Modelling of Kinetics in Multi- Component, Multi-Phase, Multi- Particle Systems: Application E....

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Modelling of Kinetics in Multi-Component, Multi-Phase, Multi-Particle Systems: Application

E. KozeschnikJ. SvobodaF.D. Fischer

Institute for Materials Science, Welding and Forming, Graz University of TechnologyMaterials Center Leoben, Austria

Academy of Sciences, Brno, Czech RepublicInstitute of Metal Physics, University of Mining, Leoben , Austria

Erich Schmid Institute of Materials Science, Austrian Academy of Sciences , AustriaInstitute of Mechanics, University of Mining, Leoben , Austria

IWS, Graz University of Technology, Austria / Materials Center Leoben E. Kozeschnik, 2005-09-02

Contents

• Model formulation

• Computer Implementation

• Algorithm flow-chart

• Application to– Nucleation, growth and coarsening of

cementite in steel– TTP Diagram for gamma_prime in Ni-base – Complex experimental tool steel


The modeling team (2001-2006) ...

• J. Svoboda– Academy of Sciences, Czech Republic, CZ

• F.D. Fischer– Institute of Mechanics, University of Leoben, A

• E. KozeschnikB. Sonderegger (2004-)– Institute for Materials Science, Welding and Forming, Graz

University of Technology, A

Task: Model development and implementation for precipitation kinetics in multi-component, multi-phase, multi-particle systems


Idea …

• System with spherical precipitates of different size, composition and phase type in multi-component matrix.

• Evolution equations from Onsager thermodynamic extremal principle: System develops with constrained maximum Gibbs Free Energy dissipation.


Model formulation: Growth …

• Gibbs Free Energy

• Maximum Gibbs Free Energy Dissipation with constraint

∑∑ ∑ ∑== = =

γπρ+⎟⎠

⎞⎜⎝

⎛μ+λ

πρ+μ=

m

1k

2k

n

1i

m

1kki

n

1ikik

3k

i0i0 4c3

4NG

kk y

Q

y

G&∂∂

−=∂∂

21

kikk cry ,=


Gibbs Free Energy dissipation …

1. Diffusion through matrix

2. Interface movement

3. Diffusion in precipitates

∑=

=m

k k

kk

MQ

1

22

2

4 ρπρ &

€

Q3 =RT

ckiDki0

ρ k

∫i=1

n

∑k=1

m

∑ 4πr2 jki2 dr =

4πRTρ k5 ˙ c ki

2

45ckiDkii=1

n

∑k=1

m

∑

∑∑∑∑∫= == =

+−≈=

m

k

n

i ii

kikikikkki

m

k

n

i

Z

ii Dc

cccRTdrJr

Dc

RTQ

k1 1 00

20

322

1 1 001

)3/)((44

&& ρρρππ

ρ


Results: Growth

• Linear system of equations in , and :r&c&

∑++

=

=pn

jijij ByA

1

1

kjkikk vcry ,, &&=

kjv


Multi-component nucleation

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

kT

GexpNZJ

*

c*

sΔ

β ⎟⎠⎞

⎜⎝⎛−=

texpJJ s

τ

2

1

*2

2

2

1

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂Δ∂−

=ir

GkT

Zπ

⎟⎟⎠

⎞⎜⎜⎝

⎛ −== ∑

=

m

i iMi

Mi

Pi

**

Dc

)cc(rA/r

1

2

2*

4 Ωπβ

22

1

Z*βτ =


Thermo-Kinetic software: MatCalc

• Equilibrium (CALPHAD)

• Diffusion (MOBILITY)

• Phase trans-formations

E. Kozeschnik, B. Buchmayr, “MatCalc – A simulation tool for multicomponent thermodynamics, diffusion and phase transformation kinetics”, in: ‘Mathematical

Modelling of Weld Phenomena 5’, Institute of Materials, London, Book 734, 2001;349.


Software implementation

Overall composition

Matrix phase Precipitate 1

Precipitate 2

Precipitate 3

…

Microstructure - f(t,T)• dislocation density• grain size• sub-grain size …

Precipitate props • , λk , Mintf

• nucleation site(s)• …

+N R XC XCr XFe …

1e12 4e-9 0.25 0.36 0.12

2e13 5e-9 0.25 0.38 0.11

8e13 6e-9 0.25 0.39 0.09

… … …

bulkdislocations

grain boundariessub-grain boundaries

other particles


Calculation: flow-chart

Pre-Proc.: Initialize and set up parameters

for

all p

reci

pita

tes

Nucleation? Add precipitate class

Growth Evaluate

Dissolution? Remove prec. classnext

tim

e st

ep

Post-Proc.: Evaluate results

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Example I

Nucleation – Growth - Coarsening


Start MatCalc ...

Live demo ...

• Cementite precipitation in Fe-0.1%C• 100 precipitate classes• Automatic interfacial energy

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Example II

TTP-diagram


γ’-precipitation in Ni-base alloy

• Ni-13at%Al

• 200 classes

• =17 mJ/m2

• Cooling rates: 0,01 – 1000 °/s


400

500

600

700

800

900

1e0 1e1 1e2 1e3 1e4

time [s]

γ’-precipitation in Ni-base alloy

0.1% 1%10% 25%

50%75%

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Example III

Complex systems ...


A Comprehensive Treatment of Precipitation Kinetics

in Complex Materials

B. Sonderegger1,6,M. Bischof2, E. Kozeschnik1 H. Leitner2, H. Clemens2, J. Svoboda4, F.D. Fischer3,5

1: Institute for Materials Science, Welding and Forming, Graz, University of Technology, Austria2: Dept. of Physical Metallurgy and Materials Testing, Montanuniversität Leoben, Austria3: Institute of Mechanics, Montanuniversität Leoben, Austria4: Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Brno, Czech Republic5: Erich Schmid Institute of Materials Science, Austrian Academy of Sciences, Leoben, Austria6: Materials Center Leoben, Leoben, Austria

Presentation given at „Solid-solid Phase Transformations in Inorganic Materials“, Phoenix, AZ, USA, 2005


Outline

Introduction

Experimental

Numerical Results

Conclusion!!


Outline Complex material

• Experimental Results

Improved Understanding of Precipitation Kinetics

• Numerical Simulations


Introduction

Precipitation Hardening in Steels

Carbides,Nitrides Intermetallic Phases(e.g maraging steels)

Testmelt


Testmelt

C Cr Mo V Ni Al Co Si Mn Fe

1.4 2.6 1.4 0.3 6.0 5.0 1.8 0.4 0.2 bal

Composition (at%)

Carbides (MC, M2C, M3C, M6C, M23C6)

Intermetallic Phases (NiAl, B2 ordering)


Experimental Investigations

1 10 1001E-5

1E-4

1E-3

0,01

0,1 A D G I

f / 1/nm

R / nm

Casting, Austenitising, HTUp to 10000min

APFIM

SANS

TEM

M. Bischof et al.: „An advanced approach to the characterisation of precipitates in steels“, 4:45pm, Room Pueblo/Sonora


Experimental - Numerical

1 10 1001E-5

1E-4

1E-3

0,01

0,1 A D G I

f / 1/nm

R / nm

Numerical Simulation:

APFIM

SANS

TEM

www.matcalc.tugraz.at

“MatCalc—a simulation tool for multicomponent thermodynamics, diffusion and phase transformation kinetics.”Kozeschnik E, Buchmayr B., Mathematical mod. of weld phenomena 5. London Institute of Materials; 2001. p. 349– 61


Simulation Starting Conditions

• Database: extended TCFE3+Mobility

• Chemical Composition (10 Elements)

• Phases: MC, M2C, M3C, M6C, M23C6, NiAl

• Matrix: Grain Size, Subgrain Size etc. (Number of Nucleation sites)

• Interfacial Energies • Chemical driving forces• Chemical potentials

• Exact Heat Treatment conditions from casting to annealing (610°C, up to 10000min (167h))

Calculated from thermodyn. Databases


Equilibrium Analysis

Precipitates after

austenitising

(990°C)

Precipitates after HT

(10000min)

(610°C)

MX MX

M6C M6C

M2C? M2C

M23C6

NiAl

No M3C

Decrease of G(M6C): G=G0-2600 [J/mol]


Calculation with improved database

Improved Database

M6C: 1,5mol%, d=580nm

MX: 0,2mol%, d=60 nm

M2C: very few primary

G(M6C)=G0-2600 [J/mol]


Identification of Precipitatescast + aust HT

NiAl

MX

M2C

Cementite

M23C6

M6C

1 10 100 10001E-6

1E-5

1E-4

1E-3

0,01

0,1

1 10 100 10001E-6

1E-5

1E-4

1E-3

0,01

0,1

f / 1/cm*Sr

R / nm

f1: 8.37+/- 0.263%f2: 1.75 +/- 0.283%f3: 1.27 +/- 0.474%

: too small

SANS (HT 10000min)


M23C6: • nucleation too fast• r stays too small• f growing too fast

Variation of G?

Correction of γ!

Increase of γ

Lower Nucleation Rate

Slower increase of f

Faster increase of r

Matrix Parameters?

cast + aust HTIdentification of Precipitates


γ(M23C6)

All numerical results agree with experimental findings

(within statistical errors)

HT


Summary and Conclusions

Simulated full heat treatment of a very complex system (10 Elements, 6 phases)

Correct Equlibrium Calculations

Very good results of kinetic simulation

Fit of 2 parameters were sufficient to meet ~ 20-30 single measurement points

Experiments get easier to interpret

Simulation results get improved

Further development of thermodynamic databases

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http://matcalc.tugraz.at

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Modelling of Kinetics in Multi- Component, Multi-Phase, Multi- Particle Systems: Application E....

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