1. Engr 2110 – Chapter 9 Dr. R. R. Lindeke
Phase Equilibrium
Engr 2110 – Chapter 9
Dr. R. R. Lindeke

2. Topics for Study Definitions of Terms in Phase studies Binary Systems
Complete solubility (fully miscible) systems
Multiphase systems
Eutectics
Eutectoids and Peritectics
Intermetallic Compounds
The Fe-C System

3. Some Definitions
System: A body of engineering material under investigation. e.g. Ag – Cu system, NiO-MgO system (or even sugar-milk system)
Component of a system: Pure metals and or compounds of which an alloy is composed, e.g. Cu and Ag or Fe and Fe3C. They are the solute(s) and solvent
Solubility Limit: The maximum concentration of solute atoms that may dissolve in the Solvent to form a “solid solution” at some temperature.

4. Definitions, cont
Phases: A homogenous portion of a system that has uniform physical and chemical characteristics, e.g. pure material, solid solution, liquid solution, and gaseous solution, ice and water, syrup and sugar.
Single phase system = Homogeneous system
Multi phase system = Heterogeneous system or mixtures
Microstructure: A system’s microstructure is characterized by the number of phases present, their proportions, and the manner in which they are distributed or arranged. Factors affecting microstructure are: alloying elements present, their concentrations, and the heat treatment of the alloy.

5. Figure (a) Schematic representation of the one-component phase diagram for H2O. (b) A projection of the phase diagram information at 1 atm generates a temperature scale labeled with the familiar transformation temperatures for H2O (melting at 0°C and boiling at 100°C).

6. Figure (a) Schematic representation of the one-component phase diagram for pure iron. (b) A projection of the phase diagram information at 1 atm generates a temperature scale labeled with important transformation temperatures for iron. This projection will become one end of important binary diagrams, such as that shown in Figure 9.19

7. Definitions, cont Definition that focus on “Binary Systems”
Phase Equilibrium: A stable configuration with lowest free-energy (internal energy of a system, and also randomness or disorder of the atoms or molecules (entropy).
Any change in Temperature, Composition, and Pressure causes an increase in free energy and away from Equilibrium thus forcing a move to another ‘state’
Equilibrium Phase Diagram: It is a “map” of the information about the control of microstructure or phase structure of a particular material system. The relationships between temperature and the compositions and the quantities of phases present at equilibrium are represented.
Definition that focus on “Binary Systems”
Binary Isomorphous Systems: An alloy system that contains two components that attain complete liquid and solid solubility of the components, e.g. Cu and Ni alloy. It is the simplest binary system.
Binary Eutectic Systems: An alloy system that contains two components that has a special composition with a minimum melting temperature.

8. With these definitions in mind: ISSUES TO ADDRESS...
• When we combine two elements...
what “equilibrium state” would we expect to get?
• In particular, if we specify...
--a composition (e.g., wt% Cu - wt% Ni), and
--a temperature (T ) and/or a Pressure (P)
then...
How many phases do we get?
What is the composition of each phase?
How much of each phase do we get?
Phase B
Phase A
Nickel atom
Copper atom

9. Gibb’s Phase Rule: a tool to define the number of phases and/or degrees of phase changes that can be found in a system at equilibrium
For any system under study the rule determines if the system is at equilibrium
For a given system, we can use it to predict how many phases can be expected
Using this rule, for a given phase field, we can predict how many independent parameters (degrees of freedom) we can specify
Typically, N = 1 in most condensed systems – pressure is fixed!

10. Looking at a simple “Phase Diagram” for Sugar – Water (or milk)

11. Effect of Temperature (T) & Composition (Co)
• Changing T can change # of phases:
See path A to B.
Changing Co can change # of phases:
See path B to D.
B (100°C,70)
1 phase
D (100°C,90)
2 phases
Adapted from Fig. 9.1,
Callister 7e.
A (20°C,70)
2 phases 70 80
100 60 40 20
Temperature (°C) Co
=Composition (wt% sugar) L
(liquid solution
i.e, syrup)
(liquid) + S
(solid
sugar)
water-
sugar
system

12. T(°C) L L (liquid) a + (FCC solid solution) • 2 phases are possible:
Phase Diagram: A Map based on a System’s Free Energy indicating “equilibuim” system structures – as predicted by Gibbs Rule
• Indicate ‘stable’ phases as function of T, P & Composition,
• We will focus on:
-binary systems: just 2 components.
-independent variables: T and Co (P = 1 atm is almost always used).
• 2 phases are possible: L
(liquid) a
(FCC solid solution)
• 3 ‘phase fields’ are observed:
L +
wt% Ni 20 40 60 80
100
1000
1100
1200
1300
1400
1500
1600
T(°C)
L (liquid)
(FCC solid
solution) +
liquidus
solidus
Phase Diagram --
for Cu-Ni system
An “Isomorphic” Phase System

13. Phase Diagrams:
• Rule 1: If we know T and Co then we know the # and types of all phases present.
wt% Ni 20 40 60 80
100
1000
1100
1200
1300
1400
1500
1600
T(°C)
L (liquid) a
(FCC solid
solution) L +
liquidus
solidus
Cu-Ni
phase
diagram
• Examples:
A(1100°C, 60):
1 phase: a B
(1250°C,35) B
(1250°C, 35):
2 phases: L + a
A(1100°C,60)

14. adapted from Phase Diagrams
• Rule 2: If we know T and Co we know the composition of each phase
wt% Ni 20
1200
1300
T(°C)
L (liquid) a
(solid) L +
liquidus
solidus 30 40 50
Cu-Ni system
• Examples: T A C o
= 35 wt% Ni
tie line 35 C o
At T A
= 1320°C:
Only Liquid (L) C L
= C o
( = 35 wt% Ni) B T 32 C L 4 C a 3
At T D
= 1190°C: D T
Only Solid ( a ) C
= C o
( = 35 wt% Ni
At T B
= 1250°C:
Both a
and L
adapted from Phase Diagrams
of Binary Nickel Alloys, P. Nash (Ed.), ASM International, Materials Park, OH, 1991. C L
= C
liquidus
( = 32 wt% Ni here) C a
= C
solidus
( = 43 wt% Ni here)

15. Figure The lever rule is a mechanical analogy to the mass-balance calculation. The (a) tie line in the two-phase region is analogous to (b) a lever balanced on a fulcrum.

16. The Lever Rule
Tie line – a line connecting the phases in equilibrium with each other – at a fixed temperature (a so-called Isotherm)
How much of each phase? We can Think of it as a lever! So to balance:
wt% Ni 20
1200
1300
T(°C)
L (liquid) a
(solid) L +
liquidus
solidus 3 4 5 B T
tie line C o S R ML M R S

17. Therefore we define
• Rule 3: If we know T and Co then we know the amount of each phase (given in wt%)
wt% Ni 20
1200
1300
T(°C)
L (liquid) a
(solid) L +
liquidus
solidus 3 4 5
Cu-Ni system T A 35 C o 32 B D
tie line R S
• Examples: C o
= 35 wt% Ni
At T A
: Only Liquid (L) W L
= 100 wt%, W a
= 0
At T D :
Only Solid ( a ) W L
= 0, W
= 100 wt%
At T B :
Both a
and L
= 27 wt% WL = S R + Wa
Notice: as in a lever “the opposite leg” controls with a balance (fulcrum) at the ‘base composition’ and R+S = tie line length = difference in composition limiting phase boundary, at the temp of interest

18. Ex: Cooling in a Cu-Ni Binary
• Phase diagram:
Cu-Ni system.
wt% Ni 20
120
130 3 4 5
110
L (liquid) a
(solid) L +
T(°C) A 35 C o
L: 35wt%Ni
Cu-Ni
system
• System is:
--binary
i.e., 2 components:
Cu and Ni.
--isomorphous
i.e., complete
solubility of one
component in
another; a phase
field extends from
0 to 100 wt% Ni.
a: 46 wt% Ni
L: 35 wt% Ni B 46 35 C 43 32 a :
43 wt% Ni
L: 32 wt% Ni D 24 36
L: 24 wt% Ni a :
36 wt% Ni E
• Consider
Co = 35 wt%Ni.
Adapted from Fig. 9.4, Callister 7e.

19. Cored vs Equilibrium Phases
• Ca changes as we solidify.
• Cu-Ni case:
First a to solidify has Ca = 46 wt% Ni.
Last a to solidify has Ca = 35 wt% Ni.
• Fast rate of cooling:
Cored structure
• Slow rate of cooling:
Equilibrium structure
First a
to solidify:
46 wt% Ni
Uniform C :
35 wt% Ni
Last
< 35 wt% Ni

20. Cored (Non-equilibrium) Cooling
Notice:
The Solidus line is “tilted” in this non-equilibrium cooled environment

21. Binary-Eutectic (PolyMorphic) Systems
has a ‘special’ composition with a min. melting Temp.
2 components
Cu-Ag
system
T(°C)
Ex.: Cu-Ag system
1200
• 3 single phase regions
L (liquid)
(L,
a, b )
1000 a
• Limited solubility: a L + L + b
800
779°C b a TE
: mostly Cu
8.0
71.9
91.2 b
: mostly Ag
600
• TE
: No liquid below TE a + b
• CE
: Min. melting TE
400
composition
200 20 40 60 CE 80
100
• Eutectic transition
L(CE) (CE) + (CE) Co ,
wt% Ag

22. EX: Pb-Sn Eutectic System (1)
• For a 40 wt% Sn-60 wt% Pb alloy at 150°C, find...
--the phases present:
a + b
Pb-Sn
system
--compositions of phases: L + a b
200
T(°C)
18.3
C, wt% Sn 20 60 80
100
300
L (liquid)
183°C
61.9
97.8
CO = 40 wt% Sn
Ca = 11 wt% Sn
Cb = 99 wt% Sn
--the relative amount of each phase (by lever rule): W a =
C - CO
C - C 59 88
= 67 wt% S
R+S
150 R 11 C 40 Co S 99 C W  =
CO - C
C - C R
R+S 29 88
= 33 wt%
Adapted from Fig. 9.8,
Callister 7e.

23. EX: Pb-Sn Eutectic System (2)
• For a 40 wt% Sn-60 wt% Pb alloy at 220°C, find...
--the phases present:
a + L
Pb-Sn
system
--compositions of phases: L + b a
200
T(°C)
C, wt% Sn 20 60 80
100
300
L (liquid)
183°C
CO = 40 wt% Sn
Ca = 17 wt% Sn
CL = 46 wt% Sn
--the relative amount of each phase:
220 17 C R 40 Co S 46 CL W a =
CL - CO
CL - C 6 29
= 21 wt% W L =
CO - C
CL - C 23 29
= 79 wt%

24. Microstructures In Eutectic Systems: I
• Co < 2 wt% Sn
• Result:
--at extreme ends
--polycrystal of a grains
i.e., only one solid phase. L + a
200
T(°C) Co ,
wt% Sn 10 2% 20
300
100 30 b
400
(room Temp. solubility limit) TE
(Pb-Sn
System)
L: Co wt% Sn a L
a: Co wt% Sn

25. Microstructures in Eutectic Systems: II
L: Co wt% Sn
• 2 wt% Sn < Co < 18.3 wt% Sn
• Result:
Initially liquid
Then liquid + 
then  alone
finally two solid phases
a polycrystal
fine -phase inclusions
Pb-Sn
system L + a
200
T(°C) Co ,
wt% Sn 10
18.3 20
300
100 30 b
400
(sol. limit at TE) TE 2
(sol. limit at T
room ) L a
a: Co wt% Sn a b

26. Microstructures in Eutectic Systems: III
• Co = CE
• Result: Eutectic microstructure (lamellar structure)
--alternating layers (lamellae) of a and b crystals.
160 m
Micrograph of Pb-Sn
eutectic
microstructure
Pb-Sn
system
L
a
200
T(°C)
C, wt% Sn 20 60 80
100
300 L a b +
183°C 40 TE
L: Co wt% Sn CE
61.9
18.3
: 18.3 wt%Sn
97.8
: 97.8 wt% Sn
45.1%  and 54.8%  -- by Lever Rule

27. Lamellar Eutectic Structure

28. Microstructures in Eutectic Systems: IV
• wt% Sn < Co < 61.9 wt% Sn
• Result: a crystals and a eutectic microstructure WL
= (1- W a )
= 50 wt% C
= 18.3 wt% Sn CL
= 61.9 wt% Sn S R + =
• Just above TE :
Pb-Sn
system L + b
200
T(°C)
Co, wt% Sn 20 60 80
100
300 a 40 TE
L: Co
wt% Sn L a L a
18.3
61.9 S R
97.8 S R
primary a
eutectic b
• Just below TE : C a
= 18.3 wt% Sn b
= 97.8 wt% Sn S R + W =
= 72.6 wt%
= 27.4 wt%

29. Hypoeutectic & Hypereutectic Compositions
300 L
T(°C) L + a a
200 L + b b
(Pb-Sn TE
System) a + b
100
from Metals Handbook, 9th ed., Vol. 9, Metallography and Microstructures, American Society for Metals, Materials Park, OH, 1985.
175 mm a
hypoeutectic: Co = 50 wt% Sn
hypereutectic: (illustration only) b
Co, wt% Sn 20 40 60 80
100
eutectic
61.9
eutectic: Co = 61.9 wt% Sn
160 mm
eutectic micro-constituent

30. “Intermetallic” Compounds
Adapted from
Fig. 9.20, Callister 7e.
An Intermetallic Compound is also an important part of the Fe-C system!
Mg2Pb
Note: an intermetallic compound forms a line - not an area - because stoichiometry (i.e. composition) is exact.

31. Eutectoid & Peritectic – some definitions
Eutectic: a liquid in equilibrium with two solids
L  + 
cool
heat
Eutectoid: solid phase in equilibrium with two solid phases
S S1+S3
  + Fe3C (727ºC)
intermetallic compound - cementite
cool
heat
Peritectic: liquid + solid 1 in equilibrium with a single solid 2 (Fig 9.21)
S1 + L S2
 + L  (1493ºC)
cool
heat

32. Eutectoid & Peritectic
Peritectic transition  + L 
Cu-Zn Phase diagram
Eutectoid transition   + 
Adapted from
Fig. 9.21, Callister 7e.

33. Iron-Carbon (Fe-C) Phase Diagram
Max. C solubility in  iron = 2.11 wt%
• 2 important
Fe3C (cementite)
1600
1400
1200
1000
800
600
400 1 2 3 4 5 6
6.7 L g
(austenite) +L
+Fe3C a +
L+Fe3C d
(Fe)
Co, wt% C
1148°C
T(°C)
727°C = T
eutectoid
points
-Eutectic (A): L Þ g +
Fe3C A S R
4.30
-Eutectoid (B): g Þ a +
Fe3C g
Result: Pearlite =
alternating layers of a
and Fe3C phases
120 mm
(Adapted from Fig. 9.27, Callister 7e.)
0.76 C
eutectoid B R S
Fe3C (cementite-hard) a
(ferrite-soft)

34. Hypoeutectoid Steel T(°C) d L +L g (austenite) Fe3C (cementite) a
1600
1400
1200
1000
800
600
400 1 2 3 4 5 6
6.7 L g
(austenite) +L
+ Fe3C a
L+Fe3C d
(Fe)
Co , wt% C
1148°C
T(°C)
727°C
(Fe-C
System) C0
0.76 g g
Adapted from Figs and 9.29,Callister 7e. (Fig adapted from Binary Alloy Phase Diagrams, 2nd ed., Vol. 1, T.B. Massalski (Ed.-in-Chief), ASM International, Materials Park, OH, 1990.) r s w a = /( + ) g
(1-
proeutectoid ferrite
pearlite
100 mm
Hypoeutectoid
steel R S a w = /( + )
Fe3C
(1- g

35. Hypereutectoid Steel Fe3C (cementite) L g (austenite) +L a d T(°C) g g
1600
1400
1200
1000
800
600
400 1 2 3 4 5 6
6.7 L g
(austenite) +L
+Fe3C a
L+Fe3C d
(Fe)
Co , wt%C
1148°C
T(°C)
(Fe-C
System) g g
adapted from Binary Alloy Phase Diagrams, 2nd ed., Vol. 1, T.B. Massalski (Ed.-in-Chief), ASM International, Materials Park, OH, 1990. s r w
Fe3C = /( + ) g
=(1-
proeutectoid Fe3C
60 mm
Hypereutectoid
steel
pearlite R S w a = /( + )
Fe3C
(1- g Co
0.76

36. Example: Phase Equilibria
For a 99.6 wt% Fe-0.40 wt% C at a temperature just below the eutectoid, determine the following:
composition of Fe3C and ferrite ()
the amount of carbide (cementite) in grams that forms per 100 g of steel
the amount of pearlite and proeutectoid ferrite ()

37. Solution: a) composition of Fe3C and ferrite ()
CO = 0.40 wt% C Ca = wt% C CFe C = 6.70 wt% C 3
a) composition of Fe3C and ferrite ()
the amount of carbide (cementite) in grams that forms per 100 g of steel
Fe3C (cementite)
1600
1400
1200
1000
800
600
400 1 2 3 4 5 6
6.7 L g
(austenite) +L
+ Fe3C a
L+Fe3C d
Co , wt% C
1148°C
T(°C)
727°C CO R S
CFe C 3 C

38. Solution, cont: the amount of pearlite and proeutectoid ferrite ()
note: amount of pearlite = amount of g just above TE
Co = 0.40 wt% C Ca = wt% C Cpearlite = C = 0.76 wt% C
Fe3C (cementite)
1600
1400
1200
1000
800
600
400 1 2 3 4 5 6
6.7 L g
(austenite) +L
+ Fe3C a
L+Fe3C d
Co , wt% C
1148°C
T(°C)
727°C CO
pearlite = 51.2 g proeutectoid  = 48.8 g R S C C
Looking at the Pearlite:
11.1% Fe3C (.111*51.2 gm = 5.66 gm) & 88.9%  (.889*51.2gm = 45.5 gm)
 total  = = 94.3 gm

39. Alloying Steel with More Elements
• Teutectoid changes:
• Ceutectoid changes:
Adapted from Edgar C. Bain, Functions of the Alloying Elements in Steel, American Society for Metals, 1939, p. 127.) T
Eutectoid
(°C)
wt. % of alloying elements Ti Ni Mo Si W Cr Mn C
eutectoid
(wt%C)

40. Looking at a Tertiary Diagram
When looking at a Tertiary (3 element) P. diagram, like this image, the Mat. Engineer represents equilibrium phases by taking “slices at different 3rd element content” from the 3 dimensional Fe-C-Cr phase equilibrium diagram
From: G. Krauss, Principles of Heat Treatment of Steels, ASM International, 1990
What this graph shows are the increasing temperature of the euctectoid and the decreasing carbon content, and (indirectly) the shrinking  phase field

41. Fe-C-Si Ternary phase diagram at about 2.5 wt% Si
This is the “controlling Phase Diagram for the ‘Graphite‘ Cast Irons

42. Summary • Phase diagrams are useful tools to determine:
-- the number and types of phases,
-- the wt% of each phase,
-- and the composition of each phase
for a given T and composition of the system.
• Alloying to produce a solid solution usually
--increases the tensile strength (TS)
--decreases the ductility.
• Binary eutectics and binary eutectoids allow for
(cause) a range of microstructures.