Development of Quantitative Finite Element based Phase-Field model for the Precipitation of δZrH in αZr

Abstract

Due to the excellent combination between its mechanical, thermal, physical, and nu- clear properties, Zirconium based alloys are the material of choice for cladding tubes in light water nuclear reactors. However, the interaction between the water coolant and the Zirconium cladding will result in a corrosion reaction. This will result in the formation of Zirconium Oxide and the release of Hydrogen. A portion of the released Hydrogen diffuses into the cladding, and as the concentration of Hydrogen increases, Zirconium hydrides will precipitate. Zirconium hydrides are brittle mate- rials which will result in a degradation of the long term structural integrity of the cladding. Several pioneering experimental investigations were implemented in the past 50 years. These important studies made considerable advancements in under- standing the properties of Zirconium and its hydride phases, and in addition, studied the effect of several important forces on the Zirconium hydride precipitation process such as stress, texture, and temperature. However, many of the precipitation induced mechanisms are still unknown, and were not clarified by experimental investigations. This is due to the fact that precipitation occurs at very fine length and time scales, which are beyond the capability of experimental investigations. Hence, a compu- tational investigation is required. Among computational techniques, the phase-field method established itself as the main continuum method for the investigation of ma- terial transformations at the miso-scale. This is due to its fundamental origins, the ability to depict specific material structural features such as grain boundaries and defects, and its intuitive approach to multi-physical coupling. In the past 15 years, several important models were proposed for Zirconium hydride precipitation. These models made considerable advancements in understanding, and clarifying key aspects associated with precipitation process, and paved the way for the development of fu- ture models. However, the inaccurate deception of the system properties, affected the clarity of the implemented studies, and in addition, the inefficient implementation re- sulted in low fidelity studies that did not reveal several aspects that are associated with the precipitation process. Hence, this study was implemented to develop a novel, quantitative, Finite Element-based, phase-field model to investigate the precipitation process of δ-Zirconium hydride in α-Zirconium. The model was built utilizing a novel nucleation model that is based on the classical nucleation theory which respects mass conservation in the system, and results in the natural and accurate determination of the equilibrium concentrations for the phases in the system. Furthermore, the model uses the modern matrix-free implementation, and adaptive mesh refinements strate- gies. These result in a very efficient implementation and relatively moderate system. This will allow the model to conduct high-fidelity investigations. The model can be applied to investigate any two phase system, and can be easily extended to inves- tigate multiple phases. Furthermore, it has the ability to investigate a wide range of phenomena including solidification, precipitation, and second-phase growth. The model was verified & validated by comparing the numerical solution of this model, and the analytical solution of the Kim-Kim-Suzuki model. Furthermore, the model was compared with current preferred software that is used for phase-field modeling, the MOOSE framework which is developed by the Idaho National Laboratory. The developed model showed a much higher runtime efficiency and at the same time, did not require an intensive hardware consumption. This allowed the developed model to run very high fidelity analysis cases on relatively moderate hardware. The model was applied to investigate the role of capillary-induced forces on the precipitation mecha- nisms of δ-Zirconium hydride in α-Zirconium, which to our best of knowledge have not been investigated before experimentally or computationally. A high-fidelity analysis in terms of particles simulated and the duration allowed the study to verify the role of precipitate size, shape, and stacking formation structure on the precipitation process. Furthermore, it clarified the role of hydride-hydride interaction as well. In addition, a very large domain size was generated preventing the development of excessive elastic energy during the precipitation process. The study was able to reveal key mechanisms associated with δ-Zirconium hydride precipitation in α-Zirconium. The shape of the precipitates changes from a spherical shape toward a disk-like shape precipitates that aligned along the basal plane of α-Zirconium as the size increases. At small precipi- tate sizes, the isotropic interfacial energy will dominate the shape formation process resulting in a spherical shape. As the size of the precipitate increases, the shape of precipitates will become more influenced by the elastic energy leading to a more disk- like shape where the precipitates will possess a larger radius along the basal plane relative to the c-axis. This is promoted by a stronger stiffness of α-Zirconium and a higher misfit strain along the c-axis relative to the basal plane. This is confirmed by the linear relationship that develops between the aspect ratio of precipitates, and the interface to area ratio. As precipitates grow and their shapes become more disk-like, the equilibrium concentration will be affected by the surface curvature of precipitates maximizing along the basal plane where the maximum curvature is located. This leads to a lower chemical potential which promotes precipitates realignment along the basal plane. The close proximity of aligned precipitates, results in an accumu- lation of hydrogen. This is promoted by capillary-induced surface diffusion. This leads to a decrease in the chemical potential between the precipitates. This promotes the migration process of precipitates toward each other and ultimately coalescence. At the onset of coalescence, a large increase in the precipitate growth rate occurs leading to the flattening of the newly formed single precipitate surface. This will lead to the development of large size variations within the local vicinity. This results in an acceleration of capillary-induced diffusion-limited coarsening. Here, larger precip- itates will keep growing and smaller precipitates will shrink and ultimately dissolve. This is promoted by a higher equilibrium concentration of hydrogen and a higher chemical in smaller precipitates which drives the diffusion of mass from smaller to larger precipitates. The sequence of precipitate realignment, migration, coalescence, and diffusion-limited coarsening keep re-emerging as the length scale of precipitates increases. These mechanisms are verified through a complete set of results that were provided by the study, they are originally proposed, and on par with fundamentals in material kinetics.