New directions in phase- -field modeling of field modeling of New - PowerPoint PPT Presentation

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

‘Quantitative’ phase-field models • Properties bulk bulk and interfaces are and interfaces are reproduced reproduced • Properties accurately in the simulations in the simulations accurately • Effect model description and model description and parameters parameters • Effect • Numerical issues issues • Numerical • Insights in the evolution Insights in the evolution of of complex complex • morphologies and grain assemblies morphologies and grain assemblies • • Effect Effect of of individual individual bulk bulk and interface and interface properties properties • Predictive ? ? • Predictive • Depends on on availability availability and and accuracy accuracy of input of input • Depends data data – Requires Requires composition and orientation composition and orientation – dependence dependence Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, 2 Sweden, May 17-18, 2010

General framework and goal Experiments Experiments, , atomistic atomistic simulations simulations and and thermodynamic thermodynamic models models Crystal structure, phase diagram, interfacial properties (energy, mobility, anisotropy), diffusion properties, … Phase- -field field simulations simulations Phase Microstructure evolution at the mesoscale Quantitative characterization characterization Quantitative Average grain size, grain size distribution, volume fractions, texture,… Basis for statistical and mean field theories Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, 3 Sweden, May 17-18, 2010

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

2-phase systems � � Double well function φ • • Field variables: Field variables: ( , ) r t c r t ( , ) k • Phase • Phase � � : : �� �� = 0 = 0 • Phase Phase � : � � = 1 = 1 • � : • Composition • Composition: : c c B B • Free energy energy • Free  ε  � 2 Interpolation function ∫ = φ + φ + ∇ φ 2 F f ( , ) c W g ( ) | | dr   chem  2  V Interfacial energy • Bulk energy energy • Bulk ( ) ( ) = φ β + − φ α   f h f ( , ) c T 1 h f ( , ) c T   chem Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, 5 Sweden, May 17-18, 2010

Decoupling bulk and interfacial energy • Interface treated treated as mixture of 2 phases as mixture of 2 phases • Interface α β → • c- -field field for for each each phase phase c c , c • c • Equal interdiffusion interdiffusion potential potential + + • Equal conservation conservation β β α α ∂ ∂ f ( c ) f ( c ) = = µ � β α ∂ ∂ c c ( ) ( ) β α = φ + − φ   c h c 1 h c   Kim et al., PRE, 6 (1999) p 7186; Tiaden Tiaden et et Kim et al., PRE, 6 (1999) p 7186; al., Physica Physica D, D, 115 (1998) p73 115 (1998) p73 al., Bulk energy energy • • Bulk ( ) ( ) β β α α ⇒ = φ +  − φ  f h f ( c ) 1 h f ( c )   chem Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, 6 Sweden, May 17-18, 2010

Decoupling bulk and interfacial kinetics • Kinetic equations equations ( (Linear Linear non non- -equilibrium equilibrium thermodynamics thermodynamics) ) • Kinetic ∂ ϕ ∂ • Allen- -Cahn Cahn • Allen F = − M ϕ ∂ ∂ ϕ t ∂ − C 1 c ∑  β α  = ∇⋅ φ + − φ ∇ µ • • Diffusion Diffusion k � h ( ) M [1 h ( )] M   ∂ kl kl l t l • • Jump in Jump in chemical chemical potential potential accross accross interface interface ∂ − ∂ φ ∇ φ C 1 c ∑ = ∇⋅ − φ ∇ µ + ∇⋅ α L k � [1 h ( )] M ∂ kl l i ∂ ∇ φ t | t | | | = l 1 Non-variational anti-trapping current • Dilute, D S =0 : A.Karma, PRL, 87, 115701 ∆ µ ∝ (2001); B. Echebarria et al., PRE, 70, 061604 � i (2004 ) Solute trapping effect ∆ µ ∝ v • Multi-comp, D S =0 : S.G. Kim, Acta Mater. i n 55, p4391 (2007) Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, 7 Sweden, May 17-18, 2010

Multi-grain and multi-phase structures • • Single phase Single phase- -field field models models - -> Multiple > Multiple phase- -field field models models phase { } η → η η η η , , ,..., 1 2 3 p η η η η = ( , ,..., ,..., ) (0,0,...,1,...,0) 1 2 i p • Model extension • Model extension F η η η ∇ η ∇ η 2 2 ( , , ,...,| | ,| | ,...) 1 2 3 1 2 – Different Different types of interfaces types of interfaces – – Triple and Triple and higher higher order order junctions junctions – η = η = 1 1 • • Numerically Numerically i j – Same Same accuracy accuracy for all interfaces for all interfaces – and phases and phases Grain j Grain i – All interfaces – All interfaces within within range of range of validity of the of the thin thin interface interface validity η = η = 0 0 asymptotics asymptotics j i → = � cte num Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, 8 Sweden, May 17-18, 2010

Multi-grain and multi-phase models: major difficulties • • Third- Third -phase contributions phase contributions � 3 � 3 • • 12 = = � 13 = 7/10 = 7/10 � � 12 � 13 � 12 � 12 3 3 1 1 2 2 • Careful choice choice of multi of multi- -well well function function • Careful and gradient contribution and gradient contribution • • Interpolation function Interpolation function • • Zero Zero- -slope slope at at equilibrium equilibrium values of the values of the phase fields fields phase • Thermodynamic consistency consistency • Thermodynamic p p ∑ ∑ = η η ⇒ η η = i ( , ,...) ( , ) ( , ,...) 1 f h f c T h chem i 1 2 i 1 2 = = i 1 i 1 Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, 9 Sweden, May 17-18, 2010

Anisotropic grain growth model � η η η η , ,..., ( , ),..., r t • Phase fields fields • Phase 1 2 i p • With grain grain i i • With η η η η = ( , ,..., ,..., ) (0,0,...,1,...,0) 1 2 i p • Free energy energy • Free     η η κ η 4 2 p p p p 1 ( ) ∑ ∑∑ ∑ ∫ = − + γ η η + + ∇ η 2 2 2 i i   F m   ( ) dV interf i j , i j i  4 2  4 2   = = < = i 1 i 1 j i i 1 V p p p p = ∑∑ ∑∑ κ η κ η η η η 2 2 2 2 ( ) i j , i j i j = < = < i 1 j i i 1 j i η η ≠ ⇒ κ η = κ 2 2 • For each each grain grain boundary boundary • For 0 ( ) i j i j , • Inclination dependence dependence • Inclination ∇ η − ∇ η ( ) ( ) ( ) γ ψ κ ψ ψ ψ = ∇ i j , , L , i j , i j , i j , i j , i j , i j , i j , η − ∇ η L.-Q. Chen and W. Yang, PRB, 50 (1994) p15752 | | i j A. Kazaryan et al., PRB, 61 (2000) p14275 Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, 10 Sweden, May 17-18, 2010

Non-variational approach – equal interface width • Ginzburg- -Landau Landau type type equations equations • Ginzburg �     ∂ η ( , ) r t ∑ ( ) = − η η − η + η γ η − κ η ∇ η 3 2 2 i     L m 2 ( ) ∂ i i i i j , j i   t     ≠ j i • Non- -variational variational with with respect to respect to � -dependence dependence of of � • Non � - � • Similar to to Monte Monte Carlo Carlo Potts Potts approach approach • Similar Definition ‘ ‘grain grain boundary boundary width width’ ’ • • Definition 1 1 = = l η η num d d | i | j | | max max dx dx High High controllability controllability of of numerical numerical accuracy ( ( l l num /R < 5 < 5) ) accuracy num /R Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, 11 Sweden, May 17-18, 2010