Gibbs Phase Rule - Materials - UC Santa Barbara |matclass/101/pdffiles/Lecture_13.pdfIron-Carbon (Fe-C) Phase Diagram Adapted from Fig. 9.21,Callister 6e. (Fig. 9.21 adapted from Binary

Gibbs Phase RuleP + F = C + 2

P: # of phasesF: Degrees of freedomC: # of components

Normally, pressure = 1 atm

P + F = C + 1

or

F = C - P + 1Apply to eutectic phase diagram

1 phase field: F = 2 – 1 + 1 = 2 Change T and C independently in phase field

2 phase field: F = 2 – 2 + 1 = 1 C depends on T – not independent

3 phase point: F = 2 – 3 +1 = 0 C and T defined only at one point(Eutectic point) (no degrees of freedom)

Ex: Pb-Sn Eutectic System (1)

• For a 40wt%Sn-60wt%Pb alloy at 150 °C, find...--the phases present:

α + β--the compositions of

the phases:

Pb-Snsystem

Adapted from Fig. 9.7, Callister 6e. (Fig. 9.7 adaptedfrom Binary Phase Diagrams, 2nd ed., Vol. 3, T.B. Massalski (Editor-in-Chief), ASM International, Materials Park, OH, 1990.)

L + α L+β

α + β

200

T(°C)

18.3

Co, wt% Sn 20 40 60 80 100 0

Co

300

100

β

L (liquid)

α 183°C 61.9 97.8

150

Ex: Pb-Sn Eutectic System (2)

• For a 40wt%Sn-60wt%Pb alloy at 150 °C, find...--the phases present: α + β--the compositions of

the phases:Cα = 11wt%SnCβ = 99wt%Sn

--the relative amountsof each phase:

W α = 59 88

= 67 wt %

W β = 29 88

= 33 wt %

Pb-Snsystem

Adapted from Fig. 9.7, Callister 6e. (Fig. 9.7 adaptedfrom Binary Phase Diagrams, 2nd ed., Vol. 3, T.B. Massalski (Editor-in-Chief), ASM International, Materials Park, OH, 1990.)

L + α L+β

α + β

200

T(°C)

18.3

Co, wt% Sn 20 40 60 80 100 0

Co

300

100

L (liquid)

α 183°C 61.9 97.8

150

11 99

R S

β

Microstructures in Eutectic Systems (1)

L + α200

T(°C)

Co, wt% Sn10

2

200Co

300

100

L

α

30

L: Cowt%Sn

αL

α: Cowt%Sn

α + β

400

(room T solubility limit)

TE(Pb-Sn System)

• Co < 2wt%Sn• Result:

--polycrystal of α grains.

Adapted from Fig. 9.9, Callister 6e.

Microstructures in Eutectic Systems (2)

• 2wt%Sn < Co < 18.3wt%Sn• Result:

--α polycrystal with fineβ crystals.

α: Cowt%SnL + α

200

T(°C)

Co, wt% Sn10

18.3

200Co

300

100

L

α

30

L: Cowt%Sn

α + β

400

(sol. limit at TE)

TE

2(sol. limit at Troom)

Lα

αβ

Pb-Snsystem

Adapted from Fig. 9.10, Callister 6e.

Microstructures in Eutectic Systems (3)

L + α200

T(°C)

Co, wt% Sn

20 400

300

100

L

α

60

L: Cowt%Sn

α + β

TE

α: 18.3wt%Sn

β

080 100

L + β

CE18.3 97.861.9

183°C

β: 97.8wt%Sn160µm

Micrograph of Pb-Sn eutectic microstructure

• Co = CE• Result: Eutectic microstructure

--alternating layers of α and β crystals.

Pb-Snsystem

Adapted from Fig. 9.11, Callister 6e.

Adapted from Fig. 9.12, Callister 6e.(Fig. 9.12 from Metals Handbook, Vol. 9, 9th ed., Metallography and Microstructures, American Society for Metals, Materials Park, OH, 1985.)

Microstructures in Eutectic Systems (4)

• 18.3wt%Sn < Co < 61.9wt%Sn• Result: α crystals and a eutectic microstructure

L + α200

T(°C)

Co, wt% Sn

20 400

300

100

L

α

60

L: Cowt%Sn

α + β

TEβ

080 100

L + β

Co18.3 61.9

Lα

Lα

primary α

97.8

S

S

RR

eutectic αeutectic β

Pb-Snsystem

• Just above TE:

WL = (1-Wα) =50wt%

Cα = 18.3wt%Sn

CL = 61.9wt%SnS

R + SWα = =50wt%

• Just below TE:Cα = 18.3wt%Sn

Cβ = 97.8wt%SnS

R + SWα = =73wt%

Wβ = 27wt%Adapted from Fig. 9.14, Callister 6e.

Microstructure in Eutectic Systems (4')

Hypoeutectic Microstructure

Microstructure50% Pb – 50% Sn alloyExpain the phases

Hypoeutectic & Hypereutectic

T(°C)

(Pb-Sn System)

L + α200

Co, wt% Sn20 400

300

100

L

α

60

α + β

TE β

080 100

L + β

18.361.9

97.8

Cohypoeutectic

Cohypereutectic

eutectic

hypereutectic: (illustration only)

160µm

eutectic: Co=61.9wt%Sn

175µm

β

ββ

ββ

β

α

α

α

αα

α

hypoeutectic: Co=50wt%Sn

eutectic micro-constituent

Adapted from Fig. 9.7, Callister 6e. (Fig. 9.7 adapted from Binary Phase Diagrams, 2nd ed., Vol. 3, T.B. Massalski (Editor-in-Chief), ASM International, Materials Park, OH, 1990.)

(Figs. 9.12 and 9.15 from Metals Handbook, 9th ed.,Vol. 9, Metallography and Microstructures, American Society for Metals, Materials Park, OH, 1985.)

Adapted from Fig. 9.15, Callister 6e. Adapted from Fig. 9.12,

Callister 6e.

Adapted from Fig. 9.15, Callister 6e. (Illustration only)

Eutectic Microstructure Formation

Elemental partitioningα-phase: Pb-rich – rejects Snβ-phase: Sn-rich – rejects Pb

Higher cooling rateLess time for elemental redistributionFiner microstructure … greater strength … less ductility …

Phase Diagrams with Intermediate Phases/Compounds

Example: Cu-Zn binary phase diagramTerminal phases/solid solutions: α and ηIntermediate phases/solid solutions: β, β’. γ, and εCheck sequence of phase fieldsIdentify all transformations and invariant points

Phase Diagrams with Intermediate Phases/Compounds

Example: Mg-Pb binary phase diagramTerminal phases/solid solutions: α and βIntermediate compound: Mg2Pb (line compound)

Phase Diagrams with Intermediate Phases/Compounds

Example: Sn-Au binary phase diagramTerminal phases/solid solutions: α and ηIntermediate compounds: β, γ, δ, ζ

Eutectoid and Peritectic Reactions

Eutectoid

δ ↔ γ + ε

Pertectic

δ + L ↔ ε

Section of Cu-Zn Phase Diagram

Congruent Phase Transformations

Congruent transformations

No composition changefor phase transformation

e.g.,

L ↔ γ(for 44.9% Ti in Ni-Ti system)

Incongruent transformations

Composition changefor at least one phasefor phase transformation

Section of Ni-Ti Phase Diagram

Fe-C Phase Diagram (Steel!)

Iron-Carbon (Fe-C) Phase Diagram

Adapted from Fig. 9.21,Callister 6e. (Fig. 9.21 adapted from Binary Alloy Phase Diagrams, 2nd ed.,Vol. 1, T.B. Massalski (Ed.-in-Chief), ASM International, Materials Park, OH, 1990.)

(Adapted from Fig. 9.24, Callister 6e. (Fig. 9.24 from Metals Handbook, 9th ed., Vol. 9, Metallography and Microstructures, American Society for Metals, Materials Park, OH, 1985.)

Result: Pearlite = alternating layers of α and Fe3C phases.

120µm

• 2 important points

-Eutectic (A):

-Eutectoid (B): L ⇒ γ + Fe3C

γ ⇒ α +Fe3C

Fe

3C

(c

em

en

tite

)

1600

1400

1200

1000

800

600

4000 1 2 3 4 5 6 6.7

L

γ (austenite)

γ+L

γ+Fe3C

α+Fe3C

α+γ

L+Fe3C

δ

(Fe) Co, wt% C0.77 4.30

727°C = Teutectoid

1148°C

T(°C)

A

B

SR

R S

γ γγγ

Fe3C (cementite-hard)α (ferrite-soft)

αC

eu

tec

toid

Hypoeutectoid Steel

Adapted from Figs. 9.21 and 9.26,Callister 6e. (Fig. 9.21 adapted from Binary Alloy Phase Diagrams, 2nd ed., Vol. 1, T.B. Massalski (Ed.-in-Chief), ASM International, Materials Park, OH, 1990.)

Adapted fromFig. 9.27,Callister6e. (Fig. 9.27 courtesy Republic Steel Corporation.)

(Fe-C System)

Co

Fe

3C

(c

em

en

tite

)

1600

1400

1200

1000

800

600

4000 1 2 3 4 5 6 6.7

L

γ (austenite)

γ+L

γ+Fe3C

α+Fe3C

L+Fe3C

δ

Co, wt% C0.7

7

727°C

1148°C

T(°C)

R S

γ γγγ

α

γγγ γ

γγ γ

γ r s

wα =s/(r+s)wγ =(1-wα)

wα =S/(R+S)wFe3C =(1-wα)

wpearlite = wγ

α

αα

αα

α pearlite

100µm Hypoeutectoid steel

Hypoeutectoic Steel

MicrostructureFe-0.38% C Steel

Hypereutectoid Steel

(Fe-C System)

Co

Fe

3C

(c

em

en

tite

)

1600

1400

1200

1000

800

600

4000 1 2 3 4 5 6 6.7

L

γ (austenite)

γ+L

γ+Fe3C

α+Fe3C

L+Fe3C

δ

Co, wt% C0.7

71148°C

T(°C)

R S

γ γγγ

αs

wFe3C =r/(r+s)wγ =(1-wFe3C)

wα =S/(R+S)wFe3C =(1-wα)

wpearlite = wγpearlite

60µm Hypereutectoid steel

rγγ

γ γ

γγγ γ

Fe3C

Adapted from Figs. 9.21 and 9.29,Callister 6e. (Fig. 9.21 adapted from Binary Alloy Phase Diagrams, 2nd ed., Vol. 1, T.B. Massalski (Ed.-in-Chief), ASM International, Materials Park, OH, 1990.)

Adapted fromFig. 9.30,Callister6e. (Fig. 9.30copyright 1971 by United States Steel Corporation.)

Pearlite Microstructure

Partitioning of carbon (to Fe3C)

Alloying Steel With More ElementsT

Eu

tec

toid

(°C

)

wt. % of alloying elements

Ti

Ni600

800

1000

1200

0 4 8 12

Mo SiW

Cr

Mn

wt. % of alloying elements

Ce

ute

cto

id (

wt%

C)

Ni

Ti

0 4 8 120

0.2

0.4

0.6

0.8

Cr

SiMnW

Mo

• Teutectoid changes: • Ceutectoid changes:

Adapted from Fig. 9.31,Callister 6e. (Fig. 9.31 from Edgar C. Bain, Functions of the Alloying Elements in Steel, American Society for Metals, 1939, p. 127.)

Adapted from Fig. 9.32,Callister 6e. (Fig. 9.32 from Edgar C. Bain, Functions of the Alloying Elements in Steel, American Society for Metals, 1939, p. 127.)

Phase Diagram:Precipitation Strengthening System

Al-Cu Alloys – 1st Precipitation-Hardenable Alloys

Homogenize

HeatPrecipitation of Θ or related phases

Cool QuicklySupersaturatedα-phase

Precipitation Strengthening In Al-Cu

Precipitation in Al-4%Cu(aged at 180 °C, 6 h)GP-Zones

Precipitation in Al-4%Cu(aged at 200 °C, 2 h)θ’ Phase

Precipitation in Al-4%Cu(aged at 450 °C, 45 min.)θ Phase