Byeong-Joo Lee calphad. Byeong-Joo Lee calphad “Numerical Treatment of Moving Interface in Diffusional Reactions,”

Byeong-Joo Lee www.postech.ac.kr/~calphad


• “Numerical Treatment of Moving Interface in Diffusional Reactions,” Byeong-Joo Lee and Kyu Hwan Oh, Z. Metallkunde 87, 195-204 (1996).

• “Numerical Procedure for Simulation of Multicomponent and Multi-Layered Phase

Diffusion,” Byeong-Joo Lee, Metals and Materials 5, 1-15 (1999).

• “Numerical Simulation of Diffusional Reactions between Multiphase Alloys with Different Matrix Phases,”

Byeong-Joo Lee, Scripta Materialia 40, 573-579 (1999).

• “Prediction of the Amount of Retained delta-ferrite and Microsegregation in an Austenitic Stainless Steel,” Byeong-Joo Lee, Z. Metallkunde 90, 522-530 (1999).

• “Evaluation of Off-Diagonal Diffusion Coefficient from Phase Diagram Information,” Byeong-Joo Lee, J. Phase Equilibria 22, 241-246 (2001).

• “Thermo-Calc & DICTRA, computational tools for materials science,” J.-O. Andersson, Thomas Helander, Lars Höglund, Pingfang Shi and Bo Sundman, CALPHAD 26, 273-312 (2002)

ReferencesReferences


Diffusional Reactions Diffusional Reactions – binary & multicomponent systems– binary & multicomponent systems


Multicomponent Diffusion Multicomponent Diffusion

Fe-3.8Si-C Fe-C

Darken’s uphill diffusionDarken’s uphill diffusionDiffusion between multiphase layersDiffusion between multiphase layersA. Engström, Scand. J. Metall. 24, 12 (1995).

B.-J. Lee, J. Phase Equilibria 22, 241 (2001).


ContentContent1. Introduction ․ Definition ․ Diffusion Mechanism: Vacancy Mechanism, Interstitial Mechanism

2. Diffusional Flux and Application of Fick's law ․ Fick's first law in two component system ․ Fick's second law Application - Steady State Solution

3. Non-Steady State Diffusion ․ Thin Film Source (Thin Layer) ․ Semi-Infinite Source (Diffusion Couple) ․ Laplace/Fourier Transformation ․ Error function ․ Homogenization/Solute penetration ․ Trigonometric-Series Solutions ․ Determination of diffusion coefficient (Grube, Boltzman-Matano method) ․ Other Examples ․ Diffusion along high diffusion paths

4. Diffusion Coefficients ․ Reference Frame of Diffusion Darken's Equation ⇒ ․ Intrinsic, Inter, Self, Trace, Impurity Trace Diffusion Coefficient ․ Reference : Smithells Metals Reference Book, Chap. 13., Reed-Hil

5. Modelling of Multicomponent Diffusion ․ Darken's experiments : Fe-Si-C ․ Mathematical Formalism for Multicomponent Diffusion Coefficient


DefinitionDefinition

Homogenization phenomena by non-convective mass transport due to chemical potential or

electrochemical potential difference in a multicomponent single phase


General Phenomenological EquationGeneral Phenomenological Equation


Fick’s 1Fick’s 1stst law law


Fick’s 2Fick’s 2ndnd law law


As a thermally activated process

for interstitial diffusion

More about Diffusion Coefficient More about Diffusion Coefficient – Thermal Activation– Thermal Activation

2

61 BBD

RTG

z m expB

RTH

RS

zD mmB

expexp

61 2

RTQ

DD DBB

exp0

mD HQ How about for substitutional diffusion?


Steady State Solution of Diffusion Steady State Solution of Diffusion


Non-Steady State Solution of Diffusion Non-Steady State Solution of Diffusion


Non-Steady State Solution of Diffusion Non-Steady State Solution of Diffusion - Superposition Principle- Superposition Principle


Non-Steady State Solution of Diffusion Non-Steady State Solution of Diffusion - Superposition Principle- Superposition Principle


Non-Steady State Solution of Diffusion Non-Steady State Solution of Diffusion – Application of Superposition Principle– Application of Superposition Principle


Non-Steady State Solution of Diffusion Non-Steady State Solution of Diffusion – Leak Test & Error Function– Leak Test & Error Function


Non-Steady State Solution of Diffusion Non-Steady State Solution of Diffusion – Semi-Infinite Source– Semi-Infinite Source












Determination of Diffusivity Determination of Diffusivity – Grube method– Grube method


Determination of Diffusivity Determination of Diffusivity – Boltzmann-Matano– Boltzmann-Matano


Non-Steady State Solution of Diffusion Non-Steady State Solution of Diffusion – Separation of Variable– Separation of Variable










Diffusion along High Diffusion Path Diffusion along High Diffusion Path – Grain Boundary Diffusion Model – Grain Boundary Diffusion Model

dx

dCD

tL

mJ L

LL

2

dx

dCLDm LL

2

dx

dCD

Lt

mJ gb

gbgb

2

dx

dCLDm gbgb 2

LD

D

dx

dCLD

dx

dCLD

m

m

L

gb

L

gb

L

gb 22

2


Diffusion Simulation – Finite Difference MethodDiffusion Simulation – Finite Difference Method

x

CC

x

C ji

ji

211

211

2

2

)(

2

x

CCC

x

C ji

ji

ji

t

CC

t

C ji

ji

1



2

2

x

CD

t

C

)2()(

)(112

1 ji

ji

ji

ji

ji CCC

x

tDCC

D

xt

2

)( 2


Diffusion Simulation – Finite Difference MethodDiffusion Simulation – Finite Difference Method implicit integer (i-n) implicit double precision (a-h,o-z) dimension U(1000), UF(1000)c write(*,'(a)',advance='NO') ' Length of Simulation (micro-m) ? ' read(*,*) XL write(*,'(a)',advance='NO') ' Initial Composition (U-fraction) ? ' read(*,*) Uini write(*,'(a)',advance='NO') ' Boundary (Left-end) Composition ? ' read(*,*) U0 write(*,'(a)',advance='NO') ' Diffusion Coefficient (cm^2/sec) ? ' read(*,*) D write(*,'(a)',advance='NO') ' Reaction Time (sec) ? ' read(*,*) Tend write(*,'(a)',advance='NO') ' number of grid ? ' read(*,*) nc D = 1.d+08 * D dx = XL / dble(n-1) dt = 0.25d0 * dx * dx / D dtdx = D * dt / dx / dxc xiter = Tend / dt nprnt = idint(xiter/10.d0)cc initial conditionc U = Uini UF = Uini time = 0.d0 iter = 0

open(unit=1,file='result.exp',status='unknown') write(1,'(a,f12.6)') '$ time = ', time write(1,'(f6.2,f12.6,a)') 0.d0, uf(1), ' M' do i = 2, n write(1,'(f6.2,f12.6)') dble(i-1)*dx, uf(i) enddocc Boundary condition U(1) = U0 UF(1) = U0 U(n+1) = U(n-1)c 1 iter = iter + 1 time = time + dt do i = 2, n uf(i) = u(i) + dtdx * ( u(i+1) - 2.d0*u(i) + u(i-1) ) enddo uf(n+1) = uf(n-1)c u = ufc if(mod(iter,nprnt) .eq. 0) then write(1,'(a,f12.6)') '$ time = ', time write(1,'(f6.2,f12.6,a)') 0.d0, uf(1), ' M' do i = 2, n write(1,'(f6.2,f12.6)') dble(i-1)*dx, uf(i) enddo endifc if(time.lt.tend) goto 1 stop end




Diffusion Coefficient Diffusion Coefficient – Inter Diffusion– Inter Diffusion


Diffusion Coefficient Diffusion Coefficient – Inter Diffusion– Inter Diffusion


Diffusion Coefficient Diffusion Coefficient – Self/Tracer Diffusion– Self/Tracer Diffusion


Diffusion Coefficient Diffusion Coefficient – Intrinsic Diffusion Coefficient– Intrinsic Diffusion Coefficient


Diffusion Coefficient Diffusion Coefficient – Inter Diffusion Coefficient– Inter Diffusion Coefficient


B

BBAAB Nd

dDNDND

ln

ln1)(

~ **

• Inter-diffusion Coefficient in a binary alloy – linked to intrinsic diffusion by the Darken’s relation

• Intrinsic diffusion Coefficient – composed of mobility term (Tracer Diffusion) and thermodynamic factor

B

BBB Nd

dDD

ln

ln1*

• Tracer diffusion Coefficient – as a function of composition & temp.

)0(* BB ND

RTNQB

oBBB

BBeNDTND /)(* )(),(

: tracer impurity diffusion coefficient

: self-diffusion of A in the given

structure

)0(* BA ND

Diffusion Coefficient Diffusion Coefficient – Modeling – Modeling

selfABB DND ** )0(


Diffusion Coefficient Diffusion Coefficient – Modeling – Modeling

assuming composition independent D o

21

2

221

1

1

221

21

21

1

**/

221

21

21

1* nn

n

Nnn

n

N

RTQnn

nQ

nn

n

oBB DDeDN

nn

nN

nn

nD

• Linear composition dependence of QB in a composition range N1 ~ N2

221

21

21

12

21

21

21

1 )()( Qnn

nQ

nn

nN

nn

nN

nn

nQNQ

Tracer diffusion Coefficient at an intermediate composition is a geometrical mean of those at both ends – from experiments

the same for the D o term RTNQNQRTNQNDBB

BBBoBBBB

oB eeeeTND /)()(/)()(ln* ),(

Both Q o & Q are modeled as a linear function of composition


Moving Boundary Problem Moving Boundary Problem – Basic – Basic EquationEquation

RLk

LRk

RLk

LLRk

R JJCvCv ////

)( // RLk

LRk CCv


Binary DiffusionBinary Diffusion


Modeling of Multi-Component Diffusion Modeling of Multi-Component Diffusion - Basic Assumption- Basic Assumption

V x Vm k kk

n

1

V Vk S

Vk 0

for k S (substitutional) for k S

V x V V xm k kk

n

S kk S

1

Cx

V

x

xV u Vk

k

m

k

jj S

S k S

/ /

u x xk k jj S

/


V Jk kk

n

0

1

Jkk S

0

Si

ikkk JuJJ~

Modeling of Multi-Component Diffusion Modeling of Multi-Component Diffusion - Reference Frame- Reference Frame


Mathematical Formalism of Multi-Component Diffusion CoefficientMathematical Formalism of Multi-Component Diffusion Coefficient


Mathematical Formalism of Multi-Component Diffusion CoefficientMathematical Formalism of Multi-Component Diffusion Coefficient


Mathematical Formalism Mathematical Formalism - Application to Binary and Ternary Solutions- Application to Binary and Ternary Solutions




Smithells Metals Reference Book, 1992



John Ågren, Scripta Metallurgica 20, 1507-10 (1986).





Multi-Component Diffusion Simulation Multi-Component Diffusion Simulation – for C in Fe-C-M ternary system– for C in Fe-C-M ternary system

J D C D CC CC C CM M

J u y Mu u

V C u y Mu u

V CC C Va CVaC

C

C

VaS C C Va CVa

C

M

C

FeS M

y yC Va 1 y yFe M 1

J y y Md

dyV C y y M

d

dyV CC C Va CVa

C

CS C C Va CVa

C

MS M

D y y Md

dyVCC C Va CVa

C

C y

S

M

D y y M

d

dyVCM C Va CVa

C

M y

S

C

D Dd

dy

d

dy

dy

dyCM CCC

M y

C

C y

C

MC M C

/ /


Multi-Component Diffusion Simulation Multi-Component Diffusion Simulation – Darken’s uphill diffusion– Darken’s uphill diffusion

Fe-3.8Si-C

and

Fe-C





Fe-3.8Si-C and

Fe-6.45Mn-C




Multi-Component Diffusion Simulation Multi-Component Diffusion Simulation – FDM approach for Fe-Si-C– FDM approach for Fe-Si-C


Moving Boundary Problem Moving Boundary Problem – Basic Equation– Basic Equation

RLk

LRk

RLk

LLRk

R JJCvCv ////

)( // RLk

LRk CCv


Binary DiffusionBinary Diffusion


Application to Interfacial Reactions Application to Interfacial Reactions – Ti/Al– Ti/Al22OO33 Reaction Reaction


Application to Interfacial Reactions Application to Interfacial Reactions – Ti/Al– Ti/Al22OO33 Reaction Reaction


Multi-Component Diffusion Simulation Multi-Component Diffusion Simulation – Case Study : Fe-Cr-Ni– Case Study : Fe-Cr-Ni


Multi-Component Diffusion Simulation Multi-Component Diffusion Simulation – Case Study : Fe-Cr-Ni– Case Study : Fe-Cr-Ni


Multi-Component Diffusion Simulation Multi-Component Diffusion Simulation – between Multi-Phase Layers– between Multi-Phase Layers

2 2

1

( )

( ' ' ' ' )

/ /

/,

/,

J J t m

u u u ukR L

kL R

kcorr

kR L

k m kL R

k m

j

jk

jk upu





Documents

Byeong-Joo Lee calphad. Byeong-Joo Lee calphad “Numerical Treatment of Moving Interface in Diffusional Reactions,”