Quantitative phase-field modeling of grain growth and coarsening in multi-component alloys N. Moelans (1), L. Vanherpe (2), A. Serbruyns (1) (1), B. B. Rodiers (3) (1) Department of metallurgy and materials engineering and (2) Department of computer science, K.U.Leuven, Belgium, (3) LMS International, Leuven, Belgium
Acknowledgements NM is financially supported by the Research Foundation - Flanders (FWO-Vlaanderen) Simulations on Pb-free soldering are partly supported by OT/07/040 (Quantitative phase field modelling of coarsening in lead-free solder joints) performed within COST MP0602 (Advanced Solder Materials for High Temperature Application) Simulations were performed on the HP-computing infrastructure of the K.U.Leuven More information on http//nele.studentenweb.org 2
Outline Introduction and goals Phase field approach Simulation results Grain growth in fiber textured materials Diffusion controlled growth and coarsening in Pb-free solder joints Conclusions 3
Introduction Grain growth Basics: driven by grain boundary energy Normal grain growth Parabolic growth law With n = 0.5 R = kt n Grain size distribution scales in time Non-ideal systems: anisotropy, segregation, solute drag, second- phase precipitates, Atomic structure, energies, mobilities of grain boundaries? Evolution connected grain structure? Mesoscale simulations 4
Introduction Ostwald ripening: Driven by surface tension Statistical theories (random distribution, spherical, ) Growth: Driven by difference in bulk energy Analytical models (usually 1D) Real materials: : non-random spatial distribution, pipe diffusion, anisotropy, Interfacial,, grain boundary energies, mobilities? Phase stabilities,, diffusion mobilities? Evolution morphology? Interaction different processes? 5
Introduction and goals Experiments, atomistic simulations and thermodynamic models Crystal structure, phase stabilities, interfacial properties (energy, mobility, anisotropy), diffusion properties, Phase-field simulations Microstructure evolution at the mesoscale Quantitative characterization Average grain size, grain size distribution, volume fractions, texture, Basis for statistical and mean field theories 6
Phase field formulation Thin interface models Grain growth in anisotropic systems Extension to multi-component systems
Sharp Interface Sharp Diffuse Thin interface models Diffuse interface Thin interface Discontinuity (Semi) 1-D 1 problems Problem specific Complex morphologies Segregation, solute drag, trapping, lattice mismatch, However, l phys (<1nm) <<< R grain (μm-mm) mm) Mostly qualitative 8 l num independent l phys << l num << R grain Karma and Rappel (1996), Tiaden et al. (1996), Kim and Kim (1999), Karma (2001), Kazaryan et al. (2000)
Grain growth model Based on Fan and Chen (1997) and Kazaryan et al. (2000) Phase field variables η, η,..., η ( rt, ),..., η 1 2 with for orientation i ( η, η,..., η,..., η ) = (0,0,...,1,...,0) 1 2 i i p p Ginzburg-Landau equation ηi ( rt, ) 3 2 2 = L( η) m ηi ηi + 2 ηi γ( η) ηj κ( η) ηi t j i 9
Misorientation dependence γ ( η), κ( η), L( η) Parameters are formulated as p p p p 2 2 2 2 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 Individual parameters Grain i Grain j η = 0 η = 0 j i γ ( η) = γ, κ( η) = κ, L( η) = i, j i, j i, j Misorientation L θi, j( ηi, ηj) 10
Calculation grain boundary properties Grain boundary energy γ = g( γ ) mκ gb, θ i, j i, j i, j g(γ i,j ) calculated numerically 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 2 11
Grain boundary width Measure of largest gradient of the phase field profiles 1 1 l = = dηi dηj max dx dx Ct width high controllability of numerical accuracy max Inclination dependence (Moelans et al., PRL 2008; PRB 2008) 12
Numerical validation Shrinking grain: daα dt = 2πμ σ αβ αβ Triple junction angles: σ = σ, μ = μ αγ βγ αγ βγ daα dt Observations Accuracy controlled by l num /Δx Diffuse interface effects for l num /R>5 Angles outside [100-140 140 ] require larger l num /Δx for same accuracy = μ σ αγ αβ 13
Extension to multi-component alloys Phase field variables: Grains ηα1, ηα2,..., ηαi ( rt, ),..., η, η,... η β1 β2 xa, xb( r, t),..., xc Composition p 1 Phase fractions: φ with φ, φ,..., φ η α β ρ ν αi αi α = ν ν ηαi + ηρi αi ρ α ρi ν = 2,4,... 2 phase polycrystalline structure 14
Free energy Free energy: F = F + F bulk interf Bulk and interfacial contribution independent Interfacial energy: Taken from grain growth model Bulk energy: F = f ( η, η ) dv interf interf i i V F = f ( x, η ) dv bulk bulk k i V Interpolates between free energies of the different phases following a thin interphase approach (Tiaden et al. (1996), Kim and Kim (1999)) ρ ρ ρ ρ Gm( xk ) fbulk ( xk, ηρi ) = φρ f ( xk ) = φρ V ρ 15 ρ m
Thin interface approach Interface consists of 2 phases Phase fractions φ, φ α β α β x x β -- x Phase composition fields x x, x,..., x α β ρ k k k k Chemical potential μ = μ =... = μ α β ρ k k k x α Steinbach, Physica D, 127 (2006) 153-160 160 Real composition x k = φ ρ x ρ ρ k 16
Thin interface approach F bulk does not contribute to interfacial energy l num, F bulk, F Int are independent 17 Kim et al., PRE, 6 (1999) p 7186
Kinetics Solute diffusion: ρ 1 xk ρ φρ Mk f = φρ( M k μk) = V t V x m ρ ρ m k ρ with M ρ k D = 2 G x ρ k ρ m 2 k Interfaces: ηiρ δf( ηiρ, xk) = L t δη iρ Between phases α and β ν 1 ν νη i αi η j L g ( ) ( ) ηα β α β α β = int( η, η) + f f ( c c ) μ 2 t ν ν ηα η + β 18
Simulation results Grain growth in fiber textured materials Diffusion controlled growth and coarsening in Pb free solder joints
Columnar films with fiber texture Grain boundary energy: Fourfold symmetry Extra cusp at θ = 37.5 Read-shockley <0 0 1> Discrete orientations η, η,..., η ( rt, ),..., η Δ θ = 1.5 1 2 i 60 Constant mobility Initially random grain orientation and grain boundary type distributions In collaboration with F. Spaepen, School of Engineering and Applied Sciences, Harvard University 20 2D simulation White: θ = 1.5 Gray: θ = 3 Red: q = 37,5 Black: θ > 3, θ 37.5
Misorientation distribution Evolves towards stead-state state misorientation distribution In agreement with previous findings (D. Kinderlehrer,J. Gruber) Read-Shockley + cusp at θ = 37.5 21
Growth kinetics Grain growth exponent PFM: n=0.81 n < 1 in agreement with previous findings (n = 0.6 1) Steady-state growth? A A = kt 0 n High-angle boundaries behave normal Read-Shockley + cusp at θ = 37.5 22
3D simulations Wire with fiber texture Evolution of set of grains with similar orientation ϑ < 6 Evolution of volume fraction of grains with specific orientation 23
Simulation results Grain growth in fiber textured materials Diffusion controlled growth and coarsening in Pb free solder joints
Coarsening in Sn(-Ag)-Cu solder joints COST MP-0602 (Advanced( Solder Materials for High Temperature Application) WG3: Study of interfacial reactions Modeling IMC formation and growth Precipitate growth Void formation Internal stresses Grain boundary diffusion SEM-image of Sn 3.8Ag 0.7 Cu alloy after annealing for 200h at 150 C (Peng 2007) 25
Cu-Sn phase diagram 26
Parabolic free energies Cu-Sn solder joint: bulk free energy ρ A f = xk xk,0 + C 2 ρ k ( ) energies: 2 ρ 27
First simulations for Sn-2at%Cu Interdiffusion coefficients: D D D = 10 ( Cu) 25 12 2 Sn Cu6Sn5 16 13 12 2 Sn ( Sn) 12 2 Sn,10 m /s = 10, 10,10 m /s = 10 m /s Initial compositions Interfacial energies: 2 0.35J/m Initial volume fraction precipitates: f = 0.04 V Interfacial reactions are diffusion controlled System size: 0.1μmx0.5 mx0.5μm 28
Concentration profiles D = D = D = 10 m /s ( Cu) Cu6Sn5 ( Sn) 12 2 Sn Sn Sn D D D = 10 m /s ( Cu) 25 2 Sn = 10 m /s Cu6Sn5 16 2 Sn = 10 m /s ( Sn) 12 2 Sn 29
Diffusion potential μ Sn Sn 30
Conclusions A thin-interface interface phase-field approach is presented for quantitative simulations of grain growth and diffusion controlled growth and coarsening Interfacial energies/mobilities Bulk Gibbs energies of the phases (CALPHAD) Diffusion coefficients/mobilities Current goal is to apply this approach to specific materials science problems Multi-component 3D simulations Thank you for your attention! 31
Solidification Cu-Ni alloy J. Heulens,, K.U.Leuven 32