1
Introduction to phase-field modeling of
microstructure evolution
Nele Moelans
Department of Materials Engineering, KU Leuven, Belgium
2
Outline
• Introduction
• Principles of phase field modeling
• Quantitative ‘thin-interface’ models
• Application examples: Phase-field simulations of grain growthand recrystallization
• Summary
2
3
Microstructures
50 µµµµm
50 µµµµm 25 µµµµm
Fe-C:
a) Ferrite (wt% C < 0.022)
b) Ferrite + cementite (wt%
C = 0.4, slowly cooled)
c) Ferrite + cementite (wt%
C = 0.4, faster cooled)
d) Martensite (wt% C =
1.4, quenched)
250 µµµµm
a) b)
d)c)
4
Role of microstructures in materials
science
Chemical composition+
Temperature, pressure, cooling rate,…
MicrostructureShape, size and orientation of the grains, mutual distribution of the phases
Material properties
Strength, deformability, hardness, toughness,fatigue…
3
Principles of phase field modeling
• Diffuse interface concept
• Phase field variables
• Thermodynamic free energy functional
• Evolution equations -- Onsager
N. Moelan, B. Blanpain, P. Wollants, Comp. Coupling of Phase Diagrams and Thermochemistry, CALPHAD, 32, p268 (2008)
6
Diffuse-interface description
• van der Waals, Cahn-Hilliard (1958), Ginzburg-Landau (1950)
• Microstructure evolution since ± 20 years
• Quantitative ‘thin interface’ models since ± 10 years
• Results independent of ℓ
• Continuous variation in properties
• Avoids numerical interface
tracking
• Complex morphologies feasible
4
7
Representation of microstructures
• Phase-field variables: continuous functions in space and time
• Local composition
• Local structure and orientation
Binary alloy A-B
•Phase αααα: ηηηη = 0
•Phase ββββ: ηηηη = 1Antiphase boundary
( , ), ( , )B Bx r t c r t� �
( , )r tη�
( , )r tφ�
8
Polycrystalline structures
1 2
1 2
, ,..., ( , ),...,
, ,...
i
p
r tα α α
β β
η η η
η η η
�
• Single phase
• Grain i of matrix-phase
• 2-phase
• Grains
• Composition
1 2( , ,..., ,..., ) (0,0,...,1,...,0)
i pη η η η =
1 2, ,..., ( , ),...,
i pr tη η η η�
Grain i Grain j
1i
η =
0j
η = 0i
η =
1j
η =
1, ( , ),...,A B Cx x r t x −
�
5
9
Thermodynamics for heterogeneous
systems
10
Thermodynamics
• Free energy functional
2 2
0
1 1
( , , ) ( ) ( )2 2
p C
bulk surface i k k i
k iV
F F F f x T x dVκ ε
η η= =
= + = + ∇ + ∇∑ ∑∫
Homogeneous free energy
(chemical, elastic, …)
Gradient free energy
(→ diffuse interfaces)
4 2
0 0 max4( )4 2
f fη η
= ∆ −
0 max( )fκ∝ ∆
0 max( )f
κ∝
∆
Interface energy
Interface width
6
11
Homogeneous free energy
• Binary two-phase system
( ) ( )01 (( ) ( , ) ),f h hf c T f c gT
β αφ φ ω φ= + − +
f0β f0
α
αααα
ββββ
( , ) m
m
Gf c T
V
ββ =with ( , ) ,m
m
Gf c T
V
αα = (CALPHAD)
12
Homogeneous free energy
• Binary two-phase system
( ) ( )0( , ) 1 ( , ) ( )h hf f c T f c T g
β α ωφ φφ= + − +
f0β f0
α
Interpolation function
7
13
Homogeneous free energy
• Binary two-phase system
( ) ( )0( ( ), ) 1 ( , )f h f c T h f gc T
β αφ φ φω= + − +
f0β f0
α
Double well function
14
Solid-solid phase transitions
• Coupling with micro-elasticity theory →
• Effect of transformation and thermal strains, applied stress/strain
• Martensitic transformation, precipitate growth
chem int elastF F F F= + +
1( , ) ( , ) ( , )
2
el el
elast ijkl ij kl
V
F C x x x dVη ε η ε η= ∫
8
15
Kinetics
• Onsager theory
• Non-conserved field variables
(→ Interface movement)
• Conserved composition fields
(→ Mass diffusion)
This image cannot currently be displayed.
1 1( ,..., , ,..., )( , )( ( , ))
( , )
C pk
k
F x xr tL r t
t r tη
η ηηξ
η
∂∂= − +
∂ ∂
��
�
1 1( ,..., , ,..., )1 ( , )( ( , ))
( , )
C pix
m i
F x xx r tM r t
V t x r t
η ηξ
∂∂= ∇ ⋅ ∇ +
∂ ∂
��� �� �
�
201 1
( ,..., , ,..., )C p k
k
fL x x η η κ η
η
∂= − − ∇
∂
20i
i
fM x
xε
∂= ∇ ⋅ ∇ − ∇
∂
�� ��
16
Numerical implementation
• Numerical solution of partial differential equations
•Discretization
•Bounding box algortihm
•Sparse data structure
•Object oriented C++(L. Vanherpe et al., K.U.Leuven)
•Discretization (Finite differences, finite elements, Fourier-spectral method, …)
•Adaptive meshing (M. Dorr et al. AMPE, LLNL)
MICRESS, commercial software for phase-field coupled with CALPHAD
9
Quantitative ‘thin-interface’ models
• What ? Why ?
• Decoupling bulk and interfacial energy
• Decoupling bulk and interfacial kinetics
• Heterogeneous interface properties
18
Quantitative ‘thin-interface’ models
• Purpose
• Results independent of interface width and interpolation function for ℓphys << ℓnum << Rgrain
• → realistic 3D simulations
• → straightforward relations for the model parameters
• Important work
• Diffusion controlled solidification in pure and dilute systems
– Karma and Rappel (1996), Karma (2001), Echebarria (2004), Folch-Plapp (2005)
• Multi-component systems
– Tiaden et al. (1996), Kim and Kim (1999), Kim and Kim
(2007)
• Polycrystalline and multi-phase structures
– Kazaryan et al. (2000), Moelans (2008), Moelans (2011)
10
19
Decoupling bulk and interfacial energy
• Interface treated as mixture of 2 phases
• c-field for each phase
• Equal interdiffusion potential + conservation
• Bulk energy
Kim et al., PRE, 6 (1999) p 7186; Tiaden et al., Physica D, 115 (1998) p73
Similar altern. : Karma, PRL (2001); Floch,Plapp, PRE (2005), Echebarria et al., PRE (2004)
( ) ( )f c f c
c c
β β α α
β αµ
∂ ∂= =
∂ ∂ɶ
( ) ( )( ) ( )1chem
f ch ff chβ β α αφ φ⇒ = + −
( ) ( )1c h c h cβ αφ φ= + −
,c c cα β→
20
Decoupling bulk and interfacial
kinetics
• Jump in chemical potential accross interface
• Partial solution
•Dilute, DS=0: A.Karma, PRL, 87, 115701 (2001); B. Echebarria et al., PRE, 70, 061604
(2004)•Multi-comp, DS=0: S.G. Kim, Acta Mater. 55, p4391 (2007)
1
1
[1 ( )]| | | |
CLkkl l i
l
ch M
t t
φ φφ µ α
φ
−
=
∂ ∂ ∇= ∇ ⋅ − ∇ + ∇ ⋅
∂ ∂ ∇∑ ɶ
i
i nv
µ
µ
∆ ∝
∆ ∝
ℓ
Solute trapping effect
Non-variational anti-trapping current
11
21
Anisotropic grain growth model
• Phase fields
• With grain i
• Free energy
• For each grain boundary
• Inclination dependence
2 2
,0 ( )
i j i jη η κ η κ≠ ⇒ =
2 2 2
1 1
,
2( )
p p p p
i j i j
i
i j
j i i j i
κ η η ηκη η= < = <
=∑∑ ∑∑
( ) ( ) ( ), , , , , , ,, , ,| |
i j
i j i j i j i j i j i j i j
i j
Lη η
γ ψ κ ψ ψ ψη η
∇ − ∇=
∇ − ∇
4 2
2 2 2
1
,
1 1
1 ( )( )
4 2 4 2
p p p p
i iinterf i j i
i i
i j
j i iV
F dVmκ
ηγη η η
η η= = < =
= − + + + ∇
∑ ∑∑ ∑∫
1 2( , ,..., , ..., ) (0,0,...,1,...,0)i pη η η η =
1 2, ,..., ( , ),...,i pr tη η η η�
L.-Q. Chen and W. Yang, PRB, 50 (1994) p15752
A. Kazaryan et al., PRB, 61 (2000) p14275
22
Non-variational approach – equal
interface width
• Ginzburg-Landau type equations
• Non-variational with respect to ηηηη-dependence of κκκκ
• Similar to Monte Carlo Potts approach
• Definition ‘grain boundary width’
( ) 2
,
3 2( , )( )2i
i i i j i
j i
i j
r t
tL m
ηη η η η ηη γ κ η
≠
∂= − − + − ∇
∂ ∑
�
max max
1 1
| | | |num
i j
ld d
dx dx
η η= =
→High controllability of numerical
accuracy (lnum/R < 5)
12
23
Grain boundary properties
• Grain boundary energy
• Grain boundary mobility
• Grain boundary width
• Iterative algorithm
,, , ,( )i jgb i j i jg mθγ γ κ=
,
,
, , 2
,( ( ))i j
i j
gb i j
i j
Lm g
θ
κµ
γ=
g(γi,j) calculated numerically
,
2
,
4
3 ( ( ))
i j
i j
lm g
κ
γ=
N. Moelans, B. Blanpain, P. Wollants, PRL, 101, 0025502
(2008); PRB, 78, 024113 (2008)
, , , , ,,[ ],[ ] ,[ ],[ ],[ ]gb gb gb i j i j i jm Lθ θγ µ κ γ→ℓ
24
Numerical validation
• Shrinking grain:
• Triple junction angles:
• Observations• Accuracy controlled by
• Diffuse interface effects for
• Angles outside [100°-140°] require larger for same accuracy
2dA
dt
ααβ αβπµ σ= −
dA
dt
ααγ αβµ σ= −
,αγ βγ αγ βγσ σ µ µ= =
/num
x∆ℓ
/ 5num
R >ℓ
/num x∆ℓ
13
25
Rotation invariance of the model
• Mathematically, the model equations are invariant to rotation, but …
• the order parameters represent orientations in a fixed reference frame.
• The precision of αααα depends on the numerical setup,
• For the model to be rotational invariant in practice, lower limit of amount of order parameters p:
L
h∆≈∆
αα
cos
1
hn
Lp
∆>
π2
grid spacing ∆∆∆∆h
physical width of domain L
rotational symmetry n
J. Heulens and N. Moelans, Scripta Mat. (2010)
26
Phase-field modelling: summary
• Complex shapes
• Multi-grain, multi-phase structures
• Thermodynamic driving forces
• Multi-component
• Transport equations• Mass and heat diffusion/convection
• Implementation for realistic length scales
• Parameter choice
→ thin-interface models + coupling with atomistic approaches and efficient implementations
...chem int elast magn
F F F F F= + + + +
Strength
Difficulties
14
Examples of applications
• Grain growth and Zener pinning
• Grain growth in thin films with fiber texture
28
Zener pinning
• Mechanism for controlling grain size
• E.g. NbC, AlN, TiN,... in HSLA-
steels
• Nano-grain structures
• Zener relation for limiting grain size
• Influence of
• Shape of the particle
• Interfacial properties of particles
• Initial distribution
• Evolution particles
lim 1b
V
RK
fr=
20 µµµµm
Fe-0.09 to 0.53 w% C-0.02 w% P with Ce2O3 inclusions (PhD. M. Guo)
15
29
Representation of a polycrystalline
structure
• Extension grain growth model D. Fan and L.-Q. Chen
• Phase field variables:
• Particles: ΦΦΦΦ=1
• Grain i of matrix-phase: ΦΦΦΦ=0
1 2( , ,..., ,..., ) (0,0,..., 1,..., 0)
i pη η η η = ±
1 2( , ,..., ,..., ) (0,0,...,0,...,0)
i pη η η η =
1 2, ,..., ( , ),...,
i pr tη η η η�
30
Simulation results: Al thin films
• Thin films with CuAl2 - precipitates
Film preparation
33, 0.12, 21ar f l= = =
lim
0.5
11.28
a
R
fr=
(exp from H.P. Longworth and C.V. Thompson)
16
31
Simulation results: effect of particle
shape and coarsening
• Ellipsoid particles • Evolving particles
lim 1
1a V
b
aKa
R r
s r f=
+ fV=0.12, L=10M
3, 0.05V
fε = =
L. Vanherpe, K.U. Leuven
3.7, 1, 3.1K b a= = =
Modified Zener relation
32
Comparison with Other Studies
17
Application examples
• Grain growth in anisotropic systems with a fiber texture
34
Grain growth in columnar films with
fiber texture
• Grain boundary energy:
• Fourfold symmetry
• Extra cusp at θθθθ = 37.5°
• Read-shockley
• Discrete orientations
• Constant mobility
• Initially random grain orientation and grain boundary type distributions
White: θ = 1.5
Gray: θ = 3Red: θ = 37,5
Black: θ > 3, θ ≠ 37.5
1 2 60, , ..., ( , ), ..., 1.5
ir tη η η η θ⇒ ∆ = °�
2D simulation
<0 0 1>
18
35
Simulations: 1 high-angle energy cusp
• High-angle grain boundaries form independent network
• Low-angle grain boundaries follow movement of high-angle grain boundaries → elongate
• No stable quadruple junctions
White: θ = 1.5θ = 1.5θ = 1.5θ = 1.5
Gray: θ = 3θ = 3θ = 3θ = 3
Black: θ > 3, θθ > 3, θθ > 3, θθ > 3, θ ≠ 37.5
Red: θθθθ = 37,5
36
Misorientation distribution function
(MDF)
• Area weigthed MDF
• Reaches a stead-state
• Low energy boundaries lengthen + their number increases
Read-Shockley + cusp at θ = 37.5°
19
37
Grain growth kinetics
Read-Shockley (θθθθm=15°) + cusp at θθθθ=37.5°
• Grain growth exponent
• PFM: steady-state growth with
• Previous findings:
• Mean field analysis:
1n ≈
0.6...1n =
0
effn
effA A k t− =
,(0)
1eff
h
nkt
tA kt
= → ∞+
→
1 '/f
f
ef
efdA
d
kk
t N N=
+=
N. Moelans, F. Spaepen, P. Wollants,
Phil. Mag., 90 p 501-523 (2010)
38
3D simulations for wires with fiber
texture
6ϑ < °
20
39
Summary
• Understanding microstructure evolution is important for material science, but processes can be very complex
• Phase-field technique + other models and experimental input
• Principles phase-field modeling
• Diffuse interface concept, conserved and non-conserved field variables
• Evolution equations derived following thermodynamic principles
• Quantitative aspects: ‘thin interface’ approach, parameter choice, specialized implementation techniques
• Applications
• Grain growth, recrystallization, coarsening, solidification, growth intermetallic phases, …
40
Thank you for your attention !
Questions ?
• Acknowledgements
• Research Foundation - Flanders (FWO-Vlaanderen)
• Flemish Supercomputing Center (VSC)
1
New directions in phase-field modeling of
microstructure evolution in polycrystalline and
multi-component alloys
Nele Moelans
Department of Materials Engineering, KU Leuven, Belgium
2
Current state of phase-field modeling
• Modeling of microstructure evolution• Meso- & micro scale (nm - mm)
• Based on thermodynamic principles
• Growing field since 20 years; current topics• Quantitative aspects
• Realistic, complex, multi component systems
• Wide field of applications• Solidification, phase transitions, diffusion, grain growth,
deformation, crack formation, ferro-electric/magnetic domains, ….
Bulk grain structures
Thin filmsSynergetic growth of nanowires
2
3
‘Quantitative’ phase-field models
• Properties bulk and interfaces are reproduced accurately in the simulations
• Effect model description and parameters
• Numerical issues
• => Insights in the evolution of complex morphologies and grain assemblies
• Effect of individual bulk and interface properties
• Predictive ?
• Depends on availability and accuracy of input data
– Requires composition and orientation dependence
4
3
5
Parameter assessment
• Different kinds of input data • Bulk phase stabilities, bulk phase diagram information
– CALPHAD
• Interfacial energy and mobility
• Elastic properties, crystal structure, lattice parameters
• Diffusion mobilities/coefficients
– DICTRA mobilities
• Orientation and composition dependence• Anisotropy, segregation, solute drag
• Very important for microstructure evolution, but difficult to measure/calculate
6
Outline
• CALPHAD (CALculation of PHAse Diagrams) – method
• Quantitative phase-field model for multi-phase systems
• Coupling phase-field with thermodynamic databases, example for Ag-Cu-Sn
• Application examples
• Diffusion controlled growth in Cu/Cu-Sn solder joints
4
CALPHAD (CALculation of PHAse Diagrams)
– method
•Computational thermodynamics: The Calphad Method, H.L. Lukas, S.G. Fries and B. Sundman, Cambridge 2007, ISBN 978-0-521-86811-2
•Thermo-Calc : www.thermocalc.se
•Calphad : www.calphad.org
8
What ?
• Phase diagrams and thermodynamic Gibbs energies are combined
• Mission: develop a technique for calculating/predicting phase diagrams for multi-component multi-phase materials
Phase diagram Free energies
5
9
What ?
10
What ?
• STEP 1: Determine G(T,p,x) expressions for all phases
• Choice of a thermodynamic model
• Determination of the parameters based on experimental and theoretical (a.o. ab-initio) input,
– Such as phase equilibria, heat of formation, vapor pressures, ….
• STEP 2: Minimization of the total Gibbs energy of a multi-phase system
• → Phase diagrams
• → Thermodynamic quantities: chemical potential, heat of transformation, reaction, …
• → Input for diffusion and microstructure simulations
6
11
Pyramidal approach
• E.g. Gibbs energies for ternary system
• Pure elements, Al, Mg, Si
• Binary systems, Al-Mg, Al-Si, Mg-Si
• Ternary system, Al-Mg-Si
12
Pure elements
• Choice of a standard element reference state (SER) for each element (reference structure, temperature and pressure)
• Temperature dependence Gibbs energy
– a,b,c,d,… fitted to experiments, ab-initio data
• Also models for magnetic ordering, effect of pressure
2 31ln( )G HSER a bT cT T dT e fT
T− = + + + + +
2
2
12 2 6 ...pC c dT e fT
T⇒ = − − − − +
7
13
Binary substitutional alloys
• General expression
• Mechanical mixing of pure elements
• Ideal mixing (Raoults solution)
• Redlich-Kister expression for excess interactions
– T-dependenc of LABv
– LABv from experiments
ref id ex physG G G G G− = + +
,
( )ex
v v
A B AB A BG x x L x x
α
= −
2 31ln( )v
ABL a bT cT T dT e fTT
= + + + + +
ln( ) ln( )id
A A B BG x x x x= +
, 0, 0,A B
ref
A BG x G x Gα α α= +
14
Regular solution model
, 0,ex LL
A B ABG x x L=
8
15
Illustration Redlich-Kister
contributions
,
( )ex v v
A B AB A BG x x L x x
α
= −T cte=
16
Illustration Redlich-Kister
contributions
,
( )ex v v
A B AB A BG x x L x x
α
= −
T cte=
9
17
Extrapolation to higher-order systems
18
Extrapolation models
• Extrapolation schemes
• Redlich-Kister-Muggianu
0 1 2( )
ex ex ex ex
AB AC BC A B C A ABC B ABC C ABCG G G G x x x x L x L x L= + + + + +
10
19
Ternary substitutional alloys
• General expression
• Mechanical mixing of pure elements
• Ideal mixing (Raoults solution)
• Excess interactions, Redlich-Kister-Muggianu
– T-dependence
– GexAB from binary optimizations
– LABCv from experiments
0 1 2( )ex ex ex ex
AB AC BC A B C A ABC B ABC C ABCG G G G x x x x L x L x L= + + + + +
ref id ex physG G G G G− = + +
ln( ) ln( ) ln( )id
A A B B C CG x x x x x x= + +
, 0, 0, 0,A B
ref
A B C CG x G x G x Gα α α α= + +
v
ABCL a bT= +
20
Sublattice models
• Phases with ordering• Mixing only allowed within certain sublattices,
represented as (A,B)a1(A,Va)a2
– Intermetallic compound, FeTi →(%Fe,Ti)(Fe,%Ti), Fe2Ti →(%Fe,Ti)2(Fe,%Ti), B2Ti →(B)2(Ti)
– Interstitial phases, (Cr,Fe,Ni,…)a1(C,N,H,Va,…)a2
• Molar fraction → lattice fractions 1yA, 2yA, 1yB, 2yVa
11
Quantitative and thermodynamically
consistent phase-field model for multi-phase
systems
22
Multi-grain and multi-phase structures
• Single phase-field models -> Multiple phase-field models
• Model extension
– Different types of interfaces
– Triple and higher order junctions
• Numerically
– Same accuracy for all interfaces and phases
– All interfaces within range of validity of the thin interface asymptotics
{ }1 2 3, , ,...,
pη η η η η→
2 2
1 2 3 1 2( , , ,...,| | , | | ,...)F η η η η η∇ ∇
num cte→ =ℓ
1 2( , ,..., , ..., ) (0,0,...,1,...,0)i pη η η η =
Grain i Grain j
1iη =
0j
η = 0i
η =
1jη =
12
23
Multi-grain and multi-phase models:
major difficulties
• Third-phase contributions
• σσσσ12 = σσσσ13 = 7/10 σσσσ12
• → Careful choice of multi-well function
and gradient contribution
• Interpolation function
• Zero-slope at equilibrium values of the
phase fields
• Thermodynamic consistency
3
1 2
ηηηη3
1 2 1 2
1 1
( , ,...) ( , ) ( , ,...) 1p p
i
chem i i
i i
f h f c T hη η η η= =
= ⇒ =∑ ∑
24
Multi-grain and multi-phase models
• Multi-phase-field model
• Phase fields
• Free energy
– Double obstacle, higher order terms, gradient term
non-variational
– Interpolation: zero-slope or thermodynamic
consistency
• Multi-order parameter models
• Order parameters
• Interfacial energy
1 2
1
, ,..., ( , ),..., , 1p
i p i
i
r tη η η η η=
≠
∑
�
1 2 3
1
, , ,... , 1p
p i
i
ϕ ϕ ϕ ϕ ϕ=
=∑
,
int
,
,
2
2
4| |
i
i j i j
i j
j i j
j
i
f φ φ φφπ
σ η
η≠
= ∇ ⋅ ∇ +
∑
Steinbach et al.
MICRESS phase-field code
H. Garcke, B.Nestler, B. Stoth, SIAM J. Appl. Math. 60 (1999) p 295.
,0 1
i jφ< <
( )4 22 2 2
int
1 1 1
,
1( )
4 2 4 2
p p p p
i ii j i
i i j i i
i jmf
η ηη η
κγ η
η
= = < =
= − + + + ∇
∑ ∑∑ ∑
�
L.-Q. Chen and W. Yang,
PRB, 50 (1994) p15752
A. Kazaryan et al., PRB,
61 (2000) p14275
13
25
Multi-grain and multi-phase models
• Vector valued model• Orientation field ( ) and phase field ( )
• Free energy
• 2-phase solidification• Phase fields
• Fifth order interpolation functions gi(φφφφ1,φφφφ2,φφφφ3)
– Zero-slope and thermodynamic consistent
– Order gi increases with number of phase-fields
• Multi-order parameter + 4th order gradient terms
• Phase field crystal and amplitude equations
int ( , | |, | |)f f φ φ θ= ∇ ∇
φ R. Kobayashi, J.A. Warren, W.C. Carter, Physica D, 119 (1998) p415
3
1 2 3
1
, , , 1i
i
ϕ ϕ ϕ ϕ=
=∑
θ
R. Folch and M. Plapp, PRE, 72 (2005) n° 011602
I.M. McKenna, M.P. Gururajan, P.W. Voorhees, J. Mater. Sci., 44 (2009)p2206
26
Extension to multi-component multi-
phase alloys
• Phase field variables:
• Grains
• Composition
• Bulk energy:
• with
( , ) ( )bulk k i k
f c f cρ ρ
ρ ρρ
η φ=∑
1, ( , ),...,A B C
c c r t c −
�
1 2
1 2
, ,..., ( , ),...,
, ,...
i
p
r tα α α
β β
η η η
η η η
�
k kx xρ
ρρ
φ=∑
2
2
, ,...
i
i
i
i
ρ
ρ
ππ α β
ηφ
η=
=∑
∑ ∑and
( )( )...
k
k
kk
k
f cf c
c cα
α αβ β
β µ∂∂
= = =∂ ∂
ɶ
N. Moelans, Acta Mater. 2011.
14
27
Extension to multi-component multi-
phase alloys
• Bulk and interface diffusion:
With and
• Interface movement:
• Between phase αααα and ββββ
( , )i i k
i
F xL
t
ρ ρ
ρ
η δ η
δη
∂= −
∂
2
2k
k
m
k
DM
f
x
ρρ
ρ=∂
∂
( )( )
2
int 22 2
2( , ) ( ) ( ) ( )
i ji L g f c f c c ct
α β α α β β α βα
α β
η ηηη η µ
η η
∂ = − ∇ + − − − ∂ +
Curvature driven Bulk energy driven
2 2
, ,
, ,
kk i j k
i j
xM
t
ρρ ρ σ
ρ ρ σ
φ η η µ≠
∂= ∇⋅ + ∇
∂ ∑ ∑
223
/
interf gb
interf mnumk
DM
f x
δ
δ
= ∂ ∂
28
Numerical validation for multi-
component multi-phase model
• Processes for which v(t)
• Conclusions for grain growth model remain
– Accuracy controlled by
– Diffuse interface effects for
– Angles outside [100°-140°] require larger resolution for
same accuracy
/num
x∆ℓ
/ 5num
R >ℓ
/num
x∆ℓ
•Growing sphere
•Coarsening
•Triple junction
•Intermediate phase
N. Moelans, Acta Mater. 2011.
15
Coupling phase-field with thermodynamics
databases
30
Phase diagram
• Ternary Ag-Cu-Sn, T = 180 °C
x(A
G)
x(SN)
0.0
0.2
0.3
0.5
0.7
0.9
0.0 0.2 0.4 0.6 0.8 1.00 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
x(SN)
x(A
G)
CU
AG
SN
AG
SB
_O
RT
HO
+C
U6S
N5_P
+B
CT_
A5
CU
3S
N+C
U6S
N5_P
+A
GS
B_O
RT
HO
CU
3S
N+FC
C_A
1+FC
C_A
1
FC
C_A
1+FC
C_A
1
(Cu)
Cu6Sn5Cu3Sn
Ag3Sn
(Sn)
16
31
Phase diagram
• Binary Cu-Sn
Annealing temperature:
180 °C
EutecticComposition:
Sn-2at%Cu
Cu Sn
32
Parabolic composition dependence
• Bulk energy
• Parabolic composition dependence
• Simplifies solution phase-field
equations
• No composition dependence
needed for D
• Difficult for higher order
systems and large
composition variations
( )2
,02
Sn Sn
Af x x C
ρρ ρ= − +
( )( , ) ( ) m Sn
bulk i Sn Sn
m
G xf x f x
V
ρ ρρ ρ
ρ ρρ ρ
η φ φ= =∑ ∑
2
2
Sn
D DM
f A
x
ρ ρρ
ρ ρ= =∂
∂
S.Y. Hu, J. Murray, H. Weiland, Z.-K. Liu, L.-Q. Chen, Comp. Coupl. Phase Diagr. Thermoch., 31 (2007) p 303
17
33
Coupling with thermodynamic
database
• Free energies over full composition range are needed in phase-field
• Stoichiometric phases
• Approximated as
• Choice of A requires ‘trial
and error’
– Numerics – Small shift in
phase equilibria
– Here: A = 1e6
( )2
,0
,0
2Sn Sn
stoich
Sn stoich
AG x x C
C G
x x
ρρ ρ
ρ
= − +
=
=
Cu-Sn
Gibbs energies from A. Dinsdale, A. Watson, A. Kroupa, J. Vřešt'ál, A. Zemanová, J. Vízdal COST 531-Lead Free Solders: Atlas of Phase Diagrams for Lead-Free Soldering, vols. 1,2 (2008) ESC-Cost office
34
Coupling with thermodynamic
database
3 3
2
( ' , ' )
( )2
Ag Sn Ag Sn
Ag Sn
Sh Cu
G G x x
Ax x
=
+ −
• Binary phases shifted into ternary, e.g.
With
1 6
0.001Sh
A e
x
=
=
' (1 )
' (1 )
Ag Sh Ag
Sn Sh Sn
x x x
x x x
= −
= −
Ag-Cu-Sn
x(A
G)
x(SN)
0.0
0.2
0.3
0.5
0.7
0.9
0.0 0.2 0.4 0.6 0.8 1.00 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
x(SN)
x(A
G)
CU
AG
SN
Ag3Sn + Cu6Sn5 + (Sn)
Ag
3S
n +
Cu3
Sn +
Cu6S
n5
FC
C_A
1 +
Cu3S
n +
FC
C_A
1
180 degree C
Cu6Sn5Cu3Sn
Ag3Sn
(Sn)
18
35
Coupling with thermodynamic
database
Ag-Sn-0.001Cu Cu-Sn-0.001Ag
•Sublattice model + parabolic extension
•Cu6Sn5-H: (Cu).545:(Cu,Sn).122:(Sn).333
•Ag3Sn: (Ag).75(Ag,Sn).25
36
Coupling with thermodynamic
database
Ag-Sn-0.001Cu Cu-Sn-0.001Ag
•Sublattice model + parabolic extension
•Cu6Sn5-H: (Cu).545:(Cu,Sn).122:(Sn).333
•Ag3Sn: (Ag).75(Ag,Sn).25
19
37
Coupling with thermodynamic
database
Ag-Sn-0.001Cu Cu-Sn-0.001Ag
•Sublattice model + parabolic extension
•Cu6Sn5-H: (Cu).545:(Cu,Sn).122:(Sn).333
•Ag3Sn: (Ag).75(Ag,Sn).25
→ Gρρρρ(xAg,xCu,xSn) over full composition range
Interdiffusion and intermetallic growth in
Cu/Sn-Cu solder joints
20
39
System properties
( ) 25
3 16 2
6 5 15 2
( ) 12 2
10
5 10 m /s
10 m /s
10 m /s
Cu
Sn
Cu Sn
Sn
Cu Sn
Sn
Sn
Sn
D
D
D
D
−
−
−
−
=
= ⋅
=
=
Annealing temperature:
180 °C
Eutectic Composition: Sn-2at%Cu
Cu Sn
20.25 J/m
gbγ =
•Equilibrium compositions • Interdiffusion coeffcients
• Interfacial energy
40
IMC-layer growth (1D)
• Effect of bulk diffusion coefficients
( ) 25 2
3 14 2
6 5 14 2
( ) 12 2
10 m /s
10 m /s
10 m /s
10 m /s
Cu
Sn
Cu Sn
Sn
Cu Sn
Sn
Sn
Sn
D
D
D
D
−
−
−
−
=
=
=
=
6
3
6
6 5
0.0073 10
0.023 10
Cu Sn
Cu Sn
k
k
−
−
⇒ = ⋅
= ⋅
( ) ( )
,0 ,
3
,0
6 5
,0
( ) ( )
,0 ,
0.01( )
0.25
0.455
0.99( )
Cu Cu
Sn Sn eq
Cu Sn
Sn
Cu Sn
Sn
Sn Sn
Sn Sn eq
x x
x
x
x x
= <
=
=
= <
( )t s
( )h m
Diffusion coeffcients Initial composition
21
41
IMC-layer growth (1D)
( ) 25 2
3 13 2
6 5 13 2
( ) 12 2
10 m /s
10 m /
10 m /s
10 m /s
Cu
Sn
Cu Sn
Sn
Cu Sn
Sn
Sn
Sn
D
D s
D
D
−
−
−
−
=
=
=
=
( ) 25 2
3 13 2
6 5 13 2
( ) 14 2
10 m /s
10 m /
10 m /s
10 m /s
Cu
Sn
Cu Sn
Sn
Cu Sn
Sn
Sn
Sn
D
D s
D
D
−
−
−
−
=
=
=
=
6
3
6
6 5
0.0301 10
0.0833 10
Cu Sn
Cu Sn
k
k
−
−
⇒ = ⋅
= ⋅
6
3
6
6 5
0.0306 10
0.0849 10
Cu Sn
Cu Sn
k
k
−
−
⇒ = ⋅
= ⋅
( ) 12 210 m /s
Sn
SnD
−=
( )t s
( )h m
42
Comparison with experimental data
T k_Cu3Sn k_Cu6Sn5
150 °C 0.0010 10-6 0.00032 10-6
180 °C 0.0032 10-6 0.0059 10-6
200 °C 0.0043 10-6 0.0071 10-6
( ) 25 2
3 15 2
6 5 15 2
( ) 12 2
10 m /s
10 m /s
10 m /s
10 m /s
Cu
Sn
Cu Sn
Sn
Cu Sn
Sn
Sn
Sn
D
D
D
D
−
−
−
−
=
=
=
=
Parabolic growth constant experiments
Cu3Sn, T = 180 °C
Parabolic growth constant in
simulations with6
3
6
6 5
0.0023 10
0.0073 10
Cu Sn
Cu Sn
k
k
−
−
⇒ = ⋅
= ⋅
Cu6Sn5, T = 180 °C
Data from J. Janckzak, EMPA
22
43
Effect of grain boundary diffusion
SnJ →�
Snx
grains
( ) 25 2
3 15 2
6 5 15 2
( ) 12 2
9 2
10 m /s
10 m /s
10 m /s
10 m /s
D 0.66 10 m /s
Cu
Sn
Cu Sn
Sn
Cu Sn
Sn
Sn
Sn
surf
Sn
D
D
D
D
−
−
−
−
−
=
=
=
=
= ⋅ SnJ ↑�
1gb nmδ =
• 2D simulations
44
Effect of grain boundary diffusion
• 3D simulations
( ) 25 2
3 15 2
6 5 15 2
( ) 12 2
9 2
2 10 m /s
2 10 m /s
2 10 m /s
2 10 m /s
D 2 10 m /s
Cu
Sn
Cu Sn
Sn
Cu Sn
Sn
Sn
Sn
interf
D
D
D
D
−
−
−
−
−
= ⋅
= ⋅
= ⋅
= ⋅
= ⋅
SnJ ↑�
1gb
nmδ =
23
45
Growth behavior Cu3Sn ?
•
( ) 25 2
3 15 2
6 5 15 2
( ) 12 2
12 2
2 10 m /s
2 10 m /s
2 10 m /s
2 10 m /s
D 2 10 m /s
Cu
Sn
Cu Sn
Sn
Cu Sn
Sn
Sn
Sn
surf
Sn
D
D
D
D
−
−
−
−
−
= ⋅
= ⋅
= ⋅
= ⋅
= ⋅
Grain structure Composition: xSn
46
IMC growth in Sn-2at%Cu
•Initial compositions
• System size: 0.1µµµµmx0.5µµµµm
• Initially fV = 0.04
• For future work: compare with experiments for nanoparticle-reinforced solders for which the grain size is known (J. Janczak, EMPA)
( ) 25 2
6 5 16 2
( ) 12 2
10 m /s
10 m /s
10 m /s
Cu
Sn
Cu Sn
Sn
Sn
Sn
D
D
D
−
−
−
=
=
=
24
47
Conclusions
• The phase-field implementation is able to reproduce thermodynamic and kinetic properties accurately
• Important to have realistic free energy functions and atomic diffusion mobilities for microstructure simulations
• But how to treat stoichiometric phases, binary phases ?
• Most general solution: Gibbs free energy over full composition range
• Powerful tool for the study of interdiffusion in joint materials
48
Thank you for your attention !
Questions ?
• Acknowledgements
• Research Foundation - Flanders (FWO-Vlaanderen)
• Flemish Supercomputing Center (VSC)
• OT/07/040
• CREA/12/012
25
Application examples
• Coarsening of Al6Mn precipitates located on a recrystallization front in Al-Mn alloys
In collaboration with A. Miroux, E. Anselmino, S. van der Zwaag, T. U. Delft
50
Jerky motion during recrystallization in
Al-Mn alloy
• In-situ EBSD observation of recrystallization in AA3103 at 400 °C
• CamScan X500 Crystal Probe
FEGSEM
• Jerky grain boundary motion
• Stopping time: 15-25 s
• Pinning by second-phase
precipitates
– Al6(Fe,Mn), αααα-Al12(Fe,Mn)3Si
• Added to phase field model
• Grain boundary diffusion
• Driving force for recrystallization20µµµµm
26
51
Phase field model
• Multiple order parameter representation:
• Mn composition field:
• Homogeneous driving pressure for recrystallization: md
• Bulk diffusion + Surface diffusion
,1 ,2 ,( , ), ( , ),..., ( , ),...m m p ir t r t r tη η η� � �
( , )Mn
x r t�
,1 ,2 ,( , ,..., ,...) (1,0,...,0,...), (0,1,...,0,...),...(0,0,...,1,...),...m m p i
η η η =
52
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
(Miroux et al.,Mater. Sci. Forum,467-470,393(2004))
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 D0,bulk = 10-2 m2/s, Qbulk = 211 kJ/mol
→Dbulk = 5.5973·10-18 m2/s
Pipe diffusion high angle boundaries, precipitate/matrix interface
D0,p = D0,bulk, Qp = 0.65Qbulk
→Dp = 1.2195·10-12 m2/s
Bulk energy density: fρ = Aρ(x-xρ0)
2 Am = 6·1011; xm0 = 0.000258
Ap = 6·1012; xp0 = 0.1429
27
53
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
0.9µm x 0.375µm
2 2
x yJ J J= +
6 s
7·10-10
54
Precipitate coarsening and unpinning
• PD <<< PZS
• Pinning: PZS = 3.6 MPa
• Rex: PD = 1.1 MPa
0.9µm x 0.375µm
• Unpinning through grain boundary diffusion
2 2
x yJ J J= +
8 s
1.7·10-10
