In the generation IV international forum, the metallic fuels for the sodium-cooled fast breeder reactor, which have been studied for more than five decades with the documented performances, were brought under renewed focus.^[1] The metallic fuel pin is mainly comprised of rod-shaped fuel alloy made of U–Zr or U–Pu–Zr, HT-9 steel cladding, and the fuel-to-clad gap filled with liquid sodium.^[2] They have been used in the experimental breeder reactor II (EBR-II) and qualified up to a specific burnup of 10 at.%.^[3] One of the major reasons for the limitation of higher burnup is fuel–cladding chemical interactions (FCCI), which occur at the fuel and cladding interface during irradiation.^[2,4–7]

FCCI is complex, including fuel–cladding interaction (e.g., U–Fe) and the interactions among fission products containing a significant mount of lanthanides (Ln) and cladding (e.g., Ln–Fe). Among them, the greatest concern of FCCI is the lanthanide–cladding interaction, as lanthanides are observed to migrate fast in a fuel and diffuse rapidly into cladding (50–150 μm),^[8] yielding intermetallic precipitation, phase changes, and melting.^[9–11] The lanthanide interaction with clad affects the mechanical integrity of the cladding, and is deemed a long-term, high-burnup cause of the cladding failures. Actually, fission product segregation to the grain boundary (GB) is a prerequisite for the formation of a brittle intermetallic phase. The segregation behavior could produce a significant effect on the GB cohesion, therefore limiting the ductility of the metallic alloys. Thus, the segregation of fission products and their effects on GB in the cladding is of concern, and a fundamental knowledge of these processes is needed.

Impurities and alloy elements are generally aggregated at GB, which substantially affects the mechanical properties, primary ductility, fracture toughness, and strength of metallic materials. Extensive experimental and theoretical researches have been performed on their influence on GB cohesion. For instance, interstitial boron and substitutional Mo can enhance the GB strength in Fe alloys,^[12,13] while H impurities weaken GB, which is one of the mechanisms of hydrogen embrittlement.^[14] On the other hand, based on the Rice–Wang model,^[15] first-principles density functional theory (DFT) calculations of intergranular cohesion in the presence of segregated impurities in Fe were pioneered by Krasko and Olson, followed by extensive calculations by Freeman, Olson, and co-workers.^[16–18] They employed the full potential linearized augmented-plane-wave (FPLAPW) method to study several different impurities and alloying elements at Σ3(111) symmetrical tilt GB in Fe.^[18] More recently, the projector augmented wave (PAW) approach was used to study the effects of impurities on the GB. To our knowledge, there are few ab initio calculations for lanthanides segregation and intergranular cohesion in Fe.

The reports of post irradiation examination (PIE) data of metallic fuels (U–Zr and U–Pu–Zr) irradiated in EBR-II reactor suggested that the major lanthanides diffusing into cladding were La, Ce, Pr, and Nd.^[8,19] Symmetry Σ3(111) [110] GB is a typical high-angle GB with remarkable excess volume for body-centered cubic (bcc) iron. In this work, combining first-principles calculations and the Rice–Wang model, the preferential site, segregation energy, and embrittlement capability of the four lanthanides at the Σ3(111) [110] GB in α-Fe will be investigated. We organize the remainder of the paper as follows. In Section 2, the computational methods and models are described. Section 3 presents and discusses our calculated results of the behavior of the lanthanides segregation and intergranular cohesion in α-Fe. We analyze the relaxed atomic structures of lanthanide elements at Σ3(111) GB and the corresponding excess volume to understand the physical origin of segregation and embrittling behavior of these impurities. Finally, a short summary is given in Section 4.

Our first-principles DFT calculations were performed using the Vienna ab initio simulation package (VASP).^[20–22] The interaction between valence electron and nuclei was obtained using the projector augmented wave (PAW) method.^[23,24] The exchange and correlation terms in DFT method were treated with generalized gradient approximation (GGA) in the Perdew–Burke–Ernzerhof (PBE) form.^[25] A conjugate gradient algorithm was used to relax the atomic positions to a local minimum in the total energy landscape. The cut-off energy for the plane wave basic set was 350 eV. For each system, geometry optimization could continue until the total energy of this system was converged to less than 10⁻⁴ eV. A 7 × 4 × 1 k-mesh in Monkhorst–Pack scheme replaced the integration over Brillouin zone in the following simulation.^[26]

We selected the Σ3(111)[ ] tilt GB as a prototype low-energy symmetric twin boundary in bcc Fe, and modeled the GB using a supercell illustrated in Fig. 1(a). One supercell was composed of two identical grains (each contains 15 atomic layers with one in common), which formed a symmetric tilt GB in between. The two-dimensional lattice constant for the stress-free systems was chosen to be the bulk value of bcc Fe, 2.83 Å, which was also reproduced in our GGA-PBE computation. Since we have used a periodic boundary condition, we separated the neighboring slabs in [111] direction by a vacuum region of at least 10 Å to minimize the interaction between the slabs. Test calculations on GB energy for cells containing different numbers of atomic layers from 17 to 49 were carried out, and the results are shown in Fig. 2(a). It can be seen that the GB formation energy converges to 1.59–1.60 J/m² when the layer of GB slabs is larger than 21. On the other hand, we also display the calculated interlayer distance and magnetic moment on Fe atoms near the GB in Figs. 2(b) and 2(c), respectively. It is clearly seen that the oscillatory is very large near the GB center, and then decays toward the bulk layers (from site 6). In comparison to the ideal grains, the relaxations cause an increase of 0.24 Å in the excess volume per unit area, which offers a larger space available for the impurity element. In Fig. 2(c), the moments exhibit a damped oscillatory convergence toward 2.20 μ_B for the deeper grain layers. These results show that the 29-layer slab is thick enough to mimic a bulk-like environment in the middle of each grain.

Fig. 1.

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	Fig. 1. Computational models: (a) Σ3(111)[ ] tilt GB with 29 layers and (b) the Fe (111) free surface (FS) with 15 layers. The violet and blue spheres represent atoms on the ( ) and ( ) planes, respectively. The numbers here are used to denote different positions near the GB and surface.

Fig. 2.

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	Fig. 2. (a) The GB formation energy for cells containing different numbers of atomic layers from 17 to 49. (b) Relaxed interlayer distance in the Fe grains near the boundary. The (111) interlayer distance in the bulk truncated Fe is 0.82 Å. (c) Magnetic moment on Fe atoms at various layers in the vicinity of the GB (cf. Fig. 1). Horizontal dash line marks the magnetic moment of the bulk Fe.

In order to assess the tendency of the fission product to segregate to the GB from the bulk environment, the segregation energy of one X atom (X = La, Ce, Pr, and Nd) at GB is defined by 1 where and are the total energies of the system with X atom near the GB and in the bulk (site 7) far away from the GB, respectively. As shown in Figs. 2(b) and 2(c), site 7 can be chosen as the substitutional site with a bulk-like atomic environment. The negative means that Ln X can segregate at the GB.

Since lanthanides La, Ce, Pr, and Nd are rare earth elements with strongly localized f electrons, it should be careful for their first-principles calculations. In transitional metal oxides or cerium oxides, Hubbard U correction to the standard DFT calculation works well for adjusting the band gap by taking into account of the orbital dependence of the intra-atomic Coulomb interaction.^[27] In fact, the empirical Hubbard U correction is controversial for the first-principles calculations. In the recent work, Chen et al. studied the electronic and optical properties of rare-earth-doped VO₂ nanoparticles,^[28] they did not apply Hubbard U correction on lanthanides but on vanadium. Furthermore, Hao proved that the on-site Coulomb interaction has little effect on the description of Ce in iron solid solutions.^[29] In view of this fact, we have selected Nd with most 4f electrons to examine the effect of the on-site Coulomb interaction on the segregation energy of Nd at GB in iron in GGA +U calculations. The results are listed in Table 1, and suggest that the on-site Coulomb interaction has only a marginal effect on the interaction between Nd and GB (0.02–0.03 eV). Therefore, we conclude that GGA can deal with the Ln–Fe alloy system well enough and we will proceed with our investigations using only GGA treatment in the work.

Table 1.

Table 1.

Table 1. Dependence of the segregation energy of Nd at the GB (in eV), defined by Eq. (1) in bcc Fe on Hubbard U, the effective on-site Coulomb interaction (in eV). . Eseg Site 0 Site 1 Site 2 Site 3 Site 4 U = 0 –1.50 –1.48 –1.48 –1.22 –0.13 U = 4 –1.53 –1.51 –1.49 –1.25 –0.15

Table 1. Dependence of the segregation energy of Nd at the GB (in eV), defined by Eq. (1) in bcc Fe on Hubbard U, the effective on-site Coulomb interaction (in eV). .

Using first-principles density calculations, we have investigated the segregation energy of Ln fission products (La, Ce, Pr, and Nd) near Σ3(111) tilt symmetric GB and their effects on the intergranular embrittlement in α-Fe. Firstly, we determined the segregation energies of La, Ce, Pr, and Nd at different substitutional sites near the GB. The results show that they all tend to segregate to Σ3(111)[ ] tilt GB from bulk environment with a large energetic driving force. For La and Ce, they tend to locate at the first layer (site 1) near the GB with segregation energies of −2.55 eV and −1.60 eV, while Pr and Nd atoms prefer to occupy the core mirror plane (site 0) of the GB with lower segregation energies of −1.41 eV and −1.50 eV. Atomic configuration analysis shows that the absorption of La and Ce to the GB could cause the slipping of the upper and lower atoms of the GB and the deviation of atoms in the mirror plane. Based on the McLean’s equation, we thermodynamically evaluated their occupation probabilities near the GB. The strong driving forces to segregate to the GB make the fission product highly concentrate at the GB at the operating environment. Besides, based on Rice and Wang model, we have obtained that La, Ce, Pr, and Nd atoms are all strong embrittlers with strengthening energies of 2.05 eV, 1.52 eV, 1.50 eV, and 1.64 eV, respectively. The embrittlement capability of these four lanthanide elements can be determined by the factors of their atomic size and magnetism.