Edits to Nucleation Benchmark

This presentation describes the edits made the nucleation benchmark problem on the PF Hub website made by José Mancias in spring 2023.

Transcript

1. Edits to the Nucleation Benchmark Can be found on PF Hub Website José Mancias CHIMAD XV August 2023 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. 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
5. 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 • EDIT: 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)
6. Part 2 Boundary Condition Study N = 300 runs Neumann n = 1.91 Periodic n = 2.03
7. Edit #2 – Effect of incubation time • Parts 2 & 3 in CMS paper and PF Hub website • r 0 /r* affects the rate of growth and Avrami Constant • Changing r0 /r* to 2.2 in part 3 yields ~3.0 values for Avrami constant • Edit: 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
8. Edit #2 – Effect of incubation time How r0/r* affects each part of the benchmark: Mancias et al. On the Effect of Nucleation Undercooling on Phase Transformation Kinetics, IMMI 2022 Part 1 Part 2 Part 3
9. 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 • New 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
10. Edit #4 – Statistical Approach • Parts 2 & 3 involve stochasticity • Large numbers of simulations provides greater statistical accuracy • Changed the number of simulations suggested for parts 2 and 3 • 5 simulations each to 10 simulations each • Different random positions of seeds and times • N = 5 obtains 1.97 ≤ n ≤ 2.03 77% of the time for pt. 2 • N = 10 obtains 1.97 ≤ n ≤ 2.03 96% of the time for pt. 2
11. Summary and Further Info • 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 • Completed edits can be viewed at: https://pages.nist.gov/pfhub/benchmarks/benchmark8.ipynb/ • More in: Mancias, José, et al. "On the Effect of Nucleation Undercooling on Phase Transformation Kinetics." Integrating Materials and Manufacturing Innovation 11.4 (2022) • Corresponding Github Repo: https://github.com/joseam2/PF_Nucleation_Benchmark_Julia