Machine learning interatomic potential with DFT accuracy for general grain boundaries: Analysis of grain boundary energy and atomic structure in α-Fe polycrystals

Machine learning interatomic potential with DFT accuracy for general grain boundaries: Analysis of grain boundary energy and atomic structure in α-Fe polycrystals | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Machine learning interatomic potential with DFT accuracy for general grain boundaries: Analysis of grain boundary energy and atomic structure in α-Fe polycrystals Kazuma Ito, Tatsuya Yokoi, Katsutoshi Hyodo, Hideki Mori This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4550958/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 13 Nov, 2024 Read the published version in npj Computational Materials → Version 1 posted 12 You are reading this latest preprint version Abstract To advance the development of high-strength polycrystalline metallic materials towards achieving carbon neutrality, it is essential to design materials in which the atomic-level control of general grain boundaries (GGBs), which govern the material properties, is achieved. However, owing to the complex and diverse structures of GGBs, there have been no reports on interatomic potentials capable of reproducing them. This accuracy is essential for conducting molecular dynamics analyses to derive material design guidelines. In this study, we constructed a machine learning interatomic potential (MLIP) with density functional theory (DFT) accuracy to model the energy, atomic structure, and dynamics of arbitrary grain boundaries (GBs), including GGBs, in α-Fe. Specifically, we employed a training dataset comprising diverse atomic structures generated based on crystal space groups. The GGB accuracy was evaluated by directly comparing with DFT calculations performed on cells cut near GBs from nano-polycrystals, and extrapolation grades of the local atomic environment based on active learning methods for the entire nano-polycrystal. Furthermore, we analyzed the GB energy and atomic structure in α-Fe polycrystals through large-scale molecular dynamics analysis using the constructed MLIP. Conventional interatomic potentials cannot accurately calculate the GB energy and atomic structure in α-Fe polycrystals. Conversely, the average GB energy of α-Fe polycrystals calculated by the constructed MLIP is 1.57 J/m 2 , exhibiting good agreement with experimental predictions. Our findings demonstrate the methodology for constructing an MLIP capable of representing GGBs with high accuracy, thereby paving the way for materials design based on computational materials science for polycrystalline materials. Physical sciences/Materials science/Structural materials/Metals and alloys Physical sciences/Materials science/Theory and computation/Atomistic models Machine learning potential density functional theory (DFT) grain boundaries grain boundary segregation steels Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 1. Introduction In recent years, there has been an increasing demand for the development of high-strength metallic materials to achieve carbon neutrality. For example, within the field of steel materials, increasing the strength of steel used for automotive steel plates can reduce CO 2 emissions by lightening the vehicle body. Thus, development of high-strength steels is being pursued intensively [ 1 – 6 ]. Most metallic materials are polycrystalline, and therefore, contain grain boundaries (GBs). GBs have a significant influence on manufacturability and material properties, either directly or indirectly, through the formation of microstructures. However, enhancing material strength increases susceptibility to GB embrittlement, a phenomenon characterized by the degradation of material properties and manufacturability. This occurs due to cracking at GBs caused by the GB segregation of certain alloying elements or impurity atoms [ 7 – 10 ]. Therefore, the suppression of hydrogen embrittlement [ 5 , 11 – 18 ], liquid metal embrittlement (LME) [ 19 – 24 ], and red hot embrittlement [ 25 – 29 ], all of which entail cracking at GBs, is a significant issue in the development of high-strength steels. Consequently, there is a critical need to design materials capable of suppressing the degradation of material properties and manufacturability associated with GB cracking. A highly promising approach to suppress GB cracking involves controlling the GB segregation of alloying and impurity elements. Recently, the addition of Mo through segregation at GBs has been demonstrated to suppress GB cracking and improve the hydrogen embrittlement resistance of high-strength martensitic steels [ 5 , 11 , 30 ]. In LME resulting from Zn plating, the susceptibility to GB cracking depends on the GB character [ 31 ]. Furthermore, the addition of trace amounts of B suppresses Zn penetration into GBs due to the segregation of B at these boundaries. This process strengthens the GBs and effectively suppresses LME [ 21 ]. Therefore, obtaining material design guidelines to suppress GB cracking necessitates a quantitative understanding of the extent of GB segregation of diverse solute elements. This understanding should account for their content, heat treatment process, and GB character. To acquire such knowledge, GB segregation using highly symmetric GBs that can be treated by density functional theory (DFT), i.e., GBs with small Σ values, has been intensively studied [ 32 – 41 ]. These studies have clarified that the amount of GB segregation of solute elements is strongly affected by the local atomic structure of each site constituting the GB. Specifically, factors such as Voronoi (or occupied) volume [ 32 ], coordination number, and interatomic distance play crucial roles [ 34 , 37 ]. However, such GB cracking primarily occurs at so-called general GBs (GGBs), which constitute most GBs in polycrystals and lack specific symmetry. Recent studies have revealed that predicting GB segregation in polycrystalline materials can be difficult to predict using highly symmetric GBs, which are amenable to analysis using DFT. This difficulty arises because of the diverse array of local atomic structures present within these boundaries [ 42 ]. Consequently, studying only highly symmetric GBs that can be analyzed using DFT is insufficient to obtain the knowledge necessary to control GB segregation. Therefore, numerous studies have recently investigated GB segregation in GGBs using interatomic potentials, such as embedded atom method (EAM) and modified EAM (MEAM) potentials, and nano-polycrystalline models [ 43 – 49 ]. However, interatomic potentials such as EAM and MEAM are rarely fitted to GBs. Consequently, they cannot accurately replicate the intricate local atomic structures found at stable GBs that affect GB segregation and the energies and dynamics of each atom that determine them. For example, the GB segregation energy of transition metal alloying elements in Fe varies approximately in proportion to the Voronoi volume of the segregation site. Moreover, the GB segregation energy exerts an exponential influence on the extent of GB segregation. Therefore, even a small change in Voronoi volume results in a significant difference in segregation levels. Furthermore, in α-Fe, interatomic potentials such as EAM and MEAM have been shown to significantly underestimate the GB energy of symmetric tilt GBs in pure Fe relative to DFT values [ 50 ]. Therefore, the local atomic structures at stable grain boundaries that affect GB segregation and the kinetics near grain boundaries that determine them also differ significantly from DFT. In recent years, highly accurate machine learning interatomic potentials (MLIPs) have been constructed [ 51 , 52 ], which have also been applied to the study of GBs. For example, Yokoi et al. have demonstrated that the GB energy of Al and its temperature dependence can be calculated with DFT accuracy using a neural network interatomic potential (NNIP) [ 53 ]. Various MLIPs for α-Fe have also been constructed [ 54 – 60 ] since the development of the Gaussian approximate potential (GAP) [ 61 ] in 2018. For example, Mori et al. have created an NNIP for α-Fe that reproduces the stability of dislocation structures with an accuracy comparable to that of DFT [ 62 ]. In particular, the NNIP for Fe–H explicitly includes symmetric tilt GBs with small Σ values in the training data, and provides accuracy comparable to that of DFT for the GB energy of symmetric tilt GBs [ 63 ]. However, there are no reports on interatomic potentials capable of reproducing the local atomic structure and GB energies of GGBs, as well as the dynamics near GBs that determine them, with DFT accuracy. These aspects are essential for the design of future metallic materials. MLIP is recognized to be inaccurate in extrapolated regions [ 64 ], and its accuracy for GGBs with more complex and diverse atomic structures warrants thorough testing. This holds even if symmetric tilt GBs are included in the training data. However, GGBs are characterized by five degrees of freedom and comprise diverse and complex atomic structures [ 65 ]. Therefore, performing DFT calculations exhaustively on all GGBs, explicitly incorporating them as training data to construct interatomic potentials, and evaluating the accuracy of these potentials with respect to GGBs is challenging. In this study, rather than explicitly incorporating GGBs into the training dataset, we employed a training dataset generated through a method recently proposed by Paul et al. [ 66 ]. This method mechanically generates various atomic structures based on crystal space groups. The moment tensor potential (MTP) was selected as the MLIP because of its excellent balance between computational cost and accuracy [ 67 , 68 ]. The accuracy of the MLIP for GGBs was verified through direct comparison with DFT calculations for cells cut out in the vicinity of GBs using randomly oriented nano-polycrystals. These nano-polycrystals provided a comprehensive sampling of the atomic environment of polycrystals with grain sizes in the micrometer range. Additionally, the accuracy was evaluated through extrapolation grades based on active learning methods for the entire nano-polycrystal. These verifications demonstrated that the constructed MLIP can calculate the atomic structure, energy, and dynamics of arbitrary GBs, including GGBs, with a level of accuracy comparable to that of DFT. As an application of the MLIP, the average GB energy of α-Fe polycrystals, which is an important yet challenging property to measure, was calculated. Finally, the properties of GGBs in α-Fe polycrystals were analyzed at the atomic level and their relationship with GB segregation discussed. Consequently, we clarified that the MLIP constructed in this study is valuable for quantitatively predicting GB segregation at GGBs. This is because the previously constructed interatomic potentials are not sufficiently accurate for this purpose. 2. Methods 2.1. Details of calculations using DFT and interatomic potentials Spin-polarized electronic structure calculations and structural optimization were performed, using the Vienna ab initio simulation package (VASP) with the projector-augmented wave (PAW) method [ 69 , 70 ] within the generalized gradient approximation (GGA) framework, utilizing Perdew–Burke–Ernzerhof (PBE) parametrization [ 71 ]. These calculations were employed to build the training dataset and verify the accuracy of the interatomic potentials. The cutoff energy for the plane-wave basis set was set to 520 eV. The k-point mesh for each atomic structure was set to an accuracy equivalent to 18 × 18 × 18 for the α-Fe conventional unit cell using Monkhorst-Pack k-mesh [ 72 ]. The Methfessel–Paxton smearing method [ 73 ] with a width of 0.1 eV was employed. The atomic positions were relaxed until reaching energy and force convergence values of 10 − 4 eV and 10 − 2 eV/Å, respectively. Ab initio molecular dynamics (AIMD) based on the Parrinello–Rahman dynamics with the Langevin thermostat [ 74 , 75 ] were performed to generate training datasets, as described in Section 2.2 . Both NVT and NPT conditions were used with a time step of 2 fs. All calculations using interatomic potentials were conducted using LAMMPS [ 76 ], while OVITO [ 77 ] was used for visualizing the atomic structure. Additionally, calculations involving the quasi-harmonic approximation were performed using PHONOPY [ 78 ]. 2.2. Training datasets Table 1 lists the training datasets, which comprise two types. The first dataset, referred to as the domain expertise (DE) dataset hereafter, considers the basic properties and lattice defects of α-Fe and was manually created. The second is a training dataset, referred to as the RANDSPG dataset hereafter, is based on atomic structures mechanically generated by the RANDSPG algorithm [ 79 ], mainly for reproducing GGBs [ 66 ]. For the RANDSPG dataset, the method used to construct the Mg training dataset [ 66 ] was modified and applied to α-Fe. The MTP constructed using this training dataset has been demonstrated to accurately reproduce physical properties and lattice defect energies, such as symmetric tilt GBs for low Σ values, for Mg. Notably, this is achieved even though the corresponding atomic structures are not explicitly included in the training dataset. Specifically, we first used the RANDSPG algorithm [ 79 ] to create the basic structure for constructing the training dataset. The main input parameters for generating the atomic structure are the maximum number of atoms, crystal space group to be considered, and allowable atomic volume per atom. Building on the study on Mg [ 66 ], 2482 basic structures (hereafter referred to as RANDSPGs) were generated. These structures were created with a maximum of 10 atoms, encompassing all crystal space groups, and allowing a tolerance of ± 10% of the equilibrium lattice constant for the atomic volume per atom. These basic structures were then subjected to stepwise structural relaxation: (1) volume-only relaxation (VOLMIN), (2) cell shape-only relaxation with constant volume (CELLMIN), and (3) atomic position relaxation (INTMIN). Each of these structures was then used as a candidate training dataset. In addition, the atomic structure of INTMIN was disturbed by (1) random triaxial strain (TRIAX, up to 80%), (2) a combination of random shear strains (SHEAR, up to 80%), and (3) random displacements of atoms combined with a small random strain tensor (RATTLE, 0.5 Å mean displacement and up to 5% strain). These modified structures were also included as candidates for the training dataset. DFT calculations were performed on these structures to construct the training dataset. However, owing to the high-spin degrees of freedom of Fe, many atomic structures with nonmagnetic, low-spin, and spin-flip magnetic states were obtained. Consequently, we supposed that including these structures in the training data would hinder the attainment of an MTP capable of reproducing ferromagnetic α-Fe with high accuracy. Therefore, for the training dataset, we exclusively included structures possessing magnetic moments with the same direction and values exceeding 1.5 µB. This value corresponds to the magnitude of stable magnetic moments in which the body-centered cubic (bcc), face-centered cubic (fcc), hexagonal close-packed (hcp), and simple cubic structures of Fe do not transition to the nonmagnetic state. The number of atomic structures and atomic environments in the final RANDSPG dataset was comparable to those included in the training dataset for Mg in previous studies [ 66 ], comprising 17460 structures and 117146 atomic environments. A DE dataset was added to increase the accuracy of the physical properties of α-Fe and lattice defects such as self-interstitial atoms (SIAs). This also serves to compensate for the compressible structure, which renders the magnetic moment unstable and is preferentially excluded in the selection process of the RANDSPG dataset described above. Specifically, AIMD was conducted on 3 × 3 × 3 supercells of α-Fe at 300, 600, 1000, 1400, 2000, and 3000 K under the NPT ensemble. AIMD was also performed at 300, 600, 1000, 1400, 2000, and 3000 K under the NVT ensemble for 3 × 3 × 3 supercells of α-Fe, with lattice constants varying within ± 2.5% from the equilibrium lattice constant. In addition, AIMD simulations were conducted at 300, 600, 1000, 1400, 2000, and 3000 K under the NPT ensemble for a 3 × 3 × 3 supercell of α-Fe containing a single vacancy. Atomic structures were extracted from these AIMDs every five steps and included in the training dataset. Snapshots of the structural relaxation process of the structure containing an SIA were also added to the training dataset. Finally, the two datasets were combined to create a training dataset for building interatomic potentials consisting of 19950 structures and 267188 atomic environments. In particular, the RANDSPG dataset contains a relatively large number of structures but consists of at most 10 atoms. Considering that, for single-element systems, the computational cost of DFT is proportional to the cube of the number of atoms, it is noteworthy that the computational cost of constructing these training datasets is very small. Specifically, the computational cost of obtaining the RANDSPG dataset is comparable to the computational cost of performing AIMD simulations at 300, 600, and 1000 K using the 3 × 3 × 3 supercells of α-Fe in the DE dataset. Table 1 Details of the training dataset for α-Fe. The training dataset for α-Fe consists of the domain expertise (DE) dataset, which is a training dataset for reproducing the basic properties and lattice defects of α-Fe, and the RANDSPG dataset, which is a training dataset for reproducing general grain boundaries. \({N}_{\text{s}\text{t}\text{r}}\) and \({N}_{\text{f}\text{o}\text{r}\text{c}\text{e}}\) are the number of atomic structures (number of total energies) and atomic forces, respectively, in the training dataset. \({N}_{\text{a}\text{t}\text{o}\text{m}}\) is the number of atoms in each atomic structure in the training dataset. The root mean squared errors (RMSEs) of the energies and forces for each training dataset are also shown. Datasets Dataset N str N atom N force Energy (meV/atom) Force (meV/Å) DE Perfect crystal 1776 54 95936 2.2 69.38 Vacancy 500 53 26500 1.33 63.88 SIA 214 129 27606 1.86 36.43 Total 2490 150042 2.03 63.6 RANDSPGs RANDSPG 691 3–10 4078 16.28 88.49 VOLMIN 1273 3–10 7918 12.72 77.98 CELLMIN 1836 3–10 11882 11.14 83.2 INTMIN 2015 3–10 13236 11.3 54 TRIAX 5187 3–10 35806 16.15 62.7 SHEAR 3209 3–10 22134 21.66 83.85 RATTLE 3249 3–10 22092 9.1 114.45 Total 17460 117146 15.14 81.94 All Total 19950 267188 14.11 71.88 2.3. Construction of MTP In this study, the MLIP was constructed using the MTP formalism [ 67 , 68 ]. The MTP can represent various atomic environments through angle-dependent many-body interactions, expressed as tensor products of atomic displacements. Moreover, it has no transcendental functions and relies solely on polynomial arithmetic operations, thereby significantly reducing the computational cost. Owing to these advantages, in certain systems, MTP can achieve accuracy levels comparable to those of GAP, which is considered one of the most accurate MLIPs, in less than a tenth of the computation time [ 80 ]. A template of level 22 potentials was employed to construct the MTP. The maximum cutoff radius was set to 6.5 Å, consistent with the NNIP for Fe–H [ 63 ], and training was conducted using the MLIP-3 package [ 80 ]. The MTP constructed in this study can be accessed via the URL provided in the Data Availability section. As shown in Supplementary Fig. 1, the RMSEs of energy for the training and validation datasets were 14.11 and 14.79 meV/atom, respectively. The RMSE of atomic force was 71.88 meV/Å for the training dataset and 75.42 meV/Å for the validation dataset. The RMSE of energy was approximately fourfold larger than that of the previously reported NNIP of Fe–H [ 63 ]. However, this discrepancy can be attributed to the diverse range of atomic structures present in the RANDSPG dataset, a trend also observed in the construction of the MTP for Mg [ 66 ]. Indeed, as shown in Table 1 , the RMSE for the DE dataset is comparable to the NNIP of Fe–H. Supplementary Table S1 and Supplementary Fig. 2 demonstrate that the constructed MTP reproduced the lattice parameter, elastic modulus, and phonon dispersion of α-Fe, along with the defect energies for SIAs and vacancy defects with the same accuracy as that of the NNIP. In addition, the two-dimensional energy profiles of screw dislocation core positions [ 62 ] and generalized stacking fault energy surfaces [ 81 ], crucial for analyzing plastic deformation and crack propagation behavior, were also reproduced with very good accuracy. These results suggest excellent transferability of the MTP, as discussed in Section 3 . 2.4. Method for evaluating the accuracy of interatomic potentials for GGBs In this study, we evaluated the accuracy of the constructed MTP for GGBs using GBs in nano-polycrystals with random orientations obtained by Voronoi tessellation [ 82 ] and relaxation through molecular dynamics calculations. Detailed analysis of the GB structure of nano-polycrystals obtained through this method suggests that the GB structure is similar to that of general polycrystals with grain sizes larger than micrometer dimensions [ 83 ]. The time evolution of the average grain size of nano-polycrystals prepared by this method via annealing exhibits similar growth behavior as polycrystals with grain sizes larger than micrometer dimensions, growing at a rate proportional to the square root of time [ 84 ]. Given grain growth behavior is strongly influenced by the nature of GBs [ 85 ], these results suggest that the GBs of nano-polycrystals created with Voronoi tessellation are similar to those found in common polycrystals. Furthermore, Wagih et al. demonstrated that the histogram of GB segregation energies, calculated for nano-polycrystals generated via Voronoi tessellation, exhibits little change for a 15 nm cubic polycrystalline model consisting of eight grains, even as the number of nano-polycrystalline grains and their model size increase [ 46 ]. They demonstrated that nano-polycrystals created through Voronoi tessellation could effectively represent the diverse local atomic environments found in real polycrystals with actual grain sizes in the micrometer range., since the GB segregation energy at each site depends on the local atomic environment of the host metal around the solute atoms [ 42 ]. The energy and force of each atom of the host metal are determined by the local atomic environment as well as GB segregation. Therefore, if the constructed MTP can accurately reproduce the local atomic environment in nano-polycrystals created through Voronoi tessellation, it should be capable of reproducing GBs, or GGBs, in polycrystals with grain sizes in the micrometer range. We evaluated the accuracy of the MTP for GGBs in two ways using nanocrystalline polycrystals created via Voronoi tessellation. The first method involved directly comparing energies and atomic forces by conducting DFT calculations on numerous DFT-calculable regions selected from the GBs within the nano-polycrystals, obtained by molecular dynamics relaxation using the constructed MTP. However, while this method provides direct verification, the computational cost makes it impractical to evaluate the accuracy of the entire nano-polycrystal. As a second method to address this limitation, all local atomic environments constituting the nano-polycrystal were evaluated. This involved calculating extrapolation grades that can numerically determine whether the constructed MTP is an interpolated region where high accuracy can be expected or an extrapolated region where accuracy is reduced [ 80 ]. Validation of the transferability of the constructed potentials, based on the evaluation of extrapolation grades, has also been employed in previous studies [ 66 ]. While not a direct accuracy assessment, this method is relatively computationally inexpensive because it relies on pre-constructed MTPs, enabling evaluation for all atomic environments within the nano-polycrystal. The initial structure of the nano-polycrystals was generated using Atomsk [ 82 ] with Voronoi tessellation to create polycrystals with random orientations. The structural relaxation of the nano-polycrystals was based on the method outlined by Van Swygenhoven et al. [ 83 ]. This involved initially relaxing the cell size and atomic positions using the conjugate gradient method, followed by annealing for 0.2 ns at 300 K in the NPT ensemble. Subsequently, the system was cooled from 300 to 0.1 K before performing another relaxation step to adjust the cell size and atomic positions using the conjugate gradient method. The annealing time was set as the time for sufficient convergence of the obtained GB energy. Eight grains were included in the nano-polycrystals, following the methodology outlined by Wagih et al. in their study of GB segregation of Mg in Al using EAM [ 46 ]. The initial dimensions of the nano-polycrystals were set to 28.3 × 28.3 × 28.3 nm 3 . As shown in Supplementary Fig. 4, the local atomic environment in nano-polycrystals created under these conditions is largely independent of the random seeding of the initial structure creation and the expansion of the model size. The method of comparison using DFT calculations is illustrated in Fig. 1 . For this analysis, we used a nano-polycrystal with initial cell dimensions of 28.3 × 28.3 × 28.3 nm 3 , hereafter denoted as (28.3 nm) 3 polycrystal. Specifically, we utilized the structure of the (28.3 nm) 3 polycrystal during annealing (at 0.1 ns) and subsequent relaxation, as described in the previous paragraph. From all the GBs (64) in each nano-polycrystal, 64 regions with dimensions of 15.0 × 15.0 × 15.0 Å 3 (~ 250 atoms) were cut out as calculation cells. Given polycrystals typically contain GB triple junctions, we also cut out 33 regions measuring 15.0 × 15.0 × 15.0 Å 3 from the GB triple junctions in each nano-polycrystal as calculation cells. Periodic boundary conditions were applied to the cells, and if the distance between the atoms was less than 2.0 Å, one of the atoms was removed from the cell boundary. As depicted in Fig. 1 (c) and (d), the boundary of the calculation cell, influenced by the periodic boundary conditions, exhibits a structure like a general GB. Consequently, the energy of the system differs from that which would be observed if this region existed within the nano-polycrystal. Nevertheless, achieving sufficient accuracy for these calculation cells suggests the potential to accurately compute the energies of atoms proximal to GBs and GB triple junctions. The atomic forces at the boundaries of the calculation cell also differ from those in the nano-polycrystal. However, the atomic forces near the center of the calculation cell, which is sufficiently far from the cell boundaries, are consistent with those observed in the nano-polycrystal. As illustrated in Supplementary Fig. 3, the atomic forces acting on atoms within 4.0 Å of the center of the calculation cell are maintained within a 10% error margin. Therefore, for the comparison of the atomic forces near the general GB, we compared the results of the DFT and interatomic potential calculations for atoms within 4.0 Å of the center of the calculation cell and in the one-atomic layer region (2.46 Å) from the GB center. For atoms near the GB triple junctions, DFT and MTP calculations were compared for atoms within 4.0 Å of the calculation cell center and 2.46 Å from the nearest and second nearest GBs, respectively. The GB center is defined as a Voronoi polyhedron when the initial structure is created by Voronoi tessellation. Remarkably, during annealing at 300 K and subsequent relaxation via the conjugate gradient method, the relative positions of the GBs exhibited minimal movement, despite changes in the cell size. For comparison, we also evaluated the NNIP for Fe–H [ 63 ] using the same methodology. Notably, this is the only MLIP that utilizes symmetric tilt GBs for α-Fe as training data. Similarly, MEAM [ 86 ] and EAM [ 87 ] were evaluated for accuracy using this method. These two interatomic potentials are identical to those used for comparison in previous studies [ 50 ]. To provide a relative interpretation of the accuracy evaluated for GGBs, the GB energies and atomic forces acting on atoms within 2.46 Å of the GB center during annealing were also compared through DFT calculations for eight and symmetric tilt GBs with Σ values ranging from 3 to 11, which can be treated by DFT calculations. In addition, we investigated whether the stable GB structure obtained by DFT relaxation can be reproduced by the interatomic potentials. For both DFT and interatomic potential calculations of GB energies, several initial structures were created with one grain rigidly shifted parallel to the GB, and the cell sizes and atomic configurations were subsequently relaxed. In these interatomic potential calculations, OpenKIM [ 88 ] was employed. AIMD was conducted at 300, 600, and 1000 K in the NVT ensemble for eight stable structures of symmetric tilt GBs derived from DFT. For the atomic structures during annealing, the atomic forces acting on atoms in the region within 2.46 Å of the GB center were calculated using interatomic potentials and compared with those obtained from DFT calculations. For the extrapolation grade calculations, the structures of the (28.3 nm) 3 polycrystal during (at 0.1 ns) and after relaxation were used, alongside direct comparisons with DFT calculations. For all atoms constituting these two structures, the extrapolation grade of the local atomic environment for the active learning method recently implemented in MLIP-3 was evaluated [ 80 ]. In the extrapolation grade calculations, extrapolation grades were evaluated for nano-polycrystals annealed at 600 and 1000 K for 0.1 ns in addition to those at 300 K. For details on extrapolation grades, see Ref. [ 80 ]. Briefly, an atom with an extrapolation grade between 0 and 1 falls into the "interpolation region" for the constructed interatomic potentials, which guarantees high calculation accuracy. Extrapolation grades between 1 and 2 indicate the "accurate extrapolation” region, while those between 2 and 10 fall into the "reliable extrapolation” region. Structures containing atoms in this last range are additionally labeled by DFT calculations during the active learning process to indicate insufficient learning of their atomic environment. If the extrapolation grade of an atom exceeds 10, it is considered to be in the "dangerous extrapolation” region and the learning process is terminated [ 80 ]. Thus, an extrapolation grade of 2 or less for all atoms in the annealed and relaxed nano-polycrystals indicates that the obtained interatomic potentials are sufficiently learned for the entire polycrystal. Consequently, no additional learning is required in active learning and high accuracy is expected. 2.5. Analysis of GB energy and atomic structure in α-Fe polycrystals The determination of the GB energy of GGBs in polycrystalline metallic materials remains a significant area of research. To date, these energies have been determined through experimentation [ 89 ]. Therefore, as an application of the MTP constructed in this study, we calculated the average GB energy of α-Fe polycrystals using the nano-polycrystals described in Section 2.4 . We then compared this energy to those of GGBs calculated based on experimental data. Furthermore, we analyzed the properties of GGBs in α-Fe polycrystals at the atomic level and discussed the relationship with GB segregation. The area of the GB required for calculating GB energies was determined by correcting the area of the Voronoi polyhedron, which was utilized to generate the initial structure of the nano-polycrystal, for the change in cell size before and after relaxation. Notably, during annealing at 300 K and subsequent relaxation through the conjugate gradient method, the relative positions of the GBs exhibit minimal movement, despite changes in the cell size. However, the average GB energy may strongly depend on the number of grains (or the effect of GB character) and the size of the grains. Therefore, fixing the number of grains at eight, we calculated the average GB energy for nano-polycrystals with cell sizes ranging from 5.66 × 5.66 × 5.66 nm 3 (20 times the lattice constant of Fe for a side length) to 28.3 × 28.3 × 28.3 nm 3 (100 times the lattice constant of Fe for a side length). This allowed us to investigate the dependence of the average GB energy on the grain size. For each cell size, the dependence on GB character was further evaluated. This was achieved by calculating the average GB energy of nano-polycrystals for three cases, in which only the random seed was varied when creating the initial nano-polycrystal structure by Voronoi tessellation. 3. Results and discussion 3.1. Accuracy of the MTP for symmetric tilt GBs To provide a relative interpretation of the accuracy for GGBs, we first present the accuracy for symmetric tilt GBs, where the GB energy is well defined and a direct comparison with the DFT calculations is possible for both energy and atomic forces. Figure 2 depicts the GB energies of the symmetric tilt GBs, obtained through relaxation using both DFT calculations and interatomic potentials. On average, the difference between the GB energies of the eight symmetric tilt GBs calculated with the constructed MTP and those calculated with DFT is 0.037 J/m 2 . This result indicates that the MTP reproduces the DFT values with high accuracy. In particular, the MTP exhibits a level of high accuracy comparable to that of the NNIP, which explicitly incorporates training data from Σ3(112), Σ3(111), Σ5(210), and Σ5(310). Conversely, the GB energies calculated using the MEAM and EAM greatly underestimate those calculated using DFT. Figure 3 presents the atomic forces near the GB center (within the one-atomic layer region from the GB center) for eight symmetric tilt GBs. The calculations were performed using both DFT and interatomic potential calculations, with GBs annealed at 300, 600, and 1000 K using AIMD under the NVT ensemble. Figure 3(a) displays the collective atomic forces acting on atoms within the eight symmetric tilt angle GBs. In contrast, Fig. 3(b) and (c) showcase the atomic forces at the symmetric tilt GBs calculated using MTP and NNIP, respectively, for each of the eight GBs. Notably, the annealing structures of Σ3(112), Σ3(111), Σ5(210), and Σ5(310) are explicitly included in the training data for the NNIP. The MTP demonstrates higher accuracy compared to the NNIP, EAM, and MEAM across both GBs and all annealing temperatures. For example, the RMSE of the atomic force for the eight symmetric tilt GBs at 300 K is 82.4 meV/Å. Notably, upon focusing on the accuracy for each GB, MTP reproduces the DFT for the GBs that are explicitly included in the NNIP training dataset with the same high accuracy as that of the NNIP. Additionally, for GBs absent in the NNIP training dataset, the MTP consistently exhibits higher accuracy than the NNIP. In particular, NNIP displays a particularly large error for structures annealing in Σ9(114) and Σ11(113), which are not included in the training dataset. This result clearly shows that even though the stable structure of GBs and their GB energies can be calculated with good accuracy, the error in the dynamics of GBs may be large. In summary, the MTP constructed in this study can accurately calculate both the GB energy of symmetric tilt GBs and their atomic forces during annealing, without explicitly including them in the training data. This suggests that the constructed MTP exhibits excellent accuracy for GGB stable structures, as well as their associated energies and dynamics. 3.2. Accuracy of the MTP for GGBs Figure 4(a) and (b) depict the energies and atomic forces near the GGB and GB triple junctions, respectively. These calculations employed cut-out calculation cells from a (28.3 nm) 3 polycrystal annealed at 300 K for 0.1 ns via MTP, using DFT and interatomic potentials. Considering that the energy basis varies depending on the interatomic potentials, the energy was compared by measuring the change from the energy of an atom of α-Fe with a lattice constant of 2.830 Å at each potential. Here, 2.830 Å represents the equilibrium lattice constant of α-Fe in DFT, MTP, and NNIP. Notably, the periodic boundary conditions result in a GB-like defect structure near the boundary of the calculation cell, which differs from that observed in the nano-polycrystal. The RMSE of the energy and the atomic force for the general GB in the constructed MTP are 2.17 meV/atom and 81.05 eV/Å, respectively. The RMSE of the atomic force on the general GB is comparable to that observed during annealing at 300 K for symmetric tilt GBs, for which the GB energy can be accurately calculated. This indicates that MTP exhibits high calculation accuracy for GGBs. The RMSE of the energy and atomic force for the constructed MTP for GB triple junctions are 2.44 meV/atom and 86.21 eV/Å, respectively, which are also comparable to those for GGBs. In particular, the force and energy RMSEs for both the general GB and GB triple junctions were approximately half those obtained using NNIP. These results clearly demonstrate that the constructed MTP accurately reproduces the dynamics near the general GB and GB triple junctions of nano-polycrystals at 300 K. Figure 5(a) and (b) depict the energies and atomic forces near the GGB and GB triple junctions, respectively. These calculations utilized cut-out calculation cells from a (28.3 nm) 3 polycrystal annealed at 300 K and relaxed via the conjugate gradient method using MTP, through DFT and interatomic potentials. The RMSEs of the energy and atomic force for the constructed MTP for a GGB are 2.44 meV and 80.34 meV/Å, respectively. Given these are the forces acting on the atoms in the nano-polycrystal after relaxation, the atomic forces tend to approach zero. The RMSEs of the energy of the GB triple junctions and atomic force in the constructed MTP were determined to be 2.44 meV and 81.55 meV/Å, respectively. These results indicate that nano-polycrystals can be relaxed using MTP with good accuracy, and stable structures and their energies can be obtained with DFT accuracy. Owing to the cost of DFT calculations, these described calculations are limited to a portion of the nano-polycrystal. Therefore, to investigate the accuracy across the entire nano-polycrystal, encompassing all local atomic environments in a typical polycrystal, we constructed histograms of extrapolation grades for all atoms in the (28.3 nm) 3 polycrystal during annealing at 300, 600 and 1000 K and after relaxation using MTP (Fig. 6). Most extrapolation grades for approximately two million sites for all structures were determined to be less than one. This indicates that most local atomic structures in the polycrystals fall within the interpolation region, thereby guaranteeing high calculation accuracy in the constructed MTP [80]. While there were a few atoms with high extrapolation grades, the maximum extrapolation grade observed was 1.72, which falls within the "accurate extrapolation” region. Thus, even with the application of active learning methods based on the local atomic environment to these nano-polycrystals, the constructed MTPs remain unchanged. Consequently, in comparison with both direct DFT calculations and extrapolation grade evaluations, the constructed MTPs demonstrate superior DFT accuracy for the stable structures of GGBs and GB triple junctions, as well as their energies and dynamics. 3.3. Average GB energy of α-Fe polycrystals using the constructed MTP Figure 7 illustrates the relationship between average GB energy and model size in nano-polycrystals. As the model size increases, the average GB energy converges towards a constant value. For example, the difference in average GB energies between the (22.6 nm) 3 and (28.3 nm) 3 polycrystals is merely 0.03 J/m 2 . The average GB energy for the three (28.3 nm) 3 polycrystals is 1.57 J/m 2 with a standard error of 0.03 J/m 2 . This result indicates that despite the presence of approximately 60 GBs, the deviation of the average GB energy among polycrystalline models in relation to the GB character is relatively small. The average GB energy of the (5.66 nm) 3 polycrystal is also relatively small. This is attributed to the presence of extremely small grains, which promote grain growth even at 300 K, thereby leading to a decrease in the actual GB area. On the other hand, in the calculation of the average GB energy, the area of the GB was evaluated based on the area of the Voronoi polyhedron when the initial structure was created. The calculated average GB energies of nano-polycrystals were compared to those of GGBs based on experimental data. For bcc metals, Li et al. investigated the scaling factor among different defect structures of the same metal and among the same defect structures across different metals [89]. Subsequently, they estimated the energy of the GGB of α-Fe at 0 K, using experimental values of the grain boundary energy of the GGB of W measured at high temperatures. The results revealed that the GGB energy of α-Fe is 1.63 J/m 2 . This value is in good agreement with the average GB energy of 1.57 J/m 2 observed for the (28.3 nm) 3 polycrystal (Fig. 7). In contrast, the EAM and MEAM significantly underestimate the average GB energy. This finding indicates that the choice of interatomic potential markedly influences the simulations of grain growth driven by GB energy. In addition, the underestimation of GB energy is related to the Voronoi volume at sites near the GB (Section 3.4), which also plays a significant role in GB segregation. Notably, the average GB energy was not calculated using the NNIP owing to limited computing resources. 3.4. GB energy and atomic structure in α-Fe polycrystals The findings presented in Section 3.3 reveal that the constructed MTP can describe α-Fe polycrystals with high accuracy. The (28.3 nm) 3 polycrystals relaxed through the constructed MTP serve as a good sampling of the local atomic environment found in polycrystals with grain sizes in the micrometer range, a common feature in steel materials. Moreover, the average GB energy closely approximates that of polycrystals. Therefore, we supposed that a detailed investigation of the nano-polycrystals relaxed through the MTP would be particularly interesting. In particular, as mentioned in the Introduction (Section 1), GB segregation is an important control target in material design. Consequently, we conducted a thorough investigation on the Voronoi volume and coordination number, both dominant factors for GB segregation, for nano-polycrystals relaxed through MTP. Figures 8(a) and (b) illustrate the Voronoi volume and coordination number of each atom, respectively, in a (28.3 nm) 3 polycrystal with an average GB energy of 1.58 J/m 2 . Additionally, Fig. 8(c) and (d) show the Voronoi volume and coordination number, respectively, as a function of distance from the GB center. These results pertain to one (28.3 nm) 3 polycrystal; however, the results for the other two (28.3 nm) 3 polycrystals are almost identical. Notably, the coordination number is defined as the number of planes obtained through Voronoi tessellation in each atomic region. Hence, the next nearest neighbor equivalent to the bcc structure is also counted as a coordination number. The Voronoi volume begins to increase in standard deviation from the GB center at approximately 8.0 Å. The mean Voronoi volume at the GB center measures 12.0 Å 3 , marking a 6.0% increase compared to the bulk volume. The coordination number increases in standard deviation from the GB center to approximately 7.0 Å; however, the mean value remains relatively unchanged and is 14.2 at the GB center. Thus, when considering the local atomic environment in terms of Voronoi volume and coordination number, the deviation from the bulk is observed at approximately 8.0 Å from the GB center. Figure 8(e) presents histograms of Voronoi volumes for all sites within 7.35 Å (three atomic layers) of the GB center and for each atomic layer from the GB center in the (28.3 nm) 3 polycrystal. In the (28.3 nm) 3 polycrystal, the Voronoi volume ranges from − 13.3–20.8% at GBs. In particular, the Voronoi volume changes significantly in one atomic layer from the GB center. Within 7.35 Å, or three atomic layers, from the GB center, 66.3% of the sites exhibit looser sites (sites with Voronoi volumes larger than that of the bulk), while 33.7% display tighter sites (sites with Voronoi volumes smaller than that of the bulk). Considering these results, we explore GB segregation in α-Fe polycrystals. For example, for transition metal elements, the solute element occupancy at each site increases exponentially with Voronoi volume from the bulk [90], indicating that the major segregation sites exist within one atomic layer from the GB center for both large and small solute elements. In particular, for substitutional solute elements, the proportion of tighter sites is approximately half that of looser sites, indicating that segregation sites for substitutional solute elements with smaller atomic sizes are more limited than those with larger atomic sizes. Figure 8(f) displays the Voronoi volume as a function of distance from the GB center for the (28.3 nm) 3 polycrystals relaxed through EAM or MEAM. For example, the ratio of the Voronoi volume at the GB center to that of the bulk is 3.3% and 2.9% for the MEAM and EAM, respectively, which is approximately half of the 6.0% observed for the MTP. This result indicates that, for example, when considering the GB segregation of a solute with a large atomic volume, the calculation of GB segregation energy using MEAM- or EAM-relaxed nano-polycrystals underestimates the energy by approximately half, which varies approximately linearly with the Voronoi volume [90]. Furthermore, the amount of GB segregation, which exhibits an exponential dependence on the GB segregation energy [91], would be significantly underestimated. Figure 9(a) and (b) present histograms of the misorientation angle and GB energy, respectively, for three (28.3 nm) 3 polycrystals obtained by MTP relaxation. Additionally, Fig. 9(c) illustrates the relationship between GB energy and misorientation angle for each of the three (28.3 nm) 3 polycrystals. The area of each GB is the same as that used for calculating the average GB energy, employing the Voronoi polyhedron used to create the initial nano-polycrystalline structure. Each atom was assigned to the nearest GB, and the energy of each GB was calculated. GBs with areas larger than 1000 Å 2 were evaluated. The average GB energies of the polycrystals, obtained by averaging the energies for each GB across the three cases, were 1.57, 1.53, and 1.60 J/m 2 . These values closely align with the average GB energies of 1.58, 1.53, and 1.59 J/m 2 , calculated by dividing the excess energy of the entire polycrystal by the GB area shown in Fig. 7. The histograms obtained for the three cases displayed a mode value of 1.65–1.70 J/m 2 , which is higher than the average value for polycrystalline materials, as discussed at the end of this section. The maximum GB energy was 1.79 J/m 2 , representing a 13.3% increase over the average GB energy value. Next, the atomic structure of a representative GB among the large-angle GBs in Case 1 was analyzed in detail. The GB with the lowest GB energy (1.26 J/m 2 ) (GBL), the one corresponding to the mode (1.67 J/m 2 ) (GBM), and that with the highest energy (1.78 J/m 2 ) (GBH) are discussed here. Figure 9(d) displays the probability density of the Voronoi volume in the one-atomic layer region near the GB center for these three GBs, alongside the polycrystal in Case 1 for comparison. The probability density of the Σ3(111) symmetric tilt GB (1.58 J/m 2 ), which is the most used GB in DFT studies, is also shown. The probability density in Σ3(111) is scaled down to 1/10 for comparison. In Fig. 9(e), the probability densities of these Voronoi volumes are shown as the difference from the probability density of the Voronoi volume in the one-atomic layer region near the GB center for the polycrystals in Case 1. The probability density of the Voronoi volume for GBM is close to that of the entire polycrystal. In contrast, in the GBH, the probability density of Voronoi volumes exceeding 12.2 Å 3 is increased relative to that of the entire polycrystal. Finally, in the GBL, the probability density of Voronoi volumes close to the bulk Voronoi volume is increased relative to the whole polycrystal. Given the correlation between GB energy and the probability density of Voronoi volumes near the GB [50], this indicates that the GB energies of these three GBs have been correctly evaluated. In Σ3(111), there are sites with three different Voronoi volumes, all falling within the range observed in polycrystals. The ratio of sites with these volumes in Σ3(111) closely resembles that observed in the polycrystal. This result indicates that when considering solute atoms for which the Voronoi volume is the dominant factor in GB segregation, the analysis using Σ3(111) GBs is effective for qualitatively understanding the GB segregation. However, because there are only three sites with different Voronoi volumes, it is not sufficient to accurately predict GB segregation in polycrystalline materials. Figure 9(f) depicts the atomic structures and energies of the atoms of the GBL, GBM, and GBH. In the GBM and GBH, the regions where the atomic energies are higher than those of the bulk are linearly distributed, and the difference between the two GBs is unclear. In contrast, in the GBL, the atoms in the GB center form periodic facets. The atoms in the faceted region have atomic energies relatively close to those of the bulk, even though the atoms are located near the center of the GB. The atomic structure of this region forms an atomic structure similar to a well-consistent (111)-terminated twist GB. Thus, even in polycrystalline materials with random orientations, instances occur where locally well-matched GBs are formed, resulting in a decrease in the GB energy. This result is one of the reasons why the histogram shown in Fig. 9(b) for misorientation angles of 15° or more does not exhibit a normal distribution but instead displays a large spread from the mode to the low-energy side. The results presented in this section, which were obtained through relaxation at near-room temperature (300 K) using a high-precision MTP, are expected to be similar to those of real α-Fe polycrystals with random orientations. As described earlier, the MTP constructed in this study proves valuable for analyzing GBs in polycrystals at the atomic level and predicting GB segregation with high accuracy. 3.5. Application of findings from this study The MTP constructed in this study reproduces the energies and acting atomic forces on arbitrary GGB and GB triple junctions in α-Fe with high accuracy both during annealing and after relaxation. Therefore, it finds applicability in numerous studies, including grain growth of polycrystals [92] and investigation of GB energy and mobility [93]. Recently, a method has also been proposed to calculate GB segregation energy using machine learning from the local atomic environment of segregation sites [48, 94, 95]. As discussed in Section 3.4, the GB structure of the host metal significantly influences the predicted GB segregation energy and amount of GB segregation. The MTP constructed in this study can serve as a highly accurate GB model for this purpose. Alternatively, by extending the MTP to binary systems based on the constructed MTP and applying a method for predicting GB segregation using nano-polycrystals, the MTP should accurately predict the segregation of solute elements at GBs without the need for experiments. The MTP constructed in this study may also find application in simulating the deformation and fracture of α-Fe polycrystals. As demonstrated in the Supplementary Note 4, the constructed MTP reproduces the two-dimensional energy profiles of the screw dislocation core position and the Peierls potential with DFT accuracy. These are necessary for accurate reproduction of plastic deformation. In addition, the generalized stacking energy curves under no strain and tensile strain, which are essential for accurately describing the crack propagation, also demonstrate accuracy comparable to that of DFT. While the applicability to the simulation of deformation and fracture of α-Fe polycrystals falls beyond the scope of this study and is not addressed herein, it will be investigated in detail in the future. At the very least, the constructed MTP should provide a good starting point for studying these issues with high precision. In this study, we demonstrated the feasibility of constructing an interatomic potential that reproduces all GBs, including GGBs, with high accuracy. This is achieved by augmenting the basic training dataset, which reproduces the bulk properties, with the RANDSPG dataset proposed by Paul et al. [66], and subsequently constructing an MTP. Given the successful construction of a generic MTP for Mg by Paul et al. using a RANDSPG dataset, the present results indicate the potential to construct MLIPs with excellent accuracy for any GB. This can be achieved for not only α-Fe but also for a variety of metallic materials by augmenting the RANDSPG dataset. Remarkably, the RANDSPG dataset comprises only 10 atoms at maximum, resulting in exceptionally low computational cost for acquiring a training dataset. Specifically, the calculation cost required to construct the RANDSPG dataset is comparable to that of performing AIMD at 300, 600, and 1000 K using a 3 × 3 × 3 supercell of α-Fe. In essence, a diverse range of GBs can be effectively incorporated at a computational cost equivalent to collecting a small fraction of the training dataset typically utilized to reproduce the bulk properties of metallic materials. Therefore, while snapshots obtained from AIMD for basic crystal structures such as bcc, fcc, and hcp are generally used as a starting point for MLIP construction [96], it is also beneficial to employ the RANDSPG dataset as one of the basic initial datasets for MLIP construction. At the very least, when the analysis includes GBs, it can serve as a strong starting point for any MLIP construction, such as those based on concurrent [96] or active [68] learning. Thus, the findings obtained in this study will significantly advance material design, benefiting not only steel materials but also any polycrystalline metallic materials through the proposed method of constructing interatomic potentials. 4. Conclusion In this study, we constructed a tailored MTP to accurately reproduce the general GB behavior of α-Fe, serving as a basic interatomic potential for exploring GB segregation control. Our training dataset comprised a DE dataset incorporating basic physical properties and lattice defects of α-Fe, alongside a training dataset containing diverse atomic structures generated mechanically based on crystal space groups using RANDSPG. We verified the accuracy of the MTP for GGBs through direct comparison with DFT calculations on calculation cells cut near GBs from nano-polycrystals relaxed using the constructed MTP. Additionally, we assessed the accuracy using extrapolation grades for the entire nano-polycrystal. Our results demonstrate that the constructed MTP exhibits high accuracy for various arbitrary GBs, including GGBs, while accurately capturing the basic properties and lattice defects of α-Fe. Furthermore, we employed the MTP to calculate the average GB energies of polycrystals through large-scale molecular dynamics simulations, obtaining values consistent with experimental estimates. The constructed MTP enables precise determination of GB energy and mobility as a function of GB character. Additionally, it can be utilized to construct a polycrystalline GB model for accurate prediction of GB segregation, which is necessary to gain knowledge on the control of GBs in α-Fe. Notably, despite the minimal computational cost to obtain the RANDSPG dataset, the accuracy for GGBs can be further enhanced. Therefore, the insights gained from this study hold significant potential for designing high-strength metallic materials through the construction of high-precision interatomic potentials for all metals, extending beyond the scope of α-Fe. Declarations Authorship contributions Kazuma Ito: Conceptualization, Methodology, Software, Data curation, Writing – original draft, Visualization, Investigation. Tatuya Yokoi: Methodology, Validation, Writing – review & editing. Katsutoshi Hyodo: Validation, Writing – review & editing. Hideki Mori: Supervision, Software, Validation, Writing – review & editing. Competing Interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work used computational resources of the Supercomputer Fugaku provided by Riken through the HPCI System Research Project (Project ID: hp230272). This work was partly supported by Accompanying User Support Program (【23Z-03, 23Z-05, 24H1-01】, Support content:【porting of application program, execution performance tuning】) performed by Research Organization for Information Science and Technology. Data availability The potential is available on the Github page https://github.com/KazumaIto0810/MTP. References C.D. Horvath, Chapter 2 - Advanced steels for lightweight automotive structures, in: P.K. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4550958","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":324733293,"identity":"e26f7c85-acb5-4447-b2ac-a6b8e0a91784","order_by":0,"name":"Kazuma Ito","email":"data:image/png;base64,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","orcid":"https://orcid.org/0000-0002-3456-1055","institution":"Nippon Steel Corporation","correspondingAuthor":true,"prefix":"","firstName":"Kazuma","middleName":"","lastName":"Ito","suffix":""},{"id":324733294,"identity":"9ee6b967-dad0-4f7c-bee5-8e7290e185b2","order_by":1,"name":"Tatsuya Yokoi","email":"","orcid":"https://orcid.org/0000-0002-5178-3952","institution":"Nagoya University","correspondingAuthor":false,"prefix":"","firstName":"Tatsuya","middleName":"","lastName":"Yokoi","suffix":""},{"id":324733295,"identity":"516d215a-79be-45ed-ae2e-5587bef522fa","order_by":2,"name":"Katsutoshi Hyodo","email":"","orcid":"","institution":"Nippon Steel Corporation","correspondingAuthor":false,"prefix":"","firstName":"Katsutoshi","middleName":"","lastName":"Hyodo","suffix":""},{"id":324733296,"identity":"64baad71-6353-41cd-b5e3-1c67f21f4c6a","order_by":3,"name":"Hideki Mori","email":"","orcid":"","institution":"College of Industrial Technology","correspondingAuthor":false,"prefix":"","firstName":"Hideki","middleName":"","lastName":"Mori","suffix":""}],"badges":[],"createdAt":"2024-06-08 14:30:20","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4550958/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4550958/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1038/s41524-024-01451-y","type":"published","date":"2024-11-13T05:00:00+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":61599066,"identity":"c12a4b61-9582-40d9-9f63-da1d226a7997","added_by":"auto","created_at":"2024-08-01 17:34:15","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":2424485,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eSchematic of the method for evaluating the accuracy of interatomic potentials for general grain boundaries. \u003c/strong\u003e(a) Two base structures for comparison: (28.3 nm)\u003csup\u003e3\u003c/sup\u003e polycrystalline structure annealed at 300 K for 0.1 ns and structure relaxed using the conjugate gradient method after annealing. (b) Magnified view of the vicinity of the general grain boundary (GGB) of the (28.3 nm)\u003csup\u003e3\u003c/sup\u003e polycrystal after relaxation. (c) Cross-section of a periodic boundary cell for density functional theory (DFT) extracted from the vicinity of the GGB. The colors in these figures represent the energy of each atom, and the energy reference is the energy of Fe atoms in the perfect crystal. A total of 64 regions near the GGB and 33 regions near the grain boundary (GB) triple junction were cut out of the calculation cell for the DFT calculation. Atomic forces for atoms within 4.0 Å of the center of the calculation cell and in the one-atomic layer region (2.46 Å) from the GB center or GB triple junction center were compared.\u003c/p\u003e","description":"","filename":"fig1.png","url":"https://assets-eu.researchsquare.com/files/rs-4550958/v1/c129e11976ba65552d70b604.png"},{"id":61598520,"identity":"adbbc545-6fc3-414e-90dc-c457d76d165e","added_by":"auto","created_at":"2024-08-01 17:34:12","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":472746,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eCalculation accuracy of grain boundary energy for symmetric tilt grain boundaries. \u003c/strong\u003eSymmetric tilt grain boundary (GB) energies of α-Fe calculated from density functional theory (DFT) and interatomic potentials. (a) \u0026lt;110\u0026gt; and (b) \u0026lt;001\u0026gt; symmetric tilt GBs.\u003c/p\u003e","description":"","filename":"fig2.png","url":"https://assets-eu.researchsquare.com/files/rs-4550958/v1/fe08ddf3c1ad2df582fcdf8e.png"},{"id":61599334,"identity":"8db2f982-f05d-46cc-ab2d-90f737ec2df8","added_by":"auto","created_at":"2024-08-01 17:34:16","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":2000860,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eCalculation accuracy of atomic forces at symmetric tilt grain boundaries. \u003c/strong\u003eForces calculated using density functional theory (DFT) and interatomic potentials, on atoms near the grain boundary (GB) center of symmetric tilt GBs during annealing at 300, 600, and 1000 K in ab initio molecular dynamics (AIMD) under NVT ensemble. (a) Collective atomic forces on the atoms in eight symmetric tilt GBs. Atomic forces at symmetric tilt GBs calculated using the (b) moment tensor potential (MTP) and (c) neural network interatomic potential (NNIP) for each of the eight GBs.\u003c/p\u003e","description":"","filename":"fig3.png","url":"https://assets-eu.researchsquare.com/files/rs-4550958/v1/c47e115b7528313d27b67547.png"},{"id":61599511,"identity":"9868b9ee-c73c-4ad5-988c-aa60c8538fc7","added_by":"auto","created_at":"2024-08-01 17:34:29","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":1245768,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eCalculation accuracy of atomic forces at general grain boundaries and grain boundary triple junctions during annealing. \u003c/strong\u003eEnergies and atomic forces near the (a) general grain boundary (GGB) and (b) grain boundary (GB) triple junctions, calculated using cut-out calculation cells from a (28.3 nm)\u003csup\u003e3\u003c/sup\u003e polycrystal annealed at 300 K for 0.1 ns using moment tensor potential (MTP), through density functional theory (DFT) and interatomic potentials. The calculation cells were cut out one by one from the 64 GBs and 33 GB triple junctions in the nano-polycrystal.\u003c/p\u003e","description":"","filename":"fig4.png","url":"https://assets-eu.researchsquare.com/files/rs-4550958/v1/75e0983cbc1e78b35394b5d4.png"},{"id":61599514,"identity":"5217701c-8363-4b0e-8cc3-10f3b65bdb9d","added_by":"auto","created_at":"2024-08-01 17:34:30","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":1142167,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eCalculation accuracy of atomic forces at general grain boundaries and grain boundary triple junctions\u003c/strong\u003e \u003cstrong\u003eafter relaxation. \u003c/strong\u003eEnergies and atomic forces near the (a) general grain boundary (GGB) and (b) grain boundary (GB) triple junctions, calculated using cut-out calculation cells from a (28.3 nm)\u003csup\u003e3\u003c/sup\u003e polycrystal fully relaxed using the moment tensor potential (MTP), through density functional theory (DFT) and interatomic potentials. The calculation cells were cut out one by one from the 64 GBs and 33 GB triple junctions in the nano-polycrystal.\u003c/p\u003e","description":"","filename":"fig5.png","url":"https://assets-eu.researchsquare.com/files/rs-4550958/v1/75a2c51d551b15c3e0550c6e.png"},{"id":61599512,"identity":"34975268-dadc-45ec-90a6-606c005ab530","added_by":"auto","created_at":"2024-08-01 17:34:29","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":417805,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eCalculation accuracy for the entire nano-polycrystal based on extrapolation grade. \u003c/strong\u003e(a) Histograms of extrapolation grades for all atoms in the (28.3 nm)\u003csup\u003e3\u003c/sup\u003e polycrystal during annealing at 300, 600, and 1000 K and after relaxation using moment tensor potential (MTP). (b) Enlarged view of the high-extrapolation-grade region.\u003c/p\u003e","description":"","filename":"fig6.png","url":"https://assets-eu.researchsquare.com/files/rs-4550958/v1/8e87b6fb949b9092f351288e.png"},{"id":61599382,"identity":"84703b50-52f4-4407-a881-78766e2ee707","added_by":"auto","created_at":"2024-08-01 17:34:19","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":73009,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eRelationship between average grain boundary energy and model size in nano-polycrystals.\u003c/strong\u003e The general grain boundary (GB) energy of α-Fe estimated from the experimental results of W is also shown for comparison [89].\u003c/p\u003e","description":"","filename":"fig7.png","url":"https://assets-eu.researchsquare.com/files/rs-4550958/v1/cc0087a7f4cb07b1cc6c2d6a.png"},{"id":61599389,"identity":"25970d21-771c-4766-b719-019346097d6a","added_by":"auto","created_at":"2024-08-01 17:34:21","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":1749608,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eAtomic structure of α-Fe polycrystals.\u003c/strong\u003e (a) Voronoi volume and (b) coordination number of each atom in a (28.3 nm)\u003csup\u003e3\u003c/sup\u003e polycrystal with an average grain boundary (GB) energy of 1.58 J/m\u003csup\u003e2\u003c/sup\u003e. (c) Voronoi volume and (d) coordination number as a function of distance from the GB center. (e) Histograms of Voronoi volumes for all sites within 7.35 Å (three atomic layers) of the GB center and for each atomic layer from the GB center in the (28.3 nm)\u003csup\u003e3\u003c/sup\u003e polycrystal. (f) Voronoi volume as a function of distance from the GB center for the (28.3 nm)\u003csup\u003e3\u003c/sup\u003e polycrystals relaxed using the moment tensor potential (MTP), embedded atom method (EAM), or modified EAM (MEAM).\u003c/p\u003e","description":"","filename":"fig8.png","url":"https://assets-eu.researchsquare.com/files/rs-4550958/v1/d66a9e23bd83f9620d0ec715.png"},{"id":61599383,"identity":"d337868b-6a76-4daf-b468-9b578f1aa753","added_by":"auto","created_at":"2024-08-01 17:34:19","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":2248558,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eGrain boundary energy and atomic structure of α-Fe polycrystalline grain boundaries. \u003c/strong\u003eHistograms of (a) misorientation angle and (b) grain boundary (GB) energy for three (28.3 nm)\u003csup\u003e3\u003c/sup\u003e polycrystals obtained by moment tensor potential (MTP) relaxation. (c) Relationship between GB energy and misorientation angle for each of the three (28.3 nm)\u003csup\u003e3\u003c/sup\u003e polycrystals. (d) Probability density of the Voronoi volume in the one-atomic layer region near the GB center for these three GBs, along with the polycrystal in Case 1 for comparison. The probability density of the Σ3(111) symmetric tilt GB (1.58 J/m\u003csup\u003e2\u003c/sup\u003e) is also shown. Note that the probability density in Σ3(111) is scaled down to 1/10 for comparison. (e) Probability densities of these Voronoi volumes are shown as the difference from the probability density of the Voronoi volume in the one-atomic layer region near the GB center for the polycrystals in Case 1. (f) Atomic structures and energies of the atoms of the GBs with lowest energy (GBL), corresponding to the mode (GBM), and with highest energy (GBH). Here, the energy reference is the energy of the bulk Fe atom. The energies of these GBs are also shown.\u003c/p\u003e","description":"","filename":"fig9.png","url":"https://assets-eu.researchsquare.com/files/rs-4550958/v1/83a12e32f326cf58156bd273.png"},{"id":68984479,"identity":"30771c46-65ed-4f92-aa64-1925fc366864","added_by":"auto","created_at":"2024-11-14 08:10:31","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":13943525,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4550958/v1/c5ca5dd7-7666-49e7-997b-20fe26191194.pdf"},{"id":61599513,"identity":"d88f160a-3a64-443c-be62-48a968040db7","added_by":"auto","created_at":"2024-08-01 17:34:30","extension":"docx","order_by":12,"title":"","display":"","copyAsset":false,"role":"supplement","size":1687270,"visible":true,"origin":"","legend":"","description":"","filename":"Supplementaryinformation20240623.docx","url":"https://assets-eu.researchsquare.com/files/rs-4550958/v1/06b344431619e9d5ce697acd.docx"}],"financialInterests":"(Not answered)","formattedTitle":"Machine learning interatomic potential with DFT accuracy for general grain boundaries: Analysis of grain boundary energy and atomic structure in α-Fe polycrystals","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eIn recent years, there has been an increasing demand for the development of high-strength metallic materials to achieve carbon neutrality. For example, within the field of steel materials, increasing the strength of steel used for automotive steel plates can reduce CO\u003csub\u003e2\u003c/sub\u003e emissions by lightening the vehicle body. Thus, development of high-strength steels is being pursued intensively [\u003cspan additionalcitationids=\"CR2 CR3 CR4 CR5\" citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e]. Most metallic materials are polycrystalline, and therefore, contain grain boundaries (GBs). GBs have a significant influence on manufacturability and material properties, either directly or indirectly, through the formation of microstructures. However, enhancing material strength increases susceptibility to GB embrittlement, a phenomenon characterized by the degradation of material properties and manufacturability. This occurs due to cracking at GBs caused by the GB segregation of certain alloying elements or impurity atoms [\u003cspan additionalcitationids=\"CR8 CR9\" citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e]. Therefore, the suppression of hydrogen embrittlement [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e, \u003cspan additionalcitationids=\"CR12 CR13 CR14 CR15 CR16 CR17\" citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e], liquid metal embrittlement (LME) [\u003cspan additionalcitationids=\"CR20 CR21 CR22 CR23\" citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e], and red hot embrittlement [\u003cspan additionalcitationids=\"CR26 CR27 CR28\" citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e], all of which entail cracking at GBs, is a significant issue in the development of high-strength steels. Consequently, there is a critical need to design materials capable of suppressing the degradation of material properties and manufacturability associated with GB cracking.\u003c/p\u003e \u003cp\u003eA highly promising approach to suppress GB cracking involves controlling the GB segregation of alloying and impurity elements. Recently, the addition of Mo through segregation at GBs has been demonstrated to suppress GB cracking and improve the hydrogen embrittlement resistance of high-strength martensitic steels [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e, \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e, \u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e]. In LME resulting from Zn plating, the susceptibility to GB cracking depends on the GB character [\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e]. Furthermore, the addition of trace amounts of B suppresses Zn penetration into GBs due to the segregation of B at these boundaries. This process strengthens the GBs and effectively suppresses LME [\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e]. Therefore, obtaining material design guidelines to suppress GB cracking necessitates a quantitative understanding of the extent of GB segregation of diverse solute elements. This understanding should account for their content, heat treatment process, and GB character.\u003c/p\u003e \u003cp\u003eTo acquire such knowledge, GB segregation using highly symmetric GBs that can be treated by density functional theory (DFT), i.e., GBs with small Σ values, has been intensively studied [\u003cspan additionalcitationids=\"CR33 CR34 CR35 CR36 CR37 CR38 CR39 CR40\" citationid=\"CR32\" class=\"CitationRef\"\u003e32\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e41\u003c/span\u003e]. These studies have clarified that the amount of GB segregation of solute elements is strongly affected by the local atomic structure of each site constituting the GB. Specifically, factors such as Voronoi (or occupied) volume [\u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e32\u003c/span\u003e], coordination number, and interatomic distance play crucial roles [\u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e34\u003c/span\u003e, \u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e37\u003c/span\u003e]. However, such GB cracking primarily occurs at so-called general GBs (GGBs), which constitute most GBs in polycrystals and lack specific symmetry. Recent studies have revealed that predicting GB segregation in polycrystalline materials can be difficult to predict using highly symmetric GBs, which are amenable to analysis using DFT. This difficulty arises because of the diverse array of local atomic structures present within these boundaries [\u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e42\u003c/span\u003e]. Consequently, studying only highly symmetric GBs that can be analyzed using DFT is insufficient to obtain the knowledge necessary to control GB segregation.\u003c/p\u003e \u003cp\u003eTherefore, numerous studies have recently investigated GB segregation in GGBs using interatomic potentials, such as embedded atom method (EAM) and modified EAM (MEAM) potentials, and nano-polycrystalline models [\u003cspan additionalcitationids=\"CR44 CR45 CR46 CR47 CR48\" citationid=\"CR43\" class=\"CitationRef\"\u003e43\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR49\" class=\"CitationRef\"\u003e49\u003c/span\u003e]. However, interatomic potentials such as EAM and MEAM are rarely fitted to GBs. Consequently, they cannot accurately replicate the intricate local atomic structures found at stable GBs that affect GB segregation and the energies and dynamics of each atom that determine them. For example, the GB segregation energy of transition metal alloying elements in Fe varies approximately in proportion to the Voronoi volume of the segregation site. Moreover, the GB segregation energy exerts an exponential influence on the extent of GB segregation. Therefore, even a small change in Voronoi volume results in a significant difference in segregation levels. Furthermore, in α-Fe, interatomic potentials such as EAM and MEAM have been shown to significantly underestimate the GB energy of symmetric tilt GBs in pure Fe relative to DFT values [\u003cspan citationid=\"CR50\" class=\"CitationRef\"\u003e50\u003c/span\u003e]. Therefore, the local atomic structures at stable grain boundaries that affect GB segregation and the kinetics near grain boundaries that determine them also differ significantly from DFT.\u003c/p\u003e \u003cp\u003eIn recent years, highly accurate machine learning interatomic potentials (MLIPs) have been constructed [\u003cspan citationid=\"CR51\" class=\"CitationRef\"\u003e51\u003c/span\u003e, \u003cspan citationid=\"CR52\" class=\"CitationRef\"\u003e52\u003c/span\u003e], which have also been applied to the study of GBs. For example, Yokoi et al. have demonstrated that the GB energy of Al and its temperature dependence can be calculated with DFT accuracy using a neural network interatomic potential (NNIP) [\u003cspan citationid=\"CR53\" class=\"CitationRef\"\u003e53\u003c/span\u003e]. Various MLIPs for α-Fe have also been constructed [\u003cspan additionalcitationids=\"CR55 CR56 CR57 CR58 CR59\" citationid=\"CR54\" class=\"CitationRef\"\u003e54\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR60\" class=\"CitationRef\"\u003e60\u003c/span\u003e] since the development of the Gaussian approximate potential (GAP) [\u003cspan citationid=\"CR61\" class=\"CitationRef\"\u003e61\u003c/span\u003e] in 2018. For example, Mori et al. have created an NNIP for α-Fe that reproduces the stability of dislocation structures with an accuracy comparable to that of DFT [\u003cspan citationid=\"CR62\" class=\"CitationRef\"\u003e62\u003c/span\u003e]. In particular, the NNIP for Fe\u0026ndash;H explicitly includes symmetric tilt GBs with small Σ values in the training data, and provides accuracy comparable to that of DFT for the GB energy of symmetric tilt GBs [\u003cspan citationid=\"CR63\" class=\"CitationRef\"\u003e63\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eHowever, there are no reports on interatomic potentials capable of reproducing the local atomic structure and GB energies of GGBs, as well as the dynamics near GBs that determine them, with DFT accuracy. These aspects are essential for the design of future metallic materials. MLIP is recognized to be inaccurate in extrapolated regions [\u003cspan citationid=\"CR64\" class=\"CitationRef\"\u003e64\u003c/span\u003e], and its accuracy for GGBs with more complex and diverse atomic structures warrants thorough testing. This holds even if symmetric tilt GBs are included in the training data. However, GGBs are characterized by five degrees of freedom and comprise diverse and complex atomic structures [\u003cspan citationid=\"CR65\" class=\"CitationRef\"\u003e65\u003c/span\u003e]. Therefore, performing DFT calculations exhaustively on all GGBs, explicitly incorporating them as training data to construct interatomic potentials, and evaluating the accuracy of these potentials with respect to GGBs is challenging.\u003c/p\u003e \u003cp\u003eIn this study, rather than explicitly incorporating GGBs into the training dataset, we employed a training dataset generated through a method recently proposed by Paul et al. [\u003cspan citationid=\"CR66\" class=\"CitationRef\"\u003e66\u003c/span\u003e]. This method mechanically generates various atomic structures based on crystal space groups. The moment tensor potential (MTP) was selected as the MLIP because of its excellent balance between computational cost and accuracy [\u003cspan citationid=\"CR67\" class=\"CitationRef\"\u003e67\u003c/span\u003e, \u003cspan citationid=\"CR68\" class=\"CitationRef\"\u003e68\u003c/span\u003e]. The accuracy of the MLIP for GGBs was verified through direct comparison with DFT calculations for cells cut out in the vicinity of GBs using randomly oriented nano-polycrystals. These nano-polycrystals provided a comprehensive sampling of the atomic environment of polycrystals with grain sizes in the micrometer range. Additionally, the accuracy was evaluated through extrapolation grades based on active learning methods for the entire nano-polycrystal. These verifications demonstrated that the constructed MLIP can calculate the atomic structure, energy, and dynamics of arbitrary GBs, including GGBs, with a level of accuracy comparable to that of DFT. As an application of the MLIP, the average GB energy of α-Fe polycrystals, which is an important yet challenging property to measure, was calculated. Finally, the properties of GGBs in α-Fe polycrystals were analyzed at the atomic level and their relationship with GB segregation discussed. Consequently, we clarified that the MLIP constructed in this study is valuable for quantitatively predicting GB segregation at GGBs. This is because the previously constructed interatomic potentials are not sufficiently accurate for this purpose.\u003c/p\u003e"},{"header":"2. Methods","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e2.1. Details of calculations using DFT and interatomic potentials\u003c/h2\u003e \u003cp\u003eSpin-polarized electronic structure calculations and structural optimization were performed, using the Vienna ab initio simulation package (VASP) with the projector-augmented wave (PAW) method [\u003cspan citationid=\"CR69\" class=\"CitationRef\"\u003e69\u003c/span\u003e, \u003cspan citationid=\"CR70\" class=\"CitationRef\"\u003e70\u003c/span\u003e] within the generalized gradient approximation (GGA) framework, utilizing Perdew\u0026ndash;Burke\u0026ndash;Ernzerhof (PBE) parametrization [\u003cspan citationid=\"CR71\" class=\"CitationRef\"\u003e71\u003c/span\u003e]. These calculations were employed to build the training dataset and verify the accuracy of the interatomic potentials. The cutoff energy for the plane-wave basis set was set to 520 eV. The k-point mesh for each atomic structure was set to an accuracy equivalent to 18 \u0026times; 18 \u0026times; 18 for the α-Fe conventional unit cell using Monkhorst-Pack k-mesh [\u003cspan citationid=\"CR72\" class=\"CitationRef\"\u003e72\u003c/span\u003e]. The Methfessel\u0026ndash;Paxton smearing method [\u003cspan citationid=\"CR73\" class=\"CitationRef\"\u003e73\u003c/span\u003e] with a width of 0.1 eV was employed. The atomic positions were relaxed until reaching energy and force convergence values of 10\u003csup\u003e\u0026minus;\u0026thinsp;4\u003c/sup\u003e eV and 10\u003csup\u003e\u0026minus;\u0026thinsp;2\u003c/sup\u003e eV/\u0026Aring;, respectively. Ab initio molecular dynamics (AIMD) based on the Parrinello\u0026ndash;Rahman dynamics with the Langevin thermostat [\u003cspan citationid=\"CR74\" class=\"CitationRef\"\u003e74\u003c/span\u003e, \u003cspan citationid=\"CR75\" class=\"CitationRef\"\u003e75\u003c/span\u003e] were performed to generate training datasets, as described in Section \u003cspan refid=\"Sec4\" class=\"InternalRef\"\u003e2.2\u003c/span\u003e. Both NVT and NPT conditions were used with a time step of 2 fs.\u003c/p\u003e \u003cp\u003eAll calculations using interatomic potentials were conducted using LAMMPS [\u003cspan citationid=\"CR76\" class=\"CitationRef\"\u003e76\u003c/span\u003e], while OVITO [\u003cspan citationid=\"CR77\" class=\"CitationRef\"\u003e77\u003c/span\u003e] was used for visualizing the atomic structure. Additionally, calculations involving the quasi-harmonic approximation were performed using PHONOPY [\u003cspan citationid=\"CR78\" class=\"CitationRef\"\u003e78\u003c/span\u003e].\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e2.2. Training datasets\u003c/h2\u003e \u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e lists the training datasets, which comprise two types. The first dataset, referred to as the domain expertise (DE) dataset hereafter, considers the basic properties and lattice defects of α-Fe and was manually created. The second is a training dataset, referred to as the RANDSPG dataset hereafter, is based on atomic structures mechanically generated by the RANDSPG algorithm [\u003cspan citationid=\"CR79\" class=\"CitationRef\"\u003e79\u003c/span\u003e], mainly for reproducing GGBs [\u003cspan citationid=\"CR66\" class=\"CitationRef\"\u003e66\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eFor the RANDSPG dataset, the method used to construct the Mg training dataset [\u003cspan citationid=\"CR66\" class=\"CitationRef\"\u003e66\u003c/span\u003e] was modified and applied to α-Fe. The MTP constructed using this training dataset has been demonstrated to accurately reproduce physical properties and lattice defect energies, such as symmetric tilt GBs for low Σ values, for Mg. Notably, this is achieved even though the corresponding atomic structures are not explicitly included in the training dataset. Specifically, we first used the RANDSPG algorithm [\u003cspan citationid=\"CR79\" class=\"CitationRef\"\u003e79\u003c/span\u003e] to create the basic structure for constructing the training dataset. The main input parameters for generating the atomic structure are the maximum number of atoms, crystal space group to be considered, and allowable atomic volume per atom. Building on the study on Mg [\u003cspan citationid=\"CR66\" class=\"CitationRef\"\u003e66\u003c/span\u003e], 2482 basic structures (hereafter referred to as RANDSPGs) were generated. These structures were created with a maximum of 10 atoms, encompassing all crystal space groups, and allowing a tolerance of \u0026plusmn;\u0026thinsp;10% of the equilibrium lattice constant for the atomic volume per atom. These basic structures were then subjected to stepwise structural relaxation: (1) volume-only relaxation (VOLMIN), (2) cell shape-only relaxation with constant volume (CELLMIN), and (3) atomic position relaxation (INTMIN). Each of these structures was then used as a candidate training dataset. In addition, the atomic structure of INTMIN was disturbed by (1) random triaxial strain (TRIAX, up to 80%), (2) a combination of random shear strains (SHEAR, up to 80%), and (3) random displacements of atoms combined with a small random strain tensor (RATTLE, 0.5 \u0026Aring; mean displacement and up to 5% strain). These modified structures were also included as candidates for the training dataset. DFT calculations were performed on these structures to construct the training dataset. However, owing to the high-spin degrees of freedom of Fe, many atomic structures with nonmagnetic, low-spin, and spin-flip magnetic states were obtained. Consequently, we supposed that including these structures in the training data would hinder the attainment of an MTP capable of reproducing ferromagnetic α-Fe with high accuracy. Therefore, for the training dataset, we exclusively included structures possessing magnetic moments with the same direction and values exceeding 1.5 \u0026micro;B. This value corresponds to the magnitude of stable magnetic moments in which the body-centered cubic (bcc), face-centered cubic (fcc), hexagonal close-packed (hcp), and simple cubic structures of Fe do not transition to the nonmagnetic state. The number of atomic structures and atomic environments in the final RANDSPG dataset was comparable to those included in the training dataset for Mg in previous studies [\u003cspan citationid=\"CR66\" class=\"CitationRef\"\u003e66\u003c/span\u003e], comprising 17460 structures and 117146 atomic environments.\u003c/p\u003e \u003cp\u003eA DE dataset was added to increase the accuracy of the physical properties of α-Fe and lattice defects such as self-interstitial atoms (SIAs). This also serves to compensate for the compressible structure, which renders the magnetic moment unstable and is preferentially excluded in the selection process of the RANDSPG dataset described above. Specifically, AIMD was conducted on 3 \u0026times; 3 \u0026times; 3 supercells of α-Fe at 300, 600, 1000, 1400, 2000, and 3000 K under the NPT ensemble. AIMD was also performed at 300, 600, 1000, 1400, 2000, and 3000 K under the NVT ensemble for 3 \u0026times; 3 \u0026times; 3 supercells of α-Fe, with lattice constants varying within \u0026plusmn;\u0026thinsp;2.5% from the equilibrium lattice constant. In addition, AIMD simulations were conducted at 300, 600, 1000, 1400, 2000, and 3000 K under the NPT ensemble for a 3 \u0026times; 3 \u0026times; 3 supercell of α-Fe containing a single vacancy. Atomic structures were extracted from these AIMDs every five steps and included in the training dataset. Snapshots of the structural relaxation process of the structure containing an SIA were also added to the training dataset.\u003c/p\u003e \u003cp\u003eFinally, the two datasets were combined to create a training dataset for building interatomic potentials consisting of 19950 structures and 267188 atomic environments. In particular, the RANDSPG dataset contains a relatively large number of structures but consists of at most 10 atoms. Considering that, for single-element systems, the computational cost of DFT is proportional to the cube of the number of atoms, it is noteworthy that the computational cost of constructing these training datasets is very small. Specifically, the computational cost of obtaining the RANDSPG dataset is comparable to the computational cost of performing AIMD simulations at 300, 600, and 1000 K using the 3 \u0026times; 3 \u0026times; 3 supercells of α-Fe in the DE dataset.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003e\u003cb\u003eDetails of the training dataset for α-Fe.\u003c/b\u003e The training dataset for α-Fe consists of the domain expertise (DE) dataset, which is a training dataset for reproducing the basic properties and lattice defects of α-Fe, and the RANDSPG dataset, which is a training dataset for reproducing general grain boundaries. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({N}_{\\text{s}\\text{t}\\text{r}}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({N}_{\\text{f}\\text{o}\\text{r}\\text{c}\\text{e}}\\)\u003c/span\u003e\u003c/span\u003e are the number of atomic structures (number of total energies) and atomic forces, respectively, in the training dataset. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({N}_{\\text{a}\\text{t}\\text{o}\\text{m}}\\)\u003c/span\u003e\u003c/span\u003e is the number of atoms in each atomic structure in the training dataset. The root mean squared errors (RMSEs) of the energies and forces for each training dataset are also shown.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"7\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDatasets\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDataset\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cem\u003eN\u003c/em\u003e\u003csub\u003estr\u003c/sub\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cem\u003eN\u003c/em\u003e\u003csub\u003eatom\u003c/sub\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003e\u003cem\u003eN\u003c/em\u003e\u003csub\u003eforce\u003c/sub\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eEnergy (meV/atom)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003eForce (meV/\u0026Aring;)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDE\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePerfect crystal\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1776\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e54\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e95936\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e2.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e69.38\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eVacancy\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e500\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e53\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e26500\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e1.33\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e63.88\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSIA\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e214\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e129\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e27606\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e1.86\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e36.43\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eTotal\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e2490\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e150042\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e2.03\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e63.6\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eRANDSPGs\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eRANDSPG\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e691\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e3\u0026ndash;10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e4078\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e16.28\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e88.49\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eVOLMIN\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1273\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e3\u0026ndash;10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e7918\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e12.72\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e77.98\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eCELLMIN\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1836\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e3\u0026ndash;10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e11882\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e11.14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e83.2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eINTMIN\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e2015\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e3\u0026ndash;10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e13236\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e11.3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e54\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eTRIAX\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e5187\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e3\u0026ndash;10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e35806\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e16.15\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e62.7\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSHEAR\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e3209\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e3\u0026ndash;10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e22134\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e21.66\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e83.85\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eRATTLE\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e3249\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e3\u0026ndash;10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e22092\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e9.1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e114.45\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eTotal\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e17460\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e117146\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e15.14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e81.94\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAll\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eTotal\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e19950\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e267188\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e14.11\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e71.88\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e2.3. Construction of MTP\u003c/h2\u003e \u003cp\u003eIn this study, the MLIP was constructed using the MTP formalism [\u003cspan citationid=\"CR67\" class=\"CitationRef\"\u003e67\u003c/span\u003e, \u003cspan citationid=\"CR68\" class=\"CitationRef\"\u003e68\u003c/span\u003e]. The MTP can represent various atomic environments through angle-dependent many-body interactions, expressed as tensor products of atomic displacements. Moreover, it has no transcendental functions and relies solely on polynomial arithmetic operations, thereby significantly reducing the computational cost. Owing to these advantages, in certain systems, MTP can achieve accuracy levels comparable to those of GAP, which is considered one of the most accurate MLIPs, in less than a tenth of the computation time [\u003cspan citationid=\"CR80\" class=\"CitationRef\"\u003e80\u003c/span\u003e]. A template of level 22 potentials was employed to construct the MTP. The maximum cutoff radius was set to 6.5 \u0026Aring;, consistent with the NNIP for Fe\u0026ndash;H [\u003cspan citationid=\"CR63\" class=\"CitationRef\"\u003e63\u003c/span\u003e], and training was conducted using the MLIP-3 package [\u003cspan citationid=\"CR80\" class=\"CitationRef\"\u003e80\u003c/span\u003e]. The MTP constructed in this study can be accessed via the URL provided in the Data Availability section.\u003c/p\u003e \u003cp\u003eAs shown in Supplementary Fig.\u0026nbsp;1, the RMSEs of energy for the training and validation datasets were 14.11 and 14.79 meV/atom, respectively. The RMSE of atomic force was 71.88 meV/\u0026Aring; for the training dataset and 75.42 meV/\u0026Aring; for the validation dataset. The RMSE of energy was approximately fourfold larger than that of the previously reported NNIP of Fe\u0026ndash;H [\u003cspan citationid=\"CR63\" class=\"CitationRef\"\u003e63\u003c/span\u003e]. However, this discrepancy can be attributed to the diverse range of atomic structures present in the RANDSPG dataset, a trend also observed in the construction of the MTP for Mg [\u003cspan citationid=\"CR66\" class=\"CitationRef\"\u003e66\u003c/span\u003e]. Indeed, as shown in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, the RMSE for the DE dataset is comparable to the NNIP of Fe\u0026ndash;H. Supplementary Table \u003cspan refid=\"MOESM1\" class=\"InternalRef\"\u003eS1\u003c/span\u003e and Supplementary Fig.\u0026nbsp;2 demonstrate that the constructed MTP reproduced the lattice parameter, elastic modulus, and phonon dispersion of α-Fe, along with the defect energies for SIAs and vacancy defects with the same accuracy as that of the NNIP. In addition, the two-dimensional energy profiles of screw dislocation core positions [\u003cspan citationid=\"CR62\" class=\"CitationRef\"\u003e62\u003c/span\u003e] and generalized stacking fault energy surfaces [\u003cspan citationid=\"CR81\" class=\"CitationRef\"\u003e81\u003c/span\u003e], crucial for analyzing plastic deformation and crack propagation behavior, were also reproduced with very good accuracy. These results suggest excellent transferability of the MTP, as discussed in Section \u003cspan refid=\"Sec8\" class=\"InternalRef\"\u003e3\u003c/span\u003e.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e2.4. Method for evaluating the accuracy of interatomic potentials for GGBs\u003c/h2\u003e \u003cp\u003eIn this study, we evaluated the accuracy of the constructed MTP for GGBs using GBs in nano-polycrystals with random orientations obtained by Voronoi tessellation [\u003cspan citationid=\"CR82\" class=\"CitationRef\"\u003e82\u003c/span\u003e] and relaxation through molecular dynamics calculations. Detailed analysis of the GB structure of nano-polycrystals obtained through this method suggests that the GB structure is similar to that of general polycrystals with grain sizes larger than micrometer dimensions [\u003cspan citationid=\"CR83\" class=\"CitationRef\"\u003e83\u003c/span\u003e]. The time evolution of the average grain size of nano-polycrystals prepared by this method via annealing exhibits similar growth behavior as polycrystals with grain sizes larger than micrometer dimensions, growing at a rate proportional to the square root of time [\u003cspan citationid=\"CR84\" class=\"CitationRef\"\u003e84\u003c/span\u003e]. Given grain growth behavior is strongly influenced by the nature of GBs [\u003cspan citationid=\"CR85\" class=\"CitationRef\"\u003e85\u003c/span\u003e], these results suggest that the GBs of nano-polycrystals created with Voronoi tessellation are similar to those found in common polycrystals. Furthermore, Wagih et al. demonstrated that the histogram of GB segregation energies, calculated for nano-polycrystals generated via Voronoi tessellation, exhibits little change for a 15 nm cubic polycrystalline model consisting of eight grains, even as the number of nano-polycrystalline grains and their model size increase [\u003cspan citationid=\"CR46\" class=\"CitationRef\"\u003e46\u003c/span\u003e]. They demonstrated that nano-polycrystals created through Voronoi tessellation could effectively represent the diverse local atomic environments found in real polycrystals with actual grain sizes in the micrometer range., since the GB segregation energy at each site depends on the local atomic environment of the host metal around the solute atoms [\u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e42\u003c/span\u003e]. The energy and force of each atom of the host metal are determined by the local atomic environment as well as GB segregation. Therefore, if the constructed MTP can accurately reproduce the local atomic environment in nano-polycrystals created through Voronoi tessellation, it should be capable of reproducing GBs, or GGBs, in polycrystals with grain sizes in the micrometer range.\u003c/p\u003e \u003cp\u003eWe evaluated the accuracy of the MTP for GGBs in two ways using nanocrystalline polycrystals created via Voronoi tessellation. The first method involved directly comparing energies and atomic forces by conducting DFT calculations on numerous DFT-calculable regions selected from the GBs within the nano-polycrystals, obtained by molecular dynamics relaxation using the constructed MTP. However, while this method provides direct verification, the computational cost makes it impractical to evaluate the accuracy of the entire nano-polycrystal. As a second method to address this limitation, all local atomic environments constituting the nano-polycrystal were evaluated. This involved calculating extrapolation grades that can numerically determine whether the constructed MTP is an interpolated region where high accuracy can be expected or an extrapolated region where accuracy is reduced [\u003cspan citationid=\"CR80\" class=\"CitationRef\"\u003e80\u003c/span\u003e]. Validation of the transferability of the constructed potentials, based on the evaluation of extrapolation grades, has also been employed in previous studies [\u003cspan citationid=\"CR66\" class=\"CitationRef\"\u003e66\u003c/span\u003e]. While not a direct accuracy assessment, this method is relatively computationally inexpensive because it relies on pre-constructed MTPs, enabling evaluation for all atomic environments within the nano-polycrystal.\u003c/p\u003e \u003cp\u003eThe initial structure of the nano-polycrystals was generated using Atomsk [\u003cspan citationid=\"CR82\" class=\"CitationRef\"\u003e82\u003c/span\u003e] with Voronoi tessellation to create polycrystals with random orientations. The structural relaxation of the nano-polycrystals was based on the method outlined by Van Swygenhoven et al. [\u003cspan citationid=\"CR83\" class=\"CitationRef\"\u003e83\u003c/span\u003e]. This involved initially relaxing the cell size and atomic positions using the conjugate gradient method, followed by annealing for 0.2 ns at 300 K in the NPT ensemble. Subsequently, the system was cooled from 300 to 0.1 K before performing another relaxation step to adjust the cell size and atomic positions using the conjugate gradient method. The annealing time was set as the time for sufficient convergence of the obtained GB energy. Eight grains were included in the nano-polycrystals, following the methodology outlined by Wagih et al. in their study of GB segregation of Mg in Al using EAM [\u003cspan citationid=\"CR46\" class=\"CitationRef\"\u003e46\u003c/span\u003e]. The initial dimensions of the nano-polycrystals were set to 28.3 \u0026times; 28.3 \u0026times; 28.3 nm\u003csup\u003e3\u003c/sup\u003e. As shown in Supplementary Fig.\u0026nbsp;4, the local atomic environment in nano-polycrystals created under these conditions is largely independent of the random seeding of the initial structure creation and the expansion of the model size.\u003c/p\u003e \u003cp\u003eThe method of comparison using DFT calculations is illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. For this analysis, we used a nano-polycrystal with initial cell dimensions of 28.3 \u0026times; 28.3 \u0026times; 28.3 nm\u003csup\u003e3\u003c/sup\u003e, hereafter denoted as (28.3 nm)\u003csup\u003e3\u003c/sup\u003e polycrystal. Specifically, we utilized the structure of the (28.3 nm)\u003csup\u003e3\u003c/sup\u003e polycrystal during annealing (at 0.1 ns) and subsequent relaxation, as described in the previous paragraph. From all the GBs (64) in each nano-polycrystal, 64 regions with dimensions of 15.0 \u0026times; 15.0 \u0026times; 15.0 \u0026Aring;\u003csup\u003e3\u003c/sup\u003e (~\u0026thinsp;250 atoms) were cut out as calculation cells. Given polycrystals typically contain GB triple junctions, we also cut out 33 regions measuring 15.0 \u0026times; 15.0 \u0026times; 15.0 \u0026Aring;\u003csup\u003e3\u003c/sup\u003e from the GB triple junctions in each nano-polycrystal as calculation cells. Periodic boundary conditions were applied to the cells, and if the distance between the atoms was less than 2.0 \u0026Aring;, one of the atoms was removed from the cell boundary. As depicted in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e(c) and (d), the boundary of the calculation cell, influenced by the periodic boundary conditions, exhibits a structure like a general GB. Consequently, the energy of the system differs from that which would be observed if this region existed within the nano-polycrystal. Nevertheless, achieving sufficient accuracy for these calculation cells suggests the potential to accurately compute the energies of atoms proximal to GBs and GB triple junctions. The atomic forces at the boundaries of the calculation cell also differ from those in the nano-polycrystal. However, the atomic forces near the center of the calculation cell, which is sufficiently far from the cell boundaries, are consistent with those observed in the nano-polycrystal. As illustrated in Supplementary Fig.\u0026nbsp;3, the atomic forces acting on atoms within 4.0 \u0026Aring; of the center of the calculation cell are maintained within a 10% error margin. Therefore, for the comparison of the atomic forces near the general GB, we compared the results of the DFT and interatomic potential calculations for atoms within 4.0 \u0026Aring; of the center of the calculation cell and in the one-atomic layer region (2.46 \u0026Aring;) from the GB center. For atoms near the GB triple junctions, DFT and MTP calculations were compared for atoms within 4.0 \u0026Aring; of the calculation cell center and 2.46 \u0026Aring; from the nearest and second nearest GBs, respectively. The GB center is defined as a Voronoi polyhedron when the initial structure is created by Voronoi tessellation. Remarkably, during annealing at 300 K and subsequent relaxation via the conjugate gradient method, the relative positions of the GBs exhibited minimal movement, despite changes in the cell size. For comparison, we also evaluated the NNIP for Fe\u0026ndash;H [\u003cspan citationid=\"CR63\" class=\"CitationRef\"\u003e63\u003c/span\u003e] using the same methodology. Notably, this is the only MLIP that utilizes symmetric tilt GBs for α-Fe as training data. Similarly, MEAM [\u003cspan citationid=\"CR86\" class=\"CitationRef\"\u003e86\u003c/span\u003e] and EAM [\u003cspan citationid=\"CR87\" class=\"CitationRef\"\u003e87\u003c/span\u003e] were evaluated for accuracy using this method. These two interatomic potentials are identical to those used for comparison in previous studies [\u003cspan citationid=\"CR50\" class=\"CitationRef\"\u003e50\u003c/span\u003e].\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eTo provide a relative interpretation of the accuracy evaluated for GGBs, the GB energies and atomic forces acting on atoms within 2.46 \u0026Aring; of the GB center during annealing were also compared through DFT calculations for eight\u0026thinsp;\u0026lt;\u0026thinsp;110\u0026thinsp;\u0026gt;\u0026thinsp;and \u0026lt;\u0026thinsp;100\u0026thinsp;\u0026gt;\u0026thinsp;symmetric tilt GBs with Σ values ranging from 3 to 11, which can be treated by DFT calculations. In addition, we investigated whether the stable GB structure obtained by DFT relaxation can be reproduced by the interatomic potentials. For both DFT and interatomic potential calculations of GB energies, several initial structures were created with one grain rigidly shifted parallel to the GB, and the cell sizes and atomic configurations were subsequently relaxed. In these interatomic potential calculations, OpenKIM [\u003cspan citationid=\"CR88\" class=\"CitationRef\"\u003e88\u003c/span\u003e] was employed. AIMD was conducted at 300, 600, and 1000 K in the NVT ensemble for eight stable structures of symmetric tilt GBs derived from DFT. For the atomic structures during annealing, the atomic forces acting on atoms in the region within 2.46 \u0026Aring; of the GB center were calculated using interatomic potentials and compared with those obtained from DFT calculations.\u003c/p\u003e \u003cp\u003eFor the extrapolation grade calculations, the structures of the (28.3 nm)\u003csup\u003e3\u003c/sup\u003e polycrystal during (at 0.1 ns) and after relaxation were used, alongside direct comparisons with DFT calculations. For all atoms constituting these two structures, the extrapolation grade of the local atomic environment for the active learning method recently implemented in MLIP-3 was evaluated [\u003cspan citationid=\"CR80\" class=\"CitationRef\"\u003e80\u003c/span\u003e]. In the extrapolation grade calculations, extrapolation grades were evaluated for nano-polycrystals annealed at 600 and 1000 K for 0.1 ns in addition to those at 300 K. For details on extrapolation grades, see Ref. [\u003cspan citationid=\"CR80\" class=\"CitationRef\"\u003e80\u003c/span\u003e]. Briefly, an atom with an extrapolation grade between 0 and 1 falls into the \"interpolation region\" for the constructed interatomic potentials, which guarantees high calculation accuracy. Extrapolation grades between 1 and 2 indicate the \"accurate extrapolation\u0026rdquo; region, while those between 2 and 10 fall into the \"reliable extrapolation\u0026rdquo; region. Structures containing atoms in this last range are additionally labeled by DFT calculations during the active learning process to indicate insufficient learning of their atomic environment. If the extrapolation grade of an atom exceeds 10, it is considered to be in the \"dangerous extrapolation\u0026rdquo; region and the learning process is terminated [\u003cspan citationid=\"CR80\" class=\"CitationRef\"\u003e80\u003c/span\u003e]. Thus, an extrapolation grade of 2 or less for all atoms in the annealed and relaxed nano-polycrystals indicates that the obtained interatomic potentials are sufficiently learned for the entire polycrystal. Consequently, no additional learning is required in active learning and high accuracy is expected.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003e2.5. Analysis of GB energy and atomic structure in α-Fe polycrystals\u003c/h2\u003e \u003cp\u003eThe determination of the GB energy of GGBs in polycrystalline metallic materials remains a significant area of research. To date, these energies have been determined through experimentation [\u003cspan citationid=\"CR89\" class=\"CitationRef\"\u003e89\u003c/span\u003e]. Therefore, as an application of the MTP constructed in this study, we calculated the average GB energy of α-Fe polycrystals using the nano-polycrystals described in Section \u003cspan refid=\"Sec6\" class=\"InternalRef\"\u003e2.4\u003c/span\u003e. We then compared this energy to those of GGBs calculated based on experimental data. Furthermore, we analyzed the properties of GGBs in α-Fe polycrystals at the atomic level and discussed the relationship with GB segregation.\u003c/p\u003e \u003cp\u003eThe area of the GB required for calculating GB energies was determined by correcting the area of the Voronoi polyhedron, which was utilized to generate the initial structure of the nano-polycrystal, for the change in cell size before and after relaxation. Notably, during annealing at 300 K and subsequent relaxation through the conjugate gradient method, the relative positions of the GBs exhibit minimal movement, despite changes in the cell size. However, the average GB energy may strongly depend on the number of grains (or the effect of GB character) and the size of the grains. Therefore, fixing the number of grains at eight, we calculated the average GB energy for nano-polycrystals with cell sizes ranging from 5.66 \u0026times; 5.66 \u0026times; 5.66 nm\u003csup\u003e3\u003c/sup\u003e (20 times the lattice constant of Fe for a side length) to 28.3 \u0026times; 28.3 \u0026times; 28.3 nm\u003csup\u003e3\u003c/sup\u003e (100 times the lattice constant of Fe for a side length). This allowed us to investigate the dependence of the average GB energy on the grain size. For each cell size, the dependence on GB character was further evaluated. This was achieved by calculating the average GB energy of nano-polycrystals for three cases, in which only the random seed was varied when creating the initial nano-polycrystal structure by Voronoi tessellation.\u003c/p\u003e \u003c/div\u003e"},{"header":"3. Results and discussion","content":"\u003cdiv id=\"Sec9\"\u003e\n \u003ch2\u003e3.1. Accuracy of the MTP for symmetric tilt GBs\u003c/h2\u003e\n \u003cp\u003eTo provide a relative interpretation of the accuracy for GGBs, we first present the accuracy for symmetric tilt GBs, where the GB energy is well defined and a direct comparison with the DFT calculations is possible for both energy and atomic forces. Figure\u0026nbsp;2 depicts the GB energies of the symmetric tilt GBs, obtained through relaxation using both DFT calculations and interatomic potentials. On average, the difference between the GB energies of the eight symmetric tilt GBs calculated with the constructed MTP and those calculated with DFT is 0.037 J/m\u003csup\u003e2\u003c/sup\u003e. This result indicates that the MTP reproduces the DFT values with high accuracy. In particular, the MTP exhibits a level of high accuracy comparable to that of the NNIP, which explicitly incorporates training data from \u0026Sigma;3(112), \u0026Sigma;3(111), \u0026Sigma;5(210), and \u0026Sigma;5(310). Conversely, the GB energies calculated using the MEAM and EAM greatly underestimate those calculated using DFT.\u003c/p\u003e\n \u003cp\u003eFigure\u0026nbsp;3 presents the atomic forces near the GB center (within the one-atomic layer region from the GB center) for eight symmetric tilt GBs. The calculations were performed using both DFT and interatomic potential calculations, with GBs annealed at 300, 600, and 1000 K using AIMD under the NVT ensemble. Figure\u0026nbsp;3(a) displays the collective atomic forces acting on atoms within the eight symmetric tilt angle GBs. In contrast, Fig.\u0026nbsp;3(b) and (c) showcase the atomic forces at the symmetric tilt GBs calculated using MTP and NNIP, respectively, for each of the eight GBs. Notably, the annealing structures of \u0026Sigma;3(112), \u0026Sigma;3(111), \u0026Sigma;5(210), and \u0026Sigma;5(310) are explicitly included in the training data for the NNIP. The MTP demonstrates higher accuracy compared to the NNIP, EAM, and MEAM across both GBs and all annealing temperatures. For example, the RMSE of the atomic force for the eight symmetric tilt GBs at 300 K is 82.4 meV/\u0026Aring;. Notably, upon focusing on the accuracy for each GB, MTP reproduces the DFT for the GBs that are explicitly included in the NNIP training dataset with the same high accuracy as that of the NNIP. Additionally, for GBs absent in the NNIP training dataset, the MTP consistently exhibits higher accuracy than the NNIP. In particular, NNIP displays a particularly large error for structures annealing in \u0026Sigma;9(114) and \u0026Sigma;11(113), which are not included in the training dataset. This result clearly shows that even though the stable structure of GBs and their GB energies can be calculated with good accuracy, the error in the dynamics of GBs may be large.\u003c/p\u003e\n \u003cp\u003eIn summary, the MTP constructed in this study can accurately calculate both the GB energy of symmetric tilt GBs and their atomic forces during annealing, without explicitly including them in the training data. This suggests that the constructed MTP exhibits excellent accuracy for GGB stable structures, as well as their associated energies and dynamics.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec10\"\u003e\n \u003ch2\u003e3.2. Accuracy of the MTP for GGBs\u003c/h2\u003e\n \u003cp\u003eFigure\u0026nbsp;4(a) and (b) depict the energies and atomic forces near the GGB and GB triple junctions, respectively. These calculations employed cut-out calculation cells from a (28.3 nm)\u003csup\u003e3\u003c/sup\u003e polycrystal annealed at 300 K for 0.1 ns via MTP, using DFT and interatomic potentials. Considering that the energy basis varies depending on the interatomic potentials, the energy was compared by measuring the change from the energy of an atom of \u0026alpha;-Fe with a lattice constant of 2.830 \u0026Aring; at each potential. Here, 2.830 \u0026Aring; represents the equilibrium lattice constant of \u0026alpha;-Fe in DFT, MTP, and NNIP. Notably, the periodic boundary conditions result in a GB-like defect structure near the boundary of the calculation cell, which differs from that observed in the nano-polycrystal. The RMSE of the energy and the atomic force for the general GB in the constructed MTP are 2.17 meV/atom and 81.05 eV/\u0026Aring;, respectively. The RMSE of the atomic force on the general GB is comparable to that observed during annealing at 300 K for symmetric tilt GBs, for which the GB energy can be accurately calculated. This indicates that MTP exhibits high calculation accuracy for GGBs. The RMSE of the energy and atomic force for the constructed MTP for GB triple junctions are 2.44 meV/atom and 86.21 eV/\u0026Aring;, respectively, which are also comparable to those for GGBs. In particular, the force and energy RMSEs for both the general GB and GB triple junctions were approximately half those obtained using NNIP. These results clearly demonstrate that the constructed MTP accurately reproduces the dynamics near the general GB and GB triple junctions of nano-polycrystals at 300 K.\u003c/p\u003e\n \u003cp\u003eFigure\u0026nbsp;5(a) and (b) depict the energies and atomic forces near the GGB and GB triple junctions, respectively. These calculations utilized cut-out calculation cells from a (28.3 nm)\u003csup\u003e3\u003c/sup\u003e polycrystal annealed at 300 K and relaxed via the conjugate gradient method using MTP, through DFT and interatomic potentials. The RMSEs of the energy and atomic force for the constructed MTP for a GGB are 2.44 meV and 80.34 meV/\u0026Aring;, respectively. Given these are the forces acting on the atoms in the nano-polycrystal after relaxation, the atomic forces tend to approach zero. The RMSEs of the energy of the GB triple junctions and atomic force in the constructed MTP were determined to be 2.44 meV and 81.55 meV/\u0026Aring;, respectively. These results indicate that nano-polycrystals can be relaxed using MTP with good accuracy, and stable structures and their energies can be obtained with DFT accuracy.\u003c/p\u003e\n \u003cp\u003eOwing to the cost of DFT calculations, these described calculations are limited to a portion of the nano-polycrystal. Therefore, to investigate the accuracy across the entire nano-polycrystal, encompassing all local atomic environments in a typical polycrystal, we constructed histograms of extrapolation grades for all atoms in the (28.3 nm)\u003csup\u003e3\u003c/sup\u003e polycrystal during annealing at 300, 600 and 1000 K and after relaxation using MTP (Fig. 6). Most extrapolation grades for approximately two million sites for all structures were determined to be less than one. This indicates that most local atomic structures in the polycrystals fall within the interpolation region, thereby guaranteeing high calculation accuracy in the constructed MTP [80]. While there were a few atoms with high extrapolation grades, the maximum extrapolation grade observed was 1.72, which falls within the \u0026quot;accurate extrapolation\u0026rdquo; region. Thus, even with the application of active learning methods based on the local atomic environment to these nano-polycrystals, the constructed MTPs remain unchanged. Consequently, in comparison with both direct DFT calculations and extrapolation grade evaluations, the constructed MTPs demonstrate superior DFT accuracy for the stable structures of GGBs and GB triple junctions, as well as their energies and dynamics.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec11\"\u003e\n \u003ch2\u003e3.3. Average GB energy of \u0026alpha;-Fe polycrystals using the constructed MTP\u003c/h2\u003e\n \u003cp\u003eFigure\u0026nbsp;7 illustrates the relationship between average GB energy and model size in nano-polycrystals. As the model size increases, the average GB energy converges towards a constant value. For example, the difference in average GB energies between the (22.6 nm)\u003csup\u003e3\u003c/sup\u003e and (28.3 nm)\u003csup\u003e3\u003c/sup\u003e polycrystals is merely 0.03 J/m\u003csup\u003e2\u003c/sup\u003e. The average GB energy for the three (28.3 nm)\u003csup\u003e3\u003c/sup\u003e polycrystals is 1.57 J/m\u003csup\u003e2\u003c/sup\u003e with a standard error of 0.03 J/m\u003csup\u003e2\u003c/sup\u003e. This result indicates that despite the presence of approximately 60 GBs, the deviation of the average GB energy among polycrystalline models in relation to the GB character is relatively small. The average GB energy of the (5.66 nm)\u003csup\u003e3\u003c/sup\u003e polycrystal is also relatively small. This is attributed to the presence of extremely small grains, which promote grain growth even at 300 K, thereby leading to a decrease in the actual GB area. On the other hand, in the calculation of the average GB energy, the area of the GB was evaluated based on the area of the Voronoi polyhedron when the initial structure was created.\u003c/p\u003e\n \u003cp\u003eThe calculated average GB energies of nano-polycrystals were compared to those of GGBs based on experimental data. For bcc metals, Li et al. investigated the scaling factor among different defect structures of the same metal and among the same defect structures across different metals [89]. Subsequently, they estimated the energy of the GGB of \u0026alpha;-Fe at 0 K, using experimental values of the grain boundary energy of the GGB of W measured at high temperatures. The results revealed that the GGB energy of \u0026alpha;-Fe is 1.63 J/m\u003csup\u003e2\u003c/sup\u003e. This value is in good agreement with the average GB energy of 1.57 J/m\u003csup\u003e2\u003c/sup\u003e observed for the (28.3 nm)\u003csup\u003e3\u003c/sup\u003e polycrystal (Fig. 7). In contrast, the EAM and MEAM significantly underestimate the average GB energy. This finding indicates that the choice of interatomic potential markedly influences the simulations of grain growth driven by GB energy. In addition, the underestimation of GB energy is related to the Voronoi volume at sites near the GB (Section 3.4), which also plays a significant role in GB segregation. Notably, the average GB energy was not calculated using the NNIP owing to limited computing resources.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec12\"\u003e\n \u003ch2\u003e3.4. GB energy and atomic structure in \u0026alpha;-Fe polycrystals\u003c/h2\u003e\n \u003cp\u003eThe findings presented in Section 3.3 reveal that the constructed MTP can describe \u0026alpha;-Fe polycrystals with high accuracy. The (28.3 nm)\u003csup\u003e3\u003c/sup\u003e polycrystals relaxed through the constructed MTP serve as a good sampling of the local atomic environment found in polycrystals with grain sizes in the micrometer range, a common feature in steel materials. Moreover, the average GB energy closely approximates that of polycrystals. Therefore, we supposed that a detailed investigation of the nano-polycrystals relaxed through the MTP would be particularly interesting. In particular, as mentioned in the Introduction (Section 1), GB segregation is an important control target in material design. Consequently, we conducted a thorough investigation on the Voronoi volume and coordination number, both dominant factors for GB segregation, for nano-polycrystals relaxed through MTP.\u003c/p\u003e\n \u003cp\u003eFigures\u0026nbsp;8(a) and (b) illustrate the Voronoi volume and coordination number of each atom, respectively, in a (28.3 nm)\u003csup\u003e3\u003c/sup\u003e polycrystal with an average GB energy of 1.58 J/m\u003csup\u003e2\u003c/sup\u003e. Additionally, Fig.\u0026nbsp;8(c) and (d) show the Voronoi volume and coordination number, respectively, as a function of distance from the GB center. These results pertain to one (28.3 nm)\u003csup\u003e3\u003c/sup\u003e polycrystal; however, the results for the other two (28.3 nm)\u003csup\u003e3\u003c/sup\u003e polycrystals are almost identical. Notably, the coordination number is defined as the number of planes obtained through Voronoi tessellation in each atomic region. Hence, the next nearest neighbor equivalent to the bcc structure is also counted as a coordination number. The Voronoi volume begins to increase in standard deviation from the GB center at approximately 8.0 \u0026Aring;. The mean Voronoi volume at the GB center measures 12.0 \u0026Aring;\u003csup\u003e3\u003c/sup\u003e, marking a 6.0% increase compared to the bulk volume. The coordination number increases in standard deviation from the GB center to approximately 7.0 \u0026Aring;; however, the mean value remains relatively unchanged and is 14.2 at the GB center. Thus, when considering the local atomic environment in terms of Voronoi volume and coordination number, the deviation from the bulk is observed at approximately 8.0 \u0026Aring; from the GB center.\u003c/p\u003e\n \u003cp\u003eFigure\u0026nbsp;8(e) presents histograms of Voronoi volumes for all sites within 7.35 \u0026Aring; (three atomic layers) of the GB center and for each atomic layer from the GB center in the (28.3 nm)\u003csup\u003e3\u003c/sup\u003e polycrystal. In the (28.3 nm)\u003csup\u003e3\u003c/sup\u003e polycrystal, the Voronoi volume ranges from \u0026minus;\u0026thinsp;13.3\u0026ndash;20.8% at GBs. In particular, the Voronoi volume changes significantly in one atomic layer from the GB center. Within 7.35 \u0026Aring;, or three atomic layers, from the GB center, 66.3% of the sites exhibit looser sites (sites with Voronoi volumes larger than that of the bulk), while 33.7% display tighter sites (sites with Voronoi volumes smaller than that of the bulk).\u003c/p\u003e\n \u003cp\u003eConsidering these results, we explore GB segregation in \u0026alpha;-Fe polycrystals. For example, for transition metal elements, the solute element occupancy at each site increases exponentially with Voronoi volume from the bulk [90], indicating that the major segregation sites exist within one atomic layer from the GB center for both large and small solute elements. In particular, for substitutional solute elements, the proportion of tighter sites is approximately half that of looser sites, indicating that segregation sites for substitutional solute elements with smaller atomic sizes are more limited than those with larger atomic sizes. Figure\u0026nbsp;8(f) displays the Voronoi volume as a function of distance from the GB center for the (28.3 nm)\u003csup\u003e3\u003c/sup\u003e polycrystals relaxed through EAM or MEAM. For example, the ratio of the Voronoi volume at the GB center to that of the bulk is 3.3% and 2.9% for the MEAM and EAM, respectively, which is approximately half of the 6.0% observed for the MTP. This result indicates that, for example, when considering the GB segregation of a solute with a large atomic volume, the calculation of GB segregation energy using MEAM- or EAM-relaxed nano-polycrystals underestimates the energy by approximately half, which varies approximately linearly with the Voronoi volume [90]. Furthermore, the amount of GB segregation, which exhibits an exponential dependence on the GB segregation energy [91], would be significantly underestimated.\u003c/p\u003e\n \u003cp\u003eFigure\u0026nbsp;9(a) and (b) present histograms of the misorientation angle and GB energy, respectively, for three (28.3 nm)\u003csup\u003e3\u003c/sup\u003e polycrystals obtained by MTP relaxation. Additionally, Fig. 9(c) illustrates the relationship between GB energy and misorientation angle for each of the three (28.3 nm)\u003csup\u003e3\u003c/sup\u003e polycrystals. The area of each GB is the same as that used for calculating the average GB energy, employing the Voronoi polyhedron used to create the initial nano-polycrystalline structure. Each atom was assigned to the nearest GB, and the energy of each GB was calculated. GBs with areas larger than 1000 \u0026Aring;\u003csup\u003e2\u003c/sup\u003e were evaluated. The average GB energies of the polycrystals, obtained by averaging the energies for each GB across the three cases, were 1.57, 1.53, and 1.60 J/m\u003csup\u003e2\u003c/sup\u003e. These values closely align with the average GB energies of 1.58, 1.53, and 1.59 J/m\u003csup\u003e2\u003c/sup\u003e, calculated by dividing the excess energy of the entire polycrystal by the GB area shown in Fig.\u0026nbsp;7. The histograms obtained for the three cases displayed a mode value of 1.65\u0026ndash;1.70 J/m\u003csup\u003e2\u003c/sup\u003e, which is higher than the average value for polycrystalline materials, as discussed at the end of this section. The maximum GB energy was 1.79 J/m\u003csup\u003e2\u003c/sup\u003e, representing a 13.3% increase over the average GB energy value.\u003c/p\u003e\n \u003cp\u003eNext, the atomic structure of a representative GB among the large-angle GBs in Case 1 was analyzed in detail. The GB with the lowest GB energy (1.26 J/m\u003csup\u003e2\u003c/sup\u003e) (GBL), the one corresponding to the mode (1.67 J/m\u003csup\u003e2\u003c/sup\u003e) (GBM), and that with the highest energy (1.78 J/m\u003csup\u003e2\u003c/sup\u003e) (GBH) are discussed here. Figure\u0026nbsp;9(d) displays the probability density of the Voronoi volume in the one-atomic layer region near the GB center for these three GBs, alongside the polycrystal in Case 1 for comparison. The probability density of the \u0026Sigma;3(111) symmetric tilt GB (1.58 J/m\u003csup\u003e2\u003c/sup\u003e), which is the most used GB in DFT studies, is also shown. The probability density in \u0026Sigma;3(111) is scaled down to 1/10 for comparison. In Fig.\u0026nbsp;9(e), the probability densities of these Voronoi volumes are shown as the difference from the probability density of the Voronoi volume in the one-atomic layer region near the GB center for the polycrystals in Case 1. The probability density of the Voronoi volume for GBM is close to that of the entire polycrystal. In contrast, in the GBH, the probability density of Voronoi volumes exceeding 12.2 \u0026Aring;\u003csup\u003e3\u003c/sup\u003e is increased relative to that of the entire polycrystal. Finally, in the GBL, the probability density of Voronoi volumes close to the bulk Voronoi volume is increased relative to the whole polycrystal. Given the correlation between GB energy and the probability density of Voronoi volumes near the GB [50], this indicates that the GB energies of these three GBs have been correctly evaluated. In \u0026Sigma;3(111), there are sites with three different Voronoi volumes, all falling within the range observed in polycrystals. The ratio of sites with these volumes in \u0026Sigma;3(111) closely resembles that observed in the polycrystal. This result indicates that when considering solute atoms for which the Voronoi volume is the dominant factor in GB segregation, the analysis using \u0026Sigma;3(111) GBs is effective for qualitatively understanding the GB segregation. However, because there are only three sites with different Voronoi volumes, it is not sufficient to accurately predict GB segregation in polycrystalline materials.\u003c/p\u003e\n \u003cp\u003eFigure\u0026nbsp;9(f) depicts the atomic structures and energies of the atoms of the GBL, GBM, and GBH. In the GBM and GBH, the regions where the atomic energies are higher than those of the bulk are linearly distributed, and the difference between the two GBs is unclear. In contrast, in the GBL, the atoms in the GB center form periodic facets. The atoms in the faceted region have atomic energies relatively close to those of the bulk, even though the atoms are located near the center of the GB. The atomic structure of this region forms an atomic structure similar to a well-consistent (111)-terminated twist GB. Thus, even in polycrystalline materials with random orientations, instances occur where locally well-matched GBs are formed, resulting in a decrease in the GB energy. This result is one of the reasons why the histogram shown in Fig.\u0026nbsp;9(b) for misorientation angles of 15\u0026deg; or more does not exhibit a normal distribution but instead displays a large spread from the mode to the low-energy side.\u003c/p\u003e\n \u003cp\u003eThe results presented in this section, which were obtained through relaxation at near-room temperature (300 K) using a high-precision MTP, are expected to be similar to those of real \u0026alpha;-Fe polycrystals with random orientations. As described earlier, the MTP constructed in this study proves valuable for analyzing GBs in polycrystals at the atomic level and predicting GB segregation with high accuracy.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec13\"\u003e\n \u003ch2\u003e3.5. Application of findings from this study\u003c/h2\u003e\n \u003cp\u003eThe MTP constructed in this study reproduces the energies and acting atomic forces on arbitrary GGB and GB triple junctions in \u0026alpha;-Fe with high accuracy both during annealing and after relaxation. Therefore, it finds applicability in numerous studies, including grain growth of polycrystals [92] and investigation of GB energy and mobility [93]. Recently, a method has also been proposed to calculate GB segregation energy using machine learning from the local atomic environment of segregation sites [48, 94, 95]. As discussed in Section 3.4, the GB structure of the host metal significantly influences the predicted GB segregation energy and amount of GB segregation. The MTP constructed in this study can serve as a highly accurate GB model for this purpose. Alternatively, by extending the MTP to binary systems based on the constructed MTP and applying a method for predicting GB segregation using nano-polycrystals, the MTP should accurately predict the segregation of solute elements at GBs without the need for experiments.\u003c/p\u003e\n \u003cp\u003eThe MTP constructed in this study may also find application in simulating the deformation and fracture of \u0026alpha;-Fe polycrystals. As demonstrated in the Supplementary Note 4, the constructed MTP reproduces the two-dimensional energy profiles of the screw dislocation core position and the Peierls potential with DFT accuracy. These are necessary for accurate reproduction of plastic deformation. In addition, the generalized stacking energy curves under no strain and tensile strain, which are essential for accurately describing the crack propagation, also demonstrate accuracy comparable to that of DFT. While the applicability to the simulation of deformation and fracture of \u0026alpha;-Fe polycrystals falls beyond the scope of this study and is not addressed herein, it will be investigated in detail in the future. At the very least, the constructed MTP should provide a good starting point for studying these issues with high precision.\u003c/p\u003e\n \u003cp\u003eIn this study, we demonstrated the feasibility of constructing an interatomic potential that reproduces all GBs, including GGBs, with high accuracy. This is achieved by augmenting the basic training dataset, which reproduces the bulk properties, with the RANDSPG dataset proposed by Paul et al. [66], and subsequently constructing an MTP. Given the successful construction of a generic MTP for Mg by Paul et al. using a RANDSPG dataset, the present results indicate the potential to construct MLIPs with excellent accuracy for any GB. This can be achieved for not only \u0026alpha;-Fe but also for a variety of metallic materials by augmenting the RANDSPG dataset. Remarkably, the RANDSPG dataset comprises only 10 atoms at maximum, resulting in exceptionally low computational cost for acquiring a training dataset. Specifically, the calculation cost required to construct the RANDSPG dataset is comparable to that of performing AIMD at 300, 600, and 1000 K using a 3 \u0026times; 3 \u0026times; 3 supercell of \u0026alpha;-Fe. In essence, a diverse range of GBs can be effectively incorporated at a computational cost equivalent to collecting a small fraction of the training dataset typically utilized to reproduce the bulk properties of metallic materials. Therefore, while snapshots obtained from AIMD for basic crystal structures such as bcc, fcc, and hcp are generally used as a starting point for MLIP construction [96], it is also beneficial to employ the RANDSPG dataset as one of the basic initial datasets for MLIP construction. At the very least, when the analysis includes GBs, it can serve as a strong starting point for any MLIP construction, such as those based on concurrent [96] or active [68] learning. Thus, the findings obtained in this study will significantly advance material design, benefiting not only steel materials but also any polycrystalline metallic materials through the proposed method of constructing interatomic potentials.\u003c/p\u003e\n\u003c/div\u003e"},{"header":"4. Conclusion","content":"\u003cp\u003eIn this study, we constructed a tailored MTP to accurately reproduce the general GB behavior of \u0026alpha;-Fe, serving as a basic interatomic potential for exploring GB segregation control. Our training dataset comprised a DE dataset incorporating basic physical properties and lattice defects of \u0026alpha;-Fe, alongside a training dataset containing diverse atomic structures generated mechanically based on crystal space groups using RANDSPG. We verified the accuracy of the MTP for GGBs through direct comparison with DFT calculations on calculation cells cut near GBs from nano-polycrystals relaxed using the constructed MTP. Additionally, we assessed the accuracy using extrapolation grades for the entire nano-polycrystal. Our results demonstrate that the constructed MTP exhibits high accuracy for various arbitrary GBs, including GGBs, while accurately capturing the basic properties and lattice defects of \u0026alpha;-Fe. Furthermore, we employed the MTP to calculate the average GB energies of polycrystals through large-scale molecular dynamics simulations, obtaining values consistent with experimental estimates. The constructed MTP enables precise determination of GB energy and mobility as a function of GB character. Additionally, it can be utilized to construct a polycrystalline GB model for accurate prediction of GB segregation, which is necessary to gain knowledge on the control of GBs in \u0026alpha;-Fe. Notably, despite the minimal computational cost to obtain the RANDSPG dataset, the accuracy for GGBs can be further enhanced. Therefore, the insights gained from this study hold significant potential for designing high-strength metallic materials through the construction of high-precision interatomic potentials for all metals, extending beyond the scope of \u0026alpha;-Fe.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eAuthorship contributions\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eKazuma Ito: Conceptualization, Methodology, Software, Data curation, Writing – original draft, Visualization, Investigation. Tatuya Yokoi: Methodology, Validation, Writing – review \u0026amp; editing. Katsutoshi Hyodo: Validation, Writing – review \u0026amp; editing. Hideki Mori: Supervision, Software, Validation, Writing – review \u0026amp; editing.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCompeting Interests\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAcknowledgments\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis work used computational resources of the Supercomputer Fugaku provided by Riken through the HPCI System Research Project (Project ID: hp230272). This work was partly supported by Accompanying User Support Program (【23Z-03, 23Z-05, 24H1-01】, Support content:【porting of application program, execution performance tuning】) performed by Research Organization for Information Science and Technology.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eData availability\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe potential is available on the Github page https://github.com/KazumaIto0810/MTP.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eC.D. Horvath, Chapter 2 - Advanced steels for lightweight automotive structures, in: P.K. Mallick (Ed.), Materials, Design and Manufacturing for Lightweight Vehicles (Second Edition), Woodhead Publishing, Cambridge, 2021, pp. 39-95.\u003c/li\u003e\n\u003cli\u003eD.-W. Suh, S.-J. Kim, Medium Mn transformation-induced plasticity steels: Recent progress and challenges, Scr. Mater. 126 (2017) 63\u0026ndash;67.\u003c/li\u003e\n\u003cli\u003eR.L. Plaut, C. Herrera, D.M. 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Commun. 253 (2020) 107206.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"npj-computational-materials","isNatureJournal":false,"hasQc":false,"allowDirectSubmit":false,"externalIdentity":"npjcompumats","sideBox":"Learn more about [npj Computational Materials](http://www.nature.com/npjcompumats/)","snPcode":"41524","submissionUrl":"https://mts-npjcompumats.nature.com/","title":"npj Computational Materials","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"ejp","reportingPortfolio":"NPJ","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"Machine learning potential, density functional theory (DFT), grain boundaries, grain boundary segregation steels","lastPublishedDoi":"10.21203/rs.3.rs-4550958/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4550958/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eTo advance the development of high-strength polycrystalline metallic materials towards achieving carbon neutrality, it is essential to design materials in which the atomic-level control of general grain boundaries (GGBs), which govern the material properties, is achieved. However, owing to the complex and diverse structures of GGBs, there have been no reports on interatomic potentials capable of reproducing them. This accuracy is essential for conducting molecular dynamics analyses to derive material design guidelines. In this study, we constructed a machine learning interatomic potential (MLIP) with density functional theory (DFT) accuracy to model the energy, atomic structure, and dynamics of arbitrary grain boundaries (GBs), including GGBs, in α-Fe. Specifically, we employed a training dataset comprising diverse atomic structures generated based on crystal space groups. The GGB accuracy was evaluated by directly comparing with DFT calculations performed on cells cut near GBs from nano-polycrystals, and extrapolation grades of the local atomic environment based on active learning methods for the entire nano-polycrystal. Furthermore, we analyzed the GB energy and atomic structure in α-Fe polycrystals through large-scale molecular dynamics analysis using the constructed MLIP. Conventional interatomic potentials cannot accurately calculate the GB energy and atomic structure in α-Fe polycrystals. Conversely, the average GB energy of α-Fe polycrystals calculated by the constructed MLIP is 1.57 J/m\u003csup\u003e2\u003c/sup\u003e, exhibiting good agreement with experimental predictions. Our findings demonstrate the methodology for constructing an MLIP capable of representing GGBs with high accuracy, thereby paving the way for materials design based on computational materials science for polycrystalline materials.\u003c/p\u003e","manuscriptTitle":"Machine learning interatomic potential with DFT accuracy for general grain boundaries: Analysis of grain boundary energy and atomic structure in α-Fe polycrystals","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-08-01 17:17:10","doi":"10.21203/rs.3.rs-4550958/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"revise","date":"2024-08-16T14:58:32+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"This content is not available.","date":"2024-08-08T15:41:43+00:00","index":3,"fulltext":"This content is not available."},{"type":"editorInvitedReview","content":"This content is not available.","date":"2024-07-23T15:12:30+00:00","index":1,"fulltext":"This content is not available."},{"type":"reviewerAgreed","content":"This content is not available.","date":"2024-07-18T18:25:44+00:00","index":3,"fulltext":"This content is not available."},{"type":"editorInvitedReview","content":"This content is not available.","date":"2024-07-16T13:26:01+00:00","index":2,"fulltext":"This content is not available."},{"type":"reviewerAgreed","content":"This content is not available.","date":"2024-07-10T07:23:32+00:00","index":2,"fulltext":"This content is not available."},{"type":"reviewerAgreed","content":"This content is not available.","date":"2024-07-09T13:20:44+00:00","index":1,"fulltext":"This content is not available."},{"type":"reviewersInvited","content":"","date":"2024-07-09T13:05:48+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2024-06-27T01:43:15+00:00","index":"","fulltext":""},{"type":"submitted","content":"npj Computational Materials","date":"2024-06-22T18:07:28+00:00","index":"","fulltext":""},{"type":"checksFailed","content":"","date":"2024-06-12T10:18:45+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2024-06-08T14:27:41+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"npj-computational-materials","isNatureJournal":false,"hasQc":false,"allowDirectSubmit":false,"externalIdentity":"npjcompumats","sideBox":"Learn more about [npj Computational Materials](http://www.nature.com/npjcompumats/)","snPcode":"41524","submissionUrl":"https://mts-npjcompumats.nature.com/","title":"npj Computational Materials","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"ejp","reportingPortfolio":"NPJ","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"edbc2edb-fc72-4963-b9c0-3dc0f487e773","owner":[],"postedDate":"August 1st, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[{"id":34348171,"name":"Physical sciences/Materials science/Structural materials/Metals and alloys"},{"id":34348172,"name":"Physical sciences/Materials science/Theory and computation/Atomistic models"}],"tags":[],"updatedAt":"2024-11-14T08:10:19+00:00","versionOfRecord":{"articleIdentity":"rs-4550958","link":"https://doi.org/10.1038/s41524-024-01451-y","journal":{"identity":"npj-computational-materials","isVorOnly":false,"title":"npj Computational Materials"},"publishedOn":"2024-11-13 05:00:00","publishedOnDateReadable":"November 13th, 2024"},"versionCreatedAt":"2024-08-01 17:17:10","video":"","vorDoi":"10.1038/s41524-024-01451-y","vorDoiUrl":"https://doi.org/10.1038/s41524-024-01451-y","workflowStages":[]},"version":"v1","identity":"rs-4550958","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4550958","identity":"rs-4550958","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}