Abstract
Antiphase boundaries (APBs) are planar defects that play a critical role in strengthening Nibased superalloys, and their sensitivity to alloy composition offers a flexible tuning parameter for alloy design. Here, we report a computational workflow to enable the development of sufficient data to train machinelearning (ML) models to automate the study of the effect of composition on the (111) APB energy in Ni_{3}Albased alloys. We employ ML to leverage this wealth of data and identify several physical properties that are used to build predictive models for the APB energy that achieve a crossvalidation error of 0.033 J m^{−2}. We demonstrate the transferability of these models by predicting APB energies in commercial superalloys. Moreover, our use of physically motivated features such as the ordering energy and stoichiometrybased features opens the way to using existing materials properties databases to guide superalloy design strategies to maximize the APB energy.
Introduction
Owing to their excellent hightemperature mechanical properties and corrosion resistance, Nibased superalloys are the materials of choice for several important technologies, including airplane turbines and landbased power generators^{1,2,3,4}. The microstructure of these alloys consists of a Nirich, disordered FCC matrix (γ phase) that envelops coherent Ni_{3}Al precipitates (\(\gamma ^{\prime}\) phase) which give them their strength^{5}. The Ni_{3}Al precipitates are ordered in the L1_{2} structure, where Al atoms are on the corners and Ni atoms are on the face centers of a simple cubic lattice. Upon the passage of a matrix \(\frac{a}{2}\langle 110\rangle\)type dislocation through the ordered precipitates, highenergy antiphase boundaries (APBs) are created on the {111} family of planes to contribute toward \(\gamma ^{\prime}\)phase strengthening in superalloys^{6,7,8}. One example of where this strengthening mechanism manifests is in the yield strength of the superalloy when dislocations shear the Ni_{3}Al precipitates. Depending on the microstructure, dislocation pairs can be weaklycoupled or stronglycoupled, and Reppich found that the yield strength (σ_{y}) reaches a peak at the transition point between the two regimes, where it can be approximated by:
where f is the volume fraction of the \(\gamma ^{\prime}\) phase, b is the magnitude of the Burgers vector, and γ_{APB} is the APB energy. Relationships such as Eq. (1) reinforce the importance of the APB energy in modulating superalloy properties and justify why it has garnered so much attention in superalloy design.
In addition, Nibased superalloys have multicomponent chemistries often with ten or more elements^{1,2}, and this alloying further improves the properties of the base Ni–Al alloy to enhance hightemperature performance^{9,10,11}. The APB energy is sensitive to alloy composition, which provides a rich design space one can explore when optimizing the composition of alloys to provide higher yield strength and creep resistance. With so many degrees of freedom, however, it would be difficult and prohibitively expensive to rely solely on experimental methods to probe the composition dependence of the APB energy^{12,13}. As a result, the community has explored the use of computational methods to understand this correlation and exploit it in alloy design.
Numerous computational investigations into the APB energy in Ni_{3}Al and Ni_{3}Albased alloys have been undertaken over the past two decades using a plethora of different methods. Ab initio approaches include density functional theory (DFT)^{14,15,16,17,18,19} and DFTbased methods such as the cluster variation method^{20,21}, coherent potential approximation (CPA)^{22,23}, and cluster expansion (CE) with Monte Carlo (MC) sampling^{24,25}. In addition, semiempirical methods include molecular dynamics^{26} and Monte Carlo simulations^{27} based on classical potentials, and the CALculation of PHAse Diagram (CALPHAD) approach^{18,28}. These studies have contributed greatly to the fundamental understanding of the APB energy in Ni_{3}Albased alloys, including its dependence on temperature, spin polarization, composition, and ordering, all of which have implications for alloy design.
Focusing on the composition dependence, an early DFT study by Chandran and Sondhi found a large strengthening effect due to the addition of Ti, Ta, and Nb atoms, where the (111) APB energy (hereafter symbolized as γ_{111}) increased from its intrinsic value of 0.181 J m^{−2} in Ni_{3}Al to over 0.6 J m^{−2} in the ternary alloys^{15}. Kumar et al. performed a more comprehensive DFT study with 16 different ternary solutes, considering a single atom near the APB in their 96atom supercell for the ternary element^{19}. That study also revealed the sensitivity of γ_{111} to sublattice occupancy of solute species in Ni_{3}Al, which must be precisely determined in order to obtain accurate calculations for γ_{111}. Crudden et al. used a combination of DFT and CALPHAD to calculate γ_{111} in several ternary Ni_{3}Albased alloys with refractory metal solutes^{18}, and they found compositional dependencies that reasonably agreed with previous studies of the APB energy and yield strength in these alloys. The use of CALPHAD methods for calculating the APB energy is possible through its connection to the ordering energy (E_{ord}), as detailed by Miodownik and Saunders^{28} and leveraged in the work by Crudden et al.^{18}. Intuitively, a correlation between these two quantities can be understood as the creation of an APB in the L1_{2}ordered Ni_{3}Al precipitates forms an interfacial defect that disrupts the preferred ordered atomic arrangement in the vicinity of the APB. The relationship was also studied by Gorbatov et al., who used the CPA method to discover a strong correlation between γ_{111} and E_{ord} in ternary Ni_{3}Albased alloys with 3d transition metal solutes at low concentrations (2.5 at.%)^{23}. The correlation between γ_{111} and E_{ord} has been extensively discussed in the community and is particularly attractive as it enables one to use materials thermodynamic databases, which are extensively developed for multicomponent Nibased superalloy systems, to model the APB energy.
Building on the previous studies reviewed above, our goal is to further expand the predictive capabilities of APB energy models by leveraging machinelearning (ML) algorithms to automatically learn the sophisticated relationships between structural and energetic properties and the APB energy from data. ML allows us to combine the ordering energy with other physically meaningful features (derived from materials properties) as inputs into a flexible model which has the potential to predict the APB energy with greater accuracy and across a wider range of superalloy chemistries. The wealth of literature on the APB energy in Ni_{3}Albased alloys suggests that having a transferable, fast, and predictive model can be impactful for alloy design in screening novel compositions. For this purpose, we require a training dataset that extends across a wider range of compositions than previously considered so that the ML model can learn the appropriate relationships for more exploratory designs with less data bias^{29}.
In this work, we create an automated workflow to calculate APB energies at scale, leveraging recent advancements in highthroughput calculations for alloy design^{30}. Using this workflow (see Fig. 1; detailed in “Methods”), we analyze the composition dependence of γ_{111} in 111 model ternary Ni_{3}Albased alloys, generating significantly more DFT data than what are currently present in the literature. We present the results and compare them with previous studies to validate known behavior and uncover insights across a wide range of ternary elements. In particular, we employ ML to analyze feature correlations and build predictive models for the APB energy that can be used to rapidly screen alloy compositions at a much lower computational cost. Finally, we discuss how the infusion of scientific domain knowledge into datadriven methodologies can enhance superalloy design going forward.
Results
Solute site preference
Previous works have demonstrated the sensitivity of the APB energy^{19} and mechanical properties^{31} of Ni_{3}Albased alloys on the solute site preference, i.e., the preference for the solute to reside on the Al or Ni sublattices of the L1_{2} structure. Therefore, we first determine the site preference of ternary species in Ni_{3}Al using the DFTbased dilutesolution thermodynamic framework implemented in PyDII^{32,33} (see “Methods” for details), which has been previously used to predict the solute site preference in intermetallics such as NiAl^{32} and NiTi^{34}. From our DFT calculations and analysis with PyDII, we compute the fractional site occupancy on the Al sublattice (f_{i}(Al) ≡ f) of 46 ternary elements in Ni_{3}Al as defined by the following equation:
where 〈c_{i}(X)〉 is the average concentration of species i on each site of the X sublattice. We perform the calculations at T = 1000 K and obtain the results shown in Fig. 2, where f is plotted as a function of the overall mole fraction (concentration) of Al in the Ni_{3}Al compound. The horizontal dashed line indicates no site preference and corresponds to \(f^\ast\) = 0.25, when atomic species Z randomly occupies the two sublattices independent of the Al concentration. By comparing the behavior of f with the threshold \(f^\ast\) in the composition range shown in Fig. 2a, we classify the sublattice preference of the ternary species into the following three categories:

I.
If \(f < f^\ast\) across the entire composition range (blue curves in Fig. 2a), then that solute prefers the Ni sublattice. For the model ternary supercells used to compute the APB energies (see below), these solutes are distributed entirely on the Ni sublattice at all concentrations. The resulting alloy has the composition Ni_{0.75−x}Al_{0.25}Z_{x}.

II.
If \(f > f^\ast\) across the entire composition range (red curves), then that solute prefers the Al sublattice. For the APB supercell calculations, these solutes are distributed entirely on the Al sublattice at all concentrations, resulting in a model ternary alloy with composition Ni_{0.75}Al_{0.25−x}Z_{x}.

III.
Otherwise, we classify the solute as having a mixed sublattice preference (purple curves). These species are distributed such that 3/4 of the solute atoms are on the Ni sublattice and 1/4 of the atoms are on the Al sublattice. The resulting alloy has a composition Ni_{0.75(1−x)}Al_{0.25(1−x)}Z_{x}.
By automating these calculations and using the above classification system, we identify patterns across the entire periodic table (Fig. 2b). We observe that most elements in groups IX, X, and XI strongly prefer the Ni sublattice (type I) when alloyed in Ni_{3}Al in dilute quantities. The exception is Ir, which—along with Fe and three group XII elements (Zn, Cd, Hg)—is a type III solute whose occupancy on the Al sublattice is strongly compositiondependent, and these elements are adjacent to the type I elements in the periodic table. The vast majority of the elements prefer the Al sublattice (type II), although that preference is weaker among the pblock posttransition metals. These results may be rationalized by the larger size of Al atoms compared to Ni atoms (metallic radii of 143 pm and 124 pm, respectively)^{1,35}, causing the vast majority of the elements to prefer the Al sublattice. The exception for elements near group X may be due to favorable dorbital hybridization with the p orbitals in Al, which is a prominent interaction in Ni_{3}Al^{36}. Additionally, we note that PyDII predicts most of the common alloying elements in Nibased superalloys^{1,2} to strongly prefer the Al sublattice in Ni_{3}Al, at least in the dilute regime in Fig. 2.
Our approach using PyDII to calculate site preference of ternary solutes in Ni_{3}Al can be validated by comparing these results to the previous studies^{37,38,39,40,41,42}. In a similar study where firstprinciples calculations were used to obtain defect formation enthalpies to parametrize a WagnerSchottky model, Jiang and Gleeson^{39} predicted the fractional sublattice occupancy in Ni_{3}Al at 1273 K and produced concentration profiles that are consistent with those in Fig. 2a. They also found a majority of their transition metal solutes to strongly prefer the Al sublattice (f ≈ 1 across the whole composition range) while elements like Ir and Fe had a strong compositiondependent site preference in their model. They calculated a stronger compositiondependent site preference for Co and Mn (PyDII predicts Ni and Al sublattice preference, respectively), but other solute profiles were consistent with ours in the composition range we considered. Liu et al. used DFT calculations to parametrize the formation energies in a dilutesolution model and also found a strong Al site preference for most transition metal elements^{41}, which they attributed to an atomic size effect when the solute radius exceeds that of Ni by more than 15%. When the size mismatch is below this threshold, they found through analysis of the differential charge density that the Al site preference for a solute Z may be correlated to the stronger Ni–Z interactions from dband electrons compared to Al–Z interactions.
APB energy
Table 1 shows the DFTcalculated values for γ_{111} in model ternary Ni_{3}Albased alloys for 37 ternary species at 1.39 at.%, 5.56 at.%, and 9.72 at.% (111 compositions in total). These elements are chosen for their known contributions to hightemperature strength, oxidation resistance, and creep resistance^{1,2}, including a few rareearth metals^{43}, or otherwise to span a diverse range of compositions for our datadriven approach. The locations (i.e., site preference) of the ternary species in the supercells are determined using the predictions from PyDII and then randomized over that sublattice using special quasirandom structures (SQSs)^{44} generated using the software developed by van de Walle et al.^{45,46}. We find that an alloy with 9.72 at.% Ta has the highest APB energy of 0.464 J m^{−2}, which exceeds the intrinsic (111) APB energy for Ni_{3}Al by nearly 0.3 J m^{−2}. The strong influence of Ta on increasing γ_{111} is corroborated by several prior studies^{15,17,18}. Notably, we find the highest value for γ_{111} at the medium solute concentration (5.56 at.%) is not for Ta (0.353 J m^{−2}), but rather for W (0.369 J m^{−2}), which appears to give rise to a strong increase in γ_{111} at lower concentrations. Our calculations show that alloying with Re produces the highest mean APB energy (0.227 J m^{−2}) at low concentration (1.39 at.%), although the variance in γ_{111} means that several other elements, including Ce, Ta, and W, are equally potent within the statistical scatter in our calculations. The contributions from Ta, W, and Re in maximizing γ_{111} at low solute concentrations have also been discussed in other DFTbased studies in the literature^{16,19}.
To better illustrate the APB energy data in Table 1, we plot in Fig. 3 a few representative trends for the dependence of γ_{111} on solute concentration. In a majority of ternary systems, γ_{111} varies approximately linearly with concentration up to 10 at.%, which matches previous studies on γ_{111} in Ni_{3}Albased alloys^{17,23,24,25}. A more comprehensive comparison of the composition dependence between this work and previously published studies may be found in Supplementary Figs. 1 and 2. Elements in groups IV and V of the periodic table, like Ti, help stabilize the \(\gamma ^{\prime}\) phase^{1} and greatly increase γ_{111} as a linear function of their concentration. A few of the elements behave similarly to Co, which has little effect on the APB energy^{23}, while others like Rh cause a roughly linear reduction in the APB energy with increasing solute concentration. We observe these nonincreasing dependencies (exemplified by Co and Rh) most often in alloys where the ternary species is the type I or type III, corresponding to Ni and mixed sublattice preference, respectively.
Perhaps the most striking dependence on concentration is displayed by the elements in groups VI and VII, with a representative example given by Mo in Fig. 3. Adding these solutes increases the value of γ_{111} up to medium concentrations (5.56 at.%), but any further addition induces a drop in the APB energy. Gorbatov et al. reported a similar nonmonotonic dependence in their CPA calculations for γ_{111} for Cr additions^{23}, while CALPHAD calculations by Crudden et al. revealed a similar trend in γ_{111} for Mo and W^{18}. When we look at the variation in γ_{111} with respect to composition across the first three periods of the dblock (Fig. 4), we see that this nonmonotonic behavior is exhibited by all of the group VI and VII transition metals. Followup DFT calculations for γ_{111} without allowing for structural relaxations result in the same qualitative trend for Mo and W, leading us to hypothesize that the origin of such behavior is predominantly electronic. Vamsi and Karthikeyan found that W and Mo increase the contributions to the APB energy from Al–Al violations across the APB^{47}, which matches the dramatic strengthening effect we observe at the lower solute concentrations.
Machinelearning modeling
To better understand the composition dependence of γ_{111}, we engineer several physically motivated features and use ML methods to analyze their predictive capabilities in modeling the APB energy. We first use DFT to calculate the ordering energy (E_{ord}) according to Eq. (6) (see “Methods”) and obtain a value of −0.118 eV atom^{−1} for Ni_{3}Al, which is consistent with the value of −0.12 eV atom^{−1} obtained by Gorbatov et al. using DFT^{23}. The trend for the ordering energy ratio \(\left({E}_{{{\mathrm{ord}}}}/{E}_{{{\mathrm{ord}}}\,}^{{{{{\rm{Ni}}}}}_{3}{{{\rm{Al}}}}}\right)\) among the 3d elements also agrees qualitatively with the CPA results obtained by Gorbatov et al., although our energy ratios are higher by ~40%. We plot γ_{111} against E_{ord} in Fig. 5 and further differentiate the data points based on the ternary solute concentration. There exists a clear negative correlation between the two variables, with a Pearson correlation coefficient of r = − 0.541, indicating a moderately strong linear correlation. This correlation appears to be present at all solute concentrations and agrees qualitatively with previous studies on the 3d transition metals^{23}.
In addition to the DFTcalculated ordering energy, density, and change in lattice parameters, we use several compositionbased features generated using the matminer Python® package (Python^{®} is a registered trademark of Python Software Foundation Corporation.)^{48} (see “Methods”). When we compute the Pearson correlation coefficient between each feature and γ_{111}, we find that four of the features have a stronger linear correlation to the APB energy than the ordering energy (Table 2). The feature with the strongest linear correlation (r = 0.692) is the weighted standard deviation of the number of unfilled d orbitals in each constituent element. The second strongest correlation (r = 0.570) is for the weighted standard deviation of the column number of each constituent element while the third (r = 0.547) is the weighted mean of the covalent radius of each constituent element. We find that the weighted mean Pettifor number^{49} also has a marginally stronger linear correlation (r = − 0.543) than the ordering energy. These features are all automatically generated from the alloy’s composition using matminer^{48} and their correlations to γ_{111} reflect the trends shown in Fig. 4. For example, adding transition metals from the left side of the dblock is shown to dramatically boost the APB energy and it increases the value of the first two features (those elements have more unfilled d orbitals than Ni (2) and Al (0) and they have a smaller column number than Ni (10) and Al (13)). We also generally observe γ_{111} to decrease across a row and increase down a group, which agrees qualitatively with the trends in elemental covalent radii.
After filtering out features with undefined values and features that are highly correlated to another feature, we apply recursive feature elimination (RFE) to further downselect the features and use grouped crossvalidation (CV)^{50} to obtain the optimal number of nine features. We note that E_{ord} is the only physical parameter of the L1_{2} structure that remains. We use these nine features to build a final random forest (RF) model and perform grouped fivefold CV to predict γ_{111}. Figure 6a shows the normalized feature importances^{51} computed by the RF algorithm for the nine features selected from RFE. The standard deviation of the elemental number of unfilled d orbitals surfaces as the most important feature, which follows from its large positive correlation to γ_{111}. The ordering energy is only the third most important feature in our model, preceded by the standard deviation of the elemental number of unfilled orbitals. The parity plot in Fig. 6b compares the crossvalidated predictions of γ_{111} using the RF model against the DFTcalculated values of γ_{111}. We again color the data points by solute concentration and observe relatively good agreement between the ML predictions and the DFT calculations. The overall RMSE from grouped fivefold CV is 0.033 J m^{−2} (R^{2} = 0.753), which is comparable in magnitude with the average experimental error of 0.027 J m^{−2} for multinary (>2 elements) compositions in Table 1 from Crudden et al.^{18} As there may be interest in an ML model without E_{ord} as a feature, particularly for multicomponent compositions where this information may not be readily available, we report the fivefold CV results for such a model in the Supplementary Methods.
Extrapolation to multicomponent chemistries
To demonstrate the generalizability of our ML model, we use our methodology to train an RF model to predict γ_{111} for real superalloy chemistries reported in the literature. Specifically, we use the compositions for the \(\gamma ^{\prime}\) phase reported by Baither et al.^{13} and Vamsi and Karthikeyan^{47}, which were obtained through experiments and thermodynamic modeling, respectively. The \(\gamma ^{\prime}\) phase in these commercial superalloys contain multicomponent chemistries ranging from 5 to 11 constituent elements. As we do not have DFTcalculated data nor access to a thermodynamic database for these alloys, we first retrain the RF model with only compositionbased features (does not include E_{ord}) for the full training set of 111 ternary Ni_{3}Albased alloys (training error of 0.011 J m^{−2}; R^{2} = 0.974). We then use this trained model to predict γ_{111} for the unseen test data (i.e., 23 commercial, multicomponent superalloys) and compare the ML predictions with the γ_{111} values from the literature (Fig. 7). While direct application of the model for ternary systems (which includes E_{ord}) is not possible, our analysis shows that models with comparable performance can be developed without E_{ord} (see Supplementary Methods). We also use several other experimental measurements from the literature^{12,52,53,54,55,56} to test our model and we include those results in Fig. 7 (black squares). The overall test RMSE is 0.038 J m^{−2}, which compares favorably to the CV error, even though the test set contains compositions that are more complex than what exist in our training set. Conversely, when we build a model that simply uses a linear combination of ternary APB energies^{18,25}, i.e., \({{\Delta }}{\gamma }^{{{\mathrm{total}}}}=\mathop{\sum }\nolimits_{i = 1}^{n}{k}_{i}{{\Delta }}{\gamma }_{i}^{\,{{\mathrm{ternary}}}\,}\), we obtain a higher extrapolation error of 0.048 J m^{−2} on the test set.
Compared with the experimental measurements by Baither et al. (black diamonds in Fig. 7), our model predicts values for γ_{111} that are consistent with those reported in that study^{13}. We find the error from our RF model reasonable given the comparison to experimental data and our model still outperforms other predictive models in the literature^{25,47} (Supplementary Fig. 3). The red points in Fig. 7 represent the APB energies from the study by Vamsi and Karthikeyan, who used an environmentdependent nearestneighbor bond (EDNNB) model that was also fit to DFT data^{47}. They used a thermodynamic database to predict the composition of the \(\gamma ^{\prime}\)phase for commercial superalloys and then used those compositions as inputs to their EDNNB model. Our ML model predicts values for γ_{111} that are consistent with their EDNNB model predictions for a majority of the superalloys, but underpredicts γ_{111} at higher values. We note, however, that our ML predictions for Waspaloy, Rene88DT, and RR1000 are in closer agreement with the predicted values from Crudden et al. (purple points)^{18}, which exemplifies the variability between different models for the APB energy.
Discussion
The APB energy is an important parameter for alloy design as it influences several mechanical properties of precipitationstrengthened superalloys and can be tuned by adjusting the chemical composition. By creating a computational workflow to automate DFT calculations, we generate a wealth of data including solute site preference, (111) APB energies, and physically meaningful features such as the ordering energy to enable a datadriven assessment of the chemical contributions to the APB energy. The synergy between highthroughput calculations and machine learning presents an opportunity to explore vast swaths of alloy composition space and accelerate the process of alloy development.
It is interesting to note PyDII’s predictions for Co and Cr, which are the two species with the greatest variability in site preference among previously published studies. For Co, we classify its behavior as preferring the Ni sublattice, but its concentration profile varies the most among type I elements, suggesting it has the possibility of occupying both sublattices^{35,39,42} when analyzed with different compositions or at different temperatures. For Cr, PyDII predicts a very strong preference for the Al sublattice, which reasonably agrees with previous computational studies, but differs from the MC study by Saito and Harada^{57} and a few atom probe tomography experiments^{58,59}. These three studies notably used multicomponent superalloy chemistries and BoothMorrison et al. found that Ta has a propensity to kick Cr out of Al sites^{59}, which could help explain the differences with our results for Cr. Competition between multiple solute species, which more closely mimics real superalloy chemistries, deserves further scrutiny in future investigations and can be incorporated in the dilutesolution thermodynamic formalism employed by PyDII. In general, validation against the literature confirms the viability of using PyDII as an automated tool to predict site preferences of solute species in ordered intermetallics, which can potentially be extended to other precipitationstrengthened alloys^{60,61}.
Our DFT results also reveal that several elements exhibit a nonmonotonic concentration dependence for γ_{111}, which can have important consequences for alloy design. As we scan across the dblock elements (Fig. 4), Ta may be the element that maximizes γ_{111} at high concentrations, but the same cannot be said at lower concentrations, where other solutes like Mo and W perform comparably or better. Furthermore, several predictive models for the APB energy in the literature use a linear regression model based on each solute’s concentration^{18,25}. Crudden et al.^{18} calculated γ_{111} for 12.5 at.% Mo and W, and they assumed that the APB energy varied linearly between 0 at.% and 12.5 at.%. Our results show that for several elements, particularly those in groups VI and VII which are currently used in superalloys, the choice of those endpoints can affect the estimate of its chemical contribution to the APB energy in the \(\gamma ^{\prime}\) phase, and we must be more cautious of making assumptions about monotonicity. These data can also be used to guide the selection of alloying elements to maximize APB energy when facing additional constraints (e.g., total solute quantity, weight, cost, processing techniques).
Building on previous analytical models for the APB energy, our nonlinear ML model uses a derived set of physically motivated features and is not constrained to a particular analytical form. Seeing ordering energy surface as one of the most important features for predicting γ_{111} agrees favorably with our physical intuition, given the known relationships between the APB energy and the ordering energy^{18,23,28}. Furthermore, the ability to extract the ordering energy of an alloy system from CALPHAD^{28} makes it a viable input feature for modeling the APB energy by integrating ML workflows with existing tools and databases from alloy design^{62,63}. Using stoichiometry alone, we can calculate several additional features that improve the generalizability and interpretability of our ML model. Evidenced by the grouped CV performance, these features are already effective at making predictions for alloy chemistries that are not present in the training dataset, albeit with room for improvement (W at medium concentration is the severely underpredicted orange data point in Fig. 6b). This predictive ability is important for exploratory alloy development, where such a model is sufficiently accurate to be able to screen for promising compositions (i.e., rank those with high APB energies as shown in Fig. 7) in new regions of chemical space and can couple with existing tools for increased predictability in the broader context of alloy design. The continued expansion of databases for multicomponent alloys presents exciting opportunities to enrich our ML models and extrapolate the methods presented here to other extended defects (e.g., stacking faults). As materials data become increasingly digitized, abundant, and accessible, we foresee the coupling of materials thermodynamic databases and datadriven methodologies to have an outsize impact in this field.
Methods
We outlined our computational workflow in Fig. 1 and elaborate on the details in the following subsections.
Solute site preference
PyDII^{32} employs the grand canonical, dilutesolution model (DSM) thermodynamic formalism based on a statistical–mechanical lowtemperature expansion following the framework described by Woodward et al.^{64} to predict equilibrium defect concentrations in intermetallics as a function of composition and temperature. As a brief review of the PyDII framework, we can define concentration variables c_{i}(p) to denote the atomic fraction of species i at site p in an ordered intermetallic. Here, p is either the Ni or Al sublattice and i can be Ni, Al, or the solute species. When we write out the expression for the grand potential Ω, we can take the firstorder lowtemperature expansion of Ω to obtain (assuming zero stress):
In Eq. (3), E^{0} and \({c}_{i}^{0}(p)\) are the energy and concentrations, respectively, in the groundstate configuration, which corresponds to binary, stoichiometric Ni_{3}Al in this study. μ_{i} is the chemical potential of species i, λ(p) denotes the multiplicity of site p, k_{B} is the Boltzmann constant and T is temperature. δE^{ϵ}(p) and \(\delta {c}_{i}^{\epsilon }(p)\) are the defect “excitation energy” and change in site concentration, respectively, associated with introducing a substitutional or vacancy point defect at site p.
Equation (3) can be differentiated to derive an equation for the average concentration:
where 〈c_{i}(p)〉 denotes the ensembleaveraged concentration of species i at site p. Combining Eqs. (3) and (4) and the condition of thermodynamic equilibrium with respect to vacancies^{64} produces a nonlinear system of equations that can be solved at a specified T to obtain a set of chemical potentials, {μ_{i}}. The {μ_{i}} are then used to determine sublattice concentrations for each species as a function of the Al concentration.
For Ni_{3}Al, the fraction of a species i on the Al sublattice can then be computed using Eq. (2) in the main text. The PyDII code^{33} is used to generate the default structures and settings for the DFT calculations of the excitation energies using the Vienna Ab initio Simulation Package (VASP^{®}) (VASP^{®} is a registered trademark of VASP Software GmbH Limited.)^{65,66,67,68} with the projector augmentedwave (PAW) method^{69,70} and the generalized gradient approximation of Perdew, Burke, and Ernzerhof (GGAPBE)^{71}. The default VASP settings include a fixed plane wave energy cutoff of 520 eV, 5000 kpoints per reciprocal atom, spin polarization, ionic relaxation, and convergence criteria for electronic selfconsistency and ionic relaxation of 1 × 10^{−6} eV and 1 × 10^{−3} eV, respectively.
We then use the DFTcalculated defect excitation energies to calculate the equilibrium sublattice concentrations at T = 1000 K and a fixed concentration of 1 at.% for the ternary species. The sublattice preference calculated from PyDII is then used to guide supercell construction for model ternary Ni_{3}Albased alloys according to the following distributions for the ternary species:

I.
Solutes with strong Ni sublattice preference are distributed entirely on the Ni sublattice at all concentrations. The resulting alloy has the composition Ni_{0.75−x}Al_{0.25}Z_{x}.

II.
Solutes with strong Al sublattice preference are distributed entirely on the Al sublattice at all concentrations. The resulting alloy has the composition Ni_{0.75}Al_{0.25−x}Z_{x}.

III.
The remaining solutes with compositiondependent site preference are distributed such that 3/4 of the solute atoms are on the Ni sublattice and 1/4 of the atoms are on the Al sublattice. The resulting alloy has a composition Ni_{0.75(1−x)}Al_{0.25(1−x)}Z_{x}.
For each composition, we randomize the placement of atoms on the appropriate sublattices using special quasirandom structures (SQSs)^{44} generated using the Alloy Theoretic Automated Toolkit (ATAT)^{45,46}. We optimize for pair and triplet correlations out to 0.55 nm, which captures the first few NN correlations for each configuration. The maximum absolute value of the correlation difference between the SQS and ideally random is 0.017. We emphasize that these solute distributions are simply chosen to create model supercells for our highthroughput workflow. The species distribution in the \(\gamma ^{\prime}\) phase in a real superalloy will differ depending on the temperature, (relative) chemical potentials of the species, processing techniques, etc. Despite this simplification, we believe our method will still capture the dominant behaviors and trends in site preference—and, correspondingly, the APB energy—which we validate against other studies in Results and Supplementary Note 1. We find that the sensitivity of the APB energy to local ordering (e.g., using a different SQS) results in differences by as much as 0.01 J m^{−2} in γ_{111}, but should not affect our conclusions (see Supplementary Methods). This is an intrinsic limitation of the SQS/DFT approach and we have ongoing work to better quantify this sensitivity by comparing across random, SQS, and DFTMC supercells.
DFT calculations for γ _{111}
We perform DFT calculations for γ_{111} using VASP with the PAW method and the GGAPBE functional. For these calculations, we set the plane wave cutoff energy for each calculation to be at least 30% greater than the highest cutoff energy (ENMAX in VASP) specified in the pseudopotential files (ENCUT is at least 350 eV). Spinpolarized calculations are performed to account for the effects of local magnetic moments on the APB energy^{14}. Calculations with spin polarization appear to give closer agreement to literature values (both computational and experimental, see Supplementary Figs.) as nonspinpolarized calculations result in values for γ_{111} that are ~30% lower.
For all APB energy calculations, we use a 288atom computational cell with the x, y, and zaxes oriented along the \([\bar{1}\bar{1}2]\), \([1\bar{1}0]\), and [111] directions, respectively (Fig. 8). This corresponds to a 2 × 2 × 3 L1_{2} supercell and nine (111) planes separating the APBs, which we deem sufficient based on convergence tests. We use Γcentered MonkhorstPack kpoint grids^{72} of 2 × 3 × 2, corresponding to 5000 kpoints per reciprocal atom, based on a test of energy convergence for L1_{2} (Supplementary Fig. 4). We apply a firstorder MethfesselPaxton smearing method^{73} with a smearing width of σ = 0.1 eV. Structural relaxations are performed using a conjugate gradient algorithm in two steps: First, only the volume is allowed to relax for the bulk structure to capture the lattice expansion due to the solute. Then, the bulk structure is copied and tilted such that the top face translates by \({{{\bf{b}}}}=a/2[1\bar{1}0]\) (Fig. 8c) to create a structure with a single (111) APB. Independent ionic relaxations are then performed on the bulk structure (without an APB) and the second structure with an APB. The convergence criteria for the electronic selfconsistency and ionic relaxation loops are 1 × 10^{−5} eV and 1 × 10^{−3} eV, respectively. Other VASP settings and DFT convergence tests may be found in Supplementary Methods.
For binary Ni_{3}Al, our DFT calculations give an equilibrium lattice constant of approximately 0.3565 nm, which agrees well with reported computational (e.g., refs. ^{19,26}) and experimental values (0.357 nm)^{1}. We find that γ_{111} for stoichiometric Ni_{3}Al is 0.188 J m^{−2}, and we observe a roughly linear increase in γ_{111} with increasing Al concentration (Supplementary Table 1), which are consistent with previous experimental^{52,74} and computational (e.g., refs. ^{14,15,23}) studies. We also observe a variability in γ_{111} depending on the species arrangements around the APB for a fixed composition. To account for these differences, we shift the (111) planes in the tilted supercell to create five different APBs to help quantify the statistical scatter. The energy of a single (111) APB, denoted by \({\gamma }_{111}^{(i)}\), is calculated according to Eq. (5):
where \({E}_{\,{{\mathrm{apb}}}\,}^{(i)}\) is the total energy of the ith structure with an APB, E_{bulk} is the total energy of the nontilted supercell, and A_{111} is the area of the APB (Fig. 8a). The mean and standard deviation are calculated for five different APB energies to obtain γ_{111} for each composition. We do this for each ternary alloy system at three different solute concentrations: low (1.39 at.%), medium (5.56 at.%), and high (9.72 at.%). We note that while these DFT calculations at 0 K do not account for the configurational entropy contributions to the free energy^{20} nor the decrease in APB energy at high temperatures (>1000 K)^{24}, they can still provide quantitatively accurate data for roomtemperature properties^{23} that can be used to build predictive models for screening new compositions.
To calculate the ordering energy E_{ord}, we perform additional DFT calculations using smaller, 80atom supercells of similar compositions and a kpoint grid of the same density as before. E_{ord} is the difference in energy between two structures: one with atoms on their preferred sublattice (which we determine using PyDII) and another with the same composition with atoms randomly distributed on all sites. In this case, all degrees of freedom are allowed to relax, but all other DFT settings remain the same as those for calculating γ_{111}. E_{ord} in units of eV atom^{−1} is then given by Eq. (6):
where \({E}_{{{\mathrm{L}}}{1}_{2}}\) and E_{random} are the total energies of the structures with correct and random sublattice distributions, respectively, and N is the total number of atoms. Eq. (6) is used to obtain the ordering energy for all ternary Ni_{3}Albased alloys except for 9.72 at.% Ce whose calculations did not converge.
Machine learning
We generate physically motivated features from the alloy composition using the matminer Python package^{48}, as a similar approach by Ling et al. was shown to be effective for featuring superalloy chemistries^{75}. Briefly, the matminer featurizer creates an equallength feature vector for every composition by calculating statistical quantities (we use the compositionweighted mean and standard deviation) over a set of elemental properties^{76} of the alloying elements. To this elemental feature vector from matminer we also include several more derived features that we believe to be influential in modeling the APB energy, including the DFTcalculated ordering energy (E_{ord}), density, change in lattice parameter, and compositionweighted Pettifor number^{49}, among others (see Supplementary Table S2). We generate a total of 70 features and remove all features with at least one undefined value and those that have a Spearman’s rank correlation^{77} greater than 0.9 with another feature.
We train a random forest (RF) regression model, as implemented in the scikitlearn Python package^{78}, to predict γ_{111} using the rootmeansquare error (RMSE) as the error metric. The model is trained with 100 decision trees grown to full depth and with a third of the features considered at each split. To further downselect the features for predictive modeling, we use recursive feature elimination to recursively remove the least important feature from our model based on the feature importance metric in random forests^{51}. The RF model’s performance is evaluated on the full dataset using grouped fivefold crossvalidation (CV)^{50}, where the data are grouped by ternary species. This means that four of the folds are used as training data and the final fold is used for validation, while alloys with the same ternary species will always appear in the same fold. This training–validation procedure is repeated such that each of the fivefold is used once as validation data and the model’s overall performance (RMSE) is computed as the average of all five RMSE scores from CV. The grouping process helps ensure that our ML model is extrapolating to new regions of chemical space and calculating more faithful validation metrics that are representative of novel alloy compositions encountered during the screening. We compare our choice of fivefold CV (a standard 80–20 training–validation split used in the community) with other techniques (e.g., tenfold, leaveonesoluteout) in Supplementary Fig. 5, but emphasize that a better demonstration of extrapolation performance is to new multicomponent chemistries as reported in Results.
Data availability
The data that support the findings of this study are available on MPContribs^{80}. Other data are available from the corresponding authors upon reasonable request.
Code availability
The DFT calculations are performed using the Vienna Ab initio Simulation Package (VASP)^{65,66,67,68}. All other software (PyDII^{33}, ATAT^{45}, and the Python packages^{48,78}) is open source.
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Acknowledgements
This work is partly performed under the auspices of the U.S. Department of Energy (DOE) by the Lawrence Livermore National Laboratory (LLNL) under Contract No. DEAC5207NA27344. The authors are grateful for project funding from the HighPerformance Computing for Materials (HPC4Mtls) Program of the DOE Vehicle Technologies Office under Cooperative Research and Development Agreement (CRADA) No. TC02309. Computing support for this work comes from the LLNL Institutional Computing facilities, and the National Energy Research Scientific Computing Center (NERSC), a DOE Office of Science User Facility operated under Contract No. DEAC0205CH11231. E.C. acknowledges a fellowship through the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE1752814. M.A. acknowledges support for his contributions by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division, under Contract No. DEAC0205CH11231 within the Materials Project program (KC23MP). All figures are produced using matplotlib^{79}.
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E.C., M.A., A.T., and T.F. conceived the research. E.C. performed the simulations and modeling and analyzed the results with feedback from M.A. and T.F. T.W. performed the CALPHAD calculations. E.C. and M.A. drafted the manuscript. All authors discussed the results and commented on the manuscript. M.A. and T.F. supervised the project.
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Chen, E., Tamm, A., Wang, T. et al. Modeling antiphase boundary energies of Ni_{3}Albased alloys using automated density functional theory and machine learning. npj Comput Mater 8, 80 (2022). https://doi.org/10.1038/s41524022007551
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DOI: https://doi.org/10.1038/s41524022007551