Data set for diffusion coefficients and relative creep rate ratios of 26 dilute Ni-X alloy systems from first-principles calculations.

The development of future generations of Ni-base superalloys will depend on a systematic understanding of how each alloying element affects the fundamental properties of Ni-base superalloys, particularly with respect to their creep behavior. First, this article presents the temperature-dependent data of all factors entering into dilute impurity diffusion for 26 Ni-X alloy systems, including atomic jump frequencies, thermodynamic parameters, and diffusivity plots. Second, this article presents the data used to calculate the relative creep rate ratios showing the effect of each of the 26 alloying elements, X, on the dilute Ni-X alloy. The dataset refers to "A comprehensive first-principles study of solute elements in dilute Ni alloys: Diffusion coefficients and their implications to tailor creep rate" by Hargather et al. [1].

Specifications table

Value of the data

Future Ni alloys or Ni-base superalloys can be designed using the diffusion coefficient and relative creep rate ratio data from this systematic study for 26 Ni-X alloy systems.

A less-computationally expensive but accurate method for calculating all factors entering into dilute solute diffusion as a function of temperature is demonstrated.

The diffusion coefficient data could be useful for future development or validation of CALPHAD-style diffusion mobility databases.

Computational methodology

Dilute solute diffusion coefficients as a function of temperature were calculated in the present work using density functional theory within the confines of the 5-frequency model [2], [3]. 26 solute atoms were studied in the present work: Al, Co, Cr, Cu, Fe, Hf, Ir, Mn, Mo, Nb, Os, Pd, Pt, Re, Rh, Ru, Sc, Si, Ta, Tc, Ti, V, W, Y, Zn, and Zr.

Total energy calculations are carried out for a 32-atom supercell using the plane wave density functional code, Vienna ab-initio Simulation Package (VASP) [4]. A constant plane wave energy cutoff of 350 eV is used for all calculations, which is 1.3 times the default plane wave energy cutoff of nickel and larger than plane wave energy cutoff of all other solute atoms considered. A Monkhorst-Pack k-mesh scheme is used for all calculations, with a sampling of 8 × 8 × 8 for each system studied. For relaxation during the VASP calculations, the Methfessel-Paxton smearing method [5] is used for the calculation of forces acting on the atoms, and a final static calculation is performed after each relaxation using the linear tetrahedron method with Blöchl׳s [6] correction for an accurate total energy calculation. Total electronic energy is converged to be at least  eV/atom. Due to the ferromagnetism of nickel up until its Curie point, [7], all calculations are performed in the present work within the spin polarized approximation. Further details of diffusion and computational theory can be found in the main article by Hargather et al. [1].

Data

0 K results

Fig. 1 shows the effect of having one solute atom and no vacancies present in the 32-atom Ni supercell at 0 K on the following properties: equilibrium volume , bulk modulus, , first derivative of bulk modulus with respect to pressure, , and spin magnetic moment, . These results are shown without the effect of zero point vibrational energy.

Fig. 1

EOS calculated equilibrium properties for the ps (X) at 0 K without the effect of zero point vibrational energy. The (a) equilibrium volume , (b) bulk modulus, , (c) first derivative of bulk modulus with respect to pressure, , and (d) spin magnetic moment are plotted as a function of atomic number along different rows in the periodic table.

Atomic jump frequencies and thermodynamic parameters for all X systems

The data in this section presents all factors relating to dilute solute diffusion as a function of temperature for all X systems studied in the present work. Data is presented at T=700 K and T=1700 K. A detailed explanation of each of the jump frequencies and importance of the thermodynamic parameters can be found in the main article by Hargather et al. [1]. Table 1 gives the Gibbs energy of migration for each jump in the 5-frequency model for each of the 26 solutes in a Ni host lattice at the designated temperatures.

Table 1

Gibbs energy of migration, for each of the five jump frequencies for dilute solute diffusion of all X systems studied in the present work.

Solute	Temp, (K)	ΔG_m0 (eV)	ΔG_m1 (eV)	ΔG_m2 (eV)	ΔG_m3 (eV)	ΔG_m4 (eV)
Al	T=700 K	1.12	1.36	0.79	1.29	1.11
	T=1700 K	1.07	1.43	0.86	1.34	1.19
Co	T=700	1.12	1.24	1.38	1.23	1.27
	T=1700	1.07	1.27	1.41	1.27	1.32
Cr	T=700	1.12	1.18	1.30	1.20	1.27
	T=1700	1.07	1.25	1.20	1.19	1.32
Cu	T=700 K	1.12	1.27	0.98	1.25	1.17
	T=1700 K	1.07	1.34	1.04	1.31	1.24
Fe	T=700	1.12	1.29	1.20	1.23	1.25
	T=1700	1.07	1.33	1.26	1.27	1.32
Hf	T=700	1.12	1.73	0.40	1.36	0.79
	T=1700	1.07	1.75	0.51	1.40	0.92
Ir	T=700	1.12	1.41	1.72	1.07	1.17
	T=1700	1.07	1.46	1.77	1.14	1.25
Mn	T=700	1.12	1.24	0.95	1.35	1.27
	T=1700	1.07	2.02	1.72	2.23	1.45
Mo	T=700	1.12	1.44	1.31	1.15	1.07
	T=1700	1.07	1.48	1.37	1.20	1.18
Nb	T=700	1.12	1.58	0.76	1.24	0.92
	T=1700	1.07	1.60	0.85	1.27	1.03
Os	T=700	1.12	1.38	1.89	1.06	1.21
	T=1700	1.07	1.42	1.95	1.14	1.32
Pd	T=700	1.12	1.44	1.11	1.19	1.04
	T=1700	1.07	1.49	1.18	1.25	1.13
Pt	T=700	1.12	1.43	1.37	1.13	1.07
	T=1700	1.07	1.47	1.42	1.18	1.15
Re	T=700	1.12	1.38	1.89	1.09	1.18
	T=1700	1.07	1.44	1.97	1.16	1.31
Rh	T=700	1.12	1.41	1.40	1.13	1.14
	T=1700	1.07	1.44	1.43	1.18	1.21
Ru	T=700	1.12	1.39	1.51	1.09	1.16
	T=1700	1.07	1.42	1.53	1.15	1.24
Sc	T=700	1.12	−0.65	0.81	1.12	0.78
	T=1700	1.07	0.15	1.65	1.89	1.93
Si	T=700 K	1.12	1.12	0.95	1.34	1.22
	T=1700 K	1.07	1.16	0.99	1.28	1.37
Ta	T=700	1.12	1.56	0.92	1.21	0.95
	T=1700	1.07	1.59	1.02	1.24	1.06
Tc	T=700	1.12	1.38	1.57	1.10	1.16
	T=1700	1.07	1.42	1.63	1.15	1.25
Ti	T=700	1.12	1.43	0.61	1.28	1.02
	T=1700	1.07	1.46	0.69	1.31	1.11
V	T=700	1.12	1.26	1.09	1.22	1.17
	T=1700	1.07	1.29	1.15	1.24	1.25
W	T=700	1.12	1.45	1.50	1.13	1.09
	T=1700	1.07	1.51	1.60	1.20	1.22
Y	T=700	1.12	2.42	0.25	0.26	−1.28
	T=1700	1.07	2.45	0.40	0.41	−0.79
Zn	T=700	1.12	1.34	0.80	1.31	1.11
	T=1700	1.07	1.41	0.89	1.38	1.18
Zr	T=700	1.12	1.82	0.27	0.14	−0.66
	T=1700	1.07	1.80	0.35	0.20	−0.60

Table 2 presents all of the thermodynamic factors entering into dilute solute diffusion for all X systems studied in the present work. The thermodynamic factors include the correlation factor, , the enthalpy of vacancy formation adjacent to the solute, , the enthalpy of migration of the solute atom moving into an adjacent vacancy, , the entropy of vacancy formation adjacent to a solute, , entropy of migration of the solute atom, , and the temperature dependence of the correlation factor, C.

Table 2

Thermodynamic parameters at 700 K and 1700 K given for all factors entering into vacancy mediated dilute solute diffusion for the X systems studied in the present work. Calculated values include the correlation factor, , the enthalpy of vacancy formation adjacent to the solute, , the enthalpy of migration of the solute atom moving into an adjacent vacancy, , the entropy of vacancy formation adjacent to a solute, , entropy of migration of the solute atom, , and the temperature dependence of , C.

Solute	Temp, (K)	f_2	ΔH_f (eV)	ΔH_m (eV)	ΔS_f (k_B)	ΔS_m (k_B)	C (eV)
Al	T=700 K	0.0006	1.62	0.75	2.13	−0.607	0.532
	T=1700 K	0.104	1.69	0.70	2.84	−1.11	0.482
Co	T=700	0.973	1.70	1.37	1.95	−0.17	−0.004
	T=1700	0.896	1.76	1.33	2.55	−0.60	−0.016
Cr	T=700 K	0.951	1.71	1.36	1.72	0.97	−0.009
	T=1700 K	0.755	1.76	1.41	2.20	1.41	−0.056
Cu	T=700	0.030	1.62	0.96	2.17	−0.38	0.273
	T=1700	0.340	1.69	0.88	2.92	−1.12	0.181
Fe	T=700	0.614	1.71	1.18	1.97	−0.35	0.020
	T=1700	0.725	1.79	1.12	2.69	−0.91	0.014
Hf	T=700	0.000	1.35	0.34	1.80	−0.98	1.022
	T=1700	0.005	1.43	0.27	2.54	−1.60	1.172
Ir	T=700	1.000	1.66	1.69	1.90	−0.38	0.000
	T=1700	0.995	1.72	1.64	2.50	−0.93	−0.003
Mn	T=700	0.010	1.87	0.60	7.35	−5.87	0.345
	T=1700	0.159	3.02	−0.10	18.21	−12.44	0.125
Mo	T=700	0.970	1.65	1.27	1.75	−0.52	−0.004
	T=1700	0.898	1.70	1.22	2.19	−1.04	−0.015
Nb	T=700	0.000	1.52	0.71	1.80	−0.76	0.554
	T=1700	0.123	1.58	0.66	2.40	−1.28	0.495
Os	T=700	1.000	1.70	1.87	1.80	−0.44	0.000
	T=1700	0.998	1.77	1.81	2.50	−1.00	−0.001
Pd	T=700	0.366	1.57	1.09	2.17	−0.41	0.079
	T=1700	0.619	1.66	1.02	3.02	−1.03	0.031
Pt	T=700	0.993	1.58	1.36	2.06	−0.26	−0.002
	T=1700	0.931	1.66	1.31	2.80	−0.76	−0.018
Re	T=700	1.000	1.71	1.86	1.51	−0.51	0.000
	T=1700	0.998	1.76	1.79	1.95	−1.21	−0.001
Rh	T=700	0.996	1.64	1.40	1.99	−0.16	−0.001
	T=1700	0.935	1.68	1.37	2.39	−0.43	−0.020
Ru	T=700	1.000	1.73	1.44	2.81	−1.06	0.000
	T=1700	0.973	1.98	1.25	4.17	−1.96	−0.013
Sc	T=700 K	1.0000	1.31	0.36	8.97	−7.34	0.000
	T=1700 K	1.0000	1.85	−0.18	14.09	−12.51	0.000
Si	T=700 K	0.06	1.57	0.95	1.91	−0.16	0.169
	T=1700 K	0.32	1.63	0.90	2.57	−0.61	0.162
Ta	T=700	0.010	1.58	0.86	1.72	−0.91	0.383
	T=1700	0.366	1.63	0.80	2.19	−1.51	0.234
Tc	T=700	1.000	1.69	1.55	1.80	−0.35	0.000
	T=1700	0.985	1.75	1.49	2.39	−0.94	−0.007
Ti	T=700 K	0.0000	1.60	0.57	1.94	−0.70	0.760
	T=1700 K	0.0410	1.68	0.50	2.68	−1.32	0.689
V	T=700 K	0.2771	1.60	1.06	1.84	−0.47	0.108
	T=1700 K	0.6244	1.75	1.01	2.35	−1.00	0.058
W	T=700	0.999	1.68	1.45	1.68	−0.84	0.000
	T=1700	0.975	1.72	1.38	2.07	−1.50	−0.009
Y	T=700	0.489	0.42	0.17	1.19	−1.41	0.001
	T=1700	0.495	0.48	0.10	1.71	−2.05	0.001
Zn	T=700	0.001	1.59	0.76	2.23	−0.65	0.533
	T=1700	0.099	1.69	0.68	3.18	−1.42	0.469
Zr	T=700	0.884	1.19	0.21	1.42	−0.83	−0.013
	T=1700	0.724	1.22	0.18	1.66	−1.08	−0.029

Dilute solute diffusivity plots

Additional plots of diffusivity as a function of 1000/T for the solute systems that were not presented in the main article [1] and have known experimental data are presented in this section. It should be noted that all of the plots are produced from data calculated directly from first-principles, and do not represent Arrhenius fits of data. The following plots in Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig. 8, Fig. 9, Fig. 10, Fig. 11, Fig. 12, Fig. 13, Fig. 14, Fig. 15 are shown for 2nd row solute elements: Si, for 3d transition row solute elements: Ti, V, Cr, Mn, Fe, and Co, for 4d transition row solute elements: Zr and Mo, and for 5d transition row solute elements: Hf, Ta, W, Re, and Pt. The corresponding plots for the following solutes are shown in the main article [1]: Al, Cu, Nb, and W.

Fig. 2

Solute diffusion coefficient Si in Ni calculated in the present work (solid line) compared to poly-crystal data of Allison et al. [8] and Swalin et al. [9].

Fig. 3

Solute diffusion coefficient Ti in Ni calculated in the present work (solid line) compared to poly-crystal data of Bergner [10] and Swalin et al. [11].

Fig. 4

Solute diffusion coefficient V in Ni calculated in the present work (solid line) compared to poly-crystal data of Murarka et al. [12].

Fig. 5

Solute diffusion coefficient Cr in Ni calculated in the present work (solid line) compared to poly-crystal data of Monma et al. [13]. Růžičková et al. [14], Tutunnik et al. [15], and Glinchuk et al. [16].

Fig. 6

Solute diffusion coefficient Mn in Ni calculated in the present work (solid line) compared to poly-crystal data of Swalin et al. [11].

Fig. 7

Solute diffusion coefficient Fe in Ni calculated in the present work (solid line) compared to single-crystal data of Bakker et al. [17], and to poly-crystal data of Guiarldenq [18] and Badia et al. [19].

Fig. 8

Solute diffusion coefficient Co in Ni calculated in the present work (solid line) compared to single-crystal data of Vladimirov et al. [20] and to poly-crystal data of Badia et al. [19], Hirano et al. [21], Hassner et al. [22]. Divya et al. [23], and McCoy et al. [24].

Fig. 9

Solute diffusion coefficient Zr in Ni calculated in the present work (solid line) compared to poly-crystal data of Allison et al. [8] and Bergner [10].

Fig. 10

Solute diffusion coefficient Mo in Ni calculated in the present work (solid line) compared to poly-crystal data of Swalin et al. [9].

Fig. 11

Solute diffusion coefficient Hf in Ni calculated in the present work (solid line) compared to the poly-crystal data of Bergner [10].

Fig. 12

Solute diffusion coefficient Ta in Ni calculated in the present work (solid line) compared to the poly-crystal data of Bergner [10].

Fig. 13

Solute diffusion coefficient W in Ni calculated in the present work (solid line) compared to the single-crystal data of Vladimirov et al. [20], and the poly-crystal data of Bergner [10], Swalin et al. [11], and Monma [25].

Fig. 14

Solute diffusion coefficient Re in Ni calculated in the present work (solid line) compared to poly-crystalline diffusion couple experimental data by [26].

Fig. 15

Solute diffusion coefficient Pt in Ni calculated in the present work (solid line) compared to poly-crystalline diffusion couple experimental data by [27].

2nd row solute elements

see Fig. 2.

3d transition row solute elements

see Fig. 3.

4d transition row solute elements

see Fig. 4.

5d transition row solute elements

see Fig. 5.

Relative creep rate ratio data table

The diffusivity data from the present work is combined with the elastic constant calculations from Shang et al. [28] and the stacking fault energy calculations from Shang et al. [29] on the same X systems to calculate a relative creep rate ratio. The relative creep rate ratio shows the effect of each solute element on the creep rate of the dilute Ni-X alloy compared to the creep rate of pure Ni. The data used for the relative creep rate ratio plots in the main article [1] is presented in Table 3.

Table 3

Elastic [28] and stacking fault energy [29] data used for calculation of the relative creep rate ratio in the main article [1].

Temp, (K)	Solute	D, m^2/sec	b, Å	G, GPa	γ_SFE, mJ/m^2	E, GPa
300 K	Al	3.08E-54	1.447	86.31	109.15	223.38
	Co	1.14E-57	1.444	92.26	113.56	237.34
	Cr	3.50E-57	1.445	89.91	100.11	232.05
	Cu	1.88E-53	1.446	87.56	115.12	226.21
	Fe	3.07E-55	1.445	90.52	109.86	232.99
	Hf	1.39E-51	1.456	81.67	68.87	212.19
	Ir	1.58E-62	1.451	88.65	102.77	229.09
	Mn	1.77E-52	1.447	88.43	110.86	228.01
	Mo	1.95E-55	1.450	85.52	62.08	223.23
	Nb	2.77E-52	1.453	82.80	64.31	215.79
	Ni	2.31E-53	1.444	92.22	128.20	236.66
	Os	4.25E-66	1.450	91.74	86.31	236.58
	Pd	4.57E-52	1.451	87.98	118.12	226.82
	Pt	2.07E-55	1.452	88.96	121.06	229.68
	Re	1.90E-66	1.449	88.00	66.57	228.02
	Rh	5.49E-57	1.450	86.10	107.33	223.08
	Ru	2.53E-59	1.449	91.78	91.12	236.18
	Sc	2.88E-34	1.454	82.39	74.82	213.94
	Si	4.62E-51	1.444	85.79	112.50	222.85
	Ta	9.51E-53	1.453	83.10	71.44	216.82
	Tc	1.06E-60	1.449	89.72	71.08	231.67
	Ti	3.63E-54	1.449	86.42	83.08	223.79
	V	2.08E-54	1.446	87.41	81.33	226.56
	W	4.16E-59	1.450	85.93	66.54	223.45
	Y	3.32E-17	1.463	73.79	48.26	193.31
	Zn	4.81E-54	1.447	82.89	111.69	215.57
	Zr	3.85E-30	1.458	79.99	60.31	208.69
600 K	Al	1.38E-29	1.447	82.19	105.85	212.01
	Co	1.33E-31	1.444	87.82	109.79	225.37
	Cr	3.32E-31	1.445	84.98	99.05	218.59
	Cu	3.31E-29	1.446	83.60	111.26	215.20
	Fe	1.96E-30	1.445	86.80	106.45	222.45
	Hf	1.41E-28	1.456	78.26	66.57	202.18
	Ir	4.50E-34	1.451	83.34	99.59	214.86
	Mn	5.01E-29	1.447	84.36	107.40	216.68
	Mo	1.37E-30	1.450	80.99	60.20	211.41
	Nb	6.70E-29	1.453	79.30	62.37	205.87
	Ni	2.41E-29	1.444	87.91	124.22	224.72
	Os	6.77E-36	1.450	87.40	84.31	224.67
	Pd	1.07E-28	1.451	84.22	114.17	216.33
	Pt	1.84E-30	1.452	85.39	117.22	219.57
	Re	3.94E-36	1.449	83.80	64.89	216.37
	Rh	3.07E-31	1.450	83.93	103.97	216.39
	Ru	1.94E-32	1.449	87.03	88.40	223.30
	Sc	6.37E-20	1.454	78.82	71.85	204.08
	Si	2.62E-28	1.444	81.42	109.21	210.97
	Ta	3.99E-29	1.453	79.53	69.48	206.83
	Tc	3.57E-33	1.449	86.31	69.43	222.09
	Ti	1.21E-29	1.449	82.19	80.49	212.04
	V	7.98E-30	1.446	83.32	78.74	215.37
	W	1.71E-32	1.450	81.49	64.59	211.30
	Y	6.40E-12	1.463	70.09	45.49	183.50
	Zn	1.83E-29	1.447	79.26	107.92	205.41
	Zr	4.14E-18	1.458	76.41	58.12	198.75
900 K	Al	2.67E-21	1.460	78.09	102.02	200.54
	Co	7.79E-23	1.458	83.25	105.42	212.92
	Cr	1.89E-22	1.460	80.55	97.81	206.30
	Cu	4.51E-21	1.459	79.40	106.78	203.60
	Fe	4.33E-22	1.459	82.79	102.51	211.17
	Hf	8.77E-21	1.469	74.45	63.95	191.38
	Ir	1.66E-24	1.464	78.43	95.93	201.56
	Mn	5.74E-21	1.460	79.93	103.39	204.32
	Mo	3.07E-22	1.463	76.52	58.03	199.39
	Nb	6.79E-21	1.467	75.43	60.16	194.99
	Ni	3.22E-21	1.458	83.41	119.62	212.43
	Os	9.62E-26	1.463	82.95	82.03	212.40
	Pd	7.51E-21	1.465	79.97	109.58	204.59
	Pt	4.64E-22	1.466	81.46	112.80	208.51
	Re	5.92E-26	1.462	79.64	62.96	204.68
	Rh	1.42E-22	1.463	81.32	100.08	208.54
	Ru	2.22E-23	1.463	81.84	85.28	209.27
	Sc	4.65E-15	1.468	74.85	68.41	193.15
	Si	1.26E-20	1.457	76.93	105.37	198.72
	Ta	4.89E-21	1.466	75.68	67.26	196.07
	Tc	6.45E-24	1.462	82.58	67.54	211.61
	Ti	2.79E-21	1.462	77.66	77.53	199.55
	V	1.24E-21	1.459	78.93	75.74	203.30
	W	1.49E-23	1.464	76.80	62.35	198.49
	Y	4.36E-10	1.478	66.07	42.28	172.76
	Zn	3.26E-21	1.461	75.37	103.55	194.51
	Zr	4.78E-14	1.471	72.51	55.63	187.98
1200 K	Al	4.01E-17	1.468	74.05	97.66	189.12
	Co	2.05E-18	1.466	78.64	100.44	200.17
	Cr	5.03E-18	1.470	76.36	96.40	194.54
	Cu	5.27E-17	1.468	75.12	101.67	191.82
	Fe	7.06E-18	1.467	78.58	98.03	199.37
	Hf	9.46E-17	1.477	70.28	61.02	179.94
	Ir	1.10E-19	1.471	74.09	91.80	189.59
	Mn	7.72E-17	1.468	75.17	98.83	191.02
	Mo	4.93E-18	1.471	72.16	55.60	187.20
	Nb	8.84E-17	1.474	71.15	57.67	183.10
	Ni	4.34E-17	1.466	78.73	114.40	199.93
	Os	1.26E-20	1.470	78.41	79.44	199.88
	Pd	6.43E-17	1.473	75.37	104.38	191.91
	Pt	8.07E-18	1.474	77.26	107.78	196.80
	Re	7.78E-21	1.470	75.64	60.79	193.23
	Rh	3.33E-18	1.471	78.20	95.68	199.36
	Ru	8.41E-19	1.470	76.45	81.76	194.62
	Sc	1.36E-12	1.476	70.61	64.51	181.48
	Si	9.78E-17	1.466	72.58	100.97	186.70
	Ta	5.90E-17	1.474	71.48	64.76	184.44
	Tc	2.99E-19	1.470	78.58	65.42	200.36
	Ti	4.45E-17	1.470	73.13	74.19	187.08
	V	1.56E-17	1.467	74.28	72.34	190.53
	W	4.69E-19	1.471	71.97	59.83	185.29
	Y	3.88E-09	1.486	61.73	38.61	161.06
	Zn	4.66E-17	1.469	71.46	98.56	183.42
	Zr	5.46E-12	1.479	68.31	52.84	176.45

Subject area	Materials Science
More specific subject area	Ni-base superalloys
Type of data	Tables, figures
How data was acquired	Density functional theory calculations using
	Vienna Ab-initio Simulation Package (VASP)
Data format	Analyzed
Experimental factors	Not applicable
Experimental features	Not applicable
Data source location	State College, PA, USA
Data accessibility	All data is available in this article