First-principles energy and vibration spectrum simulations of Cr/V interacting with H in W-based alloy in a fusion reactor

Figures & data

Table 1. The vibration frequencies (ω in unit of THz) of H at the TS (ω_TS) and OS (ω_OS) as well as transition states (ω_Tran) in W and W-based alloy at 0 K

	ωTS	ωTran	ωOS
W	46.1, 46.1, 34.9	63.0, 46.1, i25.1	75.6, i26.1, i26.1
W–Cr alloy	47.8, 39.0, 30.9	64.5, 38.5, i22.2	77.4, i21.8, i23.1
W–V alloy	48.1, 43.1, 33.2	64.9, 42.5, i23.9	77.8, i24.1, i24.8

Table 2. The ratio (a_T/a₀) of lattice parameter, the lattice parameter (a_T) and the lattice expansion strain (%) of W-based alloy at the relevant temperature

Temperature (K)	aT/a0	aT (Å)	Strain (%)
0	1.0000	3.1715	0
300	1.0014	3.1745	0.14
600	1.0028	3.1789	0.28
900	1.0042	3.1834	0.42
1200	1.0058	3.1883	0.58
1500	1.0074	3.1934	0.74
1800	1.0092	3.1991	0.92
2100	1.0111	3.2053	1.11

Figure 1. (Top view from the [001] direction) The different neighbor TS (a) and OS (b) around the substitute Cr/V (gray ball) in W (red ball) atomic structure plane supercell. The distances of 1nn, 2nn, 3nn, 4nn TS (OS) are $\sqrt{5}a/4$(a/2), $\sqrt{13}a/4$($\sqrt{2}a/2$), $\sqrt{29}a/4$($\sqrt{5}a/2$), and$\sqrt{37}a/4$(3a/2), respectively. The TS and OS are denoted by triangle and circle, respectively.

Figure 2. The formation energies of H at the TS with the increasing temperature in W and W-based alloys. (a) The total formation energy in reference to the total H chemical potential μ_H(T, P_1atm) = μ_H(T = 0 K) + μ_H(T ≠ 0 K, P_1atm). (b) The corresponding static formation energy in reference to the static H chemical potential μ_H(T = 0 K) based on Equation (13)(13) $$\begin{array}{c}\hfill G_{H\left(\mathrm{ strain }\right)}^f=E_{a_T\left(\mathrm{H}-\mathrm{W}-\mathrm{ alloy }\right)}-E_{a_T\left(\mathrm{W}-\mathrm{ alloy }\right)}-{\mu }_H(T=0K),\end{array}$$(13) and it is related to the lattice expansion. (c) Phonon vibration formation energy referring to the temperature-dependent H chemical potential μ_H(T ≠ 0 K, P_1atm) based on Equation (14)(14) $$\begin{array}{ccc}\hfill G_{H\left(\mathrm{ vibration }\right)}^f& =& F_{a_T\left(\mathrm{H}-\mathrm{W}-\mathrm{ alloy }\right)}^{vib}\left(T\right)-F_{a_T\left(\mathrm{W}-\mathrm{ alloy }\right)}^{vib}\left(T\right)\hfill \\ & & -{\mu }_H(T\ne 0K,P_{1\mathrm{ atm }}),\hfill \end{array}$$(14) .

Figure 3. The formation energies of H at the TS with the increase of temperature in W and W-based alloys. (a) Here, the total formation energy only referring to the static H chemical potential μ_H(T = 0 K). (b) The corresponding static formation energy in reference to the static H chemical potential μ_H(T = 0 K) based on Equation (15)(15) $$\begin{array}{ccc}\hfill G_{H\left(\mathrm{ strain }\right)}^f& =& E_{a_T\left(\mathrm{H}-\mathrm{W}-\mathrm{ alloy }\right)}-E_{a_T\left(\mathrm{W}-\mathrm{ alloy }\right)}\hfill \\ & & -{\mu }_H(T=0\mathrm{K}),\hfill \end{array}$$(15) and it is related to the lattice expansion. (c) Phonon vibration formation energy. Here, it does not refer to any H chemical potential since it is only the difference between H–W-alloy supercell and pure W-alloy supercell, according to Equation (16)(16) $$\begin{array}{c}\hfill G_{H\left(\mathrm{ vibration }\right)}^f=F_{a_T\left(\mathrm{H}-\mathrm{W}-\mathrm{ alloy }\right)}^{\mathrm{ vib }}\left(T\right)-F_{a_T\left(\mathrm{W}-\mathrm{ alloy }\right)}^{\mathrm{ vib }}\left(T\right).\end{array}$$(16) .

Figure 4. Solubility of H as a function of the reciprocal of temperature at one atmosphere pressure in W and W-based alloys. The experimental solubility of H in W is given from Frauenfelder [63].

Figure 5. (Top view from the [001] direction) The different neighbor (1nn, 2nn and 3nn) t→t and t→o→t diffusion routes around the substitute Cr/V (gray ball) in W (red ball) atomic structure plane supercell. The red and black arrows represent t→t and t→o→t routes, respectively. The TS and OS are denoted by triangle and circle, respectively.

Figure 6. Along the t→t diffusion route, the H-migrating energy barriers with the temperature in W and W-based alloys. (a) The total energy barriers. (b) The energy barriers from the lattice expansion contribution. (c) The energy barriers from phonon vibration contribution.

Figure 7. Along the t→o→t diffusion route, the H-migrating energy barriers with the temperature in W and W-based alloys. (a) The total energy barriers. (b) The energy barriers from the lattice expansion contribution. (c) The energy barriers from phonon vibration contribution.

Figure 8. Along the t→t route, the diffusivity (D) of H as a function of reciprocal of temperature in W and W-based alloys.

Frauenfelder R. Permeation of hydrogen through tungsten and molybdenum. J Chem Phys. 1968;48:3955–3965.

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