J. Mater. Sci. Technol. ›› 2021, Vol. 87: 133-142.DOI: 10.1016/j.jmst.2021.01.054
• Research Article • Previous Articles Next Articles
Jie Xiongb,c, San-Qiang Shib,c,*(
), Tong-Yi Zhanga,d,**()
Received:2020-11-02
Revised:2021-01-19
Accepted:2021-01-19
Published:2021-10-10
Online:2021-03-17
Contact:
San-Qiang Shi,Tong-Yi Zhang
About author:** School of Materials Science and Engineering, Harbin Institute of Technology, Shenzhen, China.E-mail addresses: zhangty@shu.edu.cn (T.-Y. Zhang).Jie Xiong, San-Qiang Shi, Tong-Yi Zhang. Machine learning of phases and mechanical properties in complex concentrated alloys[J]. J. Mater. Sci. Technol., 2021, 87: 133-142.
Fig. 1. Ashby plot of ultimate tensile strength versus hardness of CCAs with different phases, where the first phase in each label of multi-phases is the matrix, e.g., BCC is the matrix for BCC + FCC + B2.
| Description | Abb. | Definition | |
|---|---|---|---|
| Elemental parameters | Atomic Number | AN | $\bar{x}=\sum a_{i}x_{i}$ $\delta_{x}=\sqrt{\sum a_{i}(1-x_{i}/\bar{x})^{2}}$ |
| Metallic Radius | MR | ||
| Melting Point | Tm | ||
| Boiling Point | Tb | ||
| Pauling Electronegativity | XP | ||
| Electron Affinity | Eea | ||
| First Ionization Potential | I1 | ||
| Molar Heat Capacity | Cm | ||
| Thermal Conductivity | K | ||
| Valence Electron | VEC | ||
| Heat of Fusion | Hf | ||
| Thermodynamic parameters | Enthalpy of mixing | Hmix | ${{H}_{\text{mix}}}=4\underset{i=1}{\overset{N}{\mathop \sum }}\,\underset{j=1}{\overset{N}{\mathop \sum }}\,\text{ }\!\!\Delta\!\!\text{ }{{H}_{ij}}{{a}_{i}}{{a}_{j}}$ |
| Entropy of mixing | Smix | ${{S}_{\text{mix}}}=-R\underset{i=1}{\overset{N}{\mathop \sum }}\,{{a}_{i}}\text{ln}{{a}_{i}}$ | |
| Entropy of fusion | Sf | ${{S}_{f}}={}^{\overline{{{H}_{f}}}}/{}_{\overline{{{T}_{m}}}}$ | |
| Gibbs free energy of mixing | Gmix | ${{G}_{\text{mix}}}={{H}_{\text{mix}}}-{{S}_{\text{mix}}}\text{ }\!\!\cdot\!\!\text{ }\overline{{{T}_{m}}}$ | |
| The reciprocal of Ω | 1/Ω | $1/\Omega ={}^{\left| {{H}_{\text{mix}}} \right|}/{}_{\overline{{{T}_{m}}}{{S}_{\text{mix}}}}$ | |
| VEC distributions | fraction of the electrons in the s valence orbitals | fs | ${{f}_{s}}={}^{\overline{s\text{VEC}}}/{}_{\overline{\text{VEC}}}$ |
| fraction of the electrons in the p valence orbitals | fp | ${{f}_{s}}={}^{\overline{p\text{VEC}}}/{}_{\overline{\text{VEC}}}$ | |
| fraction of the electrons in the d valence orbitals | fd | ${{f}_{s}}={}^{\overline{d\text{VEC}}}/{}_{\overline{\text{VEC}}}$ |
Table 1 Feature blocks consisting of elemental parameters, thermodynamic parameters, and VEC distributions.
| Description | Abb. | Definition | |
|---|---|---|---|
| Elemental parameters | Atomic Number | AN | $\bar{x}=\sum a_{i}x_{i}$ $\delta_{x}=\sqrt{\sum a_{i}(1-x_{i}/\bar{x})^{2}}$ |
| Metallic Radius | MR | ||
| Melting Point | Tm | ||
| Boiling Point | Tb | ||
| Pauling Electronegativity | XP | ||
| Electron Affinity | Eea | ||
| First Ionization Potential | I1 | ||
| Molar Heat Capacity | Cm | ||
| Thermal Conductivity | K | ||
| Valence Electron | VEC | ||
| Heat of Fusion | Hf | ||
| Thermodynamic parameters | Enthalpy of mixing | Hmix | ${{H}_{\text{mix}}}=4\underset{i=1}{\overset{N}{\mathop \sum }}\,\underset{j=1}{\overset{N}{\mathop \sum }}\,\text{ }\!\!\Delta\!\!\text{ }{{H}_{ij}}{{a}_{i}}{{a}_{j}}$ |
| Entropy of mixing | Smix | ${{S}_{\text{mix}}}=-R\underset{i=1}{\overset{N}{\mathop \sum }}\,{{a}_{i}}\text{ln}{{a}_{i}}$ | |
| Entropy of fusion | Sf | ${{S}_{f}}={}^{\overline{{{H}_{f}}}}/{}_{\overline{{{T}_{m}}}}$ | |
| Gibbs free energy of mixing | Gmix | ${{G}_{\text{mix}}}={{H}_{\text{mix}}}-{{S}_{\text{mix}}}\text{ }\!\!\cdot\!\!\text{ }\overline{{{T}_{m}}}$ | |
| The reciprocal of Ω | 1/Ω | $1/\Omega ={}^{\left| {{H}_{\text{mix}}} \right|}/{}_{\overline{{{T}_{m}}}{{S}_{\text{mix}}}}$ | |
| VEC distributions | fraction of the electrons in the s valence orbitals | fs | ${{f}_{s}}={}^{\overline{s\text{VEC}}}/{}_{\overline{\text{VEC}}}$ |
| fraction of the electrons in the p valence orbitals | fp | ${{f}_{s}}={}^{\overline{p\text{VEC}}}/{}_{\overline{\text{VEC}}}$ | |
| fraction of the electrons in the d valence orbitals | fd | ${{f}_{s}}={}^{\overline{d\text{VEC}}}/{}_{\overline{\text{VEC}}}$ |
Fig. 2. The PCC matrix of features in the ML-A dataset.
| Dataset | PCC selected Features |
|---|---|
| ML-A | $\overline{AN},{{\delta }_{AN}},\overline{MR},{{\delta }_{MR}},{{\delta }_{Tb}},\overline{Tm},{{\delta }_{Tm}},\overline{Eea},{{\delta }_{Eea}},{{\delta }_{XP}},\overline{Hf},{{\delta }_{Hf}},$ $\overline{I 1}, \delta_{I 1}, \overline{C m}, \delta_{C m}, \overline{K}, \delta_{K}, \overline{\mathrm{VEC}}, \delta_{\mathrm{VEC}}, f_{p}, H_{\operatorname{mix}}, S_{\operatorname{mix}}, S_{f}$ |
| ML-B | $\overline{AN},{{\delta }_{AN}},\overline{MR},{{\delta }_{MR}},{{\delta }_{Tb}},\overline{Tm},{{\delta }_{Tm}},\overline{Eea},{{\delta }_{Eea}},\overline{XP},{{\delta }_{XP}},{{\delta }_{Hf}},$ $\overline{I1},{{\delta }_{I1}},\overline{Cm},{{\delta }_{Cm}},\bar{K},{{\delta }_{K}},\overline{\text{VEC}},{{\delta }_{\text{VEC}}},{{f}_{p}},{{f}_{d}},{{H}_{mix}},{{S}_{\text{mix}}},{{S}_{f}},{{G}_{\text{mix}}}$ |
| ML-C | $\overline{AN},{{\delta }_{AN}},{{\delta }_{MR}},{{\delta }_{Tb}},\overline{Tm},{{\delta }_{Tm}},\overline{Eea},{{\delta }_{Eea}},\overline{XP},{{\delta }_{XP}},{{\delta }_{Hf}},{{\delta }_{I1}},$ $\overline{Cm},{{\delta }_{Cm}},\bar{K},{{\delta }_{K}},\overline{\text{VEC}},{{\delta }_{\text{VEC}}},{{f}_{p}},{{H}_{\text{mix}}},{{S}_{\text{mix}}},{{S}_{f}},{{G}_{\text{mix}}},1/\Omega $ |
Table 2 The PCC selected features.
| Dataset | PCC selected Features |
|---|---|
| ML-A | $\overline{AN},{{\delta }_{AN}},\overline{MR},{{\delta }_{MR}},{{\delta }_{Tb}},\overline{Tm},{{\delta }_{Tm}},\overline{Eea},{{\delta }_{Eea}},{{\delta }_{XP}},\overline{Hf},{{\delta }_{Hf}},$ $\overline{I 1}, \delta_{I 1}, \overline{C m}, \delta_{C m}, \overline{K}, \delta_{K}, \overline{\mathrm{VEC}}, \delta_{\mathrm{VEC}}, f_{p}, H_{\operatorname{mix}}, S_{\operatorname{mix}}, S_{f}$ |
| ML-B | $\overline{AN},{{\delta }_{AN}},\overline{MR},{{\delta }_{MR}},{{\delta }_{Tb}},\overline{Tm},{{\delta }_{Tm}},\overline{Eea},{{\delta }_{Eea}},\overline{XP},{{\delta }_{XP}},{{\delta }_{Hf}},$ $\overline{I1},{{\delta }_{I1}},\overline{Cm},{{\delta }_{Cm}},\bar{K},{{\delta }_{K}},\overline{\text{VEC}},{{\delta }_{\text{VEC}}},{{f}_{p}},{{f}_{d}},{{H}_{mix}},{{S}_{\text{mix}}},{{S}_{f}},{{G}_{\text{mix}}}$ |
| ML-C | $\overline{AN},{{\delta }_{AN}},{{\delta }_{MR}},{{\delta }_{Tb}},\overline{Tm},{{\delta }_{Tm}},\overline{Eea},{{\delta }_{Eea}},\overline{XP},{{\delta }_{XP}},{{\delta }_{Hf}},{{\delta }_{I1}},$ $\overline{Cm},{{\delta }_{Cm}},\bar{K},{{\delta }_{K}},\overline{\text{VEC}},{{\delta }_{\text{VEC}}},{{f}_{p}},{{H}_{\text{mix}}},{{S}_{\text{mix}}},{{S}_{f}},{{G}_{\text{mix}}},1/\Omega $ |
Fig. 3. The rank of features in the (a) ML-A, (b) ML-B, and (c) ML-C dataset resulting from on the SBS + RFC (in red), SFS + RFC (in blue), and MDI/RFC (in yellow).
Fig. 4. The miss rate of each RFC model with 100 trees versus the feature number for the (a) ML-A, (b) ML-B, and (c) ML-C dataset.
| Dataset | Feature set | Method | Features sorted by the feature rank |
|---|---|---|---|
| ML-A | OS-A | SBS + RFC | $\overline{MR}$, ${{\delta }_{Tb}}$, Hmix, $\overline{Eea}$, ${{\delta }_{Cm}}$ |
| ML-B | OS-B | SFS + RFC | $\overline{\text{VEC}}$, $\overline{AN}$, Hmix, ${{\delta }_{K}}$, ${{\delta }_{Cm}}$ |
| ML-C | OS-C | SFS + RFC | $\overline{AN}$, ${{\delta }_{AN}}$, ${{\delta }_{XP}}$, $\bar{K}$, Hmix |
Table 3 Finally selected features for ML-A, ML-B, and ML-C datasets.
| Dataset | Feature set | Method | Features sorted by the feature rank |
|---|---|---|---|
| ML-A | OS-A | SBS + RFC | $\overline{MR}$, ${{\delta }_{Tb}}$, Hmix, $\overline{Eea}$, ${{\delta }_{Cm}}$ |
| ML-B | OS-B | SFS + RFC | $\overline{\text{VEC}}$, $\overline{AN}$, Hmix, ${{\delta }_{K}}$, ${{\delta }_{Cm}}$ |
| ML-C | OS-C | SFS + RFC | $\overline{AN}$, ${{\delta }_{AN}}$, ${{\delta }_{XP}}$, $\bar{K}$, Hmix |
| Dataset | n_estimators | max_features | max_depth |
|---|---|---|---|
| Preset: [ | Preset: [ | Preset: None and [ | |
| ML-A | 88 | 4 | None |
| ML-B | 86 | 1 | None |
| ML-C | 92 | 3 | 7 |
Table 4 Hyperparameters in the RFC models.
| Dataset | n_estimators | max_features | max_depth |
|---|---|---|---|
| Preset: [ | Preset: [ | Preset: None and [ | |
| ML-A | 88 | 4 | None |
| ML-B | 86 | 1 | None |
| ML-C | 92 | 3 | 7 |
Fig. 5. The confusion matrixes of the RFC models on the (a) ML-A, (b) ML-B, and (c) ML-C datasets.
Fig. 6. (a) The cross-validated NRMSE versus the feature number of each RFR model for hardness prediction, (b) ML-predicted hardness versus the measured values.
Fig. 7. (a) The cross-validated NRMSE versus the feature number of each RFR model for UTS prediction, (b) ML-predicted UTS versus the measured values.
| Dataset | n_estimators | max_features | max_depth | CV-r | |||
|---|---|---|---|---|---|---|---|
| preset | result | preset | result | preset | result | ||
| Hardness | [ | 90 | [ | 2 | None and [ | None | 0.9062 |
| UTS | [ | 74 | [ | 1 | None and [ | None | 0.9498 |
Table 5 Hyperparameters in the RFR models.
| Dataset | n_estimators | max_features | max_depth | CV-r | |||
|---|---|---|---|---|---|---|---|
| preset | result | preset | result | preset | result | ||
| Hardness | [ | 90 | [ | 2 | None and [ | None | 0.9062 |
| UTS | [ | 74 | [ | 1 | None and [ | None | 0.9498 |
Fig. 8. (a) Ranked mean absolute value of SHAP values of the 6 BSS + RFR selected features for hardness. The SHAP values (negative in blue, positive in red) of (b) $\overline{\text{VEC}}$, (c) ${{H}_{\text{mix}}}$, (d) ${{\delta }_{XP}}$, (e) $\overline{AN}$, (f) δCm, and (g) $\bar{K}$ for every one of the data.
Fig. 9. (a) Ranked mean absolute value of SHAP values of the 5 BSS + RFR selected features for UTS. The SHAP values (negative in blue, positive in red) of (b)δTb, (c) Hmix, (d) $\overline{AN}$, (f) $\bar{K}$, and (g) δK for every one of the data.
| Alloy | $\overline{\text{VEC}}$ | ${{H}_{\text{mix}}}$ | ${{\delta }_{XP}}$ | δTb | Hardness (HV) | UTS (MPa) | Phases |
|---|---|---|---|---|---|---|---|
| CoCrFeNi | 8.25 | -3.75 | 0.0531 | 0.0328 | 173.6 (129.8 [ | 527.3 (480 [ | SP-FCC (FCC [ |
| CoCrFeNiTi | 7.4 | -16.32 | 0.0802 | 0.0623 | 592.6 | 655.0 | MP-IM |
| CoCrFeNiV | 7.6 | -8.96 | 0.0646 | 0.0754 | 377.5 | 424.1 (311 [ | MP-IM (FCC + σ [ |
| CoCrFeNiMn | 8 | -4.16 | 0.0783 | 0.1101 | 175.7 (144.0 [ | 495.9 (494 [ | SP-FCC (FCC [ |
| CoCrFeNiCu | 8.8 | 3.2 | 0.0502 | 0.0474 | 152.9 | 558.3 | SP-FCC |
| CoCrFeNiZr | 7.4 | -22.72 | 0.1244 | 0.1846 | 616.5 | 1133.3 | MP-IM (BCC + C15 [ |
| CoCrFeNiNb | 7.6 | -14.88 | 0.0694 | 0.2191 | 615.2 (602 [ | 1293.1 | MP-IM (FCC + C14 [ |
| CoCrFeNiHf | 7.4 | -19.52 | 0.1313 | 0.2047 | 586.1 | 1209.0 | MP-IM (BCC + C36 [ |
Table 6 The feature values, ML-predicted mechanical properties and structures of CoCrFeNi-X (X = Ti, V, Mn, Cu, Zr, Nb, or Hf) CCAs, where mechanical properties and structures in parentheses are measured values.
| Alloy | $\overline{\text{VEC}}$ | ${{H}_{\text{mix}}}$ | ${{\delta }_{XP}}$ | δTb | Hardness (HV) | UTS (MPa) | Phases |
|---|---|---|---|---|---|---|---|
| CoCrFeNi | 8.25 | -3.75 | 0.0531 | 0.0328 | 173.6 (129.8 [ | 527.3 (480 [ | SP-FCC (FCC [ |
| CoCrFeNiTi | 7.4 | -16.32 | 0.0802 | 0.0623 | 592.6 | 655.0 | MP-IM |
| CoCrFeNiV | 7.6 | -8.96 | 0.0646 | 0.0754 | 377.5 | 424.1 (311 [ | MP-IM (FCC + σ [ |
| CoCrFeNiMn | 8 | -4.16 | 0.0783 | 0.1101 | 175.7 (144.0 [ | 495.9 (494 [ | SP-FCC (FCC [ |
| CoCrFeNiCu | 8.8 | 3.2 | 0.0502 | 0.0474 | 152.9 | 558.3 | SP-FCC |
| CoCrFeNiZr | 7.4 | -22.72 | 0.1244 | 0.1846 | 616.5 | 1133.3 | MP-IM (BCC + C15 [ |
| CoCrFeNiNb | 7.6 | -14.88 | 0.0694 | 0.2191 | 615.2 (602 [ | 1293.1 | MP-IM (FCC + C14 [ |
| CoCrFeNiHf | 7.4 | -19.52 | 0.1313 | 0.2047 | 586.1 | 1209.0 | MP-IM (BCC + C36 [ |
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