Understanding the mechanical behavior of structural metals and alloys requires modeling defect interactions at the atomic scale. Quantum mechanics-based methods like DFT are accurate but computationally expensive, limiting their applicability to large-scale simulations of phenomena like dislocation motion and fracture. Empirical interatomic potentials offer improved efficiency but often lack accuracy and transferability, particularly in reproducing crucial properties like generalized stacking fault energies (γ-lines). This limitation is especially significant for materials with complex crystal structures and deformation mechanisms, such as titanium, which exhibits three allotropes (HCP-α, BCC-β, and hexagonal-ω) and anisotropic elastic and plastic behavior. While various classical interatomic potentials (EAM, MEAM, tight-binding, bond-order) have been developed for titanium, they often fail to accurately predict γ-lines and stacking fault energies. This work addresses this challenge by employing machine learning, specifically the Deep Potential (DP) method, to develop an accurate and efficient interatomic potential for titanium that overcomes the limitations of classical approaches. The authors propose a specialization step within the DP-GEN framework to systematically train the potential, ensuring accurate reproduction of key structural, energetic, and mechanical properties.

The introduction thoroughly reviews existing interatomic potentials for titanium, highlighting their limitations. Classical potentials like EAM, MEAM, and tight-binding, while efficient, often inaccurately predict fundamental properties, particularly the γ-lines of HCP titanium, which are crucial for understanding plasticity and fracture. The bond-order potential, although showing some improvements, still exhibits inconsistencies. The paper positions its work within the context of emerging machine-learning-based interatomic potentials, emphasizing the use of the Deep Potential (DP) method and its potential for improved accuracy and transferability over classical approaches. The authors mention several other machine learning approaches for developing interatomic potentials, but focus on the DP method for its robustness and flexibility.

The methodology involves a three-step workflow: Initialization, DP-GEN Loop, and Specialization. In the Initialization step, primitive cells of BCC, FCC, and HCP titanium are constructed and relaxed using DFT calculations (VASP). Supercells are then created and subjected to uniform volume scaling and AIMD simulations at 100 K. The resulting data (atomic coordinates, forces, energies, and virial tensors) constitute the initial training set. The DP-GEN Loop iteratively trains and refines the DP model. Trial DPs are applied to perturbed bulk and surface structures in finite-temperature MD simulations (LAMMPS). Configurations with high force variance across the ensemble of trial DPs are selected as candidates. DFT calculations are then performed on these candidates, generating additional training data. This loop continues until convergence. The Specialization step introduces structures relevant to mechanical properties (e.g., sheared configurations along γ-lines) into the training set. DFT calculations are performed on these specialized structures, and the DP model is retrained with the combined “classic” (from DP-GEN Loop) and “special” training sets. The weights of the special structures in the loss function are adjusted to prioritize the accuracy of the target properties. The DeePMD-kit package is used for training the neural network potential. Specific details of DFT calculation parameters (VASP settings), DP training parameters (DeePMD-kit settings), and the selection and weighting of special training sets are provided. The specialized DP is referred to as “DPspecX”, where X represents the specialized properties, here being the mechanical response of titanium (Ti-DPspecMech).

The developed Ti-DPspecMech potential accurately reproduces a wide range of properties, both within and outside the training dataset. The lattice parameters and energies of HCP, BCC, and FCC titanium are in excellent agreement with DFT and experimental data (deviations less than 0.002 Å and 1 meV/atom). The elastic constants also match well with DFT and experimental values (deviations generally less than 10%). The potential accurately predicts surface energies and vacancy formation energies for HCP titanium. Critically, the potential correctly reproduces the ordering of stacking fault energies on various slip planes in HCP titanium (basal, prism, pyramidal I, pyramidal II), consistent with experimental observations and DFT calculations. The γ-lines calculated using the DP model match the general shape and key features of those calculated via DFT, especially for the important basal, prism, and pyramidal planes. This accurate reproduction of γ-lines, crucial for understanding dislocation behavior, is a significant improvement over existing empirical potentials. For BCC titanium, the DP model captures the negative stacking fault energies and absence of metastable points on the {110}, {112}, and {123} planes, consistent with DFT calculations. The temperature dependence of the lattice parameters and elastic constants is reasonably well reproduced by the model when compared to experimental data, with some minor discrepancies possibly attributed to differences between DFT calculations and experiments. The phase transition temperatures (HCP-BCC and BCC-liquid) predicted by the DP potential show remarkable agreement with experimental values. Finally, the study examines dislocation core structures using molecular dynamics simulations, finding that the DP potential predicts the correct ground state dissociation plane for the screw (a) dislocation in HCP titanium.

The results demonstrate the success of the DPspecX approach in creating a highly accurate and transferable interatomic potential for titanium. The accurate reproduction of γ-lines and stacking fault energies, not explicitly included in the general DP training sets, highlights the model's predictive power. The agreement with experimental data on phase transition temperatures, elastic constants, and thermal expansion further validates the potential's accuracy. While some minor discrepancies exist between the DP predictions and experimental values (e.g., in some elastic constants at finite temperatures), these are likely due to inherent limitations in DFT calculations and the experimental measurements. The comparison with existing empirical potentials (EAM and MEAM) underscores the advantages of the specialized DP model. The computational cost of the DP method is higher than that of empirical potentials, although it remains considerably faster than DFT calculations, enabling large-scale simulations previously intractable with DFT. The accuracy obtained compensates for this increased computational cost. The potential's accurate description of dislocation core structures and phase transitions suggests its use in diverse simulations of titanium's behavior.

This paper introduces a general methodology, DPspecX, for specializing machine learning potentials to accurately reproduce specific material properties. The application of this method to titanium results in a highly accurate interatomic potential (Ti-DPspecMech) suitable for large-scale simulations of a broad range of phenomena. The potential's accuracy, transferability, and efficiency make it a valuable tool for materials research and design. Future work could focus on further optimization of the DP model, applying this methodology to other materials, and exploring additional applications within the DP framework or other machine learning-based approaches.

While the DPspecMech potential shows remarkable accuracy, some limitations exist. Minor discrepancies between the DP and experimental values for some elastic constants at finite temperatures suggest possible areas for improvement. The computational cost of the DP method, although significantly lower than DFT, is higher than that of empirical potentials, which might pose a limitation for extremely large-scale simulations. The focus on mechanical properties, however, does not explore other properties, which can be extended in future studies. The transferability of the model to conditions far outside the training data range should be carefully considered.