1. Phase diagrams of unary and binary systems
Lection 4
Phase diagrams of unary and binary systems
Phase diagrams for unary systems, relations with the Gibbs energy, critical point;
Binary systems, relations with the Gibbs energy of phases, lever rule, types of phase diagrams: eutectic, peritectic, eutectoid, congruent transformation, monotectic etc., relations with microstructure development
Phase diagram is a chart showing conditions (pressure, temperature, composition , volume etc.) at which thermodynamically distinct phases occur and coexist in equilibrium.
Phase diagram consist of lines of equilibrium or phase boundaries where change of phase assemblage occurs (one phase disappears or occurs).

2. Phase diagrams of unary systems
Phase diagram of H2O
Gibbs phase rule:F=K-P+2
K=1 F=3-P
OB is univariant line Vapor + Liquid
OC – Vapor + Solid
OA – Solid + Liquid
Point O invariant equilibrium
Liquid + Solid + Vapor

3. Phase diagram of unary systems
Phase diagram of sulphur

4. Relation between the Gibbs energy of phases and phase diagram

5. Critical point
Critical point is the end of a phase equilibrium curve. The liquid-vapor boundary terminates in critical point at Tc and Pc (critical temperature and pressure). In the vicinity of the critical point the physical properties of liquid and vapor phases change dramatically and both phases become very similar. Above critical point there exists a state of matter called supercritical fluid.

6. dP = S  dT V  Slope of univariant lines
Slope of melting curve:
dP/dT > 0 (pos.), when V > 0 („normal“)
dP/dT < 0 (neg.), when V < 0 (z. B., H2O, Ge, Bi)
ΔV: V (liquid) – V (solid)
Substance
Transformation
T(K)
H (kJ/mol)
V  (cm3/mol)
V  (cm3/mol)
dT/dP (K bar-1) Bi
Melting
544.59
11.297
21.5
20.8 –
BF3
281.95
12.029
42.39
48.16
H2O
273.15
6.009
19.65
18.02 –
Evaporation
372.78
40.671
18.76
30.14

7. Binary systems According to the Gibbs phase rule: F=K-P+2 K=2 F=4-P
at fixed pressure 3 phases can coexist in equilibrium
1. Continuous solid and liquid solutions
A solid solution is a crystalline material with variable composition
Two types:
substitutional (solute atom/ion replaces solvent atom/ion;
Interstitial (solute atom/ion occupies interstitial site;
Example : Al2O3-Cr2O3 (substitutional, continuous);
Fe-C (interstitial, partial)
a) substitutional, (b) interstitial solid solutions

8. Lever rule for binary systems
Liquid
solid solution n L
a- solid solution
na – amount of phase a
nb – amount of phase b
XaB – mole fraction of component B in phase a
XbB – mole fraction of component B in phase b
XB – mole fraction of component B in alloy X

9. Phase diagrams of binary systems with maximum/minimum

10. Phase diagrams of binary systems with miscibility gap (1), ordering (2) and without mutual solubility of components (3)
(1)
(2)
(3)

11. Eutectic type of diagram
Eutectic reaction L=a+b
For eutectic solidification both phases form directly from liquid; i.e. locally one has La and Lb. Thus the necessary solute redistribution occurs in the liquid ahead of the individual interfaces, which are in close proximity. Redistribution of components occurs through diffusion in liquid.

12. Different types of eutectic microstructures
More complex arrangements of the two phases occurs if interface attachment kinetics are sluggish (usually encountered for crystals that grow from liquid with crystallographic facets). Then two solid phases grow independently from the melt with very little communication of the solute fields in the liquid. This leads to much coarser mixture of the two solid phases (divorced eutectic).
a - globular eutectic
b – acicular (needle-like) eutectic
c - lamellar eutectic
d – Chinese script

13. Eutectic type of diagrams

14. Eutectic type of diagrams
L=a+b

15. Peritectic reaction L+ba
It requires the complete disappearance of b phase, a process that involves solute diffusion in two solid phases at peritectic temperature. The kinetics is different from eutectic because the diffusion rate is very different in liquid and substitutional solids. If only interstitial diffusion is required the peritectic reaction occurs more easily.
If one assume that no diffusion occurs in the solid upon cooling, solidification merely switch from freezing of high temperature phase L b to freezing of low temperature phase L a. Then b phase usually surrounds a phase resulting in coarser two phase microstructure than eutectic one.

16. Peritectic type of diagram

17. Peritectic type of diagram
L+a=b

18. Phase diagrams with eutectoid and congruent phase transformation
 =  + 
Congruent phase transformation:
 = 
 = b

19. Types of binary diagrams
Liquidus line
Critical point
Monotectic line
Monotectic reaction
Metatectic line
Metatectic reaction
Syntectic line
Syntectic reaction

20. The types of invariant reactions in a binary equilibrium phase diagram at constant pressure

21. Typical Microstructure
Monotectic diagram
Miscibility gap in liquid state
Typical Microstructure
Phase diagram Solidification process

22. Phase diagrams with intermediate compound
Congruently melting phase v 
Incongruently melting phase v 
The system A-B contains one peritectic reaction (L +  = v) and one eutectic reaction (L = v + )
System A-B consists from two formally eutectic subsystems A-v and v-B
Phase diagrams with intermediate compound
Phase diagram with the compound and the Gibbs energy curves

23. Binary diagrams and microstructure of alloys
Microstructure of Cu-Cd alloy. Primary Cu5Cd8 (white) participating in peritectic reaction is surrounded by the second phase (CuCd3, grey). Remaining liquid crystallizes as eutectics (CuCd3 + Cd).
Different kinds of phase diagrams with compound

24. Binary system
The system A-B contains 3 intermediate phases and one polymorphic transformation
 6 invariant equilibria
1 Metatectic ( =  + L)
2 Eutectic (L =  + v1)
(L = v3 + )
2 Peritectic (L + v1 = v2)
(L + v2 = v3)
1 Eutectoid (v1 =  + v2)

25. Incorrect features of phase diagrams
Single phase fields can meet only in one point and never along a phase boundary. Two liquidus or two solidus lines (or two solvus or solvus and solidus of the same phase) must meet in one composition at invariant temperature
Two neighboring single phase fields are separated by a two-phase field
Each three-phase equilibrium produces three two-phase fields
Two three-phase isotherms can be connected through one two-phase field if they have two common phases
All metastable continuations of phase boundaries must extrapolate into two-phase field after crossing of invariant point
Methods for phase diagram determination L.C. Zhao (Ed) Elsevier 2007