Post on 03-Nov-2020
Effect of defects on microstructure evolution in Effect of defects on microstructure evolution in the the interdiffusioninterdiffusion zone in Cuzone in Cu--SnSn solder joints: A solder joints: A
phasephase--field study field study
NeleNele MoelansMoelans
Department of metallurgy and materials engineering, Department of metallurgy and materials engineering, K.U.LeuvenK.U.Leuven, , BelgiumBelgium
2Nele MoelansCOST MP0602, mid-term meetingBochum, April 15-17, 2009
Coarsening in Sn(-Ag)-Cu solder joints
•• WG3: WG3: ModelingModeling of of interfacialinterfacial reactionsreactions•• IMC formation and IMC formation and growthgrowth –– precipitateprecipitate growthgrowth –– KirkendalKirkendal voidsvoids ––
stresses stresses –– grain grain boundaryboundary diffusiondiffusion–– CALPHAD description CALPHAD description –– Diffusion coefficients, Diffusion coefficients, growthgrowth coefficient for IMCcoefficient for IMC--layerslayers
SEM-image of Sn – 3.8Ag–0.7 Cu alloy afterannealing for 200h at 150°C (Peng 2007) Cu-Sn
3Nele MoelansCOST MP0602, mid-term meetingBochum, April 15-17, 2009
Outline
•• BasicsBasics of the of the phasephase--fieldfield modelmodel
•• Effect of Effect of graingrain boundaryboundary diffusiondiffusion onon the the growthgrowth of the IMC of the IMC
•• KirkendallKirkendall voidingvoiding
•• SummarySummary + + questionsquestions forfor furtherfurther workwork
4Nele MoelansCOST MP0602, mid-term meetingBochum, April 15-17, 2009
Phase field model
( ),1 ( ),2 ,( , ,..., ,...)
(1,0,...,0,...), (0,1,...,0,...),...(0,0,...,1,...),...Cu Cu iρη η η =
3 3
6 5 6 5
( ),1 ( ),2 ( ),
,1 ,2
,1 ,2
( ),1 ( ),2 ( )
, ,..., ( , ),...,
, ,...
, ,...
, ,... ,...
Cu Cu Cu i
Cu Sn Cu Sn
Cu Sn Cu Sn
Sn Sn Sn i
r tη η η
η η
η η
η η η
•• Multiple order parameter model:Multiple order parameter model:
•• GrainsGrains and and phasesphases
–– withwith
•• CompositionComposition field:field: ( , )Snx r t
5Nele MoelansCOST MP0602, mid-term meetingBochum, April 15-17, 2009
Grain boundaries and interfaces
•• For For eacheach grain grain boundaryboundary andand
•• PropertiesProperties of of individualindividual grain grain boundariesboundaries as as functionfunction ofof–– InterfacialInterfacial energyenergy, interface diffusion , interface diffusion
2 2 0i jη η ≠
Grain i Grain j
1iη =
0jη = 0iη =
1jη =
2 2i jη η
0iη∇ ≠
6Nele MoelansCOST MP0602, mid-term meetingBochum, April 15-17, 2009
Free energy functional
•• Free Free energyenergy functionalfunctional::
•• InterfacialInterfacial energyenergy::
•• Bulk Bulk energyenergy::
( )4 22 2 2
1 1 1,
1( , ) ( )4 2 4 2
p p p pi i
interf i i i j ij
jVi i
ii i
F m dVη ηη η ηκ η
γ η η= = < =
⎡ ⎤⎛ ⎞⎛ ⎞∇ = − + + + ∇⎢ ⎥⎜ ⎟⎜ ⎟
⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦∑ ∑∑ ∑∫
interf bulkF F F= +
3 6 5( , ) [ ( )] with ( ), , , ( )bulk i Sn SnV
F x f x dV Cu Cu Sn Cu Sn Snρ ρρ
ρ
η φ ρ= =∑∫2i
i
ii
ρρ
ρ να
α α
ηφ
η=∑∑∑
withwith ‘‘phase fractionsphase fractions’’
(Chen and Yan 1994, Kazaryan et al. 2000)
(Tiaden et al. 1996, Kim et al. 1999)
( )( ) m kk
m
G xf xV
ρ ρρ ρ
ρφ=and free and free energyenergy
7Nele MoelansCOST MP0602, mid-term meetingBochum, April 15-17, 2009
Diffusion equations
•• DiffusionDiffusion flux (of the flux (of the formform ))
•• Bulk and interface/Bulk and interface/graingrain boundaryboundary diffusiondiffusion
WithWith andand
•• MassMass conservationconservation => => diffusiondiffusion equationequation forfor SnSn
223
/gbs
interf mnumMn
DMf x
δδ
⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟⎜ ⎟∂ ∂ ⎝ ⎠⎝ ⎠
2 2, ,
, ,
Sns i j Sn
i j
x M Mt
ρρ ρ σ
ρ ρ σ
φ η η μ≠
⎡ ⎤⎛ ⎞∂= ∇ ⋅ + ∇⎢ ⎥⎜ ⎟∂ ⎢ ⎥⎝ ⎠⎣ ⎦
∑ ∑
2
2
Sn
DMf
x
ρρ
ρ=∂∂
2 2 2 2, , , ,
, , , ,
( ) 1SnSn interf i j s i j Sn
i j i jSn m
f xJ M M M Mx V
ρ ρρ ρ
ρ ρ σ ρ ρ σρ ρ σ ρ ρ σ
φ η η φ η η μ≠ ≠
⎛ ⎞ ⎛ ⎞⎡ ⎤∂ −= − + ∇ = + ∇⎜ ⎟ ⎜ ⎟⎢ ⎥∂⎣ ⎦⎝ ⎠ ⎝ ⎠
∑ ∑ ∑ ∑
Sn SnJ M μ= − ∇
1gb nmδ =
8Nele MoelansCOST MP0602, mid-term meetingBochum, April 15-17, 2009
Equations for interface movement
•• Interface Interface movementmovement::
•• Grain Grain boundaryboundary betweenbetween grain grain ρρ,i,i and and ρρ,j,j
•• BetweenBetween phase phase αα and and ββ
( , )i i k
i
F xL
tρ ρ
ρ
η δ ηδη
∂= −
∂
( )( )
1
int 2( , ) ( ) ( ) ( )i ji L g f c f c c ct
ν να β α α β β α βα
ν να β
νη ηη η η μη η
−⎛ ⎞∂ ⎜ ⎟= − ∇ + − − −⎜ ⎟∂ +⎝ ⎠
Curvature driven Bulk energy driven
2, int ( , )i interf
i j ii
fL Lg
tρ
ρρ
ηκ η η η
η⎛ ⎞∂ ∂
= − − ∇ = − ∇⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠
9Nele MoelansCOST MP0602, mid-term meetingBochum, April 15-17, 2009
Cu-Sn system
( ) 25
3 16 2
6 5 15 2
( ) 12 2
10
5 10 m /s
10 m /s
10 m /s
CuSnCu SnSnCu SnSn
SnSn
D
D
D
D
−
−
−
−
=
= ⋅
=
=
Annealing temperature:
180 °C
EutecticComposition: Sn-2at%Cu
Cu Sn
20.35 J/mgbγ =
••EquilibriumEquilibrium compositionscompositions ••((Inter)DiffusionInter)Diffusion coeffcientscoeffcients
••InterfacialInterfacial energyenergy
10Nele MoelansCOST MP0602, mid-term meetingBochum, April 15-17, 2009
Cu-Sn solder joint: bulk free energy
••ParabolicParabolic free free energiesenergies:: ( )2,02 Sn Sn
Af x x Cρ
ρ ρ= − +
11Nele MoelansCOST MP0602, mid-term meetingBochum, April 15-17, 2009
IMC-layer growth (1D)
•• EffectEffect of of bulkbulk diffusion coefficientsdiffusion coefficients
( ) 25 2
3 14 2
6 5 14 2
( ) 12 2
10 m /s
10 m /s
10 m /s
10 m /s
CuSnCu SnSnCu SnSn
SnSn
D
D
D
D
−
−
−
−
=
=
=
=
63
66 5
0.0073 10
0.023 10Cu Sn
Cu Sn
k
k
−
−
⇒ = ⋅
= ⋅
( ) ( ),0 ,
3,0
6 5,0
( ) ( ),0 ,
0.01( )
0.25
0.455
0.99( )
Cu CuSn Sn eq
Cu SnSn
Cu SnSn
Sn SnSn Sn eq
x x
x
x
x x
= <
=
=
= <
( )t s
( )h m
Diffusion coeffcients Initial composition
12Nele MoelansCOST MP0602, mid-term meetingBochum, April 15-17, 2009
IMC-layer growth (1D)
( ) 25 2
3 13 2
6 5 13 2
( ) 12 2
10 m /s
10 m /
10 m /s
10 m /s
CuSnCu SnSnCu SnSn
SnSn
D
D s
D
D
−
−
−
−
=
=
=
=
( ) 25 2
3 13 2
6 5 13 2
( ) 14 2
10 m /s
10 m /
10 m /s
10 m /s
CuSnCu SnSnCu SnSn
SnSn
D
D s
D
D
−
−
−
−
=
=
=
=
63
66 5
0.0301 10
0.0833 10Cu Sn
Cu Sn
k
k
−
−
⇒ = ⋅
= ⋅
63
66 5
0.0306 10
0.0849 10Cu Sn
Cu Sn
k
k
−
−
⇒ = ⋅
= ⋅
( ) 12 210 m /sSnSnD −=
( )t s
( )h m
13Nele MoelansCOST MP0602, mid-term meetingBochum, April 15-17, 2009
Effect of grain boundary diffusion
•• 2D simulation 2D simulation withwith grain grain boundaryboundary diffusiondiffusion
SnJ →Snx
grains
( ) 25 2
3 15 2
6 5 15 2
( ) 12 2
9 2
10 m /s
10 m /s
10 m /s
10 m /s
D 0.66 10 m /s
CuSnCu SnSnCu SnSn
SnSnsurfSn
D
D
D
D
−
−
−
−
−
=
=
=
=
= ⋅
SnJ ↑
14Nele MoelansCOST MP0602, mid-term meetingBochum, April 15-17, 2009
Effect of grain boundary diffusion
•• 2D simulation 2D simulation withoutwithout grain grain boundaryboundary diffusiondiffusion
( ) 25 2
3 15 2
6 5 15 2
( ) 12 2
2
10 m /s
10 m /s
10 m /s
10 m /s
D 0m /s
CuSnCu SnSnCu SnSn
SnSnsurfSn
D
D
D
D
−
−
−
−
=
=
=
=
=grains
SnJ →
SnJ ↑
Snx
15Nele MoelansCOST MP0602, mid-term meetingBochum, April 15-17, 2009
Effect of grain boundary diffusion
•• 2D simulations2D simulations
( ) 25 2
3 15 2
6 5 15 2
( ) 12 2
10 m /s
10 m /s
10 m /s
10 m /s
CuSnCu SnSnCu SnSn
SnSn
D
D
D
D
−
−
−
−
=
=
=
=
16Nele MoelansCOST MP0602, mid-term meetingBochum, April 15-17, 2009
Effect of grain boundary diffusion
•• 3D simulations3D simulations ( ) 25 25 2
3 15 13 2
6 5 15 13 2
( ) 12 12 2
2 10 ,*2 10 m /s
2 10 ,*2 10 m /s
2 10 ,*2 10 m /s
2 10 ,*2 10 m /s
CuSnCu SnSnCu SnSn
SnSn
D
D
D
D
− −
− −
− −
− −
= ⋅ ⋅
= ⋅ ⋅
= ⋅ ⋅
= ⋅ ⋅
SnJ ↑
17Nele MoelansCOST MP0602, mid-term meetingBochum, April 15-17, 2009
Growth behavior Cu3Sn ?
••( ) 25 2
3 15 2
6 5 15 2
( ) 12 2
12 2
2 10 m /s
2 10 m /s
2 10 m /s
2 10 m /s
D 2 10 m /s
CuSnCu SnSnCu SnSn
SnSnsurfSn
D
D
D
D
−
−
−
−
−
= ⋅
= ⋅
= ⋅
= ⋅
= ⋅
Grain structureGrain structure Composition: Composition: xxSnSn
18Nele MoelansCOST MP0602, mid-term meetingBochum, April 15-17, 2009
Effect of vacancies
•• Composition:Composition:
•• withwith
•• MolarMolar Volume:Volume:
•• EquilibriumEquilibrium composition:composition:
•• Free Free energyenergy::
•• withwith
,, 1 , 1A B A A B Va A Bx x x u u u u u= − ⇒ = − −
andA BA B
A B A B
u ux xu u u u
= =+ +
2 2,0 ,0( , ) ( ) ( )
2 2A B
A B A A B BA Af u u u u u u Cρ ρ
ρ ρ⇒ = − + − +
mm
A B
VVu u
⇒+
( )202 B
Af x x Cρ
ρ ρ= − +
, ,, , ,( )(1 ), ( )B eq B eqB eq A eq A B B eq A Bx u u u x u u u x⇒ = + − = +
andA BA B A B
A AA Au u u u
ρ ρρ ρ= =
+ +
19Nele MoelansCOST MP0602, mid-term meetingBochum, April 15-17, 2009
Effect of vacancies
•• Diffusion fluxesDiffusion fluxes
•• WithWith M M relatedrelated to to intrinsicintrinsic diffusion coefficientsdiffusion coefficients
•• Mass conservationMass conservation
0A BJ J+ ≠
A AA
fJ Mu
ρρ
ρ ρρ
φ ∂= ∇
∂∑ B BB
fJ Mu
ρρ
ρ ρρ
φ ∂= ∇
∂∑
AA
A
u fMt u
ρρ
ρ ρρ
φ⎡ ⎤∂ ∂
= ∇ ⋅ ⎢ ⎥∂ ∂⎣ ⎦∑
BB
B
u fMt u
ρρ
ρ ρρ
φ⎡ ⎤∂ ∂
= ∇ ⋅ ⎢ ⎥∂ ∂⎣ ⎦∑
20Nele MoelansCOST MP0602, mid-term meetingBochum, April 15-17, 2009
2D simulations Kirkendall voiding
•• Phase Phase αα
•• PhasePhase ββ
•• Phase Phase ‘‘airair’’
, ,
,0 ,0
12 12
0.5*0.999, 0.5*0.999,
0.45*0.998, 0.55*0.998
1 10 , 1 10
A eq B eq
A B
A B
u u
u u
D D
β β
β β
β β− −
= =
= =
= ⋅ = ⋅
, ,
,0 ,0
12 12
0.1*0.999, 0.9*0.999,
0.1*0.998, 0.9*0.998
1 10 , 0.1 10
A eq B eq
A B
A B
u u
u u
D D
α α
α α
α α− −
= =
= =
= ⋅ = ⋅
, ,
,0 ,0
12 12
0.0001, 0.0001,
0.0001, 0.0001
1 10 , 1 10
Va VaA eq B eq
Va VaA B
Va VaA B
u u
u u
D D− −
= =
= =
= ⋅ = ⋅
,0 0.05Vaf =
,A eqxα,1 Va equα−
21Nele MoelansCOST MP0602, mid-term meetingBochum, April 15-17, 2009
2D simulations Kirkendall voiding
, ,
,0 ,0
12 12
0.5*0.98, 0.5*0.98,
0.5*0.998, 0.5*0.998
1 10 , 1 10
A eq B eq
A B
A B
u u
u u
D D
β β
β β
β β− −
= =
= =
= ⋅ = ⋅
, ,
,0 ,0
12 12
0.1*0.999, 0.9*0.999,
0.02*0.998, 0.98*0.998
1 10 , 1 10
A eq B eq
A B
A B
u u
u u
D D
α α
α α
α α− −
= =
= =
= ⋅ = ⋅
, ,
,0 ,0
12 12
0.0001, 0.0001,
0.0001, 0.0001
1 10 , 1 10
Va VaA eq B eq
Va VaA B
Va VaA B
u u
u u
D D− −
= =
= =
= ⋅ = ⋅
,0 0.05Vaf =
•• Phase Phase αα
•• PhasePhase ββ
•• Phase Phase ‘‘airair’’
22Nele MoelansCOST MP0602, mid-term meetingBochum, April 15-17, 2009
Conclusions
•• SummarySummary•• GrowthGrowth of of IMCIMC--layerlayer mainlymainly determineddetermined byby diffusiondiffusion coefficientscoefficients of of IMCIMC’’ss
•• ParabolicParabolic growthgrowth regimesregimes
•• CuCu33Sn Sn onlyonly starts starts growinggrowing in a later stage (in a later stage (forfor appliedapplied simulationsimulationconditionsconditions))
•• VoidVoid formationformation dependsdepends onon intrinsicintrinsic diffusiondiffusion coefficientscoefficients, , initialinitialcompositioncomposition, , growthgrowth directiondirection, , ……
•• QuestionsQuestions•• To To whichwhich extendextend is the effect of is the effect of diffusiondiffusion coefficientscoefficients onon growthgrowth behaviorbehavior
of of IMCIMC’’ss understoodunderstood ??
•• CouplingCoupling of model of model withwith vacancyvacancy diffusiondiffusion withwith CALPHAD ?CALPHAD ?
23Nele MoelansCOST MP0602, mid-term meetingBochum, April 15-17, 2009
Homogeneous free energy
•• BinaryBinary twotwo--phasephase systemsystem
ηη
αα ββ
f0αf0β
24Nele MoelansCOST MP0602, mid-term meetingBochum, April 15-17, 2009
Misorientation dependence
•• Parameters are Parameters are formulatedformulated asas
•• For For eacheach graingrain boundaryboundary
•• IndividualIndividual parameters parameters
•• InclinationInclination dependencedependence
2 2 0i jη η ≠
, , ,( ) , ( ) , ( )i j i j i jL Lγ η γ κ η κ η= = =
2 2 2
1 1,
2( )p p p p
i j i ji
i jj i i j i
κ η η ηκη η= < = <
= ∑∑ ∑∑
( ), ( ), ( )Lγ η κ η η
Grain i Grain j
1iη =
0jη = 0iη =
1jη =
( ) ( ) ( ), , , , , , ,, , ,| |
i ji j i j i j i j i j i j i j
i j
Lη η
γ ψ κ ψ ψ ψη η
∇ −∇=∇ −∇
25Nele MoelansCOST MP0602, mid-term meetingBochum, April 15-17, 2009
‘Thin’ interface models
•• E.g. MultiE.g. Multi--component component systemssystems
xx
xxαα
xxββ
••Interface Interface isis mixture of 2 phases mixture of 2 phases withwith compositioncomposition
••Local Local propertiesproperties are are averagedaveragedover the over the coexistingcoexisting phasesphases
k kx xρρ
ρ
φ= ∑
...k k kα β ρμ μ μ= = =
, ,...,k k k kx x x xα β ρ→
( )kf f xρ ρρ
ρ
φ= ∑
26Nele MoelansCOST MP0602, mid-term meetingBochum, April 15-17, 2009
Calibration grain boundary properties
•• Grain Grain boundaryboundary energyenergy
•• Grain Grain boundaryboundary mobilitymobility
•• Grain Grain boundaryboundary widthwidth
•• GivenGiven materialmaterial propertiesproperties ( , ) and ( , ) and numericalnumerical widthwidth ( ) ( ) •• →→ , , , , and and
,, , ,( )i jgb i j i jg mθγ γ κ=
,
,, , 2
,( ( ))i j
i jgb i j
i j
Lm gθ
κμ
γ=
g(g(γγi,ji,j) ) calculatedcalculated numericallynumerically
,2
,
43 ( ( ))
i j
i j
lm g
κγ
=
,, i jgb θγ,, i jgb θμ l
,i jκ ,i jγ ,i jLmMoelans et al., PRL, 101, 0025502 (2008),
PRB, 78, 024113 (2008)
27Nele MoelansCOST MP0602, mid-term meetingBochum, April 15-17, 2009
Calibration grain boundary properties
•• DefinitionDefinition ‘‘grain grain boundaryboundary widthwidth’’
max max
1 1
| | | |num
i jl d d
dx dxη η= =
BasedBased onon maximum maximum gradientgradient→→EqualEqual widthwidth resultsresults in in equalequal numericalnumerical accuracyaccuracy→→High High controllabilitycontrollability of of numericalnumerical accuracyaccuracy ((llnumnum/R /R < 5< 5))
28Nele MoelansCOST MP0602, mid-term meetingBochum, April 15-17, 2009
Acknowledgements
•• PostdoctoralPostdoctoral fellowfellow of the Research Foundation of the Research Foundation -- FlandersFlanders((FWOFWO--VlaanderenVlaanderen))
•• PartlyPartly supportedsupported byby OT/07/040 (Quantitative phase field OT/07/040 (Quantitative phase field modellingmodelling of coarsening in leadof coarsening in lead--free solder joints) free solder joints)
•• SimulationsSimulations werewere performedperformed onon the the HPHP--computingcomputinginfrastructureinfrastructure of the of the K.U.LeuvenK.U.Leuven
•• More More informationinformation onon http//http//nele.studentenweb.orgnele.studentenweb.org
29Nele MoelansCOST MP0602, mid-term meetingBochum, April 15-17, 2009
Free energy functional
•• Free Free energyenergy functionalfunctional::
•• InterfacialInterfacial energyenergy::
•• Bulk Bulk energyenergy::
( )4 22 2 2
1 1 1,
1( , ) ( )4 2 4 2
p p p pi i
interf i i i j ij
jVi i
ii i
F m dVη ηη η ηκ η
γ η η= = < =
⎡ ⎤⎛ ⎞⎛ ⎞∇ = − + + + ∇⎢ ⎥⎜ ⎟⎜ ⎟
⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦∑ ∑∑ ∑∫
interf bulkF F F= +
( ) ( ) 3 3 6 5 6( ) 3 6 5
5 ( ) ( )( )( ) ( ) ( ) ( )( , ) [ ]Cu Cu Cu Sn Cu Sn Cu Sn Cu Sn
bulk i k Cu Cu SnSn
Cu SnSn
Sn Sn SS SnV
n nF x df x f x f x f x Vη φ φ φ φ= ++ +∫
2i
i
ii
ρρ
ρ να
α α
ηφ
η=∑∑∑
withwith ‘‘phase fractionsphase fractions’’
(Chen and Yan 1994, Kazaryan et al. 2000)
(Tiaden et al. 1996, Kim et al. 1999)
( )( ) m kk
m
G xf xV
ρ ρρ ρ
ρφ=and free and free energyenergy
30Nele MoelansCOST MP0602, mid-term meetingBochum, April 15-17, 2009
IMC–layer growth (1D)
( ) 25 2
3 12 2
6 5 12 2
( ) 12 2
10 m /s
10 m /
10 m /s
10 m /s
CuSnCu SnSnCu SnSn
SnSn
D
D s
D
D
−
−
−
−
=
=
=
=
( ) 14 2
3 12 2
6 5 12 2
( ) 14 2
10 m /s
10 m /
10 m /s
10 m /s
CuSnCu SnSnCu SnSn
SnSn
D
D s
D
D
−
−
−
−
=
=
=
=
63
66 5
0.0965 10
0.2658 10Cu Sn
Cu Sn
k
k
−
−
⇒ = ⋅
= ⋅
63
66 5
0.0977 10
0.2674 10Cu Sn
Cu Sn
k
k
−
−
⇒ = ⋅
= ⋅