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General Phase Diagram Sections
Arthur D. Pelton
Centre de Recherche en Calcul Thermochimique
École Polytechnique de Montréal
Montréal, Québec, CANADA
General rules of construction of all true phase diagram sections;Proper choice of variables and constants to give a “true” phase
diagram section (with a unique equilibrium state at each point);A general algorithm for calculating all true phase diagram sections.
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CaO-MgO: phase diagram
MgO + Liquid
CaO MgO
Liquid
CaO + Liquid
CaO + MgO
Mole fraction MgO
T(K
)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11200
1400
1600
1800
2000
2200
2400
2600
2800
3000
3200
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0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.10.20.30.40.50.60.70.80.9
0.10.2
0.30.4
0.50.6
0.70.8
0.9
weight fraction CaO
SiO2
CaO Al2O3
wei
ght f
ract
ion
SiO
2
weight fraction A
l2 O3
à
à
ßCA2
C3S
C2S
Mullite
L
L + C
L + A
L +
L + S
Mullite
L+C3S+C
L+C2S
L+CA2
L+C3S
L+C2S+C3S
L+CA6+CA2
L+CA6
L+A+CA6
L+A+Mullite
CA6
CA6 = (CaO)(Al2O3)6
CA2 = (CaO)(Al2O3)2
C3S = (CaO)3(SiO2)C2S = (CaO)2(SiO2)S = SiO2
A = Al2O3
C = CaOL = Liquid
System SiO2 - CaO - Al2O3
1600 °C
Diagram: CaO-Al2O3-SiO2
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C-Fe-W System at 5 wt% W
LIQUID
LIQUID + BCCBCC
BCC
M6C + BCC
M6C + BCC + Fe2W
M6C + BCC + WC
M6C + FCC + BCC
M6C + FCC
M6C + FCC + WC
FCC + BCC + WC
BCC + WC + C
FCC + WC + C
FCC + WC
FCC
FCC +FCC + LIQUID + BCC
FCC + LIQUID
FCC + LIQUID + WC
BCC
BCC + WC
Fe - C - Wwt% W = 5%
wt% C
Tem
per
atu
re (
K)
0.0 0.5 1.0 1.5 2.0
900
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
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A Phase Diagram for the Fe-Cr-V-C System
fcc
fcc + MC
fcc + M7C3
bcc + M23C6
fcc + bcc+ M23C6
fcc + bccfcc + MC + M7C3
bcc+ fcc+ MC
+ M7C3 bcc + M7C3
bcc + MC + M7C3
bcc + fcc + MC
bcc + MC+ M23C6
– fcc + M23C6
– fcc + M7C3 + M23C6
– bcc + fcc + M7C3 + M23C6
– bcc + MC + M7C3 + M23C6
bcc + MC
+ M7C3
bcc
+ M 7
C 3 +
M23
C 6
‚
‚ƒ
„
„
ƒ
Fe - Cr - V - C SystemT = 850°C, wt.% C = 0.3, Ptot = 1 atm
<F*A*C*T>
mass fraction Cr
mas
s fr
acti
on
V
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.160.00
0.01
0.02
0.03
0.04
0.05
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Fe-Cr-V-C system at 1.5 wt% Cr and 0.1 wt% V
FCC
MC + FCC
MC + FCC + CEMENTITE
MC + BCC + CEMENTITEMC + BCC + CEMENTITE + M7C3
MC + BCC + M7C3
MC + FCC + BCC
FCC + BCC
MC + FCC + BCC + M7C3
MC + FCC + BCC + CEMENTITE BCC + CEMENTITE
BCC + CEMENTITE + FCC
MC + FCC+ M7C3
Fe-Cr-V-C systemat 1.5 wt% Cr and 0.1 wt% V
wt % C
Tem
per
atu
re (
°C)
0.0 0.2 0.4 0.6 0.8 1.0600
700
800
900
1000
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P–T diagram for Al2TiO5
Kyanite
Sillimanite
Andalusite
Al2SiO5
Temperature (°C)
Pre
ssure
(bar)
350 450 550 650 750 850 9500
2000
4000
6000
8000
10000
12000
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Fe2O3(s2)
Spinel
Slag-liqFe-liq
Fe(alpha)
Fe(gamma)
Fe(alpha)
Fe2O3(s1)
Wustite
RT lnPO2 (kJ)
Te
mp
era
ture
(K
)
-500 -400 -300 -200 -100 0 100700
800
900
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000
Fe-O2 System
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Corundum
Spinel+
Corundum
Spinel
Wustite + Spinel
Spinel + FCC
Corundum + FCC
Corundum + BCC
BCC
FCC FCC + BCC
1573 K
Mole Ratio Cr/(Fe+Cr)
log
10(P
O2)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-20
-18
-16
-14
-12
-10
-8
-6
-4
-2
0
System Fe-Cr-O2
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SO2-O2-Cu System at 1000 K
CuS
Cu2S
Cu
Cu2OCuO
(CuO)(CuSO4)Cu2SO4
CuSO4
1000 K
log10 PO2
log
10 P
SO
2
-20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0-12
-10
-8
-6
-4
-2
0
2
4
6
8
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Fe2O3-MgO-SiO2-O2 SystemSiO2/(Fe2O3+ MgO + SiO2) = 0.20 in air
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Fe-Cr-S-O System
Spinel
(Fe,Cr)S
Cr2O3
Cr2O3
Spinel + (Fe,Cr)S
bcc Alloy
Spinel + fcc Alloy
fcc Alloy+
(Fe,
Cr)
S +
Cr 2
O3
bcc Alloy +
bcc Alloy + (Fe,Cr)S
fcc Alloy + (Fe,Cr)S
Fe-Cr-S-O SystemT=1273 K , Molar ratio: nCr/nFe = 1
log10(PO2)
log
10(P
S2)
-25 -23.5 -22 -20.5 -19 -17.5 -16 -14.5 -13 -11.5 -10-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
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A Phase Diagram for the Fe-Cr-V-C System
fcc
fcc + MC
fcc + M7C3
bcc + M23C6
fcc + bcc+ M23C6
fcc + bccfcc + MC + M7C3
bcc+ fcc+ MC
+ M7C3 bcc + M7C3
bcc + MC + M7C3
bcc + fcc + MC
bcc + MC+ M23C6
– fcc + M23C6
– fcc + M7C3 + M23C6
– bcc + fcc + M7C3 + M23C6
– bcc + MC + M7C3 + M23C6
bcc + MC
+ M7C3
bcc
+ M 7
C 3 +
M23
C 6
‚
‚ƒ
„
„
ƒ
Fe - Cr - V - C SystemT = 850°C, wt.% C = 0.3, Ptot = 1 atm
<F*A*C*T>
mass fraction Cr
mas
s fr
acti
on
V
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.160.00
0.01
0.02
0.03
0.04
0.05
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The Law of Adjoining Phase Regions
«As a phase boundary line is crossed, one and only one phase appears or disappears.»
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A node in a general true phase diagram section.
1 + 2 + … + n)
1 + 2 + … + n) 1 + 2 + … + n)
1 + 2 + … + n)
+ +
++
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CaO-MgO: phase diagram
MgO + Liquid
CaO MgO
Liquid
CaO + Liquid
CaO + MgO
Mole fraction MgO
T(K
)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11200
1400
1600
1800
2000
2200
2400
2600
2800
3000
3200
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An isobaric binary temperature-composition diagram with the eutectic “opened-up” to show that this is an infinitely narrow 3-phase region.
BA
T
XB
L
+ L + + L
+ L
+
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SO2-O2-Cu System at 1000 K
CuS
Cu2S
Cu
Cu2OCuO
(CuO)(CuSO4)Cu2SO4
CuSO4
1000 K
log10 PO2
log
10 P
SO
2
-20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0-12
-10
-8
-6
-4
-2
0
2
4
6
8
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A “potential-potential” phase diagram “opened-up” to show that the lines are infinitely narrow 2-phase regions.
1
+
2
+
+
++
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• Extensive Variables(not the same for all phases at equilibrium)
Xi (composition)
V (volume)
• Potential Variables(the same for all phases at equilibrium)
T (temperature)
P (pressure)
i (chemical potential)
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If only potential variables are held constant, then all tie-lines lie in the plane of the section. In this case, the compositions of the individual phases at equilibrium can be read from the diagram and the lever rule applies.
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Zero Phase Fraction (ZPF) Lines
fcc
fcc + MC
fcc + M7C3
bcc + M23C6
fcc + bcc+ M23C6
fcc + bccfcc + MC + M7C3
bcc+ fcc+ MC
+ M7C3 bcc + M7C3
bcc + MC + M7C3
bcc + fcc + MC
bcc + MC+ M23C6
– fcc + M23C6
– fcc + M7C3 + M23C6
– bcc + fcc + M7C3 + M23C6
– bcc + MC + M7C3 + M23C6
bcc + MC
+ M7C3
bcc
+ M 7
C 3 +
M23
C 6
‚
‚ƒ
„
„
ƒ
Fe - Cr - V - C SystemT = 850°C, wt.% C = 0.3, Ptot = 1 atm
<F*A*C*T>
mass fraction Cr
mas
s fr
acti
on
V
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.160.00
0.01
0.02
0.03
0.04
0.05MCfccbccM7C3
M23C6
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Zero Phase Fraction (ZPF) Lines
System CaO - MgOT vs. (mole fraction) P = constant = 1 bar
Mole fraction XCaO
Te
mp
era
ture
, °C
0.0 0.2 0.4 0.6 0.8 1.01600
1800
2000
2200
2400
2600
2800
LIQUID
LIQUID + L+
SOLID SOLID
2 SOLIDS
()
So
lidu
s
Liquidus
So
lvu
s lin
e So
lvus lin
e
LIQUID
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Fe-S-O Predominance diagram (ZPF lines)
Fe2(SO4)3(s)FeS(s3)
FeSO4(s)
Fe(s) Fe3O4(s) Fe2O3(s)
FeS2(s)
Fe - S - O SystemPredominance diagram T = constant = 800 K
log10 PO2 , atm
log
10 P
S2 ,
atm
-35 -30 -25 -20 -15 -10 -5 0-40
-35
-30
-25
-20
-15
-10
-5
0
5
10
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Choice of variables to always give a true phase diagram (single-valued) everywhere
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Phase Diagram 26 www.factsage.com
N-Component System (A-B-C-…-N)Extensive variable Corresponding potential
qi
S T
V -PnA
A
nB
B
. .
. .
. .nN
N
0i i i iSdT VdP n d q d Gibbs-Duhem:
i i i idU TdS PdV dn dq
j
ii q
U
q
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Phase Diagram 27 www.factsage.com
Choice of variables
N-component system(1) Choose n potentials: 1, 2, … , n
(2) From the non-corresponding extensive variables (qn+1, qn+2, … ), form (N+1-n) independent ratios (Qn+1, Qn+2, …, QN+1).
Example:
[1, 2, … , n; Qn+1, Qn+2, …, QN+1] are then the (N+1) variables of which 2 are chosen as axes and the remainder are held constant.
1n N
2
1
1 1ii N
jj n
qQ n i N
q
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Phase Diagram 28 www.factsage.com
MgO-CaO Binary System
S T
V -P
nMgO MgO
nCaO CaO
1 = T
2 = -P
3
3
4
MgO
CaO
MgO CaOCaO
q nn
Qn n
q n
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CaO-MgO: phase diagram
MgO + Liquid
CaO MgO
Liquid
CaO + Liquid
CaO + MgO
Mole fraction MgO
T(K
)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11200
1400
1600
1800
2000
2200
2400
2600
2800
3000
3200
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Phase Diagram 30 www.factsage.com
Fe-Cr-S-O System
S T
V -P
nFe Fe
nCr Cr
1 = T
2 = -P
2
2
3
4
5
5
6
O
S
Cr
Cr
FeFe
q nn
Qn
q n
22
2 2
O
S
O
S
n
n
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Spinel
(Fe,Cr)S
Cr2O3
Cr2O3
Spinel + (Fe,Cr)S
bcc Alloy
Spinel + fcc Alloy
fcc Alloy+
(Fe,
Cr)
S +
Cr 2
O3
bcc Alloy +
bcc Alloy + (Fe,Cr)S
fcc Alloy + (Fe,Cr)S
Fe-Cr-S-O SystemT=1273 K , Molar ratio: nCr/nFe = 1
log10(PO2)
log
10(P
S2)
-25 -23.5 -22 -20.5 -19 -17.5 -16 -14.5 -13 -11.5 -10-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
Fe-Cr-S-O System
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Phase Diagram 32 www.factsage.com
Pressure vs. Volume diagram for H2O
This is NOT a true phase diagram.
S+L
L+G
S+G
P
V
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Phase Diagram 33 www.factsage.com
Fe-Cr-C System
S T
V -P
nC C
nFe Fe
nCr Cr
1 = T
2 = -P
3 = C
(NOT OK)
(OK)
4
4
Cr
Fe C
Cr
e
r
F Cr C
nQ
n n n
nQ
n n
Requirement: 0 3j
i
dQfor i
dq
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Phase Diagram 34 www.factsage.com
Fe-Cr-C system, T = 1300 K, XCr = nCr/(nFe+nCr+nC) vs. (carbon activity)
This is NOT a true phase diagram.
M23C6
M7C3
bcc
fcc
cementitelog(ac)
Mo
le f
ract
ion
of
Cr
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
-3 -2 -1 0 1 2
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“Corresponding” phase diagrams of the Fe-O system
-Iron
-Iron
-Iron
Liquidiron
Liquid oxide
Wustite
Magnetite
Hematite
-500 -400 -300 -200 -100 0
RT ln PO2 (kJ)
-Iron
Wustite
-Iron
Wustite
-Iron Magnetite
Wustite
Magnetite
Hematite
Magnetite
Hematite+
Oxygen
Liquid iron Liquid iron liquid oxide
Liquid oxide
Wustite
700
900
1100
1300
1500
1700
1900
0 0.50 0.54 0.58 0.62
Mole fraction XO
Te
mp
era
ture
(K
)
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Other Sets of Extensive Variables and Corresponding Potentials
Example 1: Extensive variable Corresponding potential
qi
H -1/T V/T -P nj j /T
1 0j
j i iVHd d P n d q dT T TGibbs-Duhem:
j
ii q
U
q
Example 2: Extensive variable Corresponding potential
qi
TU/P -1/T A/T -T/P nj j /P
1 0j
j i iTU A Td d n d q dP T T P PGibbs-Duhem:
j
ii q
U
q
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Mg-Si binary phase diagram
Si + Mg2SiMgHCP + Mg2Si
Liquid + Mg2Si
Liquid + Mg2Si
Liquid + Si
Liquid
Liquid + MgHCP
Mg-Si
Molar fraction Si
Tem
pera
ture
(°C
)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
300
600
900
1200
1500
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Mg-Si, H vs XSi phase diagram