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 A ̊ to 8 A ̊ at 1 A ̊ 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.

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 A ̊ , for ensuring sufficient interaction range while maintaining computational efficiency. The minimum radius ( r _min) was chosen to be 2 A ̊ . 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: w e = 1 for energy, w f = 0.01 for force, and w s = 0.001 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.

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 A ̊ , with the smoothing function commencing at 4 A ̊ , 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 3.51 ⋅ 1 0 − 8 . 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.

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 α 2 -Ti₃Al and γ -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.

Table 2 shows the equilibrium volume predicted by DFT, MTP and DeePMD for α 2 -Ti₃Al and γ -TiAl. For the α 2 -Ti₃Al, equilibrium volumes predicted by DFT, MTP, and DeePMD are respectively 15.88 A ̊ 3 , 16.40 A ̊ 3 and 14.52 A ̊ 3 . In a similar vein, for γ -TiAl phase, the corresponding values are 16.22 A ̊ 3 , 15.76 A ̊ 3 and 15.46 A ̊ 3 . 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 α 2 -Ti₃Al and γ -TiAl are compared to corresponding experimental values (PEARSON, 1958; He et al., 1997) in Table 2. For the α 2 -Ti₃Al phase, MTP predicts lattice constants of a = 5.683 A ̊ and c = 4.633 A ̊ . These closely match the DFT-calculated values of a = 5.625 A ̊ and c = 4.586 A ̊ and experimental values (PEARSON, 1958) of a = 5.770 A ̊ and b = 4.620 A ̊ . Conversely, DeePMD predicts a = 5.453 A ̊ and c = 4.446 A ̊ , slightly diverging from the DFT lattice parameters. The equilibrium lattice constants for γ -TiAl as forecasted by MTP–a = 3.910 A ̊ and c = 4.099 A ̊ – exhibit a striking resemblance to the DFT-determined values of a = 3.950 A ̊ and c = 4.140 A ̊ whereas the experimental values (He et al., 1997) are a = 3.998 A ̊ and c = 4.067 A ̊ . The DeePMD predictions, a = 3.910 A ̊ and c = 4.099 A ̊ , 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 α 2 -Ti₃Al and γ -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.

Next we assessed the elastic constants of α 2 -Ti₃Al and γ -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 α 2 -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 α 2 -Ti₃Al phase in this analysis.

In the analysis of the γ -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 α 2 -Ti₃Al and γ -TiAl at 300 K, 500 K, 700 K, and 900 K. As expected, the elastic constants for both α 2 -Ti₃Al and γ -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 γ -TiAl; (Tanaka et al., 1996) for α 2 -Ti₃Al), though it is important to note that experimental values are not available for γ -TiAl at 900 K and for α 2 -Ti₃Al at 500 K, 700 K, and 900 K. Moreover, the C₆₆ constant does not apply to the α 2 -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.

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 α 2 -Ti₃Al. Additionally, Figures 7, 8 show a similar comparison for Nb in γ -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 α 2 -Ti₃Al and γ -TiAl phases.

Notably, the RDFs for γ -TiAl from AIMD exhibited a closer match with the ML potentials compared to those for α 2 -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 α 2 -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 γ -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.

To assess the predictive capability of trained MTP and DeePMD potentials at finite temperatures, this section reports the computation of specific heat capacity ( C V ) for 3.1 at.% Nb structures of α 2 -Ti₃Al and γ -TiAl and thermal expansion coefficients ( α a and α c ) for α 2 -Ti₃Al and γ -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 C V values calculated using DeePMD and MTP against those derived from DFT for 3.1 at.% Nb of α 2 -Ti₃Al and γ -TiAl, respectively. It is observed that C V progressively increases with temperature across all methods, reflecting the system’s access to more degrees of freedom at higher temperatures. For α 2 -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, C V 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 γ -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.

In terms of thermal expansion, we computed the coefficients α a and α c corresponding to lattice parameters a and c at temperatures up to 900 K (Figure 10) for α 2 -Ti₃Al and γ -TiAl. These values were compared with experimental results for γ -TiAl (He et al., 1997) and DFT calculations for α 2 -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 α a and α c 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, α a and α c 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.

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 α 2 -Ti₃Al and γ -TiAl phases, are illustrated in Figure 11. Table 4 presents a comparative analysis of ultimate tensile strength (UTS) values for these materials.

We focused on comparing the UTS of the intermetallic phases to gauge the accuracy of mechanical property predictions. For the α 2 -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 γ -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 γ -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 α 2 -Ti₃Al and γ -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 α 2 -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 γ -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.

This section evaluates the generalized stacking fault energy (GSFE) of γ -TiAl and α 2 -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 γ -TiAl. We used atomman (atomistic manipulation toolkit) (Becker et al., 2013; Hale et al., 2018) for the computation of stacking fault energy.

For γ -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 ⟨ 1 ̄ 10 ] /2 dislocations, while SISF and APB are linked to ⟨ 1 ̄ 10 ] and ⟨ 11 2 ̄ ] /2 super-dislocations. Figures 13A–D shows the γ -lines along [ 11 2 ̄ ] direction for γ -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 SISF < CSF < APB 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 γ -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 γ -TiAl, although this aligns with the tension curves in Figures 11C, D.

Turning to the α 2 -Ti₃Al phase, characterized by hexagonal symmetry and various slip systems, our analysis primarily focuses on the fundamental {0001} plane. Figures 13E, F shows the γ -line along [ 01 1 ̄ 0 ] direction obtained using MTP and DeePMD and the corresponding energy values are listed in Table 5. The results for α 2 -Ti₃Al show less consistency with DFT and referenced values (Qi et al., 2023) compared to γ -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 γ -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 γ -TiAl.