Phase Transformation of Materials

Phase Transformation of Materials Phase Transformation of Materials

Eun Eun Soo Soo Park Park

Office: 33-316

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

2009 fall

09.08.2009

2

Gibbs Free Energy as a Function of Temp. and Pressure

- Single component system Chapter 1

= 0 dG

P V S G

T G

T P

⎟ =

⎠

⎜ ⎞

⎝

⎛

∂

− ∂

⎟ =

⎠

⎜ ⎞

⎝

⎛

∂

∂ ,

Tm

T G = LΔ Δ

Thermondynamics and Phase Diagrams

- Equilibrium

- Driving force for solidification - Classification of phase transition

2 1

0

G G G

Δ = − <

- Phase Transformation

Contents for previous class

Lowest possible value of G No desire to change ad infinitum

First order transition: CDD/Second order transition: CCD

V

T H dT

dP

eq

eq Δ

= Δ

) (

Clausius-Clapeyron Relation:

Contents for today’s class

- Chemical potential and Activity

- Gibbs Free Energy in Binary System - Binary System

Ideal solution and Regular solution

4

*

Binary System (two component)

Æ A, B

- Mixture ; A – A, B – B ; Æ 각각의 성질 유지, boundary는 존재, 섞이지 않고 기계적 혼합

A B

- Solution ; A – A – A ; Æ atomic scale로 섞여 있다. Random distribution A – B – A Solid solution : substitutional or interstitial

- compound ; A – B – A – B ; Æ A, B의 위치가 정해짐, Ordered state B – A – B – A

*

Single component system One element (Al, Fe), One type of molecule (H₂O)

:

평형 상태 압력과 온도에 의해 결정됨

:

평형 상태 압력과 온도 이외에도 조성의 변화를 고려

1.3 Binary Solutions

- Mixture ; A – A, B – B ; Æ 각각의 성질 유지, boundary는 존재, 섞이지 않고 기계적 혼합

A B

1.3 Binary Solutions

6

- Solution ; A – A – A ; Æ atomic scale로 섞여 있다. Random distribution A – B – A Solid solution : substitutional or interstitial

1.3 Binary Solutions

- Compound ; A – B – A – B ; Æ A, B 의 위치가 정해짐, Ordered state B – A – B – A

MgCNi3

1.3 Binary Solutions

ruthenium 루테늄 《백금류의 금속 원소;기호 Ru, 번호 44》 8

Solid Solution vs. Intermetallic Compounds

Crystal structure

Surface

Pt_0.5Ru_0.5 – Pt structure (fcc) PtPb – NiAS structure

partial or complete solid solution Alloying: atoms mixed on a lattice

1.3 Binary Solutions

Solid Solution vs. Intermetallic Compounds

Pt_0.5Ru_0.5 – Pt structure (fcc) PbPt – NiAS structure

1.3 Binary Solutions

10

• Solid solution:

– Crystalline solid

– Multicomponent yet homogeneous

– Impurities are randomly distributed throughout the lattice

• Factors favoring solubility of B in A

(Hume-Rothery Rules) – Similar atomic size: ∆r/r ≤ 15%

– Same crystal structure for A and B

– Similar electronegativities: |χ_A – χ_B| ≤ 0.6 (preferably ≤ 0.4) – Similar valence

• If all four criteria are met: complete solid solution

• If any criterion is not met: limited solid solution

1.3 Binary Solutions

complete solid solution limited solid solution

1.3 Binary Solutions

Binary Solutions

12

* Composition in mole fraction X_A, X_B X_A + X_B = 1

1. bring together X_A mole of pure A and X_B mole of pure B

2. allow the A and B atoms to mix together to make a homogeneous solid solution.

Gibbs Free Energy of Binary Solutions

1.3 Binary Solutions

Gibbs Free Energy of The System

In Step 1

- The molar free energies of pure A and pure B pure A;

pure B;

;X

_A

, X

_B

(mole fraction

) )

, (T P G_A

) , (T P G_B

G

₁

= X

_A

G

_A

+ X

_B

G

_B

J/mol

1.3 Binary Solutions

14

Gibbs Free Energy of The System

In Step 2

G

₂

= G

₁

+ Δ G

_mix

J/mol

Since G

₁

= H

₁

– TS

₁

and G

₂

= H

₂

– TS

₂

And putting ∆ H

_mix

= H

₂

- H

₁

∆ S

_mix

= S

₂

- S

₁

∆ G

_mix

= ∆ H

_mix

- T ∆ S

_mix

∆ H

_mix

: Heat of Solution i.e. heat absorbed or evolved during step 2

∆ S

_mix

: difference in entropy between the mixed and unmixed state

^.

How can you estimate Δ H

_mix

and Δ S

_mix

?

1.3 Binary Solutions

Mixing free energy Δ G

_mix

가정

1 ; ∆ H

_mix

=0 :

; A

와

B

가

complete solid solution ( A,B ; same crystal structure)

; no volume change

- Ideal solution

w k

S = ln

^w : degree of randomness, k: Boltzman constant Æ thermal; vibration ( no volume change )

Æ Configuration; atom 의 배열 방법 수 ( distinguishable )

Δ G

_mix

= -T Δ S

_mix

J/mol

∆ G

_mix

= ∆ H

_mix

- T ∆ S

_mix

Entropy can be computed from randomness by Boltzmann equation, i.e.,

= + S S S

1.3 Binary Solutions

16

)]

ln (

) ln

( ) ln

[(

,

,

1

! ln

!

)!

ln (

0 0

0

o B o

B o

B o

A o

A o

A o

o o

B A

B B

A A

B A

B before A

after mix

N X N

X N

X N

X N

X N

X N

N N

k

N N

N N

X N

N X N

N k N

N k N

S S

S

−

−

−

−

−

=

= +

=

=

→

+ −

=

−

= Δ

이므로 N

N N

N ! ≈ ln − ln

Excess mixing Entropy

) , _(

_

!

!

)!

(

) _

_(

_

1

B A B

A B A

config config

N N solution after

N N

N w N

pureB pureA

solution before

w

+ →

=

→

=

If there is no volume change or heat change,

Number of distinguishable way of atomic arrangement

kN

0

R =

using Stirling’s approximation and

1.3 Binary Solutions

Ideal solution

Excess mixing Entropy

) ln

ln

(

_{A A B B}

mix

R X X X X

S = − +

Δ

) ln

ln

(

_{A A B B}

mix

RT X X X X

G = +

Δ Δ G

_mix

= -T Δ S

_mix

1.3 Binary Solutions

Ideal solution

18

Since Δ H = 0 for ideal solution,

Compare G

_solution

between high and low Temp.

G₂ = G₁ + ΔG_mix

1.3 Binary Solutions

Ideal solution

G₁ ΔG_mix

Chemical potential

= μ

_{A A}

dG ' dn

The increase of the total free energy of the system by the increase of very small quantity of A, dn

_A

, will be proportional to dn

_A

.

소량 첨가에 의한 내부 에너지 변화 계산

(T, P, n

_B

: constant )

G =

H-TS = E+PV-TS

For A-B binary solution, dG' = μ

A

dn

A

+ μ

B

dn

B

For variable T and P

1) Ideal solution

1.3 Binary Solutions

μ

_A

:

partial molar free energy of A or chemical potential of A

⎛ ∂ ⎞

μ = ⎜^A ⎝ ∂ ^A ⎟⎠T, P, n_B

G' n

⎛ ∂ ⎞

μ = ⎜^B ⎝ ∂ ^B ⎟⎠T, P, n_A

G' n

20

−1

= μ

_{A A}

+ μ

_{B B}

G X X Jm ol

A A B B

dG = μ dX + μ dX

_{B A}

B

dG

dX = μ − μ

^{A B}

B

dG μ = μ − dX

( 1 )

B A B B

B

B A A B B

B

B A

B

B B

B

G dG X X

dX

X dG X X

dX dG X dX

dG X dX

⎛ ⎞

= μ − ⎜ ⎟ + μ

⎝ ⎠

= μ − + μ

= μ −

= μ − −

B A

B

G dG X μ = + dX

For 1 mole of the solution (T, P: constant )

Chemical potential 과 Free E 와의 관계 1) Ideal solution

μ

_A

’

μ

_B

’

X

_B

’

1.3 Binary Solutions

B B

B A

A

A

RT X X G RT X X

G ln ) ( ln )

( +

Chemical potential

+ +

=

22

Ideal solution : ΔH

_mix

= 0

Quasi-chemical model assumes that heat of mixing, ΔH

_mix

, is only due to the bond energies between adjacent atoms.

Structure model of a binary solution

Regular Solutions

1.3 Binary Solutions

∆

_{Gmix =}

∆

_{Hmix - T}

∆

_Smix

Bond energy Number of bond A-A ε

AA

P

_AA

B-B ε

BB

P

_BB

A-B ε

AB

P

_AB

=

_{AA AA}

ε +

_{BB BB}

ε +

_{AB AB}

ε

E P P P

Δ = ε ε = ε − 1 ε + ε

H P where ( )

Internal energy of the solution

Regular Solutions

1.3 Binary Solutions

24

Δ H

_mix

= 0

Completely random arrangement

ideal solution

AB

=

a A B a

P N zX X bonds per mole N : Avogadro ' s number

z : number of bonds per atom

∆ H

_mix

= Ω X

_A

X

_B

where Ω = N

_a

z ε

Δ H

_mix

= P

_AB

ε

Regular Solutions

= 0

ε ( )

2 1

BB AA

AB

ε ε

ε = +

↑

→

< 0 P

_AB

ε ε > 0 → P

_AB

↓

≈ 0 ε

Ω >0 인 경우

1.3 Binary Solutions

Regular Solutions

Reference state

Pure metal

G

_A⁰

= G

_B⁰

= 0

) ln

ln

(

_{A A B B}

B A B

B A

A

G X G X X RT X X X X

X

G = + + Ω + +

G

₂

= G

₁

+ Δ G

_mix

∆ G

_mix =

∆ H

_mix

- T ∆ S

_mix

∆

_{Hmix -T}

∆

_Smix

26

Phase separation in metallic glasses

−1

= μ

_{A A}

+ μ

_{B B}

G X X Jmol

For 1 mole of the solution (T, P: constant )

G

=

E+PV-TS G

=

H-TS

A B B

A B

AX X X X X

X = ² + ²

2) regular solution

B B

B B

A A

A A

X RT

X G

X RT

X G

ln )

1 (

ln )

1 (

2 2

+

− Ω

+

=

+

− Ω

+

= μ μ

) ln

ln

(

_{A A B B}

B A B

B A

A

G X G X X RT X X X X

X

G = + + Ω + +

) ln

) 1

( (

) ln

) 1

(

(

_{A A} ² _{A B B B} ² _B

A G +Ω − X + RT X + G +Ω − X + RT X

=

μ μ

μ = +

μ = +

A A A

B B B

G RT ln X G RT ln X

Chemical potential 과 Free E 와의 관계

Ideal solution

28

Activity, a : mass action 을 위해 effective concentration

ideal solution regular solution

μ

_A

= G

_A

+ RTlna

_A

μ

_B

= G

_B

+ RTlna

_B

μ = + μ = +

A A A

B B B

G RT ln X G RT ln X

B B

B B

A A

A

A = G + Ω (1 − X )² + RT ln X

μ

= G + Ω (1 − X )² + RT ln X

μ

γ =

_B ^B

B

a X

)

2

1 ( )

ln(

_B

B

B

X

RT X

a Ω −

=

Solution 에서 a 와 X 와의 관계

조성 따른 activity 변화 aB a_A

Line 1 : (a) a

_B

=X

_B

, (b) a

_A

=X

_A

ideal solution…Rault’s law Line 2 : (a) a

_B

<X

_B

, (b) a

_A

<X

_A ΔH_mix<0

Line 3 : (a) a

_B

>X

_B

, (b) a

_A

>X

_A ΔH_mix>0

γ =_B ^B ≅

B

a cons tan t (Henry ' s Law) X

a

• For a dilute solution of B in A (X_B→0)

30

μ

_A

= G

_A

+ RTlna

_A

⎛ ⎞ = Ω −

⎜ ⎟

⎝ ⎠

B

B B

ln a (1 X ) X RT

Activity는 solution의 상태를 나타내는

조성 과 Chemical potential 과 상관관계 가짐.

→

^A

A

a

degree of non-ideality ?

X

^A

= γ

_{A A}

= γ

_{A A}

A

a , a X

X

γ

_A

: activity coefficient

Chemical Equilibrium (μ, a) → multiphase and multicomponent

(μ

_i^α

= μ

_i^β

= μ

_i^γ

= …), (a

_i^α

= a

_i^β

= a

_i^γ

= …)

Contents for today’s class

- Chemical potential and Activity

- Gibbs Free Energy in Binary System - Binary System

Ideal solution

Regular solution

mixture/ solution / compound

G

₂

= G

₁

+ Δ G

_mix

J/mol G

₁

= X

_A

G

_A

+ X

_B

G

_B

J/mol

( ∆ H

_mix

=0)

) ln ln

( _{A A B B}

mix RT X X X X

G = +

Δ

_mix

=

_AB

ε

Δ

ε = ε −

_AB

1 ε + ε

_{AA BB}

H P where ( )

2

) ln

ln

(

_{A A B B}

B A B

B A

A

G X G X X RT X X X X

X

G = + + Ω + +

⎛ ∂ ⎞

μ = ⎜ G'

•

μ_A

= G

_A

+ RTlna

_A