Phase Equilibrium

Engr 2110 – Chapter 9

Dr. R. R. Lindeke

Topics for Study

� Definitions of Terms in Phase studies

� Binary Systems�Complete solubility (fully miscible) systems

�Multiphase systems

� Eutectics

� Eutectoids and Peritectics

� Intermetallic Compounds

� The Fe-C System

Some Definitions

Component of a system: Pure metals and or compounds of which an

alloy is composed, e.g. Cu and Ag or Fe and Fe3C. They are the

solute(s) and solvent

System: A body of engineering material under investigation. e.g. Ag – Cu

system, NiO-MgO system (or even sugar-milk system)

Solubility Limit: The maximum concentration of solute atoms that may

dissolve in the Solvent to form a “solid solution” at some temperature.

Phases: A homogenous portion of a system that has uniform physical and

chemical characteristics, e.g. pure material, solid solution, liquid solution,

and gaseous solution, ice and water, syrup and sugar.

Microstructure: A system’s microstructure is characterized by the number

of phases present, their proportions, and the manner in which they are

distributed or arranged. Factors affecting microstructure are: alloying

elements present, their concentrations, and the heat treatment of the alloy.

Single phase system = Homogeneous system

Multi phase system = Heterogeneous system or mixtures

Definitions, cont

Figure 9.3 (a) Schematic representation of the one-component phase

diagram for H2O. (b) A projection of the phase diagram information at 1

atm generates a temperature scale labeled with the familiar

transformation temperatures for H2O (melting at 0°C and boiling at

100°C).

Figure 9.4 (a) Schematic representation of the one-component phase

diagram for pure iron. (b) A projection of the phase diagram information

at 1 atm generates a temperature scale labeled with important

transformation temperatures for iron. This projection will become one

end of important binary diagrams, such as that shown in Figure 9.19

Phase Equilibrium: A stable configuration with lowest free-energy (internal energy of

a system, and also randomness or disorder of the atoms or molecules (entropy).

Any change in Temperature, Composition, and Pressure causes an increase in free

energy and away from Equilibrium thus forcing a move to another ‘state’

Equilibrium Phase Diagram: It is a “map” of the information about the control of

microstructure or phase structure of a particular material system. The relationships

between temperature and the compositions and the quantities of phases present at

equilibrium are represented.

Definition that focus on “Binary Systems”

Binary Isomorphous Systems: An alloy system that contains two components that attain

complete liquid and solid solubility of the components, e.g. Cu and Ni alloy. It is the

simplest binary system.

Binary Eutectic Systems: An alloy system that contains two components that has a

special composition with a minimum melting temperature.

Definitions, cont

With these definitions in mind:

ISSUES TO ADDRESS...

• When we combine two elements...what “equilibrium state” would we expect to get?

• In particular, if we specify...--a composition (e.g., wt% Cu - wt% Ni), and

--a temperature (T ) and/or a Pressure (P)then...

How many phases do we get?

What is the composition of each phase?

How much of each phase do we get?

Phase BPhase A

Nickel atomCopper atom

Gibb’s Phase Rule: a tool to define the number of phases and/or degrees of phase changes that can be

found in a system at equilibrium

� For any system under study the rule determines if the system is at equilibrium

� For a given system, we can use it to predict how many phases can be expected

� Using this rule, for a given phase field, we can predict how many independent parameters (degrees of freedom) we can specify

� Typically, N = 1 in most condensed systems – pressure is fixed!

where:

F is # degrees of freedom of the system (independent parameters)

C is # components (elements) in system

P is # phases at equil.

N is # "noncompostional" parameters in system (temp &/or Press

F C P N= − +

ure)

Looking at a simple “Phase Diagram” for

Sugar – Water (or milk)

Effect of Temperature (T) & Composition (Co)

• Changing T can change # of phases:

D (100°C,90)

2 phases

B (100°C,70)

1 phase

See path A to B.

• Changing Co can change # of phases: See path B to D.

Adapted from Fig. 9.1, Callister 7e.

A (20°C,70)

2 phases

70 80 1006040200

Temperature (°C

)

Co =Composition (wt% sugar)

L

(liquid solutioni.e, syrup)

20

100

40

60

80

0

L(liquid) + S

(solid sugar)

water-

sugar

system

Phase Diagram: A Map based on a System’s Free Energy indicating

“equilibuim” system structures – as predicted by Gibbs Rule

• Indicate ‘stable’ phases as function of T, P & Composition,

• We will focus on:

-binary systems: just 2 components.

-independent variables: T and Co (P = 1 atm is almost always used).

Phase Diagram --for Cu-Ni system

• 2 phases are possible:

L (liquid)

α (FCC solid solution)

• 3 ‘phase fields’ are observed:

L

L + α

α

wt% Ni20 40 60 80 1000

1000

1100

1200

1300

1400

1500

1600

T(°C)

L (liquid)

α

(FCC solid solution)An

“Isomorphic”

Phase System

wt% Ni20 40 60 80 10001000

1100

1200

1300

1400

1500

1600

T(°C)

L (liquid)

α(FCC solid solution)

Cu-Ni

phase

diagram

Phase Diagrams:

• Rule 1: If we know T and Co then we know the # and types of all

phases present.

• Examples:

A(1100°C, 60): 1 phase: α

B(1250°C, 35): 2 phases: L + α

B(1250°C,35)

A(1100°C,60)

wt% Ni

20

1200

1300

T(°C)

L (liquid)

α

(solid)

30 40 50

Cu-Ni

system

Phase Diagrams:

• Rule 2: If we know T and Co we know the composition of each phase

• Examples:TA

A

35Co

32CL

At TA = 1320°C:

Only Liquid (L) CL = Co ( = 35 wt% Ni)

At TB = 1250°C:

Both α and L

CL = C liquidus ( = 32 wt% Ni here)

Cα = Csolidus ( = 43 wt% Ni here)

At TD = 1190°C:

Only Solid ( α)

Cα = Co ( = 35 wt% Ni)

Co = 35 wt% Ni

adapted from Phase Diagrams

of Binary Nickel Alloys, P. Nash (Ed.), ASM

International, Materials Park, OH, 1991.

BTB

DTD

tie line

4Cα3

Figure 9.31 The lever rule is a mechanical analogy to the mass-

balance calculation. The (a) tie line in the two-phase region is

analogous to (b) a lever balanced on a fulcrum.

� Tie line – a line connecting the phases in equilibrium with

each other – at a fixed temperature (a so-called Isotherm)

The Lever Rule

How much of each phase?

We can Think of it as a lever! So to

balance:

ML Mα

R S

RMSM L ⋅=⋅α

L

L

LL

LL

CC

CC

SR

RW

CC

CC

SR

S

MM

MW

−

−=

+=

−

−=

+=

+=

α

α

α

α

α

00

wt% Ni

20

1200

1300

T(°C)

L (liquid)

α

(solid)

30 40 50

BTB

tie line

Co

CL Cαααα

SR

• Rule 3: If we know T and Co then we know the amount of each phase (given in wt%)

• Examples:

At TA: Only Liquid (L)

WL = 100 wt%, Wα = 0

At TD: Only Solid ( α)

WL = 0, Wα = 100 wt%

Co = 35 wt% Ni

Therefore we define

wt% Ni

20

1200

1300

T(°C)

L (liquid)

α

(solid)

30 40 50

Cu-Ni

system

TAA

35Co

32CL

BTB

DTD

tie line

4Cα3

R S

At TB: Both α and L

% 733243

3543wt=

−

−=

= 27 wt%

WL= S

R +S

Wα=

RR +S

Notice: as in a lever “the opposite leg” controls with a balance (fulcrum) at the ‘base composition’ and R+S = tie line length = difference in composition limiting phase boundary, at the temp of

interest

wt% Ni20

1200

1300

30 40 501100

L (liquid)

α

(solid)

T(°C)

A

35Co

L: 35wt%Ni

Cu-Ni

system

• Phase diagram:Cu-Ni system.

• System is:--binaryi.e., 2 components:

Cu and Ni.

--isomorphousi.e., complete

solubility of one

component in

another; α phase

field extends from

0 to 100 wt% Ni.

Adapted from Fig. 9.4, Callister 7e.

• ConsiderCo = 35 wt%Ni.

Ex: Cooling in a Cu-Ni Binary

4635

4332

α: 43 wt% Ni

L: 32 wt% Ni

L: 24 wt% Ni

α: 36 wt% Ni

Bα: 46 wt% Ni

L: 35 wt% Ni

C

D

E

24 36

• Cα changes as we solidify.

• Cu-Ni case:

• Fast rate of cooling:Cored structure

• Slow rate of cooling:Equilibrium structure

First α to solidify has Cα = 46 wt% Ni.

Last α to solidify has Cα = 35 wt% Ni.

Cored vs Equilibrium Phases

First α to solidify:

46 wt% Ni

Uniform Cα:

35 wt% Ni

Last α to solidify:

< 35 wt% Ni

Cored (Non-equilibrium) Cooling

Notice:The Solidus

line is “tilted” in this non-

equilibrium

cooled environment

: Min. melting TE

2 componentshas a ‘special’ composition with a min.

melting Temp.

Binary-Eutectic (PolyMorphic) Systems

• Eutectic transition

L(CE) α(CαE) + β(CβE)

• 3 single phase regions

(L, α, β)

• Limited solubility: α: mostly Cu

β: mostly Ag

• TE : No liquid below TE

• CE

composition

Ex.: Cu-Ag system

Cu-Ag

system

L (liquid)

α L + αL+β β

α + β

Co , wt% Ag

20 40 60 80 1000200

1200

T(°C)

400

600

800

1000

CE

TE 8.0 71.9 91.2779°C

L+αL+β

α+ β

200

T(°C)

18.3

C, wt% Sn20 60 80 1000

300

100

L (liquid)

α183°C

61.9 97.8

β

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

system

EX: Pb-Sn Eutectic System (1)

α + β

--compositions of phases:

CO = 40 wt% Sn

--the relative amount

of each phase (by lever rule):

150

40Co

11Cα

99Cβ

SR

Cα = 11 wt% Sn

Cβ = 99 wt% Sn

Wα=Cβ - COCβ - Cα

=99 - 40

99 - 11=59

88= 67 wt%

SR+S

=

Wβ=CO - Cα

Cβ - Cα

=R

R+S

=29

88= 33 wt%=

40 - 11

99 - 11

Adapted from Fig. 9.8, Callister 7e.

L+β

α+ β

200

T(°C)

C, wt% Sn

20 60 80 1000

300

100

L (liquid)

α β

L+α

183°C

• For a 40 wt% Sn-60 wt% Pb alloy at 220°C, find...--the phases present: Pb-Sn

system

EX: Pb-Sn Eutectic System (2)

α + L

--compositions of phases:

CO = 40 wt% Sn

--the relative amount

of each phase:

Wα =CL - CO

CL - Cα

=46 - 40

46 - 17

=6

29= 21 wt%

WL =CO - Cα

CL - Cα

=23

29= 79 wt%

40

Co

46CL

17

Cα

220SR

Cα = 17 wt% Sn

CL = 46 wt% Sn

• Co < 2 wt% Sn

• Result:

--at extreme ends

--polycrystal of α grains

i.e., only one solid phase.

Microstructures In Eutectic Systems: I

0

L+ α

200

T(°C)

Co, wt% Sn10

2%

20Co

300

100

L

α

30

α+β

400

(room Temp. solubility limit)

TE(Pb-SnSystem)

αL

L: Co wt% Sn

α: Co wt% Sn

• 2 wt% Sn < Co < 18.3 wt% Sn

• Result:� Initially liquid

� Then liquid + α

� then α alone

� finally two solid phases

� α polycrystal

� fine β-phase inclusions

Microstructures in Eutectic Systems: II

Pb-Sn

system

L + α

200

T(°C)

Co , wt% Sn10

18.3

200Co

300

100

L

α

30

α+ β

400

(sol. limit at TE)

TE

2(sol. limit at Troom)

L

α

L: Co wt% Sn

αβ

α: Co wt% Sn

• Co = CE• Result: Eutectic microstructure (lamellar structure)

--alternating layers (lamellae) of α and β crystals.

Microstructures in Eutectic Systems: III

160µm

Micrograph of Pb-Sn eutectic microstructure

Pb-Snsystem

L + β

α + β

200

T(°C)

C, wt% Sn

20 60 80 1000

300

100

L

α β

L+α

183°C

40

TE

18.3

α: 18.3 wt%Sn

97.8

β: 97.8 wt% Sn

CE61.9

L: Co wt% Sn

45.1% αααα and 54.8% ββββ-- by Lever Rule

Lamellar Eutectic Structure

• 18.3 wt% Sn < Co < 61.9 wt% Sn

• Result: α crystals and a eutectic microstructure

Microstructures in Eutectic Systems: IV

18.3 61.9

SR

97.8

SR

primary αeutectic α

eutectic β

WL = (1-Wα) = 50 wt%

Cα = 18.3 wt% Sn

CL = 61.9 wt% SnS

R + SWα= = 50 wt%

• Just above TE :

• Just below TE :

Cα = 18.3 wt% Sn

Cβ = 97.8 wt% SnS

R + SWα= = 72.6 wt%

Wβ = 27.4 wt%

Pb-Sn

systemL+β

200

T(°C)

Co, wt% Sn

20 60 80 1000

300

100

L

α β

L+α

40

α+β

TE

L: Co wt% Sn LαLα

L+αL+β

α+ β

200

Co, wt% Sn20 60 80 1000

300

100

L

α βTE

40

(Pb-Sn System)

Hypoeutectic & Hypereutectic Compositions

160 µm

eutectic micro-constituent

hypereutectic: (illustration only)

β

ββ

ββ

β

from Metals Handbook, 9th ed., Vol. 9, Metallography and

Microstructures, American

Society for Metals, Materials Park, OH, 1985.

175 µm

α

α

α

αα

α

hypoeutectic: Co = 50 wt% Sn

T(°C)

61.9

eutectic

eutectic: Co= 61.9wt% Sn

“Intermetallic” Compounds

Mg2Pb

Note: an intermetallic compound forms a line - not an area -

because stoichiometry (i.e. composition) is exact.

Adapted from Fig. 9.20, Callister 7e.

An Intermetallic Compound is also

an important part of

the Fe-C system!

� Eutectoid: solid phase in equilibrium with two solid phases

S2 S1+S3

γγγγ αααα + Fe3C (727ºC)

intermetallic compound - cementite

cool

heat

� Peritectic: liquid + solid 1 in equilibrium with a single solid

2 (Fig 9.21)

S1 + L S2

δδδδ + L γγγγ (1493ºC)cool

heat

� Eutectic: a liquid in equilibrium with two solids

L αααα + ββββ

Eutectoid & Peritectic – some definitions

coolheat

Eutectoid & Peritectic

Cu-Zn Phase diagram

Adapted from Fig. 9.21, Callister 7e.

Eutectoid transition δ γ + ε

Peritectic transition γ + L δ

Iron-Carbon (Fe-C) Phase Diagram

• 2 important points

-Eutectoid (B):γ ⇒ α +Fe3C

-Eutectic (A):L ⇒ γ +Fe3C

Fe3C (cementite)

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

(Fe) Co, wt% C

1148°C

T(°C)

α 727°C = Teutectoid

A

SR

4.30

Result: Pearlite = alternating layers of α and Fe3C phases

120 µm

(Adapted from Fig. 9.27, Callister 7e.)

γ γ

γγ

R S

0.76

Ceutectoid

B

Fe3C (cementite-hard)

α (ferrite-soft)

Max. C solubility in γ iron = 2.11 wt%

Hypoeutectoid Steel

Adapted from Figs. 9.24 and 9.29,Callister 7e. (Fig. 9.24 adapted from

Binary Alloy Phase Diagrams, 2nd ed., Vol.

1, T.B. Massalski (Ed.-in-Chief), ASM International, Materials Park, OH,

1990.)

Fe3C (cementite)

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

(Fe) Co, wt% C

1148°C

T(°C)

α727°C

(Fe-C System)

C0

0.76

proeutectoid ferritepearlite

100 µmHypoeutectoid

steel

R S

α

wα =S/(R+S)

wFe3C

=(1-wα)

w pearlite = w γpearlite

r s

wα =s/(r+s)wγ =(1-wα)

γ

γ γ

γα

αα

γγγ γ

γ γ

γγ

Hypereutectoid Steel

Fe3C (cementite)

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

(Fe) Co, wt%C

1148°C

T(°C)

α

adapted from Binary Alloy Phase Diagrams, 2nd ed., Vol. 1, T.B. Massalski

(Ed.-in-Chief), ASM International, Materials

Park, OH, 1990.

(Fe-C

System)

0.76 Co

proeutectoid Fe3C

60 µmHypereutectoid steel

pearlite

R S

wα =S/(R+S)

wFe3C =(1-wα)

wpearlite = wγpearlite

sr

wFe3C =r /(r+s)

w γ =(1-w Fe3C )

Fe3C

γγ

γ γ

γγγ γ

γγγ γ

Example: Phase Equilibria

For a 99.6 wt% Fe-0.40 wt% C at a temperature

just below the eutectoid, determine the

following:

a) composition of Fe3C and ferrite (αααα)

b) the amount of carbide (cementite) in grams that

forms per 100 g of steel

c) the amount of pearlite and proeutectoid ferrite (αααα)

Solution:

g 3.94

g 5.7 CFe

g7.5100 022.07.6

022.04.0

100xCFe

CFe

3

CFe3

3

3

=α

=

=−

−=

−

−=

α+ α

α

x

CC

CCo

b) the amount of carbide

(cementite) in grams that

forms per 100 g of steel

a) composition of Fe3C and ferrite (α)

CO = 0.40 wt% C

Cα = 0.022 wt% C

CFe C = 6.70 wt% C3

Fe3C (cementite)

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% C

1148°C

T(°C)

727°C

CO

R S

CFe C3

Cαααα

Solution, cont:c) the amount of pearlite and proeutectoid ferrite (αααα)

note: amount of pearlite = amount of γγγγ just above TE

Co = 0.40 wt% C

Cα = 0.022 wt% C

Cpearlite = Cγ = 0.76 wt% C

γ

γ + α=Co −Cα

Cγ −Cα

x 100 = 51.2 g

pearlite = 51.2 g

proeutectoid α = 48.8 g

Fe3C (cementite)

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% C

1148°C

T(°C)

727°C

CO

R S

CγγγγCααααLooking at the Pearlite:11.1% Fe3C (.111*51.2 gm = 5.66 gm) & 88.9% α (.889*51.2gm = 45.5 gm)

∴ total α = 45.5 + 48.8 = 94.3 gm

Alloying Steel with More Elements

• Teutectoid changes: • Ceutectoid changes:

Adapted from Edgar C. Bain, Functions of the Alloying Elements in Steel, American Society for Metals, 1939, p. 127.)

Adapted from Edgar C. Bain, Functions of the Alloying Elements in Steel, American Society for Metals, 1939, p. 127.)

TEutectoid(°C)

wt. % of alloying elements

Ti

Ni

MoSi

W

Cr

Mn

wt. % of alloying elements

Ceutectoid

(wt%

C)

Ni

Ti

Cr

SiMn

WMo

Looking at a Tertiary Diagram

When looking at a Tertiary (3 element) P. diagram, like this

image, the Mat. Engineer represents equilibrium phases

by taking “slices at different 3rd

element content” from the 3 dimensional Fe-C-Cr phase

equilibrium diagram

What this graph shows are the increasing temperature of the euctectoid and the decreasing carbon content, and

(indirectly) the shrinking γ phase field

From: G. Krauss, Principles of Heat Treatment of Steels, ASM International, 1990

Fe-C-Si Ternary

phase diagram at

about 2.5

wt% SiThis is the

“controlling Phase

Diagram

for the ‘Graphite‘

Cast Irons

• Phase diagrams are useful tools to determine:

-- the number and types of phases,

-- the wt% of each phase,

-- and the composition of each phase

for a given T and composition of the system.

• Alloying to produce a solid solution usually

--increases the tensile strength (TS)

--decreases the ductility.

• Binary eutectics and binary eutectoids allow for

(cause) a range of microstructures.

Summary