Particle nucleation and coarsening in aluminum...

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Delft Institute of Applied MathematicsDelft University of Technology

Particle nucleation and coarseningin aluminum alloysA literature study

D. den Ouden

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Contents

This presentation will be about:

• Alloys• Diffusion• Particle nucleation and coarsening• Elastic deformations• Conclusions• Future work

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Alloys

Alloys can be characterized by• the solvent metal, such as

• Aluminum• Lead• Iron

• the number of elements• by the apparent number of elements

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By apparent number

Binary Ternary Quaternary Complex

Quasi-Binary

Quasi-Binary

Quasi-Binary Quasi-Ternary

Quasi-Ternary

Quasi-Quaternary

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Diffusion

• Interstitial

⇒

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Diffusion

• Interstitial• Substitutional

⇒

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Nucleation of particles

Directly influenced by

• Mean concentration C̄

• Equilibrium concentration Ce

• Temperature T

• Activation energy for diffusion Qd

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Nucleation of particles

Directly influenced by

• Mean concentration C̄

• Equilibrium concentration Ce

• Temperature T

• Activation energy for diffusion Qd

Formula:

j(t) = j0 exp

(

−

(

A0

RT

)3(

1

ln(C̄/Ce)

)2)

exp

(

−Qd

RT

)

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Growth of particles

Directly influenced by

• Mean concentration C̄

• Interface concentration Ci

• Internal concentration Cp

• Diffusion coefficient D

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Growth of particles

Directly influenced by

• Mean concentration C̄

• Interface concentration Ci

• Internal concentration Cp

• Diffusion coefficient D

Formula:

v(r, t) =C̄ − Ci

Cp − Ci

D

r

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Critical particles

Particles with no growth:

v(r∗, t) = 0

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Critical particles

Particles with no growth:

v(r∗, t) = 0

or equivalent:

r∗(t) =2σVm

RT

(

ln

(

C̄

ce

))−1

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Number of particles

Directly influenced by• Growth rate v

• Nucleation rate j

• Critical radius r∗

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Number of particles

Directly influenced by• Growth rate v

• Nucleation rate j

• Critical radius r∗

Continuity equation:

∂N

∂t= −

∂Nv

∂r+ S

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Source term

Influenced by• Critical radius r∗

• Nucleation rate j

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Source term

Influenced by• Critical radius r∗

• Nucleation rate j

Definition

S(r, t) =

{

j(t) if r = r∗ + ∆r∗,0 otherwise.

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Relations

Define φ by

φ(r, t) =N(r, t)

∆r

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Relations

Define φ by

φ(r, t) =N(r, t)

∆r

Then the particle volume fraction equals

f(t) =

∫

∞

0

4

3πr3φ(r, t) dr

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Relations

Define φ by

φ(r, t) =N(r, t)

∆r

Then the particle volume fraction equals

f(t) =

∫

∞

0

4

3πr3φ(r, t) dr

and the mean concentration

C̄(t) =C0 − Cpf(t)

1 − f(t)

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Simulation

Material: Aluminum alloy AA 6082• 0.9 wt% Silicon• 0.6 wt% Magnesium

Temperature: 180◦ C

Initial condition: No particles

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Distribution function φ

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Derived quantities

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Derived quantities

100

102

104

106

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Particle volume fraction

part

icle

vol

ume

frac

tion

(%)

time (s)

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Derived quantities

100

102

104

106

0

10

20

30

40

50

60

70

80

90Mean radius vs critical radius

time (s)

radi

us (

Å)

Mean radiusCriticial radius

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Derived quantities

100

102

104

106

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7Mean concentration vs equilibrium concentration

time (s)

conc

entr

atio

n (w

t%)

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Derived quantities

100

102

104

106

10−120

10−100

10−80

10−60

10−40

10−20

100

1020

Nucleation rate

time (s)

nucl

eatio

n ra

te (

#/m

3 s)

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Time integration

Three methods:• θ-method

(

I − θ∆t

∆rAn

)

~Nn+1 =

(

I + (1 − θ)∆t

∆rAn

)

~Nn + ∆t~Sn

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Time integration

Three methods:• θ-method• First DIRK-method

~Nn+1 = ~Nn +∆t

2An(

~Nn1 + ~Nn2

)

+ ∆t~Sn

~Nn1 = ~Nn + ∆tγ(

An ~Nn1 + ~Sn)

~Nn2 = ~Nn + ∆tAn(

(1 − 2γ) ~Nn1 + γ ~Nn2

)

+ ∆t(1 − γ)~Sn

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Time integration

Three methods:• θ-method• First DIRK-method• Second DIRK-method

~Nn+1 = ~Nn + ∆tAn(

b1~Nn + b2

~Nn2 + γ ~Nn3

)

+ ∆t~Sn

~Nn2 = ~Nn + ∆tγAn(

~Nn + ~Nn2

)

+ 2∆tγ ~Sn

~Nn3 = ~Nn + ∆tAn(

b1~Nn + b2

~Nn2 + γ ~Nn3

)

+ ∆t~Sn

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Time integration

OperationMethod Vector addition Matrix addition Matrix multiplication Matrix inversion

θ-method 1 2 1 1DIRK-1 6 2 2 2DIRK-2 9 2 3 2

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Time integration

Results:• θ-method large differences

Second order for θ = 1/2

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Time integration

Results:• θ-method large differences

Second order for θ = 1/2

• DIRK-methods small differencesSecond order for all γ

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Time integration

Results:• θ-method large differences

Second order for θ = 1/2

• DIRK-methods small differencesSecond order for all γ

• θ = 1/2- and DIRK-methods small differences

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Time integration

Results:• θ-method large differences

Second order for θ = 1/2

• DIRK-methods small differencesSecond order for all γ

• θ = 1/2- and DIRK-methods small differences

Conclusion:θ = 1/2-method favorable

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Recap

Results:• Model correctly predicts nucleation and

coarsening

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Recap

Results:• Model correctly predicts nucleation and

coarsening• Second order accuracy at low costs

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Elastic deformation

Strain in a deformed block:

εij =1

2

(

∂ui

∂xj

+∂uj

∂xi

)

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Elastic deformation

Strain in a deformed block:

εij =1

2

(

∂ui

∂xj

+∂uj

∂xi

)

Stress in a deformed block:

σij = λδij

3∑

k=1

εkk + 2µεij

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Elastic deformation

Force balance:

∇ · σ = −b

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Elastic deformation

Force balance:

∇ · σ = −b

Or equivalent:

λ∂

∂x1

(∇ · u) + µ

(

∇ ·

(

∇u1 +∂u

∂x1

))

= −b1

λ∂

∂x2

(∇ · u) + µ

(

∇ ·

(

∇u2 +∂u

∂x2

))

= −b2

λ∂

∂x3

(∇ · u) + µ

(

∇ ·

(

∇u3 +∂u

∂x3

))

= −b3

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Finite Element Mesh

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Simulation

Before deformation:

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Simulation

After deformation:

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Simulation

Displacements:

−0.5

0

0.5

−0.5

0

0.5−0.5

0

0.5

x1

x2

x 3

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Simulation

Update:

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Conclusions

Conclusions:• Realistic results with both models• High accuracy• Easy to derive data

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Future work

• Coupling of the two models

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Future work

• Coupling of the two models• Non-elastic deformations

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Future work

• Coupling of the two models• Non-elastic deformations• Coupling with elastic deformations

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Future work

• Coupling of the two models• Non-elastic deformations• Coupling with elastic deformations• Coupling of the three models

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Future work

• Coupling of the two models• Non-elastic deformations• Coupling with elastic deformations• Coupling of the three models• Extension to multi-component alloys

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Questions?