Chapter 7 Phase Equilibria and Phase
Diagrams
• The one-component phase diagram
– Gibbs Phase Rule
• Phase equilibria in a two-component system
– The isomorphous diagram
– The lever rule
– Equilibrium solidification and microstructure of
isomorphous alloys
• Liquidius and solidus boundaries
• Deviations from ideal behavior
Chapter 7 Phase Equilibria and Phase
Diagrams, Continued
• Phase equilibria in a two-component system
–
–
–
–
–
The eutectic phase diagram
The peritectic phase diagram
The monotectic phase diagram
Complex diagrams
Phase equilibria involving solid-to-solid reactions
1
Why important ?
Some properties that might be difficult to predict using a
“common sense” without the knowledge of the phase diagrams
• example 1: Melting temperature of a
mixture AB (solution) of two components A
and B could be either lower or higher than
the melting point of each component (!).
This could be a failure mechanism in
electronic or mechanical components. But
could also be used to your advantage.
• example 2: Upon cooling to a lower
temperature a phase transformation of a
material could cause expansion, which
could cause internal stresses and failure
(e.g. tin food cans will crumble at low T)
Example: Chip-Solder-Joint-Failure
• example 3: No abrupt liquid-to-solid
transformation when two components are
present (solid + liquid in a temperature range)
Why important ?
Some properties that might be difficult to predict using a
“common sense” without the knowledge of the phase diagrams
• example 4: Tmelt (Sn) = 232 C, Tmelt (Pb) = 327 C……
but Tmelt(Sn0.62Pb0.38) = 183 C, so this is a common soldering alloy
• example 5: Tmelt (Au) = 1064 C, Tmelt (Si) = 2550 C……
but Tmelt(Au0.97Si0.03) = 363 C, so thin layer of gold is used to attach Si chip
to a ceramic substrate (shock protection)
• example 6: Mechanical properties (hardness and tensile strength) of an
alloy could be substantially higher than that of the individual components
(e.g. hardness (AgCu) about twice the harness of Ag or Cu)
2
One-Component Phase Diagrams
# of state variables (e.g. two: P and T)
F=C–P+2
C- Components
P- Number of phases
F- Degrees of freedom
Gibbs Phase rule:
F=2
F=1
F=0
Two-Component Phase Diagrams
Isomorphous system
( complete solubility over the
composition range)
F=1
T
F=C–P+1
If pressure is fixed (1 atm)
F=2
In a two-phase field need to
specify either the temperature
or the composition of one of the
phases.
Hume-Rothery Rules for
substitutional solution:
•
•
Xs
Xl
•
The size < ~15%.
The electronegativities and valance –
similar
The crystal structures of the two species
must be the same to form a continuous
series of solid solutions.
3
Two-Component Phase Diagrams
fl + f s = 1
X o = X l fl + X s f s
fl = 1 − f s
Temperature
X o = X l (1 − f s ) + X s f s
X o = X l − X l fs + X s fs
X o − X l = fs ( X s − X l )
( Xo − Xl )
( Xs − Xl )
( Xs − Xo )
fl =
( Xs − Xl )
fs =
Composition, XB
The Lever Rule in a Two-Component System
4
Two-Component
Phase Diagrams
Temperature, C
Two-Component Phase Diagrams
Time
Time
Composition, XB
5
Two-Component Phase Diagrams
Congruent melting maximum
EAB > 0.5 (EAA + EBB)
Deviation from ideal behavior
Two-Component Phase Diagrams
EAB > 0.5 (EAA + EBB)
Deviation from
ideal behavior
Congruent melting
minimum
6
Eutectic Phase Diagrams
TA
F=2
T
Temperature
F=1
T
Xs
Xl
F = 2, must
specify temperature
and composition
F=1
F=2
F=0
Xα
Xβ
F=2
Solvus
F=1
T X
α
A
TB
Xl
Xs
Xβ
X1
XE
Composition, XB
X2
B
F = 1, must
specify temperature
or the composition
of one of the phases
F = 0, temperature
and compositions
of the phase are
fixed.
Cooling Curves and Phase Boundaries
Temperature
Alloy 1
Time
Composition, XB
7
Cooling Curves and Phase Boundaries
Temperature
Alloy 2
Time
Composition, XB
Cooling Curves and Phase Boundaries
Temperature
Alloy 3
Time
Composition, XB
8
Various physical
properties and their
relationship to a
eutectic phase
diagram
Eutectic Phase Diagrams
Temperature
1. For the alloy composition of
0.27 % B calculate the fraction of
solid and the fraction of liquid that
forms under equilibrium cooling at
the eutectic T
Composition, in % B
2. Calculate the amount of β and α
that will form from the liquid just
below the eutectic isotherm
3. Calculate the amount of α in the
alloy at temperature just below the
eutectic T
9
Eutectic Phase Diagrams
Temperature
Just above the eutectic
temperature the fraction
of liquid and solid are:
0.37 − 0.27
0.27 − 0.20
fα ≈
fl ≈
0.37 − 0.20
0.37 − 0.20
f
≈
0.59
fl ≈ 0.41
α
The first solid that forms
is called primary α
Composition, in % B
This liquid becomes the
eutectic mixture of α and β
when the temperature
drops just below the eutectic
temperature which is composed
of:
0.37 − 0.20
0.73 − 0.37
fβ ≈
fα ≈
0.73 − 0.20
0.73 − 0.20
f β ≈ 0.32
fα ≈ 0.68
Just below the eutectic temperature
the microstructure is composed of
primary α that formed above the
eutectic temperature and α from
the eutectic mixture
fαtotal = fα primary + fαeutectic
fαtotal = 0.59 + (0.41)(0.68) = 0.87
or
fα =
0.73 − 0.27 f = 0.87
α
0.73 − 0.20
Microstructure Above and Below the Eutectic
Temperature for an Off-Eutectic Alloy
Just above TE
Just below TE
10
Microstructure Above and Below the Eutectic
Temperature for Off-Eutectic Alloys
Just below TE
Increasing primary α
Decreasing eutectic
Decreasing primary α
Increasing eutectic
Temperature
Deviation from Hume-Rothery’s Rules
Increasing deviation leads to decrease
in the maximum solid solubility of B in α.
A
Composition, XB
11
Temperature
Eutectic Phase Diagram, No Solid Solubility
Composition, XB
Eutectic Phase Diagrams
Al-Si System
12
Methods for Determining a Phase Diagram
Microstructure of an Aluminum-Silicon Alloy
Primary α-aluminum
α−aluminum / silicon eutectic
13
Phase Diagrams Containing Two Eutectics
Line
compound
• Possible to have several solid solution regions: e.g. 2 eutectic reactions and 3
solid solutions (α, β, and γ)
• Note that upon cooling from T max at the alloy composition X there is a phase
change but no composition change (CONGRUENT melting)
Peritectic Phase Diagrams
• if both the L and S phases have
a tendency to cluster, the liquidus
temperature increase and the
solidus temperature decreases
• In addition, a miscibility gap
(region of non-mixing) appears
• A progressive increase in the
clustering tendency leads to the
PERITECTIC phase diagram
l +α = β
14
The Use of Cooling Curves for Determining a
Peritectic Phase Diagram
l +α = β
Temperature
TA
TL
TP
X2
XP
X2
Composition, XB
Time
Analysis of a Peritectic Phase Diagram
Alloy 1 Alloy 2
Alloy 3
Temperature
Alloy 3 at T2
0.88 − 0.60
0.88 − 0.3
f l = 0.48
fl =
Alloy 3 at T5
0.90 − 0.60
0.90 − 0.34
fα = 0.54
fα =
Composition
15
Monotectic Phase Diagrams
• A region of immiscibility (nonmixing) develops in the L phase
L2
L1
• example: oil and water
• Liquid1 = Liquid2 + α (solid)
L2
L2
XM
Review of Invariant Binary Reactions
Eutectic Type
Eutectic
l α+β
α
Eutectoid
α
γ
α+β
Monotectic
l1
α + l2
Monotectoid
α2
α1 + β
α
α1
l
γ
l1
α2
β
Al-Si, Fe-C
β
Fe-C
l2
Cu-Pb
β
Al-Zn, Ti-V
On cooling one phase going to two phases
16
Review of Invariant Binary Reactions
Peritectic Type
Peritectic
l+β
α
Peritectoid
α+β
γ
l
α
α
β
Fe-C
β
Cu-Al
γ
On cooling two phases going to one phase
HW Questions
1. When a solid melts congruently, the liquid and solid have different /
the same composition(s).
2. At constant temperature the fraction of the phases in a two-phase field
changes / remains the same when the overall composition of the alloy is
changed, but remains in the two-phase field.
3. Why would alloys close to the eutectic composition be suitable for
castings rather than alloy compositions far from the eutectic
composition?
4. On cooling when a two-phase liquid plus solid transforms to a solid
phase the transformation is eutectic / peritectic in nature.
5. On cooling the peritectoid reaction written symbolically has
one phase going to two / two phases going to one.
17
HW Questions
•
At what T an alloy containing
88% B will start melting?
•
At what T it will completely
transform into liquid?
•
What is the composition of α
phase for this alloy @ T8?
•
What is the maximum solid solubility of B in a and A in b? Whose rules apply
here?
•
For an alloy containing 88% B, calculate the fraction of the liquid and solid
phases and their compositions at temperature T3, T4, and T5
•
At a temperature just below the eutectic temperature, how much β is primary β,
what is the total fraction of β, and what is the fraction β in the eutectic. (Alloy
composition is 88% B)
Labeling Complex Phase Diagrams
Temperature
TB
1. Label all
phase fields.
2. Identify all
invariant
reactions.
TA
A
Composition, XB
B
18
Labeling Complex Phase Diagrams
Peritectic
Eutectic
Eutectic
l =α +β
Peritectic
l +γ = β
Eutectic
l =γ +δ
Eutectic
Labeling Complex Phase Diagrams
19
Summary
• One-component phase diagrams with
temperature and pressure as the
experimental variables that affect equilibrium.
• Introduction to the Gibbs Phase Rule and its
application to one-component systems.
• Two-component systems and the rules that
govern the composition of the phases, the
number of phases and the amount of each
phase at equilibrium.
• The applications of these rules to complex,
two-component systems illustrated that
regardless of how complex the phase
diagram appeared, the rules that were
developed could be easily applied.
20