Phase Equilibrium
Engr 2110 – Chapter 9
Dr. R. R. Lindeke
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

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.
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.
Figure 9.3 (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).
Figure 9.4 (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
Definitions, cont
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.
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
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
F CPN
where:
F is # degrees of freedom of the system (independent parameters)
C is # components (elements) in system
P is # phases at equil.
N is # "noncompostional" parameters in system (temp &/or Pressure)

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!
Looking at a simple “Phase Diagram” for
Sugar – Water (or milk)
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) D (100°C,90)
1 phase
watersugar
system
Adapted from
Fig. 9.1,
Callister 7e.
Temperature (°C)
100
2 phases
L
80
(liquid)
60
L
(liquid solution
i.e, syrup)
40
20
0
0
+
S
(solid
sugar)
A (20°C,70)
2 phases
20
40
60 70 80
100
Co =Composition (wt% sugar)
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).
T(°C)
• 2 phases are possible:
1600
Phase Diagram -1500
for Cu-Ni system
L (liquid)
a (FCC solid solution)
L (liquid)
1400
1300
a
(FCC solid
solution)
1200
An
“Isomorphic”
Phase System
1100
1000
0
20
40
60
80
• 3 ‘phase fields’ are observed:
L
L+a
a
100
wt% Ni
Phase Diagrams:
• Rule 1: If we know T and Co then we know the # and types of all
phases present.
A(1100°C, 60):
1 phase: a
B(1250°C, 35):
2 phases: L + a
1600
L (liquid)
B (1250°C,35)
• Examples:
T(°C)
1500
1400
1300
1200
1100
1000
Cu-Ni
phase
diagram
a
(FCC solid
solution)
A(1100°C,60)
0
20
40
60
80
100
wt% Ni
Phase Diagrams:
• Rule 2: If we know T and Co we know the composition of each phase
• Examples:
T(°C)
Cu-Ni
system
A
TA
Co = 35 wt% Ni
tie line
1300 L (liquid)
At T A = 1320°C:
Only Liquid (L)
B
TB
CL = Co ( = 35 wt% Ni)
a
At T D = 1190°C:
(solid)
1200
D
Only Solid ( a)
TD
Ca = Co ( = 35 wt% Ni)
20
3032 35 4043
50
At T B = 1250°C:
CLCo
Ca wt% Ni
Both a and L
adapted from Phase Diagrams
CL = C liquidus ( = 32 wt% Ni here)
of Binary Nickel Alloys, P. Nash (Ed.), ASM
International, Materials Park, OH, 1991.
Ca = C solidus ( = 43 wt% Ni here)
Figure 9.31 The lever rule is a mechanical analogy to the massbalance calculation. The (a) tie line in the two-phase region is
analogous to (b) a lever balanced on a fulcrum.
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:
T(°C)
tie line
1300
L (liquid)
B
TB
a
(solid)
1200
R
20
Ma
ML
S
30C C
40 C
a
L o
R
50
S
M a S  M L R
wt% Ni
WL 
C  C0
ML
S

 a
ML  M a R  S Ca  CL
Wa 
C  CL
R
 0
R  S Ca  CL
Therefore we define
• Rule 3: If we know T and Co then we know the amount of each
phase (given in wt%)
• Examples:
Cu-Ni
system
T(°C)
Co = 35 wt% Ni
A
TA
At T A : Only Liquid (L)
W L = 100 wt%, W a = 0
At T D: Only Solid ( a)
W L = 0, Wa = 100 wt%
At T B : Both a and L
WL 
S  43  35  73 wt %
R + S 43  32
Wa 
R
= 27 wt%
R +S
1300
TB
1200
TD
20
tie line
L (liquid)
B
R S
D
3032 35
CLCo
a
(solid)
40 43
50
Ca wt% Ni
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
Ex: Cooling in a Cu-Ni Binary
• Phase diagram:
Cu-Ni system.
• System is:
--binary
i.e., 2 components:
Cu and Ni.
T(°C) L (liquid)
130 0
L: 35 wt% Ni
a: 46 wt% Ni
• Consider
Co = 35 wt%Ni.
Cu-Ni
system
A
35
32
--isomorphous
i.e., complete
solubility of one
component in
another; a phase
field extends from
0 to 100 wt% Ni.
L: 35wt%Ni
B
C
46
43
D
24
L: 32 wt% Ni
36
120 0
a: 43 wt% Ni
E
L: 24 wt% Ni
a: 36 wt% Ni
a
(solid)
110 0
20
30
Adapted from Fig. 9.4,
Callister 7e.
35
Co
40
50
wt% Ni
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
Last a to solidify:
< 35 wt% Ni
Uniform C a:
35 wt% Ni
Cored (Non-equilibrium) Cooling
Notice:
The Solidus
line is “tilted”
in this nonequilibrium
cooled
environment
Binary-Eutectic (PolyMorphic) Systems
has a ‘special’ composition with a min.
melting Temp.
Cu-Ag
T(°C)
system
2 components
Ex.: Cu-Ag system
1200
• 3 single phase regions
L (liquid)
(L, a, )
1000
a L+a
• Limited solubility:
L+ 
779°C
800
T
a: mostly Cu
E
8.0
71.9 91.2
: mostly Ag
600
• TE : No liquid below TE
a
400
• CE : Min. melting TE
composition
200
• Eutectic transition
L(CE)
0
a(CaE) + (CE)
20
40
60 CE 80
Co , wt% Ag
100
EX: Pb-Sn Eutectic System (1)
• For a 40 wt% Sn-60 wt% Pb alloy at 150°C, find...
--the phases present: a + 
T(°C)
--compositions of phases:
CO = 40 wt% Sn
Ca = 11 wt% Sn
C = 99 wt% Sn
--the relative amount
of each phase (by lever rule):
C - CO
S
= 
Wa =
R+S
C - Ca
Pb-Sn
system
300
200
L (liquid)
a
L+ a
18.3
150
61.9
R
97.8
S
a+
100
99 - 40
59
=
= 67 wt%
99 - 11
88
C - Ca
W = R = O
C - Ca
R+S
L+ 
183°C
=
=
40 - 11
29
=
= 33 wt%
99 - 11
88
0 11 20
Ca
40
Co
Adapted from Fig. 9.8,
Callister 7e.
60
80
C, wt% Sn
99100
C
EX: Pb-Sn Eutectic System (2)
• For a 40 wt% Sn-60 wt% Pb alloy at 220°C, find...
--the phases present: a + L
T(°C)
--compositions of phases:
CO = 40 wt% Sn
Ca = 17 wt% Sn
CL = 46 wt% Sn
--the relative amount
of each phase:
CL - CO
46 - 40
=
Wa =
CL - Ca
46 - 17
6
=
= 21 wt%
29
Pb-Sn
system
300
a
220
200
L+a
R
L (liquid)
L+ 
S
183°C
a+
100
0
CO - Ca
23
=
WL =
= 79 wt%
CL - Ca
29
17 20
Ca
40 46 60
Co CL
C, wt% Sn
80
100
Microstructures In Eutectic Systems: I
• Co < 2 wt% Sn
• Result:
T(°C)
L: Co wt% Sn
400
L
a
--at extreme ends
300
--polycrystal of a grains
i.e., only one solid phase.
L
a
200
L+ a
(Pb-Sn
System)
a: Co wt% Sn
TE
a+ 
100
0
Co
10
20
30
Co, wt% Sn
2%
(room Temp. solubility limit)
Microstructures in Eutectic Systems: II
L: Co wt% Sn
• 2 wt% Sn < Co < 18.3 wt% Sn 400T(°C)
• Result:




Initially liquid
Then liquid + a
then a alone
finally two solid phases
 a polycrystal
 fine -phase inclusions
L
L
a
300
L+a
a
200
TE
a: Co wt% Sn
a

100
a+ 
0
10
20
Pb-Sn
system
30
Co
Co , wt%
2
(sol. limit at T room )
18.3
(sol. limit at TE)
Sn
Microstructures in Eutectic Systems: III
• Co = CE
• Result: Eutectic microstructure (lamellar structure)
--alternating layers (lamellae) of a and  crystals.
T(°C)
L: Co wt% Sn
300
Pb-Sn
system
TE
a
200
L+ a
L
a
0
L 
183°C
100
20
18.3
40
Micrograph of Pb-Sn
eutectic
microstructure
: 97.8 wt% Sn
a: 18.3 wt%Sn
60
CE
61.9
80
100
97.8
C, wt% Sn
160 m
45.1% a and 54.8% 
-- by Lever Rule
Lamellar Eutectic Structure
Microstructures in Eutectic Systems: IV
• 18.3 wt% Sn < Co < 61.9 wt% Sn
• Result: a crystals and a eutectic microstructure
L: Co wt% Sn
T(°C)
L
a
L
300
Pb-Sn
system
a
200
a L
L+a
R
TE
L+ 
S
S
R
primary a
eutectic a
eutectic 
0
20
18.3
40
60
61.9
Ca = 18.3 wt% Sn
CL = 61.9 wt% Sn
Wa = S = 50 wt%
R+S
WL = (1- Wa) = 50 wt%
• Just below TE :
a+
100
• Just above TE :
80
Co, wt% Sn
100
97.8
Ca = 18.3 wt% Sn
C = 97.8 wt% Sn
Wa = S = 72.6 wt%
R+S
W = 27.4 wt%
Hypoeutectic & Hypereutectic Compositions
300
L
T(°C)
a
200
L+ a
L+ 
TE
a+
(Pb-Sn
System)
100
0
20
40
hypoeutectic: Co = 50 wt% Sn
from Metals Handbook, 9th
ed., Vol. 9, Metallography and
Microstructures, American
Society for Metals, Materials
Park, OH, 1985.
a
a
a
60
80
eutectic
61.9
Co, wt% Sn
hypereutectic: (illustration only)
eutectic: Co = 61.9 wt% Sn

a a

a
175 m
100
160 m
eutectic micro-constituent

 

“Intermetallic” Compounds
Adapted from
Fig. 9.20, Callister 7e.
Mg2Pb
An Intermetallic
Compound is also
an important part of
the Fe-C system!
Note: an intermetallic compound forms a line - not an area because stoichiometry (i.e. composition) is exact.
Eutectoid & Peritectic – some definitions
Eutectic: a liquid in equilibrium with two solids

L

cool
heat
a+
Eutectoid: solid phase in equilibrium with two solid phases
S2
S1+S3
intermetallic compound - cementite


cool
heat
a + Fe3C
(727ºC)
Peritectic: liquid + solid 1 in equilibrium with a single solid
2 (Fig 9.21)
S1 + L
S2
+L
cool
heat

(1493ºC)
Eutectoid & Peritectic
Peritectic transition  + L
Cu-Zn Phase diagram
Eutectoid transition 
+
Adapted from
Fig. 9.21, Callister 7e.

Iron-Carbon (Fe-C) Phase Diagram
1600

1200
  a + Fe3C
 +L

(austenite)
 
 
1000
a
800
600
120 m
Result: Pearlite =
alternating layers of
a and Fe3C phases
(Adapted from Fig. 9.27, Callister 7e.)
S
 +Fe3C
727°C = Teutectoid
R
S
1
0.76
L+Fe3C
R
B
400
0
(Fe)
A
1148°C
2
3
a+Fe3C
4
5
6
Fe3C (cementite)
L
1400
L   + Fe3C
-Eutectoid (B):
Max. C solubility in  iron = 2.11 wt%
T(°C)
C eutectoid
• 2 important
points
-Eutectic (A):
6.7
4.30
Co, wt% C
Fe3C (cementite-hard)
a (ferrite-soft)
Hypoeutectoid Steel
T(°C)
1600

L
 
 
 +L

1200
(austenite)
 
 
1000
a
a

a
 
800
1148°C
727°C
r s
aRS
a + Fe3C
1
C0
w pearlite = w 
0.76
400
0
(Fe)
pearlite
w a =S/(R+S)
w Fe3 =(1-w a )
C
L+Fe3C
 + Fe3C
w a =s/(r +s) 600
w  =(1- wa )
a
(Fe-C
System)
pearlite
2
3
4
5
6
Fe3C (cementite)
1400
Adapted from Figs. 9.24
and 9.29,Callister 7e.
(Fig. 9.24 adapted from
Binary Alloy Phase
Diagrams, 2nd ed., Vol.
1, T.B. Massalski (Ed.-inChief), ASM International,
Materials Park, OH,
1990.)
6.7
Co , wt% C
100 m
Hypoeutectoid
steel
proeutectoid ferrite
Hypereutectoid Steel
T(°C)
1600

L
Fe3C


 +L

1200
(austenite)


1000
 
 
r
800
w Fe3C =r/(r +s)
w  =(1-w Fe3C )
a R
600
400
0
(Fe)
pearlite
1148°C
L+Fe3C
 +Fe3C
0.76




(Fe-C
System)
s
S
1 Co
w pearlite = w 
w a =S/(R+S)
w Fe3C =(1-w a )
a +Fe3C
2
3
4
5
6
Fe3C (cementite)
1400
adapted from Binary Alloy
Phase Diagrams, 2nd
ed., Vol. 1, T.B. Massalski
(Ed.-in-Chief), ASM
International, Materials
Park, OH, 1990.
6.7
Co , wt%C
60 mHypereutectoid
steel
pearlite
proeutectoid Fe3C
Example: Phase Equilibria
For a 99.6 wt% Fe-0.40 wt% C at a temperature
just below the eutectoid, determine the
following:
a)
b)
c)
composition of Fe3C and ferrite (a)
the amount of carbide (cementite) in grams that
forms per 100 g of steel
the amount of pearlite and proeutectoid ferrite (a)
Solution:
CO = 0.40 wt% C
Ca = 0.022 wt% C
CFe3 C = 6.70 wt% C
a) composition of Fe3C and ferrite (a)
b) the amount of carbide
(cementite) in grams that
forms per 100 g of steel
1600

Fe3C
Co  Ca

x100
1400
Fe3C  a CFe 3C  Ca
T(°C)
Fe3C  5.7 g
a  94.3 g

 +L
L+Fe3C
1148°C
(austenite)
1000
Fe C (cementite)
0.4  0.022

x 100  5.7g
6.7  0.022
1200
L
 + Fe3C
800
727°C
R
S
a + Fe3C
600
400
0
Ca CO
1
2
3
4
Co , wt% C
5
6
6.7
CFe
3C
Solution, cont:
c)
the amount of pearlite and proeutectoid ferrite (a)
note: amount of pearlite = amount of  just above TE

C  Ca
 o
x 100  51.2 g
  a C  Ca
pearlite = 51.2 g
proeutectoid a = 48.8 g
1600

L
1400
T(°C)
1200

1000
 + Fe3C
800
600
727°C
RS
a + Fe3C
1
Ca CO C
2
11.1% Fe3C (.111*51.2 gm = 5.66 gm) & 88.9% a (.889*51.2gm = 45.5 gm)
 total a = 45.5 + 48.8 = 94.3 gm
L+Fe3C
1148°C
(austenite)
400
0
Looking at the Pearlite:
 +L
3
4
Co , wt% C
5
6
Fe C (cementite)
Co = 0.40 wt% C
Ca = 0.022 wt% C
Cpearlite = C = 0.76 wt% C
6.7
Alloying Steel with More Elements
Ti
Mo
Si
W
Cr
Mn
Ni
wt. % of alloying elements
Adapted from Edgar C. Bain, Functions of the
Alloying Elements in Steel, American Society for
Metals, 1939, p. 127.)
• Ceutectoid changes:
Ceutectoid (wt%C)
T Eutectoid (°C)
• Teutectoid changes:
Ni
Cr
Si
Ti Mo
W
Mn
wt. % of alloying elements
Adapted from Edgar C. Bain, Functions of the
Alloying Elements in Steel, American Society for
Metals, 1939, p. 127.)
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
Fe-C-Si
Ternary
phase
diagram at
about 2.5
wt% Si
This is the
“controlling
Phase
Diagram
for the
‘Graphite‘
Cast Irons
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.