DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING Concurrently Modeling Multiple Length Scales by Coupling the Phase-Field Method to Spatially-Resolved Cluster Dynamics Dong-Uk Kim, Ali Muntaha, Michael R Tonks, University of Florida Sophie Blondel, Brian Wirth, University of Tennessee-Knoxville David Bernholdt, Phillip Roth, Oak Ridge National Laboratory David Andersson, Los Alamos National Laboratory Work is funded by Joint NE/SciDAC Project on the simulation of fission gas 

DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING Phase field methods can efficiently model small features or large features but not both in the same simulation 

DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING n Nucleation and growth n Grain boundary segregation n Grain growth with small precipitates There are various examples of where microstructural features of very different sizes impact material behavior Dendritic growth of precipitates in Ni- based super alloys Henry, M. F., Yoo, Y. S., Yoon, D. Y., & Choi, J. (1993). Metallurgical Transactions A, 24(8), 1733- 1743. 

DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING Fission gas release in irradiated fuel occurs due to materials processes that occur across various length and time scales Kashibe, S., K. Une, and Kazuhiro Nogita. JNM 206, no. 1 (1993): 22-34. Rest, J., et al. JNM 513 (2019): 310-345. 

DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING n Initial bubble radius: 44 nm n Interfacial width: 30 nm n Initial bubble radius: 1.5 nm n Interfacial width: 0.7 nm Intergranular bubble analysis Phase field simulations of fission gas bubbles currently target specific length scale of interest Intragranular bubble analysis L.K. Aagesen, D. Schwen, M.R. Tonks, Y. Zhang, Computational Materials Science. 161 (2019) 35–45. M.R. Tonks, et al., In proceedings M&C 2013, May 5 – 9, 2013, Sun Valley ID 20×20×20 nm 1.2×1.04×0.45 *m 

DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING We have developed a hybrid model that uses cluster dynamics to model small bubbles and phase field for large Cluster Dynamics (Xolotl) Phase Field (MOOSE) Both codes are open source 

DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING + = n !"# !$ = ̇ '() + +) ,-.) − 0 .) , where n = 1, 2, …, >1000 n +) = 0 for n > 1 n Reactions are represented by 0 .) n 0 .) = 3) .) .4 − 3 )54 . )54 .4 + 3) 678$.) − 3 )94 678$ . )94 + 3) :6;<.) − 3 )94 :6;< . )94 n Include n Clustering: Xe1 + Xen → Xen+1 n Emission and resolution: Xen → Xe1 + Xen-1 The cluster dynamics model uses a series of coupled reaction diffusion equations to model Xe atom clusters + = … 

DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING n Solves the coupled equations using implicit finite difference n Nonlinear solves are carried out using PETSc n Scales well to 10,000’s of processors n GPU acceleration is underway Xolotl is an open source spatially-resolved cluster dynamics code developed at the University of Tennessee-Knoxville Limitations: • Cannot represent large bubbles due to high computational expense • Cannot represent interfacial motion such as grain boundary migration 

DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING n Predicts the growth and coalescence of fission gas bubbles and grain boundary migration n Grand potential model (Plapp) with multiphase field grain growth model n Xe atom and U vacancy chemical potentials (2 DOF). n Bubble order parameter (1 DOF). n Grain order parameters (1 – 20 DOF, reused for multiple grains). n The model uses 4 – 22 DOF per node. We are building our work on the most recent fission gas model in MOOSE/MARMOT (developed by Larry Aagesen) Aagesen et al. Computational Materials Science 161 (2019): 35-45. 

DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING The phase field fission gas model in MOOSE/MARMOT predicts bubble and grain boundary evolution 33 days 77 days 145 days Limitations: • Cannot represent bubbles that are much smaller than the grain size • Cannot directly represent bubble nucleation • Cannot directly represent re-solution 

DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING Xolotl describes all behavior in the grains while MARMOT models grain boundary bubbles and migration Cluster Dynamics (Xolotl) • Xe production • Intragranular diffusion • Clustering • Resolution Phase Field (MARMOT) • Intergranular bubble growth and coalescence • Grain boundary migration Gas atom arrival rate at Interfaces Interface locations 

DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING Xolotl and MARMOT are coupled using the Multiapps and Transfers systems in the MOOSE framework MARMOT mesh MOOSE-Xolotl wrapper mesh Xolotl grid Interpolation transfer Direct copy Microstructure Gas source Interface coordinates Gas source 

DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING The coupled code provides the capability to represent small intragranular bubbles and large intergranular bubbles 20 µm ̇ & = 1.09×10,- fissions/(m3s) . = 1800 K 

DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING We verified that the codes are coupled correctly by comparing a hybrid model simulation to stand-alone MARMOT 0 200 400 600 800 1000 1200 1400 Time (days) 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 Intergranular bubble fraction Stand-alone MARMOT Coupled (No Clu. & No Re-s.) ̇ " = 1.09×10)* fissions/(m3s) + = 1800 K Initial microstructure Stand-alone MARMOT Coupled 

DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING Clu. & Re-s. 4 8 16 32 64 128 # of processors 0 1000 2000 3000 4000 5000 6000 Computation time (s) Data transfer MARMOT Xolotl Clu. & No Re-s. 4 8 16 32 64 128 # of processors 0 1000 2000 3000 4000 5000 6000 Computation time (s) Data transfer MARMOT Xolotl The computational cost of Xolotl and MARMOT are similar and cost of the data transfer is negligible No Clu. & No Re-s. 4 8 16 32 64 128 # of processors 0 1000 2000 3000 4000 5000 6000 Computation time (s) Data transfer MARMOT Xolotl 

DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING The parallel scalability of the hybrid model is excellent 20 40 60 80 100 120 # of processors 10 20 30 40 50 60 Speedup Ideal Stand-alone MARMOT Coupled (No Clu. & No Re-s.) Coupled (Clu. & No Re-s.) Coupled (Clu. & Re-s.) 4 16 36 64 100 # of processors 0 0.5 1 1.5 2 Speedup Ideal Stand-alone MARMOT Coupled (No Clu. & No Re-s.) Coupled (Clu. & No Re-s.) Coupled (Clu. & Re-s.) Strong Scaling Weak Scaling 

DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING We will demonstrate the capabilities of the hybrid model by investigating three aspects of fission gas behavior Intragranular physics Grain size Temperature 200 400 600 800 1000 1200 1400 Time (days) 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Intergranular bubble fraction No clustering & no re-solution Clustering & no re-solution Clustering & re-solution Effective diffusion 200 400 600 800 1000 1200 1400 Time (days) 0.0145 0.015 0.0155 0.016 0.0165 0.017 Intergranular bubble fraction @1800K, no re-solution @1800K, re-solution @1000K, no re-solution @1000K, re-solution 200 400 600 800 1000 1200 1400 Time (days) 0.014 0.015 0.016 0.017 0.018 0.019 0.02 Intergranular bubble fraction 10 grains 20 grains 30 grains 

DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING The intragranular physics included in Xolotl has a large impact on the overall fission gas behavior 20 µm 1800 K ̇ & = 1.09×10,- fissions/(m3s) Note: Initial intergranular bubbles are used to avoid the need to model nucleation in MARMOT 

DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING The intragranular physics included in Xolotl has a large impact on the overall fission gas behavior 20 µm 1800 K ̇ & = 1.09×10,- fissions/(m3s) Note: Initial intergranular bubbles are used to avoid the need to model nucleation in MARMOT 

DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING Clustering results in as much as 70% of the fission gas being trapped within the grains 200 400 600 800 1000 1200 1400 Time (days) 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Intergranular bubble fraction No clustering & no re-solution Clustering & no re-solution Clustering & re-solution Effective diffusion 200 400 600 800 1000 1200 1400 Time (days) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Xe fraction in grains No clustering & no re-solution Clustering & no re-solution Clustering & re-solution 1800 K 

DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING Temperature results in much less gas arriving at grain boundaries and less grain growth 20 µm ̇ & = 1.09×10,- fissions/(m3s) 

DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING Temperature results in much less gas arriving at grain boundaries and less grain growth 1000 K, No re-solution 1000 K, Re-solution 1800 K, No re-solution 1800 K, Re-solution 20 µm ̇ & = 1.09×10,- fissions/(m3s) 

DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING More gas is trapped in the grains at lower temperature and the intergranular bubbles grow much slower 200 400 600 800 1000 1200 1400 Time (days) 0.0145 0.015 0.0155 0.016 0.0165 0.017 Intergranular bubble fraction @1800K, no re-solution @1800K, re-solution @1000K, no re-solution @1000K, re-solution 200 400 600 800 1000 1200 1400 Time (days) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Xe fraction in grains @1800K, no re-solution @1800K, re-solution @1000K, no re-solution @1000K, re-solution 

DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING More grain growth occurs with more grains, resulting in more sweeping up of gas by migrating grain boundaries 20 µm 1800 K, Clustering, Re-solution ̇ & = 1.09×10,- fissions/(m3s) 

DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING More grain growth occurs with more grains, resulting in more sweeping up of gas by migrating grain boundaries 20 µm 1800 K, Clustering, Re-solution ̇ & = 1.09×10,- fissions/(m3s) 

DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING Decreasing grain size accelerates bubble growth, even though we do not include fast grain boundary diffusion 200 400 600 800 1000 1200 1400 Time (days) 0.014 0.015 0.016 0.017 0.018 0.019 0.02 Intergranular bubble fraction 10 grains 20 grains 30 grains 200 400 600 800 1000 1200 1400 Time (days) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Xe fraction in grains 10 grains 20 grains 30 grains 1800 K, Clustering, Re-solution 

DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING We are currently working on 3D simulations with our hybrid model Microstructure Intragranular fission gas 

DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING Direct coupling between phase field and cluster dynamics could be used to model other phenomena Cluster Dynamics • Diffusion and atom clustering • Nucleation and growth of small precipitates Phase Field • Large precipitate evolution • Interface evolution • Grain boundary evolution 

DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING Our hybrid model provides a means of modeling the impact of small scale precipitates on large scale evolution 

DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING