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Chapter 7 Phase Equilibria and Phase Diagrams

• The one-component phase diagram– Gibbs Phase Rule

• Phase equilibria in a two-component system– The isomorphous diagram– The lever rule– Equilibrium solidification and microstructure of

isomorphous alloys • Liquidius and solidus boundaries• Deviations from ideal behavior

Chapter 7 Phase Equilibria and Phase Diagrams, Continued

• Phase equilibria in a two-component system– The eutectic phase diagram– The peritectic phase diagram– The monotectic phase diagram– Complex diagrams– Phase equilibria involving solid-to-solid reactions

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Why important ?Some properties that might be difficult to predict using a “common sense” without the knowledge of the phase diagrams

• example 1: Melting temperature of a mixture AB (solution) of two components A and B could be either lower or higher than the melting point of each component (!). This could be a failure mechanism in electronic or mechanical components. But could also be used to your advantage.

• example 2: Upon cooling to a lower temperature a phase transformation of a material could cause expansion, which could cause internal stresses and failure (e.g. tin food cans will crumble at low T)

• example 3: No abrupt liquid-to-solid transformation when two components are present (solid + liquid in a temperature range)

Example: Chip-Solder-Joint-Failure

Why important ?Some properties that might be difficult to predict using a “common sense” without the knowledge of the phase diagrams

• example 4: Tmelt (Sn) = 232 C, Tmelt (Pb) = 327 C……

but Tmelt(Sn0.62Pb0.38) = 183 C, so this is a common soldering alloy

• example 5: Tmelt (Au) = 1064 C, Tmelt (Si) = 2550 C……but Tmelt(Au0.97Si0.03) = 363 C, so thin layer of gold is used to attach Si chip to a ceramic substrate (shock protection)

• example 6: Mechanical properties (hardness and tensile strength) of an alloy could be substantially higher than that of the individual components (e.g. hardness (AgCu) about twice the harness of Ag or Cu)

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One-Component Phase Diagrams

F = C – P + 2C- ComponentsP- Number of phasesF- Degrees of freedomF = 2

F = 1

F = 0

# of state variables (e.g. two: P and T)

Gibbs Phase rule:

Two-Component Phase Diagrams

F = C – P + 1

F = 2

F = 1T

In a two-phase field need to specify either the temperature or the composition of one of the phases.

Xs Xl

Isomorphous system ( complete solubility over the composition range)

If pressure is fixed (1 atm)

Hume-Rothery Rules for substitutional solution:• The size < ~15%.• The electronegativities and valance –

similar • The crystal structures of the two species

must be the same to form a continuous series of solid solutions.

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Two-Component Phase Diagrams

Composition, XB

Tem

pera

ture ( )

( )( )( )

1

11

l s

o l l s s

l s

o l s s s

o l l s s s

o l s s l

o ls

s l

f fX X f X ff fX X f X fX X X f X fX X f X X

X Xf

X X

+ == +

= −

= − +

= − +

− = −

−=

−

( )( )

s ol

s l

X Xf

X X−

=−

The Lever Rule in a Two-Component System

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Two-Component Phase Diagrams

Two-Component Phase Diagrams

Tem

pera

ture

, C

Composition, XB

Time Time

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Two-Component Phase Diagrams

Deviation from ideal behavior

Congruent melting maximum

EAB > 0.5 (EAA + EBB)

Two-Component Phase Diagrams

Deviation from ideal behavior Congruent melting

minimum

EAB > 0.5 (EAA + EBB)

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Eutectic Phase Diagrams

F = 1, must specify temperatureor the compositionof one of the phases

F = 1

F = 1

F = 1

F = 0

F = 0, temperatureand compositionsof the phase arefixed.

Composition, XB

TATB

X1 XE X2A B

Tem

pera

ture

F = 2

F = 2

F = 2, mustspecify temperatureand composition

F = 2 Xα Xβ

TXs

Xl

TXs

Xl

T Xα Xβ

Solvus

Cooling Curves and Phase Boundaries

Tem

pera

ture

Composition, XB

Time

Alloy 1

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Cooling Curves and Phase BoundariesTe

mpe

ratu

re

Composition, XB

Time

Alloy 2

Cooling Curves and Phase Boundaries

Tem

pera

ture

Composition, XB

Time

Alloy 3

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Various physical properties and their relationship to a eutectic phase diagram

Eutectic Phase Diagrams

Tem

pera

ture

Composition, in % B

1. For the alloy composition of 0.27 % B calculate the fraction of solid and the fraction of liquid that forms under equilibrium cooling at the eutectic T

2. Calculate the amount of β and αthat will form from the liquid just below the eutectic isotherm

3. Calculate the amount of α in the alloy at temperature just below the eutectic T

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Eutectic Phase Diagrams

This liquid becomes theeutectic mixture of α and βwhen the temperature drops just below the eutectic temperature which is composedof: 0.73 0.37

0.73 0.200.68

f

f

α

α

−≈

−≈

0.37 0.200.73 0.200.32

f

f

β

β

−≈

−≈

Just above the eutectic temperature the fractionof liquid and solid are:

0.27 0.200.37 0.200.41

l

l

f

f

−≈

−≈

0.37 0.270.37 0.200.59

f

f

α

α

−≈

−≈

The first solid that formsis called primary α

Just below the eutectic temperaturethe microstructure is composed ofprimary α that formed above the eutectic temperature and α from the eutectic mixture

total primary eutecticf f fα α α= +

0.59 (0.41)(0.68) 0.87total

fα = + =

0.73 0.270.73 0.20

fα−

=−

0.87fα =or

Tem

pera

ture

Composition, in % B

Microstructure Above and Below the Eutectic Temperature for an Off-Eutectic Alloy

Just above TE Just below TE

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Microstructure Above and Below the Eutectic Temperature for Off-Eutectic Alloys

Just below TE

Increasing primary αDecreasing eutectic

Decreasing primary αIncreasing eutectic

Deviation from Hume-Rothery’s Rules

Increasing deviation leads to decrease in the maximum solid solubility of B in α.

Tem

pera

ture

Composition, XBA

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Eutectic Phase Diagram, No Solid Solubility

Tem

pera

ture

Composition, XB

Eutectic Phase DiagramsAl-Si System

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Methods for Determining a Phase Diagram

Primary α-aluminum

α−aluminum / silicon eutectic

Microstructure of an Aluminum-Silicon Alloy

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Phase Diagrams Containing Two Eutectics

• Possible to have several solid solution regions: e.g. 2 eutectic reactions and 3 solid solutions (α, β, and γ)

• Note that upon cooling from T max at the alloy composition X there is a phase change but no composition change (CONGRUENT melting)

Line compound

Peritectic Phase Diagrams

l α β+ =

• if both the L and S phases have a tendency to cluster, the liquidustemperature increase and the solidus temperature decreases

• In addition, a miscibility gap(region of non-mixing) appears

• A progressive increase in the clustering tendency leads to the PERITECTIC phase diagram

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The Use of Cooling Curves for Determining a Peritectic Phase Diagram

Tem

pera

ture

Composition, XB TimeXP X2X2

TA

TL

TP

l α β+ =

Analysis of a Peritectic Phase Diagram

Tem

pera

ture

Composition

Alloy 1 Alloy 2 Alloy 3

Alloy 3 at T20.88 0.600.88 0.3

0.48

l

l

f

f

−=

−=

Alloy 3 at T5

0.90 0.600.90 0.340.54

f

f

α

α

−=

−=

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Monotectic Phase Diagrams

• A region of immiscibility (non-mixing) develops in the L phase

• example: oil and water

• Liquid1 = Liquid2 + α (solid)

L1 L2

XM

L2

L2

Review of Invariant Binary ReactionsEutectic Type

lα β

α βγ

l2l1α

Eutectic

Eutectoid

Monotectic

Monotectoid

l α + β

γ α + β

l1 α + l2

α1 βα2

α2 α1 + β

Al-Si, Fe-C

Fe-C

Cu-Pb

Al-Zn, Ti-V

On cooling one phase going to two phases

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Review of Invariant Binary ReactionsPeritectic Type

αβl

l + β α

α + β γα β

γ

Peritectic

Peritectoid

Fe-C

Cu-Al

On cooling two phases going to one phase

HW Questions

1. When a solid melts congruently, the liquid and solid have different / the same composition(s).

2. At constant temperature the fraction of the phases in a two-phase field changes / remains the same when the overall composition of the alloy is

changed, but remains in the two-phase field.

3. Why would alloys close to the eutectic composition be suitable for castings rather than alloy compositions far from the eutectic composition?

4. On cooling when a two-phase liquid plus solid transforms to a solid phase the transformation is eutectic / peritectic in nature.

5. On cooling the peritectoid reaction written symbolically has one phase going to two / two phases going to one.

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HW Questions

• At what T an alloy containing 88% B will start melting?

• At what T it will completely transform into liquid?

• What is the composition of αphase for this alloy @ T8?

• What is the maximum solid solubility of B in a and A in b? Whose rules apply here?

• For an alloy containing 88% B, calculate the fraction of the liquid and solid phases and their compositions at temperature T3, T4, and T5

• At a temperature just below the eutectic temperature, how much β is primary β, what is the total fraction of β, and what is the fraction β in the eutectic. (Alloy composition is 88% B)

Tem

pera

ture

Composition, XB

TA

TB

A B

1. Label all phase fields.

2. Identify all invariant reactions.

Labeling Complex Phase Diagrams

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Labeling Complex Phase Diagrams

Eutectic

Eutectic

PeritecticPeritectic

Eutectic

Eutectic l α β= +

l γ β+ =

l γ δ= +

Labeling Complex Phase Diagrams

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Summary• One-component phase diagrams with

temperature and pressure as the experimental variables that affect equilibrium.

• Introduction to the Gibbs Phase Rule and its application to one-component systems.

• Two-component systems and the rules that govern the composition of the phases, the number of phases and the amount of each phase at equilibrium.

• The applications of these rules to complex, two-component systems illustrated that regardless of how complex the phase diagram appeared, the rules that were developed could be easily applied.