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





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

Documents

byeongjoo lee

uphill diffusion diffusion

interstitial diffusion

examples diffusion

substitutional diffusion

multicomponent diffusion

diagonal diffusion coefficient

multilayered phase diffusion