interdiffusion in multicomponent systems
TRANSCRIPT
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Interdiffusion in Multicomponent Systems
John Ågren
Dept of Materials Science and Engg.Royal Institute of Technology
SE-100 44 Stockholm, SWEDEN
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Content
1. Introduction2. Basics3. Multicomponent systems and
coupling effects4. Diffusion models5. Kirkendall effect6. Multi-phase diffusion couples
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• 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
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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
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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
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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
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zcDJ BA
ABA ∂∂
−=
A is the dependent concentration variable
Flux of A Concentration gradient of B
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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’)
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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
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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
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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.
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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
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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
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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
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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
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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
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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.
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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*
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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
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Reason for coupling effects
• Thermodynamic interactions• Frame of reference• Correlation effects
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zcD
zcDJ Si
CSiC
CCC ∂∂
∂∂
−−=
Coupling due to thermodynamic interactions
Example:Fe-Si-C ”Darken effect”
Si-rich steel
Si-poor steel
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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!
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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
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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:
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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.
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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.
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zVxMJ k
m
kkk ∂
µ∂−=
)/1(/)/( TTFTF
FLJ
Qii
ikik
∇=−∇=
=∑
and where
µ
Onsager’s generalization of coupling effects:
Assume generally:
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Classical examples of cross effects
FickVolta(galvanic
cell)
DufourChemical potential gradient
Electromigration
OhmPeltierVoltage
SoretThermal migration
SeebeckFourierTemperaturegradient
DiffusionElectricHeatFlux
Force
∑= ikik FLJ
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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
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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
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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
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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
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∑
∑
−−=
==
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
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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
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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?
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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:
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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)
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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.
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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
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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 ∆−=
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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
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Assume random mixture of vacancies in thermalequilibrium, i.e.
The probability that a site is vacant
0==∂∂
VaVanG µ
Vay
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⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∆−= 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
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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
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Net flux of atoms in the limit of lowDriving forces:
zRTRTGycJ BB
VaBB ∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛ ∆−−≅
µνδ 1exp*
2
BMThe mobility!
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For interstitial B the fraction of vacancies isusually large and known from composition
zRTRTGycJ BB
VaBB ∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛ ∆−−≅
µνδ 1exp*
2
BVaM
The mobility!
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behaviour! Arrhenius
have or then dependence etemperatur
linear a has and constant If
BVaB
B
MRTMRTG*2 ∆νδ
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Magnetic ordering
Para-ferromagnetictransition in pure Feat TC. Non-Arrheniusbehaviour.
TC
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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
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Cu-Zn(Cuper at al. 1956)
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A single jump creat a defectin a highly ordered structure.
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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= −' ''
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B2
Fe-Al
A2
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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
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[ ]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
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Al-Fe-NiHelander and Ågren1999
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Al-Fe-Ni B2
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Al-Fe-Ni B2
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Ionic systems
More complicated because:Complex phasesComplex defects
- defects that do not change composition- defects that change composition
Electroneutrality
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( )
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
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0
'
,,'
)(1:2
2
2
2
22
2
2
2
2
2
2
2
2
2
2
2
'''
''
'
'''
=∂
∂+
∂
∂−=
∂
∂−=
=∂
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=++−=
===
−
−
−
++
+
+
+
−
−
−
−
+
+
+
+
O
OVaOVa
m
kkVakVa
m
kk
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m
OO
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m
kk
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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
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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!
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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
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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
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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
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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
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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
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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
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Ti-Zr: Daruka et al. 1996
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Z’ position of a given lattice plane (or markers fixed to that plane) relativenumber-fixed frame.
zxDDtzJVtZvdtdZ B
ABVam ∂∂
−=== )(),(),'(/' '''
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Lattice plane velocity Lattice plane position
MatanoMatano
Kirkendall
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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
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)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
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Interdiffusion coefficient in NiAl-B2(Helander and Ågren 1999)
Example:Kirkendall effect during oxidation of NiAl B2
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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)(
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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)
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Oxidation of NiAl at 1050°C
• Vacancy flux in lattice fixed frame of reference
• Negative sign means ”toward the oxide”.
• Accumulation of vacancies towards oxide!
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Oxidation of NiAl at 1050°C
• Integrated vacancy volume flux => pore volume per square meter of metal surface
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Svensson, Petrova and Stiller 2006:
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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
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[ ]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
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( ) ( )( )
( )
( )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)
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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!
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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.
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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
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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.
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βα δ γ
In a binary diffusion couple which is heatedisothermally we can only form layers ofdifferent phases.
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Example: Oxidation or nitriding of puremetal: Only external layer possible!
Nitriding of pure Fe
εγ
γ’
α
Gas Metal
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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
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Internal oxidationof Fe-Mn-Al-C(Perez et al.)
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Virtual and real diffusion path
Both alloys in γ phase butdiffusion path in two-phase field.
Virtual path because noprecipitation of β phase.
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Real diffusion pathin two-phase field.
Precipitation of β taken into account.
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Carburizing of Ni-30%Cr alloyEngström et al. 1994
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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
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∑
∑
∑
∑
−
=
−
=
−
=
−
−
=
∂∂
=
∂∂
−=
∂
∂
∂∂
=∂∂
=
==>∂∂
−=
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
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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
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( )
path?diffusion of shapefor mean this does What
0detdet
0det
1
1=⎟
⎟⎠
⎞⎜⎜⎝
⎛
∂∂
=
⇒=⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
∑−
=
n
i
nkio
j
ieffnkj
oj
i
DccD
cc
αα
α
α
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Diffusion path
Transformation matrix = δij
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Zig-Zag diffusion path in two-phase field!
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γγ γ+βγ+β
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Different types of planar interfaces
βα
αββαβ
βααβαβα
>
>+>
+>
++
or2 type
or 1 type
0 type
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OxideB1O1
O-2
B+2
Internal and external oxidation
O2B Metal
Pure B
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uA
uO
aO at surface
No ox
B B1O1
Oxidation without B diffusion possible
Binary alloy A-B
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Oxygen diffusion in both oxide and metalstabilises the planar front (external oxidation).
B supersaturation destablizes planar front (internaloxidation).
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uA
uO
aO at surface
No ox
B B1O1
Oxidation without B diffusion possible
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uA
uO
aO at surface
No ox
B B1O1
Oxidation without B diffusion possible
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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.