CHAPTER
9
Phase Diagrams—
Equilibrium
Microstructural
Development
The microstructure of a slowly cooled “eutectic” soft solder ( ≈ 38 wt % Pb − wt % Sn)
consists of a lamellar structure of tin-rich solid
solution (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 commercially
pure molybdenum, 200 × . Although there are many
grains in this microstructure, each grain has the same,
uniform composition. (From Metals Handbook, 8th
ed., Vol. 7: Atlas of Microstructures, American Society
for Metals, Metals Park, Ohio, 1972.)
Figure 9-2 Two-phase microstructure of pearlite found in
a steel with 0.8 wt % C, 500× . This carbon content is
an average of the carbon content in each of the alternating layers of ferrite (with < 0.02 wt % C) and cementite (a compound, Fe 3 C, which contains 6.7 wt %
C). The narrower layers are the cementite phase. (From
Metals Handbook, 9th ed., Vol. 9: Metallography and
Microstructures, American Society for Metals, Metals
Park, Ohio, 1985.)
Temperature
T(°C)
Gas
Steam
100
Water
Liquid
0
Solid
1 atm
(a)
Ice
Pressure (log scale)
(b)
Figure 9-3 (a) Schematic representation of the one-component phase diagram for H 2 O.
(b) A projection of the phase diagram information at 1 atm generates a temperature
scale labeled with the familiar transformation temperatures for H 2 O (melting at 0 ◦ C
and boiling at 100 ◦ C).
T(˚C)
Temperature
Gas
Liquid
Liquid
1538
1394
910
1 atm
Pressure (log scale)
(a)
(b)
Figure 9-4 (a) Schematic representation of the one-component phase diagram for pure
iron. (b) A projection of the phase diagram information at 1 atm generates a temperature scale labeled with important transformation temperatures for iron. This projection
will become one end of important binary diagrams such as Figure 9–19.
Liquidus
Melting point
of B
Temperature
L
L + SS
Melting point
of A
Solidus
SS
A
0
100
20
80
40
60
60
40
80
20
B
100 ← wt % B
0 ←wt % A
Composition (wt %)
Figure 9-5 Binary phase diagram showing complete solid solution. The liquidphase field is labeled L and the solid solution is designated SS. Note the
two-phase region labeled L + SS.
Composition
of L at T1
State
point
L
System
temperature
T1
L + SS
Composition
of SS at T1
SS
A
X1
B
System
composition
Figure 9-6 The compositions of the phases in a two-phase region of the phase
diagram are determined by a tie line (the horizontal line connecting the phase
compositions at the system temperature).
Temperature
F=C–P+1
F =2–1+1=2
F =1–2+1=0
F =2–2+1
=1
F =2–1+1=2
A
B
Composition
Figure 9-7 Application of Gibbs phase rule (Equation 9.2) to various points
in the phase diagram of Figure 9–5.
Lsystem
Temperature
T1
All liquid (Lsystem)
Crystallites of SS1
in matrix of L1
L1
SS1
Polycrystalline solid
(SSsystem)
SSsystem
A
B
System
composition
Composition
Figure 9-8 Various microstructures characteristic of different regions in
the complete solid-solution phase diagram.
Atomic percentage nickel
˚C
1500
10
20
30
40
50
60
70
80
90
1455˚
L
1400
1300
1200
1100
1084.87˚
1000
900
800
700
600
500
Cu
10
20
40
30
50
60
70
Weight percentage nickel
80
90
Ni
Figure 9-9 Cu–Ni phase diagram. (After Metals Handbook, 8th ed., Vol. 8:
Metallography, Structures, and Phase Diagrams, 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.)
˚C
2800
L
2600
L + SS
2400
SS
2200
2000
NiO
20
40
60
80
MgO
Mole % MgO
Figure 9-10 NiO–MgO phase diagram. (After Phase
Diagrams for Ceramists, Vol. 1, American Ceramic
Society, Columbus, Ohio, 1964.)
Temperature
L
Liquidus
A+L
Eutectic
temperature
L+B
Solidus
A+B
A
B
Eutectic
Composition
Composition
Figure 9-11 Binary eutectic phase diagram showing no solid solution. This general appearance can be contrasted to the opposite case of complete solid solution illustrated in Figure 9–5.
Temperature
All liquid (Leutectic)
Crystallites of A
in matrix of L1
Crystallites of B
in matrix of L2
L1 L2
Eutectic microstructure—
fine, alternating layers of
A and B
Leutectic
A
B
Composition
Figure 9-12 Various microstructures characteristic of different regions in a binary eutectic phase diagram with no solid solution.
Atomic percentage, silicon
˚C
1500
10
20
30
40
50
60
70
80
90
1414˚
1400
1300
L
1200
1100
1000
900
800
700
660.452˚
600
1.6
577˚
12.6
500
400
300
A1
10
20
30
40
50
60
70
80
90
Si
Weight percentage, silicon
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.)
Temperature
L
A
B
Composition
Figure 9-14 Binary eutectic phase diagram with
limited solid solution. The only difference
from Figure 9–11 is the presence of solid-solution
regions α and β .
Temperature
All liquid (Leutectic)
Leutectic
L1
L2
A
B
Composition
Figure 9-15 Various microstructures characteristic of different regions in the binary eutectic phase diagram with limited solid solution. This illustration is essentially equivalent to Figure 9–12 except
that the solid phases are now solid solutions ( α and β ) rather than pure components (A and B).
Atomic percentage tin
˚C
400
10
20
30
40
50
60
327.502˚
70
80
90
L
300
231.9681˚
200
19
183˚
61.9
97.5
100
0
Pb
13˚
10
20
30
40
50
60
Weight percentage tin
70
80
90
Sn
Figure 9-16 Pb–Sn phase diagram. (After Metals Handbook, 8th ed., Vol. 8: Metallography, Structures, and Phase Diagrams, American Society for Metals, Metals Park, Ohio,
1973, and Binary Alloy Phase Diagrams, Vol. 2, T. B. Massalski, ed., American Society
for Metals, Metals Park, Ohio, 1986.)
Temperature
L
Eutectic
temperature
Eutectoid
temperature
A
B
Eutectoid
composition
Eutectic
composition
Composition
Figure 9-17 This eutectoid phase diagram contains both a eutectic reaction (Equation 9.3) and its solid-state analog, a eutectoid reaction (Equation 9.4).
Temperature
A
B
Composition
Figure 9-18 Representative microstructures for the eutectoid diagram of Figure 9–17.
˚C
1700
2
1600 1538˚
1500
1400
Atomic percentage carbon
10
15
20
5
1495˚
25
L
1394˚
1300
1227˚C
1200
L + Fe3C
1148˚
1100
4.30
2.11
6.69
1000
900 912˚
800
700
727˚
0.02 0.77
600
Fe3C
(cementite)
500
400
300
200
100
0
Fe
1
2
3
4
5
6
7
Weight percentage carbon
Figure 9-19 Fe–Fe 3 C phase diagram. Note that the composition axis is given in weight percent carbon even though Fe 3 C,
and not carbon, is a component. (After Metals Handbook,
8th ed., Vol. 8: Metallography, Structures, and Phase Diagrams, 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.)
˚C
2200
2
Atomic percentage carbon
10
15
20
5
25
2100
2000
1900
1800
1700
1600
1538˚
1500
1400
1300
L+C
1495˚
1394˚
1200
1154˚
1100
1000
900
912˚
800
700
600
4.26
2.08
738˚
0.02
0.68
C
(graphite)
500
400
300
200
100
0
Fe
1
2
3
4
5
6
Weight percentage carbon
99
100
Figure 9-20 Fe–C phase diagram. The left side of this diagram is nearly identical to that for the Fe–Fe 3 C diagram
(Figure 9–19). In this case, however, the intermediate compound Fe 3 C does not exist. (After Metals Handbook, 8th
ed., Vol. 8: Metallography, Structures, and Phase Diagrams, 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.)
Temperature
Composition of
liquid formed upon
melting of AB
L+B
L
A + L L + AB
AB + B
A + AB
A
AB
B
Composition
Figure 9-21 Peritectic phase diagram showing a peritectic reaction (Equation 9.5). For simplicity, no solid
solution is shown.
Temperature
Crystallites of B
in matrix of L1
L
Polycrystalline solid
(compound AB)
A
AB
B
Composition
Figure 9-22 Representative microstructures for the peritectic diagram of
Figure 9–21.
˚C
2200
2100
2054˚
L
2000
L + Al2O3
1900
1890˚
SiO2 (cristobalite) + L
1800
1700
1726˚
L + mullite(SS)
Al2O3 + mullite(SS)
1600
1500
1400
SiO2
mullite(SS)
1587
SiO2 (cristobalite) + mullite(SS)
10
20
30
40
50
60
70
80
90
Al2O3
Mole % Al2O3
Figure 9-23 Al 2 O 3 –SiO 2 phase diagram. Mullite is an intermediate compound with ideal stoichiometry 3Al 2 O 3 · 2SiO 2 . (After F. J. Klug, S.
Prochazka, and R. H. Doremus, J. Am. Ceram. Soc. 70, 750 (1987).)
Temperature
L
A+L
AB + L
L + AB
A
B+L
AB + B
A + AB
AB
B
Composition
(a)
L
Temperature
Figure 9-24 (a) Binary phase diagram with a congruently melting
intermediate compound, AB. This
diagram is equivalent to two simple binary eutectic diagrams (the
A–AB and AB–B systems). (b)
For analysis of microstructure for
an overall composition in the AB–
B system, only that binary eutectic
diagram need be considered.
A+L
AB + L
L + AB
AB + B
A + AB
A
B+L
AB
Composition
(b)
B
Temperature
L
A
A 2B
AB
AB2
AB4
B
AB4
B
Composition
(a)
Temperature
L
A
A 2B
AB
Composition
AB2
˚C
3000
L
2500
L + spinel (SS)
2000
Periclase (SS)
+L
L + Al2O3
Periclase (SS)
1500
1000
MgO
Spinel (SS)
Periclase (SS) + spinel (SS)
10
20
30
40
Spinal (SS) + Al2O3
50
60
70
80
90
Al2O3
Mole % Al2O3
Figure 9-26 MgO–Al 2 O 3 phase diagram. Spinel is an intermediate compound with ideal stoichiometry MgO · Al 2 O 3 . (After Phase Diagrams
for Ceramists, Vol. 1, American Ceramic Society, Columbus, Ohio,
1964.)
Atomic percentage, copper
˚C
0
1100
10
20
30
40
50
60
70
80 90 100
1084.87˚
1000
L
900
800
700
660.452˚
53.5
600
η1
548.2˚
500
5.65
32.7
567˚
52.5
400
300
Al
10
20
30
40
50
60
Weight percentage, copper
70
80
90
Cu
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.)
˚C
700
Atomic percentage, magnesium
10
20
30
40
50
60
70
80
90
660.452˚
100
650˚
600
L
35.6
500
450˚
17.1
400
36.1
455˚
59.8
66.7
437˚
87.4
δ
300
200
100
Al
10
20
30
40
50
60
70
Weight percentage, magnesium
80
90
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.)
Atomic percentage, zinc
ºC
1300
10
20
30
40
50
60
70
90
Atomic percentage Cu
1
2
3
450
L
1.7 424
1250
1200
400
1150
1100
80
1084.87º
2.7
350
L
300
1050
250
1000
200
950
903º
900
32.5
37.5
150
36.8
850
56.5
800
100
1
Zn
2
3
Weight percentage Cu
59.8
835º
750
700
73.0
700º
69.8
80.5
650
78.6 598º
600
558º
550
74.1
500
39.0
450
456º
48.9
98.3
468º
45.5
87.5
400
424º
97.3
419.58º
350
300
250
200
99.7%
at 100º
150
100
50
0
Cu
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
Zn
Weight percentage, zinc
Figure 9-29 Cu–Zn phase diagram. (After Metals Handbook, 8th ed., Vol. 8: Metallography, Structures, and
Phase Diagrams, 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.)
4
˚C
8
CaO (wt %)
12
16
20
24
28
500
0
ZrO2
Cubic
ZrO2SS + ZrCaO3
Monoclinic ZrO2SS +
Cubic ZrO2SS
1000
Cubic ZrO2SS
1500
Tetragonal ZrO2SS
2000
Tetragonal ZrO2SS + Cubic ZrO2SS
2500
10
20
30
CaO (mol %)
40
Figure 9-30 CaO–ZrO 2 phase diagram. The dashed lines
represent tentative results. (After Phase Diagrams for
Ceramists, Vol. 1, American Ceramic Society, Columbus, Ohio, 1964.)
50
Temperature
L
A
A2B
AB
Composition
AB2
AB4
B
Temperature
L
L + SS
T1
SS
0
A
30
50
80
100
Composition (wt % B)
B
mL + mSS = mtotal
0.30mL + 0.80mSS = 0.50mtotal
→mL = 0.60mtotal
mSS = 0.40mtotal
Figure 9-31 A more quantitative treatment of the tie line
introduced in Figure 9–6 allows the amount of each
phase (L and SS) to be calculated by means of a mass
balance (Equations 9.6 and 9.7).
(a)
Fulcrum
(b)
Figure 9-32 The lever rule is a mechanical analogy to the mass balance calculation. The
(a) tie line in the two-phase region is analogous to (b) a lever balanced on a fulcrum.
Temperature
Lsystem
100% liquid
(Lsystem)
L1
T1
SS1
L2
T2
10% SS1 in
matrix of L1
SS2
L3
T3
SS3
40% SS2 in
matrix of L2
90% SS3 in
matrix of L3
SSsystem
A
Composition
B
100% Solid
(SSsystem)
Figure 9-33 Microstructural development during the slow cooling of a
50% A–50% B composition in a phase diagram with complete solid
solution. At each temperature, the amounts of the phases in the microstructure correspond to a lever rule calculation. The microstructure at T2 corresponds to the calculation in Figure 9–31.
Temperature
Leutectic
100% liquid
(Leutectic)
T1
T2
A
Composition
B
*The only differences from the T1 microstructure are
the phase compositions and the relative amounts of
each phase. For example, the amount of b will be
proportional to
Figure 9-34 Microstructural development during the slow cooling
of a eutectic composition.
Temperature
100% liquid
(Lsystem = 80% B)
Lsystem
L2
L1
T2 (= Teutectic + 1 )
T3 (= Teutectic – 1 )
0
A
30
60
80
Composition (wt % B)
90 100
B
Figure 9-35 Microstructural development during the slow cooling of a hypereutectic composition.
Temperature
Lsystem
100% liquid
(Lsystem = 40% B)
L1
T2 (= Teutectic + 1 )
T3 (= Teutectic – 1 )
0
A
30 40
60
Composition (wt % B)
90 100
B
Figure 9-36 Microstructural development during the slow cooling of a hypoeutectic composition.
Temperature
Lsystem
100% liquid
(Lsystem = 10% B)
L1
0
A
10
Composition (wt % B)
(a)
Temperature
Lsystem
100% liquid
(Lsystem = 20% B)uid
(Lsystem = 20%
100
B
L1
0
A
10
20
Composition (wt % B)
(b)
100
B
Temperature
100% liquid
(3% C)
L1
0
3.0
6.7
Weight percentage carbon
Figure 9-38 Microstructural development for white cast iron (of composition 3.0 wt % C) shown with the aid of the Fe–Fe 3 C phase diagram.
The resulting (low-temperature) sketch can be compared with a micrograph in Figure 11–1a.
Temperature
0
0.77
6.7
Weight percentage carbon
Figure 9-39 Microstructural development for eutectoid steel (of
composition 0.77 wt % C). The resulting (low-temperature)
sketch can be compared with the micrograph in Figure 9–2.
Temperature
Proeutectoid cementite
+ pearlite
0
1.13
6.7
Weight percentage carbon
Figure 9-40 Microstructural development for a slowly cooled hypereutectoid steel
(of composition 1.13 wt % C).
Temperature
Proeutectoid ferrite
+ pearlite
0 0.50
6.7
Weight percentage carbon
Figure 9-41 Microstructural development for a slowly cooled hypoeutectoid steel
(of composition 0.50 wt % C).
Temperature
100% liquid
(3% C)
L1
C flakes (from eutectic
and eutectoid reactions)
in matrix of ferrite
0
3
100
Weight percentage carbon
Figure 9-42 Microstructural development for gray cast iron (of composition 3.0 wt % C) shown on the Fe–C phase diagram. The resulting
low-temperature sketch can be compared with the micrograph in
Figure 11–1b. A dramatic difference is that, in the actual microstructure, a substantial amount of metastable pearlite was formed at the
eutectoid temperature. It is also interesting to compare this sketch
with that for white cast iron in Figure 9–38. The small amount of
silicon added to promote graphite precipitation is not shown in this
two-component diagram.
The phase diagram for this alloy system is
T
A
B