Comparative analysis of ternary TiAlNb interatomic potentials: moment tensor vs. deep learning approaches

Abstract

Intermetallic titanium aluminides, leveraging the ordered -TiAl phase, attract increasing attention in aerospace and automotive engineering due to their favorable mechanical properties at high temperatures. Of particular interest are -TiAl-based alloys with a Niobium (Nb) concentration of 5–10 at.%. It is a key question how to model such ternary alloys at the atomic scale with molecular dynamics (MD) simulations to better understand (and subsequently optimize) the alloys. Here, we present a comparative analysis of ternary TiAlNb interatomic potentials developed by the moment tensor potential (MTP) and deep potential molecular dynamics (DeePMD) methods specifically for the above mentioned critical Nb concentration range. We introduce a novel dataset (TiAlNb dataset) for potential training that establishes a benchmark for the assessment of TiAlNb potentials. The potentials were evaluated through rigorous error analysis and performance metrics, alongside calculations of material properties such as elastic constants, equilibrium volume, and lattice constant. Additionally, we explore finite temperature properties including specific heat and thermal expansion with both potentials. Mechanical behaviors, such as uniaxial tension and the calculation of generalized stacking fault energy, are analyzed to determine the impact of Nb alloying in TiAl-based alloys. Our results indicate that Nb alloying generally enhances the ductility of TiAl-based alloys at the expense of reduced strength, with the notable exception of simulations using DeePMD for the -TiAl phase, where this trend does not apply.

1 Introduction

Gamma titanium aluminide (-TiAl) intermetallic alloys attract increasing attention in aerospace and automotive engineering as high-performance lightweight structural materials. This growing interest is primarily due to their unique combination of low density, remarkable oxidation resistance, and superior strength and creep resistance at elevated temperatures Appel et al. (2011). Key to these alloys are the primary intermetallic phases: -TiAl, with its ordered face-centered tetragonal structure (L1₀, P4/mmm) and -Ti₃Al, noted for its ordered hexagonal structure (D0₁₉, P6₃/mmc). The advancement of -TiAl based alloys, particularly those tailored for higher service temperatures, such as TNM (TiAl-Nb-Mo) and TNB (TiAl-Nb-B) alloys, has been significant (Appel et al., 2011; Clemens and Mayer, 2012; Li et al., 2014; Klein et al., 2016; Zhang et al., 2016). These alloys typically contain 5–10 at.% Nb, along with small amounts of other elements like Mo, B, C, Si, W, Cr, Ta. Extensive research highlights Nb’s crucial role in boosting mechanical properties, notably increasing the strength and ductility of TiAl alloys (Appel et al., 2011; Clemens and Mayer, 2012; Liu et al., 2002; Song et al., 2020; Cheng et al., 2016; Liu et al., 2022; Zhang et al., 2023). Despite these improvements, the specific contribution of Nb to the enhancement of strength and ductility in TNB and TNM alloys has remained somewhat unclear. In our previous work (Chandran et al., 2024), we addressed this gap by examining the effects of Nb on the thermo-mechanical properties of TiAl-based alloys through atomistic simulations. Utilizing Farkas’ ternary interatomic potential (Farkas and Jones, 1996), we scrutinized various TiAl-based models, ranging from single-phase structures to single lamellar interfaces, and progressing to more complex microstructure-informed atomistic models (MIAMs) with nano-polycolonies. However, it became evident that it is a limitation of Farkas’ potential that it could not adequately handle Nb concentrations above 1 at.% in MIAMs and 2 at.% in certain types of single lamellar interfaces. This limitation is significant, as higher Nb concentrations (5–10 at.%) are known to be most interesting for improving strength and ductility. To overcome this challenge, we pursue in this article the direction of developing machine learning (ML) -based interatomic potentials for molecular dynamics (MD) simulations.

Generally, MD simulations offer profound insights into the behavior of atomic systems, ranging in scale from around 10³–10⁹ atoms. These simulations are instrumental in capturing a wide array of interactions, including thermal, mechanical, chemical, and microstructural dynamics. However, the accuracy of MD simulations is contingent upon the selection of appropriate interatomic potentials or force fields, as well as the boundary conditions implemented. In recent years, the integration of machine learning with MD simulations has emerged as a rapidly evolving field. This integration primarily focuses on the modeling of interatomic potential energy surfaces (PES) using reference data derived from ab initio simulations. Various ML methodologies have made significant contributions to the study of condensed matter systems and can be broadly categorized into linear regression [e.g., moment tensor potentials (Shapeev, 2016; Podryabinkin and Shapeev, 2017; Gubaev et al., 2019)], kernel methods [e.g., gaussian approximation potential (Bartók et al., 2013; Szlachta et al., 2014; Dragoni et al., 2018), spectral neighbor analysis (Thompson et al., 2015; Chen et al., 2017; Li et al., 2018; Deng et al., 2019)], and deep neural network-based techniques (Behler and Parrinello, 2007; Behler, 2011; Zhang et al., 2018a), respectively.

Recent comparative studies have evaluated the effectiveness of these diverse techniques, as seen in the works of Zuo et al. (2020); Deringer et al. (2019); Unke et al. (2020, 2021). Among these, deep neural network-based potentials, particularly the deep potential (DP) and neural network potential (NNP), stand out due to their successes in modeling both ordered and disordered systems. The flexibility of the descriptor proposed by Zhang et al. (2018a) for DP potentials is particularly noteworthy. The DP method has demonstrated its efficacy in various systems, such as LiF and FLiBe (Rodriguez et al., 2021), MgCl₂-NaCl and MgCl₂-KCl (Xu et al., 2023), AlN (Li et al., 2024), Cu (Du et al., 2022) and several others (Niu et al., 2020; Nguyen et al., 2022). In comparison, the moment tensor potential (MTP) is notable for its efficiency, derived from a polynomial basis of interatomic distances and angles. MTP not only outpaces the other methods in terms of speed but has also demonstrated equivalent accuracy in modeling various material systems, as shown in studies by Novikov et al. (2018); Podryabinkin et al. (2019); Novoselov et al. (2019). MTP has been recognized for its optimal balance between accuracy and computational efficiency, a comparison elucidated in Zuo et al. (2020) performance analysis. Tasnádi et al. (2021) developed an MTP potential for efficiently predicting the elastic properties of Ti_0.5Al_0.5N. Furthermore, the work by Lu et al. (2023) deserves attention, where the authors generated a DeePMD potential for TiAlNb. They assert that this potential successfully validates the bulk material properties of TiAl-based alloys and provides accurate evaluations of the stacking fault energy and tensile properties of -TiAl. It is also worth mentioning that the neuroevolution potential (NEP) method, as used in the study by Zhao et al. (2024), was applied to train a general-purpose Ti-Al-Nb potential. The authors claim that this trained potential not only explains the high-temperature mechanical properties of TiAl-based alloys but also accurately reproduces the fundamental material properties.

In this paper, we initially create essential datasets for training TiAlNb-based interatomic potentials using ab initio molecular dynamics (AIMD) simulations. These datasets (TiAlNb datasets) are intended to serve as a benchmark for evaluating TiAlNb-based interatomic potentials. Utilizing them, we train the TiAlNb interatomic potentials using both MTP and DP methods. Subsequently, we conduct a comparative evaluation of these potentials through error analysis and performance metrics, in addition to calculating material properties such as elastic constants, equilibrium volume, and lattice constants. We further compute finite temperature properties, including specific heat capacity and thermal expansion. Furthermore, we assess the mechanical properties by performing simulated uniaxial tension tests and calculations of generalized stacking fault energy.

2 Materials and methods

2.1 Dataset generation

We performed a large number of AIMD simulations with the Vienna ab initio simulation package (VASP) Kresse and Furthmüller (1996a); Kresse and Furthmüller (1996b). These simulations employed projector augmented wave (PAW) Blöchl, (1994) method to intricately model the interactions between electrons and ions. We incorporated the generalized gradient approximation (GGA) (Perdew et al., 1996, Perdew et al., 1997) for addressing exchange and correlation effects, specifically using the Perdew–Burke–Ernzerhof (PBE) functional. Our computational framework was rigorously set up with a significant cut-off energy of 510 eV, which was crucial for ensuring the accuracy and precision of our calculations. Furthermore, a -centered k-point mesh of dimensions 1 was employed to efficiently sample the Brillouin zone, striking a balance between computational efficiency and temporal resolution with a time step of 0.5 femtoseconds in all AIMD simulations.

For the development of the interatomic potential, we curated a comprehensive dataset encompassing a wide spectrum of Nb concentrations in -TiAl and -Ti₃Al alloys, ranging from 1 at.% to 14 at.%. This compilation resulted in an extensive array of 28 unique structural configurations, featuring 108 atoms in -TiAl and 128 atoms in -Ti₃Al. Our primary focus was on the Nb concentration range of 1–10 at.%, although we extended our dataset to include concentrations up to 14 at.% Nb to enhance the training process. Among these 28 configurations, a subset was reserved and excluded from training to assess the potential’s predictive accuracy. The introduction of Nb atoms into titanium lattice sites in both phases was carefully executed, taking into account the preferential site occupancy of Nb in these alloys. For a more detailed exposition of this methodology, readers are referred to our previous work (Chandran et al., 2024) and other pertinent literature (Holec et al., 2016; Ouadah et al., 2020; Ouadah et al., 2021; Song et al., 2000; Wei et al., 2012). The Atomsk tool (Hirel, 2015) was utilized for the precise generation of these datasets, ensuring an accurate representation of the structural configurations. It should be noted that defect configurations were not included in the training process.

Subsequent AIMD simulations were conducted with meticulous care across all these structures, using the NVT and NPT ensemble over a period of nearly 10 ps. To comprehensively analyze thermal behaviors, these simulations spanned a range of temperatures including 1 K, 300 K, 500 K, 700 K, and 900 K. In the NVT ensemble, particular attention was given to selecting frames post-equilibration for analysis. The snapshots derived from these AIMD simulations provided a rich and diverse data source for training both MTP and DeepMD potentials. The snapshots were partitioned in an 80:20 ratio between the training and test sets to optimize the learning process. More details of the training database are provided in the Supplementary Material, in Section 1.

The TiAlNb dataset developed for this research is now available for public access. It includes the initial structural files in VASP format, essential for generating AIMD frames, alongside the training and validation datasets for DeePMD, and the training and testing datasets for MTP. Additionally, input files necessary for model training are provided, ensuring that users can replicate the results. Access to the dataset is facilitated through the link: TiAlNb dataset offering comprehensive resources for further exploration and validation.

For training the DeePMD potential, we utilized the DeePMD kit (Wang et al., 2018) along with its corresponding library designed for integration with the LAMMPS package (Thompson et al., 2022) for executing MD simulations. Similarly, the training of the MTP potential was conducted using the MLIP package (Novikov et al., 2021), employing its library specifically developed for compatibility with the LAMMPS package (Thompson et al., 2022) to facilitate MD simulations.

2.2 MTP

MTP is a type of machine learning-developed interatomic potential, implemented using the MLIP package (Novikov et al., 2021). In MTP, the energy of each atomic configuration (denoted as ) is expressed as the sum of contributions from the local atomic neighborhoods n_i for N atoms. This is mathematically represented as,where each contribution is linearly expanded via a set of basis functions ,

In the above equation, the set of parameters = are derived during the training process and N_lin is the number of these parameters. The atomic environments are characterized using moment tensor descriptors or moments, which include both radial and angular components. The moment tensor descriptor for the -th atom is defined as,where is the position of -th atom relative to -th atom and is the corresponding interatomic distance. is the number of atoms in neighborhood. The descriptor consists of the angular part, (the symbol ‘’ represents the outer product of vectors and, hence, the angular part is the tensor of order v) and the radial part, having the following form,

where {} and represents the set of radial parameters and radial basis functions, respectively.

The basis functions are constructed using the of moments:

These coefficients have been found to be optimal for many datasets Gubaev et al. (2019). The basis functions are invariant to rotations, reflections, and permutations. For defining a functional form of MTP, we choose the maximum level, lev_max, and include all the basis functions whose level is less or equal to _max, i.e., .

The MTP parameters comprise of the radial parameters and , which are obtained during the fitting procedures. The fitting and learning procedure of MTP consists of finding parameters by minimizing the optimization problem,where the training set contains configurations , and , and are the quantum mechanical energy, forces and stress tensors, respectively. is the number of atoms in configuration and , and are the non-negative weights of energies, forces and stresses during the optimization. Equations 1–6 pertains to MTP formulation.

2.3 DeePMD

The DP method used in this study was developed using DeePMD kit (Wang et al., 2018). In DeePMD, the total energy of the system is represented as sum of energies of all the atoms. Suppose a system contains atoms, then the total energy of the system according to DeePMD model can be represented as,Each atomic energy is obtained by the position of -th atom and its neighbours,where denotes the chemical specie of atom and denotes the set of neighbour atoms that are at a cut-off radius of from atom . The cut-off radius of the neighbouring atoms should be such that = . In order to map the atomic positions, descriptors that guarantee translational, rotational and permutational symmetries are used. We have used the deep potential smooth edition (DeepPot-SE) Zhang et al. (2018b) suggested in DeePMD Kit that includes the radial and angular information of atomic configurations. These symmetry preserving descriptors are set up using an embedding net and are later passed to the fitting net to obtain the energy of each atom. Parameter optimization is performed by minimizing the loss function where,where E, F and correspond to the root mean square (RMS) error of energy, force and stress, respectively. During the optimization process, the prefactors of energy , force and stress are changed. The prefactors are formulated aswhere and denotes the learning rate at the beginning and at the training step , respectively. shows the starting prefactor, which goes to at the end of the learning. Here, an exponential learning rate is considered as,where and denote the decay rate and decay steps, respectively. Equations 7–11 pertains to DeePMD formulation.

2.4 Potential training

2.4.1 MTP

Our investigation commenced with an analysis of the convergence of the MTP towards density functional theory (DFT) energy and force metrics. A key aspect of this study involved conducting a grid search to optimize the MTP parameters. This search varied the potential levels from 6 to 24 and the r_cut values from 5 to 8 at 1 intervals. This strategic approach enabled us to explore a wide range of potential configurations to ascertain the most effective one. This analysis was conducted using a selection of 1800 datasets that were not included in the fitting of the potential, as illustrated in Figure 1. A pivotal aspect of the convergence test was the assignment of greater weight to energy, reflecting its critical role in the overall accuracy of the potential.

FIGURE 1

The convergence analysis revealed that the energy per atom began showing signs of convergence at level 16. This was a key observation, indicating that the MTP was effectively capturing the energy characteristics consistent with DFT calculations from this level. Similarly, the force values also demonstrated significantly low root mean square error (RMSE) commencing at the same level, further affirming the reliability of the potential from level 16 onwards.

Given these findings, and to ensure a conservative and robust approach, we ultimately selected level 18, for the final model. This level was deemed to provide an optimal balance between complexity and accuracy. Additionally, r_cut was set to 7 , for ensuring sufficient interaction range while maintaining computational efficiency. The minimum radius (_min) was chosen to be 2 . This parameter is crucial for determining the minimum distance at which interactions are considered in the potential, thereby influencing the model’s sensitivity to shorter-range forces.

For the MTP’s training objective function, weights were thoughtfully assigned as follows: for energy, for force, and for stress, with a priority on energy precision. This weighting strategy ensured a nuanced consideration of both force and stress, albeit with a lesser priority compared to energy, essential for crafting a nuanced and precise interatomic potential. The dataset leveraged for this purpose contained 40,000 frames, distributed in an 80:20 ratio between training and validation sets, to support effective model training and subsequent validation.

2.4.2 DeePMD

In our study, the DeePOT-SE model, as implemented in the DeePMD-kit package, was employed. The cut-off radius for the model was set to 6 , with the smoothing function commencing at 4 , ensuring a smooth transition and reducing potential artifacts in the force calculations.

The model’s architecture included radial and angular embedded-atom neural networks, each featuring three hidden layers. These layers were composed of 10, 20, and 40 nodes, respectively, providing a robust framework for capturing the complex interatomic interactions. Additionally, the fitting networks were designed with three hidden layers, each containing 100 nodes.

Regarding the training parameters, the initial learning rate was set at 0.001, gradually decreasing to a final rate of . This gradual reduction in the learning rate allowed for finer adjustments as the training progressed, leading to more accurate model predictions. The weighting factors for energies and forces were also carefully calibrated. Initially, the weights for energies were set at 0.02, increasing to 1 in the final stages, whereas the weights for forces started at 1,000 and were reduced to 1. This approach prioritized force accuracy in the initial stages of training and gradually shifted the focus towards energy accuracy.

It is noteworthy that virial data was not included in the training process. This decision was made to streamline the training and focus on the most critical aspects of the potential. The training was conducted over a substantial number of epochs, totaling 2,00,0000, to ensure comprehensive learning and optimization of the model parameters. The utilized dataset comprised 3,33,340 frames, divided into a training and validation set following an 80:20 ratio, facilitating effective model training and validation.

3 Results and discussion

3.1 Error analysis

The correlation between predictions from ML models and DFT calculations is pivotal in ascertaining the models’ accuracy, particularly in simulating potential energy surfaces and material dynamics. Our study methodically evaluates this correlation for Nb-alloyed -TiAl and -Ti₃Al phases using both MTP and DP models. Figures 2, 3 are instrumental in this analysis, showcasing the comparison of MTP and DP model predictions against DFT data, respectively.

FIGURE 2

FIGURE 3

The primary metrics for this comparison are energy per atom and the three components of force. For the MTP model, the recorded root mean square errors (RMSEs) are 0.0031 eV/atom for energy and 0.1285 eV/Å for the x-direction force (fx), 0.1329 eV/Å for the y-direction force (fy), and 0.1292 eV/Å for the z-direction force (fz). Conversely, the DP model exhibits RMSEs of 0.0011 eV/atom for energy, 0.0783 eV/Å for fx, 0.0795 eV/Å for fy, and 0.0820 eV/Å for fz. The DP model’s RMSEs are notably lower than those of the MTP model, suggesting a marginally superior precision. Nevertheless, the energy and force RMSEs for both models are within acceptable ranges, confirming their effectiveness in reflecting DFT outcomes. Additionally, the RMSEs for energy per atom, fy, fy, and fz were computed for the training datasets and are presented in Table 1.

Both models’ predictions exhibit a close alignment with the y = x line, highlighting their capacity to accurately reproduce the test dataset energies and atomic forces. This congruence is a robust indicator of the models’ exceptional accuracy, which extends to untrained test data, suggesting their effectiveness in generalizing beyond the configurations they were trained on. The considered test data encompasses structures over a complete temperature range (1 K, 300 K, 500 K, 700 K and 900 K) and varying Nb concentrations (1–14 at.%). The lower RMSEs for both energy and force underscore the DP model’s superior ability to achieve DFT-level accuracy compared to the MTP model. This indicates the potential of the DP model as a more precise tool for simulating the behaviors of Nb-doped -TiAl and -Ti₃Al phases, backed by its comparative closeness to DFT calculations.

3.2 MD simulations

3.2.1 Energy volume curve

To augment the validity of the DeePMD and MTP potentials, this study meticulously examines the energy-volume relationships obtained using the Murnaghan fit in Nb-alloyed -Ti₃Al and -TiAl phases. Illustrations of these relationships are effectively depicted in Figure 4. Remarkably, the outcomes from MD simulations employing DeePMD and MTP exhibit exceptional congruence with those derived from DFT calculations. This alignment highlights the adeptness of the trained potentials in precisely capturing the energy-volume characteristics inherent to these materials.

FIGURE 4

Table 2 shows the equilibrium volume predicted by DFT, MTP and DeePMD for -Ti₃Al and -TiAl. For the -Ti₃Al, equilibrium volumes predicted by DFT, MTP, and DeePMD are respectively 15.88 , 16.40 and 14.52 . In a similar vein, for -TiAl phase, the corresponding values are 16.22 , 15.76 and 15.46 . Notably, the equilibrium volume predictions by MTP align more closely with those from DFT than do those by DeePMD for both phases.

The lattice constants predicted by DFT, MTP and DeePMD for -Ti₃Al and -TiAl are compared to corresponding experimental values (PEARSON, 1958; He et al., 1997) in Table 2. For the -Ti₃Al phase, MTP predicts lattice constants of a = 5.683 and c = 4.633 . These closely match the DFT-calculated values of a = 5.625 and c = 4.586 and experimental values (PEARSON, 1958) of a = 5.770 and b = 4.620 . Conversely, DeePMD predicts a = 5.453 and c = 4.446 , slightly diverging from the DFT lattice parameters. The equilibrium lattice constants for -TiAl as forecasted by MTP–a = 3.910 and c = 4.099 – exhibit a striking resemblance to the DFT-determined values of a = 3.950 and c = 4.140 whereas the experimental values (He et al., 1997) are a = 3.998 and c = 4.067 . The DeePMD predictions, a = 3.910 and c = 4.099 , are identical to those of MTP, showcasing their close alignment with DFT values.

Accurate energy prediction in competitive phases is vital to avert unphysical phase segregation during MD simulations. Therefore, the integration of machine learning potentials like DeePMD and MTP notably enhances the structural analysis and prediction capabilities in Nb-alloyed -Ti₃Al and -TiAl phases, underscoring their significance in the realm of advanced materials research. In terms of equilibrium volume and lattice constant prediction, MTP demonstrates superior performance compared to DeePMD.

3.2.2 Elastic constants

Next we assessed the elastic constants of -Ti₃Al and -TiAl alloys utilizing the different computational methodologies DFT, MTP, and DeePMD. The outcomes, quantified in gigapascals (GPa), provide a detailed comparison of the mechanical attributes predicted by these models against established experimental values (Tanaka and Koiwa, 1996; Tanaka, 1996) in Table 2.

For the -Ti₃Al alloy, notable discrepancies emerge in the elastic constants across the computational approaches compared to experimental standards (Tanaka and Koiwa, 1996). Specifically, the DeePMD’s C₁₁ prediction of 339 GPa significantly overshoots the figures from DFT (157 GPa), MTP (147 GPa), and experimental data (183 GPa), indicating an overestimation of longitudinal stiffness by DeePMD. The variations in C₁₂ and C₃₃, which reflect differences in predicted interatomic bond strengths and compressibility along distinct crystallographic directions, are particularly striking, with DFT (103 GPa and 219 GPa) and experimental (89 GPa and 225 GPa) benchmarks. The DeePMD’s divergence in C₁₃, which measures axial-longitudinal strain interactions, sharply contrasts with DFT’s 74 GPa and the experimental value of 63 GPa. Additionally, the C₄₄ constant, crucial for evaluating resistance to shear deformation, shows variation, with MTP predicting 34 GPa against DFT’s 46 GPa and experimental findings of 64 GPa. It is noteworthy that the C₆₆ constant is not applicable for the -Ti₃Al phase in this analysis.

In the analysis of the -TiAl alloy, the elastic constants exhibit significant variances across the computational techniques, juxtaposed with experimental findings (Tanaka, 1996). The C₁₁ constant, indicative of the material’s longitudinal rigidity, shows a broad range of values, with DFT’s prediction of 179 GPa closely mirroring the experimental value of 187 GPa, in contrast to the lower estimations of 146 GPa by both MTP and DeePMD. Conversely, the shear-related C₄₄ constant demonstrates minimal variation among the models, aligning closely with the experimental measure of 109 GPa and DFT value of 102 GPa.

Table 3 presents a comparison of the elastic constants for -Ti₃Al and -TiAl at 300 K, 500 K, 700 K, and 900 K. As expected, the elastic constants for both -Ti₃Al and -TiAl decrease with increasing temperature, consistent with findings reported in Qi et al. (2023). Our calculated values have been compared to available experimental data (He et al., 1997; Tanaka, 1996) for -TiAl; (Tanaka et al., 1996) for -Ti₃Al), though it is important to note that experimental values are not available for -TiAl at 900 K and for -Ti₃Al at 500 K, 700 K, and 900 K. Moreover, the C₆₆ constant does not apply to the -Ti₃Al phase in this analysis. The discrepancies observed in the elastic constants in Table 2 are also present here; however, the overall trend of decreasing elastic constants with increasing temperature for both phases remains consistent with the previous studies mentioned.

The observed disparities in elastic constants among the DFT, MTP, and DeePMD methodologies, relative to experimental data (Tanaka and Koiwa, 1996; Tanaka, 1996), highlight the intricate challenges of accurately simulating material behaviors. These differences can be attributed to the unique features of each approach, including DFT’s detailed electron correlation handling, MTP’s potential function structure, and DeePMD’s reliance on extensive training datasets and neural network designs. Our analysis underscores the critical importance of selecting appropriate computational strategies tailored to specific material characteristics and the need for cautious interpretation of computational findings in materials science. This study reinforces the indispensable role of experimental validation in confirming the veracity of computational predictions.

3.2.3 Radial distribution function

The radial distribution function (RDF) serves as an instrumental tool for analyzing the structural properties of materials. In our study, illustrated in Figures 5, 6, we present a comparative analysis of RDFs predicted by MTP and DeePMD against those obtained from AIMD for Nb in -Ti₃Al. Additionally, Figures 7, 8 show a similar comparison for Nb in -TiAl. It is important to note that the structural instances used for RDF analysis with DFT were not part of the training sets for either the MTP or DeePMD models. The agreement between the outcomes underscores the dependability of DeePMD and MTP models in probing the structure of Nb-alloyed -Ti₃Al and -TiAl phases.

FIGURE 5

FIGURE 6

FIGURE 7

FIGURE 8

Notably, the RDFs for -TiAl from AIMD exhibited a closer match with the ML potentials compared to those for -Ti₃Al. This could be attributed to the ML models encountering challenges in accurately replicating structures when two different phases are present. In the case of -Ti₃Al, we observed a diminishing correlation between the RDFs from AIMD and ML potentials with increasing temperature, as demonstrated in Figure 5G (6.3 at.% Nb) and 6G (6.3 at.% Nb). Furthermore, at 300 K, the RDF for 7.4 at.% Nb in -TiAl (Figure 8E) showed a discrepancy between AIMD and DeePMD, while the MTP model maintained a good correlation for the same scenario. These observations highlight the nuanced performance of ML potentials in different phase contexts and temperature conditions.

Additionally, we examined the scalability of the developed potentials with system sizes, as detailed in Supplementary Figures S1–S4 in the Supplementary Material in Section 2. For further insights, please refer to this section in the Supplementary Material. Our study also extended to comparing RDFs calculated by DFT, MTP, and DeePMD for a specific case of 14.8 at.% Nb concentration, which exceeds the training data’s Nb concentration range. These results are documented in the Supplementary Material in Section 3, providing valuable perspectives on the models’ performance beyond their initial training scope.

3.2.4 Specific heat capacity and thermal expansion

To assess the predictive capability of trained MTP and DeePMD potentials at finite temperatures, this section reports the computation of specific heat capacity for 3.1 at.% Nb structures of -Ti₃Al and -TiAl and thermal expansion coefficients ( and ) for -Ti₃Al and -TiAl as the temperature increases. For the computation of specific heat capacity we have used phonopy (Togo et al., 2023; Togo, 2023).

Figures 9A, B illustrate the comparisons of values calculated using DeePMD and MTP against those derived from DFT for 3.1 at.% Nb of -Ti₃Al and -TiAl, respectively. It is observed that progressively increases with temperature across all methods, reflecting the system’s access to more degrees of freedom at higher temperatures. For -Ti₃Al, the alignment of the curves from the three methods is notably good. Specifically, the DeePMD results closely mirror the DFT outcomes, whereas the MTP results show minor deviations between 177 K and 777 K, aligning better at temperatures beyond this range. For quantitative clarity, at 200 K, values are 0.045 J/g.K, 0.099 J/g.K, and 0.082 J/g.K for DeePMD, MTP, and DFT, respectively; at 600 K, they are 0.401 J/g.K, 0.449 J/g.K, and 0.392 J/g.K; and at 1000 K, they are 0.647 J/g.K, 0.595 J/g.K, and 0.608 J/g.K. It should be noted that the training set for these potentials only included structures up to 900 K. For -TiAl, although the MTP and DeePMD predictions initially closely match, they begin to diverge slightly beyond 288 K. Quantitative values at 200 K are 0.104 J/g.K for both DeePMD and MTP, and 0.355 J/g.K for DFT; at 600 K, the values are 0.525 J/g.K, 0.473 J/g.K, and 0.352 J/g.K; and at 1000 K, they are 0.765 J/g.K, 0.655 J/g.K, and 0.554 J/g.K.

FIGURE 9

In terms of thermal expansion, we computed the coefficients and corresponding to lattice parameters a and c at temperatures up to 900 K (Figure 10) for -Ti₃Al and -TiAl. These values were compared with experimental results for -TiAl (He et al., 1997) and DFT calculations for -Ti₃Al (Holec et al., 2019). It is important to note that low-temperature behaviors, often influenced by quantum effects, are not typically well captured by classical interatomic potentials. Both and typically increase with temperature, aiming to stabilize at higher temperatures. Experimental and DFT reference values are only available up to 750 K. At higher temperatures, and calculated using DeePMD are closer to these reference values, whereas MTP underpredicts these coefficients for both phases. Thus, DeePMD proves to be more accurate in predicting thermal expansion coefficients than MTP.

FIGURE 10

Holec et al., 2019) and experimental values (He et al., 1997) for -Ti₃Al (A, B) and -TiAl (C, D) respectively.

3.2.5 Tension test

Here, we aimed to assess the capabilities of the MTP and DeePMD potentials in capturing the thermo-mechanical characteristics of specific materials. To achieve this, we simulated uniaxial tension tests on chosen samples, applying a strain rate of 10⁹ s⁻¹. The stress-strain curves obtained from these tension tests, utilizing both MTP and DeePMD potentials for the -Ti₃Al and -TiAl phases, are illustrated in Figure 11. Table 4 presents a comparative analysis of ultimate tensile strength (UTS) values for these materials.

FIGURE 11

We focused on comparing the UTS of the intermetallic phases to gauge the accuracy of mechanical property predictions. For the -Ti₃Al phase, MTP demonstrated consistency in UTS with increase in Nb concentration. Conversely, with the DeePMD potential, the UTS decreased as the Nb concentration increased. In the -TiAl phase, an increase in Nb concentration typically led to a rise in UTS with DeePMD, whereas with MTP, a decrease in UTS was observed with increasing Nb, except for the 7.4 at.% Nb scenario. The results of tension curves for -TiAl with Nb introduction using DeePMD shows a discrepancy with that presented in Lu et al. (2023).

Further insights into the nanomechanical behavior during uniaxial tension tests in the -Ti₃Al and -TiAl phases are provided in Figure 12. These figures employ color coding to depict the centrosymmetry parameter (CSP), which helps identify local lattice distortions and hence, defects. For the -Ti₃Al phase, the strain levels at which defects appeared remained relatively constant across different Nb concentrations with MTP, but showed a decrease with increasing Nb concentration in DeePMD, except in the case of 7 at.% Nb. These findings are in agreement with the stress-strain curves of depicted in Figures 11A, B. In the -TiAl phase, the strain threshold for defect formation decreased with increasing Nb concentration using MTP, while it increased with increasing Nb concentration in DeePMD. This observation aligns with the stress-strain data shown in Figures 11C, D.

FIGURE 12

3.2.6 Generalized stacking fault energy

This section evaluates the generalized stacking fault energy (GSFE) of -TiAl and -Ti₃Al using both DeePMD and MTP potentials. Our analysis compares these findings with results derived from DFT and other referenced studies. Specifically, we examine studies such as Qi et al. (2023), which utilized the MTP potential, and (Lu et al., 2023), which employed the DeePMD method, where the values are available only for -TiAl. We used atomman (atomistic manipulation toolkit) (Becker et al., 2013; Hale et al., 2018) for the computation of stacking fault energy.

For -TiAl, dislocation glide occurs on the {111} close-packed planes, which are susceptible to three distinct types of stacking faults: intrinsic stacking fault (SISF), antiphase boundary (APB), and complex stacking fault (CSF). CSF is associated with ordinary /2 dislocations, while SISF and APB are linked to and /2 super-dislocations. Figures 13A–D shows the -lines along direction for -TiAl using both MTP and DeePMD. The energy values are documented in Table 5. MTP models generally align with the DFT and other referenced values for CSF, though MTP tends to overestimate SISF energy, in agreement with findings from Qi et al. (2023). Conversely, DeePMD tends to align the SISF values more closely with DFT, while both models underpredict APB energy. Notably, the sequence remains consistent across both computational approaches as reported in Qi et al. (2023) and Lu et al. (2023). Generally, a higher GSFE indicates increased resistance and reduced mobility of dislocations, which directly contributes to the strengthening behavior in nanomaterials, whereas a lower GSFE suggests higher mobility of dislocations and improved ductility. With an increase in Nb concentration, there is a noticeable decline in the energies for SISF, CSF, and APB for MTP, whereas DeePMD shows an increase (except for 4 at.% Nb case for CSF). Zhao et al. (2024) reported that an increase in Nb concentration reduced the stacking fault energies in the -TiAl phase. Similar findings were reported by Dumitraschkewitz et al. (2017), suggesting that MTP predicts this trend well. However, DeePMD presents a discrepancy where Nb alloying has increased the stacking fault energy for -TiAl, although this aligns with the tension curves in Figures 11C, D.

FIGURE 13

Turning to the -Ti₃Al phase, characterized by hexagonal symmetry and various slip systems, our analysis primarily focuses on the fundamental {0001} plane. Figures 13E, F shows the -line along direction obtained using MTP and DeePMD and the corresponding energy values are listed in Table 5. The results for -Ti₃Al show less consistency with DFT and referenced values (Qi et al., 2023) compared to -TiAl. MTP predictions for SISF align with DFT and referenced data (Qi et al., 2023), whereas DeePMD tends to slightly overestimate SISF and notably overpredict APB energy. However, DeePMD predictions for CSF energy are closer to DFT and reference values. With an increase in Nb concentration, SISF, APB, and CSF energies decrease with DeePMD, consistent with Figures 11A, B. However, MTP shows minimal change in APB and SISF values, with only a slight decrease in SISF upon Nb introduction, indicating a minimal impact of Nb alloying on the material’s strength, as corroborated by the data in Table 4.

We have observed that, except for DeePMD for -TiAl, Nb alloying generally reduces the stacking fault energy in both phases, indicating improved ductility. Our previous work (Chandran et al., 2024) demonstrated that Nb alloying enhances dislocation density and thus overall ductility, albeit at the expense of reduced strength. The stacking fault energy computations presented here align with this finding except for DeePMD for -TiAl.

3.3 Comparison of performance of MTP and DeePMD

Here, we conducted a thorough assessment of the computational efficiency in MD simulations, focusing on DeePMD and MTP potentials. Figure 14 provides a graphical representation of simulation performance measured in hours per nanosecond (hours/ns) for varying numbers of atoms, specifically 4, 32, 108, 256, 864, 1,372, and 2048, in systems composed of -TiAl. These evaluations were conducted through short NPT simulations over a duration of 1 picosecond.

FIGURE 14

Our results demonstrate a linear correlation between computational time and atom count for both MTP and DeePMD potentials, highlighting the scalability of each potential with increasing system size. This aspect is crucial for simulations that involve large numbers of atoms. The “hours/ns” performance metric offers crucial insights into the computational demands, quantifying the time needed to complete 1 nanosecond of MD simulation. A significant efficiency disparity is evident from Figure 14, with MTP potential-based simulations exhibiting markedly superior speed compared to those employing DeePMD potential.

For a granular quantitative comparison, Table 6 details the message passing interface (MPI) task timings in an NVT simulation with 108 atoms over a span of 4 picoseconds (ps) for both potentials. Notably, MD simulations (NVT ensemble at 300 K for 4 ps on a system of 108 atoms) with DeePMD potential required 14.978 h/ns, whereas those with MTP potential required only 3.356 h/ns. This finding indicates that the MTP potential is approximately 4.5 times more efficient than the DeePMD potential. Furthermore, Table 6 delineates the time distribution across different simulation processes such as force computation, interprocessor communication, output generation, modifications via fixes, and other related tasks. These tests were carried out on a high-performance computing (HPC) cluster utilizing a single node equipped with 48 processors. This detailed breakdown unequivocally shows that simulations with the DeePMD potential consume more computational time than those with the MTP potential for identical simulation conditions.

4 Conclusion

In our study, we conducted a comparative analysis of the interatomic potentials for TiAlNb alloys, developed through both MTP and deep learning approaches. We introduced a comprehensive dataset for TiAlNb alloys, aimed at serving as a benchmark for studying the Nb-alloyed phases of -Ti₃Al and -TiAl. Through our evaluation of the errors and performance metrics of these generated potentials, we concluded that MTP potentials offer a viable alternative in scenarios with limited computational resources. This is attributed to the lower computational cost of MD simulations relying on MTP, alongside its lower dataset requirements for effective training. However, it is important to note a slight compromise in accuracy when opting for MTP over DeePMD, despite the error margins remaining within acceptable bounds. MTP emerges as a preferable choice for applications where precise energy and force calculations are not paramount. Conversely, for endeavors requiring very high fidelity in energy and force predictions, and where it is possible to allocate more computational resources and training data, DeePMD stands out as the preferred choice. The entire workflow, encompassing dataset generation, method comparison, and the resulting conclusions, is comprehensively illustrated in Figure 15.

FIGURE 15

Our study analyzed material parameters like equilibrium volume, lattice constants, and elastic constants, revealing MTP’s equilibrium volume predictions were closer to DFT results than DeePMD’s for both -Ti₃Al and -TiAl phases. MTP also provided more accurate lattice parameters for the -Ti₃Al phase, while both methods agreed on the -TiAl phase. Elastic constant predictions of both deviated from DFT, but were more accurate for -TiAl. RDF curves from both models correlated well with DFT, especially for -TiAl. Finite temperature properties such as specific heat and thermal expansion coefficients were in good agreement with their corresponding DFT and experimental data. Simulations of uniaxial tension tests indicated that Nb alloying decreases the strength of TiAl-based alloys, with the exception of the results from DeePMD for -TiAl. Additionally, computations of stacking fault energy revealed that Nb alloying enhances the ductility of TiAl-based alloys at the expense of strength, except in simulations using DeePMD for -TiAl. These findings align with the outcomes of our prior work (Chandran et al., 2024). Moreover, the results for generalized stacking faults—except for the DeePMD simulations of -TiAl—are consistent with those reported in Dumitraschkewitz et al. (2017) for Nb-alloyed systems and in Lu et al. (2023); Qi et al. (2023) for Nb-free systems, where the sequence has been accurately reproduced. This succinct analysis highlights the advantages and limitations of MTP and DeePMD in simulating TiAlNb alloys, guiding researchers towards the most suitable computational strategy in view of a trade-off between accuracy and resource needs.

During the model training phase, we encountered difficulties in accurately capturing the -Ti₃Al phase characteristics using both models. To address this and enhance model performance, our future work will incorporate advanced learning techniques, like active learning, into our methodology. This strategy aims to significantly improve model accuracy while reducing the reliance on extensive datasets.

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