1. Potential diagrams
Chemical potential, intensive and extensive parameters
Types of phase diagrams
Unary diagrams
Binary diagrams
Ternary diagrams
Volatility diagram and Ellingham diagram
Schreinemakers rules Fe
Fe-O
Al-Ni-O

2. Equilibrium condition is G=min or dG=0 and d2G>0
dG= -SdT+VdP+SmidNi
S - entropy, V- volume, mi - chemical potential of component i, Ni - number of moles of component i
where Gm=G/N Gibbs energy
per mole of phase , xi – mole fraction Ni/N
mi=m0i+RT∙lnai, where ai is activity ai=gi∙xi
Extensive variables: S, V, ni
Intensive variables (potentials): T, P, mi

3. 1. Potential diagrams – two potentials
Types of diagrams:
1. Potential diagrams – two potentials
2. Mixed diagrams – one potential and one ratio of extensive properties
3. Extensive properties diagrams – two ratios of extensive properties
Fe-O
Examples
1. P-T diagram, T-activity Fe
Al-Mg
Al-Mg-Si
2. T-x diagram,
vertical sections
3. Isothermal sections xi-xj,
liquidus projections

4. Other classifications
Sections
2. Projections
Phase relations
2. Property diagrams
How to control chemical potential experimentally?
Equilibria in gas mixtures:
2H2+O2=2H2O DG0+RTln(pH2O)2/(pO2·pH22)=0
2CO+O2=2CO2 DG0+RTln(pCO2)2/(pO2·pCO2)=0

5. Unary systems
Relations between P – T and P – V diagram G

6. Examples: of P-T diagrams: H2O

7. Phase diagrams of Fe

8. Potential phase diagrams for binary systems and their relations with other types of phase diagrams
Liq+Fcc
T=1500 K
Reference
Ni - Fcc
Cu - Liq
Liq
Fcc Cu Ni
Reference
Ni – fcc
Cu - fcc

9. Binary systems
Example Mg-Al

10. Binary systems
Example: Fe-O
Sundman (1991), Kjellqvist et al. (2008)

11. Binary systems
Example: Fe-O
Activity of oxygen (in form of oxygen partial pressure) vs. x(O) along with isothems
T-x(O) diagram along with oxygen isoactivity lines in form of lgP(O2)

12. Ternary diagrams. Example Al-Ni-O: phase diagrams at 1100 K
(c)
Relations between activity of O and Ni
Activity of O vs. Ratio Ni/(Ni+Al);
1 - Al3Ni2, 2 – Al3Ni2+b
c. Isothermal section
1 - Al2O3+Al3Ni2, 2 – Al2O3-Al3Ni2+b

13. How to calculate potential phase diagram
1. Set reference of components and calculate isothermal section at given temperature
2. Calculate phase equilibrium at initial composition
3. Determine mi (or lnai), then instead of using xi as a condition use mi (or lnai) and calculate equilibrium again
4. Determine ranges mi or (lnai) in where phase diagram should be calculated
5. Calculate phase diagram in selected ranges of mi or lnai
m(O)=1/2RT·lnP(O2)
lnx/log10x=ln10=
m(O)=1/2RT·2.3026·log10P(O2)
log10P(O2)=2/(2.3026·RT)m(O)

14. Volatility diagram for the system Si-O
Volatility diagram is isothermal plot showing partial pressures of two gaseous species in equilibrium with several condensed phases possible in a system. Volatility diagrams are not phase diagrams. The lines simply represent thermodynamics of various gas-solid reactions.
Si(l)+O2=SiO2(l) (1)
Si(l)=Si(g) (2)
SiO2(l)=Si(g)+O2 (3)
DG10+RTln(1/PO2)=0
DG10-RTlnPO2=0
DG10/(RT)=lnPO2
DG10/(2.3026∙RT)=lgPO2
x=a
DG20+RTlnPSi=0
-DG20/(RT)=lnPSi
-DG20/(2.3026∙RT)=lgPSi
y=b
DG30+RTln(PO2∙Psi)=0
-DG30/(RT)=lnPO2+lnPSi
-DG30/(2.3026∙RT)=lgPO2+lgPSi
c=x+y
y=c-x

15. Volatility diagram for the Si-O system at 1900 K
Minor species are neglected; maximum equilibrium pressure lines
lgK
Si(l)+O2=SiO2(l) (1)
Si(l)=Si(g) (2)
SiO2(l)=Si(g)+O2 (3)
2Si(l)+O2=2SiO(g) (4) 13.86
lgK
2SiO2(l)=2SiO(g)+O2 (5)
SiO2(l)+Si(l)=2SiO(g) (6)
Si(l)+O2=SiO2(g) (7)
SiO2(l)=SiO2(g) (8)

16. Phase diagram of the Si-O system
Hallstedt (1992)
Calculated phase diagrams (T-x and log10PO2-T) at P= Pa

17. Ellingham type diagram
A major application of Ellingham diagram is the determination of the conditions required to reduce metal compounds (i.e. oxides) to obtain pure metal.

18. 2M+O2=2MO (1)
DG1=DG10+RTlna2(MO)/(a2M∙aO2)=0
-DG10/(RT)=lna2(MO)/(a2M∙aO2)
DG10/(RT)=lnPO2
2CO+O2=2CO2 (2)
DG2=DG20+RTlnP2CO2/(P2CO∙PO2)
-DG20/(RT)=2lnPCO2/PCO-lnPO2
lnPCO/PCO2=DG20/(2RT)-0.5lnPO2

19. Ellingham-type diagram and volatility diagram for the Si-O system
They are complementary diagrams.
Si(l)+O2=SiO2(l) (1)
(C) SiO2(l)=SiO2(gas)
(D) Si(l)=Si(gas)
Si(l)+O2=SiO2(l) passive oxidation
2Si(l)+O2=2SiO(g) active oxidation
P(O2)-T dependences at fixed P(SiOx)
M - melting point of Si; between lines C and D SiO is major gas species (between C and (1) gas is over SiO2(l), between (1) and D over Si(l))
SiO2(l)
Si(l)
A.H. Heuer, V.L.K. Lou (1990)

20. Schreinemakers rules
Schreinemakers method is geometric approach to determine relationships of reaction curves that intersect in invariant point. This method produces topologically correct sequence of reaction curves around invariant point
n-component system
There n+2 phases in invariant point
In general case, there are n+2 univariant curves with n+1 phases involved in univariant reaction. Univariant curves intersect in invariant point
There are n+2 divariant fields with n phases stable in each field.
Special case is degenerate reaction described with fewer components than overall system. Degenerate reactions can be a) collinear on opposite side of invariant point; b) super-imposed on top of each other and one line can represent reaction with two or more phases absent
Univariant curves intersect in invariant point and on the opposite side from invariant point the reaction is metastable and absent phase is stable
Angle between univariant curves can not be more than 180°

21. Schreinemakers rules MgO-CaO-SiO2 Al2SiO5 (Sil) And=Ky (And) Sil=Ky
(Qz) (Ak) En+Wo=Di
(En) Ak+Q=Di+Wo
(Wo) Ak+Qz+En=Di
(Di) Ak+Qz=Wo+En

22. Schreinemakers rules
Step 1