Department of Physics and Astronomy Introduction to phase ...

Department of Physics and Astronomy

William Meier

Physics 590B

Fall 2018

Introduction to phase diagrams

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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


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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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)


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


Other p-T phase diagrams

6

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Car

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_d

ioxid

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-tem

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ature

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CO2

https://commons.wikimedia.org/wiki/File:Pure_iron_phase_diagram_(EN).svg

Fe• Phase fields

• Boundaries and coexistence


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)


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


• 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


T-H phase diagram

• Magnetic field suppresses

the superconducting states

10

http://hyperphysics.phy-astr.gsu.edu/hbase/Solids/scbc.html


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


Part 2 – Fundamental concepts

• Definitions

• Reactions and transitions

• The lever rule (conservation of matter)

13


Definition

Phase

• A region of space

occupied by a

homogeneous material

• Mechanically separable?

Examples

• Water vapor

• Liquid water

• Ice

• Diamond

• Graphite

• Paramagnet

• Ferromagnet

14


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


Definition

State variables• Parameters that determine

the configuration of the

system

16

Examples

• Composition

• Temperature

• Pressure

• Electric field

• Magnetic field


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


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


Definition

Equilibrium

• Practical definition:

• The configuration is

independent of time

• Changes could be slow

• Is the equilibrium

configuration accessible?

19

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Batman (William Meier)

Physics 590B

Fall 2018


Day 2

AS

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29


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


(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


Determining the stable phases

24


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



25


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



26


2. Draw tie line



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



27


2. Draw tie line



phases



ErSb rt + L

Er5Sb3Er5Sb3(Er0.99,Sb0.01)(Er0.99,Sb0.01)



28


2. Draw tie line



phases



ErSb rt + L

(Er) +

Er5Sb3



29


2. Draw tie line



phases



ErSb rt + L

(Er) +

Er5Sb3

L +

Er5Sb3



30


2. Draw tie line



phases



ErSb rt + L

(Er) +

Er5Sb3

L +

Er5Sb3

L + ErSb rt



31


2. Draw tie line



phases



ErSb rt + L

(Er) +

Er5Sb3

L +

Er5Sb3

Er 5

Sb

3+

ErS

b r

t

L + ErSb rt

ErSb ht + LErSb ht + L



32


2. Draw tie line



phases



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


Stable phases

33


Stable phases

34


Stable phases

35


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


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


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


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


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


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


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%


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


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


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


Invariant reactions – Eutectic

46


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


Invariant reactions – Peritectic

47


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


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


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


William Meier

Physics 590B

Fall 2018


Day 3

AS

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29


Review

• Stable phases

• Lever rule

• Eutectic

• Peritectic

51

© ASM International 2006. Diagram No. 900834

a b

Tie line


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



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



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



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


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



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



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



59

T

(°C)Phases

at%

NaPhase%

Overall

at% Na

-30Liquid 20 50

10Cs 0 50

-30°C

Micrograph

-30°C


Liquid

Cs0.80Na0.20

Liquid

Cs0.80Na0.20

Cs Na

CsN

a2

lt

(Cs)

(Na

) rt

Overall:

Cs0.9Na0.1

Overall:

Cs0.9Na0.1


CsN

a2

lt

(Cs)


60

T

(°C)Phases

at%

NaPhase%

Overall

at% Na

-32

Liquid 20 -

10Cs 0 -

CsNa2 67 -

-32°C

Micrograph

-32°C


Liquid

Cs0.80Na0.20

Liquid

Cs0.80Na0.20

Eutectic reaction

(Ca0.80Na0.20)liq → Cs + CsNa2

Eutectic reaction


Cs Na

(Na

) rt

Overall:

Cs0.9Na0.1

Overall:

Cs0.9Na0.1


CsN

a2

lt

(Cs)


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


Cs metalCs metal

Eutectic reaction


Eutectic reaction


Cs Na

(Na

) rt

Overall:

Cs0.9Na0.1

Overall:

Cs0.9Na0.1


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



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



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



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



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


Cs Na

CsN

a2

lt

(Cs)

(Na

) rt



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


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



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


CsN

a2

lt

(Cs)


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


CsN

a2

lt

(Cs)


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


Cs + CsNa2


Cs Na

(Na

) rt


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


CsNa2CsNa2

Cs Na

LNaNa

Cs

Rate limited by

solid state diffusion

Rate limited by

solid state diffusion


CsN

a2

lt

(Cs)

(Na

) rt


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


CsNa2CsNa2

Liquid

Cs0.78Na0.22

Liquid

Cs0.78Na0.22

Cs Na


CsN

a2

lt

(Cs)

(Na

) rt


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


CsNa2CsNa2

Cs + CsNa2


Cs + CsNa2


Cs Na


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


CsNa2CsNa2

Cs + CsNa2


Cs + CsNa2

Eutectic microstructureSolidified

Cu0.8Sn0.2 melt

Solidified

Cu0.8Sn0.2 melt


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



76

500

700

900

1100

1300

1500

0 20 40 60 80 100

Tem

pera

ture


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



Solid δ

• Peritectic decomposition

δ → Liquid + γ


77

500

700

900

1100

1300

1500

0 20 40 60 80 100

Tem

pera

ture


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



• γ dissolves into liquid


78

500

700

900

1100

1300

1500

0 20 40 60 80 100

Tem

pera

ture


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%


950°C950°C




• Homogenous liquid


79

500

700

900

1100

1300

1500

0 20 40 60 80 100

Tem

pera

ture


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%


1000°C1000°C

LiquidLiquid



• γ precipitates from liquid


80

500

700

900

1100

1300

1500

0 20 40 60 80 100

Tem

pera

ture


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 δ


Liquid + γ→ δ

Incomplete


81

500

700

900

1100

1300

1500

0 20 40 60 80 100

Tem

pera

ture


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 δSolid δ




• δ crystallizes


Liquid + δ → β

Incomplete


82

500

700

900

1100

1300

1500

0 20 40 60 80 100

Tem

pera

ture


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 δSolid δ


Solid βSolid β



• β crystallizes


Liquid → α + β


83

500

700

900

1100

1300

1500

0 20 40 60 80 100

Tem

pera

ture


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




Solid δSolid δ

Solid βSolid β


• β crystallizes


Liquid → α + β


84

500

700

900

1100

1300

1500

0 20 40 60 80 100

Tem

pera

ture


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



Solid γSolid γ

Solid δSolid δ

Solid βSolid β


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


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 %


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


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

https://www.asminternational.org/home/-/journal_content/56/10192/15469013/DATABASE

https://ceramics.org/publications-resources/phase-equilibrium-diagrams


Key concepts

• Equilibrium phases as a function of state variables

• Lever rule

• Invariant reactions (e.g. eutectic and peritectic)

• Equilibrium vs non-equilibrium processes

89

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