재 료 상 변 태

Phase Transformation of Materials

2008.09.09.

박 은 수

서울대학교 재료공학

Contents for previous class

• 상, 상태도, 상변태에대한이해

• 열역학계와열역학법칙에대한이해

• Gibbs 와 Helmholtz Free Energy의유도

• 평형의개념

• 단일성분계의이해와변태거동

• 1차변태, 2차변태와변태시압력효과에관한이해

• 응고구동력과과냉의역할에대한이해

Contents for today’s class

• Ideal solution과 regular solution의이해

• 이상용액과규칙용액에서 Gibbs Free Energy

• Chemical potential과 Activity 의이해

• Real solution의이해

* Binary System (two component) A, B

- Mixture ; A – A, B – B ; 각각의성질유지, boundary는존재, 섞이지않고기계적혼합

A B

- Solution ; A – A – A ; atomic scale로섞여있다. Random distribution

A – B – A Solid solution : substitutional or interstitial

- compound ; A – B – A – B ; A, B의위치가정해짐, Ordered state

B – A – B – A

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

: 평형상태온도와압력이외에도조성의변화를고려

- Mixture ; A – A, B – B ; 각각의성질유지, boundary는존재, 섞이지않고기계적혼합

A B

- Solution ; A – A – A ; atomic scale로섞여있다. Random distribution

A – B – A Solid solution : substitutional or interstitial

- compound ; A – B – A – B ; A, B의위치가정해짐, Ordered state

B – A – B – A

MgCNi3

ruthenium 루테늄《백금류의금속원소;기호 Ru, 번호 44》

• 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: |χA – χB| ≤ 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

: 합금에대한열역학기본개념도입위해 2원고용체, 1기압의고정된압력조건고려

* Composition in mole fraction XA, XB XA + XB = 1

1. bring together XA mole of pure A and XB 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 SystemIn Step 1

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

;XA, XB (mole fraction)

),( PTGA

),( PTGB G1 = XAGA + XBGB J/mol

Free energy of mixture

Gibbs Free Energy of The System

In Step 2 G2 = G1 + ΔGmix J/mol

Since G1 = H1 – TS1 and G2 = H2 – TS2And putting ∆Hmix = H2 - H1 ∆Smix = S2 - S1

∆Gmix = ∆Hmix - T∆Smix

∆Hmix : Heat of Solution i.e. heat absorbed or evolved during step 2∆Smix : difference in entropy between the mixed and unmixed state.

How can you estimate ΔHmix and ΔSmix ?

Mixing free energy ΔGmix

가정1 ; ∆Hmix=0 :; A와 B가 complete solid solution

( A,B ; same crystal structure); no volume change

- Ideal solution

wkS ln= w : degree of randomness, k: Boltzman constant

thermal; vibration ( no volume change )Configuration; atom 의배열방법수 ( distinguishable )

ΔGmix = -TΔSmix J/mol

∆Gmix = ∆Hmix - T∆Smix

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

= +th configS S S

)]ln()ln()ln[(

, ,

1ln!!)!(ln

000

oBoBoBoAoAoAooo

BABBAA

BA

BAbeforeaftermix

NXNXNXNXNXNXNNNk

NNNNXNNXN

kNNNNkSSS

−−−−−=

=+==→

−+

=−=Δ

이므로

NNNN −≈ ln!ln

Excess mixing Entropy

),_(_ !!)!(

)__(_ 1

BABA

BAconfig

config

NNsolutionafterNNNNw

pureBpureAsolutionbeforew

→+

=

→=If there is no volume change or heat change,

Number of distinguishable way of atomic arrangement

0kNR =using Stirling’s approximation

and

Excess mixing Entropy

)lnln( BBAAmix XXXXRS +−=Δ

)lnln( BBAAmix XXXXRTG +=ΔΔGmix = -TΔSmix

Since ΔH = 0 for ideal solution,

Compare Gsolutionbetween high and low Temp.

G2 = G1 + ΔGmix

Chemical potential

= μA AdG' dn

⎛ ⎞∂μ = ⎜ ⎟∂⎝ ⎠ B

AA T, P, n

G'n

The increase of the total free energy of the system by the increase of very small quantity of A, dnA, will be proportional to dnA.

(T, P, nB: constant )

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

⎛ ⎞∂μ = ⎜ ⎟∂⎝ ⎠ A

BB T, P, n

G'n

Chemical potential

= μ + μA A B BdG' dn dn

= − + + μ + μA A B BdG' SdT VdP dn dn

for A-B binary solution, dG’

for variable T and P dG = -SdT + VdP

Chemical potential

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂= μ = = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠j j j j

ii i i iT,P,n S,V,n S,P,n T,V,n

G U H Fn 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 + μA A B BG X X Jmol

For 1 mole of the solution = μ + μA A B BdG' dn dn

μ = +μ = +

A A A

B B B

G RTln XG RTln X

μ = +

μ = +A A A

B B B

G RTln XG RTln X

μA’

μB’

XB’

Ideal solution : ΔHmix= 0

Quasi-chemical model assumes that heat of mixing, ΔHmix,is only due to the bond energies between adjacent atoms.

Structure model of a binary solution

Regular Solutions

∆Gmix = ∆Hmix - T∆Smix

Bond energy Number of bond

A-A εAA PAAB-B εBB PBBA-B εAB PAB

= ε + ε + εAA AA BB BB AB ABE P P P

Δ = ε ε = ε − ε + εmix AB AB AA BB1H P where ( )2

Internal energy of the solution

Regular Solutions

Δ =mixH 0

Completely random arrangement

ideal solution

=AB a A B

a

P N zX X bonds per moleN : Avogadro's numberz : number of bonds per atom

∆Hmix = ΩXAXB where Ω = Nazε

Δ = εmix ABH P

0

Regular Solutions

=ε )(21

BBAAAB εεε +=

↑→< ABP0ε ↓→> ABP0ε

0≈ε

Ω >0 인경우

Regular Solutions

Reference state

Pure metal 000 == BA GG

)lnln( BBAABABBAA XXXXRTXXGXGXG ++Ω++=

G2 = G1 + ΔGmix ∆Gmix = ∆Hmix - T∆Smix

Phase separation in metallic glasses

Regular Solutions

)lnln( BBAABABBAA XXXXRTXXGXGXG ++Ω++=

BBBB

AAAA

XRTXG

XRTXG

ln)1(

ln)1(2

2

+−Ω+=

+−Ω+=

μ

μ

Free Energy Change on Mixing

ABBABA XXXXXX 22 +=

1−= μ + μA A B BG X X Jmol

Activity, aideal solution regular solution

• μA = GA + RTlnaA μB = GB + RTlnaB

⎛ ⎞ Ω= −⎜ ⎟

⎝ ⎠B

BB

aln (1 X )X RT

γ = BB

B

aX

μ = +μ = +

A A A

B B B

G RTln XG RTln X

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

γ = ≅

γ = ≅

BB

B

AA

A

a cons tant (Henry 's Law)Xa 1 (Rault 's Law)X