wt %)

L + SS

Liquidus

Solidus

SS

L

Figure 9-5 Binary phase diagram showing complete solid solution. The liquid-phase field is labeled L and the solid solution is designated SS. Note thetwo-phase region labeled L + SS.

A B

Compositionof SS at T1

Compositionof L at T1

Statepoint

Systemcomposition

L + SS

Systemtemperature

X1

SS

L

T1

Figure 9-6 The compositions of the phases in a two-phase region of the phasediagram are determined by a tie line (the horizontal line connecting the phasecompositions at the system temperature).

Composition

Temperature

A B

F = C – P + 1

F = 2 – 1 + 1 = 2 F = 1 – 2 + 1 = 0

F = 2 – 1 + 1 = 2

F = 2 – 2 + 1 = 1

Figure 9-7 Application of Gibbs phase rule (Equation 9.2) to various pointsin the phase diagram of Figure 9–5.

A B

Systemcomposition

Temperature

T1L1

Lsystem

Composition

SS1

SSsystem

All liquid (Lsystem)

Crystallites of SS1in matrix of L1

Polycrystalline solid(SSsystem)

Figure 9-8 Various microstructures characteristic of different regions inthe complete solid-solution phase diagram.

Cu 20 4010

10

1084.87˚

˚C

1455˚1500

1400

1300

1200

1100

1000

900

800

700

600

500

20 30 40 50 60 70 80 90

30 50Weight percentage nickel

Atomic percentage nickel

70 9060 80 Ni

L

Figure 9-9 Cu–Ni phase diagram. (After Metals Handbook, 8th ed., Vol. 8:Metallography, Structures, and Phase Diagrams, American Society forMetals, Metals Park, Ohio, 1973, and Binary Alloy Phase Diagrams, Vol.1, T. B. Massalski, ed., American Society for Metals, Metals Park, Ohio,1986.)

2800

2600

2400

2200

2000

˚C

NiO 20 40 60

L + SS

SS

L

80 MgO

Mole % MgO

Figure 9-10 NiO–MgO phase diagram. (After PhaseDiagrams for Ceramists, Vol. 1, American CeramicSociety, Columbus, Ohio, 1964.)

A

L

A + LL + B

A + B

B

Eutectictemperature

EutecticComposition

Tem

pera

ture

Composition

Liquidus

Solidus

Figure 9-11 Binary eutectic phase diagram showing no solid so-lution. This general appearance can be contrasted to the op-posite case of complete solid solution illustrated in Figure 9–5.

A BComposition

Leutectic

Temperature

Crystallites of Ain matrix of L1

All liquid (Leutectic)

Crystallites of Bin matrix of L2

Eutectic microstructure—fine, alternating layers ofA and B

L1 L2

Figure 9-12 Various microstructures characteristic of different regions in a binary eutectic phase di-agram with no solid solution.

300

400

500

600

1.6 12.6577˚

1414˚

˚C

700660.452˚

800

900

1000

1100

1200

1300

1400

1500

A1 10 20 30 40 50 60 70 80 90

10 20 30 40 50 60 70 80 90

Si

Weight percentage, silicon

Atomic percentage, silicon

L

Figure 9-13 Al–Si phase diagram. (After Binary Alloy Phase Diagrams, Vol.1, T. B. Massalski, ed., American Society for Metals, Metals Park, Ohio,1986.)

L

A B

Tem

pera

ture

Composition

Figure 9-14 Binary eutectic phase diagram withlimited solid solution. The only differencefrom Figure 9–11 is the presence of solid-solutionregions α and β .

Leutectic

A B

Temperature

Composition

All liquid (Leutectic)

L1 L2

Figure 9-15 Various microstructures characteristic of different regions in the binary eutectic phase di-agram with limited solid solution. This illustration is essentially equivalent to Figure 9–12 exceptthat the solid phases are now solid solutions (α and β) rather than pure components (A and B).

10400˚C

327.502˚

300

200

100

0Pb 10 20 30 40

L

50 60 70 80 90 Sn

20 30 40 50 60

Atomic percentage tin

Weight percentage tin

70 80 90

19 183˚ 61.9 97.5

231.9681̊

13˚

Figure 9-16 Pb–Sn phase diagram. (After Metals Handbook, 8th ed., Vol. 8: Metallogra-phy, Structures, and Phase Diagrams, American Society for Metals, Metals Park, Ohio,1973, and Binary Alloy Phase Diagrams, Vol. 2, T. B. Massalski, ed., American Societyfor Metals, Metals Park, Ohio, 1986.)

L

A B

Temperature

Eutectoidtemperature

Eutectictemperature

Eutectoidcomposition

Eutecticcomposition

Composition

Figure 9-17 This eutectoid phase diagram contains both a eutectic reaction (Equa-tion 9.3) and its solid-state analog, a eutectoid reaction (Equation 9.4).

A B

Temperature

Composition

Figure 9-18 Representative microstructures for the eutectoid diagram of Figure 9–17.

Atomic percentage carbon˚C 2 5

L

10 15 20 251700

1538˚1495˚

1394˚

1148˚

2.11 4.30

L + Fe3C

Fe

Weight percentage carbon

1 2 3 4 5 6 7

727˚

0.02 0.77

6.69

1227˚C

912˚

1600

1500

1400

1300

1200

1100

1000

900

800

700

600

500

400

300

200

100

0

Fe3C(cementite)

Figure 9-19 Fe–Fe3C phase diagram. Note that the composi-tion axis is given in weight percent carbon even though Fe3C,and not carbon, is a component. (After Metals Handbook,8th ed., Vol. 8: Metallography, Structures, and Phase Di-agrams, American Society for Metals, Metals Park, Ohio,1973, and Binary Alloy Phase Diagrams, Vol. 1, T. B. Mas-salski, ed., American Society for Metals, Metals Park, Ohio,1986.)

Atomic percentage carbon

1538˚1495˚

1394˚

1154˚

4.26

L + C

Weight percentage carbon

738˚

2.08

0.02 0.68

912˚

C(graphite)

2˚C2200

2100

2000

1900

1800

1700

1600

1500

1400

1300

1200

1100

1000

900

800

700

600

500

400

300

200

100

0Fe 1 2 3 4 5 6 99 100

5 10 15 20 25

Figure 9-20 Fe–C phase diagram. The left side of this dia-gram is nearly identical to that for the Fe–Fe3C diagram(Figure 9–19). In this case, however, the intermediate com-pound Fe3C does not exist. (After Metals Handbook, 8thed., Vol. 8: Metallography, Structures, and Phase Dia-grams, American Society for Metals, Metals Park, Ohio,1973, and Binary Alloy Phase Diagrams, Vol. 1, T. B.Massalski, ed., American Society for Metals, Metals Park,Ohio, 1986.)

A AB

A + AB

AB + B

L + B

L + ABA + L

L

B

Temperature

Composition ofliquid formed upon

melting of AB

Composition

Figure 9-21 Peritectic phase diagram showing a peritec-tic reaction (Equation 9.5). For simplicity, no solidsolution is shown.

A AB

Crystallites of Bin matrix of L1

Polycrystalline solid(compound AB)

L

B

Temperature

Composition

Figure 9-22 Representative microstructures for the peritectic diagram ofFigure 9–21.

2200

2100L

2000

1900

1726˚

1587

1890˚

2054˚

1800

1700

1600

1500

140010 20 30 40 50 60 70 80 90 Al2O3

Mole % Al2O3

˚C

SiO2 (cristobalite) + L

SiO2 (cristobalite) + mullite(SS)

SiO2

mul

lite(

SS)

Al2O3 + mullite(SS)

L + Al2O3

L + mullite(SS)

Figure 9-23 Al2O3–SiO2 phase diagram. Mullite is an intermediate com-pound with ideal stoichiometry 3Al2O3 · 2SiO2. (After F. J. Klug, S.Prochazka, and R. H. Doremus, J. Am. Ceram. Soc. 70, 750 (1987).)

Figure 9-24 (a) Binary phase dia-gram with a congruently meltingintermediate compound, AB. Thisdiagram is equivalent to two sim-ple binary eutectic diagrams (theA–AB and AB–B systems). (b)For analysis of microstructure foran overall composition in the AB–B system, only that binary eutecticdiagram need be considered.

Tem

pera

ture

A

A + L

A + ABAB + B

L + ABAB + L B + L

AB

L

B

Composition

(a)

Tem

pera

ture

A

AB + B

AB + L B + L

AB

L

B

Composition

(b)

L + ABA + L

A + AB

Temperature

Composition(a)

A A2B AB AB2 AB4 B

L

Temperature

Composition

A A2B AB AB2 AB4 B

L

10 30 5020 40 60 8070 90MgO Al2O3

Spinal (SS) + Al2O3

L + Al2O3

Mole % Al2O3

3000

2500

Periclase (SS)+ L

Periclase (SS)

Periclase (SS) + spinel (SS)

L + spinel (SS)

L

2000

1500

1000

˚C

Spinel (SS)

Figure 9-26 MgO–Al2O3 phase diagram. Spinel is an intermediate com-pound with ideal stoichiometry MgO · Al2O3. (After Phase Diagramsfor Ceramists, Vol. 1, American Ceramic Society, Columbus, Ohio,1964.)

Atomic percentage, copper

Weight percentage, copper

300

400

500

600

700

800

900

1000

1100

Al 10

0 10 20 30 40 50 60 70 80 90 100

20 30 40 50 60 70 80 90 Cu

L

660.452˚

5.65 32.7548.2˚

53.5

52.5567˚

1084.87˚

˚C

η1

Figure 9-27 Al–Cu phase diagram. (After Binary Alloy Phase Diagrams, Vol. 1, T. B. Massalski,ed., American Society for Metals, Metals Park, Ohio, 1986.)

Atomic percentage, magnesium

Weight percentage, magnesium

100

200

300

400

500

600

700

Al 10

10

660.452˚ 650˚

87.436.1

17.1

450˚35.6

59.8437˚

66.7

20 30 40 50 60 70 80 90 100

20 30 40 50 60 70 80 90

˚C

L

δ

455˚

Mg

Figure 9-28 Al–Mg phase diagram. (After Binary Alloy Phase Diagrams, Vol. 1, T. B.Massalski, ed., American Society for Metals, Metals Park, Ohio, 1986.)

4501 2 3

400

350

300

250

200

150

1001

Cu

130010 20 30 40 50 60 70 80 90

1250

1200

1150

1100 1084.87º

37.5

59.8

80.5

56.5

69.8

36.832.5

39.0

ºC

L

1050

1000

950

900

850

800

750

700

650

600

550

500

450

400

350

300

250

200

150

100

50

010 205 15 25 35 45 55 65 75 85 9530 40 50

Weight percentage, zinc

Atomic percentage, zinc

60 70 80 90 Zn

Weight percentage Cu

Atomic percentage Cu

Zn 2 3

2.7

L 1.7 424

903º

835º

73.0

74.1

87.5

98.3

78.6 598º

456º468º48.9

45.5

99.7%at 100º

419.58º

424º97.3

700º

558º

Figure 9-29 Cu–Zn phase diagram. (After Metals Hand-book, 8th ed., Vol. 8: Metallography, Structures, andPhase Diagrams, American Society for Metals, MetalsPark, Ohio, 1973, and Binary Alloy Phase Diagrams,Vol. 1, T. B. Massalski, ed., American Society for Met-als, Metals Park, Ohio, 1986.)

CubicZrO2SS + ZrCaO3

Cub

ic Z

rO2S

S

Tetr

agon

al Z

rO2S

S

Tetr

agon

al Z

rO2S

S +

Cub

ic Z

rO2S

S

Mon

oclin

ic Z

rO2S

S +

Cub

ic Z

rO2S

S

4

2500

2000

1500

1000

500

0

8 12CaO (wt %)

16 20 24 28˚C

ZrO2 10 20 30 40 50CaO (mol %)

Figure 9-30 CaO–ZrO2 phase diagram. The dashed linesrepresent tentative results. (After Phase Diagrams forCeramists, Vol. 1, American Ceramic Society, Colum-bus, Ohio, 1964.)

Tem

pera

ture

Composition

A A2B AB AB2 AB4 B

L

Temperature

A

0 30 50 80 100

L

T1

L + SS

SS

BComposition (wt % B)mL + mSS = mtotal0.30mL + 0.80mSS = 0.50mtotal

→mL = 0.60mtotalmSS = 0.40mtotal

Figure 9-31 A more quantitative treatment of the tie lineintroduced in Figure 9–6 allows the amount of eachphase (L and SS) to be calculated by means of a massbalance (Equations 9.6 and 9.7).

(a)

(b)Fulcrum

Figure 9-32 The lever rule is a mechanical anal-ogy to the mass balance calculation. The(a) tie line in the two-phase region is analo-gous to (b) a lever balanced on a fulcrum.

Temperature

A

T1

T2

T3

SSsystem

SS3

SS2

SS1

Lsystem

L3

L2

L1

BComposition

100% liquid(Lsystem)

10% SS1 inmatrix of L1

40% SS2 inmatrix of L2

90% SS3 inmatrix of L3

100% Solid(SSsystem)

Figure 9-33 Microstructural development during the slow cooling of a50% A–50% B composition in a phase diagram with complete solidsolution. At each temperature, the amounts of the phases in the mi-crostructure correspond to a lever rule calculation. The microstruc-ture at T2 corresponds to the calculation in Figure 9–31.

100% liquid(Leutectic)

Leutectic

Temperature

T1

T2

CompositionA B

*The only differences from the T1 microstructure arethe phase compositions and the relative amounts ofeach phase. For example, the amount of b will beproportional to

Figure 9-34 Microstructural development during the slow coolingof a eutectic composition.

100% liquid(Lsystem = 80% B)

Temperature

T2 (= Teutectic + 1 )T3 (= Teutectic – 1 )

0 30 60 9080Composition (wt % B)A

100B

Lsystem

L2 L1

Figure 9-35 Microstructural development during the slow cooling of a hypereutectic com-position.

100% liquid(Lsystem = 40% B)

Temperature

T2 (= Teutectic + 1 )T3 (= Teutectic – 1 )

0 30 60 9040Composition (wt % B)A

100B

Lsystem

L1

Figure 9-36 Microstructural development during the slow cooling of a hypoeutectic com-position.

100% liquid(Lsystem = 10% B)

100% liquid(Lsystem = 20% B)uid

(Lsystem = 20%

Temperature

0 10

(a)

Composition (wt % B)A

100

B

Lsystem

L1

Temperature

0 10 20

(b)

Composition (wt % B)A

100

B

Lsystem

L1

Figure 9 37 Mi t t l d l t f t iti th t id th

Temperature

100% liquid(3% C)

0 3.0

Weight percentage carbon

6.7

L1

Figure 9-38 Microstructural development for white cast iron (of compo-sition 3.0 wt % C) shown with the aid of the Fe–Fe3C phase diagram.The resulting (low-temperature) sketch can be compared with a mi-crograph in Figure 11–1a.

Temperature

0 0.77 6.7Weight percentage carbon

Figure 9-39 Microstructural development for eutectoid steel (ofcomposition 0.77 wt % C). The resulting (low-temperature)sketch can be compared with the micrograph in Figure 9–2.

Temperature

Proeutectoid cementite+ pearlite

Weight percentage carbon

0 1.13 6.7

Figure 9-40 Microstructural development for a slowly cooled hypereutectoid steel(of composition 1.13 wt % C).

Temperature

Proeutectoid ferrite+ pearlite

0 0.50

Weight percentage carbon

6.7

Figure 9-41 Microstructural development for a slowly cooled hypoeutectoid steel(of composition 0.50 wt % C).

Temperature

0 3 100

Weight percentage carbon

C flakes (from eutecticand eutectoid reactions)in matrix of ferrite

L1

100% liquid(3% C)

Figure 9-42 Microstructural development for gray cast iron (of compo-sition 3.0 wt % C) shown on the Fe–C phase diagram. The resultinglow-temperature sketch can be compared with the micrograph inFigure 11–1b. A dramatic difference is that, in the actual microstruc-ture, a substantial amount of metastable pearlite was formed at theeutectoid temperature. It is also interesting to compare this sketchwith that for white cast iron in Figure 9–38. The small amount ofsilicon added to promote graphite precipitation is not shown in thistwo-component diagram.

T

The phase diagram for this alloy system is

A B