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Delft Institute of Applied MathematicsDelft University of Technology
Mathematical modeling of particlenucleation and growth in metallicalloys
Dennis den Ouden
14-12-2009
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Contents
• Something about metallurgy• A model for nucleation• A model for deformations• Combining the models• Results• Conclusion
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Something about alloys
Binary Ternary Quaternary Complex
Quasi-Binary
Quasi-Binary
Quasi-Binary Quasi-Ternary
Quasi-Ternary
Quasi-Quaternary
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Something about alloys
Binary Ternary Quaternary Complex
Quasi-Binary
Quasi-Binary
Quasi-Binary Quasi-Ternary
Quasi-Ternary
Quasi-Quaternary
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Something about deformations
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Something about deformations
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Models for nucleation
Two models, with small differences• Myhr and Grong (2000)• Robson et al. (2003)
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Models for nucleation
Two models, with small differences• Myhr and Grong (2000)• Robson et al. (2003)
Comparison:• Basic model == Myhr and Grong (2000)• Adapted using Robson et al. (2003)
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Governing DE
Changein time
=Change
by growth+
Productionrate
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Governing DE
Changein time
=Change
by growth+
Productionrate
⇓ ⇓ ⇓
∂N
∂t= −
∂ (Nv)
∂r+ S
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Unknowns
• Growth rate v
• Production term S
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Growth of spherical particles
Directly influence by
• Mean concentration C
• Interface concentrationCi
• Internal concentrationCp
• Diffusion coefficient D
Cp C
Ci
Matrix
Particle
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Growth of spherical particles
v(r, t) =C − Ci
Cp − Ci
D
r Cp C
Ci
Matrix
Particle
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A special particle
A particle with radius r∗ that will neither grow ordissolve:
v(r∗) = 0
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A special particle
A particle with radius r∗ that will neither grow ordissolve:
v(r∗) = 0
Solved for r∗:
r∗ =2γαβVm
RT
(
ln
(
C
Ce
))−1
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A special particle
A particle with radius r∗ that will neither grow ordissolve:
v(r∗) = 0
Solved for r∗:
r∗ =2γαβVm
RT
(
ln
(
C
Ce
))−1
Definition: Critical particle radius
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Production term
• Indicates the number of particles thatnucleate over the whole domain
• Influenced by critical radius r∗
• Influenced by nucleation rate j
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Production term
• Indicates the number of particles thatnucleate over the whole domain
• Influenced by critical radius r∗
• Influenced by nucleation rate j
Kampmann et al. (1987):
S(r, t) =
{
j(t) if r = r∗ + ∆r∗,0 otherwise.
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Nucleation rate
The number of particles that nucleate with radiusr∗ + ∆r∗:
• Influenced by diffusion• Only if some barrier has been overcome
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Nucleation rate
The number of particles that nucleate with radiusr∗ + ∆r∗:
• Influenced by diffusion• Only if some barrier has been overcome
j = j0 exp
(
−∆G∗
het
RT
)
exp
(
−Qd
RT
)
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Nucleation energy barrier
• Chemical composition• Misfit strain energy
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Nucleation energy barrier
• Chemical composition• Misfit strain energy
∆G∗het =
A30
(∆Gv + ∆Gm
s)2
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Model overview
• Governing DE:
∂N
∂t= −
∂ (Nv)
∂r+ S
• Source term:
S(r, t) =
{
j(t) if r = r∗ + ∆r∗,0 otherwise.
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Model overview
• Governing DE:
∂N
∂t= −
∂ (Nv)
∂r+ S
• Growth rate:
v =C − Ci
Cp − Ci
D
r
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Elastic deformations
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Elastic deformations
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Assumptions
Rotation symmetry:• No deformations in tangential direction• No deformation at center axis in radial
direction• All derivatives in tangential direction vanish
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Assumptions
Rotation symmetry:• No deformations in tangential direction• No deformation at center axis in radial
direction• All derivatives in tangential direction vanish
uθ = 0 uη(0, θ, z) = 0∂(.)
∂θ= 0
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Strain and deformation
Chau and Wei (2000):
εηη =∂uη
∂ηεθθ =
uη
η
εzz =∂uz
∂zεηθ = 0
εηz =1
2
(
∂uη
∂z+
∂uz
∂η
)
εθz = 0
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Stress and strain
Hook’s Law:
σαβ = δαβλ (εηη + εθθ + εzz) + 2µεαβ
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Stress and strain
Hook’s Law:
σαβ = δαβ λ (εηη + εθθ + εzz) + 2 µ εαβ
Stiffness matrix Shear modulus
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Force balance
Jaeger et al. (2007):
∂σηη
∂η+
∂σηz
∂z+
σηη − σθθ
η+ bη = 0
∂σηz
∂η+
∂σzz
∂z+
σηz
η+ bz = 0
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Boundary conditions
• Symmetry condition:
uη(0, θ, z) = 0
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Boundary conditions
• Symmetry condition:
uη(0, θ, z) = 0
• Fixed boundaries:
uα(η, θ, z) = 0
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Boundary conditions
• Symmetry condition:
uη(0, θ, z) = 0
• Fixed boundaries:
uα(η, θ, z) = 0
• Moving boundaries:(
σ(η, θ, z))
α· n = fα(η, θ, z)
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Coupling the models
Remember the nucleation energy barrier:
∆G∗het =
A30
(
∆Gv + ∆Gms
)2
Misfit strain energy
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Coupling the models
Remember the nucleation energy barrier:
∆G∗het =
A30
(
∆Gv + ∆Gms
)2
Misfit strain energy
Question:
Is there something like elastic strain energy?
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Coupling the models (2)
Answer:YES!!!
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Coupling the models (2)
Answer:YES!!!
Solution:
∆Gels =
1
2σ : ε
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Coupling the models (2)
Answer:YES!!!
Solution:
∆Gels =
1
2σ : ε
and:
∆G∗het =
A30
(
∆Gv + ∆Gms +∆Gel
s
)2
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Coupling the models (3)
Question:Is there also a reverse coupling?
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Coupling the models (3)
Question:Is there also a reverse coupling?
Answer:YES !!!
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Coupling the models (3)
Question:Is there also a reverse coupling?
Solution by Pal (2005):
µ = µm +
(
15(1 − νm)(µp − µm)
2µp(4 − 5νm) + µm(7 − 5νm)
)
µm f
E = Em + (10β1(1 + νm) + β2(1 − 2νm)) Em f
λ = µE − 2µ
3µ − E
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Recap
Two models:• Nucleation model• Elastic model
Two couplings:• From elastic to nucleation• From nucleation to elastic
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Numerical methods
Nucleation model:• Upwind scheme
• IMEX-θ method with θ = 1
2
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Numerical methods
Nucleation model:• Upwind scheme
• IMEX-θ method with θ = 1
2
(
I −1
2
∆t
∆rAn
)
~Nn+1 =
(
I +1
2
∆t
∆rAn
)
~Nn + ∆t~Sn
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Numerical methods (2)
Elastic model:• Finite Element Method• Linear elements• Use of rotation symmetry
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Numerical methods (2)
0 0.5 1 1.5 2 2.5 3
x 10−3
0
0.005
0.01
0.015
0.02
0.025
0.03z
(m)
η (m)
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Numerical methods (2)
Equation:[
Sηη Sηz
Szη Szz
][
uη
uz
]
=
[
qη
qz
]
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Numerical methods
Algorithm:
1. Set all constants;
2. Set all initial values;
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Numerical methods
Algorithm:
1. Set all constants;
2. Set all initial values;
3. For each time step:(a) Calculate elastic parameters;(b) Build matrices for elastic deformation;(c) Calculate elastic deformations;(d) Calculate elastic strain energy;
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Numerical methods
Algorithm:. . .
3. For each time step:. . .
(e) For each point:i. Calculate nucleation parameters;ii. Calculate matrices for nucleation;iii. Calculate nucleation.
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Simulation
Material:• Aluminum alloy AA 6082• Mg2Si particles
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Simulation
Material:• Aluminum alloy AA 6082• Mg2Si particles
Shape:• Cylindrical• Height 30 millimeter• Radius 3 millimeter
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Simulation (2)
Time• Total of 3000 seconds• Time step of 0.5 seconds
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Simulation (2)
Time• Total of 3000 seconds• Time step of 0.5 seconds
Test:• Tensile test• Bottom axial and radial fixed• Top radial fixed
• Axial force at top of 6 million N/m2
• Sides free
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Typical deformations: Axial
0 0.5 1 1.5 2 2.5 3
x 10−3
0
0.005
0.01
0.015
0.02
0.025
0.03
0
1
2
3
4
5
6
7
8x 10
−6Deformation uz (m)z
(m)
η (m)
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Typical deformations: Radial
0 0.5 1 1.5 2 2.5 3
x 10−3
0
0.005
0.01
0.015
0.02
0.025
0.03
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3x 10
−7
z(m
)
η (m)
Deformation uη (m)
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Typical deformations: Energy
0 0.5 1 1.5 2 2.5 3
x 10−3
0
0.005
0.01
0.015
0.02
0.025
0.03
2
4
6
8
10
12
14x 10
4Elastic strain energy (N/m2)z
(m)
η (m)
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Nucleation results: Nucleation rate
0
1
2
3
x 10−3
0
0.01
0.02
0.0310
0
101
102
103
0.5
1
1.5
2
2.5
3
x 10−7Particle nucleation rate
time
(s)
Per
cent
age
(%)
z (m) η (m)
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Nucleation results: Number density
0
1
2
3
x 10−3
0
0.01
0.02
0.0310
0
101
102
103
0.5
1
1.5
2
2.5
3
x 10−7Particle number density
time
(s)
Per
cent
age
(%)
z (m) η (m)
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Nucleation results: Concentration
0
1
2
3
x 10−3
0
0.01
0.02
0.0310
0
101
102
103
−2
−1.8
−1.6
−1.4
−1.2
−1
−0.8
−0.6
−0.4
−0.2
0x 10
−7Mean concentration
time
(s)
Per
cent
age
(%)
z (m) η (m)
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Reflection
Are the results anomalies during simulation?
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Reflection
Are the results anomalies during simulation?
Increase force to test for similar behavior.
F = 6 × 109N
m2
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Reflection
Are the results anomalies during simulation?
Increase force to test for similar behavior.
F = 6 × 109N
m2
Physically no longer elasticity
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New elastic stain energy
0 0.5 1 1.5 2 2.5 3
x 10−3
0
0.005
0.01
0.015
0.02
0.025
0.03
1
2
3
4
5
6
7
8
9
10
11
12x 10
10Elastic strain energy (N/m2)z
(m)
η (m)
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Nucleation results: Number density
0
1
2
3
x 10−3
0
0.01
0.02
0.0310
0
101
102
103
0.005
0.01
0.015
0.02
0.025
0.03
0.035
Particle number density
time
(s)
Per
cent
age
(%)
z (m) η (m)
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Conclusions
• Two separate nucleation models combined
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Conclusions
• Two separate nucleation models combined• Formulated model for elastic deformations
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Conclusions
• Two separate nucleation models combined• Formulated model for elastic deformations• Coupling between nucleation and
deformations
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Conclusions
• Two separate nucleation models combined• Formulated model for elastic deformations• Coupling between nucleation and
deformations• Simulations show influence of deformations
on nucleation
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Recommended future work
• Extension to multiple particle configurations
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Recommended future work
• Extension to multiple particle configurations• Adaption to other alloys
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Recommended future work
• Extension to multiple particle configurations• Adaption to other alloys• Improving numerical techniques
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Recommended future work
• Extension to multiple particle configurations• Adaption to other alloys• Improving numerical techniques• Comparison with experimental data
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Recommended future work
• Extension to multiple particle configurations• Adaption to other alloys• Improving numerical techniques• Comparison with experimental data• Including plastic deformations
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Recommended future work
• Extension to multiple particle configurations• Adaption to other alloys• Improving numerical techniques• Comparison with experimental data• Including plastic deformations• Including homogeneous nucleation
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Recommended future work
• Extension to multiple particle configurations• Adaption to other alloys• Improving numerical techniques• Comparison with experimental data• Including plastic deformations• Including homogeneous nucleation• Including grain prediction models
References
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jected to arbitrary surface load. Part I – analytic solution.
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Jaeger, J.C. , Cook, N.G.W. , and Zimmerman, R.W. .
Fundamentals of rock mechanics. Wiley, Blackwell, 2007.
Kampmann, R. , Eckerlebe, H. , and Wagner, R. . In:
Materials Research Society Symposium Proceedings, vol-
ume 57, page 525. MRS, 1987.
Myhr, O. R. and Grong, Ø. . Modelling of non-isothermal trans-
formations in alloys containing a particle distribution. Acta
Materialia, 48(7):1605–1615, 2000.
Pal, R. . New models for effective Young’s modulus of partic-
ulate composites. Composites Part B: Engineering, 36(6-7):
513–523, 2005.
Robson, J.D. , Jones, M.J. , and Prangnell, P.B. . Extension of
the N-model to predict competing homogeneous and hetero-
geneous precipitation in Al-Sc alloys. Acta Materialia, 51(5):
1453–1468, 2003.
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