“Phase Equilibria

Eun Soo Park

Office: 33-313

Telephone: 880-7221 Email: espark@snu.ac.kr

Office hours: by an appointment

2015 Fall

09. 23. 2015

1

“Phase Equilibria in Materials”

Equilibrium in Heterogeneous Systems

-RT lna

_B^β

-RT lna

_B^α

-RT lna

_A^α

-RT lna

_A^β

a

_A^α

=a

_A^β

a

_B^α

=a

_B^β

G _e

In X

⁰

, G

₀^β

> G

₀^α

> G

₁

α + β separation unified chemical potential

μ _B μ _A =

μ _B μ _A

Contents for previous class

3

- Two-Phase Equilibrium

1) Simple Phase Diagrams

Assumption: (1) completely miscible in solid and liquid.

(2) Both are ideal soln.

(3) T

_m

(A) > T

_m

(B) ^{∆ = 0}

S

H

mix

= 0

∆ H

_mix^L

①

4

→ Fig. 23f

* Consider the free energy curves for liquid and α phase at a temperature T, where T

_A

>T>T

_B

. The standard states are pure solid A and pure liquid B at temperature T. → Derive the free energy curves for the liquid and α phases.

X-T relationship

in A-rich composition

X-T relationship

in B-rich composition

② X-T relationship in A-rich and B-rich compositions

5

③ ΔH effect for G curvature

: initial slope of solidus

and liquidus curve

* Consider actual (or so-called regular) solutions

in which

④ Temperature effect for G variation

: role of ΔS

7

1) Simple Phase Diagrams a) Variation of temp.: G

^L

> G

^s

b) T ↓→ Decrease of curvature of G curve

(∵ decrease of -TΔS

_mix

effect)

1) Simple Phase Diagrams Assumption:

(1) completely miscible in solid and liquid.

(2) Both are ideal soln.

(3) T

_m

(A) > T

_m

(B)

(4) T

₁

> T

_m

(A) >T

₂

> T

_m

(B) >T

₃

1.5 Binary phase diagrams

a) Variation of temp.: G

^L

> G

^s

b) T ↓→ Decrease of curvature of G curve

(∵ decrease of -TΔS_mix effect)

Tie line

9

Contents for today’s class

- Equilibrium in Heterogeneous Systems

- Binary phase diagrams

1) Simple Phase Diagrams

G

₀^β

> G

₀^α

> G

₀^α+β

α + β separation unified chemical potential

Assume: (1) completely miscible in solid and liquid.

(2) Both are ideal soln.

= 0

∆ H

_mix^S

= 0

∆ H

_mix^L

2) Variant of the simple phase diagram

> 0

∆

>

∆ H

_mix^α

H

_mix^l

∆ H

_mix^α

< ∆ H

_mix^l

< 0

miscibility gab Ordered phase

3.2.6 Pressure-Temperature-Composition phase diagram for a system with continuous series of solutions

α

L

V

11

3.2.6 Pressure-Temperature-Composition phase diagram for a system with continuous series of solutions

α

L

V

3.2.6 Pressure-Temperature-Composition phase diagram for a system with continuous series of solutions

α

L

V

13

3.2.6 Pressure-Temperature-Composition phase diagram

for a system with continuous series of solutions

Contents for today’s class

* Three-Phase Equilibrium : Eutectic Reactions a) Structural Factor: Hume-Rothery Rules

b) The eutectic reaction

c) Limiting forms of eutectic phase diagrams d) Retrogade solidus curves

CHAPTER 4

Binary Phase Diagrams

Three-Phase Equilibrium Involving Limited Solubility of the Components in the Solid State but Complete Solubility in the Liquid State

complete solid solution limited solid solution

Empirical rules for substitutional solid-solution

Similar atomic radii, the same valency and crystal structure

15

• Three-Phase Equilibrium Involving Limited Solubility of the Compo- nents in the Solid State but Complete Solubility in the Liquid State

* Simple Eutectic Systems ∆ H _mix ^α > ∆ H _mix ^l > 0

Eutectic horizontal

• ΔH

_m

>>0 and the miscibility gap extends to the melting temperature.

( when both solids have the same structure.)

* Simple Eutectic Systems

17

(when each solid has the different crystal structure.)

(c) d = 15 ^mm

45 mm (b) d = 10 mm

(a) d = 6 mm

t T R

_c

T

^{i g}

∆

= −

* Cooling curves measured at the center of the three transverse cross sections

bottom position : 137 K/s middle position : 64K/s top position : recalescence

5 10 15 20 25 30 35 40 45

300 400 500 600 700 800 900 1000

Mg

₆₅

Cu

_7.5

Ni

_7.5

Zn

₅

Ag

₅

Y

₅

Gd

₅

Te m pe ra tur e ( K)

(Cu mold casting in air atmosphere)

(a) (b) (c)

T

_g

T

_l

time (sec)

Measurement of R _c in Mg BMG ( D _max =14 mm)

recalescence

Nucleation Theory as Applied to solidification

 The recalescence process is illustrated by a temperature versus time plot,

showing the temperature rise on nucleation due to release of the heat of fusion.

 A sudden glowing in a undercooled liquid of metal caused by liberation of the latent heat of transformation

 The higher the recalescence temperature, the larger the microstructural scale in

the solid.

21

1.5 Binary phase diagrams

= 0

∆ H

_mix^L

∆ H

_mix^S

>> 0

1.5 Binary phase diagrams

Alloy II

23

1.5 Binary phase diagrams

Alloy II

4.3.2 Eutectic Solidification: L→α + β

D.A. Porter and K.E. Eastering, "Phase Transformations in Metals and Alloys"

25

4.3.2 Eutectic Solidification

Normal eutectic Anomalous eutectic

Fig. 4.29 Al-CuAl₂ lamellar eutectic normal to the growth direction (x 680).

The microstructure of the Pb-61.9%Sn (eutectic) alloy presented a coupled growth of the (Pb)/βSn eutectic.

There is a remarkable change in morphology increasing the degree of undercooling with transition

from regular lamellar to anomalous eutectic.

http://www.matter.org.uk/solidification/eutectic/anomalous_eutectics.htm

During solidification both phases grow simultaneously behind an essentially planar solid/liquid interface.

both phases have low entropies of fusion. One of the solid phases is capable of faceting, i.e., has a high entropy or melting.

D.A. Porter and K.E. Eastering, "Phase Transformations in Metals and Alloys"

Divorced Eutectic Eutectic

D.A. Porter and K.E. Eastering, "Phase Transformations in Metals and Alloys"

27

1.5 Binary phase diagrams

Alloy I :

1.5 Binary phase diagrams

Alloy I

29

1.5 Binary phase diagrams

Alloy IlI

(4)

1.5 Binary phase diagrams: Hypoeutectic

Alloy IlI

31

1.5 Binary phase diagrams

Alloy IV

(4)

1.5 Binary phase diagrams : Hypereutectic

Alloy IV

33

4.2.3. Limiting forms of eutectic phase diagram

1) Complete immiscibility of two metals does not exist.

: The solubility of one metal in another may be so low (e.g. Cu in Ge <10

^-7

at%.)

that it is difficult to detect experimentally, but there will always be a measure of solubility.

* Effect of T on solid solubility

B B

B B

o

B B

o B B

o B o B B

B B

B o B

X RT

X G

G G

G G

G

X RT

X G

ln )

1 (

ln )

1 (

2 0

2

−

− Ω

−

=

−

−

=

−

=

−

=

∆

+

− Ω +

=

→ α α

α α

β α

β α

α β

α α

µ

µ µ

µ

) exp(

ln

) 1 ,

(

) 1

( ln

ln )

1 (

2 2

RT X G

G X

RT X here

X G

X RT

X RT

X G

e B B

B e

B e B

B B

B

B B

B

Ω +

− ∆

=

>>

Ω

−

∆

−

=

<<

− Ω

−

∆

−

=

−

− Ω

−

=

∆

→

→

→

→

α β α β α β α

β

) exp(

) exp(

RT H R

X S

S T H

G

B e B

B

B B

B

Ω +

− ∆

= ∆

∆

−

∆

=

∆

→

→

→

→

→

α β α

β

α β α

β α

β

이므로

↑

↑ X

_B^e

T

 



 

−

= Q

A X

_B^e

exp

A is virtually insoluble in B.

β

µ

_Bβ

~ G

_B

= ^_

Stable β form Unstable α form A is virtually insoluble in B

↑

↑ X

_B^e

T

35

* Limiting forms of eutectic phase diagram

The solubility of one metal in another may be so low.

 



 

−

= RT

A Q X

_B^e

exp

It is interesting to note that, except at absolute zero,

X_B^e can never be equal to zero, that is, no two compo

-nents are ever completely insoluble in each other.

↑

↑ X

_B^e

T

a)

b)

37

39

4.2.5. 2) Retrograde solidus curves

dX/dT=0 at a temperature T, where T_E< T < T_A

Intensive Homework 3: Understanding of retrograde solidus curves from a thermodynamic standpoint

: A maximum solubility of the solute at a temperature between the melting point of the solvent and an invariant reaction isothermal

Solidus curve in the systems with low solubility

Ex) semiconductor research using Ge and Si as solvent metals

A high value of ΔHBS (or a large difference in the melting points of the components) is associate with a significant difference in atomic radii for A and B, which can lead to a large strain energy contribution to the heat of solutions.

4.2.5. Disposition of phase boundaries at the eutectic horizontal

41

4.2.5. Disposition of phase boundaries at the eutectic horizontal

ϴ between solidus and solubility curves must be less than 180°.

This is a general rule applicable to all curves which meet at an invariant reaction horizontal in a binary diagram, whether they be eutectic, peritectics, eutectoid, etc., horizontals.

* Three-Phase Equilibrium : Eutectic Reactions a) Structural Factor: Hume-Rothery Rules

b) The eutectic reaction

c) Limiting forms of eutectic phase diagrams d) Retrogade solidus curves

CHAPTER 4

Binary Phase Diagrams

Three-Phase Equilibrium Involving Limited Solubility of the Components in the Solid State but Complete Solubility in the Liquid State

complete solid solution limited solid solution

Empirical rules for substitutional solid-solution

Similar atomic radii, the same valency and crystal structure

Contents for previous class

43

- Binary phase diagrams 1) Simple Phase Diagrams

3) Simple Eutectic Systems

* Pressure-Temperature-Composition

phase diagram for a system with

continuous series of solutions

Contents for today‘s class

By plotting a series of the free energy-composition curves at different temperatures we established the manner in which the phase compositions changes with temperature. In other words, we determined the phase limits or phase boundaries as a function of temperature. A phase diagram is nothing more than a presentation of data on the position of phase boundaries as a function of temperature.

45

different crystal structure

same crystal structure

47

4.2.3. Limiting forms of eutectic phase diagram

1) Complete immiscibility of two metals does not exist.

: The solubility of one metal in another may be so low (e.g. Cu in Ge <10

^-7

at%.)

that it is difficult to detect experimentally, but there will always be a measure of solubility.

4.2.5. 2) Retrograde solidus curves

dX/dT=0 at a temperature T, where T_E< T < T_A

: A maximum solubility of the solute at a temperature between the melting point of the solvent and an invariant reaction isothermal

Solidus curve in the systems with low solubility

Ex) semiconductor research using Ge and Si as solvent metals

A high value of ΔHBS (or a large difference in the melting points of the components) is associate with a significant difference in atomic radii for A and B, which can lead to a large strain energy contribution to the heat of solutions.

49

3) ϴ between solidus and solubility curves must be less than 180°.

This is a general rule applicable to all curves which meet at an invariant reaction horizontal in a binary diagram, whether they be eutectic, peritectics, eutectoid, etc., horizontals.

4.2.5. Disposition of phase boundaries at the eutectic horizontal

ϴ