PHASE DIAGRAMS
H. P. R. Frederikse
A phase is a structurally homogeneous portion of matter. Regardless of the number of chemical constituents of a gas, there is only one vapor phase. This is true also for the liquid form of a pure substance, although a mixture of several liquid substances may exist as one or several phases, depending on the interactions among the substances. On the other hand a pure solid may exist in several phases at different temperatures and pressures because of differences in crystal structure (Reference 1). At the phase transition temperature, Ttr> the chemical composition of the solid remains the same, but a change in the physical properties often will take place. Such changes are found in ferroelectric crystals (example BaTi03) that develop a spontaneous polarization below T f in superconductors (example Pb) that lose all electrical resistance below the transition point, and in many other classes of solids.
In quite a few cases it is difficult to bring about the phase transition, and the high- (or low-) temperature phase persists in its metastable form. Many liquids remain in the liquid state for shorter or longer periods of time when cooled below the melting point (supercooling). However, often the slightest disturbance will cause solidification. Persistence of the high temperature phase in solid-solid transitions is usually of much longer duration. An example of this behavior is found in white tin; although gray tin is the thermo-dynamically stable form below T (286.4 K), the metal remains in its undercooled, white tin state all the way to T = 0 K, and crystals of gray tin are very difficult to produce.
A phase diagram is a map that indicates the areas of stability of the various phases as a function of external conditions (temperature and pressure). Pure materials, such as mercury, helium, water, and methyl alcohol are considered one-component systems and they have unary phase diagrams. The equilibrium phases in two-component systems are presented in binary phase diagrams. Because many important materials consist of three, four, and more components, many attempts have been made to deduce their multi-component phase diagrams. However, the vast majority of systems with three or more components are very complex, and no overall maps of the phase relationships have been worked out.
It has been shown during the last 20 to 25 years that very useful partial phase diagrams of complex systems can be obtained by means of thermodynamic modeling (References 2, 3). Especially for complicated, multicomponent alloy systems the CALPHAD method has proved to be a successful approach for producing valuable portions of very intricate phase diagrams (Reference 4). With this method thermodynamic descriptions of the free energy functions of various phases are obtained that are consistent with existing (binary) phase diagram information and other thermodynamic data. Extrapolation methods are then used to extend the thermodynamic functions into a ternary system. Comparison of the results of this procedure with available experimental data is then used to fine-tune the phase diagram and add ternary interaction functions if necessary. In principle this approximation strategy can be extended to four, five, and more component systems.
The nearly two dozen phase diagrams shown below present the reader with examples of some important types of single and multicomponent systems, especially for ceramics and metal alloys. This makes it possible to draw attention to certain features like the kinetic aspects of phase transitions (see Figure 22, which presents a time-temperature-transformation, or TTT, diagram for the precipitation of a-phase particles from the (5-phase in a Ti-Mo alloy; Reference 1, pp. 358-360). The general references listed below and the references to individual figures contain phase diagrams for many additional systems.
General References
1. Ralls, K. M., Courtney, T. H., and Wulff, J., Introduction to Materials Science and Engineering, Chapters 16 and 17, John Wiley & Sons, New York, 1976.
2. Kaufman, L., and Bernstein, H., Computer Calculation of Phase Diagrams, Academic Press, New York, 1970.
3. Kattner, U. R., Boettinger, W. J. B., and Coriell, S. R., Z. Metallkd., 87, 9, 1996.
4. Dinsdale, A. T., Ed., CALPHAD, Vol. 1-20, Pergamon Press, Oxford, 1977-1996 and continuing.
5. Baker, H., Ed., ASM Handbook, Volume 3: Alloy Phase Diagrams, ASM International, Materials Park, OH, 1992.
6. Massalski, T. B., Ed., Binary Alloy Phase Diagrams, Second Edition, ASM International, Materials Park, OH, 1990.
7. Roth. R. S., Ed., Phase Diagrams for Ceramists, Vol. I (1964) to Volume XI (1995), American Ceramic Society, Waterville, OH.
References to Individual Phase Diagrams
Figure 1.    Carbon: Reference 7, Vol. X (1994), Figure 8930. Reprinted
with permission. Figure 2.    Si-Ge : Ref. 5, p. 2.231. Reprinted with permission. Figure 3.    H20 (ice): See figure.
Figure 4. Si02: Reference 7, Vol. XI (1995), Figure 9174. Reprinted with permission.
Figure 5.    Fe-O: Darken, L.S., and Gurry, R.W.,/. Am. Chem. Soc, 68,798,
1946. Reprinted with permission. Figure 6.    Ti-O: Reference 5, p. 2.324. Reprinted with permission. Figure 7.    BaO-Ti02: Reference 7, Vol. Ill (1975), Figure 4302. Reprinted
with permission.
Figure 8. MgO-Al203: Reference 7, Vol. XI (1995), Figure 9239. Reprinted with permission.
Figure 9. Y203-Zr02: Reference 7, Vol. XI (1995), Figure 9348. Reprinted with permission.
Figure 10. Si-N-Al-O (Sialon): Reference 7, Vol. X (1994), Figure 8759.
Reprinted with permission. Figure 11.  PbO-Zr02-Ti02 (PZT): Reference 7, Vol. Ill (1975), Figure
4587. Reprinted with permission. Figure 12. Al-Si-Ca-O: Reference 7 (1964), Vol. I, Figure 630. Reprinted
with permission.
Figure 13. Y-Ba-Cu-O: Whitler, J.D., and Roth, R.S., Phase Diagrams for High T Superconductors, Figure S-082, American Ceramic Society, Waterville, OH, 1990. Reprinted with permission.
Figure 14. Al-Cu: Reference 5, p. 2.44. Reprinted with permission.
Figure 15. Fe-C: Ralls, K.M., Courtney, T.H., and Wulff, J., Introduction to Materials Science and Engineering, Figure 16.13, John Wiley & Sons, New York, 1976. Reprinted with permission.
Figure 16.  Fe-Cr: Reference 5, p. 2.152. Reprinted with permission.
Figure 17.  Cu-Sn: Reference 5, p. 2.178. Reprinted with permission.
Figure 18.  Cu-Ni: Reference 5, p. 2.173. Reprinted with permission.
Figure 19. Pb-Sn (solder): Reference 5, p. 2.335. Reprinted with permission.
Figure 20. Cu-Zn (brass): Subramanian, PR., Chakrabarti, D.J., and Laughlin, D.E., Eds., Phase Diagrams of Binary Copper Alloys, p. 487, ASM International, Materials Park, OH, 1994. Reprinted with permission.
Figure 21. Co-Sm: Reference 5, p. 2.148. Reprinted with permission. Figure 22. Ti-Mo: Reference 5, p. 2.296; Reference 1, p. 359. Reprinted with permission.
Figure 23. Fe-Cr-Ni: Reference 5, Figure 48. Reprinted with permission.
12-187
12-188
Phase Diagrams
200
150
4000
r/K
FIGURE 1. Phase diagram of carbon. (A) Martensitic transition: hex graphite -> hex diamond. (B) Fast graphite-to-diamond transition. (C) Fast diamond-to-graphite transition.
Phase
(Ge,Si)
High-pressure phases Gell Sill
Composition, mass % Si 0 to 100
Pearson symbol
cFS
tI4 tI4
Space group
Fd3 m
H lamd Hlamd
Phase Diagrams
12-189
P/MRa
FIGURE 3. Diagram of the principal phases of ice. Solid lines are measured boundaries between stable phases; dotted lines are extrapolated. Ice IV is a metastable phase that exists in the region of ice V. Ice IX exists in the region below -100 °C and pressures in the range 200-400 MPa. Ice X exists at pressures above 44 GPa. See Table 1 for the coordinates of the triple points, where liquid water is in equilibrium with two adjacent solid phases.
TABLE 1. Crystal Structure, Density, and Transition Temperatures for the Phases of Ice
Crystal
Phase	system		Cell parameters	Z	n	p/g cm3	Triple points
Ih	Hexagonal	a	= 4.513; c = 7352	4	4	0.93	I-III: -21.99 °C, 209.9 MPa
Ic	Cubic	a	= 6.35	8	4	0.94	
II	Rhombohedral	a	= 7.78; a = 113.1°	12	4	1.18	
III	Tetragonal	a	= 6.73; c = 6.83	12	4	1.15	III-V: -16.99 °C, 350.1 MPa
IV	Rhombohedral	a	= 7.60; a = 70.1°	16	4	1.27	
V	Monoclinic	a c	= 9.22; b = 7.54, = 10.35; ß = 109.2°	28	4	1.24	V-VI: 0.16 °C, 632.4 MPa
VI	Tetragonal	a	= 6.27; c = 5.79	10	4	1.31	VI-VII: 82 °C, 2216 MPa
VII	Cubic	a	= 3.41	2	8	1.56	
VIII	Tetragonal	a	= 4.80; c = 6.99	8	8	1.56	
IX	Tetragonal	a	= 6.73; c = 6.83	12	4	1.16	
X	Cubic	a	= 2.83	2	8	2.51	
References
1. Wagner, W., Saul, A., and Pruss, A., / Phys. Chem. Ref. Data, 23, 515, 1994.
2. Lerner, R.G. and Trigg, G.L., Eds., Encyclopedia of Physics, VCH Publishers, New York, 1990.
Donnay, J.D.H. and Ondik, H.M, Crystal Data Determinative Tables, Third Edition, Volume 2, Inorganic Compounds, Joint Committee on Powder Diffraction Standards, Swarthmore, PA, 1973. Hobbs, PV, Ice Physics, Oxford University Press, Oxford, 1974. Glasser, L., / Chem. Edu., 81, 414, 2004.
12-190
Phase Diagrams
- Liq-,,--'	1   '      1          1 1	i     i     i     i i	i
"j_Crist ] ßQuartz			
1190° A I 1.43 kbars// Trid   /' / : / /		•9" If	
-./ /	Coesite		-
. "Quartz 1 ,	■     i     i i	/ Stishovite i   .' \     i     i i	
n I_I_\_I_I_I_I_I_I_£_1_I_I_I_
u0       20       40       60       80       100 120
P/kbar
FIGURE 4.        Si02 system. Crist = cristobalite; Trid = tridymite.
FIGURE 5.        Fe-O system.
Phase Diagrams
12-191
Point	trc	%o
A	1539	
B	1528	0.16
C	1528	22.60
G	1400"	22.84
H	1424	25.60
I	1424	25.31
1	1371	23.16
L	911"	23.10
N	1371	22.91
3 Values for pure iron.
	Point	trc
	Q	560
0.209	R	1583
0.209	R'	1583
0.263	S	1424
16.2	V	1597
16.2	Y	1457
0.282	Z	1457
0.447	Z'	
0.282		
PcoJPco PoJatm
23.26 1.05 28.30 1 28.07 1 27.64 16.2 27.64 0.0575 28.36 1 30.04 1 30.6
Mass Percent Oxygen 10 20
FIGURE 6.
Ti-O system.
Phase
(ßTi) (aTi)
Ti30 Ti20 yTiO
pTiO aTiO
ßTi^O
aTij xO ßTiÄ «Ti203 PTi3Os
aTi305
YTiP7 ßTi407 aTi407
ytía'
ßTiA, TiA3
TiA5 TiA
Rutile Ti02 Metastable phases
Anatase
Brookite High-pressure phases
Ti02-II
tkviii
20 30 40
Atomic Percent Oxygen
Composition, mass % O
Oto 3 0 to 13.5 ~8 to -13 -10 to 14.4 15.2 to 29.4 -18
-24 to -29.4 -25.0 -29.5 -29.5
33.2 to 33.6
33.2 to 33.6 35.8
35.8
35.8
36.9 36.9 36.9
37.6
38.0
38.3 38.5
38.7
40.1
Pearson symbol
cI2
hP2
hP~16
hP3
cF8
hP~5
mC16
0/12
ř/18
hR30
hR30
nf*
mC32
mC32
aPii
aPii
aPii
aP28
aC68
aP40
aC92
«P52
tP6
tin
oP24
oP12 W-48
Space group
ImSm
P63/mmc
P3c
P3ml
FmSm
P6/mmm
A2/m or B"ř
7222
14,/m
R3c
R3c
C2/m
Cc
PÍ
PI
PI
PI
Al
PI
Al
PÍ
P4Jmnm
I4jamd Pbca
Pbcn
12-192
Phase Diagrams
1600
FIGURE 7.       BaO-Ti02 system.
2800	i	1                      1                     1                      1 1	i
2600			-
2400		Periclase ss        n.            Spinel ss + Liquid              N.      T\+ Liquid	-
2200			2105°
2000	Periclase " ss	\                1995° \^>S	Spinel \ ss
1800		Periclase ss + Spinel ss	\ -
1600		-1500°	\-
		Periclase + Spinel	
1400	1	i         i         i         i i	i
I_I_I_I_I_I_l_
0        10       20       30       40       50 60
MgO Mol % Al203-►
FIGURE 8.
MgO-Al203 system.
Phase Diagrams
12-193
12-194
Phase Diagrams
3(Si02) AI6Si4013ss 2(AI203)
100 1-7-1-7—r—r-1-1-1-1-r—1-7-1-1-—si
Si3N4 Mol % 4(AIN)
FIGURE 10. 3(Si02)-Si3N4-4(AlN)-2(Al203) system. "Behavior" diagram at 1700 °C. The labels 8H, 15R, 12H, 21R, 27R, 2H6 indicate defect A1N polytypes. fT = 3-sialon (Si6 Al O N ); O' = sialon of Si2ON2 type; X = SiA102N ("nitrogen mullite"). ALON ss = aluminum oxynitride ss extending from approximately A1?09N to A1303N.
PbO
FIGURE 11.      PbO-Zr02-Ti02 (PZT) system, subsolidus at 1100 °C. P = PbO; T = TiQ2; Z = ZrQ2.
Phase Diagrams
12-195
Crystalline Phases Notation Oxide formula
Cristobalite    1 Si02 Tridymite •*
Pseudowollastonite CaOSi02
Rankinite 3Ca02Si02
Lime CaO
Corundum AljOj
Mullite 3Al203-2Si02
Anorthite CaO-Al203-2Si02
Gehlenite 2CaO-Al203-Si02
Temperatures up to approximately 1550 °C are on the Geophysical Laboratory Scale; those above 1550 °C are on the 1948 International Scale.
12-196
Phase Diagrams
Atomic Percent Copper
0 10 20 30 40      50      60    70   80 90100
FIGURE 14.      Al-Cu system.
Phase Diagrams
12-197
o
	Composition,	Pearson	Space
Phase	wt%Cu	symbol	group
(Al)	0 to 5.65	cF4	Fm3m
B	52.5 to 53.7	tin	I4lmcm
1,	70.0 to 72.2	oP16 or oC16	Pban or Cmmm
n2	70.0 to 72.1	mC20	C2/W
C	74.4 to 77.8	hP42	P6lmmm
c2	74.4 to 75.2	(a)	-
	77.5 to 79.4	(b)	-
E2	72.2 to 78.7	hP4	P63/mmc
8	77.4 to 78.3	(c)	R3m
Yo	77.8 to 84	(d)	-
Yi	79.7 to 84	cP52	P43m
P.	83.1 to 84.7	(d)	-
P	85.0 to 91.5	cI2	Im3m
	88.5 to 89	(e)	-
(Cu)	90.6 to 100	cF4	Fm3m
Metastable phases			
B'	-	tP6	-
P'	-	cH6	Fm3m
Al3Cu2	61 to 70	hp5	P3m\
(a) Monoclinic? (b) Cubic? (c) Rhombohedral. (d) Unknown, (e) DO			type long-period
superlattice.			
	Atomic Percent Carbon		
10	20	30	
2000
1500
1394°C
Ü
" 1000
912°C
500
0 2 4 6 8 10 12
Mass Percent Carbon
FIGURE 15.      Fe-C system.
Composition,
Phase mass % C
(8Fe) 0 to 0.09
(yFe) 0 to 2.1
(aFe) 0 to 0.021
(C) 100 Metastable/high-pressure phases
(eFe) 0
Martensite < 2.1
Fe4C 5.1
Fe3C (8) 6.7
Fe5C2(X) 7.9
Fe7C3 8.4
Fe7C3 8.4
Fe2C (n) 9.7
Fe2C (e) 9.7
Fe2C 9.7
(C) 100
Pearson symbol
cI2 cF4 cI2 hP4
hP2
tI4
cP5
oP16
mC28
hP20
oP40
oP6
hP*
hP*
cFS
Space group
Im3m Fm3m Im3m Pdjmmc
Pdjmmc
I4lmmm
P43m
Pnma
C2/c
P6jnc
Pnma
Pnnm
P6322
P3wl
Fd3m
12-198
Phase Diagrams
Atomic Percent Chromium
0      10       20       30       40       50       60       70       80       90 100
Mass Percent Chromium
FIGURE 16.      Fe-Cr system.
	Composition,	Pearson	Space
Phase	mass % Cr	symbol	group
(aFe, Cr)	0 to 100	cI2	Imim
(yFe)	Oto 11.2	cF4	Fmim
a	42.7 to 48.2	tP30	Pijmnm
Atomic Percent Tin
1001-'-1-1-U-1-1-111-1-1-1-1
0      10      20     30      40      50      60      70      80      90 100
Cu Mass Percent Tin Sn
FIGURE 17.
Cu-Sn system.
Phase Diagrams
12-199
Phase	Composition, mass % Sn	Pearson symbol	Space group
a	0 to 15.8	cF4	Fm3m
ß	22.0 to 27.0	ell	I m'-> m
y	25.5 to 41.5	cF16	Fm3m
5	32 to 33	cF416	F43m
c	32.2 to 35.2	hP26	«>3
e	27.7 to 39.5	oC80	Cmcm
1	59.0 to 60.9	hP4	P6Jmmc
1'	44.8 to 60.9	(a)	-
(ßSn)	-100	ÜA.	HJamd
(aSn)	100	cFS	Fd3m
(a) Hexagonal; superlattice based on NiAs-type structure.
Composition,      Pearson Space Phase mass % M        symbol group
(Cu, Ni) (above 354.5 °C) 0 to 100 cF4 Fm3m
12-200
Phase Diagrams
Composition, mass % Sn 0 to 18.3 97.8 to 100 100
Pearson symbol
cF4
m
cFS
Space group
Fm3m I4Jamd Fd3m
Phase
(Pb) ((3Sn) (aSn)
High-pressure phases
e(a) 52 to 74 hPl Pdlmmm
e'(b) 52 hP2 P6.Jmmc
(a) From phase diagram calculated at 2500 MPa. (b) This phase was claimed for alloys at 350 °C and 5500 MPa.
Phase Diagrams
12-201
Mass Percent Zinc
0        10 20        30 40        50 60 70 80        90 100
Cu Atomic Percent Zinc Zn
FIGURE 20.      Cu-Zn system.
	Composition,	Pearson	Space
Phase	mass % Zn	symbol	group
a or (Cu)	0 to 38.95	cF4	Fm3m
P	36.8 to 56.5	cI2	Imim
P'	45.5 to 50.7	cPl	Pnßm
Y	57.7 to 70.6	cIS2	I43m
S	73.02 to 76.5	hP3	P6
E	78.5 to 88.3	hP2	P6Jmmc
r| or (Zn)	97.25 to 100	hP2	P6Jmmc
12-202
Phase Diagrams
°0 10 20 30 40 50 60 70 80 90 100 ^° Mass Percent Samarium ®m
FIGURE 21.      Co-Sm system.
	Composition,	Pearson	Space
Phase	mass % Sm	symbol	group
(aCo)	0 to -3.7	cF4	Fm3m
(eCo)	~0	hP2	Pdjmmc
ßCo17Sm2	-23.0	hP38	Pdjmmc
aCo17Sm2	-23.0	hR19	R3m
		hPS	Pdlmmm
Cos + >Sm	-33 to 34	-	-
Cos Sm	-34 to 35	-	-
Co19Sm5	-40.1	hR24	R3m
		hP48	Pdjmmc
aCo7Sm2	-42.1	hR18	R3m
ßCo7Sm2	-42.1	hP36	Pdjmmc
Co3Sm	46	hRVl	R3m
Co2Sm	56.0	hR4	R3m
		cF24	Fd3m
Co4Sm9	-85.1	0**	-
CoSm3	88	oP16	Pnma
(ySm)	-100	cI2	Im3m
(ßSm)	-100	hP2	Pdjmmc
(aSm)	-100	hR3	R3m
Other reported phases			
CosSm	-33.8	hP6	Pdlmmm
Co2Sm5	-86.4	mC28	Cllc
Phase Diagrams
12-203
o
2600 2400 2200 2000 1800 1600 1400 1200 1000 800 600 400
1670°C
882°C
Atomic Percent Molybdenum 20 30        40 50
70   80   90 100
(°Ti,Mo)
30       40       50       60 70 Mass Percent Molybdenum
90
100 Mo
FIGURE 22.      Ti-Mo system.
Composition, Pearson Space
Phase mass % Mo symbol group
(|3Ti, Mo) 0 to 100 ell Im3m
(aTi) 0 to 0.8 hP2 Pdjmmc
a' (a) hP2 Pdjmmc
a" (a) oC4 Cmcm
Co (a) hP3 Pdlmmm
(a) Metastable.
4001-—-1-1-1-1
0.1 1.0 10 100 1000
Time in Minutes
Experimental time-temperature-transformation (TTT) diagram for Ti-Mo. The start and finish times of the isothermal precipitation reaction vary with temperature as a result of the temperature dependence of the nucleation and growth processes. Precipitation is complete, at any temperature, when the equilibrium fraction of a is established in accordance with the lever rule. The solid horizontal line represents the athermal (or nonthermally activated) martensitic transformation that occurs when the (3 phase is quenched.
12-204
Phase Diagrams
Fe        10        20        30        40        50        60        70        80 90 Ni
Mass Percent Nickel
FIGURE 23. The isothermal section at 900 °C (1652 °F) of the iron-chromium-nickel ternary phase diagram, showing the nominal composition of 18-8 stainless steel.