Phase-field simulations of grain growth in materials containing second-phase particles

Transcription

1 Phase-field simulations of grain growth in materials containing second-phase particles Presentation PhD-research Promotors: Patrick wollants Bart Blanpain Nele Moelans 10 April 2006

2 Incentives Interest in the phase-field method Simulation technique for microstructure evolution PhD research of M. Guo Experimental study on Zener pinning in steel containing Ce 2 O 3 and CeS-particles Phase-field simulations of grain growth in materials containing second-phase particles 2

3 Contents Introduction on the phase-field method Introduction on grain growth and zener pinning Phase-field model for zener pinning Simulation results Conclusions + outlook 3

4 Introduction on the phase-field method 4

5 General introduction Moving boundary problems on a mesoscale Microstructure evolution Solidification Solid-state transformations Grain growth and coarsening Crack propagation, dislocations, thin films, rheological systems, Phenomenological Diffuse-interface interface description 5

6 Sharp-interface versus diffuseinterface Sharp-interface Within each domain Differential equations At the interfaces Flux condition Constitutional condition Diffuse-interface interface Field variables and differential equations defined over the whole system Complex grain morphologies No constitutional assumptions 6

7 Field variables Composition Molar fraction fields x (,) rt Mass conservation Structure i Order parameter fields (symmetry) η (,) rt i Phase-fields (phase fraction) φ (,) rt i 7

8 Thermodynamics for heterogeneous systems 8

9 Thermodynamic free energy functional 2 2 (, ) ε ( ) κ F = F ( ) bulk + Fsurf = f0 xb η + xb + η dr V 2 2 Homogeneous free energy density (J/m 3 ) Reflects equilibrium bulk properties of coexisting domains Conserved composition fields common tangent Structural fields degenerate minima 9

10 Thermodynamic free energy functional 2 2 (, ) ε ( ) κ F = F ( ) bulk + Fsurf = f0 xb η + xb + η dr V 2 2 Gradient free energy density Gradient energy coefficients: ε, κ > 0 Diffuse interfaces Surface tension 0 10

11 Kinetic equations Structural field variables Ginzburg-Landau equation η F f 0 2 = L = L κ η t η η Composition field variables Cahn-Hilliard equation x F f V t x x 1 B 0 2 = M = M ε xb m B B Monotonic decrease of the free energy in time + mass conservation 11

12 Example of microstructure evolution Spinodal decomposition Decomposition F bulk Coarsening F int Mass conservation lever rule 12

13 Introduction on grain growth and Zener pinning 13

14 Normal grain growth Reduction of interfacial free energy Curvature driven P g ασ = R gb Mechanical equilibrium at vertices Spacefilling 14

15 Normal grain growth Isotropic,, pure, single-phase materials Parabolic growth law: : n=2 1/ R = k* t n Grain size distribution time-invariant: invariant: f ( R/ R) = cte Von Neumann for 2D- structures Equal-sized hexagons Real materials Anisotropy, impurity drag, Zener pinning, triple junction drag,, 15

16 Zener pinning MnS-particle in low-c steel Dimple-shape Mobility 16

17 Zener analysis Final grain size Pinning force of particle max at β=π/4 F max Z = πσ r Pinning pressure of particle distribution 3 fvσ gb PZ = = zσ gb 2r Grain growth stagnation when P g gb Zener type relation R = P Z lim 1 r = K f b V 17

18 Kinetic aspects Rate of grain growth dr dt ασ gb 3 fvσ gb = µ *( Pg PZ) = µ *( ) R 2r Sharpening of grain size distribution Temporal evolution of particles Dissolution Ostwald ripening Precipitation 18

19 Phase-field model 19

20 Representation of a polycrystalline structure Model of L.-Q. Chen and W. Yang (1994) for normal grain growth Order parameter fields: η, η,..., η,..., η 1 2 i p Particles : Φ=1 ( η, η,..., η,..., η ) = (0,0,...,0,...,0) 1 2 i Grain i of matrix-phase : Φ=0 p ( η, η,..., η,..., η ) = (0,0,...,1,...,0) 1 2 i p 20

21 Representation of a particle Evolution of η i across particle 21

22 Free energy and kinetic equations Free energy functional F p p p p p κ = m ( ηi ηi ) + η V iηj + εφ ηi + ηi i= i= 1 j i i= 1 i= 1 2 ( ) 2 dv Equilibrium Φ=0 : Φ=1 : ( η, η,..., η ) = (1,0,...,0),(0,1,...,0),...(0,0,...,1),( 1,0,...,0), p ( η, η,..., η ) = (0,0,...,0) p Kinetic equations (Ginzburg-Landau) ηi (,) rt F f ( η, η,...) L L t ηi(,) rt ηi(,) rt = = κ ηi (,) rt 22

23 Homogeneous free energy density Matrix phase Second-phase particle 23

24 Properties of diffuse grain boundary Analysis diffuse interfaces (Cahn( and Hilliard,, Allen and Cahn) Grain boundary energy Grain boundary width 0.58 κm κ m Grain boundary mobility * 1 1 v= Lκ + = µσ gb + = µ + ρ1 ρ2 ρ1 ρ2 ρ1 ρ2 Interfacial energy particle-matrix f ( ε ) κm 24

25 Properties of diffuse grain boundaries Interfacial width κ m Numerical accuracy 25

26 Properties of diffuse grain boundaries Interfacial energy κm Excess homogeneous free energy density Gradient energy density 26

27 Local interactions: spherical grain Grain boundary passing a particle 3-D simulation κ = 0.5, m= 1, L= 1, ε = 1 r R0 dc p = 8, = 90, = 76 Dimple-shape Break-free at β > π/4 27

28 Temporal evolution spherical grain 28

29 Simulation results 29

30 Large-scale 2-D simulations Isotropic interfacial properties κ = 0.5, L= 1, m= 1, ε = 1 Round particles r = 2.5, r = 3 Area fraction f a = System size: : 256x256, 512x512 and 1024x1024 Initial microstructure Grain nucleation in presence of particles (R 0 =0) Grain nucleation and initial grain growth without particles (R 0 >0) Random particle distribution 30

31 Initial microstructure Grain nucleation in the presence of particles R 0 = 0 Most particles on grain boundaries r = 3, f a =

32 Initial microstructure Grain nucleation and initial growth without particles 32

33 Initial microstructure Grain nucleation and initial growth without particles Addition of particles when R 0 > 0 Many particles within grains r = f = R = 3, a 0.02,

34 2-D simulations R 0 = 0: R 0 > 0 r f R = 3, a = 0.04, 0 = 0 r fa R0 = 3, = 0.04, =

35 Simulation data for final grain size R lim r = f 0.5 V 35

36 Role initial grain size Fraction of particles on grain boundaries: : temporal evolution 36

37 Comparison with experiment Thin Al-films Columnar grains CuAl 2 -particles Many particles within the grains Other effects Surface grooving Surface energy anisotropy Semi-coherent particle-matrix interface Data from H.P. Longworth and C.V. Thompson 37

38 3-D simulations for thin films Columnar grain structure Interaction particle-grain boundary is 3-D 3 curvature out of the plane Particles in the middle of the film are more effective Film thickness r = 3, f = 0.05, l = 21 a 38

39 Conclusions and outlook 39

40 Conclusions + outlook Phase-field model and simulations for Zener pinning Realistic dimensions and volume fractions for thin films Role of initial grain structure 2-D D versus 3-D 3 grain growth and pinning behavior Future research Particle-matrix interface + stability of the particles Surface energy for thin films Multi-grain structure with composition field 40

41 End 41

42 Thermodynamic free energy functional 2 2 (, ) ε ( ) κ F = F ( ) bulk + Fsurf = f0 xb η + xb + η dr V 2 2 Homogeneous free energy density (J/m 3 ) Multiple field variables Binary two-phase structure 42

43 Pinning in 2D Restraining force: F = 2σ sinβ Maximum force at: F 2D Z max = 2σ gb gb β = 90 Grain boundaries become straight between particles => Very strong pinning effect 43

44 Experimental studies Rlim 1 = K b with r f V b = 1 (3D, low f V ) b = 1/3 (3D, high f V ) b = 1/2 (2D) Manohar et al. (1998) : K = 0.17, b = 1 for f V <

45 Properties of diffuse grain boundary Analysis of flat grain boundary σ κ dη d i κ η j = f (...,0, η, η,0,...) f + dx 2 + dx 2 dx gb 0 i j 0,min κ dη dη i κ j = f dx 2 dx dx Interfacial energy 0.58 κm Interfacial width κ m Interfacial mobility * 1 1 v= Lκ + = µσ gb + = µ + ρ1 ρ2 ρ1 ρ2 ρ1 ρ2 45

46 Interaction energy Energetic consideration Geometry Theoretical interaction energy 2 rσ (2 D) Diffuse grain boundaries gb 2 πr σ (3 D) gb Interaction energy slightly too negative Lower limit on particle size 46

47 Interaction energy 47

48 Comparison with theory High scatter for low f a Fitting: R r lim = b f β V Theory: β = 0.5 Phase field (R 0 =0): β = 0.48, b = 1.32 Monte Carlo: β = 0.5, b = 1.7 β = 0.54, b = 1.2 Front-tracking tracking: β = 0.46 β =

49 Computational considerations 2D: R lim /r could be reproduced High f a : system size 256, time steps => 10 hours Low f a : system size 512, >60000 time steps => 10 days 3D: R lim /r: x10 => system size: : x10 Third power of system size Computer requirements: : x