Nickel (Ni), a magnetic transition metal, exists in stable face-centered cubic (FCC) and metastable hexagonal close-packed (HCP) phases. Its widespread use in structural applications necessitates understanding its mechanical properties and deformation mechanisms, particularly at the nanoscale where inverse Hall-Petch behavior and transformation-induced plasticity are significant. While density functional theory (DFT) offers high accuracy, its computational cost restricts its use for large systems and long timescales needed for studying these phenomena. Existing empirical interatomic potentials, such as embedded-atom method (EAM) and modified embedded-atom method (MEAM) potentials, exhibit limited accuracy and transferability, especially for predicting properties of metastable phases like HCP Ni. The paper addresses this limitation by developing a machine-learning (ML) based interatomic potential that accounts for the influence of magnetism on non-magnetic properties without explicitly including magnetic moment degrees of freedom. This approach aims to improve the accuracy and transferability of simulations involving large-scale atomistic simulations for phase transformations.

Numerous interatomic potentials for Ni have been developed over the years, primarily based on EAM and MEAM. However, these potentials often struggle to accurately predict the properties of both FCC and HCP Ni, exhibiting significant discrepancies in elastic constants and other properties. Machine-learning (ML) potentials offer a potential solution, but simply ignoring magnetism in the training data can lead to unreliable results. Existing ML methods, such as those based on Gaussian approximation potentials and neural network potentials (NNPs), often lack sufficient transferability across diverse properties. The authors reviewed existing potentials and found considerable inaccuracies in their prediction of metastable HCP properties, leading to the need for a more accurate and transferable potential.

The researchers developed a "magnetism-hidden" machine-learning Deep Potential (DP) model for Ni. The model was trained using a supervised ML technique on a dataset derived from spin-polarized DFT calculations. The training labels included atomic coordinates, total energy, atomic forces, and virial tensors. The DP-GEN framework and DeepPot-SE were employed for training. A "specialization" strategy was used to improve accuracy, focusing on configurations along the cohesive energy line. The training process involved an iterative loop (DP-GEN) where initial datasets from distorted BCC, FCC, and HCP structures, generated through ab initio molecular dynamics (AIMD) simulations, were refined by adding DFT data from selected configurations obtained from DP-based MD (DPMD) simulations. The final training dataset consisted of 2020 entries. The performance of the resulting DP-Ni model was benchmarked against several widely-used empirical/semi-empirical interatomic potentials, including EAM, MEAM_2021, MEAM_2015, and ML qSNAP, across a range of properties.

The DP-Ni model demonstrated exceptional accuracy and transferability in predicting various properties of both FCC and HCP Ni. Comparisons with DFT calculations, experimental data, and other interatomic potentials revealed the following key findings: 1. **Basic Crystal Properties:** DP-Ni showed excellent agreement with DFT and experimental values for lattice parameters, cohesive energies, and elastic constants of both FCC and HCP Ni. Significant deviations were observed in other potentials, particularly for HCP Ni elastic constants, showcasing DP-Ni's superior accuracy. 2. **Phonon Spectra:** DP-Ni accurately reproduced the phonon spectra of both FCC and HCP Ni, unlike other potentials which exhibited noticeable deviations. 3. **Surface Energies and Point Defects:** DP-Ni accurately predicted surface energies and point defect (vacancy and interstitial) formation energies, demonstrating its ability to accurately predict these properties even without explicit inclusion in the training dataset. Other potentials showed significant discrepancies. 4. **Cohesive and Decohesion Energies/Stresses:** DP-Ni accurately reproduced the cohesive and decohesion energy curves and stresses compared to DFT, while other potentials showed large deviations, especially EAM, highlighting the smoothness and accuracy of DP-Ni in describing atomic plane separation. 5. **Ideal Strength:** DP-Ni accurately predicted the ideal strength of FCC Ni under various loading conditions, showing better agreement with DFT than other potentials. 6. **Finite-Temperature Properties:** DP-Ni accurately predicted the temperature dependence of lattice parameters and elastic constants, showing better agreement with experimental data than other potentials which exhibited abnormal behavior. 7. **Stacking Fault and Dislocation Core:** DP-Ni accurately predicted stacking fault energies and dislocation core structures, including partial dislocation separation distances, which are crucial for understanding plastic deformation. Other potentials showed significant deviations. 8. **Grain Boundary Energies:** DP-Ni accurately predicted the energies and structures of various grain boundaries, demonstrating its applicability to simulating grain boundary behavior. 9. **Allotropic Transformation:** Simulation of uniaxial tensile loading revealed an FCC → HCP phase transformation at a high critical strain with an atypical orientation relationship, highlighting the model's ability to predict complex phase transformations.

The results demonstrate the superior accuracy and transferability of the developed DP-Ni model compared to existing empirical and ML potentials. The model's ability to accurately predict a wide range of properties, including those not explicitly included in the training dataset, underscores its robustness and reliability. The accurate prediction of the FCC-HCP phase transition, including the atypical orientation relationship, demonstrates its potential for studying complex phase transformation behavior in Ni. The small size of the training dataset relative to other ML potentials further highlights the efficiency of this approach. The use of spin-polarized DFT data in the training, despite not explicitly using magnetic moments as descriptors, was crucial in obtaining accurate results for non-magnetic properties.

This work presents a highly accurate and transferable machine learning Deep Potential model for nickel, successfully capturing both FCC and HCP phases and a wide range of material properties. The model's superior performance over existing potentials makes it a powerful tool for large-scale atomistic simulations, especially those involving complex phase transformations and defect behavior. The provided DP-Ni model and training datasets establish a strong foundation for developing accurate ML potentials for nickel-based alloys and other multi-component alloys.

The DP-Ni model, while highly accurate, still exhibits some minor discrepancies with experimental data in certain properties, such as the melting point. This might be attributed to limitations in DFT calculations or the lack of specific solid-liquid interface configurations in the training dataset. The study focused primarily on bulk properties and defects; further validation might be needed for more complex scenarios involving surface effects or external fields.