modeling of grain growth and coarsening in multi-component alloys

Transcription

1 Quantitative phase-field modeling of grain growth and coarsening in multi-component alloys N. Moelans (1) Department of metallurgy and materials engineering, K.U.Leuven, Belgium (2) Condensed Matter & Materials division, Lawrence Livermore National al Laboratories, California, USA

2 Acknowledgements Postdoctoral fellow of the Research Foundation - Flanders (FWO-Vlaanderen) Simulations were performed on the HP-computing infrastructure of the K.U.Leuven More information on http//nele.studentenweb.org 2

3 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 and further goals 3

4 Introduction Microstructures of multi- component, multi-phase, polycrystalline microstructures Important processes Grain growth Ostwald ripening Phase growth Basics Curvature or bulk energy driven boundary motion Triple junction equilibrium Solute diffusion Low C-steel, ferritic grain structure 4

5 Introduction High complexity Anisotropy, segregation, solute drag, second-phase precipitates, pipe diffusion, mutual distribution of phases, Many material properties: : Crystal structure, Gibbs energy, diffusion coefficient of different phases? Structure, energy, mobility of grain boundaries? Evolution connected grain structure? Mesoscale simulations Importance Material development: heat treatment, alloying Reliability 5

6 Introduction and goals Experiments, atomistic simulations and thermodynamic models Crystal structure, phase diagram, 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

7 Phase field formulation Thin interface models Grain growth in anisotropic systems Extension to multi-component systems

8 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 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) 8

9 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 Interfacial energy i i p F f κ dv 2 interf = ( η1, η2,..., ηi,...) + ( ηi ) ) 2 V i p Ginzburg-Landau equations ηi ( rt, ) = L( η) m ηi ηi + 2 ηi γ( η) ηj κ( η) ηi t j i 9

10 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 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

11 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

12 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

13 Numerical validation Shrinking grain: daα dt = 2πμ σ αβ αβ Triple junction angles: σ = σ, μ = μ αγ βγ αγ βγ daα dt = μ σ αγ αβ Observations Accuracy controlled by l num /Δx Diffuse interface effects for l num /R>5 Angles outside [ ] require larger l num /Δx for same accuracy 13

14 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 2 phase polycrystalline structure 14

15 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 Thin-interface interface approach of Tiaden et al. (1996) and Kim et al. (1999) 15

16 Bulk energy: thin interface formulation Interpolates free energies of the different phases ρ ρ ρ ρ Gm( xk ) fbulk ( xk, ηρi ) = φρ f ( xk ) = φρ V ρ F bulk does not contribute to interfacial energy l num, F bulk, F Int are independent Phase composition fields x x, x,..., x Equal diffusion potential ρ α β ρ k k k k μ = μ =... = μ α β ρ k k k m Real xk = φ x ρ ρ k Kim et al., PRE, 6 (1999) p 7186 ρ 16

17 Thin interface approach Interface consists of 2 phases x x β -- x x k = φ ρ x ρ ρ k x α Steinbach, Physica D, 127 (2006) φ, φ α β α β φ η ν αi αi α = ν ν ηαi + ηρi αi ρ α ρi ν = 2,4,... 17

18 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 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 18

19 Simulation results Grain growth in fiber textured materials Diffusion controlled growth and coarsening in Pb free solder joints

20 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 = i 60 2D simulation White: θ = 1.5 Gray: θ = 3 Red: θ = 37,5 Black: θ > 3, θ 37.5 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

21 Misorientation distribution Read-Shockley + cusp at θ = 37.5 Evolves towards stead-state state misorientation distribution In agreement with previous findings (D. Kinderlehrer,J.. Gruber) 21

22 Growth kinetics n 0 = n 1 A A k t Grain growth exponent Steady-state growth Previous findings: n = Mean field analysis High-angle boundaries behave normal A h = kt For steady-state state MDF h Read-Shockley + cusp at θ = 37.5 k = k h 1+ N ' N 22

23 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

24 Simulation results Grain growth in fiber textured materials Diffusion controlled growth and coarsening in Pb free solder joints

25 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

26 Cu-Sn phase diagram 26

27 Parabolic free energies Cu-Sn solder joint: bulk free energy ρ A f = xk xk,0 + C 2 ρ k ( ) energies: 2 ρ 27

28 First simulations for Sn-2at%Cu Interdiffusion coefficients: D D D = 10 ( Cu) Sn Cu6Sn Sn ( Sn) 12 2 Sn,10 m /s = 10, 10,10 m /s = 10 m /s Initial compositions Interfacial energies: J/m Initial volume fraction precipitates: f = 0.04 V Interfacial reactions are diffusion controlled System size: 0.1μmx0.5 mx0.5μm 28

29 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 Cu6Sn Sn = 10 m /s ( Sn) 12 2 Sn 29

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 (MD, exp) Bulk Gibbs energies of the phases (CALPHAD) Diffusion coefficients/mobilities (ab initio, exp) Current goals Apply the approach to specific materials science problems Develop thin-interface interface approach further for multi-component polycrystalline alloys 3D simulations Thank you for your attention! Questions? 30

31 Solidification Cu-Ni alloy J. Heulens,, K.U.Leuven 31

32 Diffusion potential μ Sn Sn 32