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Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

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Page 1: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

Phase Equilibrium

Engr 2110 – Chapter 9

Dr. R. R. Lindeke

Page 2: 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

Page 3: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

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.

Page 4: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

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

Page 5: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

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).

Page 6: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

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

Page 7: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

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

Page 8: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

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

Page 9: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

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)

Page 10: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

Looking at a simple “Phase Diagram” for Sugar – Water (or milk)

Page 11: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

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

Tem

pera

ture

(°C

)

Co =Composition (wt% sugar)

L (liquid solution

i.e, syrup)

20

100

40

60

80

0

L (liquid)

+ S

(solid sugar)

water-sugarsystem

Page 12: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

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)

(FCC solid solution)

• 3 ‘phase fields’ are observed: LL +

wt% Ni20 40 60 80 10001000

1100

1200

1300

1400

1500

1600T(°C)

L (liquid)

(FCC solid solution)

L + liquidus

solid

us

An “Isomorphic” Phase System

Page 13: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

wt% Ni20 40 60 80 10001000

1100

1200

1300

1400

1500

1600T(°C)

L (liquid)

(FCC solid solution)

L +

liquidus

solid

us

Cu-Niphase

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:

B(1250°C, 35): 2 phases: L +

B (

1250

°C,3

5) A(1100°C,60)

Page 14: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

wt% Ni20

1200

1300

T(°C)

L (liquid)

(solid)L +

liquidus

solidus

30 40 50

L +

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

C = C solidus ( = 43 wt% Ni here)

At TD = 1190°C:

Only Solid ( ) C = 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

4C3

Page 15: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

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.

Page 16: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

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 M

R S

RMSM L

L

L

LL

LL CC

CC

SR

RW

CC

CC

SR

S

MM

MW

00

wt% Ni

20

1200

1300

T(°C)

L (liquid)

(solid)L +

liquidus

solidus

30 40 50

L + B

TB

tie line

CoCL C

SR

Page 17: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

• 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)

W L = 100 wt%, W = 0At TD: Only Solid ( )

W L = 0, W = 100 wt%

Co = 35 wt% Ni

Therefore we define

wt% Ni20

1200

1300

T(°C)

L (liquid)

(solid)L +

liquidus

solidus

30 40 50

L +

Cu-Ni system

TAA

35Co

32CL

BTB

DTD

tie line

4C3

R S

At TB: Both and L

% 733243

3543wt

= 27 wt%

WL S

R +S

W R

R +SNotice: 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

Page 18: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

wt% Ni20

1200

1300

30 40 501100

L (liquid)

(solid)

L +

L +

T(°C)

A

35Co

L: 35wt%Ni

Cu-Nisystem

• Phase diagram: Cu-Ni system.

• System is: --binary i.e., 2 components: Cu and Ni. --isomorphous i.e., complete solubility of one component in another; phase field extends from 0 to 100 wt% Ni.

Adapted from Fig. 9.4, Callister 7e.

• Consider Co = 35 wt%Ni.

Ex: Cooling in a Cu-Ni Binary

4635

4332

: 43 wt% Ni

L: 32 wt% Ni

L: 24 wt% Ni

: 36 wt% Ni

B: 46 wt% NiL: 35 wt% Ni

C

D

E

24 36

Page 19: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

• C changes as we solidify.• Cu-Ni case:

• Fast rate of cooling: Cored structure

• Slow rate of cooling: Equilibrium structure

First to solidify has C = 46 wt% Ni.

Last to solidify has C = 35 wt% Ni.

Cored vs Equilibrium Phases

First to solidify: 46 wt% Ni

Uniform C:

35 wt% Ni

Last to solidify: < 35 wt% Ni

Page 20: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

Cored (Non-equilibrium) Cooling

Notice:The Solidus line is “tilted” in this non-equilibrium cooled environment

Page 21: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

: Min. melting TE

2 componentshas a ‘special’ composition with a min.

melting Temp.

Binary-Eutectic (PolyMorphic) Systems

• Eutectic transition

L(CE) (CE) + (CE)

• 3 single phase regions (L, ) • Limited solubility: : mostly Cu : mostly Ag • TE : No liquid below TE

• CE

composition

Ex.: Cu-Ag system Cu-Agsystem

L (liquid)

L + L+

Co , wt% Ag

20 40 60 80 1000200

1200T(°C)

400

600

800

1000

CE

TE 8.0 71.9 91.2779°C

Page 22: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

L+L+

+

200

T(°C)

18.3

C, wt% Sn20 60 80 1000

300

100

L (liquid)

183°C 61.9 97.8

• 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)

+ --compositions of phases:

CO = 40 wt% Sn

--the relative amount of each phase (by lever rule):

150

40Co

11C

99C

SR

C = 11 wt% SnC = 99 wt% Sn

W=C - CO

C - C

=99 - 4099 - 11

= 5988

= 67 wt%

SR+S

=

W =CO - C

C - C=R

R+S

=2988

= 33 wt%=40 - 1199 - 11

Adapted from Fig. 9.8, Callister 7e.

Page 23: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

L+

+

200

T(°C)

C, wt% Sn

20 60 80 1000

300

100

L (liquid)

L+

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)

+ L--compositions of phases:

CO = 40 wt% Sn

--the relative amount of each phase:

W =CL - CO

CL - C=

46 - 40

46 - 17

= 6

29= 21 wt%

WL =CO - C

CL - C=

23

29= 79 wt%

40

Co

46

CL

17

C

220SR

C = 17 wt% SnCL = 46 wt% Sn

Page 24: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

• Co < 2 wt% Sn• Result: --at extreme ends --polycrystal of grains i.e., only one solid phase.

Microstructures In Eutectic Systems: I

0

L+ 200

T(°C)

Co, wt% Sn10

2%

20Co

300

100

L

30

+

400

(room Temp. solubility limit)

TE

(Pb-SnSystem)

L

L: Co wt% Sn

: Co wt% Sn

Page 25: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

• 2 wt% Sn < Co < 18.3 wt% Sn• Result:

Initially liquid Then liquid + then alonefinally two solid phases

polycrystal fine -phase inclusions

Microstructures in Eutectic Systems: II

Pb-Snsystem

L +

200

T(°C)

Co , wt% Sn10

18.3

200Co

300

100

L

30

+

400

(sol. limit at TE)

TE

2(sol. limit at Troom)

L

L: Co wt% Sn

: Co wt% Sn

Page 26: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

• Co = CE • Result: Eutectic microstructure (lamellar structure) --alternating layers (lamellae) of and crystals.

Microstructures in Eutectic Systems: III

160 m

Micrograph of Pb-Sn eutectic microstructure

Pb-Snsystem

L

200

T(°C)

C, wt% Sn

20 60 80 1000

300

100

L

L+ 183°C

40

TE

18.3

: 18.3 wt%Sn

97.8

: 97.8 wt% Sn

CE61.9

L: Co wt% Sn

45.1% and 54.8% -- by Lever Rule

Page 27: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

Lamellar Eutectic Structure

Page 28: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

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

Microstructures in Eutectic Systems: IV

18.3 61.9

SR

97.8

SR

primary eutectic

eutectic

WL = (1-W) = 50 wt%

C = 18.3 wt% Sn

CL = 61.9 wt% SnS

R + SW = = 50 wt%

• Just above TE :

• Just below TE :C = 18.3 wt% Sn

C = 97.8 wt% SnS

R + SW = = 72.6 wt%

W = 27.4 wt%

Pb-Snsystem

L+200

T(°C)

Co, wt% Sn

20 60 80 1000

300

100

L

L+

40

+

TE

L: Co wt% Sn LL

Page 29: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

L+L+

+

200

Co, wt% Sn20 60 80 1000

300

100

L

TE

40

(Pb-Sn System)

Hypoeutectic & Hypereutectic Compositions

160 m

eutectic micro-constituent

hypereutectic: (illustration only)

from Metals Handbook, 9th ed., Vol. 9, Metallography and Microstructures, American Society for Metals, Materials Park, OH, 1985.

175 m

hypoeutectic: Co = 50 wt% Sn

T(°C)

61.9eutectic

eutectic: Co = 61.9 wt% Sn

Page 30: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

“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!

Page 31: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

Eutectoid: solid phase in equilibrium with two solid phases

S2 S1+S3

+ Fe3C (727ºC)

intermetallic compound - cementite

cool

heat

Peritectic: liquid + solid 1 in equilibrium with a single solid 2 (Fig 9.21)

S1 + L S2

+ L (1493ºC)

cool

heat

Eutectic: a liquid in equilibrium with two solids

L +

Eutectoid & Peritectic – some definitions

coolheat

Page 32: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

Eutectoid & Peritectic

Cu-Zn Phase diagram

Adapted from Fig. 9.21, Callister 7e.

Eutectoid transition +

Peritectic transition + L

Page 33: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

Iron-Carbon (Fe-C) Phase Diagram• 2 important points

-Eutectoid (B):

+Fe3C

-Eutectic (A): L + Fe3C

Fe 3

C (

cem

entit

e)

1600

1400

1200

1000

800

600

4000 1 2 3 4 5 6 6.7

L

(austenite)

+L

+Fe3C

+Fe3C

+

L+Fe3C

(Fe) Co, wt% C

1148°C

T(°C)

727°C = Teutectoid

A

SR

4.30

Result: Pearlite = alternating layers of and Fe3C phases

120 m

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

R S

0.76

Ceu

tect

oid

B

Fe3C (cementite-hard) (ferrite-soft)

Max. C solubility in iron = 2.11 wt%

Page 34: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

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.)

Fe 3

C (

cem

entit

e)

1600

1400

1200

1000

800

600

4000 1 2 3 4 5 6 6.7

L

(austenite)

+L

+ Fe3C

+ Fe3C

L+Fe3C

(Fe) Co , wt% C

1148°C

T(°C)

727°C

(Fe-C System)

C0

0.76

proeutectoid ferritepearlite

100 m Hypoeutectoidsteel

R S

w =S/(R+S)wFe3C =(1-w)

wpearlite = wpearlite

r s

w =s/(r+s)w =(1- w)

Page 35: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

Hypereutectoid Steel

Fe 3

C (

cem

entit

e)

1600

1400

1200

1000

800

600

4000 1 2 3 4 5 6 6.7

L

(austenite)

+L

+Fe3C

+Fe3C

L+Fe3C

(Fe) Co , wt%C

1148°C

T(°C)

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)

0.7

6 Co

proeutectoid Fe3C

60 mHypereutectoid steel

pearlite

R S

w =S/(R+S)wFe3C =(1-w)

wpearlite = wpearlite

sr

wFe3C =r/(r+s)w =(1-w Fe3C )

Fe3C

Page 36: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

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 ()

b) the amount of carbide (cementite) in grams that forms per 100 g of steel

c) the amount of pearlite and proeutectoid ferrite ()

Page 37: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

Solution:

g 3.94

g 5.7 CFe

g7.5100 022.07.6

022.04.0

100xCFe

CFe

3

CFe3

3

3

x

CC

CCo

b) the amount of carbide (cementite) in grams that forms per 100 g of steel

a) composition of Fe3C and ferrite ()

CO = 0.40 wt% C

C = 0.022 wt% C

CFe C = 6.70 wt% C3

Fe 3

C (

cem

entit

e)

1600

1400

1200

1000

800

600

4000 1 2 3 4 5 6 6.7

L

(austenite)

+L

+ Fe3C

+ Fe3C

L+Fe3C

Co , wt% C

1148°C

T(°C)

727°C

CO

R S

CFe C3C

Page 38: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

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

note: amount of pearlite = amount of just above TE

Co = 0.40 wt% C

C = 0.022 wt% C

Cpearlite = C = 0.76 wt% C

Co CC C

x 100 51.2 g

pearlite = 51.2 gproeutectoid = 48.8 g

Fe 3

C (

cem

entit

e)

1600

1400

1200

1000

800

600

4000 1 2 3 4 5 6 6.7

L

(austenite)

+L

+ Fe3C

+ Fe3C

L+Fe3C

Co , wt% C

1148°C

T(°C)

727°C

CO

R S

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

total = 45.5 + 48.8 = 94.3 gm

Page 39: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

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.)

TE

ute

cto

id (

°C)

wt. % of alloying elements

Ti

Ni

Mo SiW

Cr

Mn

wt. % of alloying elements

Ce

ute

cto

id (

wt%

C)

Ni

Ti

Cr

SiMn

WMo

Page 40: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

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 phase field

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

Page 41: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

Fe-C-Si Ternary phase diagram at about 2.5 wt% SiThis is the “controlling Phase Diagram for the ‘Graphite‘ Cast Irons

Page 42: Phase Equilibrium Engr 2110 – Chapter 9 Dr. R. R. Lindeke

• 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