† Corresponding author. E-mail:
Project supported by the Research Key Project of Science and Technology of Education Bureau of Henan Province, China (Grant Nos. 14A140030, 15A140032, 15B150010, and 15A430037) and the Innovation Talents Program of Science and Technology of Institution of Higher Education of Henan Province, China (Grant No. 14HASTIT044).
The distribution of He in η-Fe2C has been studied by first-principles calculations. The formation energies of interstitial He and substitutional He (replacing Fe) are 3.76 eV and 3.49 eV, respectively, which are remarkably smaller than those in bcc Fe, indicating that He is more soluble in η-Fe2C than in bcc Fe. The binding potencies of both a substitutional-interstitial He pair (1.28 eV) and a substitutional-substitutional He pair (0.76 eV) are significantly weaker than those in bcc Fe. The binding energy between the two He atoms in an interstitial–interstitial He pair (0.31 eV) is the same as that in bcc Fe, but the diffusion barrier of interstitial He (0.35 eV) is much larger than that in bcc Fe, suggesting that it is more difficult for the interstitial He atom to agglomerate in η-Fe2C than in bcc Fe. Thus, self-trapping of He in η-Fe2C is less powerful than that in bcc Fe. As a consequence, small and dense η-Fe2C particles in ferritic steels might serve as scattered trapping centers for He, slow down He bubble growth at the initial stage, and make the steel more swelling resistant.
Reduced-activation ferritic/martensitic (RAFM) steels are one of the most important candidates for the first-wall structural materials in fusion reactors,[1] partly due to their low swelling rates.[2] However, in a fusion reactor the energetic He atoms enter the first-wall structural material, while at the same time large amounts of He are produced from transmutation reactions. Due to the strong self-trapping,[3] and strong binding to vacancies,[4] hydrogen,[5] grain boundaries,[6–8] and dislocations,[9] high concentration He will form He bubble and induce void swelling.[10,11] Thus these steels still suffer He-induced embrittlement.[12]
Great effort has been made to suppress He bubble formation to improve the mechanical properties of the materials, and some alloying additions appear to work this way. For example, an experiment has demonstrated that the segregated Cr at the He bubble surface in a Fe-9Cr% steel slows down the Brownian motion of a small He bubble.[13] Hao et al., using first-principles density functional theory (DFT) calculations, revealed that the segregated Cr increases the charge density at the bubble/Fe interface which prevents further He atoms passing through, and thus slows down the growth of He bubbles.[14] DFT calculations also demonstrated that Au[15] and Ce[16] have a more pronounced effect than Cr. C is one of the elementary chemical elements in ferritic steels, whose effect on the He migration and agglomeration in bcc Fe have also been investigated.[17] It is found that C inhibits the formation of higher order clusters composed of He and vacancies, i.e., HenVm, and thus increases the amount of He at substitutional positions at room temperature.
Due to its low solubility in ferritic steels, the excess carbon will precipitate in the form of carbides in quenched steels.[18,19] In the very initial stage, the C atoms occupied the octahedral sites in bcc Fe.[20] When the C content exceeds the equilibrium solubility, the cementite, θ-Fe3C, is formed. A further increase of the C content, the η-Fe2C and χ-Fe5C2 will also appear. Recently, the structure and stabilization of the iron carbides, especially, θ-Fe3C, η-Fe2C, and χ-Fe5C2, have been investigated by first-principles studies.[21–26] Faraoun et al.[23] using DFT calculation pointed out that the energy needed for the formation of the χ-Fe5C2 is three times larger than that to form η-Fe2C. Denisov et al. studied the influence of heat treatment on the interaction between hydrogen and RUSFER-EK-181 Russian ferritic–martensitic steel, using the method of thermodesorption mass spectrometry, and they found that after soaking of martensitic steels at low-temperature tempering, clusters of carbon atoms are observed and metastable carbides (Fe2C) appear.[27] Despite this, η-Fe2C is not stable at any temperature and has been scarcely investigated experimentally in the modified martensitic steel.[28] However, previous investigations have showed that it may be stabilized by impurity elements[22,29] and may be produced by the irradiation in extreme conditions in connection with realization of international projects ITER.[30] Furthermore, Mazumder et al.[31] examined He implantation of an F82H tempered martensitic steel at different temperatures, and suggested that the carbide precipitates prevent He bubbles formation by trapping the diffusing He at low temperature. Our previous studies have declared that θ-Fe3C in ferritic steels might mitigate He bubble growth[32] just as dispersed oxide particles[33,34] and smaller grains do.[35] Thus, η-Fe2C is one of the most frequently observed carbide-precipitate phases, its effect on He bubble growth in bcc iron is in urgent need to understand. Due to the lattice mismatch at the interface leaves abundant free volumes (low electron density region) which are strong traps for He, the carbide/Fe interface will be as crucial as, if not more than, the inside of the carbide particles. As a first step to tackle the η-Fe2C effect on He accumulation, in this work, we focus on the properties of He inside the grain of a η-Fe2C particle in micro-meter scale, which can be well represented by a bulk system. Our calculations demonstrate that He is more soluble in η-Fe2C than in bcc Fe and the He self-trapping in η-Fe2C is less powerful than that in bcc Fe, indicating that η-Fe2C particles in ferritic steels will trap more He atoms and might slow down He bubble growth at the initial stage, which may make the steel more swelling resistant.
Our spin-polarized first-principles calculations were performed by the Vienna ab initio simulation package (VASP).[36–38] The electron–ion interaction was described using the projector augmented wave (PAW) method.[39,40] The electron exchange and correlation were treated within the generalized gradient approximation (GGA), using the Perdew Burke Ernzerhof (PBE) functional.[41] The cutoff energy for the plane wave basis set was taken as 400 eV. The stopping criterion for ionic relaxations was convergence of the total energy to within 10−4 eV.
In the primitive cell of η-Fe2C, there are four Fe4g and two C2a atoms. The optimized lattice parameters are a = 4.71 Å, b = 4.28 Å, and c = 2.82 Å, which are consistent with the experimental results.[42] The calculated magnetic moments on Fe and C atoms are 1.66μB and −0.14μB, respectively, which is in agreement with the previous previous theoretical values.[23]
A 2×2×3 supercell with the fixed cell axes based on the optimized lattice parameter was used to trap the He atom. The sketch map for He at different interstitial sites is shown in Fig.
The binding energies between the two He atoms in an interstitial–interstitial (int–int) He pair, a substitutional–substitutional (sub–sub) He pair and a substitutional-interstitial (sub–int) He pair are defined as
In order to make a comparison, a 4×4×4 supercell of the bcc Fe was used to calculate the formation energy of the interstitial and substitutional He atoms, as well as the binding energy of a He pair.
Due to its closed 1s shell, He can only interact repulsively with other elements under compression, thus He almost always prefers larger free volume, i.e., low electron density space. In the unit cell of η-Fe2C, two octahedral (O1 and O2) and three tetrahedral interstices (T1, T2, and T3) have been investigated (Fig.
The formation energies of the He atoms at different interstices were calculated according to Eq. (
![]() | Table 1. The formation energies (in eV) of He at interstitial and substitutional sites in η-Fe2C, as well as the formation energies of He at the tetrahedral interstice (T) and the substitutional site (Fe) in bcc Fe. For an explanation of He positions in η-Fe2C, please see Fig. |
We have also calculated the spin magnetic moments of the systems with He at O1 or O2 sites, as shown in Table
The formation energies for He substitution of a Fe and a C atoms were calculated according to Eq. (
![]() | Table 2. Spin magnetic moments (M) of Fe atoms neighboring the He atom implanted into η-Fe2C and their distances (dO1−X and dO2−X) away from the He atom, which is located at interstitial site O1 or O2, before and after relaxation. For an explanation of lattice sites A–L, and the interstitial sites O1 and O2, see Fig. |
The formation of a He pair is the initial stage of He bubble growth. Our objective in the present work is to elucidate the accumulation of He at the very initial state, i.e., pairing of He, and compare its easiness with that in the bcc Fe matrix. The binding energies between the two He atoms of a He pair can be considered as the reduction of the total energy when the two favorable interstitial (or substitutional), as well as one substitutional and one interstitial He atoms come nearby. The calculated binding energies are listed in Table
![]() | Table 3. The binding energies (in eV) of int–int, sub–int, and sub–sub He pairs in η-Fe2C. The values in the parentheses are the binding energies of the most stable int–int, sub–int, and sub–sub He pairs in bcc Fe. For an explanation of He positions, please see Fig. |
From the numerical results above, we know that the formation energies of substitutional He are smaller than those of the interstitial He in both η-Fe2C and bcc Fe. The occupation probabilities of He at the interstitial and substitutional sites in η-Fe2C and bcc Fe could be estimated by McLean’s equation as
From the calculations of the He pairs, we know that the binding energies between the two He atoms of a sub–sub He pair (0.76 eV) and a sub–int He pair (1.28 eV) are much smaller than those in bcc Fe (1.18 eV and 1.86 eV). In order to find the reason for this, we provided the charge densities of He at the substitutional site, a sub–int He pair and a sub–sub He pair in η-Fe2C in Fig.
![]() | Fig. 4. The calculated occupation probability of He as a function of the energy difference at different temperatures and bulk concentrations based on the McLean’s equation. |
In summary, we have systematically investigated the stability of He inside the η-Fe2C bulk by first-principles calculations based on the density functional theory. The formation energies of the He atoms at the substitutional and interstitial sites in η-Fe2C are 3.49 eV and 3.76 eV, respectively, which is much smaller than those in bcc Fe. The binding energies between two interstitial He atoms is as large as that in bcc Fe, 0.31 eV. However, the diffusion barrier of the interstitial He is 0.35 eV, which is much larger than that in bcc Fe. Furthermore, the binding potencies between two substitutional He atoms and one substitutional and one interstitial He atoms (0.76 eV and 1.28 eV) are much weaker than those in bcc Fe. The larger solubility of a He atom and weaker binding potency between two He atoms in η-Fe2C than in bcc Fe might may make the small and dense η-Fe2C particles in ferritic steels serve as scattered trapping centers for He and mitigate He bubble growth at the initial stage. We hope that this study will promote further experimental and theoretical efforts to design more swelling resistant steels taking advantage of carbides.
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