interdiffusion in multicomponent systems

Industrial Engineering and ManagementIndustrial Engineering and Management 1

Summer School onAdvanced ThermostructuralMaterialsUCSB Aug 7-18, 2006

Interdiffusion in Multicomponent Systems

John Ågren

Dept of Materials Science and Engg.Royal Institute of Technology

SE-100 44 Stockholm, SWEDEN



Content

1. Introduction2. Basics3. Multicomponent systems and

coupling effects4. Diffusion models5. Kirkendall effect6. Multi-phase diffusion couples



• Structural changes at elevated temperatures involve changes in composition.

• Such changes require diffusion.• Thus: Diffusional processes play a

major role in- Heat treating and processing- Degradation at elevated temperatures- ...

1. Introduction



TBCs: Multi-functionalitythrough Multi-layer DesignTBCs: Multi-functionalitythrough Multi-layer Design

≤1050°C

~1150°C

Coo

ling

Load Bearing at High TemperatureInternally cooled superalloyoptimized for structural function

Supply Al to sustain TGO formationSurface alloy containing aluminidephases: β (+γ’ or γ)

Oxidation ProtectionAdherent, thermally grown oxide: Alumina

Thermal InsulationLow conductivity porous oxide with suitable mechanical and chemical integrity: 7YSZ



2. Basics

dtdmB

zx

VD

zcDJ B

mB

BBB ∂

∂−≅

∂∂

−=1

:law (first) Fick´s

timeareamoleAdt

mdJ BB //1

==

:Flux

volumemoleVx

Vmc

m

BBB /

:

===

ionConcentrat

A



The value of the flux depends on frame of reference:• Crystal lattice (lattice-fixed frame of

reference)In a binary system A – B we can evaluate JA and JB.

• Net flow of atoms: Jtot= JA + JB≠0

zcDJ

zcDJ

BABBB

BAABA

∂∂

−=

∂∂

−=

:write toprefer we Usually

zcDJ

zcDJ

BBB

AAA

∂∂

−=

∂∂

−=

law (first) Fick´s



zcDJ BA

ABA ∂∂

−=

A is the dependent concentration variable

Flux of A Concentration gradient of B



Boltzmann–Matano geometry for a diffusion couple

x

"M atano Interface"

time= const .

C

C R

C L

′x

A 1A 2

A 3

A 4

A 5

C' (x)

C=0x M

“Matano Interface” at zM

time = constant

Interdiffusion (mixing of A and B)

Matano’s Interface

∫ =−R

L

c

cM dczz 0)(

zM z’ z

c(z’)



It appears as A and B only exchange places.

...tcoefficien sioninterdiffu

tcoefficien diffusion chemical:tcoefficien diffusion one Only

:law first sFick'

:reference of frame this in atoms of flow net No

or

ABABB

AAB

BABBB

BAABA

BAtotBA

DDD

zcDJ

zcDJ

JJJJJ

~

0

=−=⇒

∂∂

−==⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−−=−

=+=−=

reference. of frame fixed-Number



01'''

')(''

''''

''

=+⇒=++=

+−+=+−−−−−−−−−−−

−=−=

BABA

BAtot

totBABABA

totBBB

totAAA

BA

JJxxJJJ

JxxJJJJ

JxJJJxJJ

JJ

where

:reference of frame fixed-number to tionTransformareference. of frame

fixed-lattice in given and Suppose:fixed-number to fixed-lattice from tionTransforma



We thus find:

ABABBB

AABA

ABBB

AABB

ABBB

ABB

AABB

ABB

ABB

ABBA

AABA

ABB

AABA

AAB

AAB

DDxDx

DDDD

DxDx

DDxDD

DxDx

DDxDD

~'')1(

')1(

)''('

')1(

)''('

=+−

⇒−==

−−=

+−=

−−=

+−=

and

:have we that Observe

The D´ coefficients are called individual diffusion coefficients.



Change of dependent concentration variable:Binary system A - B

etc

const

,

//,1//

zx

VD

zx

VD

zx

VDJ

zxzxxxVVxcVxc

B

m

AABB

m

BAAA

m

BAA

A

BABA

mmBBmAA

∂∂

∂∂

∂∂

∂∂∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛−−=−=

−==+≅==

BAA

AAB DD −=

0====

−=BBB

BAB

ABA

AAA

BBA

ABB

DDDD

DD



Diffusion coefficients and dependent concentration binary systems Fick´s law :

Ax dependent variable Bx dependent variable relations

zx

VDJ

zx

VDJ

B

m

AAB

A

B

m

ABB

B

∂∂

∂∂

−=

−=

zx

VDJ

zx

VDJ

A

m

BAA

A

A

m

BBA

B

∂∂

∂∂

−=

−=

BAA

AAB

BBA

ABB

DD

DD

−=

−=

0==== BBB

BAB

ABA

AAA DDDD



Chemical potential and force

zL

zcddD

zcDJ

zc

cdd

z

cf

BBB

B

B

BB

BBB

B

B

BB

BB

∂µ∂

∂µ∂µ

∂∂

µµµ

φφ

−=⎟⎟⎠

⎞⎜⎜⎝

⎛−=−=

∂∂

=∂∂

=

−=

/

)(

:written be may law sFick'

potential chemicalFor

Force :thatproperty the have potentials ive)(conservat All

grad



v

B

BBBB

BB

B

B

BBB

BBBB

BBBB

BBBB

cddMcD

zcD

zc

cddMc

zMcJ

vcJz

MFMv

µ

∂∂

∂∂µ

∂µ∂

∂µ∂

=

−=−=−=

=−== ,

Motion in system with friction:

Mobility

Force

Thermodynamic factor



B

AAA

AAB xd

dxMD µ=

zVxMJ A

m

AAA ∂

µ∂−=

Relation between mobility and diffusivity binary system A-B

B

BBB

ABB xd

dxMD µ=

zVxMJ B

m

BBB ∂

µ∂−=

2

2

B

m

B

B

B

A

xdGd

xdd

xdd

to alproportion are and µµ



Ideal solution or dilute solution:

( ) ( )

BABBB

B

B

ABBBBABBBB

RTMDxRTdxd

LxRTxLxRT

=⇒=

++≈−++=

/

ln)1(ln 2

µµµµ oo

Radioactive isotopes ”tracers” added in very low amountsfullfill this condition.

RTDM BB /*=

Tracer diffusion coefficients give approximately mobilities to be used also when there is a driving force.



Summary - basics

• Fick’s first law• Value of flux depends on frame of

reference – different diffusion coefficients• Lattice-fixed frame – individual• Number-fixed frame – chemical, interdiffusion

• Gradient in chemical potential a force to move species

• Relation between diffusion coefficient, mobility and thermodynamics

• RT M = D*



3. Multicomponent systems and coupling effects

Main difference compared to binary systems: diffusion is coupled due to several reasons

∑ ∂

∂−=

zc

DJ jkjk

Works in dilute solution but notin general.z

cDJ kkk ∂∂

−=

Coupled diffusion.

Lars OnsagerNobel prize in Chemistry 1968



Reason for coupling effects

• Thermodynamic interactions• Frame of reference• Correlation effects



zcD

zcDJ Si

CSiC

CCC ∂∂

∂∂

−−=

Coupling due to thermodynamic interactions

Example:Fe-Si-C ”Darken effect”

Si-rich steel

Si-poor steel



Ternary system Fe-Si-C:

),( SiCC

CCCC

ccfz

cMJ

=

−=

µ∂µ∂

zcD

zcDJ

zc

czc

ccMJ

CCSi

BCCC

Si

Si

SiC

C

CCCC

∂∂

∂∂

∂∂

∂µ∂

∂∂

∂µ∂

−−=

⇒⎭⎬⎫

⎩⎨⎧

+−=

Thermodynamic interactions predict coupling!



zVxMJ A

m

AAA ∂

µ∂−=

zVxMJ B

m

BBB ∂

µ∂−=

zVxMJ C

m

CCC ∂

µ∂−=

B

AAA

AAB x

xMD∂µ∂

=

C

CCC

ACC x

xMD∂µ∂

=

B

BBB

ABB x

xMD∂µ∂

=

C

AAA

AAC x

xMD∂µ∂

=

C

BBB

ABC x

xMD∂µ∂

=

B

CCC

ACB x

xMD∂µ∂

=

Relation between mobility and diffusivity Ternary system A-B-C



Coupling due to Frame of Reference

∑=

=n

kkJ

10

km

kk xV

JJ ϑ−= ' ϑ

r

')(''11

ik

n

iik

n

iikkk JxJxJJ ∑∑

==

−=−= δ

∑=

=n

iix

11

Measure flux relative lattice - lattice fixed frame

Change to number-fixed frame:

Defined by:



Number fixed frame of reference:

'...')1(' CCBBAAB JxJxJxJ −−+−=

I.e. the flux of B will depend on the flux of A and C!

Coupling in one frame of reference but not in another.



Coupling due to correlation effects• If the diffusive jumps are not

independent but jumps correlated.Tracer diffusion jumps are uncorrelatedDiffusion in chemical potential gradients

jumps correlated – vacancy wind.

• Small effect – difficult to estimate.



zVxMJ k

m

kkk ∂

µ∂−=

)/1(/)/( TTFTF

FLJ

Qii

ikik

∇=−∇=

=∑

and where

µ

Onsager’s generalization of coupling effects:

Assume generally:



Classical examples of cross effects

FickVolta(galvanic

cell)

DufourChemical potential gradient

Electromigration

OhmPeltierVoltage

SoretThermal migration

SeebeckFourierTemperaturegradient

DiffusionElectricHeatFlux

Force

∑= ikik FLJ



Vector-matrix notation

( )

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

=

⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

∂∂∂∂∂∂

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

CCCBCA

BCBBBA

ACABAA

CBA

C

B

C

B

A

DDDDDDDDD

JJJ

z

z

z

JJJ

D

JFJ T

A

...

µ

µ

µ



ion.concentrat dependent as chosen is and

and

:written be then may law sFick'

A

ACC

ACB

ABC

ABB

AAC

AAB

C

B

c

DDDDDD

zcz

cz

c

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

∂∂∂∂∂∂

=

−=

000

Dcgrad

cgradDJ

A



Change in frame of reference

lost! been has ninformatio some Butshould). they (as dependent are fluxes new The

det

:written be may

01

11

'

')(''11

=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−−−−−−−

=

=

−=−= ∑∑==

A

A

AJJ

CCC

BBB

AAA

ik

n

iik

n

iikkk

xxxxxxxxx

JxJxJJ δ



Another interesting choice of processes

'''')1('''')1('

CBAtot

CBBBACC

CBBBABB

JJJJJxJxJxJJxJxJxJ

++=−+−−=−−+−=

atoms of flow net and diffusion-inter processes theconsider then may We processes. tindependen of

number the change to want do we Suppose

0

1111

1

≠

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−−−−−−

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

Adet

AJ CCC

BBB

tot

C

B

xxxxxx

JJJ



∑

∑

−−=

==

ii

ii

dnVdPdUTdS

FJ

µ:micsthermodyna of law

second and first combined the to accordingi.e chosen, properly are forces the if balance

entropy an from coming expression general A

production Entropy FJT

Entropy production



Onsager reciprocal relations

TT LLFJ

LFJ

=⇔=⇔

=⇔=

∑∑

ikki

ii

ikik

i

k

LL

FJ

FLJ

FnJ

then

by given is production entropy the If

:)potentials micthermodyna in

(gradients forces driving tindependen

of functionlinear are fluxes The



The reciprocal relations have been controversial but seem accepted.

• The relations are claimed to be very useful because:- Consistency check of experimental data.- More efficient use of experimental data.

• But: Are they true? • If they are true, are they trivial? Do they

lead to any consequences?• Perhaps they are true but physically

meaningless?



frames. all in symmetricsensetrivial a in symmetric vanish tscoefficien

coupling all tindependen are processesthe where reference of frame a is there If

:invariant) production entropy (leavingto according dtransforme are forces

and fluxes that provided all in valid are theyreference of frame one in valid are they If

⇒⇔

=

=

=

T

-1T

ALAL'FAF'

AJJ´)(

Support of reciprocal relations:



Experimental evidence:• Many investigations over the years

confirm reciprocal relations, e.g.- Diffusion in ternary aqueous solutions of

salts (e.g. Miller 1960, Wendt and Shanim 1970)

- Diffusion in ternary alloys (Ziebold and Ogilvie 1967)

- Electro-osmosis (Beddiar et al. 2002)

• Some investigations confirm with a ”but...”- Diffusion in ternary alchohols and

hydrocarbons (Medvedev and Shapiro 2003)



Difficult to test reciprocal relations in diffusion:

D=LΨ L=D Ψ-1

• Precise evaluation of multicomponent diffusion coefficient matrix needed.

• Good thermodynamic descriptions needed, i.e. second derivatives of the Gibbs energy as function of composition.



Summary - Multicomponent systems and coupling effects

• Fick’s first law contains coupling coefficients.

• Coupling stems from- Thermodynamic interactions- Frame of reference- Correlation effects

• Vector matrix notation convenient • Use gradients in thermodynamic potentials• Entropy production• Onsager reciprocal relations



4. Diffusion models• The dominating diffusion mechanism in metals,

intermetallic and ionic crystals is an atom or ion exchanging place with a vacancy.

• The probability for a thermally activated ”jump”of an atom to neighboring vacant site is given by

where ∆G is the change in Gibbs energy caused by jump and k is Boltzmann’s constant.

)/exp( kTGp ∆−=



If there is a driving force ∆G will be different inthe two directions.

δµµ

δµµ

zGG

zGG

VaBB

VaBB

∂−∂

−∆=∆

∂−∂

+∆=∆

)(21

)(21

*

*

s

r

δµµz

VaB

∂−∂

−)( *

BG∆

δAbsolute reaction rate theory



Assume random mixture of vacancies in thermalequilibrium, i.e.

The probability that a site is vacant

0==∂∂

VaVanG µ

Vay



⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∆−= δµz

GkT

yp BBVa 2

11exp *r

Probability for jump in forward direction:

Probability for jump in backward direction:

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∆−= δµz

GkT

yp BBVa 2

11exp *s



Net flux of atoms (in lattice-fixed frame):

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛ ∆−−=

=⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂−

⎟⎟⎠

⎞⎜⎜⎝

⎛ ∆−

=−=

δµδ

δµδµ

δ

δ

zRTRTGvyc

zRTzRT

RTGvyc

ppvcJ

BBVaB

BB

BVaB

BB

21sinh2*exp

21exp

21exp

*exp

)(

*

*

sr



Net flux of atoms in the limit of lowDriving forces:

zRTRTGycJ BB

VaBB ∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛ ∆−−≅

µνδ 1exp*

2

BMThe mobility!



For interstitial B the fraction of vacancies isusually large and known from composition

zRTRTGycJ BB

VaBB ∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛ ∆−−≅

µνδ 1exp*

2

BVaM

The mobility!



behaviour! Arrhenius

have or then dependence etemperatur

linear a has and constant If

BVaB

B

MRTMRTG*2 ∆νδ



Magnetic ordering

Para-ferromagnetictransition in pure Feat TC. Non-Arrheniusbehaviour.

TC



Chemical ordering

800

1000

1200

1400

1600

1800

2000

TEM

PER

ATU

RE_

KEL

VIN

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

MOLE_FRACTION NI

B2γ’(L12)

Al-Ni



Cu-Zn(Cuper at al. 1956)



A single jump creat a defectin a highly ordered structure.



Girifalco (1964)Do not look at the microscopic behaviour.Look at the statistical behaviour!

BAksQQ

A,BA,B

kdiskk ,),1( 2 =+= α

))((

s y yB B= −' ''



B2

Fe-Al

A2



Interdiffusion

[ ]

2

2

'

B

mAB

B

BB

B

BBABBA

ABB

xdGdxx

xx

xxMxMxD

=

+=

∂µ∂

∂µ∂

(Schematic)

Two effects;- Increased activation energy – lower

mobility- Strong variation in thermodynamic

factor



[ ]y y y x x sA B B A A B' '' ' '' + y − =2 1

22

[ ]Q Q Q y y y x xk kdiss

korder

A B B A A B= + −∆ ' '' ' '' + y 2

∆Q Qkorder

kdis

k= 2 α

[ ][ ] nn

ijrandomnn

ijjiijjiij

i jijiji

orderkij

disskk

ppxxyyyys

xxyyQQQ

−=−+=

−∆+= ∑∑≠

2''''''

'''

BAksQQ kdiskk ,),1( 2 =+= α

:systems nentMulticompo



Al-Fe-NiHelander and Ågren1999



Al-Fe-Ni B2



Ionic systems

More complicated because:Complex phasesComplex defects

- defects that do not change composition- defects that change composition

Electroneutrality



( )

0

0

)()()1(211

)(1

0)1(2),)(,,,(

''

'

''

2

=++

=

=−−−−−−=

=++−=

=−−++

−+++

CBA

O

CBVa

CBAVa

VaCBA

cba

cJbJaJ

J

yacyabya

yyyy

ycybyayVaOVaCBA

:flux charge No

:reference of frame fixed Anion

:tralityElectroneu

:consider example,For



0

'

,,'

)(1:2

2

2

2

22

2

2

2

2

2

2

2

2

2

2

2

'''

''

'

'''

=∂

∂+

∂

∂−=

∂

∂−=

=∂

∂−=

=++−=

===

−

−

−

++

+

+

+

−

−

−

−

+

+

+

+

O

OVaOVa

m

kkVakVa

m

kk

OVaOVa

m

OO

kVakVa

m

kk

VaCBAVa

Jz

MyVx

zMy

Vx

J

zMy

Vx

J

CBAkz

MyVx

J

yyyyycba

µµ

µ

µ

:frame fixed-anion to tionTransforma

:frame fixed lattice in model Vacancypresent! are vacancies Schottky ofnumber small the Only

case theConsider



Requirements on driving forces:

• Should represent uncharged quantities• be independent

Two relations: The 4 driving forces in present case may be reduced to 2 independent driving forces.

Many possibilities!



y!diffusivittracer and mobility between relation simple No

and type the of forces two obtain to Duhem Gibbs Use

zz

zMx

zM

Vxy

J

zMMx

z

JJJ

Jz

MyVx

zMy

Vx

J

ACAB

CBA

kVakk

kVak

m

kVak

CBA

k

VaO

VakkO

CBA

O

OVaOVa

m

kkVakVa

m

kk

∂−∂∂−∂

⎥⎦

⎤⎢⎣

⎡∂

∂−

∂

∂−=

∂

∂=

∂

∂=++

=∂

∂+

∂

∂−=

∑

∑

+

++

+

+

+

+

+

−

++−

−

−

−

++

+

+

+

/)(/)(

2

2

:0

0

,,

,,

'''

2

22

2

2

2

2

2

2

222

2

2

2

22

2

2

2

µµµµ

µµ

µµ

µµ



zMy

Vx

J

zMyy

VxJ

yyy

VaOVaCBA

OVaOVa

m

OO

AAVaCVa

m

AA

CVaVa

∂

∂−=

∂∂

+−=

+=

−

−

−

−

−+++

2

2

2

2''

''

'''

2322

'

)2/('

2/

),)(,,,(

µ

µetc...

!sublattice cation on contentvacancy the increase ydramticall will C of addition The

:tralityElectroneu

:Consider



Summary - diffusion models

• Vacancy mechanism in metals, intermetallic and ionic crystals.

• Absolute reaction rate theory gives flux proportional to force and Arrhenius behaviour.

• Magnetic and chemical ordering yield a deviation from Arrhenius behaviour and increased activation energy with increased ordering.

• Ionic phases complex because complex defect structures. One needs to conisder- Electroneutrality- Charge transfer- Complex relation between mobility and tracer

diffusivity- Doping of ions with different valrncy has a strong effect



70-30 Brass

2.5mm Mo wires0.13mm diameter

Cu

h(t)

Schematic of the Smigelskas–Kirkendall diffusion couple

tK−=− h(0)h(t)

It seems as the Mo-wire has moved!

Cu

5. Kirkendall effect



In lattice-fixed frame of reference a net flow of atoms. In a frame of reference with no net-flow (number-fixed) the lattice points (the markers) thus appear to move.

( ) ( )

zVxMMJJ

zx

zx

zVxM

zVxM

zx

VDD

zx

VDDJJ

B

m

BABBA

BB

AA

B

m

BB

A

m

AA

B

m

ABB

AAB

B

mABBA

∂µ∂

∂µ∂

∂µ∂

∂µ∂

∂µ∂

)(0

1''1''

''

''

−−=+⇒=+

−−=

∂∂

+−=∂∂

−−=+

:Duhem-Gibbs

:atoms of flow-Net



Kirkendall velocity (velocity of lattice planes in number-fixed frame

( )

)(1

/1,

''2

''

BAmm

mBAm

JJVV

VJJVv

+=

=+−=

div&

:change )( density of Rate ρ

)()1(

)(1

''2

''332211

BAmp

p

p

BAmmm

JJVf

f

f

JJVVV

+−=−

+==++

⇒

div

div

&

&&&&

:) fraction (volume porosity Only

:rate Strain porosity No

εεε



Ti-Zr: Daruka et al. 1996



Z’ position of a given lattice plane (or markers fixed to that plane) relativenumber-fixed frame.

zxDDtzJVtZvdtdZ B

ABVam ∂∂

−=== )(),(),'(/' '''



Lattice plane velocity Lattice plane position

MatanoMatano

Kirkendall



The Kirkendall plane is the lattice plane correspondingto the original joint.

It is the only lattice plane that moves parabolicallyrelative the Matano plane.

Lattice plane velocity

Matano

Line with slope 1/2

Kirkendall plane



)1/( VaVap yyf += :pores to condensevacancies all if pores fraction Volume

dtzJyt

VaVa ∫ ∂−∂=0

/)(

atoms of onAccumulati

vacancies of onAccumulati

:0/':0/'

>∂∂<∂∂

zJzJ

Va

Va



Interdiffusion coefficient in NiAl-B2(Helander and Ågren 1999)

Example:Kirkendall effect during oxidation of NiAl B2



Al diffusion fasterNi diffusion faster

Ratio between tracer diffusivity of Ni and Al in Al-Ni B2

( )

∫=

∂∂

−−==+−=

t

VaVa

mtot

Va

BAB

mVaBA

m

dtJVAV

zxDD

VJJJ

Vv

0

1)(



Oxidation of NiAl at 1050°C

• Simplified treatment: Mole fraction Al on metal surface kept constant (~0.40, from EDS at CTH)

• No diffusion in oxide!• Assessment of mobilities in

NiAl made by Helander and Ågren.

(Hallström et al. 2006)




• Vacancy flux in lattice fixed frame of reference

• Negative sign means ”toward the oxide”.

• Accumulation of vacancies towards oxide!




• Integrated vacancy volume flux => pore volume per square meter of metal surface



Svensson, Petrova and Stiller 2006:



Kirkendall effect as a cross effect of interdiffusion

totBAm

BBABBA

JJJVvJxJxJJ

BA

−=−−=−+−==−

−

''/')1('

:shift marker) l(Kirkendal plane lattice and diffusion-inter processes theconsider

to want we Suppose . system binaryConsider



[ ]xxxxFxF

xx

xx

BBAB

AB

BB

BB

∂∂+∂∂−−=∂−∂=

⎟⎟⎠

⎞⎜⎜⎝

⎛−−−

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛−−−−

=

−

//)1(/)(

)1(11

)(

11)1(

2

1

1

µµµµ

forces new the yields thus rule tiontransforma The

TA

A



( ) ( )( )

( )

( )x

MMxxVy

Vv

xMxMxxx

VyJ

FMxMxMMxxMMxxMxMxxx

Vy

ABBABB

m

Va

m

ABBBABBB

m

VaB

BBABBABB

BABBBBABBB

m

Va

∂−∂

−−−=

∂−∂

−+−−=

=

⎟⎟⎠

⎞⎜⎜⎝

⎛+−−−

−−−+−=

)()1(

)()1()1(

0)1()1(

)1()1()1(

2

µµ

µµ:Duhem Gibbs to due

L

Kirkendall shift as a cross effect (Hillert 2005)



Kirkendall effect during interdiffusion between two polysterene films with different chain lengths (Kramer et al. 1984).

Thermal migration: Kirkendall effectcaused by heat flow?

Electromigration: Kirkendall effectcaused by electric current?

170 °C

Very general effect!



Summary – Kirkendall effect

• Unequal mobilities of different elements yield a net flow of atoms.

• Net flow of atoms will give porosity, stresses or deformation.

• In a diffusion couple the Kirkendall plane of the initial joint is the only lattice plane that moves parabolically.

• Kirkendall effect may give porosity at TGO/BC interface during growth of TGO.

• Kirkendall effect is a cross effect of interdiffusion.



6. Multi-phase diffusion couples

• Gibbs phase rule and diffusion• Virtual and real diffusion path• Diffusion in dispersed systems – effective

diffusivity• Different types of planar interfaces• Internal – external oxidation



Gibbs phase rule and diffusion couples

• The number of degrees freedom f is the number of pontials that can be varied independently:

f = 2+c-p

At given P and T: f = c-p

Binary system c = 2 and for one-phase system (p=1):f = 1; 1 independent chemical potentials gradient. two-phase system (p=2): f = 0, No independent chemical potential gradient. No diffusion possible.

Diffusion only possible if supersaturation has been created at a different temperature.



βα δ γ

In a binary diffusion couple which is heatedisothermally we can only form layers ofdifferent phases.



Example: Oxidation or nitriding of puremetal: Only external layer possible!

Nitriding of pure Fe

εγ

γ’

α

Gas Metal



Ternary system c = 3 and for one-phase system (p=1):f = 2; 2 independent chemical potentials gradient. Two-phase system (p=2): f = 1, 1 independent chemical potential gradient. Diffusion possible in a two-phase mixture.

γγ γ+βγ+β



Internal oxidationof Fe-Mn-Al-C(Perez et al.)



Virtual and real diffusion path

Both alloys in γ phase butdiffusion path in two-phase field.

Virtual path because noprecipitation of β phase.



Real diffusion pathin two-phase field.

Precipitation of β taken into account.



Carburizing of Ni-30%Cr alloyEngström et al. 1994



Diffusion in dispersed systems –effective diffusivity

Assumptions:

Diffusion takes place in the matrix phase only.

Equilibrium holds locally in each volume element.

Carburisation of high-temperaturealloysInternal oxidation

Interdiffusion in composite materialscoating/substrate systemsweldments between steelsjoints of dissimilar steels

Gradient sintering of cementedcarbide work-tool pices



∑

∑

∑

∑

−

=

−

=

−

=

−

−

=

∂∂

=

∂∂

−=

∂

∂

∂∂

=∂∂

=

==>∂∂

−=

1

1

1

1

1

1

1321

1

1

)...,,(

n

i

nkio

j

ieffnkj

oj

n

j

effnkjk

n

j

oj

oj

ii

on

oooii

n

i

inkik

DccD

xc

DJ

xc

cc

xc

cccccc

xcDJ

x

αα

α

αα

αα

αα

ααα

α

i.e.

i.e.

element volume each in locally mEquilibriu

: in along diffusion Phase One

1994) Morral and (Hopfe

Matrix tion Transforma oi

i

cc

∂

∂ αα



singular! is matrix tiontransforma The

t!independen not

:system phase-2 Ternary

βαβα // , CB xx

0det =⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂

∂∂

=∂∂

∂∂

=∂∂

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

∂∂

∂∂

∂∂

∂∂

ok

j

oB

BoC

CoB

BoC

B

oC

CoB

C

oC

BoB

B

cc

cc

cc

cc

cc

cc

cc

cc

cc

α

αααα

αα

αα

const and const



( )

path?diffusion of shapefor mean this does What

0detdet

0det

1

1=⎟

⎟⎠

⎞⎜⎜⎝

⎛

∂∂

=

⇒=⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

∑−

=

n

i

nkio

j

ieffnkj

oj

i

DccD

cc

αα

α

α



Diffusion path

Transformation matrix = δij



Zig-Zag diffusion path in two-phase field!



γγ γ+βγ+β



Different types of planar interfaces

βα

αββαβ

βααβαβα

>

>+>

+>

++

or2 type

or 1 type

0 type



OxideB1O1

O-2

B+2

Internal and external oxidation

O2B Metal

Pure B



uA

uO

aO at surface

No ox

B B1O1

Oxidation without B diffusion possible

Binary alloy A-B



Oxygen diffusion in both oxide and metalstabilises the planar front (external oxidation).

B supersaturation destablizes planar front (internaloxidation).



uA

uO

aO at surface

No ox

B B1O1




Summary – multiphase diffusion couples• In contrast to binary diffusion couples,

where interdiffusion only yields new phases as layers parallell with the initial joint, multi component systems may give dispersed phases.

• Virtual and real diffusion path.• Real diffusion path i zig-zag in a ternary

two-phase system.• Can be understood from effective model.• Internal versus external oxidation.

interdiffusion in multicomponent systems

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