Phase- -field field simulations simulations of of grain grain growth growth in in Phase materials containing containing second second- -phase phase particles particles materials Nele Moelans Moelans Nele Department of metallurgy and materials engineering Department of metallurgy and materials engineering Katholieke Universiteit Leuven, Belgium Belgium Katholieke Universiteit Leuven, 28 February February 2008 2008 28
Scientific context Part I
Outline part I: Scientific context • Grain growth growth in in polycrystalline polycrystalline materials materials • Grain • Pinning effect of effect of second second- -phase phase particles particles • Pinning • • Phase Phase- -field field method method for for simulating simulating microstructural microstructural evolution evolution • Incentives of the research of the research • Incentives 3
Role of microstructures in materials science Chemical composition Chemical composition + + Temperature, , pressure pressure, , cooling cooling rate rate, ,… … Temperature Microstructure Microstructure Shape Shape, , size size and and orientation orientation of the of the grains grains, , mutual mutual distribution distribution of the of the phases phases Material properties properties Material Strength, deformability, hardness, toughness,fatigue… 4
Polycrystalline microstructure with second-phase particles • Mechanism for for controlling controlling the the • Mechanism grain size size of a of a material material grain • E.g. microalloyed microalloyed steels steels • E.g. – Small Small grain grain size size required required for for – high strength high strength – Addition – Addition of of small small amounts amounts of of Nb, Ti, Al, V,… … Nb, Ti, Al, V, – Formation Formation of of NbC NbC, , AlN AlN, , TiN TiN,... ,... – – Pinning Pinning of of grain grain boundaries boundaries – during heat heat treatments treatments or or during welding welding 5
Normal grain growth • Surface tension tension • Surface P g P g = = driving driving pressure pressure for for grain grain boundary boundary movement movement ασ = gb P g R P g => P decreases in time in time => g decreases 6
Normal grain growth • Surface tension tension + + topological topological considerations considerations • Surface Smaller grains grains shrink shrink Smaller Isotropic: : Isotropic and and α 1 α 2 α 3 α = α = α = 120° ° 1 = 2 = 3 = 120 larger grains grains grow grow larger 7
Zener pinning • Grain boundary boundary area area is is reduced reduced • Grain MnS precipitate precipitate in in low low- -C C steel steel MnS when a a particle particle is is located located on on a a when grain grain boundary boundary • • Particles exert Particles exert a back a back force force on on moving grain grain boundaries boundaries moving = π σ max F r Z gb • Dimple- -shape shape • Dimple 8
Final grain size – Zener relation • Calculation of the of the total total pinning pinning • Calculation pressure pressure of the of the particles particles P P Z Z requires requires • Number of of particles particles in contact in contact • Number with a with a grain grain boundary boundary • • Angle at Angle at which which grain grain boundary boundary meets the the particle particle meets • Grain growth growth stops stops when when P P g =P Z • Grain g =P Z R 1 lim = K f b r V From P.A. P.A. Manohar Manohar (1998) (1998) From 9
Pinning effect: experimental observation • Fe- -0.09 to 0.53 w% C 0.09 to 0.53 w% C- -0.02 w% P 0.02 w% P • Fe containing Ce Ce 2 O 3 inclusions containing 2 O 3 inclusions • PhD – – work work M. M. Guo Guo • PhD • • Pinned austenite Pinned austenite grain grain boundaries boundaries 20 μ m 10
Phase-field simulations of microstructural evolution Experiments Experiments, , atomistic atomistic simulations simulations and thermodynamic thermodynamic models models and Crystal structure, phase stabilities, interfacial properties (energy, mobility, structure,anisotropy), diffusion properties Phase- -field field simulations simulations Phase Morphological evolution of the grains at the mesoscale during solidification, precipitation, solid-state phase transformations, grain growth,… Models that that predict predict macroscopic macroscopic material material properties properties Models Strength, deformability, hardness, toughness, fatigue… 11
Phase field method • van der Waals, Cahn Cahn- -Hilliard Hilliard (1958), (1958), Ginzburg Ginzburg- -Landau Landau (1950) , (1950) , • van der Waals, Hohenberg and and Halperin Halperin (1977) (1977) Hohenberg • Microstructure evolution evolution ( (started started ± ± 20 20 years years ago ago) ) • Microstructure • Solidification • Solidification • Ordering reactions reactions • Ordering • Martensitic transformation transformation • Martensitic • Nowadays • Nowadays • Wide range of applicabilities applicabilities • Wide range of • Quantitative aspects aspects • Quantitative – Parameter Parameter determination determination – – Numerical Numerical implementation implementation – 12
Representation of microstructures in the phase-field method • • Phase- Phase -field field variables: variables: continuous continuous functions functions in in space space and time and time � • Local composition composition x ( , ) r t • Local B � η ( , ) r t • Local structure structure and and orientation orientation • Local Binary alloy Binary alloy A A- -B B Phase α α : : η η = 0 • Phase = 0 • Phase β β : : η η = 1 • Phase = 1 Antiphase boundary boundary • Antiphase 13
Diffuse-interface description • • Sharp interface Sharp interface • • Diffuse interface Diffuse interface • Discontinuous variation variation in in properties properties • Continuous variation variation in in properties properties • Discontinuous • Continuous • Requires tracking tracking of the interfaces of the interfaces • Interfaces implicitly implicitly given given by by local local • Requires • Interfaces variations in in phase phase- -field field variables variables variations • • Complex Complex grain grain morphologies morphologies • Simplified grain grain morphologies morphologies • Simplified 14
Phase-field simulation technique � x ( , ) r t � B • Microstructural representation representation: : • Microstructural η ( , ) r t • Thermodynamic and and kinetic kinetic equations equations • Thermodynamic • Phase stabilities stabilities • Phase • Interfaces • Interfaces • Elastic energy energy due due to volume to volume effects effects • Elastic • Orientation dependence dependence • Orientation • Solute diffusion diffusion • Solute • Parameter determination determination • Parameter • • Numerical Numerical solution solution 15
Intermediate conclusions Incentives of the research • Pinning effect of effect of second second- -phase phase particles particles on on grain grain boundaries boundaries • Pinning and final and final grain grain size size still still not not understood understood • Mesoscale grain grain growth growth simulations simulations can can give give important important • Mesoscale insights insights • • Phase- Phase -field field method method for for simulating simulating microstructure microstructure evolution evolution • General technique technique based based on on nonequilibrium nonequilibrium thermodynamic thermodynamic • General principles principles • • Complex phenomena Complex phenomena and and morphologies morphologies ⇒ Phase ⇒ Phase- -field field simulations simulations of of grain grain growth growth in in materials materials containing containing second- -phase phase particles particles second 16
Part II Research work Model description Model description and and simulation simulation results results
Outline part II: Results • • Phase Phase- -field field model model for for grain grain growth growth and and Zener Zener pinning pinning • Study of the of the pinning pinning mechanism mechanism • Study • • Simulation Simulation results results for for polycrystalline polycrystalline materials materials • Ongoing/ /future future research research • Ongoing 18
Representation of a polycrystalline structure • Extension grain grain growth growth model model • Extension D. Fan and L.- -Q. Q. Chen Chen (1997) (1997) D. Fan and L. • Phase field variables: field variables: • Phase � η η η η , ,..., ( , ),..., r t 1 2 i p : Φ Φ =1 • Particles: =1 • Particles η η η η = ( , ,..., ,..., ) (0,0,...,0,...,0) 1 2 i p Φ =0 : Φ • Grain i of i of matrix matrix- -phase phase: =0 • Grain η η η η = ± ( , ,..., ,..., ) (0,0,..., 1,...,0) 1 2 i p 19
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