Modeling IMC growth in leadfree solder joints using the phase...

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Modeling IMC growth in leadfree solder jointsusing the phase-field method coupled with the

COST-531 thermodynamic database

Nele Moelans, A. Durga, Yuanyuan Guan

Bo Sundman, Alan Dinsdale, Suzana Fries

Final COST MP0602 meeting, June 22 - 24, BRNO

IMC growth and interdiffusion in leadfree solder joints

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[J. Mater Sci: Mater. Electron (2007) 18:155 ]

◆ IMC-layers, IMC precipi-tates, voids, stresses, cracks. . . during solidification andcoarsening

■ GP4: Modeling of microstucture evolution in the interdiffusion zone

■ Use COST-531 thermodynamic database in phase-field simulations

Outline

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■ Phase-field model

■ COST - 531 database

■ 4 approaches to model phases with low solubility

◆ Model I: Stoichiometric

◆ Model II: Parabolic composition dependence

◆ Model II: Order-disorder model

◆ Model IV: Extended sublattice representation

■ Concluding remarks

Phase-Field Model: Variables

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■ Grains and phases

ηbct,1, ηbct,2, . . . , ηbct,i(x, y, z, t), . . .

ηCu6Sn5,1, . . .

ηCu3Sn,1, . . .

ηfcc,1, . . .

ηAg3Sn,1, . . .

with (ηbct,1, ηbct,1, . . . , ηρ,i, . . .) =(1, 0, . . . , 0, . . .), (0, 1, . . . , 0, . . .), . . . , (0, 0, . . . , 1, . . .)

■ Composition: xCu, xSn, (xAg = 1 − xCu − xSn)

Phase-Field Model: Diffuse interface

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■ ηbct,i, ηCu6Sn5,i, ηCu3Sn,i, ηfcc,i, ηAg3Sn,i

Grain i Grain j

1ih =

0jh = 0ih =

1jh =

Grain i Grain j

1ih =

0jh = 0ih =

1jh =

Grain i Grain j

1ih =

0jh = 0ih =

1jh =

Grain i Grain j

1ih =

0jh = 0ih =

1jh =


6 / 25

■ Interface : Mixture of 2 phases

phase a

cac

cb

phase b

c= c + cf fb


6 / 25

■ Interface : Mixture of 2 phases

phase a

cac

cb

phase b

c= c + cf fb

◆ Phase fractions φρ =

∑

iηρ,i

∑

σ

∑

iησ,i

◆ Phase compositions xρk(xk, φρ)

Phase-Field Model: Free energy functional

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Ftotal =

∫

Vfint(ηρ,i,∇ηρ,i)dV +

∫

Vfbulk(ηρ,i, xk)dV


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Ftotal =

∫


∫


■ Bulk contribution

fbulk =∑

ρ

φρfρ(xρk) =

∑

ρ

φρGρm(x

ρk)

Vm


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Ftotal =

∫


∫



fbulk =∑

ρ

φρfρ(xρk) =

∑

ρ

φρGρm(x

ρk)

Vm


∑

iηρ,i

∑

σ

∑

iησ,i


7 / 25

Ftotal =

∫


∫



fbulk =∑

ρ

φρfρ(xρk) =

∑

ρ

φρGρm(x

ρk)

Vm


∑

iηρ,i

∑

σ

∑

iησ,i

◆ Phase compositions xρk(xk, φρ)

Phase field model: Evolution equations

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■ Diffusion

∂xk

∂t= ∇ ·

[

∑

l

[(

∑

ρ

φρMρkl

)

∇

(

∂fρ

∂xρl

)]]

◆ Within each phase: ∂xk∂t

= ∇ ·

[

∑

l

[(

Mρkl

Vm

)

∇ (µl − µAg)

]]

◆ Link with atomic mobilities βρ:

Mρkk = x

ρk(1 − x

ρk)β

ρ, Mρkl,k 6=l = −x

ρkx

ρl β

ρ

◆ Link with interdiffusion coefficients: Mρkl =D

ρkl

∂2fρ

∂xρk

xρl

=VmD

ρkl

∂2Gρ

∂xρk

xρl

Phase field model: Evolution equations

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■ Diffusion

∂xk

∂t= ∇ ·

[

∑

l

[(

∑

ρ

φρMρkl

)

∇

(

∂fρ

∂xρl

)]]

■ Interface movement

∂ηρ,i

∂t= −L

δF (ησ,j , xk)

δηρ

COST-531 tdb: Cu-Sn

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■ Phase diagram ■ Molar Gibbs Energies at 450 K

0 0.2 0.4 0.6 0.8 1-3

-2.5

-2

-1.5

-1

-0.5x 10

4

Molar fraction SnM

ola

r G

ibbs e

nerg

y [J/m

ol]

T = 450K

fcc

bct

Cu6Sn

5-L

Cu3Sn

liquid

Cu6Sn

5-H

Cu Sn

■ Bulk contribution phase-field energy: fbulk =∑

ρ φρG

ρm(x

ρk)

Vm

■ Diffusion equation: µρk − µρAg =

∂Gρ

∂xk, ∂

2Gρ

∂xkxl, k, l = Cu, Sn

COST-531 tdb: Ag-Sn

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■ Phase diagram

200

400

600

800

1000

1200

1400

0 0.2 0.4 0.6 0.8 1.0

MOLE_FRACTION SN

liquid

Ag Sn

fcc

Ag Sn+bct3

Ag Sn+liquid3

AgS

b_O

rtho

■ Molar Gibbs energies at 450 K

0 0.2 0.4 0.6 0.8 1-2.8

-2.6

-2.4

-2.2

-2

-1.8

-1.6

-1.4

-1.2x 10

4

Molar fraction Sn

Mo

lar

Gib

bs e

ne

rgy

T = 450 Kfcc

bct

Ag3Sn

liquid

Ag Sn

■ Sublattice representation Ag3Sn: (Ag)0.75(Ag, Sn)0.25

Model I: Stoichiometric phase

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■ Bulk contribution phase-field energy

fbulk =∑

ρ

φρGρm(x

ρk)

Vm+ φstoich

GstoichmVm

◆ Parallel tangent for solution phases

∂fρ(xρk)

∂xρk

=∂fσ(xσk)

∂xσk,

∀k, ρ 6= σ solution phases

◆ Mass balance

xk − xstoichk =

∑

ρ=sol

φρ(xρk − x

stoichk ), ∀k

■ Dstoichkl = Mstoichkl = 0

Model I: Stoichiometric phase

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■ Simulation precipitation IMC

Phase fractions

Cu Sn6 5 bct

m

Molar fractions

Cu Sn6 5 bct

m

■ Growth IMC layer

◆ Needs diffusion through

IMC

substrate IMC solder

JkÑmk

⇒ Gρ(xk) needed to model ∇µkthe driving force for diffusion

Model II: Parabolic composition dependence

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Vmfstoich =

Astoich

2(xSn − xstoich,Sn)

2 + Gstoichm (T = 450K) (1)

0 0.2 0.4 0.6 0.8 1

-2.5

-2

-1.5

-1

-0.5x 10

4

Molar fraction Sn

Mo

lar

Gib

bs E

ne

rgy [

J/m

ol]

T = 450 K

Fcc

Bct

Cu6Sn

5(A=1e6)

Cu3Sn (A=1e6)

Liquid

Cu6Sn

5(A=1e9)

Cu3Sn (A=1e9)

Cu Sn

◆ Steepness Astoich

◆ Induces small shift in equi-

librium

◆ M stoichSn =Dstoich

Sn

Astoich

(Composition independent)

[S.Y. Hu, J. Murray, H. Weiland, Z.-K. Liu, L.-Q. Chen, Comp. Coupl. Phase Diagr. Thermoch., 31 (2007) p 303 ]

Model II: Results

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( ) 25 2

3 13 2

6 5 13 2

( ) 12 2

10 m/s

10 m/

10 m/s

10 m/s

Cu

Sn

CuSn

Sn

CuSn

Sn

Sn

Sn

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

Cu

Sn

CuSn

Sn

CuSn

Sn

Sn

Sn

D

D s

D

D

-

-

-

-

=

=

=

=

6

6

0.0301 10

0.0833 10

Cu3Sn

Cu6Sn5

k

k

-

-

Þ = ×

= ×

6

6

0.0306 10

0.0849 10

Cu3Sn

Cu6Sn5

k

k

-

-

Þ = ×

= ×

( ) 12 210 m/sSnSn

D -=

Model II: Results

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■ With grain boundary diffusion Grain structure

Vertical Flux of Sn

−15

−10

−5

0

5x 10

−10

■ DfccSn = 2 · 10

−25 m2/s; DCu3SnSn = 2 · 10−15 m2/s; DCu6Sn5Sn = 2 · 10

−15

m2/s; DbctSn = 2 · 10−12 m2/s;

■ Dinterf = 2 · 10−9 m2/s, δgb = 1nm

Model II: Parabolic composition dependence

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Vmfstoich =

Astoich

2(xSn − xstoich,Sn)

2

+Astoich

2(xAg − xshift)

2 + Gstoichm (T = 450K)

x(A

G)

x(SN)

0.0

0.2

0.3

0.5

0.7

0.9

0.0 0.2 0.4 0.6 0.8 1

x(SN)

x(A

G)

Cu

Ag

Sn

Ag3Sn + Cu6Sn5 + (Sn)

Ag3S

n +

Cu3S

n +

Cu6S

n5

FC

C_A

1 +

Cu3S

n +

FC

C_A

1

◆ Steepness Astoich

◆ Small shift in equilibrium

◆ M stoichkl =Dstoich

kl

Astoich

◆ xshift, e.g. = 0.001

Model III: Order-disorder CALPHAD description

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■ Sublattice representation Cu3Sn: (Ag,Cu, Sn)0.75(Ag,Cu, Sn)0.25

■ Molar Gibbs Energy

GCu3Snm = xAgG0

Ag + xCuG0

Cu + xSnG0

Sn

+xAgxCuLAg,Cu + xAgxSnLAg,Sn

+xCuxSnLCu,sn

+(y1Cuy2

Sn − xCuxSn)GordCu3Sn

+RT[

0.75(y1Ag ln(y1

Ag) + y1

Cu ln(y1

Cu)

+y1Sn ln(y1

Sn))

+0.25(y2Ag ln(y2

Ag) + y2

Cu ln(y2

Cu)

+y2Sn ln(y2

Sn))]

■ G0Ag = GHSER + 5000 and LAg,Cu,LAg,Sn,LCu,Sn and GordCu3Sn optimized

[N. Dupin, I Ansara, B. Sundman, Comp. Coupl. Phase Diagr. Thermoch., 25 (2001) p279]

Model III: Order-disorder CALPHAD description

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■ Cu-Sn, T=450K

0 0.2 0.4 0.6 0.8 1-3

-2.5

-2

-1.5

-1

-0.5x 10

4

Molar fraction Sn

Mola

r G

ibbs E

nerg

y [J/m

ol]

T = 450 KFcc

Bct

Cu6Sn

5

Cu3Sn

Liquid

Cu Sn

■ Ag-Sn-0.01%Cu, T= 450 K

0 0.2 0.4 0.6 0.8-3

-2.5

-2

-1.5

-1

-0.5x 10

4

Molar fraction Sn

Mola

r G

ibbs E

nerg

y [J/m

ol]

T = 450 K, xCu

=0.0001Fcc

Bct

Cu6Sn

5

Cu3Sn

Ag3Sn

Liquid

Ag Sn

■ Miscibility gap is inherent

Model IV: Extended sublattice CALPHAD description

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■ Sublattice representation Cu3Sn: (Ag,Cu, Sn)0.75(Ag,Cu, Sn)0.25

■ Molar Gibbs Energy

GCu3Snm = y1Agy

2AgG

0Ag + y

1Cuy

2CuG

0Cu + y

1Sny

2SnG

0Sn

+y1Cuy2SnG

0Cu3Sn + y

1Sny

2CuG

0Sn3Cu

+RT[

0.75(y1Ag ln(y1Ag) + y

1Cu ln(y

1Cu)

+y1Sn ln(y1Sn))

+0.25(y2Ag ln(y2Ag) + y

2Cu ln(y

2Cu)

+y2Sn ln(y2Sn))

]

using G0Sn3Cu = G0Cu + G

0Sn − G

0Cu3Sn

Model IV: Extended sublattice CALPHAD description

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■ Cu-Sn, T=450K

0 0.2 0.4 0.6 0.8 1-3

-2

-1

0

1

2x 10

4

Molar fraction Sn

Mola

r G

ibbs E

nerg

y [J/m

ol]

T = 450 K

Fcc

Bct

Cu6Sn

5

Cu3Sn

Liquid

Cu Sn

■ Ag-Sn-0.01%Cu, T= 450 K

0 0.2 0.4 0.6 0.8-3

-2

-1

0

1

2x 10

4

Molar fraction Sn

Mo

lar

Gib

bs E

ne

rgy [

J/m

ol]

T = 450 K, xCu

=0.0001

Fcc

Bct

Cu6Sn

5

Cu3Sn

Ag3Sn

Liquid

Ag Sn

Comparison: Growth Cu 6Sn5 from supersaturated Bct

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■ Comparison stoichiometric (I) –

parabolic (II)

m

BctCu Sn6 5

■ Comparison stoichiometric (I) –

sublattice (II)

Cu Sn6 5 Bct

m

■ MCu6Sn5 = 0

■ M bctkk = βbctxbctk (1 − x

bctk ), M

bctkl,k 6=l = −β

bctxbctk xbctl

Model IV: Growth Cu 6Sn5 between Fcc and Bct

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BctFcc

m

Cu Sn6 5

■ Diffusion equation

∂xk

∂t= ∇·

[

∑

l

[(

∑

ρ

φρMρ

kl

)

∇

(

∂fρ

∂xρ

l

)

]]

■ Mbct/fcckk = β

bct/fccxk(1 − xk); Mbct/fcckl,k 6=l = −β

bct/fccxkxl

■ MCu6Sn5 = βCu6Sn5(0.545y1k(1 − y1

k) + 0.455y2

k(1 − y2

k)),MCu6Sn5kl,k 6=l = −β

Cu3Sn(0.545y1ky1

l + 0.455y2

ky2

l )

■ βρ estimated based on interdiffusion coefficients of [A. Paul, C. Ghosh and W.J.Boettinger, Metall. Mater. Trans. A, 42A (2011) p952].

Model IV: Growth rate Cu 3Sn and Cu 6Sn5

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■ 1D system

Cu Sn6 5

Cu Sn3bct

xSn

xCu

fcc

Cu3Sn grows faster than Cu6Sn5 !

■ 2D system

bct

fcc

Cu Sn6 5

Conclusions

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■ Form of the Gibbs energies influences microstructure simulations of

interdiffusion phenomena at interfaces (diffusion couples)

◆ Determines which IMC grows fastest/first

■ For a general coupling of phase-field with CALPHAD, an extended

sublattice model is most suitable

◆ Or order-disorder model if based on physics

■ Presented approaches also valuable for sharp interface diffusion

techniques, e.g. DICTRA

■ Databases with extended sublattice representations can be improved

iteratively

IMC growth and interdiffusion in leadfree solder jointsOutlinePhase-Field Model: VariablesPhase-Field Model: Diffuse interfacePhase-Field Model: Diffuse interfacePhase-Field Model: Free energy functionalPhase field model: Evolution equationsPhase field model: Evolution equationsCOST-531 tdb: Cu-SnCOST-531 tdb: Ag-SnModel I: Stoichiometric phaseModel I: Stoichiometric phaseModel II: Parabolic composition dependenceModel II: ResultsModel II: ResultsModel II: Parabolic composition dependenceModel III: Order-disorder CALPHAD descriptionModel III: Order-disorder CALPHAD descriptionModel IV: Extended sublattice CALPHAD descriptionModel IV: Extended sublattice CALPHAD descriptionComparison: Growth Cu6Sn5 from supersaturated BctModel IV: Growth Cu6Sn5 between Fcc and BctModel IV: Growth rate Cu3Sn and Cu6Sn5Conclusions

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