“Phase Transformation in Materials”

“Phase Transformation in Materials”

Eun Soo Park

Office: 33-313

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

Office hours: by an appointment

2021 Fall

09.29.2021

1

• Effect of Temperature on Solid Solubility

• Equilibrium Vacancy Concentration

• Influence of Interfaces on Equilibrium

• Gibbs-Duhem Equation:

: Be able to calculate the change in chemical potential that result from a change in alloy composition.

r G 2 γ V

^m

=

∆ Gibbs-Thomson effect

2 2

ln ln

1 1

ln ln

A B

A B

A B

d G d d

X X RT RT

dX d X d X

γ γ

   

=  +  =  + 

   

합금조성의 미소변화 (dX)로 인한 화학퍼텐셜의 미소변화

(dμ)

를 계산

• Gibbs Phase Rule F = C − P + 1

(constant pressure) Gibbs' Phase Rule allows us to construct phase diagram to represent and interpret phase equilibria in heterogeneous geologic systems.

2

A A B B

0

X d µ + X d µ =

Contents for previous class:

n V

m

G G per unit volume of

V β

∆ = ∆

0

V e

G X where X X X

∆ ∝ ∆ ∆ = −

Total Free Energy Decrease per Mole of Nuclei

Driving Force for Precipitate Nucleation

∆G

₀

=-V∆G

_V

+A γ + V∆G

_s

β α

β

α

µ

µ

_A

X

_{A B}

X

_B

G = +

∆

₁

β β β

β

µ

µ

_A

X

_{A B}

X

_B

G = +

∆

₂

1

2

G

G

G

_n

= ∆ − ∆

∆

For dilute solutions,

T X

G

_V

∝ ∆ ∝ ∆

∆

: 변태를 위한 전체 구동력/핵생성을 위한 구동력은 아님

: Decrease of total free E of system

by removing a small amount of material with the nucleus composition (X_B^β) (P point)

: Increase of total free E of system

by forming β phase with composition X_B^β (Q point)

: driving force for β precipitation

∝undercooling below T_e (length PQ)

∆G

_V

Chapter 5.1

3

• Ternary Phase Diagram

• Measurement of multi-component phase diagram

• Computation of Phase Diagram

Contents for today’s class

4

Q1: “Ternary Phase Diagram”?

5

What are ternary phase diagram?

www.sjsu.edu/faculty/selvaduray/page/phase/ternary_p_d.pdf

Diagrams that represent the equilibrium

between the various phases that are formed between three components, as a function of temperature.

Normally, pressure is not a viable variable in ternary phase diagram construction, and is therefore held constant at 1 atm.

6

Gibbs Phase Rule for 3-component Systems

7

Gibbs Triangle

An Equilateral triangle on which the pure components are represented by each corner.

Concentration can be expressed as either “wt. %” or “at.% = molar %”.

X

_A

+X

_B

+X

_C

= 1

Used to determine

the overall composition

8

Overall Composition

9

Overall Composition

10

Ternary Isomorphous System

Isomorphous System: A system (ternary in this case) that has only one solid phase. All components are totally soluble in the other components. The ternary system is therefore made up of three binaries that exhibit total solid solubility.

The Liquidus surface: A plot of the temperatures above which a homogeneous liquid forms for any given overall composition.

The Solidus Surface: A plot of the temperatures below which a (homogeneous) solid phase forms for any given overall composition.

11

Ternary Isomorphous System

T_A

T_B T_C

T_A

T_A T_A

T_C

T_C T_B

T_B

12

Ternary Isomorphous System

T_B

T_C

T_A

13

Ternary Isomorphous System

T_B

T_C

T_A

T_B

T_C

T_A

14

Ternary Isomorphous System

T_B

T_C

T_A

T_B T_C

T_A

15

Ternary Isomorphous System

Isothermal section → F = C - P

T_B

T_C

T_A

16

Ternary Isomorphous System

Isothermal section

T_B

T_C

T_A

17

Ternary Isomorphous System

Isothermal section → F = C - P

Fig. 1.41 (a) Free energy surface of a liquid and three solid phases of a ternary system.

(b) A tangential plane construction to the free energy surfaces defined equilibrium between s and l in the ternary system

(c) Isothermal section through a ternary phase diagram

18

Ternary Isomorphous System

Locate overall composition using Gibbs triangle

19

20

Ternary Eutectic System

(No Solid Solubility)

T_A

T_B

T_C T_A

T_A

T_B

T_C T_A

T_B

T_C

21

Ternary Eutectic System

Liquidus projection

(No Solid Solubility)

T_A

T_B

T_C

22

Ternary Eutectic System

(No Solid Solubility)

T_A

T_B

T_C

23

Ternary Eutectic System

(No Solid Solubility)

T_A

T_B

T_C

24

Ternary Eutectic System

(No Solid Solubility)

T_A

T_B

T_C

25

Ternary Eutectic System

(No Solid Solubility)

T_A

T_B

T_C

26

Ternary Eutectic System

(No Solid Solubility)

T_A

T_B

T_C

27

Ternary Eutectic System

(No Solid Solubility)

T_A

T_B

T_C

28

Ternary Eutectic System

(No Solid Solubility)

T= ternary eutectic temp.

A C

B L+A+C

L+A+B L+B+C

29

Ternary Eutectic System

(with Solid Solubility)

30

Ternary Eutectic System

(with Solid Solubility)

T_A

T_B T_C

T_C

T_B T_A

31

32

Ternary Eutectic System

(with Solid Solubility)

T_A

T_B T_C

32

Ternary Eutectic System

(with Solid Solubility)

T_A T_B

T_C

33

Ternary Eutectic System

(with Solid Solubility)

T_A T_B

T_C

34

Ternary Eutectic System

(with Solid Solubility)

T_A T_B

T_C

35

36

Ternary Eutectic System

(with Solid Solubility)

T_A

T_B T_C

36

Ternary Eutectic System

(with Solid Solubility)

T_A T_B

T_C

37

Ternary Eutectic System

(with Solid Solubility)

T= ternary eutectic temp.

C

L+ β+γ

L+ α+γ L+ α+β

38

Ternary Eutectic System

(with Solid Solubility)

정해솔 학생 제공 자료 참조: 실제 isothermal section의 온도에 따른 변화 http://www.youtube.com/watch?v=yzhVomAdetM

T_A T_B

T_C

39

Ternary Eutectic System

3) Solidification Sequence: liquidus surface

40

Ternary Eutectic System

* The horizontal lines are not tie lines.

(no compositional information)

* Information for equilibrium phases at different tempeatures

* Vertical section

41

Vertical section Location of vertical section

10.1. THE EUTECTIC EQUILIBRIUM (l = α + β + γ)

42

Vertical section Location of vertical section

10.1. THE EUTECTIC EQUILIBRIUM (l = α + β + γ)

44

Vertical section Location of vertical section

10.1. THE EUTECTIC EQUILIBRIUM (l = α + β + γ)

45

< Quaternary phase Diagrams >

Four components: A, B, C, D

A difficulty of four-dimensional geometry

→ further restriction on the system Most common figure:

“ equilateral tetrahedron “ Assuming isobaric conditions, Four variables: X_A, X_B, X_C and T

4 pure components 6 binary systems 4 ternary systems A quarternary system

a+b+c+d=100

%A=Pt=c,

%B=Pr=a,

%C=Pu=d,

%D=Ps=b

* Draw four small equilateral tetrahedron

→ formed with edge lengths of a, b, c, d

46

Q2: “Measurement of multi-component phase diagram”?

47

Construction of pseudo-binary phase diagram for multi-component alloy system

1MAR2017 / TMS 2017

0 10 20 30 40 50 60 70 80 90 100

600 700 800 900 1000 1100 1200 1300 1400 1500 1600

Temperature ( ℃ )

Cu composition (at. %)

α β

0 10 20 30 40 50 60 70 80 90 100

600 700 800 900 1000 1100 1200 1300 1400 1500 1600

Temperature ( ℃ )

Cu composition (at. %)

α β

0 10 20 30 40 50 60 70 80 90 100

600 700 800 900 1000 1100 1200 1300 1400 1500 1600

Temperature ( ℃ )

Cu composition (at. %)

α β

Phase diagram was expected to optimize composition and microstructure of phase separating HEA

Thermodynamic calculation X-ray diffraction

TGA/DSC FE-EPMA

• Expecting approximation of phase diagram

• Determination of phases

• Finding out temperatures of phase transformations

• Confirming invariant reaction points

• Investigation of equilibrium composition at each

temperature

48

Pseudo-binary phase diagram of PS-HEA

1MAR2017 / TMS 2017

Pseudo-binary system between FeCoCrNi and Cu shows monotectic reaction having liquid separation region.

Calculation courtesy of J.H.Kim et al.

Experiment Calculation

L

L 1 + L 2 FCC 1 + L 2

FCC 1 + FCC 2 FCC 2

Dendrite → Droplet → Dendrite

0 10 20 30 40 50 60 70 80 90 100

400 600 800 1000 1200 1400 1600

Te m pe ra tu re ( ℃ )

Cu content (at%)

^Cu

FeCoCrNi

Posted on L-88 (HEA V: Poster session)

49

MoVNbTiZr: Construction of pseudo-ternary phase diagram Nb,Ti

Mo,V Zr

bcc IC+bc

c

bcc Mo, V, Nb, Ti : Complete S.S.

V-Zr, Mo-Zr Ti-Zr, Nb-Zr

TiNbMoVZr: Construction of pseudo-ternary phase diagram

1500K 1300K

1100K

Calculated pseudo-ternary isothermal sections of the MoNbTiVZr system

MoVNbTiZr: Construction of pseudo-ternary phase diagram

X-ray diffraction analysis of the as-cast samples

Construction of pseudo-ternary phase diagram

Find single phase region without intermetallic compounds BCC1+BCC2+IC

BCC1+IC / BCC2+IC BCC1+BCC2

Single BCC

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1.0 0.0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

20

25 26

23 24

22

21

19 11

10

1213

14 15

16 17

18 9 7

8 5

6 4

2 3

MoV Zr

NbTi

Nb Ti Mo

V

Zr

1

Q3: “Computation of Phase Diagram”?

54

55

<Computation of Phase Diagram>

i ) Pure Stoichiometric Substances

Original Calculation of Enthalpy & Entropy

Debye Suggest Heat Capacity as a function of T!

θ : Debye Temperatrue

ω

D : Debye Frequency

(Maximum Frequency of lattice vibration)

Enthalpy and Entropy isState Function!

Substract Initial state to final state, Integral of dH/dT, dS/dT

After Mayer Kelley, more accurate to experiment result of specific heat capacity at Cp

In the phase diagram, we found many differences between the predicted values and the experimental results! Therefore, it is necessary to calibrate the G value!

Here,

considering contribution of lattice & magnetic & pressure

<Magnetic Contribution>

T

c = Curie temperature

(to lose magnetic property)

β

: magnetic moment per atom

f

structure-dependent function of

T

<Pressure Contribution>

α(T) = thermal expansion of volume

n is pressure derivative of bulk

This polynomial Function fit into real graph! ₅₆

ii) Calculating G for Solid Solution

57

Method of Redlich & Kister

Further multi-component system

59

Homework1 :

Please explain more detail how to calculate multi-component

phase diagram reflecting excess Gibbs energy (within 5 pages PPT)

60

* Incentive Homework 1

Please submit ternary phase diagram model which can clearly express 3D structure of ternary system by October 15 in Bldg. 33-313.

You can submit the model individually or with a small group under 2 persons.

* Homework 2 : Exercises 1 (pages 61-63) Good Luck!!

61

“Phase Transformation in Materials”

Eun Soo Park

Office: 33-313

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

Office hours: by an appointment

2021 Fall

2021.10.04.

1

2

• Ternary Equilibrium: Ternary Phase Diagram Contents for previous class

1) Gibbs Triangle

Used to determine

the overall composition

X

_A

+X

_B

+X

_C

= 1

2) Isothermal section → F = C - P

Tangential plane

2 phases region

Free energy surface

Fig. 1.41 (a) Free energy surface of a liquid and three solid phases of a ternary system.

(b) A tangential plane construction to the free energy surfaces defined equilibrium between s and l in the ternary system

(c) Isothermal section through a ternary phase diagram

3

Ternary Eutectic System

(with Solid Solubility)

3 phases region: tie triangle

4

Ternary Eutectic System

3) Solidification Sequence: liquidus surface

5

Ternary Eutectic System

* The horizontal lines are not tie lines.

(no compositional information)

* Information for equilibrium phases at different temperatures

* Vertical section

6

< Quaternary phase Diagrams >

Four components: A, B, C, D

A difficulty of four-dimensional geometry

→ further restriction on the system Most common figure:

“ equilateral tetrahedron “ Assuming isobaric conditions, Four variables: X_A, X_B, X_C and T

4 pure components 6 binary systems 4 ternary systems A quarternary system

At constant T, D

A

B

C

7

a+b+c+d=100

%A=Pt=c,

%B=Pr=a,

%C=Pu=d,

%D=Ps=b

* Draw four small equilateral tetrahedron

→ formed with edge lengths of a, b, c, d

8

1.10 The kinetics of phase transformations

exp (-ΔG^a/kT ) k: Boltzmann’s constant ( R/N_a )

Kinetic theory:

probability of an atom reaching the activated state

Activation free energy barrier

Transformation rate ∝ exp (-ΔG

^a

/k T )

Putting ΔG^a= ΔH^a - TΔS^a and changing from atomic to molar quantities

∝ exp (-ΔH

^a

/R T )

- First derived empirically from the observed temp.

dependence of the rate of chemical reactions - Apply to a wide range of processes and

transformation in metals and alloys

- “Simplest of these is the process of diffusion”

→ Arrhenius rate equation

9

Contents in Phase Transformation

(Ch1) Thermodynamics and Phase Diagrams (Ch2) Diffusion: Kinetics

(Ch3) Crystal Interface and Microstructure

(Ch4) Solidification: Liquid → Solid

(Ch5) Diffusional Transformations in Solid: Solid → Solid (Ch6) Diffusionless Transformations: Solid → Solid

Background to understand phase

transformation

Representative Phase

transformation

10

Contents for today’s class

• Diffusion

• Interstitial Diffusion – Fick’s First Law

- Effect of Temperature on Diffusivity

- Nonsteady-state diffusion – Fick’s Second Law

• Solutions to the diffusion equations

• Substitution Diffusion

1. Self diffusion in pure material 2. Vacancy diffusion

3. Diffusion in substitutional alloys

11

Q1. What is the driving force for diffusion?

⇒ a concentration gradient (x)

⇒ a chemical potential gradient (o)

Diffusion Movement of atoms to reduce its chemical potential μ.

Diffusion

12

13

Diffusion: THE PHENOMENON

• Interdiffusion: in a solid with more than one type of element (an alloy), atoms tend to migrate from regions of large concentration.

Initially After some time

< Diffusion couple between Cu and Ni >

14

Mechanism of Solid-State Diffusion

R.E. Reed-Hill, Physical Metallurgy Principles

Substitutional diffusion Interstitial diffusion

15

Diffusion : Movement of atoms to reduce its chemical potential μ.

Driving force: Reduction of G Down-hill diffusion

movement of atoms from a high concentration region to low

concentration region.

Up-hill diffusion

movement of atoms from a low concentration region to high concentration region

2 1

2 1

B B

A

A

µ

µ

µ µ

>

<

^{1 2}

1 2

A A

B B

µ µ

µ µ

>

<

‘down-hill’ diffusion

‘up-hill’ diffusion

16

Down-hill diffusion

movement of atoms from a high concentration region to low concentration region.

17

Diffusion

Movement of atoms to reduce its chemical potential μ.

Driving force: Reduction of G Down-hill diffusion

movement of atoms from a high concentration region to low

concentration region.

Up-hill diffusion

movement of atoms from a low concentration region to high concentration region

2 1

2 1

B B

A

A

µ

µ

µ µ

>

<

^{1 2}

1 2

A A

B B

µ µ

µ µ

>

<

‘down-hill’ diffusion

‘up-hill’ diffusion

18

5.5.5 Spinodal Decomposition

Spinodal mode of transformation has no barrier to nucleation

2) If the alloy lies outside the spinodal,

small variation in composition

leads to an increase in free energy and the alloy is therefore

metastable

.

2

2

0

d G dX <

The free energy can only be decreased if nuclei are formed with a composition very different from the matrix.

→ nucleation and growth How does it differ between

inside and outside the inflection point of Gibbs free energy curve?

1) Within the spinodal

: phase separation by small fluctuations in composition/

“up-hill diffusion”

: “down-hill diffusion”

Fig. 5.38 Alloys between the spinodal points are unstable and can decompose into two coherent phasees α1and α2without overcoming an activation energy barrier. Alloys between the coherent miscibility gaps and the spinodal are metastable and can decompose only after nucleation of the other phase.

: describing the transformation of a system of two or more components in a metastable phase into two stable phases

At T₂,

19

b) Normal down-hill diffusion outside the spinodal

a) Composition fluctuations within the spinodal

up-hill diffusion

interdiffusion coefficient

D<0

down-hill diffusion

Fig. 5.39 & 5.40 schematic composition profiles at increasing times in (a) an alloy quenched into the spinodal region (X₀ in Figure 5.38) and (b) an alloy outside the spinodal points (X₀’ in Figure 5.38)

20

Diffusion

Diffusion : Mechanism by which matter transported through matter What is the driving force for diffusion?

But this chapter will explain with

“concentration gradients for a convenience”.

(Down-hill diffusion)

⇒ a concentration gradient (x)

⇒ a chemical potential gradient (o)

21

Q2. Interstitial diffusion vs Substitutional diffusion

Fick’s First Law of Diffusion

22

Atomic mechanisms of diffusion

The mean vibrational energy is 3kT , therefore the amplitude of thermal oscillation of an atom increases markedly with temperature . If a neighboring atom site is vacant, then the oscillation atom can change site with a certain probability ( atom jumping ). The probability consists of creation of the vacant site and movement of a neighboring atom into the site.

Substitutional vs. Interstitial diffusion

Substitutional diffusion

FCC

# of vacant site

23

Interstitial diffusion

FCC (0.74) BCC (0.68)

Fig. 2.3 (a) Octahedral interstices (O) in an fcc crystal. (b) Octahedral interstices in a bcc crystal.

24

How interstitial diffusion differs from substitutional diffusion?

Interstitial diffusion

25

Assume that there is no lattice distortion and also that there are always six vacant sites around the diffusion atom.

J_B : Net flux of B atom

Γ_B : Average jump rate of B atoms n₁ : # of atoms per unit area of plane 1 n₂ : # of atoms per unit area of plane 2

2

1

6

1 6

1 n n

J

_B

= Γ

_B

− Γ

_B

Interstitial diffusion

Random jump of solute B atoms in a dilute solid solution of a simple cubic lattice

( )

α

− = − ∂ ∂

B B B

C (1) C (2) C x

Fig. 2.5 Interstitial diffusion by random jumps in a concentration gradient.

26

α α

= =

B B

C (1) n

₁

, C (2) n

₂

(atoms m^-2 s^-1)

( n

₁

⁻ n

₂

) ^{= α} ( C

B

(1) ⁻ C

B

(2) )

D_B: Intrinsic diffusivity or Diffusion coefficient of B

⇒ depends on microstructure of materials

Fick’s First Law of Diffusion

(

_{1 2}

)

2

1

6

1 6

1 6

1 n n n n

J

_B

= Γ

_B

− Γ

_B

= Γ

_B

−

( ) ^{ α  ∂ ∂}

= Γ − = −   Γ  ∂  = − ∂

2

1 2

1 1

6 6

B B

B B B B

C C

J n n D

x x

( )

α

− = − ∂ ∂

B B B

C (1) C (2) C x

α: Jump distance

Magnitude of D in various media Gas : D ≈ 10^-1 cm²/s

Liquid : D ≈ 10^-4 ~ 10^-5 cm²/s

Solid : Materials near melting temp. D ≈ 10^-8 cm²/s Elemental semiconductor (Si, Ge) D ≈ 10^-12 cm²/s

Concentration varies with position.

D

_B

, in general, is concentration-dependent.

For example, in γ-Fe at 1000

^o

C,

D

_c

= 2.5 x 10

^-11

m

²

s

^-1

, when C

_c

= 0.15 wt%

D

_c

= 7.7 x 10

^-11

m

²

S

^-1

, when C

_c

= 1.4 wt%

2

6

1

_B

α D

B

= Γ

* Estimate the jump frequency of

a carbon atom in γ-Fe (FCC) at 1000

^o

C .

lattice parameter of γ-Fe : ~0.37 nm

nm .

. 37 2 0 26

0 =

α = D

_C

= 2 . 5 × 10

⁻¹¹

m

²

s

⁻¹

Γ = × 2 10

⁹

jumps s

−¹

the vibration frequency of carbon : ~ 10¹³

Only about one attempt in 10⁴ results in a jump from one site to another.

↑

↑→

_C

c

D

C

* If the crystal structure is not cubic, then the probability of jump is

anisotropic (different α)

. For example the probability of jumps along the basal direction and the axial direction of hcp crystal are different.

∵C atoms stain the Fe lattice

thereby making diffusion easier as amount of strain increases.

28

Q3. What is the radial distance, r from the origin in random work?

( ) ( )

α

= Γ =

r t f D = 2.4 Dt

29

( ) ( )

α

= Γ =

r t f D

During random walk,

an atom will move in time (t)

a radial distance (r) from the origin For random walk in 3 dimensions, after n steps of length α

α

→ n

Dt D r

6 6

2

→ =

=

Γ α



In 1 s, each carbon atom will move a total distance of ~ 0.5 m

→ a net displacement :

= 2.4 Dt

~ 10 µm.

2

6 1 _Bα DB = Γ Net distance from its original position

Very few of the atom jumps provide a useful contribution to the total diffusion distance.

The direction of each new jump is independent of the direction of the previous jump.

30

Q4. What is the effect of the temperature on diffusivity?

0

exp

^ID

B B

D D Q

RT

= −

31

EFFECT OF TEMPERATURE on Diffusivity

How D varies with T?

Thermal Activation

How varies with T? Γ

Fig. 2.6 Interstitial atom, (a) in equilibrium position, (b) at the position of maximum lattice distortion.

(c) Variation of the free energy of the lattice as a function of the position of interstitial.

32

0

exp

^ID

B B

D D Q

RT

= −

m m m

G H T S

∆ = ∆ − ∆ ¹

²

B

6

B

D = Γ α

B

? jump frequency Γ

Thermally activated process

Ζ : nearest neighbor sites ν : vibration frequency

∆G_m : activation energy for moving

( ) ( )

ID m

m m

B

Q H

RT H

R S

D

≡

∆

∆

 −

 

 Ζ ∆

= exp / exp /

6

1 α

₂

ν

( ^G

_m

^RT )

B

= Ζ exp − ∆ /

Γ ν

33

 

 



− 

= . R T

D Q log D

log 1

3

0

2

Temperature Dependence of Diffusion

=

₀

exp −

^ID

B B

D D Q

RT

Therefore, from the slope of the D-curve in an log D vs 1/T coordinate, the activation energy may be found.

How to determine Q

_ID

experimentally?

Fig. 2.7 The slope of log D v. 1/T gives the activation energy for diffusion Q.

34

Q5. Steady state diffusion vs Non-steady state diffusion?

Fick’s Second Law

∂ = ∂

∂ ∂

2 2

B B

B

C C

t D x

Fick’s first Law

Concentration varies with

“position” “Both position and time”

35

Steady-state diffusion

The simplest type of diffusion to deal with is when the concentration at every point does not change with time.

Apply Fick’s First Law:

the slope, dC/dx, is constant (does not vary with position)!

Steady State

Concentration, C, in the box

then

36

Steady-state diffusion

• Steel plate at 700℃

Q: How much carbon is transferring from the rich to deficient side?

The simplest type of diffusion to deal with is when the

concentration at every point does not change with time.

37

Nonsteady-state diffusion

In most practical situations steady-state conditions are not established, i.e. concentration varies with both distance and time, and Fick’s 1st law can no longer be used.

How do we know the variation of C

_B

with time? → Fick’s 2nd law

The number of interstitial B atoms that diffuse into the slice across plane (1) in a small time interval dt :

→ J

1

A dt Likewise : J

₂

A dt

Sine J₂ < J₁, the concentration of B within the slice will have increased by

( ) ^δ

δ δ

=

¹

−

²

B

J J A t

C A x

Due to mass conservation

( J

1

− J

2

) A t δ = = ? δ C A x

_B

δ

38

Nonsteady-state diffusion

B B

C J

t x

∂ ∂

∂ = − ∂

Fick’s Second Law

substituting Fick's 1st law gives

 

 





∂

∂

∂

= ∂

∂

∂

x D C

x t

C

_B

B B

∂ = ∂

∂ ∂

2 2

B B

B

C C

t D x

x D C

J

_B ^B

∂

− ∂

=

If variation of D with concentration can be ignored, ( ) ^δ

δ δ

= ¹− ²

B

J J A t

C A x

39

x ?

C

_B

=

∂

∂

2 2

∂ ∂

∂ = ∂

2 2

B B

B

C C

t D x

Fick’s Second Law

Concentration varies with time and position.

Note that is the curvature

₂ of the C_B versus x curve.

2

x C

_B

∂

∂

2 0

2 >

∂

∂ x

C_B

2 0

2 <

∂

∂ x

C_B

>0

∂

∂ t C_B

<0

∂

∂ t C_B

All concentration increase with time All concentration decrease with time

40

Q6. How to solve the diffusion equations?

: Application of Fick’s 2

^nd

law

homogenization, carburization, decarburization, diffusion across a couple

41

Solutions to the diffusion equations (Application of Fick’s 2

^nd

law) Ex1. Homogenization

of sinusoidal varying composition

in the elimination of segregation in casting

l : half wavelength

l C x

C = + β

₀

sin π

: the mean composition C　

0 : the amplitude of the initial concentration profile β

Initial or Boundary Cond.?

at t=0

Fig. 2.10 The effect of diffusion on a sinusoidal variation of composition.

∂ = ∂

∂ ∂

2 2

B B

B

C C

t D x

42

Rigorous solution of

for

2 2

x D C t

C

∂

= ∂

∂

∂ ( )

l C x

x

C ,0 = +β₀sinπ

( )

( )

D l

t n

n

n n

n n

Dt

l e C x

C

A B

C A

l C x

C t

x B

x A

A C

e x B

x A

C

2 2

2

/ 0

0 1

0

0 1

0

sin , 0 ,

;

sin 0

cos sin

sin cos

π

λ

β π

β β π

λ λ

λ λ

−

∞

=

−

+

≡

∴

=

=

=

+

≡

→

=

+ +

=

∴

+

=

∴

∑

τ : relaxation time

 

 

 +  −

= τ

β π ^t

l C x

C

₀

sin exp

D l

2 2

τ = π

) 2 /

0 exp(

x l at

t =

−

=

β τ

β

2 2

2 2 2

1 1

Let

λ

−

≡

=

=

=

dx X d X dt

dT DT

dx X DT d

dt X dT

XT C

x B

x A

X dx X

X d

λ λ

λ

sin cos

2 0

2 2

+ ′

= ′

= +

Using a method of variable separation

l C β sin π x X(x,0) ≡ +

₀

e Dt

T T

dt D T d

dt D dT T

2

0

2 2

ln 1

λ

λ λ

= −

−

=

−

=

l λ =π

2 2

x D C t C

∂

= ∂

∂

∂

(A_n = 0 for all others)

43

Solutions to the diffusion equations

Ex1. Homogenization

of sinusoidal varying composition in the elimination of segregation in casting

l C x

C = + β

₀

sin π

^{at t=0}

τ : relaxation time

 

 

 +  −

= τ

β π ^t

l C x

C

₀

sin exp

D l

2 2

τ = π

) 2 /

0

exp(

x l at

t =

−

= β τ

β

The initial concentration profile will not usually be sinusoidal, but in general any con- centration profile can be considered as the sum of an infinite series of sine waves of varying wavelength and amplitude, and each wave decays at a rate determined by its

own “τ”. Thus, the short wavelength terms die away very rapidly and the homogenization will ultimately be determined by τ for the longest wavelength component.

decide homogenization rate

Fig. 2.10 The effect of diffusion on a sinusoidal variation of composition.

Amplitude of the concentration profile (β) decreases exponentially with time,

44

Solutions to the diffusion equations

Ex2. Carburization ^{of Steel}

The aim of carburization is to increase the carbon concentration in the surface layers of a steel product in order to achiever a harder wear-resistant surface.

1. Holding the steel in CH₄ and/or Co at an austenitic temperature.

2. By controlling gases the concentration of C at the surface of the steel can be maintained at a suitable constant value.

3. At the same time carbon continually diffuses from the surface into the steel.

45

Carburizing of steel

The error function solution:

( ) ( ) ^{0 , , t C}

₀

C

C t

C

B

s B

=

∞

=

( )

( )

^{Ζ =}

∫

^{Ζ −}



 



− 

−

=

0 0

2 2

2 dy e

erf

Dt erf x

C C

C C

y s

s

π

Depth of Carburization?

• Since erf(0.5)≈0.5, the depth at which the carbon concentration is midway between C

_s

and C

₀

is given

that is

( / 2 x Dt ) ≅ 0.5 x ≅ Dt

erf(0.5)≈0.5

 

 



= 

−

−

Dt erf x

C C

C C

s s

0

2

0

2 C

s

C

C +

=

→ Depth of Carburization

* Concentration profile : using boundary conditions

Solution of Fick’s 2^nd law

1) Specimen ~ infinitely long.

2) Diffusion coefficient of carbon in austenite increases with increasing concentration, but

an approximate solution: Taking an average value

Fig. 2.11 Concentration profiles at successive times (t₃>t₂>t₁) for diffusion into a semi-infinite bar when the surface concentration Cs is maintained constant.

∂ ∂

∂ = ∂

2 2

B B

B

C C

t D x