New directions in phase-field modeling of microstructure evolution in polycrystalline and multi-component alloys

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1 New directions in phase-field modeling of microstructure evolution in polycrystalline and multi-component alloys Liesbeth Vanherpe,, Jeroen Heulens,, Bert Rodiers K.U. Leuven, Belgium

complex morphologies and grain assemblies Effect of individual bulk and interface properties

2 Quantitative phase-field models Properties bulk and interfaces are reproduced accurately in the simulations Effect model description and parameters Numerical issues Insights in the evolution of complex morphologies and grain assemblies Effect of individual bulk and interface properties Predictive? Depends on availability and accuracy of input data Requires composition and orientation dependence 2

3 General framework and goal 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 3

4 Some aspects of model formulation 2-phase systems (single phase-field field) Multi-grain/phase systems (multiple phase-field field)

5 2-phase systems Field variables: φ( G G r, t) c ( r, t) k Double well function Phase D: I= = 0 Phase E: I = 1 Composition: c B Free energy 2 2 F = fchem( c, φ) + Wg( φ) + φ d r 2 V ε Interfacial energy G Interpolation function Bulk energy β ( φ) 1 ( φ) α fchem = h f ( c, T ) + h f ( c, T ) 5

Decoupling bulk and interfacial energy Interface treated as mixture of 2 phases α β c c, c c-field for each phase Equal interdiffusion potential + conservation β β α α f ( c ) f ( c

6 Decoupling bulk and interfacial energy Interface treated as mixture of 2 phases α β c c, c c-field for each phase Equal interdiffusion potential + conservation β β α α f ( c ) f ( c ) = = c c β c = h( φ) c + 1 h( φ) c β α µ α Bulk energy ( φ) 1 h( φ) fchem = h f β ( c β ) + f α ( c α ) Kim et al., PRE, 6 (1999) p 7186; Tiaden et al., Physica D, 115 (1998) p73 6

7 Decoupling bulk and interfacial kinetics Kinetic equations (Linear non-equilibrium thermodynamics) Allen-Cahn Diffusion ϕ = M t F ϕ Jump in chemical potential accross interface ϕ C 1 ck β α = h( φ) M kl + [1 h( φ )] M kl µ l t l φ φ φ C 1 ck L = [1 h( φ )] M kl µ l + αi t l= 1 t Non-variational anti-trapping current Solute trapping effect µ A i µ i vn Dilute, D S =0: A.Karma, PRL, 87, (2001); B. Echebarria et al., PRE, 70, (2004) Multi-comp, D S =0: S.G. Kim, Acta Mater. 55, p4391 (2007) 7

8 Multi-grain and multi-phase structures Single phase-field models -> > Multiple phase-field models η η1, η2, η3,..., η p { } ( η, η,..., η,..., η ) = (0,0,...,1,...,0) 1 2 Model extension i p F( η, η, η,..., η, η,...) Different types of interfaces Triple and higher order junctions Numerically Same accuracy for all interfaces and phases All interfaces within range of validity of the thin interface asymptotics A = num cte η i = 1 Grain i Grain j η = 0 η = 0 j η j = i 1 8

Multi-grain and multi-phase models: major difficulties Third-phase contributions V 12 = V 13 = 7/10 V 12 K 3 3 Careful choice of multi-well function and gradient contribution 1 2

9 Multi-grain and multi-phase models: major difficulties Third-phase contributions V 12 = V 13 = 7/10 V 12 K 3 3 Careful choice of multi-well function and gradient contribution 1 2 Interpolation function Zero-slope at equilibrium values of the phase fields Thermodynamic consistency p p i chem i η1 η2 i η1 η2 i= 1 i= 1 f = h (,,...) f ( c, T ) h (,,...) = 1 9

10 Phase fields With grain i Free energy F Anisotropic grain growth model η η 1 κ( η ) p 4 2 p p p i i interf = m + γ i, jηi η j + + ( ηi ) i i 1 j i 4 2 V = = < i= G η, η,..., η ( r, t),..., η 1 2 ( η, η,..., η,..., η ) = (0,0,...,1,...,0) i p p p p p κi, j i j ηi j i= 1 j< i i= 1 j< i κ ( η) η η η = i p dv For each grain boundary η η 0 κ ( η) = κ 2 2 i j i, j Inclination dependence ( ) ( ) L ( ) γ ψ, κ ψ, ψ, ψ i, j i, j i, j i, j i, j i, j i, j ηi η j = η η i j 10 L.-Q. Chen and W. Yang, PRB, 50 (1994) p15752 A. Kazaryan et al., PRB, 61 (2000) p14275

11 Non-variational approach equal interface width Ginzburg-Landau type equations G ηi ( r, t) = L( η) m ηi ηi + 2η i γ i, jη j κ ( η) ηi t j i Non-variational with respect to K-dependence of N Similar to Monte Carlo Potts approach Definition grain boundary width l num 1 1 = = dηi dη j max dx dx max High controllability of numerical accuracy (l num /R < 5) 5 11

12 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 m( g( γ i, j )) Grain boundary width l = 4 κi, j 3 m( g( γ )) i, j 2 g(γ i,j ) calculated numerically Iterative algorithm A,[ γ ],[ µ ] m,[ κ ],[ γ ],[ L ] gb gb, θ gb, θ i, j i, j i, j N. Moelans, B. Blanpain, P. Wollants, PRL, 101, (2008); PRB, 78, (2008) 12

Observations Accuracy controlled by num Diffuse interface effects for /

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

Variational approach interface width varies

t) = L( η) m η 3 2 2 2 i ηi + ηi γ i, jη j κ

( ηi ) i= 1 4 2 i= 1 j< i 4 2 i= 1 dv θ =

14 Variational approach interface width varies with interface energy Anisotropy only through γ i,j, κ=cte Energy Kinetic equations η η 1 κ G F = m V G ηi( r, t) = L( η) m η i ηi + ηi γ i, jη j κ ηi t j i p 4 2 p p p i i γ i, jηi η j ( ηi ) i= i= 1 j< i 4 2 i= 1 dv θ = wetting 1 σ = σ = σ σ13 = σ 23 = 5σ 12 η1 14

15 Rotation invariance of the model Mathematically,, the model equations are invariant to rotation, but the order parameters represent orientations in a fixed reference frame. The precision of D depends on the numerical setup, 1 h α cos α L For the model to be rotational invariant in practice, lower limit of amount of order parameters p: J. Heulens and N. Moelans, Scripta Mat. (2010) p > 2πL n h grid spacing 'h physical width of domain L rotational symmetry n 15

.. = ck k µ β k c α k = φ ρ x ρ ρ k G η, η,..., η ( r, t),..., α1 α 2 αi ηβ 1, ηβ 2,.

16 Extension to multi-component multiphase alloys Phase field variables: Grains Composition Bulk energy: x with β β α α f ( c ) f ( ck ) k = =... = ck k µ β k c α k = φ ρ x ρ ρ k G η, η,..., η ( r, t),..., α1 α 2 αi ηβ 1, ηβ 2,... η p G ca, cb( r, t),..., cc ρ ρ fbulk ( ck, ηρi ) = φρ f ( ck ) and φ ρ ρ = 1 i π = α, β,... η 2 ρi i η 2 πi 16

17 Extension to multi-component multiphase alloys Bulk and interface diffusion: x k ρ 2 2 = φρ M k + ηρ, iη σ, j µ k t ρ ρ, i σ, j With M ρ k D = ρ k 2 ρ fm x 2 k and M interf D δ 2 m f / x δ k num interf gb = 3 2 Interface movement: ηiρ δ F( ηiρ, xk ) = L t δη iρ Between phase D and E 2 2η i αiη ( ) ( ) ηα β j α α β β α β = L gint( η, η ) + f ( c ) f ( c ) ( c c ) µ t ηα η + β Curvature driven Bulk energy driven 17

Conclusions for grain growth model remain

18 Numerical validation for multicomponent multi-phase model Triple junction Intermediate phase Growing sphere Coarsening Processes for which v(t) Conclusions for grain growth model remain Accuracy controlled by A num Diffuse interface effects for / x A / R > num 5 Angles outside [ ] require larger resolution A for num / x same accuracy 18

Bounding Box Implementation Basic elements A grain is set of connected grid points r where eta_i(r eta_i(r) > epsilon For each grain, the corresponding bounding box is the smallest cuboid containing

19 Bounding Box Implementation Basic elements A grain is set of connected grid points r where eta_i(r eta_i(r) > epsilon For each grain, the corresponding bounding box is the smallest cuboid containing the grain Algorithm Solve the equations only locally, inside bounding boxes Only values inside boxes are kept in memory Boxes grow or shrink with grain Object Oriented C++ implementation In collaboration with L. Vanherpe and S. Vandewalle,, K.U. Leuven (Vanherpe( et al., PRE, 76, n n (2007)) 19

20 Application examples Grain growth in anisotropic systems with a fiber texture In collaboration with F. Spaepen, Harvard University

Grain growth in columnar films with fiber texture Grain boundary energy: Fourfold symmetry Extra cusp at T = 37.5 Read-shockley <0 0 1> 2D simulation Discrete G orientations η, η,.

21 Grain growth in columnar films with fiber texture Grain boundary energy: Fourfold symmetry Extra cusp at T = 37.5 Read-shockley <0 0 1> 2D simulation Discrete G orientations η, η,..., η ( r, t),..., η θ = i 60 Constant mobility Initially random grain orientation and grain boundary type distributions White: θ = 1.5 Gray: θ = 3 Red: θ = 37,5 Black: θ > 3, θ

22 Simulations: 1 high-angle energy cusp High-angle grain boundaries form independent network Low-angle grain boundaries follow movement of high-angle grain boundaries elongate No stable quadruple junctions White: T Gray: T Black: T!T 37.5 Red: T = 37,5 22

23 Misorientation distribution function (MDF) Area weigthed MDF Reaches a stead-state state Read-Shockley + cusp at θ = 37.5 Low energy boundaries lengthen + their number increases 23

24 Grain growth kinetics A A = k t 0 eff n eff Grain growth exponent PFM: steady-state state growth with n 1 Previous findings: n = Mean field analysis: n eff = kt 1, t A (0) + kt h k eff = da ef f k = dt 1 + N / N Read-Shockley (T m =15 ) ) + cusp at T=37.5 N. Moelans, F. Spaepen,, P. Wollants, Phil. Mag., 90 p (2010) 24

25 Application examples Coarsening of Al 6 Mn precipitates located on a recrystallization front in Al-Mn alloys In collaboration with A. Miroux,, E. Anselmino,, S. van der Zwaag,, T. U. Delft

Jerky motion during recrystallization in Al-Mn alloy In-situ EBSD observation of recrystallization in AA3103 at 400 C CamScan X500 Crystal Probe FEGSEM Jerky grain boundary motion

26 Jerky motion during recrystallization in Al-Mn alloy In-situ EBSD observation of recrystallization in AA3103 at 400 C CamScan X500 Crystal Probe FEGSEM Jerky grain boundary motion Stopping time: s Pinning by second-phase precipitates Al 6 (Fe,Mn), D-Al 12 (Fe,Mn) 3 Si Al 12 Added to phase field model Grain boundary diffusion Driving force for recrystallization 20Pm 26

27 Phase field model Multiple order parameter representation: Mn composition field: Homogeneous driving pressure for recrystallization: m d G G G η ( r, t), η ( r, t),..., η ( r, t),... ( G x, ) Mn r t m,1 m,2 p, i ( η, η,..., η,...) = (1,0,...,0,...),(0,1,...,0,...),...(0,0,...,1,...),... m,1 m,2 p, i Bulk diffusion + Surface diffusion 27

28 Material properties at 723K Grain boundary energy high angle Interfacial energy Al 6 Mn precipitates Mobility high angle grain boundary At solute content 0.3w% Mn Equilibrium composition of matrix Actual composition of matrix (supersaturated) Mn diffusion in fcc Al Pipe diffusion high angle boundaries, precipitate/matrix interface Bulk energy density: f ρ = A ρ (x-x ρ 0 )2 γ h = J/m 2 γ pr = 0.3 J/m 2 M h = m 2 s/kg (Miroux et al.,mater. Sci.. Forum, ,393(2004)) 470,393(2004)) c Mn,eq = w% ( at%) (PhD thesis Lok 2005) c Mn = 0.3 w% ( at%) (PhD thesis Lok 2005) D 0,bulk bulk = 10-2 m 2 /s, Q bulk = 211 kj/mol D bulk = m 2 /s 0,p = D 0,bulk, Q p = 0.65Q bulk D p = m 2 /s D 0,p A m = ; x m 0 = A p = ; x p 0 =

29 Precipitate coarsening and unpinning P D < P ZS (P D P ZS ZS ) Pinning: : P ZS =3.6 MPa Rex: P D =3.1 MPa Unpinning mainly through surface diffusion around precipitates J = J + J 2 2 x y 0.9µm x 0.375µm 6 s 29

30 Precipitate coarsening and unpinning P D <<< P ZS Pinning: : P ZS = 3.6 MPa Rex: P D = 1.1 MPa Unpinning through grain boundary diffusion J = J + J 2 2 x y µm x 0.375µm 8 s 30

31 Application examples Coarsening and diffusion controlled growth in lead- free solder joints Within the framework of COST MP-0602 (Advanced Solder Materials for High Temperature Application), Chairs A. Kroupa and A. Watson

Coarsening in (-Ag)-Cu solder joints IMC formation and growth precipitate growth Kirkendal voids stresses grain boundary diffusion CALPHAD description Diffusion coefficients,

32 Coarsening in (-Ag)-Cu solder joints IMC formation and growth precipitate growth Kirkendal voids stresses grain boundary diffusion CALPHAD description Diffusion coefficients, growth coefficient for IMC-layers Eutectic Composition: -2at%Cu Annealing temperature: 180 C SEM-image of 3.8Ag 0.7 Cu alloy after annealing for 200h at 150 C (Peng 2007) Cu 32

33 Phase field model Multiple order parameter model: Grains and phases G η, η,..., η ( r, t),..., 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 ρ, i (1,0,...,0,...),(0,1,...,0,...),...(0,0,...,1,...),... G Composition field: x ( r, t) 33

34 CALPHAD Gibbs energies COST 531-v3 v3-0 database (+ parabolic extensions) Cu- T=180 C A. Dinsdale, et al. COST 531-Lead Free Solders: Atlas of Phase Diagrams for Lead-Free Soldering, vols. 1,2 (2008) ESC-Cost office 34

35 IMC-layer growth (1D) Effect bulk diffusion coefficient h( m) ( ) 12 2 D = 10 m /s t( s) D D D D = 10 m /s ( Cu) 25 2 Cu = Cu = ( ) 12 2 = 10 m /s 10 m / s 10 m /s D D D D = 10 m /s ( Cu) 25 2 Cu = Cu = ( ) 14 2 = 10 m /s 10 m / s 10 m /s 35 k = k Cu3 Cu65 Cu3 Cu65 = k = k =

36 Comparison with experimental data Cu3, T = 180 C Cu65, T = 180 C T Parabolic growth constant experiments 150 C 180 C 200 C k_cu k_cu Parabolic growth constant in simulations with D D D D = 10 m /s ( Cu) 25 2 = 10 m /s Cu = 10 m /s Cu = 10 m /s ( ) 12 2 k = k Cu3 Cu65 = J. Janckzak,, EMPA

15 13 2 = ( ) 12 12 2 = 2 10,*2 10 m /s 2 10,*2

37 Effect of grain boundary diffusion D D D D = 2 10,*2 10 m /s ( Cu) Cu = Cu = ( ) = 2 10,*2 10 m /s 2 10,*2 10 m /s 2 γ gb = 0.25 J/m 2 10,*2 10 m /s G J 37

10 m /s ( Cu) 25 2 Cu3 15 2 = Cu65 15 2 = (

38 Growth behavior Cu 3? Grain structure Composition: x D D D D = 2 10 m /s ( Cu) 25 2 Cu = Cu = ( ) 12 2 = 2 10 m /s surf 12 2 = 2 10 m /s 2 10 m /s D 2 10 m /s 38

39 Conclusions What do we need next to improve the quantitative accuracy of phase-field models? [100 - Accurate representation of triple-junction angles outside [ ] CALPHAD Gibbs energies over full composition domain, also for stoichiometric phases and metastable regions Composition and orientation dependent expressions for diffusion and interfacial properties 39

40 Thank you for your attention! Acknowledgements Postdoctoral fellow of the Research Foundation - Flanders (FWO- Vlaanderen) Simulations were performed on the Flemisch Super Computer (VSC) Projects OT/07/040 (Quantitative phase field modelling of coarsening in lead- free solder joints) IUAP Program DISCO (Dynamical( Systems, Control, and Optimization IWT grant SB (Phase( Phase-Field Modelling of the Solidification of Oxidic Systems) More information on 40

41 Webpage: /conference/multiscale11/ 41

42 Multi-grain and multi-phase models Multi-phase phase-field model p ϕ, ϕ, ϕ,... ϕ, ϕ = 1 Phase fields 2 Free energy 4σ i, j η i, j fint = φ 2 i φ j + φφ i j i j ηi, j π 0 < φ i, j < 1 Double obstacle, higher order terms,, gradient term non-variational Interpolation: zero-slope or thermodynamic consistency Multi-order parameter models G p Order parameters η1, η2,..., η ( r, t),..., η, η 1 Interfacial energy f int p i= 1 i Steinbach et al. MICRESS phase-field code H. Garcke, B.Nestler, B. Stoth, SIAM J. Appl. Math. 60 (1999) p 295. i p i i= 1 p 4 2 p p p ηi η κ ( η i ) 2 γ i, jηi η j ( ηi ) i= i= 1 j< i 4 2 i= 1 = m G L.-Q. Chen and W. Yang, PRB, 50 (1994) p15752 A. Kazaryan et al., PRB, 61 (2000) p

43 Multi-grain and multi-phase models Vector valued model Orientation field θ Free energy field ( ) and phase field ( ) fint = f ( φ, φ, θ ) φ R. Kobayashi, J.A. Warren, W.C. Carter, Physica D, 119 (1998) p415 2-phase solidification Phase fields ϕ, ϕ, ϕ, ϕ = Fifth order interpolation functions g i (I 1,I 2,I 3 ) Zero-slope and thermodynamic consistent Order g i increases with number of phase-fields Multi-order parameter + 4th order gradient terms Phase field crystal and amplitude equations 3 i= 1 i R. Folch and M. Plapp, PRE, 72 (2005) n I.M. McKenna, M.P. Gururajan, P.W. Voorhees, J. Mater. Sci., 44 (2009) p

Bounding Box Algorithm Initialization by

44 Bounding Box Algorithm Initialization by random nucleation Generate sphere-shaped grain uniformly over microstructure Generate particles uniformly over microstructure Set-up sparse data structure Determine bounding box for every grain Create object for every grain 44

modeling of grain growth and coarsening in multi-component alloys

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 &