Computational determination of energy datasets
It is imperative to conduct a computational reliability analysis before evaluating any set of data derived from theoretical calculations. The reliability of DFT in calculating energies depends on the choice of the exchange-correlation functional that approximates the complex many-body interactions of electrons. Perdew-Burke-Ernzerhof (PBE) is one of the standard functionals for appreciated functionals for a broad range of applications41. Recently, a new functional of quasi-non-uniform gradient-level approximation (QNA) is developed, which also provides a well balance between computational efficiency and accuracy42,43,44.
The choice of potential may influence the precision of calculated properties. In this work, we performed the computational reliability analysis on a variety of elemental solids and L12 binary compounds by employing both PBE and QNA potentials. Figure 1 plot the calculated results of lattice constants (a) and formation energies (Hf) as a function of the available experimental and theoretical data42,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60. And the results of PBE shown in Supplementary Fig. 1. For convenience, the detailed values of a and Hf are summarized in Table 1. It is notable that for the systems containing Ni, Co, and Cr, their room-temperature magnetic states were evaluated from the Curie temperature (Tc) using the mean-field approximation61. Ni3Al and Cr are treated as paramagnetic (PM) phase with calculated Tc of 36 K (42 K from experiment62) and 76 K, below room temperature, Ni, Co, Fe, Co3Al and CrPt3 show in ferromagnetic (FM) nature with calculated Tc exceeding room temperature, i.e., 456 K (631 K from experiment63), 1511 K (1388 K from experiment64), 1479 K (1043 K from experiment65), 424 K and 304 K. Other systems are all treat as non-magnetic (NM) state. It clearly shows from Table 1 that the computational results using PBE and QNA potentials are closely aligned. For the lattice constants, the error range for PBE is 0~4%, whereas for QNA the error range is 0.02 ~ 2% when comparing our calculation with the experiments and other theoretical data. In the case of formation energy, due to the scarcity of experimental data, we primarily compare our results with the theoretical data found in the Materials Project (MP) database (https://next-gen.materialsproject.org/). Compared to the data from MP, the error range for PBE is 0.7 ~ 15%, whereas for QNA the error range is 0.4 ~ 18%. The average calculated error of ~7% demonstrates the reliability of the present calculation. On the other hand, from Fig. 1 one can find that the results from QNA seem to be more robust than those from PBE (shown in the Supplementary Fig. 1) when compared with the available experimental data, in which QNA shows an average deviation of 1% lower than PBE (~1.2%) for the lattice constants and QNA shows an average deviation of 5% lower than PBE (~18%) for the formation energy. Therefore, in subsequent energy calculations, we have chosen QNA as the exchange-correlation potential to ensure the rationality of computational data.
Fig. 1: Comparison of data calculated by QNA potential with experimental and other theoretical data.
Red circles are labeled as the data from Materials Project, blue triangles represent experimental measurement data, and green triangles represent computational data from other works. a Lattice parameters a (Å). b formation energy Hf (eV/atom).
Table 1 Lattice parameters (Å), Curie temperature Tc (K), and formation energy Hf (eV/atom) for various phases, together with the available data from experiments47,48,49,50,51,52,53,54,55,56,57,58,59,60 and other calculations (https://next-gen.materialsproject.org/)42,46
We focus on the existence of critical γ’ phase within the ternary and quaternary Co-based alloys. Eleven alloying elements commonly considered in the superalloys, including Co, Ni, Fe, Cr, Al, W, Ti, Ta, V, Mo, and Nb, had selected to construct a series of L12 Co-containing ternary and quaternary phase. Figure 2a, b presents the schematic diagram of our structure constructing ideas. The Co-rich A3B ternary phase was defined in two types: Co3(B1a,B2b) and (Com,An)3B where A corresponds to Ni, Fe, Cr and B (including B1/B2) corresponds to Al, Nb, Mo, Ti, Ta, V, W, Cr. The subscript parameters of a and b were constrained as a, b = 0.25, 0.5, 0.75 and a + b = 1; while m and n were constrained as m + n = 1 and range from 0.5–0.9. Similarly, Co-rich A3B quaternary phase was defined in three types: Co3(B1a,B2b,B3c), (Com,An)3(B1a,B2b) and (Com,A1n,A2k)3B where A (including A1/A2) corresponds to Ni, Fe, Cr and B (including B1/B2/B3) corresponds to Al, Nb, Mo, Ti, Ta, V, W, Cr. In order to improve the data generalizability, the parameters of a, b, and c in the Co3(B1a,B2b,B3c) were uniformly taken in the Co3B1-Co3B2-Co3B3 pseudo-ternary system, and the parameters of m, n, and k in the (Com,A1n,A2k)3B were uniformly taken in the Co-rich region of Co3B-A13B-A23B pseudo-ternary system as shown in Fig. 2b. The parameters of a, b, m, and n in the (Com,An)3(B1a,B2b) were determined with reference to those in the aforementioned Co-rich A3B ternary phase. According to such construction, we obtained 1674 L12 Co-containing phases including 8 Co3B, 84 Co3(B1a,B2b), 115 (Com,An)3B, 224 Co3(B1a,B2b,B3c), 88 (Com,A1n,A2k)3B and 1155 (Com,An)3(B1a,B2b).
Fig. 2: Composition design of ternary and quaternary A3B-type γ’ phases.
a A3B-type crystal structure. Gray spheres represent the positions of A atoms, and purple spheres represent the positions of B atoms. b Composition values of phases marked as solid point in the Co-rich region of the pseudo-ternary phase diagram, which are also listed in the follow tables. Purple tables indicate the values of the components at the B sites, and gray tables indicate the values of the components at the A sites. c Calculated formation (Hf) and decomposition energies (Hd) of designed ternary and quaternary A3B-type γ’ phase from both PBE and QNA. Purple dots represent the calculated results from MP-1, and blue dots represent the calculated results from MP-2. The top and right sides show the statistical distribution of energy. The blue histograms on the top represent the statistical distribution of formation energies, while the blue and pink histograms on the right represent the energy distribution of decomposition energies from MP-2 and MP-1, respectively. The green, orange, yellow, and blue highlighted areas in the figure represent Area 1, 2, 3, and 4, respectively, the Pie chart inserted in (c) shows the distribution ratio of energies in the different Area.
The phase stability predictions performed were governed by formation (Hf) and decomposition (Hd) energies. In terms of high-throughput DFT calculations, the Hf values for 1674 phases were firstly acquired by employing Eq. (1). According to the Eq. (2), then the Hd values were acquired by compares the Hf of an arbitrary phase with a linear combination of energies of set of most competing phases within a given system. It is notable that a set of competing phases was determined by constructing linear combinations of phases from the considered system which produces the same composition, i.e., if one evaluates the Hd of A3B1C1 in the A-B-C system, the relevant space of competing phases includes the elements (A, B, and C), all binary phases in the A-B, A-C, and B-C spaces, and all ternary phases in the A-B-C space, such as A + A2B + C, A3B + C. The most competitive set was determined as the linear combination that yields the lowest energy66. We screened all the observed competing phases in basis of binary and ternary phase diagrams for the 1674 constructed systems and obtained their computed Hf values using GGA-PBE and a modified strongly constrained and appropriately normed (R2SCAN)67 from Materials Project (MP) (https://next-gen.materialsproject.org/) (see Supplementary Table 1). The competitive phase energy database from GGA-PBE is defined as MP-1, and the competitive phase energy database from R2SCAN is defined as MP-2. We determined Hd and the stability-relevant decomposition reaction for each system and reported the distribution of energy data in Fig. 2c. For these 1674 systems, the distribution of computed Hf values adheres to a normal distribution, aligning with the uniformity requirements of ML models. Within the distribution, 37% of the phases exhibit negative Hf values whereas 63% have positive energies. Similarly, the distribution of computed Hd values also follows a normal distribution, with 11% of the cases in average presenting negative values. The distribution of Hf-Hd condition was counted and illustrated in Fig. 2c as well. The variance in the exchange-correlation functional leads to differences in Hd calculated by MP-1 and MP-2, with MP-2 generally producing lower values compared to those of MP-1. For these 1,674 systems, Area-1 condition (Hf > 0, Hd > 0) is found to be the largest in quantity (52% and 59.8% of phases by MP-1 and MP-2), followed by Area-2 condition (Hf < 0, Hd > 0) (19% and 30.1%), Area-3 condition (Hf < 0, Hd < 0) (9.9% and 21%), and Area-4 condition (Hf > 0, Hd < 0) (0.2% and 8%). Positive energy values within the conditions specified for Area-1 imply theoretical instability for the phases, while the phases within the Area-3 condition, characterized by negative energy values, are theoretically stable and may potentially be realized experimentally. For the Area-2, the scenario suggests that a fraction of these phases could be metastable and possibly exist under certain conditions. Conversely, a minimal quantity of energy appears in Area-4, a scenario that might represent a paradoxical condition, indicative of anomalous data. Upon further analysis of Area-4, comparing MP-1 and MP-2 computational outcomes, the incidence of anomalous data is considerably higher with MP-2 as shown in Fig. 2c, suggesting that the reliability of data derived from PBE may be superior.
Feature engineering and ML model training
Upon conducting a review of literatures, we judiciously selected a suite of 38 features (see Table 2) to initiate the training and assessment of an array of 13 conventional machine learning algorithms, including Random Forest (RF), AdaBoost, XGBoost, Gradient Boosting Decision Tree (GBDT). The models were rigorously applied to datasets comprising formation and decomposition energy metrics to ascertain their predictive prowess. The accuracy of these ML models was rigorously evaluated and benchmarked by employing the metrics of Mean Absolute Error (MAE) and the coefficient of determination (R2). This was accomplished through a robust five-fold cross-validation approach, wherein 80% of the data was allocated for training purposes, while the remaining 20% served as the test set.
Table 2 Formulas for 27 feature parameters excluding the chemical composition
Figure 3 depicts the compiled results, inclusive of both MAE and R2 from the cross-validation process. For the Hf’s dataset, it clearly shows in Fig. 3a that, among the array of models evaluated, XGBoost, RF, GBDT, and DT models distinguished themselves with superior testing accuracy. The R2 values for these models were notably high, recorded at 99.0%,98.3%, 98.8%, and 95.8%, respectively. Conversely, models such as SVR, CNN, KNN, and GPR demonstrated suboptimal performance with R2 values of 83.6%, 83.6%, 76.5%, and 82.1%, respectively. Furthermore, an analysis based on the MAE metric in Fig. 3a indicates that RF, XGBoost, and GBDT models achieved exceptionally low testing MAEs of 11mev, 10mev, and 9mev, respectively. We trained the ML models in basis of Hd’s dataset (see Fig. 3b) as well. Notably, these ML models, in general, showed a diminished accuracy in handling Hd data when compared to their performance on Hf data, indicating that the inherent characteristics of Hd dataset might pose a greater challenge for ML algorithms. Nevertheless, certain models continued to demonstrate a commendable level of precision. Specifically, the RF, XGBoost, GBDT, and DT models again emerged as top performers with the highest test accuracies, boasting R2 values of 96.0%, 96.0%, 95.9%, and 90.9%, respectively. Further insight from the MAE values depicted in Fig. 3b indicates that the RF, XGBoost, and GBDT models exhibited remarkably low MAEs, hovering around the 12mev mark. This metric reaffirms the superior predictive abilities of these models, highlighting their potential as reliable tools for the estimation of decomposition energies. Contrastingly, models such as SVR, CNN, KNN, and GPR persisted in their struggle to achieve satisfactory predictive accuracy, with R2 below 70%. This consistency in underperformance may suggest a fundamental limitation in these models’ architecture when applied to the nuanced task of energy prediction within our scientific discipline.
Fig. 3: Comparison of accuracy (R2) and error (MAE) of 13 machine learning models.
Orange and blue histograms represent the accuracy and error of the training set, while green and pink histograms represent the accuracy and error of the test set. The black error bars indicate the bias of five-fold cross-validation. Yellow background denotes (a) Formation energy (Hf). Blue background denotes (b) Decomposition energy (Hd).
Therefore, our study’s empirical evidence establishes RF, XGBoost, and GBDT as frontrunners in the predictive modeling of energy, with their high accuracy and reliability across various energy datasets. Considering the algorithmic complexity of XGBoost and GBDT models68, the RF model was chosen as the benchmark for modeling and prediction ascribed to its straightforward structure coupled with robust generalization capability68,69,70.
The starting set includes 38 high-dimensional features as detailed in Table 2, which may negatively affect model stability and its ability to generalize. This complexity may also hinder the understanding of the model’s decisions. To improve upon this, feature engineering was implemented to pare down dimensions, selecting only the most relevant features to enhance predictive accuracy. The workflow for this dimensionality reduction through feature engineering is presented in Supplementary Fig. 2. Notably, feature filtering is performed through a coupled analysis of Spearman correlation (ρ) and feature importance weights derived from the RF model. Specifically, we begin with a Spearman correlation analysis to identify feature combinations with a ρ > 0.9. Then, based on the ranking of feature importance weights from the RF model, we eliminate the lower-ranked features within the strongly correlated Spearman feature combinations. The RF weight ranking was illustrated in Fig. 4 and the Spearman correlation coefficient was shown in Supplementary Fig. 3. With respect to formation energy, features such as as with d and Smix with Co have been identified as strongly correlated as presented in Supplementary Fig. 3a. Importance weight analysis in Fig. 4a has revealed that the top five most critical features are Cr, DVEC, Hmix, as, and Tm. Guided by the importance rankings from the RF model, a total of 13 lower-ranked features within the strongly correlated Spearman feature combinations were eliminated: Smix, DAtom, AF, AW, AM, AR, AE, s, p, d, f, ap, af. Similarly, for the decomposition energy, features such as AR with AW, and AE with AF have been identified as strongly correlated. Importance weight analysis in Fig. 4b reveals that the five most significant features are VEC, Ni, Elect, AF, and DHmix. Following the ranking of feature importance by the RF model, also a total of 13 lower-ranked features within the strongly correlated Spearman feature combinations were eliminated: AW, ad, AM, d, ap, Atom, AE, Smix, af, s, AN, p, f.
Fig. 4: Importance weights analysis of features from Random Forest and feature elimination by using RFECV5.
a Importance weights for formation energy (Hf). b Importance weights for decomposition energy (Hd). c Feature elimination for Hf. d Feature elimination for Hd, in which the feature ranked as 1 is selected as the threshold value indicated by the red line.
Finally, to enhance the model’s predictive performance and interpretability, a 5-fold recursive feature elimination (RFECV5) method was utilized, systematically pruning away features that minimally affect the prediction outcomes, using a ranking of 1 as the criterion for selection. The RFECV5 feature importance ranking was also plotted in Fig. 4. Based on the results of RFECV5 analysis in Fig. 4c, d, for formation energy, there are 8 out of 25 features that meet the selection criteria: Hmix, DHmix, Tm, DVEC, Ni, Ti, Cr, and as; whereas for decomposition energy, 16 features are selected: Ni, Al, Cr, VEC, Elect, AF, DHmix, DTm, DBulk, as, Hmix, DVEC, AR, DAtom, Tm, DElect. The features after filtering are all listed in Supplementary Table 2. It intriguing that after dimensionality reduction, the two energy sets share 7 common features: Cr, DHmix, Ni, Hmix, DVEC, Tm and as. When one delves into the correlation between these shared features and the magnitudes of the two energy values as plotted in Supplementary Fig. 4, an interesting pattern emerges: DHmix, Ni, DVEC, and as are inversely correlated with both formation and decomposition energies, exhibiting a trend where the energies decrease as these feature values increase. Conversely, the energies tend to rise with increasing values of Cr, Hmix, and Tm. This fascinating relationship might hold profound physical meaning, deserving further microscopic analysis, but it falls beyond the scope of this study.
To demonstrate the efficacy of feature dimensionality reduction, we trained and compared the accuracy of RF models tested with initial versus filtered features for both energy datasets. The comparative results are illustrated in Fig. 5, in which the model training for formation energy is labeled as FEmodel and that for decomposition energy is labeled as DEmodel. Herein, we employed 90% of the data for 5-fold cross validation (used 80% of the data for training and 20% for testing, respectively) and reserved a separate 10% as an independent validation set to prevent overfitting in the dimensionality-reduced models and to verify their generalization capability. The flow chart of model training is plotted in Supplementary Fig. 2.
Fig. 5: Comparison of model accuracy before and after feature engineering.
Orange and blue histograms represent the accuracy and error of the training set, while green and pink histograms represent the accuracy and error of the test set. The red error bars indicate the bias of five-fold cross-validation. In the right-hand figure, the top and right sides show the statistical distribution of energy. The blue histograms on the top represent the statistical distribution of validation set, while the blue and pink histograms on the right represent the energy distribution of predicted values from orinal features and after feature engineering, respectively. a FEmodel. b DEmodel.
Figure. 5a, b illustrate the training and testing accuracies of the FEmodel and DEmodel, respectively. For the FEmodel (Fig. 5a), the model trained and tested with initial features achieved an impressive R2 accuracy of 99.77% and 98.35%, consistent with aforementioned results (Fig. 3). After feature dimensionality reduction to eight, the model still maintained a high R2 accuracy, with a marginal average decrease of 0.04% in training and 0.3% in testing accuracies compared to that with the original features. Correspondingly, the MAE values increased by an average of 1 meV. It suggests the effectiveness of feature dimensionality reduction in this study, as it achieved significant reduction without substantial loss. In the case of the DEmodel (Fig. 5b), compared to the FEmodel, there is a notable decrease in overall model accuracy, particularly evident in the test set with R2 accuracy of about 96% which is 2% lower than FEmodel (98%). After feature dimensionality reduction, with training and testing R2 accuracies declining by 0.01% and 0.06%, respectively. The corresponding MAE values increased by 0.06 meV and 0.23 meV. Despite this decrease, the DEmodel still exhibited an accuracy of approximately 96% (99.41% for training, 95.95% for testing), which also confirms the effectiveness of feature dimensionality reduction. Furthermore, we assessed the generalization capability of the dimensionality-reduced models using an independent validation set. As depicted in Fig. 5a, the FEmodel demonstrated prediction ability on the validation set that is highly consistent with the pre-dimensionality reduction performance, achieving an R2 accuracy of 98.07%. Similarly, the DEmodel displayed comparable prediction ability on the validation set as seen in Fig. 5b, maintaining an R2 accuracy of 97.05% with minimal deviation from the pre-reduction results. The high predictive accuracy on the validation set confirms that both the FEmodel and DEmodel retained excellent generalization capabilities even after feature dimensionality reduction. Therefore, in view of the exceptional accuracy and robust generalization of our trained models, we have confidence that the carefully chosen 8 and 16 features adequately support the RF model in maintaining a strong and consistent predictive capability for both the formation and decomposition energies of the Co-based γ’ phase in the present study.
Screening unknown γ’-containing systems
The A3B-type (CoaNibFecCrd)A-site(AleWfTigTahViMojNbk)B-site family was chosen as the system to be screened. We focus on the potential ternary and quaternary γ’ phase. The composition space (in at.%) was defined as follows: 50% ≤ a, 0% ≤ b ≤ 30%, 0% ≤ c ≤ 10%, 0% ≤ d ≤ 10%, 0% ≤ e, f, g, h, I, j, k ≤ 25%, and the composition is constrained by e + f + g + h + i + j + k = 25%, and a + b + c + d = 75%. In this composition space, we calculated the number of candidates that satisfy these two constraints exhaustively. A composition variation step of 0.5% for each element gives a total of 151,796 unknown systems to be evaluated. The preliminary analysis of the compositions of these unknown systems, as illustrated in Supplementary Fig. 5, reveals the inclusion of 1673 ternary and 150,123 quaternary cases. Within the ternary systems, Co3(B1,B2) types constitute 59.4%, (Co,Ni)3B types account for 24.7%, and both (Co,Fe)3B and (Co,Cr)3B types each represent 7.9%. In the quaternary systems, Co3(B1,B2,B3) types make up 26%, (Co,Ni)3 Co3(B1,B2) types yield 36.1%, each of (Co,Fe)3(B1,B2) and (Co,Cr)3(B1,B2) types individually accounts for 12.9%, each of (Co,Ni,Fe)3B and (Co,Ni,Cr)3B types individually account for 5.2%, while (Co,Fe,Cr)3B types are at 1.7%. This analysis highlights the diverse compositional landscape of these systems.
By utilizing our well-trained RF model, we obtained the Hf and Hd values of these unknown systems. These energies are depicted in Fig. 6a, b. The distribution of energy follows a normal distribution, in which the data from Area-2 (Hf < 0, Hd > 0) constitutes the largest proportion (ternary 52.7%, quaternary 48.3%), followed by those from Area-1 (ternary 38.3%, quaternary 41%) and Area-3 (ternary 9%, quaternary 10.7%), respectively. An interesting observation is the complete absence of energy data in Area-4, indicating that the well-trained model has effectively circumvented the generation of “abnormal” data, underscoring the robustness of our model.
Fig. 6: Screening results of unknown γ’-containing systems.
a Energy distribution of >150,000 unknown γ’ phases and (b) distribution ratio of energies in the different Area. The orange circle represents the Screened region. c Process of component screening based on domain knowledge. In the first step, >89,000 systems with Hf < 0 eV/atom were selected from >150,000. In the second step, 23,000 systems without Ni/Fe/W were selected from >89,000 systems. Finally, 1049 systems with Hd < 0.04 eV/atom were selected. Energy distribution of Co-based system after screening: (d) Al-containing system; (e) Al-free system. Different shapes and colors represent different systems.
Employing the dataset of predicted energies, the domain knowledge is applied to refine our results, aiming to narrow the compositional space and uncover novel γ/γ’ W-free Co-based systems. During the screening process, we initially focused on the systems with negative Hf, as this is a prerequisite for the stable existence of a phase structure. In addition to excluding high-density W element, Fe is also eliminated based on the developmental trends of superalloys. Furthermore, we intentionally excluded Ni to ensure a clearer and more accurate identification of the original γ/γ’ W-free Co-based system, as it is believed that the presence of Ni can significantly enhance the stability of γ’ phase and expand the composition range of γ’71. Finally, we referenced the theoretical Hd value of the γ’ phase in Co-Al-W (64 meV72) and set a tighter screening threshold of 40 meV for decomposition energies. A schematic representation of such screening process is provided in Fig. 6c. Under these thresholds, 1049 L12 phase compositions that met the threshold criteria from over 150,000 unknown compositions are identified. A detailed analysis reveals that these 1049 compositions are primarily distributed across 25 Al-free systems as shown in Fig. 6e and 11 Al-containing systems (see Fig. 6d). The systems and their corresponding compositional ranges are counted and listed in Table 3. More detailed data are listed in Supplementary Dataset. Within these Al-containing systems, two are ternary systems, whereas the remainder are quaternary systems. In contrast, of the Al-free systems, nine are ternary, with the rest being quaternary. The experimental competitive phases within the convex hull of these systems are also detailed in Table 3, encompassing a range of compounds: Co, CoAl, CoTi, CoTa2, Co3V, Co2AlV, Co6Nb7, Cr2Al, Co3Mo, and CoCr. Integrating the energy distribution analysis presented in Fig. 6d, e, the decomposition energy distribution for Al-containing systems is concentrated between 20 and 40 meV, whereas for Al-free systems, it is more dispersed, ranging from 0 to 40 meV. Notably, all these systems contain either Ti or Ta. Among them, systems containing Ti exhibit lower formation energies, while those incorporating Ta have somewhat higher Hf.
Table 3 Numbers and composition ranges (at.%) of the screened ternary and quaternary L12 γ’ phases, as well as the corresponding set of most competing phases. Notable that X1, X2, and X3 represent the rest of elements in the system
On the other hand, we performed a statistical analysis on the post-screening decomposition energy distribution for both Al-containing and Al-free systems, with the Pareto diagram of results depicted in Supplementary Fig. 6. Within the Al-containing systems, over 50 compositional points belong to the Co-Al-Ti-V system, indicating the potential ubiquity of the L12 phase across a broad compositional spectrum within this system. The number of compositional points in other systems does not exceed 20, considerably lower than that in the Co-Al-Ti-V system. This disparity indirectly suggests that the Co-Al-Ti-V may possess significant potential as a candidate γ/γ’ Co-based alloys. Similarly, examining the Al-free systems, seven systems, i.e., Co-Ti-V-Cr, Co-Nb-Ta-Cr, Co-Mo-Ti-V, Co-Nb-Ta-Ti, Co-Nb-Ta-V, Co-Nb-Ti-V, and Co-Mo-Nb-Ta that each have over 60 compositional points were identified, which is notably superior compared to other systems under consideration (as shown in the boxes in Table 3). This observation also underscores the substantial candidate potential of these eight systems.
Experimental verification
In existing research on machine learning-based predictions, most results remain at the theoretical stage. To ascertain the predictive accuracy of our ML model, we embarked on the experimental characterization of two selected candidate alloys. These were the Al-free Co-Ti-V-Cr alloy and the Al-containing Co-Al-Ti-V alloy, each distinguished by achieving the highest Pareto rank within their respective categories as delineated in Supplementary Fig. 6. Drawing upon the compositional distribution ranges for these two systems presented in Table 3, the CALPHAD (Calculation of Phase Diagrams) method was utilized to evaluate the pseudo-binary phase diagrams of Co-(21-x)Ti-xV-2Cr and Co-3Al-(20-x)Ti-xV to analyze the phase relationship of the respective alloys. The results are depicted in Fig. 7a, b. As indicated by the calculations, for the Co-Al-Ti-V system, a two-phase region, composed of equilibrium γ and γ’ phases, is confined to a temperature range extending from 900–1200 °C. This two-phase region expands with increasing V content. In the case of the Co-Ti-V-Cr system, the two-phase region is more extensive, spanning temperatures from 700–1200 °C. Increasing V expands the existing temperature range of γ-γ’ phases. Based on the phase diagram insights, the compositions for two alloys was roughly identified, designated as Co-16Ti-5V-2Cr (U01) and Co-3Al-15Ti-5V (U02) for subsequent observation.
Fig. 7: Experimental verification of the U01 and U02 includes CALPHAD evaluation, X-ray diffraction (XRD), and electron microscopy images (SEM).
CALPHAD evaluation of (a) Co-(21-x)Ti-xV-2Cr and (b) Co-3Al-(20-x)Ti-xV (x: 0 ~ 7). The red region is marked as the γ/γ’ two-phase region. c X-ray diffraction (XRD) patterns of the as-cast and heat-treated U01 and U02 alloys. The red squares indicate the positions of the main peaks. d, e SEM micrographs of the U01 and U02 alloys after aging at 900 °C for 96 h. The blue box indicates the zoomed-in region, the yellow dashed line represents the grain boundary, and the white line marks the dimension.
Then we carried out the experiment to the U01 and U02 alloys. The bulk samples are obtained through arc melting (see Methods). Figure. 7c presents the X-ray diffraction (XRD) patterns of the as-cast and heat-treated U01 and U02 alloys. As shown, the XRD peaks of the as-cast and heat-treated microstructures are nearly identical for both the U01 and U02. At all temperatures, they exhibit four major peaks at 2θ of 44°, 51°, 75°, and 90°, which correspond respectively to the (111), (200), (220), and (311) planes of an FCC and/or L12 structure. No additional extraneous peaks are detected, indicating the absence of any secondary phases in the U01 and U02 alloys under either as-cast or heat-treated condition. This observation is consistent with the predictions from CALPHAD calculations.
To visually inspect the microstructure and presence of secondary phases, the heat-treated samples were mechanically polished and etched, followed by Scanning Electron Microscopy (SEM) imaging. Figure. 7d, e plots the representative SEM micrographs of the U01 and U02 alloys after heat treatment (1150 °C/24 h + 900 °C/96 h). As clearly indicated, abundant fine-scale precipitated phase, namely the γ’ phase, can be observed through the whole micrograph. The γ’ phase in the U01 alloy exhibits an elongated rectangular morphology with the mean particle sizes around 3 × 1 µm, and a high level of γ’ phase area fraction of 84.73 ± 8% is revealed. Whereas the γ’ phase in the U02 alloy takes on a typical cubic blocky morphology with the mean particle sizes around 2 µm, and the γ’ phase area fraction of 82.77 ± 4% is estimated. Clear grain boundaries are evident, free from the precipitation of other secondary phases. The observation of clear grain boundaries, free from the precipitation of other secondary phases, suggests a homogenous microstructure and efficient distribution of the γ’ phase within the γ matrix. It is known that the γ’ phase has an ordered L12 structure and is coherent with the FCC γ matrix, exhibiting very similar major diffraction peaks as those of the FCC phase in XRD patterns. Thus, the diffraction peaks reflections in Fig. 7c should result from both γ and γ’ phases.
By using Energy Dispersive X-ray Spectroscopy (EDS) equipped on SEM, the average composition of U01 and U02 alloys are confirmed as Co-12.6Ti-6.2V-2.4Cr and Co-2.6Al-16Ti-5V, respectively. The mass density of U01 and U02 alloys are respectively determined to be 8.06 g/cm³ and 7.90 g/cm³, highly lower than the earliest reported γ/γ’ Co-Al-W-based alloys (9.54 ~ 10.1 g/cm³11,73). The reduction of alloy density has emerged as a pivotal trend in the development of γ/γ’ Co-based superalloys in recent years. The notably low-density characteristic of our newly discovered γ/γ’ U01 and γ/γ’ U02 alloys makes them as promising candidates for lightweight industrial applications. However, preliminary Vickers hardness tests have revealed that U01 and U02 exhibit relatively low hardness values of 244.15 and 174.6 HV, respectively, which are significantly lower than those of γ/γ’ Co-Al-W-based alloys (400 HV7). To address such limitation, in the future research it is imperative to conduct subsequent investigations involving the alloying treatment (such as Ni, Ta, and Mo) as well as process optimization to enhance mechanical properties of the newly alloys for demanding applications. In summary, the experimental observation validates the predictable of our well-trained ML model, indicating that ML combined with DFT calculations can efficiently accelerate the discovery of novel γ/γ’ Co-based alloys.