Thermodynamics, Bratislava, April 6, 2010. Outline. ⢠Introduction: modeling of microstructure evolution. ⢠Principles of phase field modeling. ⢠Application ...
Phase-field modeling of microstructure evolution in multi-component alloys Nele Moelans Department of Metallurgy and Materials Engineering (MTM) K.U.Leuven, Belgium Liesbeth Vanherpe, Jeroen Heulens, Bert Rodiers COST MP0602 – Industrial Seminar on Thermodynamics, Bratislava, April 6, 2010
Outline
• Introduction: modeling of microstructure evolution • Principles of phase field modeling • Application examples • • • •
•
Anisotropic grain growth Precipitates on grain boundaries Lead-free solder joints Solidification
Summary
COST MP0602 Industrial Seminar on Thermodynamics, Bratislava, April 6, 2010
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Role of microstructures in materials science Chemical composition + Temperature, pressure, cooling rate, deformation, …
Low-C steel: wt% C < 0.022 Ferrite + small carbides 250 Pm Steel: wt% C = 0.4, slow cooling rate
Microstructure Shape, size and orientation of the grains, mutual distribution of the phases
50 Pm
Ferrite + cementite
Material properties
Steel: wt% C = 0.4, fast cooling rate
Strength, deformability, hardness, toughness, fatigue…
Ferrite needles+ perlite
COST MP0602 Industrial Seminar on Thermodynamics, Bratislava, April 6, 2010
50 Pm
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Microstructure evolution in multicomponent alloys •
Connected grain structures
•
Complex morphologies
•
Concurrent processes • Solidification - grain growth Ostwald ripening – diffusion phase growth
•
Materials’ properties are largely related to microstructure • •
Low C-steel, ferritic grain structure
Material development Reliability
COST MP0602 Industrial Seminar on Thermodynamics, Bratislava, April 6, 2010
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Microstructure evolution in multicomponent alloys •
High complexity • Anisotropy, segregation, solute drag, second-phase precipitates, pipe diffusion, mutual distribution of phases, … • Many material properties: Crystal structure, Gibbs energy, diffusion coefficient of different phases? Structure, energy, mobility of grain boundaries? • Evolution connected grain structure ? – Mesoscale simulations
•
Importance • Material development: heat treatment, alloying • Reliability
COST MP0602 Industrial Seminar on Thermodynamics, Bratislava, April 6, 2010
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Phase-field modelling Strength • • •
Complex shapes Multi-grain, multi-phase structures Thermodynamic driving forces
F = Fchem + Fint + Felast + Fmagn + ... • •
Multi-component Transport equations •
Mass and heat diffusion/convection
Difficulties • •
Implementation for realistic length scales Parameter choice
COST MP0602 Industrial Seminar on Thermodynamics, Bratislava, April 6, 2010
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Principles of phase field modeling • • • • • •
Diffuse interface concept Phase field variables Thermodynamic free energy functional Evolution equations Parameter assessment Numerical implementation
Diffuse-interface description •
Sharp interface
• • •
•
Discontinuous variation in properties Requires tracking of the interfaces Simplified grain morphologies
Diffuse interface
• • •
Continuous variation in properties Interfaces implicitly given by local variations in phase-field variables Complex grain morphologies
• Diffuse interface approach (van der Waals, CahnCahn-Hilliard (1958), GinzburgGinzburg-Landau (1950)) • Microstructure evolution started ± 20 years ago COST MP0602 Industrial Seminar on Thermodynamics, Bratislava, April 6, 2010
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Representation of microstructures •
Phase-field variables: continuous functions in space and time • •
Local composition
G G xB ( r , t ), cB ( r , t )
Local structure and orientation
G η (Gr , t ) φ ( r, t )
Binary alloy A-B • Phase D: K = 0 • Phase E: K = 1
Antiphase boundary COST MP0602 Industrial Seminar on Thermodynamics, Bratislava, April 6, 2010
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Thermodynamics and kinetics •
Free energy functional F = Fbulk + Fsurface
κ p ε C 2 = ∫ f 0 ( xi ,ηk , T ) + ∑ (∇ηk ) + ∑ (∇xi )2 dV 2 k =1 2 i =1 V
Homogeneous free energy (chemical, elastic, …)
•
Gradient free energy (
diffuse interfaces)
Evolution of field variables • Non-conserved field variables ( G ∂F ( x1 ,..., xC ,η1 ,...,η p ) ∂ηk ( r, t ) G = −L ∂t ∂ηk ( r, t )
• Conserved composition fields (
Interface movement) G ( +ξ ( r, t ))
Mass diffusion)
G JG ∂F ( x1 ,..., xC ,η1 ,...,η p ) 1 ∂xi ( r, t ) JG G = ∇M ⋅ ∇ Vm ∂t ∂xi ( r, t ) COST MP0602 Industrial Seminar on Thermodynamics, Bratislava, April 6, 2010
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G ( +ξ ( r, t ))
Homogeneous free energy functional •
Anti-phase boundary
Boundary energy:
∝ κ ( ∆f 0 ) max
Boundary width:
∝
Boundary velocity: COST MP0602 Industrial Seminar on Thermodynamics, Bratislava, April 6, 2010
η4 η2 f 0 = 4( ∆f 0 ) max − 2 4
κ ( ∆f 0 ) max
∝κL 11
Homogeneous free energy •
Binary two-phase system
f 0β E D
COST MP0602 Industrial Seminar on Thermodynamics, Bratislava, April 6, 2010
f 0α
f 0 = h (φ ) f β (c, T ) + 1 − h (φ ) f α ( c, T ) + ω g (φ ) with f α ( c, T ) = 12
α m
G , Vm
Gmβ f ( c, T ) = Vm β
(CALPHAD)
Homogeneous free energy •
Binary two-phase system
f 0β
f 0α
f 0 = h (φ ) f β (c, T ) + 1 − h (φ ) f α ( c, T ) + ω g (φ ) Double well function COST MP0602 Industrial Seminar on Thermodynamics, Bratislava, April 6, 2010
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Homogeneous free energy •
Binary two-phase system
f 0β
f 0α
f 0 = h (φ ) f β (c, T ) + 1 − h (φ ) f α ( c, T ) + ω g (φ ) Interpolation function COST MP0602 Industrial Seminar on Thermodynamics, Bratislava, April 6, 2010
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Effect of elastic stresses and strain •
Coupling with micro-elasticity theory
F = Fchem + Fint + Felast
• Effect of transformation and thermal strains, applied stress/strain • Martensitic transformation, precipitate growth
COST MP0602 Industrial Seminar on Thermodynamics, Bratislava, April 6, 2010
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Polycrystalline structures •
Polycrystalline microstructure
G η1 ,η2 ,...,ηi (r , t ),...,η p •
Grain i of matrix-phase (η1 ,η2 ,...,ηi ,...,η p ) = (0,0,...,1,...,0)
ηi = 1
Grain i ηj = 0
ηj =1
Grain j
• ηi = 0
Free energy functional
F = Fsurface
COST MP0602 Industrial Seminar on Thermodynamics, Bratislava, April 6, 2010
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κ p = ∫ f 0 (η1 ,η2 ,...η p ) + ∑ (∇ηk ) 2dV 2 k =1 V
Multi-phase and multi-component alloys •
Phase field variables: • Grains
G ηα 1 ,ηα 2 ,...,ηαi ( r, t ),...,
ηβ 1 ,ηβ 2 ,...η p • Composition
G x A , xB ( r, t ),..., xC −1 •
Free energy functional f bulk ( xk ,ηρ i )
= ∑ hρ (ηρ i ) f ρ ( xkρ )
2 phase polycrystalline structure
ρ
Gmρ ( xkρ ) = ∑ hρ (ηρ i ) Vm ρ COST MP0602 Industrial Seminar on Thermodynamics, Bratislava, April 6, 2010
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Parameter assessment •
Different kinds of input data (calculated and/or measured) • • • •
•
Phase stabilities, phase diagram information – CALPHAD (ab-initio, experiments) Interfacial energy and mobility – ab-initio, (MD, MC ab-initio), experimental Elastic properties, crystal structure, lattice parameters – ab-initio, experimental Atomic diffusion mobilities – CALPHAD ((MC ab-initio), experimental)
Orientation and composition dependence •
Anisotropy, segregation, solute drag
COST MP0602 Industrial Seminar on Thermodynamics, Bratislava, April 6, 2010
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COST MP0602 Industrial Seminar on Thermodynamics, Bratislava, April 6, 2010
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Quantitative aspects •
Numerical solution of partial differential equations • Bounding box algortihm • Sparse data structure • Object oriented C++ (L. Vanherpe et al., K.U.Leuven)
• Discretization (Finite differences, finite elements, Fourier-spectral method
• Discretization MICRESS, commercial software forphasefield coupled with CALPHAD COST MP0602 Industrial Seminar on Thermodynamics, Bratislava, April 6, 2010
• Adaptive meshing (M. Dorr et al. AMPE, LLNL) 20
Examples of applications • • • •
Anisotropic grain growth Precipitates on grain boundaries Lead-free solder systems Solidification
Columnar films with fiber texture •
Grain boundary energy: • Fourfold symmetry • Extra cusp at T = 37.5° • Read-shockley
2D simulation White: θ = 1.5 Gray: θ = 3 Red: θ = 37,5 Black: θ > 3, θ 37.5
•
DiscreteG orientations
• •
Constant mobility Initially random grain orientation and grain boundary type distributions In collaboration with F. Spaepen, School of Engineering and Applied Sciences, Harvard University
η1 ,η2 ,...,ηi ( r, t ),...,η60 ⇒ ∆θ = 1.5°
COST MP0602 Industrial Seminar on Thermodynamics, Bratislava, April 6, 2010
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3D simulations for wires with fiber texture
ϑ < 6° COST MP0602 Industrial Seminar on Thermodynamics, Bratislava, April 6, 2010
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Microstructure analysis: films •
•
Misorientation distribution function (MDF)
Evolution mean grain area
A − A0 = keff t
• Evolution towards stead-state regime Simple mean field models COST MP0602 Industrial Seminar on Thermodynamics, Bratislava, April 6, 2010
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neff
neff =
keff =
kt → 1, t → ∞ Ah (0) + kt
dAeff dt
=
k 1 + N ’/ N
Examples of applications • • • •
Anisotropic grain growth Precipitates on grain boundaries Lead-free solder systems Solidification
Zener pinning •
Mechanism for controlling grain size • •
•
E.g. NbC, AlN, TiN,... in HSLAsteels Nano-grain structures
Zener relation for limiting grain size
R lim 1 =K b fV r
•
Influence of • • • •
Shape of the particle Interfacial properties of particles Initial distribution Fe-0.09 to 0.53 w% Evolution particles
COST MP0602 Industrial Seminar on Thermodynamics, Bratislava, April 6, 2010
C-0.02 w% P with Ce2O3 inclusions (PhD. M. Guo)
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20 Pm
Simulation results: Al thin films Film preparation
•
Thin films with CuAl2 - precipitates
R lim 1 = 1.28 0.5 fa r
r = 3, f a = 0.12, l3 = 21 (exp from H.P. Longworth and C.V. Thompson) COST MP0602 Industrial Seminar on Thermodynamics, Bratislava, April 6, 2010
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Simulation results: effect of particle shape and coarsening •
•
Ellipsoid particles
Evolving particles
ε = 3, fV = 0.05 Modified Zener relation R lim r 1 =K a s 1 + ara fVb
K = 3.7, b = 1, a = 3.1 L. Vanherpe, K.U. Leuven COST MP0602 Industrial Seminar on Thermodynamics, Bratislava, April 6, 2010
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fV=0.12, L=10M
Jerky motion during recrystallization in Al-Mn alloy •
In-situ EBSD observation of recrystallization in AA3103 at 400 °C •
•
Jerky grain boundary motion • •
•
CamScan X500 Crystal Probe FEGSEM
Stopping time: 15-25 s Pinning by second-phase precipitates – Al6(Fe,Mn), D-Al12(Fe,Mn)3Si
Added to phase field model • •
Grain boundary diffusion Driving force for recrystallization
20Pm In collaboration with A. Miroux, E. Anselmino, S. van der Zwaag, T. U. Delft
COST MP0602 Industrial Seminar on Thermodynamics, Bratislava, April 6, 2010
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Material properties at 723K Grain boundary energy high angle
γh = 0.324 J/m2
Interfacial energy Al6Mn precipitates
γpr = 0.3 J/m2
Mobility high angle grain boundary At solute content 0.3w% Mn
Mh = 2.94·10-11 m2s/kg
Equilibrium composition of matrix
cMn,eq = 0.0524 w% (0.02456 at%) (PhD thesis Lok 2005)
Actual composition of matrix (supersaturated)
cMn = 0.3 w% (0.1474 at%) (PhD thesis Lok 2005)
Mn diffusion in fcc Al
-2 2 D0,bulk 0,bulk = 10 m /s, Qbulk = 211 kJ/mol Dbulk = 5.5973·10-18 m2/s
(Miroux et al.,Mater. Sci. Forum,467-470,393(2004))
Pipe diffusion high angle boundaries, D0,p = D0,bulk 0,bulk, Qp = 0.65Qbulk precipitate/matrix interface Dp = 1.2195·10-12 m2/s Bulk energy density: fρ = Aρ(x-xρ0)2 COST MP0602 Industrial Seminar on Thermodynamics, Bratislava, April 6, 2010
Am = 6·1011; xm0 = 0.000258 Ap = 6·1012; xp0 = 0.1429 30
Precipitate coarsening and unpinning •
PD < PZS (PD § PZS)
•
• Pinning: PZS=3.6 MPa • Rex: PD=3.1 MPa
Unpinning mainly through surface diffusion around precipitates 7·10-10
J = J x2 + J y2
6s
0.9µm x 0.375µm
COST MP0602 Industrial Seminar on Thermodynamics, Bratislava, April 6, 2010
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Precipitate coarsening and unpinning •
•
PD
