Phase prediction of Ni-base superalloys via high-throughput experiments and machine learning

Figures & data

Figure 1. Schematic workflow of the proposed method.

Table 1. Measured composition of multicomponent alloys in wt.%.

	Co	W	Mo	Cr	Al	Ti	Ta	Nb	Hf	Ni
W1	13.0	3.0	2.9	12.2	3.03	3.96	3.1	–	0.21	Bal.
W2	27.8	3.0	2.9	12.0	3.05	4.06	3.0	–	0.21	Bal.
W3	13.0	6.1	2.9	11.8	3.05	3.92	3.0	–	0.22	Bal.
W4	13.1	3.0	6.0	12.1	3.05	4.08	3.1	–	0.19	Bal.
W5	13.0	2.9	3.0	12.0	3.07	6.01	3.0	–	0.19	Bal.
W6	13.0	3.0	2.9	12.0	3.10	4.04	8.1	–	0.20	Bal.
W7	13.0	3.0	3.0	11.9	2.98	4.12	3.0	4.0	0.22	Bal.

Figure 2. (A-G) SEM images of W1-W7 alloys; (H, I) the magnified images of σ, µ and η phases, respectively; (J, K, L) the electron backscattered diffraction Kikuchi patterns of µ, σ, and η, respectively; (M) labeled and (N) montaged SEM images of NiX-6W-4Nb diffusion triple.

Figure 3. Pearson correlation map for features of (A) alloy composition, and (B) 12 physical parameters.

Table 2. Physic parameters.

	Symbol & Formula	Meaning & Parameter	Ref.
1	$\mathrm{\Delta }{S}_{conf}=-R{\displaystyle \sum _{i=1}^n{c}_i\ln c_i}$	Structural entropy. R, gas constant, 8.314 J/(mol·K); ci, mole fraction of the ith element.	[22]
2	$\delta =100\times \sqrt{{\displaystyle \sum _{i=1}^n{c}_i{\left(1-\frac{r_i}{\overline{r}}\right)}^2}}$	Atomic size difference in multi-component alloys $\overline{r}\left(={\displaystyle \sum _{i=1}^n{c}_i{r}_i}\right)$, average atomic radius; $r_i$, atomic radius of the ith element.	[23,24]
3	$\mathrm{\Delta }{H}_{\text{mix}}={\displaystyle \sum _{i=1,i\ne j}^n{\mathrm{\Omega }}_{ij}{c}_i{c}_j}$	Mixing enthalpy of a solid solution ${\mathrm{\Omega }}_{ij}\left(=4\mathrm{\Delta }{H}_{AB}^{\text{mix}}\right)$, regular melt-interaction parameter between ith and jth elements; $\mathrm{\Delta }{H}_{AB}^{\text{mix}}$, mixing enthalpy of binary liquid alloy.	[25]
4	$\mathrm{\Delta }\chi =\sqrt{{\displaystyle \sum _{i=1}^n{c}_i{({\chi }_i-\overline{\chi })}^2}}$	Electronegativity difference between two alloy components $\overline{\chi }\left(={\displaystyle \sum _{i=1}^N{c}_i{\chi }_i}\right)$, average electronegativity; ${\chi }_i$, Pauling electronegativity for ith component.	[26]
5	$VEC={\displaystyle \sum _{i=1}^n{c}_i\left(VE{C}_i\right)}$	Valence electron concentration $VE{C}_i$, valence electron concentration of ith.	[27]
6	$T_m={\displaystyle \sum _{i=1}^n{c}_i(T_m{)}_i}$	Melting point of n-elements alloy, rule of mixtures $(T_m{)}_i$, melting point of ith.	[28]
7	$\mathrm{\Omega }={\displaystyle \frac{T_m\mathrm{\Delta }{S}_{conf}}{\left|\mathrm{\Delta }{H}_{mix}\right|}}$	Geometrical parameter—predicting the solid-solution formation	[28]
8	$\mathrm{\Lambda }={\displaystyle \frac{\mathrm{\Delta }{S}_{conf}}{{\delta }^2}}$	Purely geometrical parameter	[29]
9	$\gamma ={\omega }_S/{\omega }_L=\left[1-\sqrt{{\displaystyle \frac{\left[{\left(r_s+\overline{r}\right)}^2-{\overline{r}}^2\right]/{\left(r_s+\overline{r}\right)}^2}{1-\sqrt{\left[{\left(r_L+\overline{r}\right)}^2-{\overline{r}}^2\right]/{\left(r_L+\overline{r}\right)}^2}}}}\right]$	The ratio between the solid angles of the smallest and largest atoms $\overline{r}\left(={\displaystyle \sum _{i=1}^N{c}_i{r}_i}\right)$, average atomic radius; $r_L$ and $r_S$ are the radius of the largest and smallest atoms;	[30]
10	$S_E=S_E(r_i,c_i,\xi )$	The excessive entropy of mixing $\xi$, the atomic packing fraction of the n-element alloy range from 0.68 (BCC);	[31]
11	$\phi ={\displaystyle \frac{S_c-S_H}{\left|S_E\right|}}={\displaystyle \frac{S_c-\mathrm{\Delta }{H}_{mix}/T_m}{\left|S_E\right|}}$	A single dimensionless thermodynamic parameter $S_C$, the configurational entropy of mixing for an ideal gas.	[31]
12	$\overline{\mathrm{\Delta }V/V}={\displaystyle \frac{{\displaystyle {\sum }_{i,j}{c}_i{c}_j\left|V_i-V_j\right|}}{\left(V_i+V_j\right)/2}}$	The parameters of a novel two-dimensional structure map $V_i={\displaystyle \frac{4}{3}}\pi {r_i}^3$, atomic percentage of the ith component;	[10]

Figure 4. Recursive feature elimination results showing (A) the feature importance and (B) AUC value of TCP phases; (C) the feature importance and (D) AUC value of GCP illustrating predicted by models with different feature sets; the SHAP values of (E) TCP and (F) GCP.

Figure 5. The predicted results showing (A) the distribution and (B) ZPF line of TCP phases by ML and (C) Thermo-Calc; (D) the distribution and (E) ZPF line of GGP phases by ML and (F) Thermo-Calc; and the phase diagram by (G) experimental, (H) ML and (I) Thermo-Calc in NiX-6W-4Nb diffusion triple.

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