“Phase Transformation in Materials”

Office: 33-313

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

Office hours: by an appointment

“Phase Transformation in Materials”

Eun Soo Park 2016 Fall

09.19.2016

Gibbs Free Energy as a Function of Temp. and Pressure

a. Single component system Chapter 1

 0 dG

P V S G

T G

T P

 



 







 

 



 







 ,

Thermondynamics and Phase Diagrams

- Equilibrium

2 1

0

G G G

   

- Phase Transformation

Contents for previous class

Lowest possible value of G

V T

H dT

dP

eq

eq



  ) (

Clausius-Clapeyron Relation

- Classification of phase transition

3

The First-Order Transitions

G

T S

T

S = L/T

T C_P

N P

P T

T S C

,



 







  Latent heat

Energy barrier

Discontinuous entropy, heat capacity

The Second Order Transition

G

T S

T

S=0

Second-order transition

C_P

No Latent heat

Continuous entropy



 



 







 

N P

P T

T S C

,

Liquid Undercooled Liquid Solid

<Thermodynamic>

Solidification:

• Interfacial energy ΔT_N

Melting:

• Interfacial energy

SV LV

SL

 

  

No superheating required!

No ΔT_N T_m

vapor

Melting and Crystallization are Thermodynamic Transitions

T

7

Contents for today’s class

- Chemical potential and Activity

- Gibbs Free Energy in Binary System

Ideal solution and Regular solution

- Binary System mixture/ solution / compound Hume-Rothery Rules for Alloys

Chapter 1 Thermondynamics and Phase Diagrams

Q1: What are binary systems?

“Mixture vs. Solution vs. Compound”

b. Multi-component system:

Q2: What is “Alloying”?

Ordered Compounds or Solid Solutions

9

Q3: “Solution vs. Intermetallic compound”?

Solid Solution vs. Intermetallic Compound

Crystal structure

Surface

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

solid solution and ordered compound Alloying: atoms mixed on a lattice

11

Solid Solution vs. Intermetallic Compounds

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

: in order to introduce some of the basic concepts of the thermodynamics of alloys

Assumption: a simple physical model for “binary solid solutions”

• Solid solution of B in A plus particles of a new phase (usually for a larger amount of B)

Second phase particle --different composition --often different structure.

Particles of New Phase in Solid-Solution Alloys

Solid Solution phase: B atoms in A

13

c. Mechanism of Precipitation

(1)

(2) (3)

(1)

(2)

(3)

(1) (2) (3)

Atomic diffusion Precipitate

Matrix atom

Alloying atom

Composition

Temperature

5) Microstructure control : ② Secondary phase control

Q4: How can we classify “Solubility”?

15

Solubility

• Unlimited Solubility

– Hume Rothery’ Conditions

• Similar Size

• Same Crystal Structure

• Same Valance

• Similar Electronegativity

– Implies single phase

• Limited Solubility

– Implies multiple phases

• No Solubility

- oil and water region

Cu-Ag Alloys Cu-Ni Alloys

complete solid solution limited solid solution

17

* 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.

19

21

Q5: Can we roughly estimate

what atoms will form solid solutions?

“Hume‐Rothery Rules”

Hume-Rothery Rules for Alloys (atoms mixing on a lattice)

Will mixing 2 (or more) different types of atoms lead to a solid-solution phase?

Empirical observations have identified 4 major contributors through :

+1 +2

Atomic Size Factor, Crystal Structure, Electronegativity, Valences

23

Hume-Rothery Rules for Mixing

Empirical rules for substitutional solid-solution formation were identified from experiment that are not exact, but give an expectation of formation.

Briefly,

1) Atomic Size Factor The 15% Rule

If "size difference" of elements are greater than ±15%, the lattice distortions (i.e. local lattice strain) are too big and solid-solution will not be favored.

DR%= < ±15% will not disallow formation.

2) Crystal Structure Like elemental crystal structures are better

For appreciable solubility, the crystal structure for metals must be the same.

3) Electronegativity ΔE ~ 0 favors solid-solution.

The more electropositive one element and the more electronegative the other, then "intermetallic compounds" (order alloys) are more likely.

4) Valences Higher in lower, alright. Lower in higher, it’s a fight.

A metal will dissolve another metal of higher valency more than one of lower valency.

r_solute  r_solvent

r_solvent x100%

Valence band: 절대영도에서 전자가 존재하는 가장 높은 전자 에너지 범위, 금속의 경우 전도띠 (conduction band)

Is solid-solution favorable, or not?

 Cu-Ni Alloys

Rule 1: r_Cu = 0.128 nm and r_Ni= 0.125 nm.

DR%= = 2.3% favorable √

Rule 2: Ni and Cu have the FCC crystal structure. favorable √ Rule 3: E_Cu = 1.90 and E_Ni= 1.80. Thus, ΔE%= -5.2% favorable √ Rule 4: Valency of Ni and Cu are both +2. favorable √

Expect Ni and Cu forms S.S. over wide composition range.

At high T, it does (helpful processing info), but actually phase separates at low T due to energetics (quantum mechanics).

Hume-Rothery Empirical Rules in Action

r_solute

 r

r_solvent x100%

25

Is solid-solution favorable, or not?

 Cu-Ag Alloys

Rule 1: r_Cu = 0.128 nm and r_Ag= 0.144 nm.

DR%= = 9.4% favorable √

Rule 2: Ag and Cu have the FCC crystal structure. favorable √ Rule 3: E_Cu = 1.90 and E_Ni= 1.80. Thus, ΔE%= -5.2% favorable √ Rule 4: Valency of Cu is +2 and Ag is +1. NOT favorable

Expect Ag and Cu have limited solubility.

In fact, the Cu-Ag phase diagram (T vs. c) shows that a solubility of only 18% Ag can be achieved at high T in the Cu-rich alloys.

r_solute  r_solvent

r_solvent x100%

Hume-Rothery Empirical Rules in Action

Cu-Ag Alloys Cu-Ni Alloys

complete solid solution limited solid solution

27

(1) Thermodynamic : high entropy effect (2) Kinetics : sluggish diffusion effect (3) Structure : severe lattice distortion effect (4) Property : cocktail effect

Major element 2

Major element 3 Major

element 4, 5..

Major element 1

Minor element 2 Minor element 3

Major element

Traditional alloys

Minor element 1

Conventional alloy system High entropy alloy system

Yong Zhang et al., Adv. Eng. Mat. P534‐538, 2008

Ex) 304 steel ‐ Fe74Cr18Ni8 Ex) Al²⁰Co²⁰Cr²⁰Fe²⁰Ni²⁰

Severe lattice distortion → Sluggish diffusion & Thermal stability

High entropy alloy (HEA)

High entropy alloy (HEA)

Q6: How to calculate

“Gibbs Free Energy of Binary Solutions”?

G

= G

+ ΔG

J/mol

29

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

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

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

2) Gibbs Free Energy of Binary Solutions

: binary solid solution/ a fixed pressure of 1 atm 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

, X

(mole fraction

, ( PT G_A

) , ( PT G_B

G

= X

G

+ X

G

J/mol

1.3 Binary Solutions

Constant T, P

31

Gibbs Free Energy of The System

In Step 2 G

= G

+ ΔG

J/mol

Since G

= H

– TS

and G

= H

– TS

And putting ∆H

= H

- H

∆S

= S

- S

∆G

= ∆H

- T∆S

∆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 ΔH

and ΔS

?

1.3 Binary Solutions

Q7: How can you estimate

“ΔG _mix of ideal solid solution”?

Gibbs Free Energy of Binary Solutions

) ln

ln

(

mix

RT X X X X

G  



ΔG

= -TΔS

33

Mixing free energy, ΔG

Assumption 1; ∆ H

=0 :

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

; no volume change

- Ideal solution

w k

S  ln

 Thermal; vibration ( no volume change )

 Configuration; number of distinguishable ways of arranging the atoms

ΔG

= -TΔS

J/mol

∆G

= ∆ H

- T ∆ S

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





S S S

1.3 Binary Solutions

)]

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























 









N N

N

N !  ln  ln

Excess mixing Entropy

) , _(

_

!

!

)!

(

) _

_(

_

1

B A B

A B config A

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

R 

using Stirling’s approximation and

1.3 Binary Solutions

Since we are dealing with 1 mol of solution,

(the universal gas constant)

35

Excess mixing Entropy

) ln

ln

(

mix

R X X X X

S   



) ln

ln

(

mix

RT X X X X

G  



ΔG

= -TΔS

1.3 Binary Solutions

Fig. 1.9 Free energy of mixing for an ideal solution

Q8: How can you estimate

“Molar Free energy for ideal solid solution”?

Gibbs Free Energy of Binary Solutions

G

= G

+ ΔG

G = X

G

+ X

G

+ RT(X

lnX

+ X

lnX

)

37

Since ΔH_mix = 0 for ideal solution,

Compare G_solution between high and low Temp.

G

= G

+ ΔG

G

ΔG

1) Ideal solution

1.3 Binary Solutions

Fig. 1.10 The molar free energy (free energy per mole of solution) for an ideal solid solution.

A combination of Figs. 1.8 and 1.9.

+ =

G = X

G

+ X

G

+ RT(X

lnX

+ X

lnX

)

a fixed pressure of 1 atm

Q9: How the free energy of a given phase will change when atoms are added or removed?

“ Chemical potential ”

Gibbs Free Energy of Binary Solutions

39

Chemical potential

 

dG' dn

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

, will be proportional to μ

.

dn_A~ small enough

(∵ _A depends on the composition of phase)

(T, P, n

: constant )

G

= H-TS = E+PV-TS

For A-B binary solution, dG' ^{ }

dn

^{ }

dn

    

 

dG' SdT VdP dn dn

For variable T and P

1) Ideal solution

1.3 Binary Solutions



:

  

  ^A   ^A T, P, n_B

G' n

  

  ^B   ^B T, P, n_A

G' n

41

Chemical potential

 

dG' dn

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

, will be proportional to μ

.

dn_A~ small enough

(∵ _A depends on the composition of phase)

(T, P, n

: constant )

G

= H-TS = E+PV-TS

For A-B binary solution, dG' ^{ }

dn

^{ }

dn

    

 

dG' SdT VdP dn dn

For variable T and P

1) Ideal solution

1.3 Binary Solutions



:

  

  ^A   ^A T, P, n_B

G' n

  

  ^B   ^B T, P, n_A

G' n

Q10: “Correlation between chemical potential and free energy”?

Gibbs Free Energy of Binary Solutions

43

1

 

 

G X X Jm ol

A A B B

dG   dX   dX

B A

B

dG

dX    

A B

B

dG

    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

 

      

 

    

  

   

B A

B

G dG X

   dX

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

1) Ideal solution

Correlation between chemical potential and free energy

1

 

 

G X X Jmol

1) Ideal solution For 1 mole of the solution

Correlation between chemical potential and free energy

G_A G_B

μ_A

μ_B

X_A X_B X_B

B A

B

G dG X

   dX

) 1

(

B

B

X

dX

G  dG 

 

B B

A

B

dX

X dG

X )

( 



 

G

A B C

X

A B

B

dG

    dX

D

X

DA DC

CA  



X₀

45

A B

B

dG

    dX

1

 

 

G X X Jm ol

B A

B

G dG X

   dX

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

Correlation between chemical potential and free energy

1) Ideal solution

Fig. 1.11 The relationship between the free energy curve for a solution and the chemical potentials of the components.

1) Ideal solution 1.3 Binary Solutions

μ_B’

μ_A’ X_B’

 

 

G

X X

B B

A A

A

RT X X G RT X X

G ln ) ( ln )

(   



Q1: What is “Regular Solution”?

∆G

= ∆H

- T∆S

1.3 Binary Solutions

47

Ideal solution : H

= 0

Quasi-chemical model assumes that heat of mixing, H

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

Structure model of a binary solution

Regular Solutions

∆G

= ∆H

- T∆S

1.3 Binary Solutions

Assumption: the volumes of pure A and B are equal and do not change during mixing so that the interatomic distance and bond energies are independent of composition.

This type of behavior is exceptional in practice and usually mixing is endothermic or exothermic.

Q2: How can you estimate

“ΔH _mix of regular solution”?

Gibbs Free Energy of Regular Solutions

∆H

= ΩX

X

where Ω = N

zε

49

Bond energy Number of bond A-A 

P

B-B 

P

A-B 

P



 

 



E P P P

       1   

Internal energy of the solution

Regular Solutions

1.3 Binary Solutions

 H

 0

Completely random arrangement

ideal solution

AB



a

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

z : number of bonds per atom

∆H

= ΩX

X

where Ω = N

zε

 H

 P



Regular Solutions

 0

 ( )

2 1

BB AA

AB

 

  





 0 P

  ^{ 0  P}

^

 0



If Ω >0,

1.3 Binary Solutions

(1) (2)

(3)

Fig. 1.14 The variation of ΔH_mixwith composition for a regular solution.

51

Q3: How can you estimate

“Molar Free energy for regular solution”?

) ln

ln

(

B A B

B A

A

G X G X X RT X X X X

X

G      

G

= G

+ ΔG

Regular Solutions

Reference state

Pure metal

G

 G

 0

) ln

ln

(

B A B

B A

A

G X G X X RT X X X X

X

G      

G

= G

+ ΔG

∆G

= ∆H

- T∆S

∆H_mix -T∆S_mix

53

Q4: “Correlation between chemical potential and free energy”?

Gibbs Free Energy of Binary Solutions

1

 

 

G X X Jmol

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

G = E+PV-TS G = H-TS

2) regular solution

A B B

A B

A B A B

AX X X X X X X X X

X  (  ) ²  ²

B B

B B

A A

A A

X RT

X G

X RT

X G

ln )

1 (

ln )

1 (

2 2

























) ln

ln

(

B A B

B A

A

G X G X X RT X X X X

X

G      

) ln

) 1

( (

) ln

) 1

(

( _{A A} ² _{A B B B} ² _B

A G X RT X X G X RT X

X       



  

  

A A A

B B B

G RTln X G RTln X

Correlation between chemical potential and free energy

Ideal solution

복잡해졌네 --;;

Regular solution

55

Q5: What is “activity”?

Gibbs Free Energy of Binary Solutions

Activity, a : effective concentration for mass action

ideal solution

μ

= G

+ RTlna

μ

= G

+ RTlna

  

  

A A A

B B B

G RTln X G RTln X

B B

B B

A A

A

A  G   (1  X )²  RT ln X





 

B

a X

)

1 ( )

ln(

B

B

X

RT X

a  



57

Q6: “Chemical equilibrium of multi‐phases”?

Gibbs Free Energy of Binary Solutions

59

2

Q5: How can we define equilibrium in heterogeneous systems?

61

Equilibrium in Heterogeneous Systems

-RT lna_B^β -RT lna_B^α -RT lna_A^α

-RT lna_A^β

a

=a

a

=a

G

In X

, G

> G

> G

α + β

μ

μ

=

μ

μ

63

- Chemical potential ^and Activity

- Gibbs Free Energy in Binary System - Binary System

Ideal solution

Regular solution

mixture/ solution / compound

G

= G

+ ΔG

J/mol G

= X

G

+ X

G

J/mol

(∆H

=0) ^{ G}

^{ RT ( X}

^{ln X}

^{ X}

^{ln X}

⁾





    

1   

H P where ( )

2

) ln

ln

(

B A B

B A

A

G X G X X RT X X X X

X

G      

  

  ^A   ^A _{T, P, nB} G'

n

• μ

= G

+ RTlna

μ는 조성에 의해 결정되기 때문에 dn_A가 매우 작아서 조성변화 없어야

- Chemical equilibrium

Topic proposal for materials design

Please submit 3 materials that you want to explore for materials design and do final presentations on in this semester. Please make sure to thoroughly discuss why you chose those materials (up to 1 page on each topic). The proposal is due by September 21

on eTL.

Ex) stainless steel/ graphene/ OLED/

Bio-material/ Shape memory alloy

Bulk metallic glass, etc.