Abstract
The diffusion mechanism of Cs in high energy grain boundaries (HEGBs) of silicon carbide (SiC) remains unsolved due to the lack of reliable and computationally efficient for long time-scale diffusion simulations interatomic potentials. Constructing machine learning interatomic potentials (MLIPs) for complex HEGB structures is challenging as their sizes exceed ab initio computational capacity. Therefore, we proposed a workflow to develop moment tensor potentials (MTPs), which facilitates the extraction of unknown regions from HEGBs and enables efficient active training. Our developed MTPs allowed us to perform simulations of Cs diffusion in amorphous SiC and HEGBs. Simulations show that Cs diffuses following a cage-breaking mechanism. Radial distribution functions, Voronoi analysis and electronic structure calculations further elucidate local atomic environments and weak interactions governing Cs mobility. This work provides a generalizable workflow to train MLIPs for complex structures and atomic insights for modeling fission product behaviors.
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Data availability
Files with parameterized MTP potentials, parameterized ACE potential, fine-tuned MACE-OMAT-0, training sets, and computational models can be obtained from the public repository: https://github.com/JiaxuanLi-THU/MTP-SiC-Cs.
References
Locatelli, G., Mancini, M. & Todeschini, N. Generation IV nuclear reactors: current status and future prospects. Energy Policy 61, 1503–1520 (2013).
Kania, M. J., Nabielek, H., Verfondern, K. & Allelein, H.-J. Testing of HTR UO2 TRISO fuels in AVR and in material test reactors. J. Nucl. Mater. 441, 545–562 (2013).
Demkowicz, P. A., Liu, B. & Hunn, J. D. Coated particle fuel: historical perspectives and current progress. J. Nucl. Mater. 515, 434–450 (2019).
Malherbe, J. B. Diffusion of fission products and radiation damage in SiC. J. Phys. D Appl. Phys. 46, 473001 (2013).
Simon, P.-C. A. et al. Multiscale, mechanistic modeling of cesium transport in silicon carbide for triso fuel performance prediction. npj Comput. Mater. 11, 238 (2025).
Chen, Y. & Schuh, C. A. Diffusion on grain boundary networks: percolation theory and effective medium approximations. Acta mater 54, 4709–4720 (2006).
Kirchhofer, R., Hunn, J. D., Demkowicz, P. A., Cole, J. I. & Gorman, B. P. Microstructure of triso coated particles from the AGR-1 experiment: SiC grain size and grain boundary character. J. Nucl. Mater. 432, 127–134 (2013).
Cancino-Trejo, F., López-Honorato, E., Walker, R. C. & Ferrer, R. S. Grain-boundary type and distribution in silicon carbide coatings and wafers. J. Nucl. Mater. 500, 176–183 (2018).
Rabone, J., Lopez-Honorato, E. & Van Uffelen, P. Silver and cesium diffusion dynamics at the β-SiC σ5 grain boundary investigated with density functional theory molecular dynamics and metadynamics. J. Phys. Chem. A 118, 915–926 (2014).
Ko, H., Szlufarska, I. & Morgan, D. Cs diffusion in SiC high-energy grain boundaries. J. Appl. Phys. 122, 105901 (2017).
Keblinski, P., Phillpot, S., Wolf, D. & Gleiter, H. Thermodynamic criterion for the stability of amorphous intergranular films in covalent materials. Phys. Rev. Lett. 77, 2965 (1996).
Szlufarska, I., Nakano, A. & Vashishta, P. A crossover in the mechanical response of nanocrystalline ceramics. Science 309, 911–914 (2005).
Rabone, J. & López-Honorato, E. Density functional theory metadynamics of silver, caesium and palladium diffusion at β-SiC grain boundaries. J. Nucl. Mater. 458, 56–63 (2015).
Ko, H., Deng, J., Szlufarska, I. & Morgan, D. Ag diffusion in SiC high-energy grain boundaries: Kinetic monte carlo study with first-principle calculations. Comput. Mater. Sci. 121, 248–257 (2016).
Chen, N. et al. Analytical bond-order potential for silver, palladium, ruthenium and iodine bulk diffusion in silicon carbide. J. Phys. Condens. Matter 32, 085702 (2019).
Wang, Q. et al. Diffusion and thermo-driven migration of silver, palladium, and ruthenium nanoparticles in cubic SiC matrix using molecular dynamics. Int. J. Heat. Mass Transf. 197, 123359 (2022).
Kubo, A. & Umeno, Y. Machine-learning-based atomistic model analysis on high-temperature compressive creep properties of amorphous silicon carbide. Materials 14, 1597 (2021).
Podryabinkin, E. V. & Shapeev, A. V. Active learning of linearly parametrized interatomic potentials. Comput. Mater. Sci. 140, 171–180 (2017).
Wood, B. M. et al. Uma: a family of universal models for atoms. Preprint at https://doi.org/10.48550/arXiv.2506.23971 (2025).
Batatia, I. et al. A foundation model for atomistic materials chemistry. J. Chem. Phys. 163, 184110 (2025).
Sosso, G. C., Deringer, V. L., Elliott, S. R. & Csányi, G. Understanding the thermal properties of amorphous solids using machine-learning-based interatomic potentials. Mol. Simul. 44, 866–880 (2018).
Deringer, V. L. & Csányi, G. Machine learning based interatomic potential for amorphous carbon. Phys. Rev. B 95, 094203 (2017).
Rosset, L. A., Drabold, D. A. & Deringer, V. L. Signatures of paracrystallinity in amorphous silicon from machine-learning-driven molecular dynamics. Nat. Commun. 16, 2360 (2025).
Zuo, Y. et al. Performance and cost assessment of machine learning interatomic potentials. J. Phys. Chem. A 124, 731–745 (2020).
Shapeev, A. V. Moment tensor potentials: A class of systematically improvable interatomic potentials. Multiscale Modeling Simul 14, 1153–1173 (2016).
Podryabinkin, E., Garifullin, K., Shapeev, A. & Novikov, I. Mlip-3: active learning on atomic environments with moment tensor potentials. J. Chem. Phys. 159, 084112 (2023).
Podryabinkin, E. V. et al. Nanohardness from first principles with active learning on atomic environments. J. Chem. Theory Comput. 18, 1109–1121 (2022).
Lysogorskiy, Y., Bochkarev, A., Mrovec, M. & Drautz, R. Active learning strategies for atomic cluster expansion models. Phys. Rev. Mater. 7, 043801 (2023).
Shuang, F. et al. Modeling extensive defects in metals through classical potential-guided sampling and automated configuration reconstruction. npj Comput. Mater. 11, 118 (2025).
Zhang, L., Csányi, G., Van Der Giessen, E. & Maresca, F. Atomistic fracture in bcc iron revealed by active learning of Gaussian approximation potential. npj Comput. Mater. 9, 217 (2023).
Hodapp, M. & Shapeev, A. In operando active learning of interatomic interaction during large-scale simulations. Mach. Learn. Sci. Technol. 1, 045005 (2020).
Mismetti, L. & Hodapp, M. Automated atomistic simulations of dissociated dislocations with ab initio accuracy. Phys. Rev. B 109, 094120 (2024).
Kong, L. et al. Overcoming the size limit of first principles molecular dynamics simulations with an in-distribution substructure embedding active learner. Preprint at https://doi.org/10.48550/arXiv.2311.08177 (2023).
Erhard, L. C., Rohrer, J., Albe, K. & Deringer, V. L. Modelling atomic and nanoscale structure in the silicon–oxygen system through active machine learning. Nat. Commun. 15, 1927 (2024).
Novikov, I. S., Gubaev, K., Podryabinkin, E. V. & Shapeev, A. V. The MLIP package: moment tensor potentials with MPI and active learning. Mach. Learn. Sci. Technol. 2, 025002 (2020).
Bartók, A. P., Kondor, R. & Csányi, G. On representing chemical environments. Phys. Rev. B-Condens. Matter Mater. Phys. 87, 184115 (2013).
Himanen, L. et al. Dscribe: library of descriptors for machine learning in materials science. Comput. Phys. Commun. 247, 106949 (2020).
Lim, J. et al. Molecular dynamics study of silicon carbide using an ab initio-based neural network potential: effect of composition and temperature on crystallization behavior. J. Phys. Chem. C. 127, 22692–22703 (2023).
Wang, Y., Fan, Z., Qian, P., Ala-Nissila, T. & Caro, M. A. Structure and pore size distribution in nanoporous carbon. Chem. Mater. 34, 617–628 (2022).
Drautz, R. Atomic cluster expansion for accurate and transferable interatomic potentials. Phys. Rev. B 99, 014104 (2019).
Batatia, I., Kovacs, D. P., Simm, G. N. C., Ortner, C. & Csanyi, G. MACE: higher order equivariant message passing neural networks for fast and accurate force fields. Adv. Neural Inform. Process. Syst. https://openreview.net/forum?id=YPpSngE-ZU (2022).
Batatia, I. et al. The design space of E(3)-equivariant atom-centered interatomic potentials. Nat. Mach. Intell. 7, 56–67 (2025).
Minato, K. et al. Release behavior of metallic fission products from HTGR fuel particles at 1600 to 1900 °C. J. Nucl. Mater. 202, 47–53 (1993).
Amian, W. & Stöver, D. Diffusion of silver and cesium in silicon-carbide coatings of fuel particles for high-temperature gas-cooled reactors. Nucl. Technol. 61, 475–486 (1983).
Allelein, H., Biedermann, P. & Stoever, D. Cs Release from TRISO fuel particles. Tech. Rep., Kernforschungsanlage Juelich GmbH, Germany, Institute fuer Reaktorentwicklung (1980).
Fukuda, K., Yamagishi, S., Ikawa, K. & Shiba, K. Research and development of HTGR fuel at JAERI. Tech. Rep., Japan Atomic Energy Research Institution (1989).
Moormann, R. & Verfondern, K. Methodik umfassender probabilistischer sicherheitsanalysen fur zukunftige htr-anlagenkonzepte-ein statusbericht (stand 1986). band 3: Spaltproduktfreisetzung. Tech. Rep., Institut für Nukleare Sicherheitsforschung (1987).
Verfondern, K. & Müller, D. Modelling of fission product release behavior from HTR spherical fuel elements under accident conditions. Tech. Rep., Forschungszentrum Jülich GmbH, Germany, Institute für Sicherheitsforschung und Reaktortechnik (1991).
Hart, E. On the role of dislocations in bulk diffusion. Acta Metall 5, 597 (1957).
Stukowski, A. Computational analysis methods in atomistic modeling of crystals. JOM 66, 399–407 (2014).
Shrader, D., Szlufarska, I. & Morgan, D. Cs diffusion in cubic silicon carbide. J. Nucl. Mater. 421, 89–96 (2012).
Wei, K. et al. Interactions between silver-palladium alloy and silicon carbide cladding layer in triso fuel: An in-depth analysis. J. Eur. Ceram. Soc. 44, 4362–4375 (2024).
Wen, H., van Rooyen, I. J., Hunn, J. D. & Gerczak, T. J. Electron microscopy study of Pd, Ag, and Cs in carbon areas in the locally corroded SiC layer in a neutron-irradiated TRISO fuel particle. J. Eur. Ceram. Soc. 38, 4173–4188 (2018).
Jiang, C., Ke, J.-H., Simon, P.-C. A., Jiang, W. & Aagesen Jr, L. K. Atomistic and mesoscale simulations to determine effective diffusion coefficient of fission products in SiC. Report no. INL/EXT-21-64633. Tech. Rep., Idaho National Laboratory (INL) (2021).
Ishimaru, M., Bae, I.-T., Hirotsu, Y., Matsumura, S. & Sickafus, K. E. Structural relaxation of amorphous silicon carbide. Phys. Rev. Lett. 89, 055502 (2002).
Ishimaru, M. et al. Direct observations of thermally induced structural changes in amorphous silicon carbide. J. Appl. Phys. 104, 033503 (2008).
Grimme, S., Antony, J., Ehrlich, S. & Krieg, H. A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu. J. Chem. Phys. 132, 154104 (2010).
Becke, A. D. & Edgecombe, K. E. A simple measure of electron localization in atomic and molecular systems. J. Chem. Phys. 92, 5397–5403 (1990).
Tang, W., Sanville, E. & Henkelman, G. A grid-based bader analysis algorithm without lattice bias. J. Phys. Condens. Matter 21, 084204 (2009).
Vineyard, G. H. Frequency factors and isotope effects in solid state rate processes. J. Phys. Chem. Solids 3, 121–127 (1957).
de Souza, V. K. & Wales, D. J. Energy landscapes for diffusion: analysis of cage-breaking processes. J. Chem. Phys. 129, 164507 (2008).
He, H., Deshpande, M., Brown, R. E., Pandey, R. & Pernisz, U. C. Molecular modeling of water diffusion in amorphous SiC. J. Appl. Phys. 98, 023519 (2005).
Mistry, M. et al. Modelling the interactions and diffusion of NO in amorphous SiO2. Model. Simul. Mater. Sci. Eng. 29, 035008 (2021).
Christensen, R., Persson, K. A. & Smedskjaer, M. M. Structural origin of disorder-induced ion conduction in NaFePO4 cathode materials. J. Mater. Chem. A 13, 24619–24632 (2025).
Sørensen, S. S., Smedskjær, M. M. & Micoulaut, M. Evidence for complex dynamics in glassy fast ion conductors: the case of sodium thiosilicates. J. Phys. Chem. B 127, 10179–10188 (2023).
Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 54, 11169–11186 (1996).
Blöchl, P. E. Projector augmented-wave method. Phys. Rev. B 50, 17953–17979 (1994).
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).
Monkhorst, H. J. & Pack, J. D. Special points for Brillouin-zone integrations. Phys. Rev. B 13, 5188–5192 (1976).
Parrinello, M. & Rahman, A. Crystal structure and pair potentials: a molecular-dynamics study. Phys. Rev. Lett. 45, 1196 (1980).
Parrinello, M. & Rahman, A. Polymorphic transitions in single crystals: a new molecular dynamics method. J. Appl. Phys. 52, 7182–7190 (1981).
Tersoff, J. Chemical order in amorphous silicon carbide. Phys. Rev. B 49, 16349–16352 (1994).
Heera, V. et al. Density and structural changes in SiC after amorphization and annealing. Appl. Phys. Lett. 70, 3531–3533 (1997).
Rino, J. P. et al. Short-and intermediate-range structural correlations in amorphous silicon carbide: A molecular dynamics study. Phys. Rev. B-Condens. Matter Mater. Phys. 70, 045207 (2004).
Vashishta, P., Kalia, R. K., Nakano, A. & Rino, J. P. Interaction potential for silicon carbide: A molecular dynamics study of elastic constants and vibrational density of states for crystalline and amorphous silicon carbide. J. Appl. Phys. 101, 103515 (2007).
Hoover, W. G. Canonical dynamics: equilibrium phase-space distributions. Phys. Rev. A 31, 1695–1697 (1985).
Nosé, S. A unified formulation of the constant temperature molecular dynamics methods. J. Chem. Phys. 81, 511–519 (1984).
Martínez, L., Andrade, R., Birgin, E. G. & Martínez, J. M. Packmol: a package for building initial configurations for molecular dynamics simulations. J. Comput. Chem. 30, 2157–2164 (2009).
Cancino-Trejo, F. et al. Grain boundary complexions in silicon carbide. J. Am. Ceram. Soc. 101, 1009–1013 (2018).
Thompson, A. P., Aktulga, H. M. & Berger, R. et al. Lammps - a flexible simulation tool for particle-based materials modeling at the atomic, meso, and continuum scales. Comput. Phys. Commun. 271, 108171 (2022).
Stukowski, A. Visualization and analysis of atomistic simulation data with ovito–the open visualization tool. Model. Simul. Mater. Sci. Eng. 18, 015012 (2009).
Hjorth Larsen, A., Jørgen Mortensen, J. & Blomqvist, J. et al. The atomic simulation environment-a python library for working with atoms. J. Phys. Condens. Matter 29, 273002 (2017).
Dwaraknath, S. & Was, G. The diffusion of cesium, strontium, and europium in silicon carbide. J. Nucl. Mater. 476, 155–167 (2016).
Acknowledgements
This work was financially support by the National S&T Major Project (Grant No. ZX060901), CNNC's young talents research project, and the grant for research centers in the field of AI provided by the Ministry of Economic Development of the Russian Federation in accordance with the agreement 000000C313925P4F0002 and the agreement with Skoltech No.139-10-2025-033. J.L. acknowledge financial support from China Scholarship Council. This work made use of the resources of the SuperComputing Network and Center of High Performance Computing, Tsinghua University. J.L. thanks Olga Chalykh and Evgeny Podryabinkin for valuable discussions on MTP training.
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J.L. and N.R. conducted all the computational works. J.L., N.R., and T.W. drafted the paper. A.S., X.C., and B.L. acquired funding and supervised the entire project. All authors reviewed and edited the manuscript.
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Li, J., Rybin, N., Wang, T. et al. Revealing the diffusion mechanism of Cs in amorphous and polycrystalline SiC by actively trained moment tensor potentials. npj Comput Mater (2026). https://doi.org/10.1038/s41524-025-01944-4
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DOI: https://doi.org/10.1038/s41524-025-01944-4
