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

Transcription

1 Phase field simulations for grain growth in materials containing second-phase particles N. Moelans, B. Blanpain,, P. Wollants Research group: Thermodynamics in materials engineering Department of Metallurgy and Materials Engineering K.U. Leuven, Belgium

2 Contents Introduction on Zener pinning Phase field model Grain boundary/particle interactions Large-scale 2D-simulations Conclusions

3 Zener pinning Grain growth: P 1 1 = σ gb + R1 R2 Zener pinning: F r D max Z = πσ (3 ) gb Dimple shape Limiting grain size: R lim r = b f β V

4 Phase field model Extension of model of D. Fan and L.-Q. Chen for normal grain growth Phase field variables: η, η,..., η,..., η 1 2 Particles : Φ=1 ( η, η,..., η,..., η ) = (0,0,...,0,...,0) 1 2 Grain i of matrix-phase : Φ=0 ( η, η,..., η,..., η ) = (0,0,...,1,...,0) 1 2 i i i p p p

5 Phase field model Free energy F p p p p p α 4 β κi = ( ηi ηi ) + γ η V iηj + εφ ηi + ηi i= i= 1 j i i= 1 i= 1 2 Equilibrium Φ=0 : Φ=1 : ( η, η,..., η ) = (1,0,...,0),(0,1,...,0),...(0,0,...,1),( 1,0,...,0), Kinetic equations (Ginzburg-Landau) p ( η, η,..., η ) = (0,0,...,0) p ηi (,) rt F f ( η1, η2,...) 2 = L i = Li κi ηi(,) r t t ηi(,) rt ηi(,) rt ( ) 2 dv

6 Model parameters Equations with Grain boundary energy (expressed in J/g.p.) κ = 0.5, m = 1 σ = 0.40 Interfacial energy particles: Interfacial thickness : Growth rate : ηi (,) rt f ( η1, η2,...) 2 = Li κi ηi (,) r t t ηi (,) r t η η f = m + + Φ p 4 2 p p p i i ( ) ηiηj ε ηi i= i= 1 j i i= 1 κ = 4, m = 1 σ = 1.16 gb κ = 1, m = 0.25 σ = 0.29 gb 1 2 gb κ m 1 1 κ L( + ) R R m /2< ε < 2.5

7 Grain boundary/particle interaction Energetic consideration: Geometry: Interaction energy: 2 rσ (2 D) gb 2 πr σ (3 D) gb Diffuse grain boundaries Interaction energy slightly too negative Lower limit on particle size Evolution spherical grain boundary pinned by single particle dimple shape

8 Large-scale 2D-simulations Isotropic grain boundary properties: κ = 0.5, L = 1, m= 1, ε = 1 i Round particles: Area fraction: System size: : 256x256 and 512x512 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 i r = 2.5 r = 3 f = a

9 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 = 0.46, b = 1.38 Monte Carlo: β = 0.5, b = 1.7 β = 0.54, b = 1.2 Front-tracking tracking: β = 0.46 β = 0.5

10 Role initial grain size

11 Role initial grain size R 0 = 0: R 0 > 0 r f R = 3, a = 0.04, 0 = 0 r fa R0 = 3, = 0.04, = 13.6

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

13 Comparison with experiment Thin Al-films Columnar grains 2D- structure CuAl 2 -particles Data from H.P. Longworth and C.V. Thompson

14 Conclusions Zener pinning was studied as a function of Particles size Number of particles Initial microstructure Long computation times even for simple materials Further study of dimple shape and dynamic behavior of the grain boundaries is necessary

15 References N. Moelans, B. Blanpain,, P. Wollants,, "A" A phase field model for the simulation of grain growth in materials containing finely dispersed incoherent second-phase particles", Acta Mater., 2005, in press. N. Moelans, B. Blanpain,, P. Wollants,, "Phase" field simulations of grain growth in 2-dimensional 2 polycrystalline systems containing finely dispersed incoherent second-phase particles", in preparation. Available on ://

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