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Structure and Thermodynamics
of Liquid Oxidesof Liquid Oxides
Arthur D. Pelton,CRCT (Center for Research in Computational Thermochemistry),
École Polytechnique (University of Montréal),P.O. Box 6079, Station "Downtown",Montréal, Québec, H3C 3A7, Canada
www.crct.polymtl.ca
May 18, 2010
CRCT
Ionic (“Basic”) Oxides
Ca2+ Ca2+ Ca2+O2- O2-
Ca2+ Ca2+O2- O2- O2-
2
CaO, MgO, FeO, MnO, …
Na2O, K2O, …
Ca2+ Ca2+O2- O2- O2-
Ca2+ Ca2+ Ca2+O2-O2-
CRCT
Acidic Oxides (network formers)
• Tetrahedral:SiO2, GeO2
3
SiO2, GeO2
• Trigonal:B2O3
• Complex:P2O5
CRCT
Structure of SiO2
4
O
SiO O
O
SiO O
O
SiO OSi Si
O
SiO O
O O
O
SiO O
O
O
SiO O
O
O
SiO O
OO
SiO O
O
O O
SiO O
O
O
SiO O
O
OO
O
O O
O
Si
Si Si
SiSi Si
O
SiO O
O
CRCT
Network Modification by Basic OxidesQ4 structural units (black)Q3 structural units (green)
O0 + O2- = 2 O-
O
SiO O
O
SiO O
O
SiOO
Si Si
O
SiO O
OO O
5
O
SiO
O
SiO O
O
O
SiO O
O
O
SiO O
OO
SiO O
O
SiO O
O O
SiO O
O
O
SiO O
O
SiO O
O
SiO
O
O
O O
O
Si
Si Si
Si
SiSi
Si
Si
O
SiO O
O
O
Ca2+
Na+
Na+
SiO
CRCT
Structure of CaO-SiO2 Liquid Solutions
CompositionCaO Ca2SiO4 SiO2
mainly Ca2+, O2-, SiO4
4-mainly Ca2+, SiO 4- (orthosilicate ions)
O
Si
OSi
O
Si
OSi
6
SiO44- (orthosilicate ions)
linear + branchedchain polymers
ring polymers
3-dimensionalnetwork
Ca2+ Ca2+
Ca2+
O2-
Ca2+O2-
SiO-
O-
O-
O-
SiO-
O-
O-
O-
Si
Si
Si Si SiSiO O O O
O
O
OSi
O
CRCT
Fractions of Q Species as Measured by NMR
7
CRCT
Q4 by NMR, Halter and Mysen, 2004 [17]
Q4 by RAMAN, Yano et al., 2006 [18]
Q4 by MD, Machacek and Gedeon, 2003 [19]
Q4 by NMR, Maekawa et al., 1991 [16]
Q4
-spe
cies
0.6
0.7
0.8
0.9
1
The System NaO0.5-SiO2
Measured Fractions of Q4 Species
8
Q3
Q1
Q2
Q0
Mole fraction SiO 2 / (SiO2+NaO0.5)
Fra
ctio
n of
Qi -s
peci
es
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
CRCT
Thermodynamic Properties as Functions of Composition
SiO2 + 2MO = M2SiO4 (M = Ca, Mg, Fe, Mn, …)
9
SiO2 + 2M2O = M4SiO4 (M = Li, Na, K, …)
Si – O – Si + O2- = Si — O- + -O — Si
O0 + O2- = 2O-
∆H « 0∆G « 0
CRCT
Enthalpy of Mixing in Liquid Alkaline-earth Silicates
10
MgO-SiO2
-40
-20
0
Ent
halp
y of
mix
ing
(kJ/
mol
of o
xide
s)
BaO-SiO2
SrO-SiO2
CaO-SiO2
Mole fraction SiO2
0 0.20 0.40 0.60 0.80 1.00-120
-100
-80
-60
Ent
halp
y of
mix
ing
(kJ/
mol
of o
xide
s)
CRCT
Entropy of Mixing in Liquid Alkaline-earth Silicates
11
MgO-SiO
CaO-SiO2
2
4
6
8
10
Ent
ropy
of m
ixin
g (J
/mol
of o
xide
s)
MgO-SiO2
BaO-SiO2
SrO-SiO2
Mole fraction SiO2
0 0.20 0.40 0.60 0.80 1.00-10
-8
-6
-4
-2
0
Ent
ropy
of m
ixin
g (J
/mol
of o
xide
s)
CRCT
CaO-SiO2 System:∆G vs XSiO2
at 1750°C2
-20
-10
0
12
XSiO2
Del
ta G
(kJ)
0 0.20 0.40 0.60 0.80 1.00-70
-60
-50
-40
-30
CRCT
CaO-SiO2 System:aSiO2
and aCaO vs XSiO2at 1750°C
13
aCaO
aSiO2
0.7
0.8
0.9
1.0
XSiO2
Act
ivity
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0
0.1
0.2
0.3
0.4
0.5
0.6
CRCT
ASlag-liqCaO(s) + Ca2SiO4(s3)
ASlag-liq + CaO(s)
ASlag-liq + ASlag-liq#2
(s3)
2154o
2017o
2572o
0.2961895
o
1723o
1799o
Tem
pera
ture
(C
o )
1800
2100
2400
2700
CaO – SiO2 Phase Diagram
14
ASlag-liq + SiO2(s6)
CaSiO3(s) + SiO2(s4)
SiO2(s4) + CaSiO3(s2)
SiO2(s2) + CaSiO3(s)
CaSiO3(s) + SiO2(s)
CaS
iO3
Ca 2
SiO
4(s)
Ca 2
SiO
4(s2
)
Ca 3
Si 2O
7(s)
Ca 2
SiO
4(s3
)
Ca 3
SiO
5(s)
0.423
1463o
847o
1125o
575o
867o
1125o
1437o
1465o
0.988
1300o
1437o 1464
o
0.613
0.7181689
o
1540o
mole SiO2/(CaO+SiO2)
Tem
pera
ture
(C
0 0.2 0.4 0.6 0.8 1300
600
900
1200
1500
CRCT
ASlag-liq + SiO2(s6, cristobalite)
0.895
1722o
ASlag-liq
FactSage
1300
1500
1700
1900
Na2O – SiO2 Phase Diagram
15
Na 4
SiO
4(s) N
a 6S
i 2O7(
s)
Na 2
SiO
3(s)
Na 2
Si 2O
5(s)
ASlag-liq + SiO2(s4)
0.167
0.211
0.359 0.452
0.629
0.741
0.757
970o
909o
750o
1077o 1114
o
1087o
868o
402o
868o
793o
707o
1130o
700o
678o
575o
838o
Na 2
O(s
3)N
a 2O
(s2)
Na 2
O(s
)
1043o
1040o
SiO
2(s)
SiO
2(s2
, qua
rtz-h
)
Na2Si2O5(s3)
Na2Si2O5(s2)
Na6Si8O19(s)
mole SiO2/(Na2O+SiO2)
T(C
)
0 0.2 0.4 0.6 0.8 1
300
500
700
900
1100
1300
CRCT
Systems with Al2O3 (amphoretic)
The Charge Compensation effectAl3+ Na+
16
Al3+ can replace Si4+
Na+ or K+ compensates the charge2 Al3+ replace 2 Si4+
Ca2+ or Mg2+ compensates the charge
Bridging oxygen
Non-Bridging oxygen
Si4+
CRCT
ASlag-liq
Mullite + ASlag-liq
ASlag-liq + Al2O3(s4)
Mullite + SiO (s6)
1890o
0.325
1594o
1465o
1723o
0.957
2054o
1580
1940
2300
Al2O3 – SiO2 Phase Diagram
17
Mullite + Al2O3(s4)
Mullite + SiO2(s6)
Mullite + SiO2(s4)
Mullite + SiO2(s2)
Mullite + SiO2(s)
1465o
867o
575o
mole SiO2/(Al2O3+SiO2)
T(C
)
0 0.2 0.4 0.6 0.8 1
500
860
1220
CRCT
Na2O – SiO2 - Al2O3
Phase Diagram (liquidus projection)
18
CRCT
Albite(NaAlSi3O8) – SiO2
19
CRCT
0.7
0.8
0.9 0.1
0.2
0.3
SiO2(s6)
ASlag-liq#2
CaAl2Si
2O
8(s2)
SiO2
T(min) = 1184.29 oC, T(max) = 2571.99 oC
Four-Phase Intersection Points with ASlag-liq
AMonoxide / Ca2SiO4_alpha(s3) / Ca3SiO5_hatrurite(s)Al2O3_corundum(alpha(s4) / CaAl2Si2O8_anorthite(s2) / MulliteCa2Al2SiO7_gehlenite(s) / CaAl2O4(s) / CaAl4O7(s)Ca2Al2SiO7_gehlenite(s) / Ca2SiO4_alpha(s3) / CaAl2O4(s)Mullite / SiO2_cristobalite(h)(s6) / SiO2_tridy mite(h)(s4)Mullite / SiO2_cristobalite(h)(s6) / SiO2_tridy mite(h)(s4)Mullite / SiO2_cristobalite(h)(s6) / SiO2_tridy mite(h)(s4)AMonoxide / Ca3Al2O6(s) / Ca3SiO5_hatrurite(s)Al2O3_corundum(alpha(s4) / CaAl12O19(s) / CaAl2Si2O8_anorthite(s2)Ca2Al2SiO7_gehlenite(s) / CaAl12O19(s) / CaAl4O7(s)Ca2SiO4_alpha(s3) / Ca3Al2O6(s) / Ca3SiO5_hatrurite(s)Ca2SiO4_alpha(s3) / Ca2SiO4_alpha-prime(s2) / Ca3Al2O6(s)Ca2SiO4_alpha(s3) / Ca2SiO4_alpha-prime(s2) / Ca3Si2O7_rankinite(s)Ca2Al2SiO7_gehlenite(s) / Ca2SiO4_alpha(s3) / Ca2SiO4_alpha-prime(s2)Ca2SiO4_alpha(s3) / Ca2SiO4_alpha-prime(s2) / CaAl2O4(s)Ca2Al2SiO7_gehlenite(s) / CaAl12O19(s) / CaAl2Si2O8_anorthite(s2)CaAl2Si2O8_anorthite(s2) / Mullite / SiO2_tridy mite(h)(s4)Ca2Al2SiO7_gehlenite(s) / Ca2SiO4_alpha-prime(s2) / Ca3Si2O7_rankinite(s)Ca2SiO4_alpha-prime(s2) / Ca3Al2O6(s) / CaAl2O4(s)Ca2Al2SiO7_gehlenite(s) / Ca3Si2O7_rankinite(s) / CaSiO3_ps-wollastoni(s2)Ca2Al2SiO7_gehlenite(s) / CaAl2Si2O8_anorthite(s2) / CaSiO3_ps-wollastoni(s2)CaAl2Si2O8_anorthite(s2) / CaSiO3_ps-wollastoni(s2) / SiO2_tridy mite(h)(s4)
1:2:3:4:5:6:7:8:9:
10:11:12:13:14:15:16:17:18:19:20:21:22:
2200
2450
2600
FactSage
CaO – SiO2 - Al2O3 Phase Diagram
(liquidus projection)
20
0.1
0.2
0.3
0.4
0.5
0.6
0.10.20.30.40.50.60.70.80.9
0.4
0.5
0.6
0.7
0.8
0.9
AMonoxide
Ca2SiO
4(s3)
Ca2Al
2SiO
7Al
2O
3(s4)
CaAl4O
7
Mullite CaAl2Si
2O
8(s2)
CaSiO3(s2)
Al 2O3 CaOmole fraction
A = SiO2, B = Al2O3, C = CaOX(A) X(B) X(C) oC
0.19519 0.10447 0.70035 1726.550.47779 0.29691 0.22530 1546.560.08289 0.42591 0.49120 1528.670.14285 0.26914 0.58801 1473.680.80684 0.12036 0.07280 1466.660.80594 0.12062 0.07343 1465.340.80489 0.12093 0.07419 1463.790.10406 0.19616 0.69978 1446.510.37459 0.28096 0.34445 1446.500.33102 0.28353 0.38545 1442.810.10677 0.19492 0.69831 1440.700.10420 0.19883 0.69697 1436.850.41814 0.01877 0.56309 1436.850.36868 0.09825 0.53307 1436.850.07917 0.30656 0.61427 1436.850.36977 0.25181 0.37842 1391.940.75011 0.13231 0.11758 1367.110.39348 0.08162 0.52490 1332.880.03612 0.31096 0.65292 1331.630.42135 0.09637 0.48228 1273.450.43677 0.11142 0.45181 1257.430.63835 0.08208 0.27958 1184.29
1:2:3:4:5:6:7:8:9:
10:11:12:13:14:15:16:17:18:19:20:21:22:
1200
1450
1700
1950
T oC
CRCT
Species in Borate Melts
21
CRCT
900
1100
Na2O – B2O3 System:The Na2O – B2O3 Phase Diagram
22
Na 3
B
Na 2
B
NaB
NaB
2
NaB
3
NaB
4
NaB
5
NaB
9
mole B2O3/(Na2O+B2O3)
T(C
)
0.0 0.2 0.4 0.6 0.8 1.0300
500
700
CRCT
Na2O – B2O3 System:
Enthalpies of Mixing in the Na0.5O – BO1.5 System
-30
-20
-10
0
Hervig & Navrotsky, 1985 (700 C)
Fan, 1991 (1027 C)
Shartsis, 1954 (24 C)
Calculated (1027 C)
Yoshio & Takahashi, 1976 (25 C)
23
(kJ/
mol
of c
atio
ns)
-80
-70
-60
-50
-40
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1mole fraction BO 1.5
)()( −+ 333 BONa NaBO2 tetraborate
∆Hm
ix(k
J/m
ol o
f cat
ions
)
CRCT
Enthalpy of Mixing in Liquid Alkaline-earth Borates
24
MgO-B2O3
-50
-30
-10
10
Ent
halp
y of
mix
ing
(kJ/
mol
of o
xide
s)
BaO-B2O3
SrO-B2O3
CaO-B2O3
Mole fraction B2O3
0 0.20 0.40 0.60 0.80 1.00-130
-110
-90
-70
-50
Ent
halp
y of
mix
ing
(kJ/
mol
of o
xide
s)
CRCT
Entropy of Mixing in Liquid Alkaline-earth Borates
25
SrO-B2O3
BaO-B2O3
MgO-B2O3
6
8
10
12
14
Ent
ropy
of m
ixin
g (J
/mol
of o
xide
s)
CaO-B2O3
Mole fraction B2O3
0 0.20 0.40 0.60 0.80 1.00-6
-4
-2
0
2
4
6
Ent
ropy
of m
ixin
g (J
/mol
of o
xide
s)
CRCT
Slag +
Rockett -1965_Liquid
Rockett -1965_Liq+Sol
Rockett -1965_Solids
Pichavant -1978_Liquid
Pichavant -1978_Liq+Sol
Charles -1968
Cristobalite
1400
1600
1800
2000
(liq-liq Miscibility Gap)
B2O3 –SiO2 System: The BO1.5 – SiO2 Phase Diagram
26
Slag + Quartz(H)
Slag + Tridymite
B2O3 + Quartz(L)
Slag
Charles -1968
Slag + Quartz(L)
mole fraction SiO2
T(C
)
0.0 0.2 0.4 0.6 0.8 1.0200
400
600
800
1000
1200(liq-liq Miscibility Gap)
CRCT
B2O3 –SiO2 System: Enthalpy of Mixing in the BO1.5 – SiO2 System at 700°C
27
-1
0
1
2
3
0 0.2 0.4 0.6 0.8 1mole fraction SiO2
(kJ/
mol
)
-10
-9
-8
-7
-6
-5
-4
-3
-2
Hm
ix, k
J/m
ol Hervig & Navrotsky, 1985
Calculated
Shul'ts et al., 1986 (25 C)
∆Hm
ix(k
J/m
ol)
CRCT
1.5 1.5 2
0.6
0.7
0.8
0.9
1
B2O3 –SiO2 System: Activity of BO1.5 in the BO1.5 – SiO2 Melts by Mass-spectroscopy
28
0
0.1
0.2
0.3
0.4
0.5
0.6
0 0.2 0.4 0.6 0.8 1mole fraction SiO 2
Act
ivity
Boike, Hilpert and Muller, 1993 (1202 C)Calculated (1202 C)IdealShul'ts et al., 1986 (1315 C)Activity of SiO2
CRCT
SiO2-B2O3-Na2O Liquidus Projection
29
CRCT
Binary miscibility gaps Reciprocal miscibility gap
SiO2-B2O3-Na2O Subsolidus Miscibility Gap
30
Metastable miscibility gaps
B2O3SiO2
CRCT
Crystobalite
Rockett - 1981Morey - 1951Ghanbari-Ahari - 1993
Slag +
1400
1600
1800
2000
Na2O-B2O3-SiO2 System:Liquidus along the (NaBO2)/2-SiO2 section(close to an ideal liquid solution)
31
Slag + Tridymite
Slag + Quartz(H)
Na2Si2O5 + Na2B4O7 + Quartz(L)Na2B4O7 + NaBO2
Slag + NaBO2
Slag + Quartz(L)
Slag
Na2B4O7 + NaBO2 Slag + Na2B
4O
7
Slag +
Na2Si2O5 +
Na2B
4O
7 + NaBO
2
Na2SiO
3 +
mole fraction SiO2
T(C
)
0 0.2 0.4 0.6 0.8 1
200
400
600
800
1000
1200
1400
CRCT
SiO2-CaO-FeO
Liquidus Projection
32
CRCT
Binary miscibility gaps Reciprocal miscibility gap
SiO2-B2O3-Na2O Subsolidus Miscibility Gap
33
Metastable miscibility gaps
B2O3SiO2
CRCT
P2O5
CaO-FeO-P2O5 System
34
C1
C2
FeO
FeO·P2O5
3CaO·P2O5
CaO20 40 60 80
CRCT
The simplest example:A "regular" solution in which the molecules of each component are randomly distributed on a quasilattice:
Thermodynamic Modeling of Solutions
35
where: xi = mole fraction of component i
g°i = Gibbs function of pure component i
αij = empirical binary parameter of the model
α123 = empirical ternary parameter of the model
......
)lnlnln(
...)()(
321123133132232112
332211
033
022
011
+++++++++
+++=
xxxxxxxxx
xxxxxxRT
gxgxgxmolarg
αααα
CRCT
αij represents the energy of forming i-j nearest neighbour bonds upon mixing.
αij may be expanded as an empirical power series:
36
k
ji
k
ijijxxq )( −∑=α
The empirical parameters are found by fitting experimentaldata.
k
ijq
CRCT
• For molten oxides a more sophisticated model is required.
• We use the Modified Quasichemical Model (MQM).
SiO2 – CaO – MgO – AlO1.5 – FeO – FeO1.5 – …
• Consider a random distribution of second-nearest-neighbor cation pairs.
• Model parameters are the Gibbs energies of the pair-exchange reactions such as:
[Ca-Ca]pair + [Si-Si]pair = 2 [Ca-Si]pair ∆gCaSi < 0
( )0 0G n G n G= + +L
37
( )( )
2 2
0 0
0
2
, = number of moles and Gibbs energy of component i in solution
SiO SiO CaO CaO
configmn mn
n m
i i
G n G n G
T S n g
n G>
= + +
− ∆ + ∆∑
L
where :
nmn = number of moles of [m-n] pairs at equilibrium∆Sconfig = (Ising) entropy for random distribution of pairs = function of ni and nmn
∆gmn = empirical binary model parameters (which are functions of composition and temperature)
(The equilibrium values of nmn are obtained by setting ∂G/∂nmn= 0 at constant ni)Binary model parameters are optimized by fitting experimental data.
CRCT
Structure of CaO-SiO2 Liquid Solutions
CompositionCaO Ca2SiO4 SiO2
mainly Ca2+, O2-, SiO4
4-mainly Ca2+, SiO 4- (orthosilicate ions)
O
Si
OSi
O
Si
OSi
38
SiO44- (orthosilicate ions)
linear + branchedchain polymers
ring polymers
3-dimensionalnetwork
Ca2+ Ca2+
Ca2+
O2-
Ca2+O2-
SiO-
O-
O-
O-
SiO-
O-
O-
O-
Si
Si
Si Si SiSiO O O O
O
O
OSi
O
CRCT
Network Modification by Basic OxidesQ4 structural units (black)Q3 structural units (green)
O0 + O2- = 2 O-
O
SiO O
O
SiO O
O
SiOO
Si Si
O
SiO O
OO O
39
O
SiO
O
SiO O
O
O
SiO O
O
O
SiO O
OO
SiO O
O
SiO O
O O
SiO O
O
O
SiO O
O
SiO O
O
SiO
O
O
O O
O
Si
Si Si
Si
SiSi
Si
Si
O
SiO O
O
O
Ca2+
Na+
Na+
SiO
CRCT
Si-Si p
airSi-S
i pair
T = 1600°C
Pai
r fr
actio
n 0.6
0.7
0.8
0.9
1
Pair Fractions in the Systems
CaO-SiO2 and NaO0.5-SiO2
40
Ca-Si pairNa-Si pair
Na-N
a pairC
a-Ca pair
NaO0.5 - SiO2 system
CaO - SiO2 system
Mole fraction SiO 2 / (SiO2+MeOx)
Pai
r fr
actio
n
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
CRCT
Modeling and Phase Diagram Calculation in Oxide Systems
41
Modeling and Phase Diagram Calculation in Oxide Systems
CRCT
Several thermodynamic software/database packages
with applications in materials science have been developedover the last 30 years.
• These packages all contain large critically evaluateddatabases for thousands of compounds and hundreds ofsolution phases, as well as user interfaces of varying degrees
42
solution phases, as well as user interfaces of varying degreesof user-friendliness.
▫ HSC Chemistry
▫ MTS-NPL
▫ Thermo-Calc
▫ Thermodata
▫ FactSage
CRCT
• Thermochemical databases contain parameters giving the
Gibbs energy, G, of all compounds as functions of T (and P)
and of all solutions as functions of T, (P) and composition. This
is a complete description because all the other thermodynamic
properties (H, Cp, µ, etc.) can be calculated by taking the
appropriate derivatives of the G functions.
43
• For a given set of constraints (such as temperature, total
pressure, total mass of each element) the software calculates
the equilibrium conditions by minimizing the total Gibbs
energy of the system. This is mathematically equivalent to
solving all the equilibrium constant equations simultaneously.
• Data are automatically extracted as required from the
databases.
CRCT 44
CRCT
There are many kinds of chemical thermodynamic data for compounds and solutions:
▫ Calorimetric data:
� Heat capacity
� Solution calorimetry
� Enthalpy of mixing
� …
45
…
▫ Vapour pressures
▫ Phase equilibria:
� Solid/liquid/gas
� Phase diagrams (T-P-Composition)
▫ Chemical potentials or activities
� From electrochemical cells
� From phase equilibria (vapour pressures, isopiestic, …)
� …
▫ …(and so on)
CRCT
These diverse kinds of data are not independent of each
other, but are related through the GIBBS FUNCTIONS of
the phases.
46
the phases.
For each phase (compound or solution):
G = G(T, P, Composition)
CRCT
( )( )
,
( " ")GT
T P Composition
H enthalpy or heat ∂
= ∂ 1
( )pP
dHC heat capacity
dT =
( )G
S entropyT
∂ = − ∂
Then:
47
When phases are in equilibrium:µi (in phase α) = µi (in phase β) for all components i
= µi (in phase γ)= …
…(and so on)
,P compositionT ∂
, , ji T P n
G
n
∂= ∂
(chemical potential of component i of a solution)
µi = (where ni = moles of i)
CRCT
• Therefore, in developing a database for a multicomponent chemical system, one assessesand evaluates ALL the data SIMULTANEOUSLY in order to obtain an optimal Gibbs function, G(T, P, Composition), for each phase.
• The resultant database is then thermodynamically self-
48
• The resultant database is then thermodynamically self-consistent.
• The optimized Gibbs functions are stored in the database(as sets of parameters). All thermodynamic properties and phase equilibria canthen be calculated from these functions.
CRCT
• Step 1): Develop a mathematical model for each solution phase based upon the structureof the solution.The simplest example:a “regular” solution in which the atoms or molecules of each component are randomly distributed.
( )° ° °= + + +
49
( )( )
( )
ln ln
G molar X G X G X G
RT X X X X
X X X X X Xα α α
° ° °= + + +
+ + ++ + + +
1 1 2 2 3 3
1 1 2 2
12 1 2 23 2 3 31 3 1
L
L
L
Where: i
i
ij
X mole fraction of component i
G Gibbs function of pure component i
empirical parameters of the modelα
°
=
==
CRCT
• Step 2):Obtain the parameters of the models by simultaneous critical evaluation/optimization of all available data of all kinds (thermodynamic data, phase equilibria data, structural data) (generally for 2- and 3-component systems.)
• Step 3):Store parameters and use models to estimate properties of N-component phases using optimized 2- and 3-component
50
N-component phases using optimized 2- and 3-component parameters.
• Step 4):Calculate complex phase equilibria by minimization of the Gibbs energy of the total system.
G(total system) = nααααGαααα + + + + nββββGββββ + + + + nγγγγGγγγγ + + + + …Where: nα α α α , , , , nβ β β β , , , , nγ γ γ γ , , , , … = numbers of moles of phases α, β, γ, α, β, γ, α, β, γ, α, β, γ, … at equilibrium
Gαααα, , , , Gββββ, , , , Gγγγγ, , , , …= molar Gibbs function of each phase
CRCT
The FACT OXIDE DATABASEComponents
• Major:▫ (completely evaluated and modeled at all compositions and
temperatures)Al2O3 – CaO – FeO – Fe2O3 – MgO – SiO2 –
• Secondary:
51
• Secondary:▫ (extensively evaluated, particularly with the major
components, and particularly over composition ranges of practical importance)B2O3 – CrO – Cr2O3 – MnO – Na2O – NiO – PbO –Ti2O3 – TiO2 – ZnO – ZrO2 –
• Minor:▫ (evaluated for some combinations with other components)
As2O3 – Cu2O – K2O – SnO
CRCT
The FACT OXIDE DATABASE
Liquid Solution
52
• Modeled for all oxide componentsAlso: Non-oxide components (in dilute solution)
S, SO4, PO4, CO3, H2O, OH, F, Cl, Br, I
> 150 Solid Stoichiometric Compounds
CRCT
The FACT Oxide DatabaseMajor Oxide Solid Solutions• Spinel: (Al, Co2+, Co3+, Cr2+, Cr3+, Fe2+, Fe3+, Mg, Ni2+, Zn)
[Al, Co2+, Co3+, Cr3+, Fe2+, Fe3+, Mg, Ni2+, Zn, ���� ]2 O4
• Pyroxenes: (Ca, Fe2+, Mg)M2 (Fe2+, Fe3+, Mg, Al )
M1 (Fe3+, Al, Si)B SiA O6
• Olivine: (Ca, Fe2+, Mg, Mn, Ni, Co, Zn) [Ca, Fe2+, Mg, Mn, Ni, Co, Zn] SiO4
• Melilite: (Ca)2 [Mg, Fe2+, Fe3+, Al, Zn] {Fe3+, Al, Si}2O7
• Monoxide: CaO - MgO - MnO - CoO - NiO - FeO
53
• Monoxide: CaO - MgO - MnO - CoO - NiO - FeO
(+ Fe2O3 - Al2O3 - ZnO - Cr2O3)
• αααα-Ca2SiO4: αααα-Ca2SiO4 ( + Fe2SiO4, Mg2SiO4, Mn2SiO4)
• αααα’-Ca2SiO4: αααα’-Ca2SiO4 ( + Fe2SiO4, Mg2SiO4, Mn2SiO4 , Pb2SiO4 , Zn2SiO4)
• Wollastonite:CaSiO3 ( + FeSiO3, MgSiO3, MnSiO3)
• Corundum: Al2O3 - Cr2O3 - Fe2O3
• Ilmenite: (Fe2+, Mg, Mn, Ti3+) (Ti4+, Ti3+)O3
• Pseudobrookite: (Fe2+, Mg, Mn, Ti3+) (Ti4+, Ti3+)2O5
• 26 other solid solutions
CRCT
Sublattice Model – Compound Energy Formalism
• Used for solid solutions
▫ Example:
Spinel Solution (A2+, A3+, B2+, C3+, …)[A2+, A3+, B2+, C3+, …]O4config E
i j iji j
G Y Y G TS G′ ′′= − +∑∑
54
: :
: ,
ln 2 ln
i j
configi i j j
i j
Ei j k ij k k i j k ij
i j k i j k
where Y Y site fractions on first and second sublattices
S R Y Y Y Y configurational entropy
G Y Y Y L Y Y Y L
′ ′′ =
′ ′ ′′ ′′= − + =
′ ′ ′′ ′ ′′ ′′= +
∑ ∑
∑∑∑ ∑∑∑
(usually GE = 0 or contains only a very few small parameters Lijk )
Gij = “end-member Gibbs energies” (one for each ion pair)
The Gij are the main parameters of the formalism.
CRCT
(Sublattice Model – Compound Energy Formalism)
• End-member Gibbs energies Gij are the formalism parameters.• Some may be equal to Gibbs energies of real compounds,• In other cases, certain linear combinations of the Gij have physical
significance, and these combinations are the model parameters.▫ Example:
Fe3O4 spinel: (Fe2+, Fe3+) [Fe2+, Fe3+]2O4
� Four end-member Gibbs energies:
55
2 3
3 3 3 2 2 3
2 2 3 3 3 2 2 3
3 4Gibbs energy of hypothetical normal Fe O
2
model parameter related to degree of inversion
=
model parameter related to reciprocal exchan
Fe Fe
Fe Fe Fe Fe Fe Fe
Fe Fe Fe Fe Fe Fe Fe Fe
G
I G G G
G G G G
+ +
+ + + + + +
+ + + + + + + +
=
= + −
=∆ + − −
=
( )3 2 2 3
ge of
nearest-neighbor pairs
can be arbitrarily set equal toFe Fe Fe Fe
G G+ + + +
2 2 2 3 3 2 3 3, , ,Fe Fe Fe Fe Fe Fe Fe Fe
G G G G+ + + + + + + +
� Four end-member Gibbs energies:
CRCT
Quasichemical Model for Short-Range Ordering
• Used for the liquid solutionSiO2 – CaO – MgO – AlO1.5 – FeO – FeO1.5 – …
- Consider a random distribution of second-nearest-neighbor cation pairs.- Model parameters are the Gibbs energies of the pair-exchange reactions such as:
[Ca-Ca]pair + [Si-Si]pair = 2 [Ca-Si]pair ∆gCaSi < 0
( )0 0G n G n G= + +
56
( )( )
2 2
0 0
0
2
, = number of moles and Gibbs energy of component i in solution
SiO SiO CaO CaO
configmn mn
n m
i i
G n G n G
T S n g
n G>
= + +
− ∆ + ∆∑
L
where :
nmn = number of moles of [m-n] pairs at equilibrium∆Sconfig = (Ising) entropy for random distribution of pairs = function of ni and nmn
∆gmn = binary model parameters (which may be functions of composition and T)
(The equilibrium values of nmn are obtained by setting ∂G/∂nmn= 0 at constant ni)
CRCT
Some of the data critically evaluated and used in the modeling of the
57
and used in the modeling of the SiO2 – Al2O3 – CaO – FeO – Fe2O3
system, and comparison with calculations from the resultant
optimized database.
CRCT
Calculated Fe-Si-O
phase diagram in equilibrium with iron
Slag1 + Slag2
Slag + Cristobalite
Zhao et al.
AllenSchurmann
SchuhmanBowen
Greig
1371oC
1465oC
1670oC
1723oC
0.970.53
0.44
Tem
pera
ture
, o C
1400
1600
1800
58
Slag + Tridymite
Olivine + Tridymite
Olivine + Quartz
Wustite + Olivine
Wustite + Slag
Slag
1188oC
1371oC
1205oC
1187oC
867oC
0.370.22
Fe 2
SiO
4
mass SiO2/(FeO+SiO2)
Tem
pera
ture
,
0 0.2 0.4 0.6 0.8 1800
1000
1200
1400
CRCT
Calculated Fe-Si-O
phase diagram in equilibrium with air
Slag1 + Slag2
Slag
Muan
Greig
1672oC
1723oC
o
0.330.96
o C
1600
1700
1800
59
Slag + Cristobalite
Slag + Tridymite
Spinel + Tridymite
Hematite + Tridymite
Spinel + Slag
1389oC
1444oC
1465oC
1595oC
0.17
mass SiO2/(Fe2O3+SiO2)
Tem
pera
ture
, o
0 0.2 0.4 0.6 0.8 11300
1400
1500
1600
CRCT
Calculated Ca-Fe-O
phase diagram in equilibrium with iron
Lime + Slag
Zhao et al.Abbatista
Allen and SnowLarson
TakedaTimucin
o
o C
1400
1600
1800
60
Slag
Lime
Wustite
Ca2Fe2O5 + WustiteLime + Ca 2Fe2O5
Lime+Wustite
Lime + Slag
Ca 2
Fe 2
O5
1059oC
1125oC
1371oC
0.760.16
0.13
0.67
0.72
mass FeO/(CaO+FeO)
Tem
pera
ture
, o
0 0.2 0.4 0.6 0.8 1800
1000
1200
1400
CRCT
Calculated Ca-Fe-O
phase diagram in equilibrium with air
Lime + Slag
Slag
Phillips and MuanHaraTakeda
+ SpinelSlag
Lim
e 1438oC
1443oC
1595oC
0.580.9o C 1400
1500
1600
1700
61
Lime+Ca 2Fe2O5
Ca2Fe2O5
CaFe2O4+Hematite
CaFe4O7
Slag+Hematite
+ Spinel
Ca 2
Fe 2
O5
CaF
e 2O
4
+ Hematite
Ca2Fe2O5 + Slag
+ CaFe2O4
CaF
e 4O
7
Lim
e
1157oC
1220oC
1216oC 0.79
1389oC
0.9
mass Fe2O3/(CaO+Fe2O3)
Tem
pera
ture
, o
0 0.2 0.4 0.6 0.8 1900
1000
1100
1200
1300
1400
CRCT
Calculated FeO-Al2O3
phase diagram in equilibrium with iron
Slag
Slag + CorundumElrefaieTurnockAtlas
Fisher and HoffmanNovokhatskii et al.
Hay et al.Rosenbakh et al.
Oelsen and Heynert
1783oC0.70
2054oC
0.48
o C 1600
1800
2000
2200
62
Slag + Spinel
Wustite + Spinel Spinel + CorundumS
pine
l
1335oC 0.43
0.04
mole Al2O3/(FeO+Al2O3)
Tem
pera
ture
, o
0 0.2 0.4 0.6 0.8 1600
800
1000
1200
1400
1600
CRCT
Calculated Al-Fe-O
phase diagram in equilibrium with air
Slag + Corundum
Slag
SpinelSpinel + Corundum
Slag + Spinel
Muan and GeeRichardsTurnockAtlas
1703oC
0.750.48
2054oC
1595oC
0.96
Tem
pera
ture
, o C 1600
1800
2000
63
Spinel
Spinel + Al 2Fe2O6
Corundum
Hematite
Hematite + Corundum
Corundum + Al 2Fe2O6Hematite + Al 2Fe2O6
1318oC
1382oC
1414oC
0.17 0.87
0.880.22
0.16
mole Al2O3/(Fe2O3+Al2O3)
Tem
pera
ture
,
0 0.2 0.4 0.6 0.8 1600
800
1000
1200
1400
CRCT
Calculated Al-Fe-O
phase diagram at 1500°C
Spinel
Corundum + Spinel
Cor
undu
m
, atm
)
-4
-2
0
64
Spinel + Slag Slag
Corundum + Fe
Spinel + Fe
MayersRoiterMuan and Gee (after Roiter)Darken and Gurry (after Roiter)
mole Fe/(Al+Fe)
log 10
(PO
2, atm
)
0 0.2 0.4 0.6 0.8 1-14
-12
-10
-8
-6
CRCT
T, K
1400
1500
1600
1700
1800
1900
2000
Mag
netit
e
Hem
atite
SlagFe-liq+Slag
Fe-bcc+Slag
Fe-fcc+Slag
Fe-fcc+Wustite
1702
1801
1667
1644
1818
1863
10-210-4
10-6
10-8
1
10-4
1
10-2
10-10
10-12
10-6
10-8
FeO-Fe2O3 phase diagram
65
Weight % Fe2O3
0 10 20 30 40 50 60 70 80 90 100
T
800
900
1000
1100
1200
1300Wustite
FeO Fe2O3
827
10-6
10-12
10-8
10-10
10-16
10-28
10-24
10-12
10-16
10-20
[59]
[57]
[61]
[58]
[63][62]
[64][65][66][67]
[60]
Selected experimental points and calculated lines and invariant temperatures. Dashed lines are calculated oxygen isobars (bar).
CRCT
Fe–O System
66
Oxygen partial pressure for two-phase equilibria with magnetite in the Fe-O system.
CRCT 67
Experimental and calculated oxygen partial pressure over single-phase magnetite as a function of composition.
CRCT
Liquidus of the Ca-Fe-Si-O
system in equilibrium with iron
0.8
0.9 0.1
0.2
SiO2
Zhao et al.
Allen and Show
SiO2
Calculated
Bowen et al.
SiO2SiO2
Slag + SiO
Temperatures between 1200°C and 1650°C
68
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.10.20.30.40.50.60.70.80.9
0.3
0.4
0.5
0.6
0.7
0.8
0.9
CaO FeOweight fraction
CaO FeOweight fraction
CaO FeOweight fraction
CaO FeOweight fraction
Slag + Lime
Slag + Ca 2SiO 4
Slag + SiO2
1650 oC
1400 oC
1300 oC
1200oC
1300 oC
1650 oC
Slag
Ca 3SiO5
Ca 2SiO4
Ca3Si 2O7
CaSiO3
Fe 2SiO 4
Görl et al.
CRCT
Liquidus of the Ca-Fe-Si-O
system in equilibrium with air
0.8
0.9 0.1
0.2
SiO2
Phillips and Muan
SiO2SiO2
Zhao et al.BurdickZhang et al.
Calculated
Temperatures between 1300°C and 1450°C
69
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.10.20.30.40.50.60.70.80.9
0.3
0.4
0.5
0.6
0.7
0.8
0.9
CaO Fe2O3weight fractionCaO Fe2O3weight fractionCaO Fe2O3weight fraction
Slag
1300 oC
1350 oC
Ca3SiO5
Ca2SiO4
Ca3Si2O7
CaSiO3
Ca2Fe2O5 CaFe2O4 CaFe4O7
1450 oC
CRCT
Liquidus of the Al-Ca-Fe-O
system in equilibrium with air
0.8
0.9 0.1
0.2
Al 2O3Al 2O3Al 2O3Al 2O3
Calculated
Dayal and Glasser
Swayze
Newkirk and Thwaite
Al 2O3Al 2O3
CaAl 4O7
CaAl 12O19
At 1400°C and 1500°C
70
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.10.20.30.40.50.60.70.80.9
0.3
0.4
0.5
0.6
0.7
0.8
0.9
CaO Fe2O3weight fractionCaO Fe2O3weight fractionCaO Fe2O3weight fractionCaO Fe2O3weight fractionCaO Fe2O3weight fractionCaO Fe2O3weight fraction
Slag1500 o
C
1400 oC
1500 oC
Ca2Fe2O5CaFe 2O4 CaFe4O7
Ca 3Al 2O6
CaAl 2O4
1400 oC
CRCT 71
CRCT
LOliv+LSp+LSp+Oliv+LSp+Oliv+MW
Muan and Osborn[49]MW+L L
Oliv+MW
Oliv+MW+Sp
Oliv
+S
p
Oliv+Sp+L
Oliv+MW+L
Tem
pera
ture
, o C
1500
1600
1700
1800
72
Calculated section of the Fe2O3-MgO-SiO2-O2 phasediagram in air at SiO2/(MgO+Fe2O3+SiO2) = 20 weight %
Oliv+MW+LSp+Oliv+MWOliv+L
Ambruz et al.[50]
Correia and White[48]Oliv+MWSp+Oliv+MWOliv+MW+LSp+Oliv+LSp+Oliv
Oliv
+S
p
Sp+L
Sp+Py
Sp+T
r
Sp+T
r+H
em
Sp+Py+Tr
1375
1339
1247
Py+Hem Py+Tr+Hem
weight percent Fe2O3
Tem
pera
ture
,
0 10 20 30 40 50 60 70 801200
1300
1400
1500
CRCT
• Inputs to the FactSage module required to generate the preceding diagram
• (data retrieved automatically from database)
73
CRCT
(1) Components
74
CRCT
(2) Species Selection
75
CRCT
(3) Defining Variables
76
CRCT
CaO – MgO – SiO2 , 1400°C
0.7
0.8
0.9 0.1
0.2
0.3
SiO2SiO2(s4) (tridymite)
CaO - MgO - SiO 2
1400oC
77
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.10.20.30.40.50.60.70.80.9
0.3
0.4
0.5
0.6
0.7
0.8
0.9
CaO MgOmole fraction
Orthopyroxene
Protopyroxene
Clinopyroxene (pigeonite)
Olivine (forsterite)
Monoxide (periclase)
Monoxide (lime)
ASlag-liq
ASlag-liq
CaSiO3(s2)(pseudowollastonite)
Ca3Si2O7
(rankinite)
Aa'Ca2SiO4
(alpha - Ca2SiO4)
Ca3MgSi2O
8 (merwinite)
Olivine(monticellite)
Ca2MgSi2O7
(akermanite)
CRCT
Calculations using the Oxide Database in Conjunction with other Databases (such as Databases for Metallic Phases, Gases, Sulfides, etc.)
78
CRCT
Pyrrhotite
Pyr
rhot
ite +
Pyrrhotite
Spinel +
M 2O
3(c
orun
dum
)
+ Spinel
Fe - Cr - S2 - O21000°C, mole Fe/Cr = 1
) (a
tm)
-6
-4
79
Spinel
fcc +
Spinel + Monoxide
Pyrrhotite + fcc
Pyrrhotite
bcc +
fcc +
Spinel +
M2O3(corundum)
M2O3+ bcc
Liquid_Sulfide
Spinel
bcc (corundum)
log 10P(O2) (atm)
log
10P
(S2)
(at
m)
-25 -20 -15 -10-12
-10
-8
CRCT
Equilibrium phase composition generated by placing cursor at the
point P(S2) = 10-7 , P(O2) = 10-15 in the preceding diagram and clicking
log10(p(S2)) (atm) = -7.00, log10(p(O2)) (atm) = -15 1000.00 C=================================================
0.00 mol (0.99294E-07 S2+ 0.33664E-07 SO2+ 0.49582E-08 SO+ ·
··
+ 0.10000E-14 O2+ ·
··
80
··
(1 atm, gas)
+ 0.32583 mol Spinel Mole fraction of sublattice constituents in Spinel:Fe[2+] 0.87607 Stoichiometry = 1.0000Fe[3+] 0.12392Cr[3+] 0.30106E-05Cr[2+] 0.34690E-05------------------------Fe[2+] 0.61961E-01 Stoichiometry = 2.0000Fe[3+] 0.17195Va[0] 0.19186E-08Cr[3+] 0.76609
+ 0.21232E-01 mol ( 0.94037 FeO+ 0.41724E-01 Fe2O3+ 0.17901E-01 Cr2O3 ( Monoxide)
CRCT
Conclusions
• Thermodynamic databases have been developed which contain
critically evaluated and modeled data for thousands of compounds and
hundreds of multicomponent solutions.
• The evaluated data are generally reproduced by the models within the
experimental error limits.
81
• The models permit extrapolation into regions of temperature and
composition where data are unavailable.
• The databases are automatically accessed by user-friendly software that
calculates complex multiphase equilibria in large multicomponent
systems for a wide variety of input/output constraints.
• The databases contain parameters of models specifically developed for
different types of solutions involving sublattices, ordering, etc.
