Effect of defects on microstructure evolution in the interdiffusion zone in Cu-Sn

Transcription

1 Effect of defects on microstructure evolution in the interdiffusion zone in Cu- solder joints: phase-field study epartment of metallurgy and materials engineering, K.U.Leuven, Belgium

2 Coarsening in (-g)-cu solder joints WG3: Modeling of interfacial reactions IMC formation and growth precipitate growth Kirkendal voids stresses grain boundary diffusion CLPH description iffusion coefficients, growth coefficient for IMC-layers SEM-image of 3.8g 0.7 Cu alloy after annealing for 200h at 150 C (Peng 2007) Cu- 2

3 Outline Basics of the phase-field model Effect of grain boundary diffusion on the growth of the IMC Kirkendall voiding Summary + questions for further work 3

4 Phase field model Multiple order parameter model: Grains and phases η, η,..., η ( rt, ),..., η η ( Cu),1 ( Cu),2 ( Cu), i, η,... Cu,1 Cu,2 3 3, η,... Cu,1 Cu, η, η,... η,... ( ),1 ( ),2 ( ) i with ( η, η,..., η,...) = ( Cu),1 ( Cu),2, i (1,0,...,0,...),(0,1,...,0,...),...(0,0,...,1,...),... Composition field: x (,) r t 4

5 Grain boundaries and interfaces For each grain boundary ηη 2 2 i j 0 and η i 0 η = 1 i η = 1 j Grain i Grain j η = 0 η = 0 j i Properties of individual grain boundaries as function of Interfacial energy,, interface diffusion η η 2 2 i j 5

6 Free energy functional Free energy functional: F = F + F interf bulk Interfacial energy: (Chen and Yan 1994, Kazaryan et al. 2000) p 4 2 p p p ηi ηi κ η 2 Finterf ( ηi, ηi ) = m γ i, jηη i j ( ηi ) dv V i= i= 1 j< i 4 2 i= 1 ( ) Bulk energy: F ( η, x ) = [ φ f ( x )] dv with = ( Cu), Cu, Cu,( ) bulk i V with phase fractions φ = i α η αi 2 i η ν αi (Tiaden et al. 1996, Kim et al. 1999) and free energy f Gm( xk ) ( xk ) = φ V m 6

7 iffusion equations iffusion flux (of the form J = M μ ) Bulk and interface/grain boundary diffusion J M M f ( x ) 1 M M = φ + interfη, iησ, j s, i σ, j, i σ, j x = φ + η η μ Vm, i σ, j With M = 2 f x 2 and δ 3 f / x δ s gb = 2 2 m Mn num Mass conservation => diffusion equation for M interf x 2 2 = φm + Msη, iησ, j μ t, i σ, j δ = 1nm gb 7

8 Equations for interface movement Interface movement: ηi δf( ηi, xk) = L t δη i Grain boundary between grain,i and,j η t i f interf 2 = L κ (, ) i, j ηi = Lgint η η η i Between phase α and β ν 1 ν νη i αi η ( ) ( ) ηα β j α α β β α β = L gint( η, η) + f ( c ) f ( c ) ( c c ) μ 2 t ν ν ηα η + β Curvature driven Bulk energy driven 8

9 Cu- system Equilibrium compositions (Inter)iffusion coeffcients = 10 ( Cu) 25 = 510 m/s Cu Cu ( ) 12 2 Eutectic Composition: -2at%Cu nnealing temperature: 180 C Interfacial energy γ gb = J/m Cu 9

10 Parabolic free energies Cu- solder joint: bulk free energy f = x x,0 + C 2 ( ) energies: 2 10

11 IMC-layer growth (1) Effect of bulk diffusion coefficients hm ( ) ts () iffusion coeffcients 10 m /s 10 m /s ( Cu) 25 2 Cu = Cu = ( ) 12 2 Initial composition x = 0.01( < x ) x x x ( Cu) ( Cu),0, eq Cu3,0 Cu65,0 = 0.25 = = 0.99( < x ) ( ) ( ),0, eq 11 k = k Cu3 Cu65 =

12 IMC-layer growth (1) hm ( ) ( ) 12 2 = 10 m /s ts () ( Cu) 25 2 = 10 m / s Cu Cu ( ) 12 2 ( Cu) 25 2 = 10 m / s Cu Cu ( ) 14 2 k = k Cu3 Cu65 Cu3 Cu65 = k = k =

13 Effect of grain boundary diffusion 2 simulation with grain boundary diffusion ( Cu) 25 2 Cu Cu ( ) 12 2 surf 9 2 = m /s grains J x 13 J

14 Effect of grain boundary diffusion 2 simulation without grain boundary diffusion ( Cu) 25 2 Cu Cu ( ) 12 2 surf 2 = 0m /s grains J x 14 J

15 Effect of grain boundary diffusion 2 simulations ( Cu) 25 2 Cu Cu ( )

16 Effect of grain boundary diffusion 3 simulations = 210,*210 m/s ( Cu) = 210,*210 m/s Cu = 210,*210 m/s Cu = 210,*210 m/s ( ) J 16

17 Growth behavior Cu 3? Grain structure Composition: x = 210 m/s ( Cu) 25 2 Cu = Cu = ( ) 12 2 = 210 m/s surf 12 2 = 210 m/s 210 m/s 2 10 m /s 17

18 Effect of vacancies Composition: with Molar Volume: Equilibrium composition: with x x, x = 1 x u, u u = 1 u u B B, Va B u ub = and xb = u + u u + u Vm Vm u + u B B Free energy: ( ) 2 f = xb x0 + C 2 2 B 2 f ( u, ub) = ( u u,0 ) + ( ub ub,0) + C 2 2 B x u = ( u + u )(1 x ), u = ( u + u ) x Beq, eq, B Beq, B = and B = u + u u + u B B B, eq B, eq 18

19 Effect of vacancies iffusion fluxes J + J B 0 J φ M f u = J B φ M f u = B B With M related to intrinsic diffusion coefficients Mass conservation u t u t B = = φ M φ M B f u f u B 19

20 2 simulations Kirkendall voiding Phase α u u α α eq, Beq, α α,0 B,0 Phase β β β u = 0.5* 0.999, u = 0.5* 0.999, u α eq, Beq, β β,0 B,0 β Phase air Va u u = 0.1* 0.999, u = 0.9* 0.999, = 0.1* 0.998, u = 0.9* = 110, = α 12 B = 0.45*0.998, u = 0.55*0.998 = 110, = β 12 B = , u = , Va eq, Beq, = , u = Va Va,0 B,0 Va x α 1 u α, eq Va, eq f Va,0 = 0.05 = 110, = Va 12 B 20

21 2 simulations Kirkendall voiding Phase α u u α α eq, Beq, α α,0 B,0 u u α Phase β = 0.5* 0.98, u = 0.5* 0.98, β β eq, Beq, = 0.5* 0.998, u = 0.5* β β,0 B,0 β = 0.1* 0.999, u = 0.9* 0.999, = 0.02*0.998, u = 0.98*0.998 = 110, = α 12 B = 110, = β 12 B Phase air f Va,0 = 0.05 u u = , u = , Va Va eq, Beq, = , u = Va Va,0 B,0 Va = 110, = Va 12 B 21

22 Conclusions Summary Growth of IMC-layer mainly determined by diffusion coefficients of IMC s Parabolic growth regimes Cu 3 only starts growing in a later stage (for( applied simulation conditions) Void formation depends on intrinsic diffusion coefficients, initial composition, growth direction, Questions To which extend is the effect of diffusion coefficients on growth behavior of IMC s understood? Coupling of model with vacancy diffusion with CLPH? 22

23 Homogeneous free energy Binary two-phase system α β f 0 β f 0 α η 23

24 Misorientation dependence γ ( η), κ( η), L( η) Parameters are formulated as p p p p κi, j i j ηi j i= 1 j< i i= 1 j< i κ( η) η η η = η = 1 i η = 1 j For each grain boundary ηη 2 2 i j 0 Grain i Grain j Individual parameters γ ( η) = γ, κ( η) = κ, L( η) = L i, j i, j i, j η = 0 η = 0 j i Inclination dependence ( ) ( ) L ( ) γ ψ, κ ψ, ψ, ψ i, j i, j i, j i, j i, j i, j i, j ηi ηj = η η i j 24

25 Thin interface models E.g. Multi-component systems x x β Interface is mixture of 2 phases with composition x x, x,..., x α β k k k k μ = μ =... = μ α β k k k x α Local properties are averaged over the coexisting phases x k = φ x k f = φ f ( x k ) 25

26 Calibration grain boundary properties Grain boundary energy γ = g( γ ) mκ gb, θ i, j i, j i, j Grain boundary mobility μ = L κ i, j gb, θi, j i, j 2 mg ( ( γ i, j)) Grain boundary width l 4 κi, j = 3 mg ( ( γ )) i, j Given material properties ( gb θ, gb θ ) and numerical width ( ) κ i, j, γ i, j, and L i, j 2 m γ, i, j μ, i, j g(γ i,j ) calculated numerically Moelans et al., PRL, 101, (2008), PRB, 78, (2008) l 26

27 Calibration grain boundary properties efinition grain boundary width l num 1 1 = = dηi dηj max dx dx max Based on maximum gradient Equal width results in equal numerical accuracy High controllability of numerical accuracy (l num /R < 5) 5 27

28 cknowledgements Postdoctoral fellow of the Research Foundation - Flanders (FWO-Vlaanderen) Partly supported by OT/07/040 (Quantitative phase field modelling of coarsening in lead-free solder joints) Simulations were performed on the HP-computing infrastructure of the K.U.Leuven More information on http//nele.studentenweb.org 28

29 Free energy functional Free energy functional: F = F + F interf bulk Interfacial energy: (Chen and Yan 1994, Kazaryan et al. 2000) p 4 2 p p p ηi ηi κ η 2 Finterf ( ηi, ηi ) = m γ i, jηη i j ( ηi ) dv V i= i= 1 j< i 4 2 i= 1 ( ) Bulk energy: (Tiaden et al. 1996, Kim et al. 1999) F ( η, x ) = [ φ f ( x ) + φ f ( x ) + φ f ( x ) + φ f ( x )] dv ( Cu ) ( Cu ) Cu 3 Cu 3 Cu 6 5 Cu 6 5 ( ) ( ) bulk i k ( Cu) Cu3 Cu65 ( ) V with phase fractions φ = α i η αi 2 i η ν αi and free energy f Gm( xk ) ( xk ) = φ V m 29

30 IMC layer growth (1) ( Cu) 25 2 = 10 m / s Cu Cu ( ) 12 2 ( Cu) 14 2 = 10 m / s Cu Cu ( ) 14 2 k = k Cu3 Cu65 Cu3 Cu65 = k = k =