Department of Physics and Astronomy
William Meier
Physics 590B
Fall 2018
Introduction to phase diagrams
AS
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William Meier November 2, 2018
Outline• Part 1 – Introduction and basics
• Part 2 – Fundamental concepts
• Part 3 – Using phase diagrams
2
William Meier November 2, 2018
A phase diagram is a map
• Regions on a phase diagram
indicate the stable phases for
a set of parameters.
3
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William Meier November 2, 2018
Water phase diagram at 1 atm
4
• Phases of H2O vs temperature
• Single phase regions
• Co-existence
https://commons.wikimedia.org/wiki/File:Ice_Water_(5685106294).jpg
Solid
(ice)
Liquid
Vapor
(steam)373
273
0
T (K)
William Meier November 2, 2018
Water p-T phase diagram
5
Solid
(ice)
Liquid
Vapor
(steam)373
273
0
T (K)
http://www1.lsbu.ac.uk/water/water_phase_diagram.html
William Meier November 2, 2018
Other p-T phase diagrams
6
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Car
bon
_d
ioxid
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ure
-tem
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ature
_phas
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m.s
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CO2
https://commons.wikimedia.org/wiki/File:Pure_iron_phase_diagram_(EN).svg
Fe• Phase fields
• Boundaries and coexistence
William Meier November 2, 2018
T-x phase diagrams
7
• Temp v. overall composition
• Liquid solution
• Solid solution
• Boundaries
https://commons.wikimedia.org/wiki/File:Face-centered-cubic-unitcell.png
Liquid solution
(Ag,Au)
Solid solution
Liquid + Solid (Ag,Au)
William Meier November 2, 2018
T-x phase diagrams
8
• Limited solid solubility
• Liquid stable below Tmelt
of elements
• Eutectic reaction
• (Ag) + (Cu) → Liquid
Liquid + Solid (Cu)
Liquid + Solid (Ag)
Solid (Ag) + Solid (Cu)
Liquid
William Meier November 2, 2018
• Compound (KNa2)
• “Line compound”
• Peritectic reaction
• KNa2 → Liquid + (Na)
T-x phase diagrams
9
Solid (K) + KNa2
KNa2 +
Solid (Na)
Liquid
Liquid +
Solid (Na)
Solid (K) + Liquid
Liquid + KNa2
William Meier November 2, 2018
T-H phase diagram
• Magnetic field suppresses
the superconducting states
10
http://hyperphysics.phy-astr.gsu.edu/hbase/Solids/scbc.html
William Meier November 2, 2018
Application
Steel (Fe-C)
11
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• Carbon modifies phase
stability
• Heat treatments
https://www.e-education.psu.edu/matse81/node/2121
α + Fe3C
γ + Fe3C
BCC αFCC γ
William Meier November 2, 2018
Part 2 – Fundamental concepts
• Definitions
• Reactions and transitions
• The lever rule (conservation of matter)
13
William Meier November 2, 2018
Definition
Phase
• A region of space
occupied by a
homogeneous material
• Mechanically separable?
Examples
• Water vapor
• Liquid water
• Ice
• Diamond
• Graphite
• Paramagnet
• Ferromagnet
14
William Meier November 2, 2018
Definition
Phase• Distinguished by
• Properties
• Structure
• Composition
• “Phase field”
15
https://commons.wikimedia.org/wiki/File:Pure_iron_phase_diagram_(EN).svg
https://commons.wikimedia.org/wiki/File:Carbon_dioxide_pressure-temperature_phase_diagram.svg
CO2
Fe
William Meier November 2, 2018
Definition
State variables• Parameters that determine
the configuration of the
system
16
Examples
• Composition
• Temperature
• Pressure
• Electric field
• Magnetic field
William Meier November 2, 2018
Definition
Components
• The chemical constituents
needed to describe the
compositions of interest
• Often elements
• Sometimes compounds
17
Al Si
O
Al2O3SiO2
Mullite
Al2O3 SiO2
Mullite
William Meier November 2, 2018
Definition
Equilibrium
• System configuration with
minimum free energy
Remember
• Equilibrium is a
thermodynamic concept
• Kinetics plays an
important role
18
Unstable
(balanced) Meta-stable
Stable
William Meier November 2, 2018
Definition
Equilibrium
• Practical definition:
• The configuration is
independent of time
• Changes could be slow
• Is the equilibrium
configuration accessible?
19
Nar
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15
30
3 (
20
15
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Department of Physics and Astronomy
Batman (William Meier)
Physics 590B
Fall 2018
Introduction to phase diagrams
Day 2
AS
M P
has
e D
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m D
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Dia
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m N
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90
12
29
William Meier November 2, 2018
Review
• Phase diagrams are maps
• Equilibrium phases as a function of state variables
21
William Meier November 2, 2018 22
1. Find single phase regions
Determining the stable phases 80% Er and 20% Sb
@ 1800°C80% Er and 20% Sb
@ 1800°C
William Meier November 2, 2018 23
1. Find single phase regions
(Er)
(Sb)
Liquid (L)
Er5Sb3
ErSb rt (room temperature)
ErSb ht (high temperature)
ErSb2 hp
Determining the stable phases 80% Er and 20% Sb
@ 1800°C80% Er and 20% Sb
@ 1800°C
Phases:
Er0.80Sb0.20 Liquid
Phases:
Er0.80Sb0.20 Liquid
William Meier November 2, 2018
Determining the stable phases
24
1. Find single phase regions
2. Draw tie line
Constant temp. line to single
phase regions on left and right
30% Er and 70% Sb
@ 1000°C30% Er and 70% Sb
@ 1000°C
William Meier November 2, 2018
Determining the stable phases
25
1. Find single phase regions
2. Draw tie line
3. Endpoints identify the
composition of stable
phases
30% Er and 70% Sb
@ 1000°C30% Er and 70% Sb
@ 1000°C
ErSb rtErSb rt
Liquid
Er0.09,Sb0.91
Liquid
Er0.09,Sb0.91LiquidLiquid
William Meier November 2, 2018
Determining the stable phases
26
1. Find single phase regions
2. Draw tie line
3. Endpoints identify the
composition of stable
phases
• Same phases in whole field
• Different compositions
ErSb rtErSb rt
Liquid
Er0.09,Sb0.91
Liquid
Er0.09,Sb0.91
ErSb rtErSb rt
Liquid
Er0.20,Sb0.80
Liquid
Er0.20,Sb0.80
William Meier November 2, 2018
Determining the stable phases
27
1. Find single phase regions
2. Draw tie line
3. Endpoints identify the
composition of stable
phases
• Same phases in whole field
• Different compositions
ErSb rt + L
Er5Sb3Er5Sb3(Er0.99,Sb0.01)(Er0.99,Sb0.01)
William Meier November 2, 2018
Determining the stable phases
28
1. Find single phase regions
2. Draw tie line
3. Endpoints identify the
composition of stable
phases
• Same phases in whole field
• Different compositions
ErSb rt + L
(Er) +
Er5Sb3
William Meier November 2, 2018
Determining the stable phases
29
1. Find single phase regions
2. Draw tie line
3. Endpoints identify the
composition of stable
phases
• Same phases in whole field
• Different compositions
ErSb rt + L
(Er) +
Er5Sb3
L +
Er5Sb3
William Meier November 2, 2018
Determining the stable phases
30
1. Find single phase regions
2. Draw tie line
3. Endpoints identify the
composition of stable
phases
• Same phases in whole field
• Different compositions
ErSb rt + L
(Er) +
Er5Sb3
L +
Er5Sb3
L + ErSb rt
William Meier November 2, 2018
Determining the stable phases
31
1. Find single phase regions
2. Draw tie line
3. Endpoints identify the
composition of stable
phases
• Same phases in whole field
• Different compositions
ErSb rt + L
(Er) +
Er5Sb3
L +
Er5Sb3
Er 5
Sb
3+
ErS
b r
t
L + ErSb rt
ErSb ht + LErSb ht + L
William Meier November 2, 2018
Determining the stable phases
32
1. Find single phase regions
2. Draw tie line
3. Endpoints identify the
composition of stable
phases
• Same phases in whole field
• Different compositions
ErSb rt + L
(Er) +
Er5Sb3
L +
Er5Sb3
Er 5
Sb
3+
ErS
b r
t
L + ErSb rt
L + ErSb htL + ErSb ht ErSb ht + LErSb ht + L
William Meier November 2, 2018
The lever rule
36
ErSb rt + L
30% Er
70% Sb
30% Er
70% Sb
Use the lever rule to find the
fractions of stable phases
William Meier November 2, 2018
The lever rule
37
ErSb rt + L
30% Er
70% Sb
30% Er
70% SbT
(°C)Phases
at%
SbPhase%
Overall
at% Sb
2000 Liquid 70 100 70
Liquid
Er0.30Sb0.70
Liquid
Er0.30Sb0.70
Liquid
Er0.30Sb0.70
Liquid
Er0.30Sb0.70
2000°C
Micrograph
2000°C
Micrograph
William Meier November 2, 2018
Liquid
Er0.30Sb0.70
Liquid
Er0.30Sb0.70
The lever rule
38
ErSb rt + L
T
(°C)Phases
at%
SbPhase%
Overall
at% Sb
2000 Liquid 70 100 70
1600ErSb rt 50 23
70Liquid 76 77
ErSb rtErSb rt
Liquid
Er0.24Sb0.76
Liquid
Er0.24Sb0.76
2000°C
Micrograph
2000°C
Micrograph
William Meier November 2, 2018
Liquid
Er0.30Sb0.70
Liquid
Er0.30Sb0.70
The lever rule
39
ErSb rt + L
T
(°C)Phases
at%
SbPhase%
Overall
at% Sb
2000 Liquid 70 100 70
1600ErSb rt 50 23
70Liquid 76 77
ErSb rtErSb rt
Liquid
Er0.24Sb0.76
Liquid
Er0.24Sb0.76
2000°C
Micrograph
2000°C
Micrograph
𝑓ErSb rt =𝑏
𝑎 + 𝑏composition
differences a, b
𝑓ErSb rt =𝑏
𝑎 + 𝑏composition
differences a, b
Lever Rule
a b
Liquid
Er0.24Sb0.76
Liquid
Er0.24Sb0.76
ErSb rtErSb rt1600°C
Micrograph
1600°C
Micrograph
William Meier November 2, 2018
The lever rule
40
30% Er
70% Sb
30% Er
70% SbT
(°C)Phases
at%
SbPhase%
Overall
at% Sb
2000 Liquid 70 100 70
1600ErSb rt 50 23
70Liquid 76 77
1200ErSb rt 50 46
70Liquid 87 54
ErSb rtErSb rt
Liquid
Er0.13Sb0.87
Liquid
Er0.13Sb0.87
𝑓ErSb rt =𝑏
𝑎 + 𝑏𝑓ErSb rt =
𝑏
𝑎 + 𝑏
Lever Rule
a b
ErSb rtErSb rt
Liquid
Er0.13Sb0.87
Liquid
Er0.13Sb0.87
1600°C
Micrograph
1600°C
Micrograph
1200°C
Micrograph
1200°C
Micrograph
William Meier November 2, 2018
The lever rule
41
30% Er
70% Sb
30% Er
70% SbT
(°C)Phases
at%
SbPhase%
2000 Liquid 70 100
1600ErSb rt 50 23
Liquid 76 77
1200ErSb rt 50 46
Liquid 87 54
ErSb rtErSb rt
Liquid
Er0.13Sb0.87
Liquid
Er0.13Sb0.87
1200°C
Micrograph
1200°C
Micrograph
Liquid
Er0.24Sb0.76
Liquid
Er0.24Sb0.76
1600°C
Micrograph
1600°C
Micrograph
Liquid
Er0.30Sb0.70
Liquid
Er0.30Sb0.70
2000°C
Micrograph
2000°C
Micrograph
LiquidusLiquidus
William Meier November 2, 2018
The lever rule
42
30% Er
70% Sb
30% Er
70% Sb
𝑓ErSb rt =𝑏
𝑎 + 𝑏
composition differences a, b
𝑓ErSb rt =𝑏
𝑎 + 𝑏
composition differences a, b
Lever Rule
ErSb
Liquid
Er0.24Sb0.76
a b
23 at%
77 at%
William Meier November 2, 2018
Phase transitions
43
• Congruent melting
Compound melts directly to a
liquid of the same composition
ErSb rt + L
(Er) +
Er5Sb3
L +
Er5Sb3
Er 5
Sb
3+
ErS
b r
t
L + ErSb rt
ErSb ht + LErSb ht + LL + ErSb htL + ErSb ht
William Meier November 2, 2018
Phase transitions
44
• Congruent melting
• Solid-solid
ErSb rt + L
(Er) +
Er5Sb3
L +
Er5Sb3
Er 5
Sb
3+
ErS
brt
L + ErSb rt
ErSb ht + LErSb ht + LL + ErSb htL + ErSb ht
William Meier November 2, 2018
Invariant reactions – Eutectic
45
• Consider 13 at% Sb at
1200°C and 1400°C
(Er) +
Er5Sb3
L +
Er5Sb3
ErSb rt + L
Er 5
Sb
3+
ErS
brt
William Meier November 2, 2018
Invariant reactions – Eutectic
46
• Consider 13 at% Sb at
1200°C and 1400°C
• Eutectic reaction:(Er0.97Sb0.03)sol + Er5Sb3 → (Er0.87Sb0.13)liq
• 3 phases only coexist at
1350°C
(Er) +
Er5Sb3
L +
Er5Sb3
ErSb rt + L
Er 5
Sb
3+
ErS
brt
William Meier November 2, 2018
Invariant reactions – Peritectic
47
• Consider 67 at% Sb at
500°C and 800°C
• Peritectic reaction:
ErSb2 → ErSbrt + (Er0.03Sb0.97)liq
• 3 phases only coexist at
700°C
• Incongruent melting
(Er) +
Er5Sb3
L +
Er5Sb3
ErSb rt + L
Er 5
Sb
3+
ErS
brt
William Meier November 2, 2018
Invariant reactions – “Tics”Reactions involving liquids
48
http://www1.asminternational.org/asmenterprise/apd/help/intro.aspx
Eutecticα + γ → L1
Eutecticα + γ → L1
Peritecticε → L + δ’
Peritecticε → L + δ’
Monotecticγ + L4 → L3
Monotecticγ + L4 → L3
William Meier November 2, 2018
Invariant reactions – “Toids”Reaction involving only solids
49
http://www1.asminternational.org/asmenterprise/apd/help/intro.aspx
Eutectoidη + δ→ γ
Eutectoidη + δ→ γ
Peritectoidη → α + γ
Peritectoidη → α + γ
Department of Physics and Astronomy
William Meier
Physics 590B
Fall 2018
Introduction to phase diagrams
Day 3
AS
M P
has
e D
iagra
m D
atab
ase
Dia
gra
m N
o.
90
12
29
William Meier November 2, 2018
Review
• Stable phases
• Lever rule
• Eutectic
• Peritectic
51
© ASM International 2006. Diagram No. 900834
a b
Tie line
William Meier November 2, 2018
Conservation of matter
52
30% Er
70% Sb
30% Er
70% SbT
(°C)Phases
at%
SbPhase%
2000 Liquid 70 100
1600ErSb rt 50 23
Liquid 76 77
1200ErSb rt 50 46
Liquid 87 54
ErSb rtErSb rt
Liquid
Er0.13Sb0.87
Liquid
Er0.13Sb0.87
1200°C
Micrograph
1200°C
Micrograph
Liquid
Er0.24Sb0.76
Liquid
Er0.24Sb0.76
1600°C
Micrograph
1600°C
Micrograph
Liquid
Er0.30Sb0.70
Liquid
Er0.30Sb0.70
2000°C
Micrograph
2000°C
Micrograph
William Meier November 2, 2018
Conservation of matter
53
30% Er
70% Sb
30% Er
70% SbT
(°C)Phases
at%
SbPhase%
2000 Liquid 70 100
30 Er
70 Sb
30 Er
70 Sb
William Meier November 2, 2018
Conservation of matter
54
30% Er
70% Sb
30% Er
70% SbT
(°C)Phases
at%
SbPhase%
1600ErSb rt 50 23
Liquid 76 77
30 Er
70 Sb
30 Er
70 Sb
ErSb rtErSb rt
William Meier November 2, 2018
Conservation of matter
55
30% Er
70% Sb
30% Er
70% Sb
30 Er
70 Sb
30 Er
70 Sb
T
(°C)Phases
at%
SbPhase%
1200ErSb rt 50 46
Liquid 87 54
ErSb rtErSb rt
William Meier November 2, 2018
Equilibrium solidification
56
Overall:
Cs0.9Na0.1
Overall:
Cs0.9Na0.1
Liquid
Cs0.90Na0.10
20°C
Micrograph
20°C
Micrograph
T
(°C)Phases
at%
NaPhase%
Overall
at% Na
20 Liquid 10 100 10
LiquidusLiquidus
Cs Na
CsN
a2
lt
(Cs)
(Na
) rt
William Meier November 2, 2018
Equilibrium solidification
57
Liquid
Cs0.90Na0.10
0°C
Micrograph
0°C
Micrograph
T
(°C)Phases
at%
NaPhase%
Overall
at% Na
0Liquid 10 100
10Cs 0 ~0
Cs metalCs metal
Cs Na
CsN
a2
lt
(Cs)
(Na
) rt
Overall:
Cs0.9Na0.1
Overall:
Cs0.9Na0.1
William Meier November 2, 2018
Equilibrium solidification
58
T
(°C)Phases
at%
NaPhase%
Overall
at% Na
-20Liquid 17 59
10Cs 0 41
-20°C
Micrograph
-20°C
MicrographCs metalCs metal
Liquid
Cs0.83Na0.17
Liquid
Cs0.83Na0.17
Cs Na
CsN
a2
lt
(Cs)
(Na
) rt
Overall:
Cs0.9Na0.1
Overall:
Cs0.9Na0.1
William Meier November 2, 2018
Equilibrium solidification
59
T
(°C)Phases
at%
NaPhase%
Overall
at% Na
-30Liquid 20 50
10Cs 0 50
-30°C
Micrograph
-30°C
MicrographCs metalCs metal
Liquid
Cs0.80Na0.20
Liquid
Cs0.80Na0.20
Cs Na
CsN
a2
lt
(Cs)
(Na
) rt
Overall:
Cs0.9Na0.1
Overall:
Cs0.9Na0.1
William Meier November 2, 2018
CsN
a2
lt
(Cs)
Equilibrium solidification
60
T
(°C)Phases
at%
NaPhase%
Overall
at% Na
-32
Liquid 20 -
10Cs 0 -
CsNa2 67 -
-32°C
Micrograph
-32°C
MicrographCs metalCs metal
Liquid
Cs0.80Na0.20
Liquid
Cs0.80Na0.20
Eutectic reaction
(Ca0.80Na0.20)liq → Cs + CsNa2
Eutectic reaction
(Ca0.80Na0.20)liq → Cs + CsNa2
Cs Na
(Na
) rt
Overall:
Cs0.9Na0.1
Overall:
Cs0.9Na0.1
William Meier November 2, 2018
CsN
a2
lt
(Cs)
Equilibrium solidification
61
T
(°C)Phases
at%
NaPhase%
Overall
at% Na
-40Cs 0 85
10CsNa2 67 15
-40°C
Micrograph
-40°C
Micrograph
Cs + CsNa2
Eutectic microstructure
Cs + CsNa2
Eutectic microstructure
Cs metalCs metal
Eutectic reaction
(Ca0.80Na0.20)liq → Cs + CsNa2
Eutectic reaction
(Ca0.80Na0.20)liq → Cs + CsNa2
Cs Na
(Na
) rt
Overall:
Cs0.9Na0.1
Overall:
Cs0.9Na0.1
William Meier November 2, 2018
Eutectic microstructures• Liquid rapidly solidifies
into thin regions of solid
phases on cooling
62
Sn-InSn-In Al-SiAl-Si
Al-CuAl-Cu Mg-SnMg-Sn
https://matdata.asminternational.org/apd/homepagefiles/grantami_apd/#
Sn-PbSn-Pb Al-SiAl-Si
William Meier November 2, 2018
Equilibrium solidification
63
Overall:
Cs0.5Na0.5
Overall:
Cs0.5Na0.5
Liquid
Cs0.50Na0.50
100°C
Micrograph
100°C
Micrograph
T
(°C)Phases
at%
NaPhase%
Overall
at% Na
100 Liquid 50 100 50
Cs Na
CsN
a2
lt
(Cs)
(Na
) rt
William Meier November 2, 2018
Equilibrium solidification
64
Overall:
Cs0.5Na0.5
Overall:
Cs0.5Na0.5
Liquid
Cs0.50Na0.50
50°C
Micrograph
50°C
Micrograph
T
(°C)Phases
at%
NaPhase%
Overall
at% Na
50Liquid 50 100
50Na 100 ~0
LiquidusLiquidus
Na metalNa metal
Cs Na
CsN
a2
lt
(Cs)
(Na
) rt
William Meier November 2, 2018
Equilibrium solidification
65
Overall:
Cs0.5Na0.5
Overall:
Cs0.5Na0.5
Liquid
Cs0.62Na0.38
T
(°C)Phases
at%
NaPhase%
Overall
at% Na
20Liquid 38 81
50Na 100 19
20°C
Micrograph
20°C
MicrographNa metalNa metal
Cs Na
CsN
a2
lt
(Cs)
(Na
) rt
William Meier November 2, 2018
Equilibrium solidification
66
Overall:
Cs0.5Na0.5
Overall:
Cs0.5Na0.5
Liquid
Cs0.69Na0.31
T
(°C)Phases
at%
NaPhase%
Overall
at% Na
-5Liquid 31 72
50Na 100 28
-5°C
Micrograph
-5°C
MicrographNa metalNa metal
Cs Na
CsN
a2
lt
(Cs)
(Na
) rt
William Meier November 2, 2018
Equilibrium solidification
67
Overall:
Cs0.5Na0.5
Overall:
Cs0.5Na0.5
Liquid
Cs0.70Na0.30
T
(°C)Phases
at%
NaPhase%
Overall
at% Na
-8
Liquid 30 -
50CsNa2 67 -
Na 100 -
-8°C
Micrograph
-8°C
MicrographNa metalNa metal
CsNa2CsNa2
Peritectic reaction
(Ca0.70Na0.30)liq + Na → CsNa2
Peritectic reaction
(Ca0.70Na0.30)liq + Na → CsNa2
Cs Na
CsN
a2
lt
(Cs)
(Na
) rt
William Meier November 2, 2018
Equilibrium solidification
68
Overall:
Cs0.5Na0.5
Overall:
Cs0.5Na0.5
T
(°C)Phases
at%
NaPhase%
Overall
at% Na
-10Liquid 29 45
50CsNa2 67 55
-10°C
Micrograph
-10°C
Micrograph
Liquid
Cs0.71Na0.29
Liquid
Cs0.71Na0.29
CsNa2CsNa2
Cs Na
CsN
a2
lt
(Cs)
(Na
) rt
William Meier November 2, 2018
CsN
a2
lt
(Cs)
Equilibrium solidification
69
Overall:
Cs0.5Na0.5
Overall:
Cs0.5Na0.5
T
(°C)Phases
at%
NaPhase%
Overall
at% Na
-30Liquid 22 38
50CsNa2 67 62
-30°C
Micrograph
-30°C
Micrograph
Liquid
Cs0.78Na0.22
Liquid
Cs0.78Na0.22
CsNa2CsNa2
Cs Na
(Na
) rt
William Meier November 2, 2018
CsN
a2
lt
(Cs)
Equilibrium solidification
70
Overall:
Cs0.5Na0.5
Overall:
Cs0.5Na0.5
T
(°C)Phases
at%
NaPhase%
Overall
at% Na
-40Cs 0 25
50CsNa2 67 75
-40°C
Micrograph
-40°C
Micrograph
CsNa2CsNa2
Cs + CsNa2
Eutectic microstructure
Cs + CsNa2
Eutectic microstructure
Cs Na
(Na
) rt
William Meier November 2, 2018
CsN
a2
lt
(Cs)
(Na
) rt
Non-equilibrium solidification
71
Overall:
Cs0.5Na0.5
Overall:
Cs0.5Na0.5
Liquid
Cs0.70Na0.30
T
(°C)Phases
at%
NaPhase%
Overall
at% Na
-8
Liquid 30 -
50CsNa2 67 -
Na 100 -
-8°C
Micrograph
-8°C
MicrographNa metalNa metal
CsNa2CsNa2
Cs Na
LNaNa
Cs
Rate limited by
solid state diffusion
Rate limited by
solid state diffusion
William Meier November 2, 2018
CsN
a2
lt
(Cs)
(Na
) rt
Non-equilibrium solidification
72
Overall:
Cs0.5Na0.5
Overall:
Cs0.5Na0.5
T
(°C)Phases
at%
NaPhase%
Overall
at% Na
-30
Liquid 22 -
50CsNa2 67 -
Na 100 -
-30°C
Micrograph
-30°C
MicrographNa metalNa metal
CsNa2CsNa2
Liquid
Cs0.78Na0.22
Liquid
Cs0.78Na0.22
Cs Na
William Meier November 2, 2018
CsN
a2
lt
(Cs)
(Na
) rt
Non-equilibrium solidification
73
Overall:
Cs0.5Na0.5
Overall:
Cs0.5Na0.5
T
(°C)Phases
at%
NaPhase%
Overall
at% Na
-30
Liquid 22 -
50CsNa2 67 -
Na 100 -
-30°C
Micrograph
-30°C
MicrographNa metalNa metal
CsNa2CsNa2
Cs + CsNa2
Eutectic microstructure
Cs + CsNa2
Eutectic microstructure
Cs Na
William Meier November 2, 2018
Peritectic microstructure
• Incomplete peritectic
reaction are common
74
Cam
pb
ell,
F. C
. P
has
e D
iagra
ms:
Und
erst
and
ing t
he
Bas
ics.
AS
M I
nte
rnat
ional
, 2
01
2.
Reaction
layer
Reaction
layerPrimary
phase
Primary
phase
-30°C
Micrograph
-30°C
MicrographNa metalNa metal
CsNa2CsNa2
Cs + CsNa2
Eutectic microstructure
Cs + CsNa2
Eutectic microstructureSolidified
Cu0.8Sn0.2 melt
Solidified
Cu0.8Sn0.2 melt
William Meier November 2, 2018
Scanning calorimetry
75
500
700
900
1100
1300
1500
0 20 40 60 80 100
Tem
pera
ture
Hypothetical Phase Diagrams
35% 45%60% 75% 80%
a db g
a+bb+d d+g
a+liquidb+liquid
d+liquid
g+liquidliquid
550 650 750 850 950 1050
75%Solid δ
Figures from Kevin Dennis
William Meier November 2, 2018
Scanning calorimetry
76
500
700
900
1100
1300
1500
0 20 40 60 80 100
Tem
pera
ture
Hypothetical Phase Diagrams
35% 45%60% 75% 80%
a db g
a+bb+d d+g
a+liquidb+liquid
d+liquid
g+liquidliquid
550 650 750 850 950 1050
75%Solid δ
600°C600°C
First heatingFirst heating
Figures from Kevin Dennis
William Meier November 2, 2018
Solid δ
• Peritectic decomposition
δ → Liquid + γ
Scanning calorimetry
77
500
700
900
1100
1300
1500
0 20 40 60 80 100
Tem
pera
ture
Hypothetical Phase Diagrams
35% 45%60% 75% 80%
a db g
a+bb+d d+g
a+liquidb+liquid
d+liquid
g+liquidliquid
550 650 750 850 950 1050
75%
First heatingFirst heating
900°C900°C
Solid γSolid γLiquidLiquid
Figures from Kevin Dennis
William Meier November 2, 2018
• γ dissolves into liquid
Scanning calorimetry
78
500
700
900
1100
1300
1500
0 20 40 60 80 100
Tem
pera
ture
Hypothetical Phase Diagrams
35% 45%60% 75% 80%
a db g
a+bb+d d+g
a+liquidb+liquid
d+liquid
g+liquidliquid
550 650 750 850 950 1050
75%
First heatingFirst heating
950°C950°C
Solid γSolid γLiquidLiquid
Figures from Kevin Dennis
William Meier November 2, 2018
• Homogenous liquid
Scanning calorimetry
79
500
700
900
1100
1300
1500
0 20 40 60 80 100
Tem
pera
ture
Hypothetical Phase Diagrams
35% 45%60% 75% 80%
a db g
a+bb+d d+g
a+liquidb+liquid
d+liquid
g+liquidliquid
550 650 750 850 950 1050
75%
First heatingFirst heating
1000°C1000°C
LiquidLiquid
Figures from Kevin Dennis
William Meier November 2, 2018
• γ precipitates from liquid
Scanning calorimetry
80
500
700
900
1100
1300
1500
0 20 40 60 80 100
Tem
pera
ture
Hypothetical Phase Diagrams
35% 45%60% 75% 80%
a db g
a+bb+d d+g
a+liquidb+liquid
d+liquid
g+liquidliquid
550 650 750 850 950 1050
75%
First coolingFirst cooling
950°C950°C
Solid γSolid γLiquidLiquid
Figures from Kevin Dennis
William Meier November 2, 2018
Solid δ
• Peritectic reaction
Liquid + γ→ δ
Incomplete
Scanning calorimetry
81
500
700
900
1100
1300
1500
0 20 40 60 80 100
Tem
pera
ture
Hypothetical Phase Diagrams
35% 45%60% 75% 80%
a db g
a+bb+d d+g
a+liquidb+liquid
d+liquid
g+liquidliquid
550 650 750 850 950 1050
75%
900°C900°C
Solid γSolid γLiquidLiquid
Solid δSolid δ
First coolingFirst cooling
Figures from Kevin Dennis
William Meier November 2, 2018
• δ crystallizes
• Peritectic reaction
Liquid + δ → β
Incomplete
Scanning calorimetry
82
500
700
900
1100
1300
1500
0 20 40 60 80 100
Tem
pera
ture
Hypothetical Phase Diagrams
35% 45%60% 75% 80%
a db g
a+bb+d d+g
a+liquidb+liquid
d+liquid
g+liquidliquid
550 650 750 850 950 1050
75%
800°C800°C
Solid γSolid γLiquidLiquid
Solid δSolid δ
First coolingFirst cooling
Solid βSolid β
Figures from Kevin Dennis
William Meier November 2, 2018
• β crystallizes
• Eutectic reaction
Liquid → α + β
Scanning calorimetry
83
500
700
900
1100
1300
1500
0 20 40 60 80 100
Tem
pera
ture
Hypothetical Phase Diagrams
35% 45%60% 75% 80%
a db g
a+bb+d d+g
a+liquidb+liquid
d+liquid
g+liquidliquid
550 650 750 850 950 1050
75%
600°C600°C
First coolingFirst cooling
Figures from Kevin Dennis
Solid γSolid γLiquidLiquid
Solid δSolid δ
Solid βSolid β
William Meier November 2, 2018
• β crystallizes
• Eutectic reaction
Liquid → α + β
Scanning calorimetry
84
500
700
900
1100
1300
1500
0 20 40 60 80 100
Tem
pera
ture
Hypothetical Phase Diagrams
35% 45%60% 75% 80%
a db g
a+bb+d d+g
a+liquidb+liquid
d+liquid
g+liquidliquid
550 650 750 850 950 1050
75%
600°C600°C
α + β eutectic
microstructure
α + β eutectic
microstructure
First coolingFirst cooling
Figures from Kevin Dennis
Solid γSolid γ
Solid δSolid δ
Solid βSolid β
William Meier November 2, 2018
Ce
Sb
2
Batch
Melt
Crystallize
Decant
Target
CeSb2 Liquidus
CeSb2-Sb Eutectic
Crystal growth from solution
85
P. C
. C
anfi
eld
et
al.,
Phil
os.
Mag
., 9
6(1
), 8
4–
92
(2
01
6)
P.C
. C
anfi
eld
and
Z.
Fis
k,
Phil
os.
Mag
. B
., 6
5(6
), 1
11
7–
11
23
. (
19
92
)
Ce Sb
William Meier November 2, 2018
Compositions
86
Spinel
MgAl2O4
• Mole % vs Atom %
• Spinel (MgAl2O4 ~ MgO•Al2O3)
• 50 mol% MgO – 50 mol% Al2O3
• 29 at% Mg0.5O0.5 – 71 at% Al0.4O0.6
Fro
m A
Cer
SP
has
e E
quil
ibri
a D
iagra
ms
Onli
ne
–N
o.
11
06
6
MgO Al2O3mol %
William Meier November 2, 2018
Compositions
87
• Atom % vs Weight %
• W and C have significantly
different atomic masses
• 60 at% ~ 9 wt% carbon
https://www.totalmateria.com/page.aspx?ID=CheckArticle&LN=TR&site=ktn&NM=364
William Meier November 2, 2018
More reading
• Introduction to phase
equilibria in ceramics
(Bergeron)
Where to find phase diagrams
• ASM Alloy Phase Diagram
Database (Metals)https://www.asminternational.org/home/-
/journal_content/56/10192/15469013/DATA
BASE
• ACerS-NIST Phase Equilibria
diagrams (Oxides)https://ceramics.org/publications-
resources/phase-equilibrium-diagrams
• Also in book form
88
https://www.amazon.com/Introduction-Equilibria-Ceramics-Clifton-Bergeron/dp/1574981773