On the one hand, multi-principal element alloys (MPEAs) have created a paradigm shift in alloy design due to large compositional space, whereas on the other, they have presented enormous computational challenges for theory-based materials design, especially density functional theory (DFT), which is inherently computationally expensive even for traditional dilute alloys. In this paper, we present a machine learning framework, namely PREDICT (**PR**edict properties from **E**xisting **D**atabase **I**n **C**omplex alloys **T**erritory), that opens a pathway to predict elastic constants in large compositional space with little computational expense. The framework only relies on the DFT database of binary alloys and predicts Voigt–Reuss–Hill Young’s modulus, shear modulus, bulk modulus, elastic constants, and Poisson’s ratio in MPEAs. We show that the key descriptors of elastic constants are the A–B bond length and cohesive energy. The framework can predict elastic constants in hypothetical compositions as long as the constituent elements are present in the database, thereby enabling property exploration in multi-compositional systems. We illustrate predictions in a FCC Ni-Cu-Au-Pd-Pt system.

## I. INTRODUCTION

Multi-principal element alloys (MPEAs) present a paradigm shift in alloy design by presenting possibilities of adding multiple elements in large proportions that may consist of exciting new properties.^{1–10} This concept spreads beyond alloys to ceramics, including oxides, borides, and nitrides. However, the possibilities in the large phase space are quickly overshadowed by the computational expense that existing theory-based methods bear in materials design. The density functional theory (DFT) based materials exploration, which is highly successful in conventional/dilute alloys but is inherently computationally expensive, is particularly affected. Consequently, this challenge constrains the down selection of “compositional regions of interest,” which is a hot pursuit in MPEAs. Thus, the capability to bypass computational expense of accurate DFT calculations while simultaneously tracing the MPEA phase space for down-selection remains largely elusive.

Data science and machine learning (ML) have made rapid inroads in materials research, including in mechanical properties of materials research.^{11–24} In this paper, we present a ML framework that can trace and predict elastic constants in a multi-elemental phase space while relying simply on the database of binary alloys. Consequently, there is a reduction in the computational expense as the DFT calculations for computing elastic constants in MPEAs are bypassed. An additional benefit of this framework is that it can predict elastic constants in hypothetical face-centered cubic (FCC) alloy compositions as long as the constituent elements are present in the database. Consequently, the property exploration in multi-compositional dimensions (i.e., compositional regions of interest) becomes possible. The overarching idea is schematically shown in Fig. 1(a). The atomic level patterns and properties of the binary alloys extracted from DFT are used to train a machine learning model to make predictions in MPEAs. The model specifically learns the nearest neighbor environments, both at atomic and electronic levels. Previously, our group successfully predicted point defect energies,^{21} vibrational entropy,^{22} and stacking fault energies^{23} in MPEAs using this approach. In this paper, we demonstrate that the elastic constants in complex MPEAs can be predicted from a DFT database of binary alloys in the FCC Ni-Cu-Au-Pd-Pt system shown in Fig. 1(b). The Voigt–Reuss–Hill Young’s modulus (*E*_{VRH}), shear modulus (*G*), bulk modulus (*B*), elastic constants (*C*_{ij}), and Poisson’s ratio (*ν*) are predicted.

## II. BACKGROUND

In a pure metal, the Young’s modulus (*E*) can be extracted from the second derivative of energy (*U*) vs displacement (*r*) plot, $E\u221d\u22022U\u2202r2$.^{25} At an equilibrium bond length *r*_{0}, *E* is related to bond length as

where *k* is the bond stiffness.^{26} In our recent work, we had shown in FCC alloys that the stiffness of a bond is essentially a function of only bond length, as shown in Fig. 2.^{21} The stiffness constants shown in Fig. 2 are the stretching force constants transformed from the calculated force constant matrices through the scheme implemented in Alloy Theoretic Automated Toolkit (ATAT).^{27} Further explanation of these calculations can be found in the work performed by Manzoor and Aidhy.^{21} The data show that the stiffness of a bond is determined from the bond length for a given A–A or A–B bond. The correlation holds true even in different alloy compositions, i.e., irrespective of the chemical nearest neighbor environments of the two atoms forming a bond, a given bond length has a fixed bond stiffness [see Fig. 2(c)]. Thus, in MPEAs, the bond stiffnesses can be predicted simply from the bond lengths. This correlation had enabled us to predict vibrational entropy in MPEAs from a stiffness database of binary alloys per the approach shown in Fig. 1(a), thereby completely bypassing expensive phonon calculations needed in the conventional supercell approach method.^{21,28} We use this correlation to learn the patterns in the binary alloys and predict elastic constants in the FCC Ni-Cu-Au-Pd-Pt system MPEAs.

The PREDICT (**pr**edict properties from **e**xisting **d**atabase **i**n **c**omplex alloys **t**erritory) ML framework follows the depiction shown in Fig. 1(a), where it extracts and learns the property patterns from the existing database of constituent binary alloys and then applies ML methods to predict the same properties in their MPEAs.

## III. METHODS

### A. First-principles calculations

The Vienna *ab initio* simulation package (VASP) is used to perform DFT calculations. The projector-augmented wave (PAW) method is employed with the exchange–correlation energy evaluated by generalized-gradient approximation (GGA) using the Perdew–Burke–Ernzerhof (PBE) functional. A plane-wave cutoff of 500 eV and Monkhorst–Pack k-point grid of 6 × 6 × 12 are used for all calculations. The electronic self-consistency energy convergence criterion is chosen as 10^{−8} eV. The energy convergence criterion of the ionic relaxation is chosen as −10^{−4} eV. All DFT calculations are performed on a randomized 16-atom, 2 × 2 × 1 FCC supercell from unary to quinary compositions. The elastic stiffness matrix calculations are performed using the stress–strain method as per Page and Saxe,^{29} starting with a fully relaxed random structure. This is achieved by solving the linear equation

where *σ*_{i} (*i* = 1, 6) is the stress caused by strain, *ɛ*_{j} (*j* = 1, 6), that is applied to the structures. The resulting stiffness matrix, *C*_{ij}, is then calculated.^{29} All FCC lattices fulfill the mechanical stability criteria (*C*_{11} + 2*C*_{12} > 0, *C*_{11} > *C*_{12}, and *C*_{44} > 0).^{2,30} Similar to Ye *et al.*, our structures exhibit lower symmetry than the theoretical cubic ones due to the limited supercell size of 16 atoms.^{7} From the stiffness matrix and its inverse, the compliance matrices *S*_{ij}, *B*, *G*, *E*, and *ν* are calculated based on the Voigt–Reuss–Hill (VRH) approximation with the following equations:

where the subscripts *V* and *R* denote the Voigt and Reuss approximations.^{11,31} Due to anisotropy, defined by $A=C11\u2212C122C44\u22601$, the *E* in different crystallographic orientations is calculated using

where *α*, *β*, and *γ* are the directional cosines of the [*hkl*] directions and the [100], [010], and [001] crystallographic directions, respectively.^{26} The *E*_{VRH} approximation is comparable to the arithmetic average of the *E*s in the various crystallographic orientations, respectively.

### B. Machine learning model

We investigate several ML models from the scikit-learn Python library, including linear regression (LR), gradient boosted regression (GBR), and random forest regression (RFR), to determine the best model for the problem.^{11,14,15,18,20,32–35} The models are trained on the constituent binary descriptors with an 80%–20% train-test split and then predict on the out-of-sample (not included in training or testing data) ternary, quaternary, and quinary structures. The validation criteria for the models are the coefficient of determination (R^{2}) and the root-mean squared error (RMSE) defined as

where $y\u0304$ is the average true value, *y*_{i} is the *i-th* true value, $y\u0302i$ is the *i-th* predicted value, and *n* is the number of samples. As R^{2} goes to 1 and the RMSE decreases to 0, the model’s predictions become closer to the values calculated through DFT. Tables I and II list the R^{2} and RMSE values for the quinary Ni-Cu-Au-Pd-Pt structures’ *E*_{VRH} and *C*_{11} predictions for various ML models, respectively, illustrating that the three models perform equally. In this work, we choose to use GBR model. Each elastic constant, the *E*_{VRH}, bulk and shear moduli, and Poisson’s ratio have their own model. As demonstrated in Figs. S2(c) and S2(d), since the standard deviations of the R^{2} and RMSE are insignificant for a test of 20 random GBR models while predicting *E*_{VRH}, any of the random models will perform with similar validation criteria results. Thus, the same random_state is used for all models for reproducibility. In addition, based on Figs. S2(a) and S2(b), to ensure the models do not overfit, the GBR used in this work contained n_estimators = 50.

Model . | R^{2}
. | RMSE . |
---|---|---|

LR | 0.89 | 9.55 |

GBR | 0.87 | 10.24 |

RFR | 0.83 | 11.89 |

Model . | R^{2}
. | RMSE . |
---|---|---|

LR | 0.89 | 9.55 |

GBR | 0.87 | 10.24 |

RFR | 0.83 | 11.89 |

### C. Database

To train the ML model, the database is built from DFT calculations in a 16-atom FCC supercell of various binary alloys shown in Fig. 1(b). They are Ni-Cu, Ni-Au, Ni-Pd, Ni-Pt, Cu-Au, Cu-Pd, Cu-Pt, Au-Pd, Au-Pt, and Pd-Pt. For each binary alloy, 120 supercells are created by randomly distributing atoms ranging from 1 A to 15 B atoms to 15 A and 1 B atoms supercells to trace the compositional range. However, some of the structures fail to relax in FCC after the DFT calculations, resulting in a total of 1177 binary structures for training and testing the ML models. OVITO’s common neighbor analysis module is implemented to ensure that the structures formed in the FCC crystal structure via DFT.^{36} If the structure does not form in the FCC crystal structure, it is not included in the database. The database also consists of Ni-Cu-Au, Ni-Cu-Pd, Ni-Au-Pd, and Cu-Au-Pd ternary alloys; Ni-Cu-Au-Pd quaternary alloys; and Ni-Cu-Au-Pd-Pt quinary alloys for testing the ML models. The bond lengths (96), bond types (96), cohesive energies, elastic constants, *E*s (*E*_{VRH} and *E*s in various crystallographic directions), *B*, *G*, and *ν* of these structures are stored in the database. The ML model is trained with an 80%–20% train-test split, respectively, of the binary structures’ data and is tested on 326, 48, and 30 out-of-sample structures for the ternary, quaternary, and quinary structures, respectively. The atomic percentages for the binary, ternary, quaternary, and quinary alloys can be found in Tables VI–IX, respectively, in the Appendix.

From our analysis, we find that the cohesive energy is a key descriptor of elastic constants and is thus included within the database. It is calculated using Eq. (12),^{37} where *N* is the number of atoms in the structure, *E*_{iso} is the energy of a single atom in a vacuum of 15 Å, and *E*_{tot} is the energy of the relaxed bulk structure,

The absolute values of the elemental cohesive energies are compared to Schimka *et al.*^{37} as shown in Table III, demonstrating a decent agreement. The improvement of the model by including cohesive energy is shown in Fig. 3. The prediction of *C*_{11} without cohesive energy has an R^{2} of 0.8 and an RMSE of 9.81 GPa, as shown in Fig. 3(a). In contrast, with cohesive energy, the R^{2} increases to 0.89 and the RMSE decreases to 7.13 GPa, as shown in Fig. 3(b), illustrating a significant improvement in the predictive capability of the model. A strong correlation between *C*_{11} and cohesive energy in various Ni-Cu-Au constituting alloys is shown in Fig. 3(c). The importance of the cohesive energy as a descriptor, as well as the bond lengths, can also be seen in Fig. S3 in the supplementary material. Therefore, the bond lengths (96), bond types (96 × 2) to describe the two atoms in the given bond, and cohesive energies are used as descriptors in the ML models for this FCC 16 atom Ni-Cu-Au-Pd-Pt system.

Element . | This work PBE . | PBE^{37}
. | Experiment^{37}
. |
---|---|---|---|

Ni | 4.89 | 4.75 | 4.48 |

Cu | 3.50 | 3.50 | 3.52 |

Au | 3.04 | 3.05 | 3.83 |

Pd | 3.75 | 3.76 | 3.94 |

Pt | 5.54 | 5.51 | 5.86 |

## IV. RESULTS

We first demonstrate the reliability of the data by comparing elastic constants between our DFT calculations and available literature in Ni-Cu,^{38} Cu-Au,^{39} and Ni-Au^{40} binary alloys, as a function of composition. The results are shown in Fig. 4. The experimental data are shown by gray lines, whereas the DFT data are shown in color symbols. The distribution in our DFT data is due to multiple supercells used for a given composition to generate sufficient statistical distribution for training the ML models. The DFT data are in good agreement both qualitatively and quantitatively, especially in Ni-Cu and Cu-Au. We also conclude from this agreement that a system size of 16 atoms is sufficient to capture the elastic constants to create the DFT database. The DFT-computed elastic constants *C*_{11}, *C*_{12}, and *C*_{44} as a function of composition for all binaries (Cu-Ni, Cu-Au, Cu-Pd, Cu-Pt, Ni-Au, Ni-Pd, Ni-Pt, Au-Pd, Au-Pt, and Pd-Pt) in this work are shown in Fig. 5. Figure S1 represents the average bond length, *r*_{avg}, for the binary systems as a function of composition. Tables S1–S6 containing the range of *C*_{11}, *C*_{12}, *C*_{44}, and *r*_{avg} of a given composition for the ternary, quaternary, and quinary systems can be found in the supplementary material.

Training on this database, the ML predictions of *E*_{VRH} in the Ni-Cu-Au ternary system, corresponding DFT results, and their comparison are shown in Fig. 6. The dots in the two ternary plots represent the compositions used in the database. The ternary plots have similar contours over similar range of values illustrating that the ML model can predict *E*_{VRH} by solely training on the binary constituents. A good R^{2} of 0.9 and a narrow RMSE of 10.86 GPa shown in Fig. 6(c) further illustrates good predictive capability of the model. Figures 6(a) and 6(b) illustrate that the higher Ni concentration increases *E*_{VRH} whereas the higher Au concentration decreases it.

The composition also has a significant effect on the anisotropy. Figures 7(a)–7(c) show a 3D visualization^{41} of the *E* in different crystallographic directions in Ni-dominated (Ni_{87.5}-Cu_{6.25}-Au_{6.25}), Cu-dominated (Ni_{6.25}-Cu_{87.5}-Au_{6.25}), and Au-dominated (Ni_{6.25}-Cu_{6.25}-Au_{87.5}) Ni-Cu-Au systems from DFT calculations, respectively. Figure 7(d) shows the *E* in different directions graphically, with the highest *E* in the [111] direction for each of the three materials, illustrating high anisotropy particularly in Ni (blue line) and Cu (red line) alloys. In contrast, the curve for Au (yellow line) alloy is flatter. The *E*s increases as the composition changes from high Au to high Cu to high Ni concentrations. The comparison between DFT and ML predictions of the *E*_{VRH} found in Fig. 7 is shown in Table IV illustrating a good agreement. Here, the ML results are slightly higher than the DFT results for the Cu (120.4 GPa), Au-dominant (72.1 GPa), and Ni-dominant (205.7 GPa) structures. This is also noticed in Fig. 6(c) for the remaining Ni-Cu-Au structures, where the ML predictions of the *E*_{VRH} are slightly higher than the DFT results when the DFT moduli are lower than about 130 GPa, and the ML predictions are both slightly higher and lower when the DFT results show moduli above 130 GPa.

Structure . | DFT (GPa) . | ML (GPa) . |
---|---|---|

Ni_{87.5}-Cu_{6.25}-Au_{6.25} | 205.7 | 212.1 |

Ni_{6.25}-Cu_{87.5}-Au_{6.25} | 120.4 | 146.3 |

Ni_{6.25}-Cu_{6.25}-Au_{87.5} | 72.1 | 76.8 |

Structure . | DFT (GPa) . | ML (GPa) . |
---|---|---|

Ni_{87.5}-Cu_{6.25}-Au_{6.25} | 205.7 | 212.1 |

Ni_{6.25}-Cu_{87.5}-Au_{6.25} | 120.4 | 146.3 |

Ni_{6.25}-Cu_{6.25}-Au_{87.5} | 72.1 | 76.8 |

The ML predictions of various elastic constants for all quaternary compositions are shown in Figs. 8 and 9. Note that the predictions are made from training solely on the binary constituents. The R^{2} values are above 0.85 for all but the off diagonal (*C*_{12}, *C*_{13}, and *C*_{23}) elastic constants. Furthermore, the RMSE values are all below 10.02 GPa for the elastic properties and below 0.01 for the Poisson’s ratio predictions, demonstrating that the ML models’ predictions are fairly accurate. While analyzing the results, we find that the R^{2} values are influenced by the range of values. For example, in Fig. 10, if the values for the quinary *C*_{11} constants only ranged from 175 to 225 GPa, the R^{2} value would be 0.63 in comparison to the R^{2} value of 0.96 for the original data that ranges from 169 to 281 GPa. When interpreting the results for the off-diagonal elements (*C*_{12}, *C*_{13}, and *C*_{23}) in the quaternary system shown in Fig. 8 based solely on the R^{2} values, the smaller range of stiffness constant values (40 GPa instead of the 100 GPa range for the other six tensor elements) influenced the R^{2} values. As a result, the R^{2} values are lower than those of the main diagonals.

Similarly, the comparison of elastic constants of Ni-Cu-Au-Pd-Pt quinary structures between DFT and ML is shown in Figs. 10 and 11. The ML predictions are made by exclusively training on their binary constituents. Within these plots, the R^{2} values are above 0.82 and the RMSE values are below 10.34 GPa for the elastic properties and below 0.007 for the Poisson’s ratio predictions, demonstrating good predictions in the quinary system.

## V. DISCUSSION

Others have successfully computed elastic constants within MPEAs using DFT but mention the difficulty of these simulations when sampling the phase space.^{6,7,10,19} These properties have also been predicted within MPEAs and other materials with some limitations in the models due to the lack of data to validate the predictions on structures containing more than two or three elements or as their models are effective for materials only included in the models.^{11–20} For example, the model from Revi *et al.* was trained on and developed for binary alloys only.^{11} They reported an R^{2} of 0.779 ± 0.02.^{11} Grant *et al.* developed a machine learning model specifically for the FeNiCrCoMn alloy to predict the Young’s modulus in the [100], [110], and [111] directions.^{13} In this work, we combine the use of DFT to create a database of materials and ML to show that the elastic constants of a larger phase space of MPEAs can be predicted by training on their simpler binary constituent alloys. We had recently demonstrated that the stiffness, *k*, of a bond is essentially a function of bond length and bond type.^{21} This relationship was used to predict the vibrational entropy of FCC solids using ML, thereby bypassing expensive phonon calculations.^{28} In this work, we explore the relationship from Eq. (2) between the stiffness, bond length, and *E* of a material, showing that the *E*, as well as other elastic constants, can be predicted by using the bond lengths of the material via ML, forgoing the computationally expensive DFT stress–strain calculations.^{26,29}

The predictive capability of our model stands very well against that in literature. For example, we report an R^{2} of 0.98 and an RMSE of 3.08 GPa for our bulk moduli predictions. This compares well to the bulk moduli predictions with an R^{2} of 0.86 and an RMSE of 18.75 GPa by Furmanchuk *et al.* and an R^{2} of 0.94 and an RMSE of 17.2 GPa by Tehrani *et al.*^{16,18} Similarly, we report an R^{2} of 0.9 and an RMSE of 4.61 GPa for our *C*_{44} predictions, which is in good agreement with that of Vasquez *et al.* who reported an R^{2} of 0.88 and an RMSE of 7.34 GPa.^{14}

A significant improvement in computational time is obtained using the ML approach as demonstrated in Table V. The total wall time for the DFT simulations consider the average wall time for a quaternary or quinary elastic constant simulations only (it does not include the initial relaxation time to obtain relaxed structures), multiplied by the number of simulations (systems) that are run for each system. The wall time of the ML predictions includes the time taken to train the model on the binary constituents’ data and predict on the quaternary or quinary structures. The ML prediction time shown in Table V does not show the time taken to obtain the binary data to train the model. Although it could be argued that the ML database relies on the DFT calculations of binary alloys, which has its own computational expense, the benefit of the approach is that this computational expense is a one-time investment that enables predictions in new compositions and tracing of the compositional phase space, while completely negating the need to perform any new DFT calculations, at least in this material system. We continue to expand our database to include more elements.

System . | No. of Structures . | DFT (h) . | ML (s) . |
---|---|---|---|

Ni-Cu-Au-Pd | 48 | 2112 | 1.63 |

Ni-Cu-Au-Pd-Pt | 30 | 1380 | 1.63 |

System . | No. of Structures . | DFT (h) . | ML (s) . |
---|---|---|---|

Ni-Cu-Au-Pd | 48 | 2112 | 1.63 |

Ni-Cu-Au-Pd-Pt | 30 | 1380 | 1.63 |

^{a}

Note: ML time does not include DFT time for binary database calculations, only training the model and predicting.

We also note that there are other limitations to this model, such as it has not been tested on body-centered cubic (BCC) structures. Thus, a need to update the models may arise when adapting them to a BCC system. In addition, the Ni-Cu-Au-Pd-Pt system as chosen is paramagnetic and does not pose as a roadblock to the PREDICT framework. However, if Cr and/or Mn were present in the system, these may introduce magnetic properties into the materials, which may affect the mechanical stability as well as the magnetic moments of the structures, creating some difficulty interpreting the DFT results and further the effectiveness of the ML models.^{2,42,43}

## VI. CONCLUSION

In this work, we develop a machine learning approach that can predict Young’s moduli, bulk and shear moduli, elastic constants, and Poisson’s ratios in MPEAs from the DFT database of only their constituent binary compositions. The model is reliant only on three descriptors, i.e., cohesive energy, bond type, and bond length of the bonds. The approach can trace the whole compositional phase space in a given material system, thereby bypassing the need to perform DFT altogether. In addition, it substantially reduces the computational time needed to calculate these elastic constants. This approach opens a possibility to explore the MPEA phase space merely from the data of binary alloys.

## SUPPLEMENTARY MATERIAL

The supplementary material contains tables with the range of the *C*_{11}, *C*_{12}, and *C*_{44} elastic constants, and *r*_{avg} for the ternary, quaternary, and quinary systems reported in this work. Images of the average bond length for the binary alloys, model evaluations, and model feature importance are also found here.

## ACKNOWLEDGMENTS

This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Science, Mechanical Behavior and Radiation Effects program.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Nathan Linton**: Conceptualization (equal); Data curation (lead); Formal analysis (equal); Methodology (equal); Validation (equal); Visualization (lead); Writing – original draft (equal); Writing – review & editing (equal). **Dilpuneet S. Aidhy**: Conceptualization (equal); Formal analysis (equal); Funding acquisition (lead); Investigation (equal); Methodology (equal); Project administration (lead); Supervision (lead); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data from this project is available on GitHub at https://github.com/nlinton272/PREDICT_Ni-Cu-Au-Pd-Pt_ML and https://github.com/Materials-Computation-Data-Science-MCDC/DFT_Elastic_Constants.

### APPENDIX: ATOMIC PERCENTAGES FOR THE BINARY, TERNARY, QUARTERNARY, AND QUINARY ALLOYS

Table VI provides binary compositions in atomic percentages. Table VII provides ternary compositions in atomic percentages. Table VIII provides quaternary compositions in atomic percentages. Table IX provides compositions quinary compositions in atomic percentages.

Binary Compositions (at. %) . | |
---|---|

A . | B . |

93.75 | 6.25 |

87.5 | 12.5 |

81.25 | 18.75 |

75 | 25 |

68.75 | 31.25 |

62.5 | 37.5 |

56.25 | 43.75 |

50 | 50 |

43.75 | 56.25 |

37.5 | 62.5 |

31.25 | 68.75 |

25 | 75 |

18.75 | 81.25 |

12.5 | 87.5 |

6.25 | 93.75 |

Binary Compositions (at. %) . | |
---|---|

A . | B . |

93.75 | 6.25 |

87.5 | 12.5 |

81.25 | 18.75 |

75 | 25 |

68.75 | 31.25 |

62.5 | 37.5 |

56.25 | 43.75 |

50 | 50 |

43.75 | 56.25 |

37.5 | 62.5 |

31.25 | 68.75 |

25 | 75 |

18.75 | 81.25 |

12.5 | 87.5 |

6.25 | 93.75 |

Ternary Compositions (at. %) . | ||
---|---|---|

A . | B . | C . |

37.5 | 31.25 | 31.25 |

31.25 | 37.5 | 31.25 |

25 | 37.5 | 37.5 |

37.5 | 25 | 37.5 |

37.5 | 37.5 | 25 |

6.25 | 37.5 | 56.25 |

56.25 | 6.25 | 37.5 |

56.25 | 37.5 | 6.25 |

6.25 | 43.75 | 50 |

12.5 | 43.75 | 43.75 |

12.5 | 50 | 37.5 |

18.75 | 43.75 | 37.5 |

18.75 | 37.5 | 43.75 |

43.75 | 37.5 | 18.75 |

43.75 | 18.75 | 37.5 |

50 | 37.5 | 12.5 |

50 | 12.5 | 37.5 |

75 | 12.5 | 12.5 |

12.5 | 75 | 12.5 |

12.5 | 12.5 | 75 |

62.5 | 12.5 | 25 |

62.5 | 25 | 12.5 |

12.5 | 25 | 62.5 |

12.5 | 62.5 | 25 |

25 | 62.5 | 12.5 |

25 | 12.5 | 62.5 |

81.25 | 12.5 | 6.25 |

81.25 | 6.25 | 12.5 |

12.5 | 81.25 | 6.25 |

12.5 | 6.25 | 81.25 |

6.25 | 81.25 | 12.5 |

6.25 | 12.5 | 81.25 |

87.5 | 6.25 | 6.25 |

6.25 | 87.5 | 6.25 |

6.25 | 6.25 | 87.5 |

Ternary Compositions (at. %) . | ||
---|---|---|

A . | B . | C . |

37.5 | 31.25 | 31.25 |

31.25 | 37.5 | 31.25 |

25 | 37.5 | 37.5 |

37.5 | 25 | 37.5 |

37.5 | 37.5 | 25 |

6.25 | 37.5 | 56.25 |

56.25 | 6.25 | 37.5 |

56.25 | 37.5 | 6.25 |

6.25 | 43.75 | 50 |

12.5 | 43.75 | 43.75 |

12.5 | 50 | 37.5 |

18.75 | 43.75 | 37.5 |

18.75 | 37.5 | 43.75 |

43.75 | 37.5 | 18.75 |

43.75 | 18.75 | 37.5 |

50 | 37.5 | 12.5 |

50 | 12.5 | 37.5 |

75 | 12.5 | 12.5 |

12.5 | 75 | 12.5 |

12.5 | 12.5 | 75 |

62.5 | 12.5 | 25 |

62.5 | 25 | 12.5 |

12.5 | 25 | 62.5 |

12.5 | 62.5 | 25 |

25 | 62.5 | 12.5 |

25 | 12.5 | 62.5 |

81.25 | 12.5 | 6.25 |

81.25 | 6.25 | 12.5 |

12.5 | 81.25 | 6.25 |

12.5 | 6.25 | 81.25 |

6.25 | 81.25 | 12.5 |

6.25 | 12.5 | 81.25 |

87.5 | 6.25 | 6.25 |

6.25 | 87.5 | 6.25 |

6.25 | 6.25 | 87.5 |

Quaternary Compositions (at. %) . | |||
---|---|---|---|

A . | B . | C . | D . |

12.5 | 12.5 | 12.5 | 62.5 |

12.5 | 12.5 | 62.5 | 12.5 |

12.5 | 62.5 | 12.5 | 12.5 |

62.5 | 12.5 | 12.5 | 12.5 |

25 | 25 | 25 | 25 |

18.75 | 18.75 | 18.75 | 43.75 |

18.75 | 18.75 | 43.75 | 18.75 |

18.75 | 43.75 | 18.75 | 18.75 |

43.75 | 18.75 | 18.75 | 18.75 |

6.25 | 6.25 | 6.25 | 81.25 |

6.25 | 6.25 | 81.25 | 6.25 |

6.25 | 81.25 | 6.25 | 6.25 |

81.25 | 6.25 | 6.25 | 6.25 |

31.25 | 31.25 | 31.25 | 6.25 |

31.25 | 31.25 | 6.25 | 31.25 |

31.25 | 6.25 | 31.25 | 31.25 |

6.25 | 31.25 | 31.25 | 31.25 |

Quaternary Compositions (at. %) . | |||
---|---|---|---|

A . | B . | C . | D . |

12.5 | 12.5 | 12.5 | 62.5 |

12.5 | 12.5 | 62.5 | 12.5 |

12.5 | 62.5 | 12.5 | 12.5 |

62.5 | 12.5 | 12.5 | 12.5 |

25 | 25 | 25 | 25 |

18.75 | 18.75 | 18.75 | 43.75 |

18.75 | 18.75 | 43.75 | 18.75 |

18.75 | 43.75 | 18.75 | 18.75 |

43.75 | 18.75 | 18.75 | 18.75 |

6.25 | 6.25 | 6.25 | 81.25 |

6.25 | 6.25 | 81.25 | 6.25 |

6.25 | 81.25 | 6.25 | 6.25 |

81.25 | 6.25 | 6.25 | 6.25 |

31.25 | 31.25 | 31.25 | 6.25 |

31.25 | 31.25 | 6.25 | 31.25 |

31.25 | 6.25 | 31.25 | 31.25 |

6.25 | 31.25 | 31.25 | 31.25 |

Quinary Compositions (at. %) . | ||||
---|---|---|---|---|

Ni . | Cu . | Au . | Pd . | Pt . |

18.75 | 18.75 | 18.75 | 18.75 | 25 |

18.75 | 18.75 | 18.75 | 25 | 18.75 |

18.75 | 18.75 | 25 | 18.75 | 18.75 |

18.75 | 25 | 18.75 | 18.75 | 18.75 |

25 | 18.75 | 18.75 | 18.75 | 18.75 |

6.25 | 6.25 | 6.25 | 6.25 | 75 |

6.25 | 6.25 | 6.25 | 75 | 6.25 |

6.25 | 6.25 | 75 | 6.25 | 6.25 |

6.25 | 75 | 6.25 | 6.25 | 6.25 |

75 | 6.25 | 6.25 | 6.25 | 6.25 |

18.75 | 25 | 25 | 18.75 | 12.5 |

Quinary Compositions (at. %) . | ||||
---|---|---|---|---|

Ni . | Cu . | Au . | Pd . | Pt . |

18.75 | 18.75 | 18.75 | 18.75 | 25 |

18.75 | 18.75 | 18.75 | 25 | 18.75 |

18.75 | 18.75 | 25 | 18.75 | 18.75 |

18.75 | 25 | 18.75 | 18.75 | 18.75 |

25 | 18.75 | 18.75 | 18.75 | 18.75 |

6.25 | 6.25 | 6.25 | 6.25 | 75 |

6.25 | 6.25 | 6.25 | 75 | 6.25 |

6.25 | 6.25 | 75 | 6.25 | 6.25 |

6.25 | 75 | 6.25 | 6.25 | 6.25 |

75 | 6.25 | 6.25 | 6.25 | 6.25 |

18.75 | 25 | 25 | 18.75 | 12.5 |

## REFERENCES

_{x}CoCrFeNi (0 ≤ x ≤ 2) high-entropy alloys