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General Phase Diagram Sections

Arthur D. Pelton

Centre de Recherche en Calcul Thermochimique

École Polytechnique de Montréal

Montréal, Québec, CANADA

General rules of construction of all true phase diagram sections;Proper choice of variables and constants to give a “true” phase

diagram section (with a unique equilibrium state at each point);A general algorithm for calculating all true phase diagram sections.

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CaO-MgO: phase diagram

MgO + Liquid

CaO MgO

Liquid

CaO + Liquid

CaO + MgO

Mole fraction MgO

T(K

)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11200

1400

1600

1800

2000

2200

2400

2600

2800

3000

3200

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0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.10.20.30.40.50.60.70.80.9

0.10.2

0.30.4

0.50.6

0.70.8

0.9

weight fraction CaO

SiO2

CaO Al2O3

wei

ght f

ract

ion

SiO

2

weight fraction A

l2 O3

à

à

ßCA2

C3S

C2S

Mullite

L

L + C

L + A

L +

L + S

Mullite

L+C3S+C

L+C2S

L+CA2

L+C3S

L+C2S+C3S

L+CA6+CA2

L+CA6

L+A+CA6

L+A+Mullite

CA6

CA6 = (CaO)(Al2O3)6

CA2 = (CaO)(Al2O3)2

C3S = (CaO)3(SiO2)C2S = (CaO)2(SiO2)S = SiO2

A = Al2O3

C = CaOL = Liquid

System SiO2 - CaO - Al2O3

1600 °C

Diagram: CaO-Al2O3-SiO2

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C-Fe-W System at 5 wt% W

LIQUID

LIQUID + BCCBCC

BCC

M6C + BCC

M6C + BCC + Fe2W

M6C + BCC + WC

M6C + FCC + BCC

M6C + FCC

M6C + FCC + WC

FCC + BCC + WC

BCC + WC + C

FCC + WC + C

FCC + WC

FCC

FCC +FCC + LIQUID + BCC

FCC + LIQUID

FCC + LIQUID + WC

BCC

BCC + WC

Fe - C - Wwt% W = 5%

wt% C

Tem

per

atu

re (

K)

0.0 0.5 1.0 1.5 2.0

900

1000

1100

1200

1300

1400

1500

1600

1700

1800

1900

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A Phase Diagram for the Fe-Cr-V-C System

fcc

fcc + MC

fcc + M7C3

bcc + M23C6

fcc + bcc+ M23C6

fcc + bccfcc + MC + M7C3

bcc+ fcc+ MC

+ M7C3 bcc + M7C3

bcc + MC + M7C3

bcc + fcc + MC

bcc + MC+ M23C6

– fcc + M23C6

– fcc + M7C3 + M23C6

– bcc + fcc + M7C3 + M23C6

– bcc + MC + M7C3 + M23C6

bcc + MC

+ M7C3

bcc

+ M 7

C 3 +

M23

C 6

‚

‚ƒ

„

„

ƒ

Fe - Cr - V - C SystemT = 850°C, wt.% C = 0.3, Ptot = 1 atm

<F*A*C*T>

mass fraction Cr

mas

s fr

acti

on

V

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.160.00

0.01

0.02

0.03

0.04

0.05

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Fe-Cr-V-C system at 1.5 wt% Cr and 0.1 wt% V

FCC

MC + FCC

MC + FCC + CEMENTITE

MC + BCC + CEMENTITEMC + BCC + CEMENTITE + M7C3

MC + BCC + M7C3

MC + FCC + BCC

FCC + BCC

MC + FCC + BCC + M7C3

MC + FCC + BCC + CEMENTITE BCC + CEMENTITE

BCC + CEMENTITE + FCC

MC + FCC+ M7C3

Fe-Cr-V-C systemat 1.5 wt% Cr and 0.1 wt% V

wt % C

Tem

per

atu

re (

°C)

0.0 0.2 0.4 0.6 0.8 1.0600

700

800

900

1000

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P–T diagram for Al2TiO5

Kyanite

Sillimanite

Andalusite

Al2SiO5

Temperature (°C)

Pre

ssure

(bar)

350 450 550 650 750 850 9500

2000

4000

6000

8000

10000

12000

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Fe2O3(s2)

Spinel

Slag-liqFe-liq

Fe(alpha)

Fe(gamma)

Fe(alpha)

Fe2O3(s1)

Wustite

RT lnPO2 (kJ)

Te

mp

era

ture

(K

)

-500 -400 -300 -200 -100 0 100700

800

900

1000

1100

1200

1300

1400

1500

1600

1700

1800

1900

2000

Fe-O2 System

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Corundum

Spinel+

Corundum

Spinel

Wustite + Spinel

Spinel + FCC

Corundum + FCC

Corundum + BCC

BCC

FCC FCC + BCC

1573 K

Mole Ratio Cr/(Fe+Cr)

log

10(P

O2)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

System Fe-Cr-O2

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SO2-O2-Cu System at 1000 K

CuS

Cu2S

Cu

Cu2OCuO

(CuO)(CuSO4)Cu2SO4

CuSO4

1000 K

log10 PO2

log

10 P

SO

2

-20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0-12

-10

-8

-6

-4

-2

0

2

4

6

8

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Fe2O3-MgO-SiO2-O2 SystemSiO2/(Fe2O3+ MgO + SiO2) = 0.20 in air

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Fe-Cr-S-O System

Spinel

(Fe,Cr)S

Cr2O3

Cr2O3

Spinel + (Fe,Cr)S

bcc Alloy

Spinel + fcc Alloy

fcc Alloy+

(Fe,

Cr)

S +

Cr 2

O3

bcc Alloy +

bcc Alloy + (Fe,Cr)S

fcc Alloy + (Fe,Cr)S

Fe-Cr-S-O SystemT=1273 K , Molar ratio: nCr/nFe = 1

log10(PO2)

log

10(P

S2)

-25 -23.5 -22 -20.5 -19 -17.5 -16 -14.5 -13 -11.5 -10-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

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A Phase Diagram for the Fe-Cr-V-C System

fcc

fcc + MC

fcc + M7C3

bcc + M23C6

fcc + bcc+ M23C6

fcc + bccfcc + MC + M7C3

bcc+ fcc+ MC

+ M7C3 bcc + M7C3

bcc + MC + M7C3

bcc + fcc + MC

bcc + MC+ M23C6

– fcc + M23C6

– fcc + M7C3 + M23C6

– bcc + fcc + M7C3 + M23C6

– bcc + MC + M7C3 + M23C6

bcc + MC

+ M7C3

bcc

+ M 7

C 3 +

M23

C 6

‚

‚ƒ

„

„

ƒ

Fe - Cr - V - C SystemT = 850°C, wt.% C = 0.3, Ptot = 1 atm

<F*A*C*T>

mass fraction Cr

mas

s fr

acti

on

V

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.160.00

0.01

0.02

0.03

0.04

0.05

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The Law of Adjoining Phase Regions

«As a phase boundary line is crossed, one and only one phase appears or disappears.»

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A node in a general true phase diagram section.

1 + 2 + … + n)

1 + 2 + … + n) 1 + 2 + … + n)

1 + 2 + … + n)

+ +

++

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CaO-MgO: phase diagram

MgO + Liquid

CaO MgO

Liquid

CaO + Liquid

CaO + MgO

Mole fraction MgO

T(K

)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11200

1400

1600

1800

2000

2200

2400

2600

2800

3000

3200

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An isobaric binary temperature-composition diagram with the eutectic “opened-up” to show that this is an infinitely narrow 3-phase region.

BA

T

XB

L

+ L + + L

+ L

+

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SO2-O2-Cu System at 1000 K

CuS

Cu2S

Cu

Cu2OCuO

(CuO)(CuSO4)Cu2SO4

CuSO4

1000 K

log10 PO2

log

10 P

SO

2

-20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0-12

-10

-8

-6

-4

-2

0

2

4

6

8

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A “potential-potential” phase diagram “opened-up” to show that the lines are infinitely narrow 2-phase regions.

1

+

2

+

+

++

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• Extensive Variables(not the same for all phases at equilibrium)

Xi (composition)

V (volume)

• Potential Variables(the same for all phases at equilibrium)

T (temperature)

P (pressure)

i (chemical potential)

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If only potential variables are held constant, then all tie-lines lie in the plane of the section. In this case, the compositions of the individual phases at equilibrium can be read from the diagram and the lever rule applies.

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Zero Phase Fraction (ZPF) Lines

fcc

fcc + MC

fcc + M7C3

bcc + M23C6

fcc + bcc+ M23C6

fcc + bccfcc + MC + M7C3

bcc+ fcc+ MC

+ M7C3 bcc + M7C3

bcc + MC + M7C3

bcc + fcc + MC

bcc + MC+ M23C6

– fcc + M23C6

– fcc + M7C3 + M23C6

– bcc + fcc + M7C3 + M23C6

– bcc + MC + M7C3 + M23C6

bcc + MC

+ M7C3

bcc

+ M 7

C 3 +

M23

C 6

‚

‚ƒ

„

„

ƒ

Fe - Cr - V - C SystemT = 850°C, wt.% C = 0.3, Ptot = 1 atm

<F*A*C*T>

mass fraction Cr

mas

s fr

acti

on

V

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.160.00

0.01

0.02

0.03

0.04

0.05MCfccbccM7C3

M23C6

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Zero Phase Fraction (ZPF) Lines

System CaO - MgOT vs. (mole fraction) P = constant = 1 bar

Mole fraction XCaO

Te

mp

era

ture

, °C

0.0 0.2 0.4 0.6 0.8 1.01600

1800

2000

2200

2400

2600

2800

LIQUID

LIQUID + L+

SOLID SOLID

2 SOLIDS

()

So

lidu

s

Liquidus

So

lvu

s lin

e So

lvus lin

e

LIQUID

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Fe-S-O Predominance diagram (ZPF lines)

Fe2(SO4)3(s)FeS(s3)

FeSO4(s)

Fe(s) Fe3O4(s) Fe2O3(s)

FeS2(s)

Fe - S - O SystemPredominance diagram T = constant = 800 K

log10 PO2 , atm

log

10 P

S2 ,

atm

-35 -30 -25 -20 -15 -10 -5 0-40

-35

-30

-25

-20

-15

-10

-5

0

5

10

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Choice of variables to always give a true phase diagram (single-valued) everywhere

Phase Diagram 26 www.factsage.com

N-Component System (A-B-C-…-N)Extensive variable Corresponding potential

qi

S T

V -PnA

A

nB

B

. .

. .

. .nN

N

0i i i iSdT VdP n d q d Gibbs-Duhem:

i i i idU TdS PdV dn dq

j

ii q

U

q

Phase Diagram 27 www.factsage.com

Choice of variables

N-component system(1) Choose n potentials: 1, 2, … , n

(2) From the non-corresponding extensive variables (qn+1, qn+2, … ), form (N+1-n) independent ratios (Qn+1, Qn+2, …, QN+1).

Example:

[1, 2, … , n; Qn+1, Qn+2, …, QN+1] are then the (N+1) variables of which 2 are chosen as axes and the remainder are held constant.

1n N

2

1

1 1ii N

jj n

qQ n i N

q

Phase Diagram 28 www.factsage.com

MgO-CaO Binary System

S T

V -P

nMgO MgO

nCaO CaO

1 = T

2 = -P

3

3

4

MgO

CaO

MgO CaOCaO

q nn

Qn n

q n

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CaO-MgO: phase diagram

MgO + Liquid

CaO MgO

Liquid

CaO + Liquid

CaO + MgO

Mole fraction MgO

T(K

)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11200

1400

1600

1800

2000

2200

2400

2600

2800

3000

3200

Phase Diagram 30 www.factsage.com

Fe-Cr-S-O System

S T

V -P

nFe Fe

nCr Cr

1 = T

2 = -P

2

2

3

4

5

5

6

O

S

Cr

Cr

FeFe

q nn

Qn

q n

22

2 2

O

S

O

S

n

n

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Spinel

(Fe,Cr)S

Cr2O3

Cr2O3

Spinel + (Fe,Cr)S

bcc Alloy

Spinel + fcc Alloy

fcc Alloy+

(Fe,

Cr)

S +

Cr 2

O3

bcc Alloy +

bcc Alloy + (Fe,Cr)S

fcc Alloy + (Fe,Cr)S

Fe-Cr-S-O SystemT=1273 K , Molar ratio: nCr/nFe = 1

log10(PO2)

log

10(P

S2)

-25 -23.5 -22 -20.5 -19 -17.5 -16 -14.5 -13 -11.5 -10-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

Fe-Cr-S-O System

Phase Diagram 32 www.factsage.com

Pressure vs. Volume diagram for H2O

This is NOT a true phase diagram.

S+L

L+G

S+G

P

V

Phase Diagram 33 www.factsage.com

Fe-Cr-C System

S T

V -P

nC C

nFe Fe

nCr Cr

1 = T

2 = -P

3 = C

(NOT OK)

(OK)

4

4

Cr

Fe C

Cr

e

r

F Cr C

nQ

n n n

nQ

n n

Requirement: 0 3j

i

dQfor i

dq

Phase Diagram 34 www.factsage.com

Fe-Cr-C system, T = 1300 K, XCr = nCr/(nFe+nCr+nC) vs. (carbon activity)

This is NOT a true phase diagram.

M23C6

M7C3

bcc

fcc

cementitelog(ac)

Mo

le f

ract

ion

of

Cr

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-3 -2 -1 0 1 2

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“Corresponding” phase diagrams of the Fe-O system

-Iron

-Iron

-Iron

Liquidiron

Liquid oxide

Wustite

Magnetite

Hematite

-500 -400 -300 -200 -100 0

RT ln PO2 (kJ)

-Iron

Wustite

-Iron

Wustite

-Iron Magnetite

Wustite

Magnetite

Hematite

Magnetite

Hematite+

Oxygen

Liquid iron Liquid iron liquid oxide

Liquid oxide

Wustite

700

900

1100

1300

1500

1700

1900

0 0.50 0.54 0.58 0.62

Mole fraction XO

Te

mp

era

ture

(K

)

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Other Sets of Extensive Variables and Corresponding Potentials

Example 1: Extensive variable Corresponding potential

qi

H -1/T V/T -P nj j /T

1 0j

j i iVHd d P n d q dT T TGibbs-Duhem:

j

ii q

U

q

Example 2: Extensive variable Corresponding potential

qi

TU/P -1/T A/T -T/P nj j /P

1 0j

j i iTU A Td d n d q dP T T P PGibbs-Duhem:

j

ii q

U

q

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Mg-Si binary phase diagram

Si + Mg2SiMgHCP + Mg2Si

Liquid + Mg2Si

Liquid + Mg2Si

Liquid + Si

Liquid

Liquid + MgHCP

Mg-Si

Molar fraction Si

Tem

pera

ture

(°C

)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

300

600

900

1200

1500

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Mg-Si, H vs XSi phase diagram