Derivation of TTT diagrams with kinetic Monte-Carlo simulations
A. Gupta, B. Dutta, T. Hickel, J. Neugebauer
Department of Computational Materials DesignDüsseldorf, Germany
15. September, 2016
BRITS Workshop, Dresden, 12.–15. September 2016
TTT-Diagram : Blueprint for heat treatment
Precipitation hardening
Heat Treatment
Solid solution annealing
Finer precipitates
Coarse precipitates
ResultingMicrostructure
Al-based Alloys
Final Product
Precipitation
Sketch: Fe-C TTT-diagramHeat
Treatment
Solid solution annealing
TTT-Diagrams
Precipitation Kinetics
kMC Methodology
3
While 𝑡 < 𝑡#$%
Update simulation time 𝑡 = 𝑡 + ∆𝑡
Execute and update configuration
Choose a jump k
Assign a transition rate r to each jump
List all the possible jumps
Starting configuration
)𝒓𝒏 ≤ 𝝆𝟏𝑹 ≤ )𝒓𝒏
𝒌
𝒏1𝟏
𝒌2𝟏
𝒏1𝟏
𝑹 = ) 𝒓𝒏
𝑵
𝒏1𝟏
∆𝒕 = −𝐥𝐧(𝝆𝟐)𝑹
𝒓𝐡𝐓𝐒𝐓 = 𝚪𝟎𝒆 2𝑬𝒃 𝒌𝑩𝑻⁄
𝑬𝒃 = 𝑬𝟎 + 𝒏𝑬𝐛𝐨𝐧𝐝Linear Bond-Cutting Model
Harmonic Transition State Theory
4
Precipitate formation (T = 400K, supercell 25x25x25, Ebond 0.1 eV)
t2
𝑡J ≈ 0.1µs
t3
𝑡Q ≈ 1µs
t4
𝑡R ≈ 3µs
𝑡T = 0
t1
How does this precipitation kinetics varies with temperature?
Size distribution of precipitates (Low T)
5
Simulation Parameters: 400 K, 25x25x25 supercellEbond = 0.1 eV, E0 = 0.3 eV
Initial stage of transformation
q Decrease in gas phase concentration
q Smearing towards larger particle sizes
Later stage of transformation
q Smaller precipitates dissolves andleads to the growth of large precipitate
q Ostwald ripening
Size distribution of precipitates (High T)
6
Simulation Parameters: 700-750 K, 25x25x25 supercellEbond = 0.1 eV, E0 = 0.3 eV
700 KØ Gas phase concentration remains constantØ No precipitate growth observed
750 K
How does this transition from low to high Tvaries?
Evolution of largest precipitate (nmax)
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Simulation Parameters: 25x25x25 supercellEbond = 0.1 eV, E0 = 0.3 eV
Temperature (K)
𝒏 𝒆𝒒(𝑻)
-10 -9 -8 -7 -6 -5 -4 -3 log(t) (seconds)
0
0.2
0.4
0.6
0.8
1
400 500 600 700 800
Sudden drop (T ≈ 700 K)
𝑻𝒄q As the temperature ↑, nmax ↓q The fluctuations increase with the temperature
Nucleation stage
8
Simulation Parameters:25x25x25 supercellEbond = 0.1 eV, E0 = 0.3 eV
q As T increases: - the duration of nucleation stage increases.- the critical nuclei size increases.
We know precipitate size à remaining gas phase
Evolution of the gas phase
ØWith increasing temperature, gas phase amount increase à Entropy dominates at high temperatures
ØFluctuations increase with increasing temperature
9
Simulation Parameters:25x25x25 supercellEbond = 0.1 eV, E0 = 0.3 eV
Advancement Factor X(t) : Degree of Transformation
10
Ø X(t) = 1 means end of transformation process
Ø Equilibrium value of 1 is reached Ɐ T.
Ø As T increases, total time for completion decreases àsignifies the speed up with increasing temperature.
• 𝒄𝒎 𝒕 = 𝟎 initial conc. of solute in the matrix
• 𝒄𝒎 𝒕 remaining solute conc. at timet
• 𝒄𝒎 𝒕 → ∞ equilibrium conc. of solute atoms
Gas Phase Concentration: Analytical vs Calculated
11
ØFinite size effect observed (real system CN = 6)
ØShaded area : no stable precipitates observed
Simulation Parameters:25x25x25 supercellEbond = 0.1 eV, E0 = 0.3 eV
=
Obtaining Time Scale : Statistics
12
ØProcess repeated for different T and Nc.
Simulation Parameters:25x25x25 supercellEbond = 0.1 eV, E0 = 0.3 eV
TTT Diagrams
Calculated TTT Diagrams
13
Simulation Parameters:25x25x25 supercellEbond = 0.1 eV, E0 = 0.3 eV
ØNo clusters observed at temperatures beyond 675 K.Ø Is there a relation between Nose T
and Tasym ?
Ebond = 0.1 eV
-8.5 -8 -7.5 -7 -6.5 -6 log(t) (seconds)
-10 -9.5 -9 -8.5 -8 -7.5 log(t) (seconds)
Ebond = 0.3 eV
-10 -9.5 -9 -8.5 -8 -7.5 log(t) (seconds)
Kin
etic
s
Driving force
Ebond = 0.2 eV
Relationship between nose temperature Tn and Tasymp
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Ø Ignoring the intercept (150), 𝑦 = 0.63𝑥 reflects the asymmetric nature of C-shape of the TTT diagrams around the nose.
Ø Entropy only dominates at high temperatures which might explain this asymmetry
Ø Linear relation is obtained between Tnand Tasymp
Conclusionsq Equilibrium size of the precipitate decreases with
temperature due to entropy domination
q Finite-size effects observed
q Upper TTT curve challenging-no stable precipitates
q Relationship b/w nose and asymptotic temperature
Outlookq Extension to realistic binding energies and Al-Sc
alloy
q Effect of strain on the precipitation kinetics within a combined EAM-cluster expansion-kMC formalism (in collaboration with Prof. Mark Asta, UC Berkeley)
THANK YOU FOR YOUR ATTENTION
q Financial support from DFG under SPP-1713 Priority Program “Strong coupling of thermo-chemical and thermo-mechanical states in applied materials” is highly acknowledged.
