Effect of defects on microstructure evolution in the...

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


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


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


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∇ ≠


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


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 =


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⎛ ⎞∂ ∂

= − − ∇ = − ∇⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠


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


Cu-Sn solder joint: bulk free energy

••ParabolicParabolic free free energiesenergies:: ( )2,02 Sn Sn

Af x x Cρ

ρ ρ= − +


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


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


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 ↑



•• 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



•• 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

−

−

−

−

=

=

=

=



•• 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 ↑


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


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= =

+ +


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⎡ ⎤∂ ∂

= ∇ ⋅ ⎢ ⎥∂ ∂⎣ ⎦∑


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−


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’’


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 ?


Homogeneous free energy

•• BinaryBinary twotwo--phasephase systemsystem

ηη

αα ββ

f0αf0β


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

∇ −∇=∇ −∇


‘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= ∑


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)


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))


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


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


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

−

−

⇒ = ⋅

= ⋅

Effect of defects on microstructure evolution in the...

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