1. 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. 1

2. CaO-MgO: phase diagram2

3. Diagram: CaO-Al2O3-SiO2 3

4. C-Fe-W System at 5 wt% W 4

5. A Phase Diagram for the Fe-Cr-V-C System5

6. Fe-Cr-V-C system at 1.5 wt% Cr and 0.1 wt% V6

7. P–T diagram for Al2TiO5 7

8. Fe-O2 System 8

9. System Fe-Cr-O2 9

10. SO2-O2-Cu System at 1000 K 10

11. Fe2O3-MgO-SiO2-O2 System SiO2/(Fe2O3+ MgO + SiO2) = 0.20 in air11

12. Fe-Cr-S-O System 12

13. A Phase Diagram for the Fe-Cr-V-C System13

14. The Law of Adjoining Phase Regions
«As a phase boundary line is crossed, one and only one phase appears or disappears.» 14

15. (a1 + a2 + … + an) (a1 + a2 + … + an) (a1 + a2 + … + an) + b + g
A node in a general true phase diagram section. 15

16. CaO-MgO: phase diagram16

17. T XB B A L b + L a + b + L a + L a b a + b
An isobaric binary temperature-composition diagram with the eutectic “opened-up” to show that this is an infinitely narrow 3-phase region. 17

18. SO2-O2-Cu System at 1000 K 18

19. f1
a + b f2
b + g
a + g
a + b + g a b g
A “potential-potential” phase diagram “opened-up” to show that the lines are infinitely narrow 2-phase regions. 19

20. Potential Variables Extensive Variables T (temperature) P (pressure)
(the same for all phases at equilibrium)
T (temperature)
P (pressure)
mi (chemical potential)
Extensive Variables
(not the same for all phases at equilibrium)
Xi (composition)
V (volume) 20

21. 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. 21

22. Zero Phase Fraction (ZPF) Lines
MC fcc bcc M7C3 M23C6 22

23. Zero Phase Fraction (ZPF) Lines
LIQUID
LIQUID + a
L+b
SOLID a
SOLID b
2 SOLIDS (a + b)
Solidus
Liquidus
Solvus line
a LIQUID b 23

24. Fe-S-O Predominance diagram (ZPF lines)24

25. Choice of variables to always give a true phase diagram (single-valued) everywhere25

26. N-Component System (A-B-C-…-N)
Gibbs-Duhem:
Phase Diagram 26

27. Choice of variables N-component system
(1) Choose n potentials: f1, f2, … , fn
(2) From the non-corresponding extensive variables (qn+1, qn+2, … ), form (N+1-n) independent ratios (Qn+1, Qn+2, …, QN+1).
Example:
[f1, f2, … , fn; 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.
Phase Diagram 27

28. MgO-CaO Binary System S T V -P nMgO mMgO nCaO mCaO f1 = T f2 = -P
Phase Diagram 28

29. CaO-MgO: phase diagram29

30. Fe-Cr-S-O System S T V -P nFe mFe nCr mCr f1 = T f2 = -P
Phase Diagram 30

31. Fe-Cr-S-O System 31

32. Pressure vs. Volume diagram for H2O
S+L
L+G
S+G P V
This is NOT a true phase diagram.
Phase Diagram 32

33. Fe-Cr-C System S T V -P nC mC nFe mFe nCr mCr Requirement: f1 = T
f2 = -P
f3 = mC
(NOT OK)
(OK)
Requirement:
Phase Diagram 33

34. This is NOT a true phase diagram.
Fe-Cr-C system, T = 1300 K, XCr = nCr/(nFe+nCr+nC) vs. (carbon activity)
M23C6
M7C3
bcc
fcc
cementite
log(ac)
Mole fraction of Cr
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0 -3 -2 -1 1 2
This is NOT a true phase diagram.
Phase Diagram 34

35. “Corresponding” phase diagrams of the Fe-O system
a-Iron
g-Iron
d-Iron
Liquid
iron
Liquid oxide
Wustite
Magnetite
Hematite
-500
-400
-300
-200
-100
RT ln PO2 (kJ) +
a-Iron + Magnetite
Oxygen
Liquid iron
Liquid iron + liquid oxide
700
900
1100
1300
1500
1700
1900
0.50
0.54
0.58
0.62
Mole fraction XO
Temperature (K) 35

36. Other Sets of Extensive Variables and Corresponding Potentials
Gibbs-Duhem:
Gibbs-Duhem: 36

37. Mg-Si binary phase diagram37

38. Mg-Si, DH vs XSi phase diagram38