재 료 상 변 태

재 료 상 변 태

Phase Transformation of Materials

2008.09.09.

박 은 수

서울대학교 재료공학

Contents for previous class

•

상

,

상태도

,

상변태에 대한 이해

•

열역학 계와 열역학 법칙에 대한 이해

• Gibbs

와

Helmholtz Free Energy

의 유도

•

평형의 개념

•

단일성분계의 이해와 변태거동

• 1

차 변태

, 2

차 변태와 변태시 압력효과에 관한 이해

•

응고 구동력과 과냉의 역할에 대한 이해

Contents for today’s class

• Ideal solution 과 regular solution 의 이해

• 이상 용액과 규칙 용액에서 Gibbs Free Energy

• Chemical potential 과 Activity 의 이해

• Real solution 의 이해

*

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 온도와 압력 이외에도 조성의 변화를 고려

- 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

MgCNi3

ruthenium 루테늄 《백금류의 금속 원소;기호Ru, 번호44》

• Solid solution:

– Crystalline solid

– Multicomponent yet homogeneous

– Impurities are randomly distributed throughout the lattice

• Factors favoring solubility of B in A

– 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 valency

• If all four criteria are met: complete solid solution

• If any criterion is not met: limited solid solution

: 합금에 대한 열역학 기본 개념 도입 위해 2원 고용체, 1기압의 고정된 압력 조건 고려

* 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

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

Free energy of mixture

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

?

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.,

=

_th

+

_config

S S S

)]

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

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

Since Δ H = 0 for ideal solution,

Compare G

_solution

between high and low Temp.

G₂ = G₁ + ΔG_mix

Chemical potential

= μ

_{A A}

dG ' dn

⎛ ∂ ⎞

μ = ⎜

^A

⎝ ∂

^A

⎟ ⎠

T, P, n_B

G' n

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

.

(T, P, n

_B

: constant )

μ

_A

: partial molar free energy of A or chemical potential of A

⎛ ∂ ⎞

μ = ⎜

^B

⎝ ∂

^B

⎟ ⎠

T, P, n_A

G'

n

Chemical potential

= μ

_{A A}

+ μ

_{B B}

dG ' dn dn

= − + + μ

_{A A}

+ μ

_{B B}

dG ' SdT VdP dn dn

for A-B binary solution, dG’

for variable T and P

dG = -SdT + VdP

Chemical potential

⎛ ∂ ⎞ = μ = ⎛ ∂ ⎞ = ⎛ ∂ ⎞ = ⎛ ∂ ⎞

⎜ ∂ ⎟ ⎜ ∂ ⎟ ⎜ ∂ ⎟ ⎜ ∂ ⎟

⎝

ⁱ

⎠

T,P,n_j ⁱ

⎝

ⁱ

⎠

S,V,n_j

⎝

ⁱ

⎠

S,P,n_j

⎝

ⁱ

⎠

T,V,n_j

G U H F

n n n n

• The chemical potential is the change in a characteristic thermodynamic state function (depending on the

experimental conditions), the characteristic thermodynamic state function is either: internal energy (U), enthalpy (H), Gibbs free energy (G), or Helmholtz free energy (F)) per change in the number of molecules.

From Wikipedia, the free encyclopedia

= − + μ

= + + μ

= − − + μ

= − + + μ

∑ ∑

∑ ∑

i i

i i

i i

i i

dU TdS PdV dn

dH TdS VdP dn

dF SdT PdV dn

dG SdT VdP dn

−1

= μ

_{A A}

+ μ

_{B B}

G X X Jmol

For 1 mole of the solution dG' = μ

_A

dn

_A

+ μ

_B

dn

_B

μ = + μ = +

A A A

B B B

G RT ln X G RT ln X

μ = + μ = +

A A A

B B B

G RT ln X G RT ln X

μ

_A

’

μ

_B

’

X

_B

’

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

∆ G

_mix =

∆ H

_mix

- T ∆ S

_mix

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

_mix

=

_AB

ε ε = ε −

_AB

1 ε + ε

_{AA BB}

H P where ( )

2

Internal energy of the solution

Regular Solutions

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 0

Regular Solutions

ε = ( )

2 1

BB AA

AB

ε ε

ε = +

↑

→

< 0 P

_AB

ε ε > 0 → P

_AB

↓

≈ 0 ε

Ω >0 인 경우

Regular Solutions

Reference state

Pure metal

G

⁰_A

= 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

Phase separation in metallic glasses

Regular Solutions

) 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 + +

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

Free Energy Change on Mixing

A B B

A B

AX X X X X

X = ² + ²

−1

= μ

_{A A}

+ μ

_{B B}

G X X Jmol

Activity, a

ideal solution regular solution

• μ

_A

= G

_A

+ RTlna

_A

μ

_B

= G

_B

+ RTlna

_B

⎛ ⎞ = Ω −

⎜ ⎟

⎝ ⎠

B

B B

ln a (1 X )

X RT

γ =_B ^B

B

a X

μ = + μ = +

A A A

B B B

G RT ln X G RT ln X

• For a dilute solution of B in A (X

_B→

0)

γ = ≅ γ = ≅

B B

B A A

A

a cons tan t (Henry ' s Law) X

a 1 (Rault ' s Law)

X