Potential Edits to the Nucleation Benchmark

In this presentation, I suggested some changes to the nucleation benchmark as it is written in PF-Hub (https://pages.nist.gov/pfhub/benchmarks/benchmark8.ipynb/).

The work that we did to develop these suggestions was published in a paper:
https://doi.org/10.1007/s40192-022-00284-1
The paper is also available on arxiv: https://arxiv.org/abs/2210.16866

A brief version of just the potential edits described in the powerpoint presentation are included below:

-Use periodic boundary conditions for parts 2 and 3
-Change critical radius (r*) in part 3 to r* = 1.0
-Calculate the avrami slope using a fitting range in the y-axis (log[-log{1-Y}]) in parts 2 and 3, the range -2 < log[-log{1-Y}] < 0 worked well for us
-Potentially increase the amount of simulations to increase statistical accuracy in parts 2 and 3 (this must be weighted against longer simulation times)

The code is available on GitHub: https://github.com/joseam2/PF_Nucleation_Benchmark_Julia

This paper can be cited as such: Mancias, J., Attari, V., Arróyave, R. et al. On the Effect of Nucleation Undercooling on Phase Transformation Kinetics. Integr Mater Manuf Innov (2022). https://doi.org/10.1007/s40192-022-00284-1

Transcript

1. Potential Edits to the Nucleation Benchmark José Mancias CHIMAD XIV October 2022 Dr. Raymundo Arroyave1, Dr. Vahid Attari1, Dr. Damien Tourret2 1 2
2. Avrami (JMAK) Theory • Johnson-Mehl-Avrami-Kolmogorov Theory (JMAK) • Exact solution for first order phase transitions under certain strong assumptions • Random nucleation events with constant rates of nucleation and growth in an infinite domain • n is referred to as the Avrami exponent • Found from the slope of log(-log(1-Y)) vs log(t) • n = d + 0 (site saturation) n = d + 1 (continuous nucleation)
3. Nucleation Benchmark Formulation • Formulation • Free Energy: • Time evolution of Φ given by Allen-Cahn Equation: • Initial Condition: • Driving force for solidification: https://pages.nist.gov/pfhub/benchmarks/benchmark8.ipynb/ 𝜖2 = 1,𝜔 = 1, 𝑀 = 1 • Double Well • Interpolation Function • Definition of Constants
4. Formulation Part 1 • Initial Conditions: • System size = 100x100 • r* = 5.0 • Run three different initial nucleus radii • 𝑟0 = 0.99𝑟∗, 1.00𝑟∗, 1.01𝑟∗ • Time evolution of the system: 100 units • Goals: • Plot the volume fraction of the solid • Plot the total free energy • Both as a function of time • Plot convergence test Wu et al. Phase Field Benchmark Problems for Nucleation, CMS 2021
5. Part 1 Initial Volume Discrepancy • Initial Volume Fraction r0 = rc • Our & PF Hub Solution: 0.0083 • Wu et al. Solution: ~0.0065 • Theoretical (Circle in Square): 0.00785 •Unresolved Wu et al. Phase Field Benchmark Problems for Nucleation, CMS 2021
6. Nucleation Benchmark Pt. 2 and Pt. 3 • Initial Conditions • Shared Goals: • Plot the 2d representation of phi • Plot the volume fraction of the solid • Plot the total free energy • Plot the Avrami Plot Wu et al. Phase Field Benchmark Problems for Nucleation, CMS 2021 Part Domain Size r* r0 Seeds Part 2 500 x 500 1.0 2.2 25 Part 3 1000 x 1000 2.0 2.2 100 • Part 2 - Site Saturation • All seeds placed at t = 0 • Seeds are randomly located • Expected Avrami constant: n = 2 • Part 3 – Continuous Nucleation • Seeds are randomly placed in location and time • Expected Avrami constant: n = 3
7. Edit #1 – Boundary Conditions • CMS Paper described using Neumann BCs for parts 2 & 3 • PF Hub Website does not mention BCs in parts 2 & 3 • Suggestion: Periodic BCs fit the assumptions of Avrami theory better • Infinite domain, spatiotemporal homogeneity of nucleation and growth • Simulations have been performed showing how BCs affect the Avrami constant • With this change part 2 yields ~2.0 Avrami constant (as expected)
8. Part 2 Boundary Condition Study N = 300 runs Neumann n = 1.91 Periodic n = 2.03
9. Edit #2 – Effect of incubation time • Parts 2 & 3 in CMS paper and PF Hub website • This ratio affects the rate of growth and Avrami Constant • Changing this ratio to 2.2 in part 3 yields ~3.0 values for Avrami constant • Suggestion: Change r* in part 3 to r* = 1.0 Part Initial Radius (r0 ) Critical Radius (r*) r0 / r* Part 2 2.2 1.0 2.2 Part 3 2.2 2.0 1.1
10. Edit #2 – Effect of incubation time How r*/r0 affects each part of the benchmark: Mancias et al. On the Effect of Nucleation Undercooling on Phase Transformation Kinetics, IMMI 2022 (Accepted) Part 1 Part 2 Part 3
11. Edit #3 – Effect of fitting range • In CMS paper finding the Avrami constant uses a method of linear fitting in a range of log(t) • Part 3: 1.5 < log(t) < 2.5 • Suggested approach: • Linear fit in range of log(-log(1-Y)) • -2 < log(-log(1-Y) < 0 • Corresponds to 0.02 < Y < 0.9 • In cases of late first seed nucleation in pt. 3, 1st approach yields no value of slope • 2nd approach yields a consistent fitting range in part 2 and part 3 Wu et al. Phase Field Benchmark Problems for Nucleation, CMS 2021
12. Edit #4 – Statistical Approach • Parts 2 & 3 involve stochasticity • Large numbers of simulations provides greater statistical accuracy • N = 300 simulations carried out, each with a different random number seed • Enabled in-part using GPU parallelization • Downside is more computationally demanding problems • N = 5 obtains 1.97 ≤ n ≤ 2.03 77% of the time for pt. 2
13. Conclusion and Perspectives • Initial volume discrepancy in part 1 • Recommended edits to the benchmark • Boundary condition -> Periodic in parts 2 and 3 • r* -> 1.0 in part 3 •r0 /r* ratio and transient growth • Post-processing using limits in the log(-log(1-Y)) axis • Use of statistics • Perspectives • Quantifying incubation time and the time cone analysis More in: Mancias et al. On the Effect of Nucleation Undercooling on Phase Transformation Kinetics, IMMI 2022 (Accepted)