Transforming machine learning model knowledge into material insights for multi-principal-element superalloy phase design

Machine learning model construction

Algorithm selection

There are many ML classification algorithms, and different algorithms vary in their applicability to data. This study selects suitable algorithms from nine classification algorithms to quickly and efficiently construct ML classification models with high accuracy for predicting the phase formation of MPESAs, as shown in Table 1. The results of 10-fold cross-validation showed that the gradient boosting classification (GBC) algorithm has the highest accuracy in predicting the presence or absence of the L1₂ phase (Target-A), and the extreme gradient boosting classification (XGBC) was more conducive to predicting the presence or absence of other phases besides L1₂ phase (Target-B). Therefore, we applied GBC and XGBC to the subsequent model optimization and application of Target-A and Target-B, respectively.

Feature selection and model determination

The essence of supervised machine learning is using algorithms to construct a mapping relationship between features (inputs) and target properties (outputs)^40,41,42. This mapping relationship is called a model. Generally, the more features in the input, the more complex the model becomes, and it is more prone to overfitting^43,44. Furthermore, some features may be redundant and noisy, which can impact the accuracy of the model. Therefore, it is essential to perform feature selection on candidate features. On one hand, key features that significantly impact the target properties are selected, and redundant features are eliminated. This process can reduce model complexity and enhance accuracy. On the other hand, the computational time and the overfitting risk of the model can be reduced, and the model’s generalization ability can be improved. It should be noted that having fewer features is not necessarily better, as fewer features imply a reduction in the amount of information available, which can significantly decrease the diversity of the data and thereby reduce the model’s predicted ability for new samples.

This work performs two rounds of iterations for screening to obtain the most representative feature subset using the self-defined feature selection method introduced in Method section. As shown in Fig. 1, firstly, the importance scores of all features were output and ranked based on the selected algorithms. RFA was then used for initial screening. Features reduced from 42 to 28 and 34 for Target-A and Target-B, respectively. Subsequently, SEIFE was used for further screening, and the features were reduced from 28 and 34 to 27 and 31, respectively. The importance scores of 27 and 31 features were re-ranked and then entered the next round of screening. In the end, 24 features were retained for Target-A and 20 for Target-B.

The confusion matrices in Fig. 2a, b show the detailed 10-fold cross-validation classification results of Target-A and Target-B on the best feature subsets, respectively. The prediction accuracy of the GBC model for Have-L1₂-phase and None-L1₂-phase binary classification in the Target-A dataset reached 97.42% and 92.95%, respectively, and the overall accuracy achieved 95.42%. XGBC model classified Have-other-phases and None-other-phases in the Target-B dataset with accuracies of 76.02% and 91.35%, for an overall accuracy of 85.82%. The ROC curve of the 10-fold cross-validation for Target-A is shown in Fig. 2c, and the average AUC value reached 0.98, indicating that the model has good comprehensive performance. The average AUC value of Target-B also reached 0.9, indicating that the model has sufficient reliability (Fig. 2d). The results show that the GBC and XGBC models are robust and reliable for predicting the phase formation in MPESAs.

Determination and evaluation of the design strategy

Outputting model knowledge to determine phase formation rules

ML can help us accelerate the optimization design of alloys as an auxiliary tool. However, our ultimate goal is not only to obtain reliable ML models. More importantly, we want to obtain a generic design strategy that can be shared. Therefore, the SHAP analysis method was used to output the knowledge learned by the models and transform them into understandable material insights, thereby obtaining a generic strategy that can achieve the same purpose as the model prediction. Initially, SHAP values for each feature of each sample were output using SHAP analysis based on the two models constructed. Subsequently, the relationship between features and SHAP values was plotted, as shown in Figs. 3 and 4. The blue and orange colors are the SHAP values for Target-A, where orange indicates a positive impact, signifying that the feature value in orange promotes the precipitation of L1₂. The gray and yellow are the SHAP values for Target-B, with gray indicating a negative impact, implying that features in gray tend to lead to the precipitation of other phases. The common region between orange and yellow, representing the optimal range of the “FCC + L1₂” dual-phase microstructure, can be delineated by overlaying the SHAP values of the Target-A and Target-B models.

Co, Ni, Fe, Al, Cr, and Ti are elements with high occurrence frequency, as shown in Supplementary Fig. S1. Therefore, we analyzed the impact of these six elements on the phase formation of MPESAs. Figure 3 shows that the content of each element needs to be within a reasonable range to obtain the “FCC + L1₂” dual-phase microstructure. For example, Co and Ni require a high content because they are the main elements of form the L1₂ phase. Al and Ti have a narrow range of 1.5 ~ 8 at% and 3 ~ 6 at%, respectively, or appropriately relaxed to 1.5 ~ 8.5 at% and 1 ~ 9 at%. This result is consistent with experimental reports. The dataset has 172 MPESAs that contain Al and with the microstructure of “FCC + L1₂” dual-phase, among 88.95% of which MPESAs exhibit Al content within the range of 1.5–8.5 at%. Excessive Al can form the brittle (Ni, Co)₂ Al Heusler phase⁴⁵. Among the 133 MPESAs containing Ti and with the microstructure as “FCC + L1₂”, the Ti content of 96.99% of MPESAs ranges from 1 to 9 at%. Excessive Ti content leads to Ti combining with Co, Ni, or Fe to form Co₂Ti Laves phase, η-Ni₃Ti hexagonal phase, or Fe₂Ti Laves phase^46,47. The Cr and Fe are also frequent elements in MPESAs. The results output from the models show that the content of Cr is preferably less than 15 at%. The content of Cr less than 15 is beneficial to the formation of the L1₂ phase, but overhigh Cr content can lead to a decrease in the volume fraction of the L1₂ phase⁴⁸. Zhao et al.⁴⁹ also showed that the Cr content should be controlled to less than 15 at%, because excessive Cr can lead to the formation of brittle phases such as σ, μ, Laves, etc. The Fe content should be less than 10 at% if Fe is added. Further increasing the Fe content can result in the precipitation of Laves and NiAl phases^50,51.

The empirical parameters commonly used to determine the phase formation of alloys show an optimal range on the “FCC + L1₂” dual-phase microstructure. As shown in Fig. 4a, \(\overline{{VEC}}\) should be greater than 8 to facilitate the formation of the “FCC + L1₂” dual-phase microstructure. Gue et al.⁵² also showed that FCC solid solutions are stable at \(\overline{{VEC}}\) ≥ 8. The ∆H_mix should be controlled in the range of −16.0 to −9.7 kJ∙mol⁻¹. The optimal range of δ_r within 3.5 ~ 4.7, or appropriately relaxed to 3 ~ 5.4. Zhang et al.⁵³ proposed that δ_r less than 6.5 and ∆H_mix in the range of −15 to 5 kJ∙mol⁻¹ is beneficial to the formation of solid solutions for multi-component alloys. This study demonstrates that achieving the “FCC + L1₂” dual-phase microstructure necessitates a more constrained range of δ_r and ∆H_mix. The parameter ∆S_mix does not exhibit a highly desirable range but is relatively suitable in the range of 5.3 to 13.4 J∙mol⁻¹ ∙ K⁻¹. The parameter \(\bar{{T}_{m}}\) is best at 1723–1822 K, which can be appropriately extended to 1671 ~ 1822 K. ∆χ is better controlled at less than 0.12. New MPESAs with ideal “FCC + L1₂” dual-phase microstructure can be quickly designed using these rules reasonably.

Determination and evaluation of the design strategy

We used these rules to evaluate the phase formation of original datasets to verify their availability. Target-A in the original dataset contains 983 samples, including 543 Have-L1₂-phases and 440 None-L1₂-phases (44.76%). After screening by the criteria of \(\overline{{VEC}}\) > 8 and −16.0 < ∆H_mix < −9.7 J∙mol⁻¹ ∙ K⁻¹, 247 samples were retained, comprising 223 Had-L1₂-phase and 24 None-L1₂-phase. Subsequently, a more refined screening with 1671 < \(\bar{{T}_{m}}\) < 1822 K, left 204 samples, consisting of 192 Had-L1₂ phase and 12 None-L1₂-phase (5.88%). The Co content of the 12 None-L1₂-phase alloys is not between 31 and 72 at%. The dataset comprises 543 samples of Have-L1₂-phase, among which 347 are categorized as None-other-phases and 196 as Have-other-phases (constituting 36.1%). Following screening based on the combination of the above three empirical parameters, 185 samples remain, consisting of 145 None-other-phases and 40 Have-other-phases (21.62%). Within these 40 Have-other-phases alloys, 32 exhibit Co content less than 31 at%. Among the remaining 8 alloys, two has a Fe content exceeding 10 at%, and two alloys have an Al content of 14.2 at%, surpassing 9 at%. The screening of the two experimental datasets reveals that the content of elements, especially the content of Co, strongly influences the formation of the “FCC + L1₂” dual-phase structure.

To further evaluate these rules, we generated 10,000 candidate MPESAs and then screened them using these rules. The prediction results of the Target-A model showed that 9,412 candidates were predicted as the Have-L1₂-phase (94.12%) and 588 as the None-L1₂-phase. The prediction results of the Target-B model showed that these 9412 candidates were classified 9,021 as the None-other-phases (95.84%) and 391 as the Have-other-phases. A significant proportion (95.84%) of the generated candidate MPESAs meet the specified requirement after setting the optimal content ranges for some major elements. Following the screening criteria of \(\overline{{VEC}}\) > 8, −16.0 < ∆H_mix < −9.7 J∙mol⁻¹ ∙ K⁻¹ and 1671 <\(\bar{{T}_{m}}\) < 1822 K, 3821 candidate MPESAs were retained. The prediction results of the Target-A model showed that 3,761 of them belong to the Had-L1₂-phase. For the Target-B model predictions, 3760 (98.40%) of these Had-L1₂-phase candidates were classified as None-other-phases.

The results indicate that combination of the rules \(\overline{{VEC}}\) > 8, −16.0 < ∆H_mix < −9.7 J∙mol⁻¹ ∙ K⁻¹, and 1671 < \(\bar{{T}_{m}}\) < 1822 K enables the rapid and highly accurate (>98%) design of MPESAs with the “FCC + L1₂” dual-phase microstructure. This design strategy is generic, allowing any material designer to reasonably use the design strategy derived from this study for rapid design MPESAs with “FCC + L1₂” dual-phase microstructure. More importantly, the method of obtaining a generic design strategy by transforming model knowledge into material knowledge can also be generalized to the design of other materials, thus circumventing the issue of model non-sharability.

Design and screening of new alloys

We designed new alloys and used the design strategy for screening. Then, some candidate alloys were selected for experiments to validate the accuracy and reliability of the design strategy. Here, 17 elements were selected from the dataset. To ensure that the designed alloys can keep the advantages of MPEAs, the elemental contents did not strictly adhere to the optimal ranges obtained above, but were appropriately adjusted. The selected elements and their contents are shown in Supplementary Table S2 in the Supplementary Information. A total of 10,000 virtual alloys were randomly generated with the following rules: a total composition of 100.0 at% for each alloy, the total content of Co and Ni ≤80.0 at% if the alloy contains both Co and Ni, and the number of elements in the alloy is 4 ~ 7. These 10,000 alloys cover 17 selected elements, and the elemental occurrence frequencies are shown in Supplementary Fig. S3. Moreover, the elemental compositions of these alloys cover the setting range (Supplementary Fig. S4), indicating that the 10,000 candidate alloys are evenly distributed and possess sufficient representativeness. The design rule \(\overline{{VEC}}\) > 8, −16.0 < ∆H_mix < −9.7 J∙mol⁻¹ ∙ K⁻¹, and 1671 < \(\bar{{T}_{m}}\) < 1822 K was used for screening. After screening, 3,760 candidates met the requirements for the “FCC + L1₂” dual-phase microstructure. To ensure that the experimental results are more representative, 12 candidates with different elemental compositions were selected for experimental preparation. The compositions and empirical parameters of the 12 candidate alloys are shown in Table 2.

Experimental results of candidate alloys

Figure 5 shows the microstructures of the 12 alloys after heat treatment. It can be readily observed that only one type of precipitate is present in all the alloys, and no other precipitated phases are detected at the matrix and grain boundaries. High-density near cuboidal-shaped nanoparticles are uniformly distributed in the matrix of these 12 alloys. The XRD patterns of Fig. 6 show that only peaks of FCC structure are present in these 12 alloys. Combined with the SEM and the XRD results, it can be known that the matrix of all the designed alloys exhibits FCC structure, and the peaks of the precipitated phase overlap with those of the matrix. The peaks of the precipitated phase coincide with the FCC matrix, indicating the extremely small lattice misfit between the matrix and the precipitated phase. These experimental results provide preliminary evidence that the designed alloys are all “FCC + L1₂” dual-phase microstructures.

The solvus temperature of the L1₂ phase is closely related to the temperature-bearing capacity of superalloys, while lightweight is a demand for further development in aerospace engineering. Therefore, the density and the L1₂-phase solvus temperature of these 12 alloys were tested, and the results are shown in Fig. 7a, b. All alloys have L1₂-phase solvus temperatures above 1000 °C, and the densities are all around 8.0 g‧cm⁻³. Among them, No. 9 alloy exhibits the best combination of density and L1₂-phase solvus temperature. The densities and L1₂-phase solvus temperatures of L1₂-strengthened MPEAs, L1₂-strengthened cobalt-based superalloys, and traditional nickel-based superalloys are compared in Fig. 7c. It can be observed that MPEAs have lower densities and L1₂-phase solvus temperatures. L1₂-strengthened cobalt-based and nickel-based superalloys exhibit high L1₂-phase solvus temperatures but have higher densities, which do not meet the demands for further development in aerospace engineering. In contrast, our developed No. 9 alloy combines a 1218 °C L1₂-phase solvus temperature with a low density of 7.77 g‧cm⁻³, which shows good application potential. Therefore, we selected No. 9 alloy for further characterization.

We calculated the phase diagram of No. 9 alloy using the Thermo-calc software with the TCHEA5 database. The phase diagram shows that the L1₂ phase completely dissolves after temperature 1215 °C, as shown in Supplementary Fig. S2. The ideal microstructure consisting of an FCC matrix and L1₂ phase precipitates can be obtained within the range of 535–1215 °C. This result indicates that the alloy has a broad aging window, with only trace amounts of other phases forming at lower temperatures.

The TEM analysis was performed to better understand the precipitates of the No. 9 alloy. Figure 8a, b shows bright-field (BF) and locally magnified dark-field (DF) images of No. 9 alloy, respectively. High-density near cuboidal-shaped nano-precipitates are uniformly distributed in the FCC matrix, consistent with the SEM results. The average diameter of the nano-precipitates, calculated based on the area-equivalent diameter (size = \(2\sqrt{{area}/\pi }\)), is 225 ± 13 nm. The inset in Fig. 8b shows the corresponding selected-area diffraction pattern (SADP) along the [001] zone axis, which consists of the bright Bragg reflections of the FCC phase and the faint superlattice reflections of the L1₂ phase (indicated by the yellow dashed circles). The high-resolution TEM (HRTEM) image of the interface between L1₂ nanoparticles and the FCC matrix, along with the corresponding fast Fourier transformation (FFT) images of both phases, is shown in Fig. 8c. The results reveal that the L1₂ nanoparticles are highly coherent with the FCC matrix.

Figure 8d shows the elemental distributions of both phases detected by TEM-EDS. The EDS mapping reveals that the L1₂ nanoparticles are enriched with Ni, Al, and Ti, while the FCC matrix is rich in Co, Fe, and Cr. Table 3 shows the detailed chemical compositions of each phase, and the partitioning coefficients K_i were calculated to illustrate the elemental partitioning behavior quantitatively. Partitioning coefficients Ki describe the element partitioning behavior of the L1₂ and FCC phases, where elements with K > 1 are considered L1₂ formers, while others are L1₂ destabilizing elements. According to the compositional analyses, Ti shows the strongest tendency to partition into the L1₂ precipitates (K_Ti = 3.13), while Al exhibit a relatively marginal tendency (K_Al = 1.75). Our ML model also captured this tendency. In Fig. 1a, the feature importance ranking provided by the model predicting the presence or absence of the L1₂ phase shows that Al and Ti are at the top among all elements, suggesting their significant role in promoting the formation of the L1₂ phase. Meanwhile, Cr, Fe, and Co are strongly partitioned to the FCC matrix rather than the L1₂ particles (K_Cr = 0.32, K_Fe = 0.36, and K_Co = 0.54).

In summary, we combined an ML model with the SHAP method to develop a generic design strategy for rapidly and accurately determining the phase formation of MPESAs. Firstly, two classification models were constructed to predict the presence or absence of the L1₂ phase and other phases, respectively. The 10-fold cross-validation accuracies of the two models reached 95.42% and 85.82%, respectively. Then, the knowledge learned from these two models was extracted using the SHAP method and then transformed into comprehensible material rules. Based on these material rules, a generic design strategy that can rapidly and accurately determine the phase formation of MPESAs was obtained, specifically \(\overline{{VEC}}\) > 8, −16.0 < ∆H_mix < −9.7 J∙mol⁻¹ ∙ K⁻¹, and 1671 < \(\bar{{T}_{m}}\) < 1822 K. Based on the obtained material rules, 10,000 candidate MPESAs were designed. These candidates were then filtered using the design strategy, resulting in 12 candidates with different element combinations being selected for preparation. The characterization results of XRD, SEM, and TEM showed that these 12 new MPESAs are all “FCC + L1₂” dual-phase microstructures without any other precipitated phases. The experimental results validated the accuracy of the ML model and design strategy and indicated that ML can accelerate the design and discovery of new materials. In this work, the knowledge learned from ML models was transformed into material rules to obtain design strategy. Then, the design and screen of new materials used the comprehensible design strategy instead of the traditional ML model prediction. Therefore, this method directly avoids the non-shareability problem of the models.