Journal of Nuclear Materials 496 (2017) 157e162
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Stability and kinetics of helium interstitials in boron carbide from first principles Anton Schneider a, *, Guido Roma a, *, Jean-Paul Crocombette a, Vianney Motte b, Dominique Gosset b a b
DEN-Service de Recherches de M etallurgie Physique, CEA, Universit e Paris-Saclay, F-91191 Gif sur Yvette, France DEN-Service de Recherches de M etallurgie Appliqu ee, CEA, Universit e Paris-Saclay, F-91191 Gif sur Yvette, France
a r t i c l e i n f o
a b s t r a c t
Article history: Received 8 June 2017 Received in revised form 7 September 2017 Accepted 13 September 2017 Available online 18 September 2017
When boron carbide is used in nuclear reactors as a neutron absorber, helium concentrations on the order of a few atomic percent can be attained. It is thus of primary importance to know the distribution and kinetics of helium atoms in boron carbide. In spite of a variety is of experimental works devoted to the characterization of the microstructure and He bubbles in boron carbide irradiated in reactor, there is a serious lack of knowledge concerning the basic mechanisms governing helium kinetics. This study is devoted to the stability and mobility of helium interstitial atoms in carbon rich boron carbide. The lowest energy He insertion sites were screened through density functional theory and the most probable migration paths and energy barriers were investigated using the nudged elastic bands (NEB) approach. The results suggest that in a wide range of temperatures He interstitials undergo 2D diffusion confined between two 〈111〉 planes. The onset of 3D diffusion is expected, according to our calculations, with an activation energy close to 2 eV. Our result is in qualitative agreement with the observation of flat bubbles with 〈111〉 orientation, although a quantitative comparison with He diffusion data is hindered by discrepancies and microstructure issues in available experimental results. © 2017 Elsevier B.V. All rights reserved.
Keywords: Density functional theory Boron carbide Helium Stability Kinetics Nudged elastic bands
1. Introduction Due to boron high neutron absorption cross section, boron carbide is used in shutdown and control rods of various types of nuclear reactors; moreover, due to the absence of a resonance region in the neutron absorption cross section, it is the best candidate for control rods of fast breeder reactors. The nuclear reaction taking place when neutrons are absorbed by the 10B isotope leads to the productions of large quantities of helium and lithium. In most cases the lifetime of the rods is limited by the swelling and microcracking occurring in the material, in particular due to the accumulation of helium bubbles [1]. Flat He bubbles preferentially oriented in the 〈111〉 planes in irradiated boron carbide were reported already in the early seventies [2,3]. A transition from flat to equiaxed bubbles was reported at temperatures around 1500 C [4]. Other works reported
* Corresponding author. E-mail address:
[email protected] (G. Roma). https://doi.org/10.1016/j.jnucmat.2017.09.020 0022-3115/© 2017 Elsevier B.V. All rights reserved.
spherical bubbles [5e7] even at relatively low temperature, probably due to different irradiation and annealing conditions. In the following decades additional works provided further evidence for 〈111〉-oriented flat bubbles, with a more refined quantitative analysis of bubbles sizes as a function of temperature and discussed the conditions in which they are expected [8e10]. The earliest studies did not observe bubbles at grain boundaries [2], and concluded that He diffusion was accelerated along them, but this was due to open porosity of the samples. In more recent works He bubbles were detected at grain boundaries and it was then considered that He is trapped there [10,11]. Whether single He atoms also tend to segregate and get trapped at grain boundaries, or trapping occurs only for He clusters above a given threshold size, is not clearly understood yet. It is clear that, in order to establish more reliable models for He clustering and release from boron carbide under irradiation, reliable data on diffusion activation energy are desirable. Diffusion of He in boron carbide has first been discussed by Clayton [12] in the framework of He release from B4C specimens irradiated in reactor at room temperature: with strong assumptions concerning the
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spatial distribution of He, a diffusion activation energy of 1.26 ± 0.13 eV has been estimated. However, considering the results of later works on dose dependences and bubbles formation, one would rather suppose that a large portion of He was trapped in grain boundaries and the measured activation energy might be associated with the release of single He atoms from trapped bubbles. More involved diffusion models [13] considered indeed both He diffusion in the matrix and He detrapping from so called “accumulation centers”, it is however unclear which are the activation energy values used. The activation energy given by Clayton was used in another model of He release [14], but the diffusion prefactor was significantly reduced in order to reproduce experimental results. In this paper we investigate the elementary diffusion mechanisms and activation energy for interstitial He atoms from first principles calculations based on density functional theory. We focus mainly on the B4C stoichiometry dusing the B11C-CBC structure described laterd but, for the sake of comparison, the boron richer B12-CBC configuration was also tackled. We predict the relative stability of substitutional and interstitial He insertion sites and find minimum energy paths between the most stable ones. The paper is organised as follows: section 2 is devoted to the technical details of our approach; section 3 is subdivided in subsection 3.1, which describes the stability of insertion sites, subsection 3.2 which discusses the diffusion mechanisms, and subsection 3.3 on the role of the electronic charge density. In section 4 we discuss the results in connection with experimental results. The main results are summarized in the final section 5. 2. Model structures and methods Boron carbide has a rhombohedral crystalline structure, which is maintained in a large range of stoichiometry [15]. Here we focus mainly on the carbon rich side, with stoichiometry B4C for which we assume the structure shown in Fig. 1, B11C-CBC, where B11C icosahedra are connected by CBC chains. The latter structure is considered to be the most stable structure for this stoichiometry [16]; however some disorder, at least in the placement of the icosahedral carbon, is expected [15,17]. In absence of disorder, the icosahedral carbon atom, sitting on a polar site, should induce a monoclinic distortion; only recently an experimental evidence of such distortions was found in the presence of asymmetric twin boundaries [18], while X-ray diffraction (XRD) studies reported a rhombohedral structure. This distortion is very small according to our calculations (less than 12 arcminutes, or 0.2 on the rhombohedral angle in the unit cell). The corresponding lowering of the energy due to the distortion is less than 0.015 eV/ atom. Moreover, a test for a He interstitial, reveals a lowering of only 0.05 eV in the formation enthalpy with the distorted cell; this check was made on the interstitial site called type 2 site in the following, which is the only one not aligned with the rhombohedral axis. We then decided to keep the rhombohedral structure in our defect calculations, which were performed at the equilibrium volume of the B11C-CBC structure. We have also checked that full volume relaxation leads to negligible changes in the solution energy for the 2 2 2 supercell; in the worst case a difference of 0.11 eV was found and in most other cases it is below 0.04 eV. Almost all the calculations in this work are based on density functional theory as implemented in the Vienna Ab-initio Simulation Package (VASP) [19,20]. A few test calculations, for comparison, were performed with the Quantum-Espresso (QE) package [21]. We chose, for the exchange-correlation, the optB86b þ vdW functional [22], containing a Van-der-Waals long-range non-local term and an optimized version of the Bethe 86 exchange term. The projector augmented wave (PAW) method was used for the electronic
Fig. 1. Rhombohedral structure of B11 C CBC. Carbon atoms are yellow, boron are gray. The C-B-C chain, connecting six of the eight B11C icosahedra is clearly visible in the center. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
structure. Following the relaxation tests that were carried out, the energy cutoff were set to 400 eV for VASP calculations. A 3 3 3 Monkhorst-Pack k-point mesh was used for the 15atom unit cell for integration over the Brillouin zone. With these parameters the total energy of the unit cell is converged at 0.05 eV. The 2 2 2 supercell (120 atoms) and the 3 3 3 supercell (405 atoms) were sampled with the G point. Atomic relaxations at constant volume were performed down to a maximum atomic force of 0.05 eV/Å. For comparison, we were led to use the hybrid functional HSE06 [23] with a G centered k-point grid in a few cases. Unless stated otherwise, all calculations presented here were performed in a 2 2 2 supercell, which turned out to be reasonably well converged for solution and migration energies. The first step in the systematic study of the helium insertion sites in boron carbide was the study of its possible substitutional sites (i.e., the 15 distinct atomic sites of the defect free boron carbide system). We calculated both the solution energy and incorporation energy for each site. To write the helium incorporation energy in a substitutional site we need the total energies of the vac ), the one of the system containing a system with a vacancy (Etot He ) in the vacant site and the He substitutional helium atom (Etot chemical potential mHe ; we take the latter as the energy of an isolated He atom (the binding energy of the He-dimer is negligible [24] with respect to formation energies). The incorporation energy is then: He He vac Einc ¼ Etot Etot mHe
(1)
The results are presented for carbon rich thermodynamical
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conditions. Thus the carbon chemical potential mC , used for the calculation of vacancy formation energies is taken as the energy per atom in graphite. The boron chemical potential is then derived from the calculated formation energy of boron carbide (0.59 eV/formula unit) and the calculated energy of alpha boron. The solution energy (for a B substitutional, for instance) can be written in two ways: He He id He Esol ¼ Etot þ mB Etot mHe ¼ Einc þ Efvac
(2)
Helium can also be inserted in intersitial sites. In this case the id and the incorenergy of the system prior to incorporation is Etot poration energy coincides with the solution energy. We systematically screened all possible interstitial sites by putting He atoms on a regular three dimensional cubic grid of 1 Å side. From these positions, those closer than 0.9 Å (0.7 Å) from boron (carbon) atoms were discarded. We ended up with 40 symmetry inequivalent positions which were all fully relaxed. After relaxation, all these insertions lead to only three final He interstitial positions (see below). Once the most probable positions for helium insertion were determined, we performed Nudged Elastic Band (NEB) [25] calculations between interstitial configurations in order to find the lowest migration barriers. Atomic configurations are visualized here with the open-source software Xcrysden [26]. The software Atomsk was also used in order to generate some atomic structures [27].
3. Results 3.1. Stability In the B11C-CBC structure eleven insertion sites can be found which are non-equivalent by symmetry (see Fig. 2). Table 1 presents the results of vacancy formation energies, He incorporation and solution energies in these eleven insertion sites. Solution energies are quite large. Vacancy formation energies on equatorial and polar sites are also very large. One can notice that two incorporation energies are negative; further investigations showed that, in those two cases, the substitutional He has escaped from the starting position to an interstitial one, leaving a vacancy behind. Incidentally, we also observe that for all equatorial sites (B5 to B8), the vacancies are practically unstable, because overcoming very tiny barriers they convert to a B1 configuration. Finally, the easiest substitutional defect occurs on the chain boron site, which has also the lowest vacancy formation energy, 1.45 eV. The latter is slightly lower than previous estimations obtained with the LDA
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functional [28]. Concerning the interstitial defects, our systematic search revealed only three distinguishable configurations. The first He site lies inside the icosahedron, the second lies around the C-B-C chain while the third is situated in the prolongation of it. The last two positions are presented on Fig. 3. From here on, interstitial sites located all around the C-B-C chain (three on each pole) as shown in blue on Fig. 3 are named ”Type 2 sites” whereas the sites aligned with the C-B-C chain on each side (shown in green on Fig. 3) are named ”Type 3 sites”). If the icosahedra were made only of boron atoms, the six sites of Type 2 would be indistinguishable by symmetry and the same would hold for Type 3 sites; but due to the presence of one carbon atom, the symmetry is broken in the B11C-CBC structure dhence the small energy variation of Type 2 and Type 3 sites, as presented in Table 2. One notices that the formation energy of the Type 1 site is quite higher than the others. This site, also called endohedral He, was previously studied in the context of calculation of noble gas doped closo-borane clusters [29] (Ng@B12H2 12 ). Our solution energy agrees very well with the insertion energy obtained from those cluster calculations (7.20 eV). This insertion site is thus highly unfavorable and will not be considered in the following. Before presenting the results for the migration paths, we stress the fact that He interstitials induce very small geometrical changes of the boron carbide structure. For example, C-B-C chain angles change by 1e3 at most, B-C distances on the chain by 0.01e0.02 Å for Type 3 interstitials, less than that for the other insertion sites. As far as rhombohedral angles are concerned, full volume and shape relaxation leads to an appreciable change (0.3 or 18 arcminutes) only for Type 3 interstitials.
3.2. Mobility Starting from the sites 2 and 3 previously determined, NEB calculations were performed and the most favorable paths between them were found (see Table 3). All migration paths reduce to combinations of the following paths: path 1 (Fig. 4, energy barrier 1.2 eV) connects two Type 2 sites located around the same C-B-C chain path 2 (Fig. 5, energy barrier 0.1e0.35 eV) connects a Type 2 and a Type 3 site located near different C-B-C chains path 3 (Fig. 6, energy barrier 2.2 eV) connects two Type 2 sites located near different C-B-C chains;
Table 1 Vacancy formation energies, He incorporation and solution energies for the 11 inequivalent substitutional sites. i indicate icosahedral sites, while c refers to sites on the C-B-C chains connecting icosahedra to each other. The labels e and p refer to “equatorial” and “polar” sites, respectively. The site labels B1-8 and C1-3 are explained on Fig. 2.
Fig. 2. Visualization of the different substitutional sites. On the icosahedron, sites C3, B2, B3 and B4 are so called “polar sites”, while B5, B6, B7 and B8 are “equatorial sites”.
Site
He [eV] Einc
Efvac [eV]
He [eV] Esol
B1 B2 B3 B4 B5 B6 B7 B8
2.12 1.66 2.14 1.68 1.03 0.97 e 2.34
1.46 4.05 5.56 4.98 5.04 5.01 unstable 4.19
3.58 5.72 7.71 6.66 4.01 4.04 unstable 6.52
2.13 2.19 1.36
2.74 2.58 4.42
4.87 4.77 5.78
(c) (i-p) (i-p) (i-p) (i-e) (i-e) (i-e) (i-e)
C1 (c) C2 (c) C3 (i-p)
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however the migration barrier is always much lower than that for the jump in Fig. 4, around the chain, and at most of 0.35 eV. In order to attain 3D diffusion the third jump, shown in Fig. 6, should be activated, with a higher energy barrier of 2.2 eV. A visual sketch of 2D and 3D diffusion paths, according to the two activation barriers that we have just described is shown in Fig. 7. 3.3. Helium insertion and electron density
Fig. 3. Visualization of the different intersitial sites around a C-B-C chain: Type 2 sites in blue, Type 3 sites in green. The radius of helium atoms is exaggerated for rendering purposes. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Table 2 Formation energies of the three kinds of helium interstitial defects, whose structure is represented in Fig. 3. For Type 2 the value calculated with the hybrid functional is given between parentheses. Site
He Etot
Type 1 Type 2 Type 3
7.23 eV 2.89 (3.05)e3.01 eV 2.96e3.25 eV
Table 3 Migration energies of the three kinds of helium interstitial defects migrations. For the migration between two Type 2 sites of the same chain, the value calculated with the hybrid functional is given between parentheses Migration start/end sites
He Em
Type 2 sites - same chain Type 2 site to type 3 site Type 2 sites - different chains
1.21 (1.26) eV 0.13e0.35 eV 2.22 eV
The study of He migration revealed paths that were not fully intuitive at first sight. In particular, concerning the migration between two Type 2 positions, we could not find, in spite of several attempts with NEB calculations and constrained relaxations, a direct path connecting two adjacent positions (i.e., two with the same z-coordinate in Fig. 3). We then investigated in more detail the role of the electron density, following the idea that He, a noble gas, is not supposed to modify too much its electron cloud upon insertion in the carbide. We remarked that for insertion sites and saddle points with lowest energy, the electron density was also lower. After such a qualitative observation we decided to systematically compare the energy along a He migration path with the charge density along the same path in the perfect crystal (i.e., without He insertion). The outcome, shown on Fig. 8, reveals a very clear, in a wide range pretty linear, relationship, proving that electrostatic effects are driving the energetics of He insertion in boron carbide. This fact suggests that empirical potentials designed with the embedded atom method (EAM) could grasp the essential physics needed to describe larger systems containing several He atoms. We found a fully analogous density profile along the He migration path for the boron-rich B12-CBC, where no carbon is present on the icosahedron. Consistently with the reduced number of electrons, the charge density along the path is slightly lower than for B11C-CBC, although the trend associated with electron deficiency can be reversed on certain chain bonds, as previously reported [30].
Through jumps along path 1 the He atom is confined in one unit cell, turning around a C-B-C chain. We re-calculated this barrier with a hybrid functional and obtained a value of 1.26 eV. Combining this jump with path 2, long range diffusion is granted; the He atom is however confined in two dimensions, between two 〈111〉 planes of the rhombohedral structure. The energy barrier for path 2 depends on the position of the C atom on the nearby icosahedron,
Fig. 4. Visualization of the migration between two Type 2 sites around a C-B-C chain. The energy barrier of this migration is 1.2 eV.
Fig. 5. Visualization of the migration path between a Type 3 site and a Type 2 site located near different C-B-C chains. The energy barrier of this migration is 0.1e0.3 eV depending on the position of the C atom in the icosahedra nearby.
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Fig. 8. The energy of stable configurations (red squares) and along a migration path (black line) vs the charge density at the helium site in the perfect crystal. The zero of energy is at the starting position of the migration path (a Type 2 interstitial site). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 6. Visualization of the migration between two Type 2 sites located near different C-B-C chains. The energy barrier of this migration is 2.2 eV.
4. Discussion The picture suggested by our results supports the idea that the formation of He flat bubbles aligned along the 〈111〉 direction, observed experimentally, might be favored by a two-dimensional He diffusion regime. Indeed we find that migration energies for a 2D diffusion are lower than the ones for a 3D motion of helium. However, it is clear that a kinetic model able to give quantitative results would need several parameters in order to lead to a sensible comparison with experiments. In particular, one should know the thermodynamic parameters associated to He bubbles, which are commonly considered under high pressure. We refer to parameters like the binding energy of He clusters of various sizes, the dissociation energy of He atoms from those clusters, the mobility, if any, of
small He clusters, the segregation energy of He atoms on grain boundaries, their diffusion activation energy along them, etc. It is nevertheless worth discussing how our activation energy compares with the available informations coming from experiments. An early value of 1.26 eV [12] as a diffusion activation energy seems to fit very well our calculated activation energy for 2D diffusion. However, in spite of what the authors claim, the apparent presence of bubbles in their micrographs and the irradiation conditions suggest that the measured activation energy might pertain to other mechanisms than simply diffusion, like He dissociation from bubbles or emission from grain boundaries. In a more recent work it is found that diffusion occurs with much lower activation energy [10], however in this preliminary work the grain size was probably still too small to disentangle the role played by grain boundaries. We further checked the possible limitations of our model in two ways. First, by performing hybrid functional calculations instead of simple local or semi-local functional calculations. As indicated in the result section the migration energy of Path 1 goes from 1.21 to 1.26 eV when turning on the hybrid functional. Second, we investigated the role of stoichiometry: we performed analogous
Fig. 7. A sketch of possible diffusion paths when type 3 jumps (activation barrier: 2.2 eV) are not yet activated (on the left), and when they are activated (on the right). In the first case the diffusion is clearly two dimensional, while in the right panel a 3D diffusion path is obtained.
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calculations for the B12-CBC structure (with stoichiometry B13 C2 ). The results are qualitatively the same as those for the B4C stoichiometry presented in section 3. The migration energies are slightly lower, namely 1.1 eV instead of 1.2 eV and 1.9 eV instead of 2.2 eV for 2D and 3D diffusion, respectively. This slight lowering of barriers is associated with a lowering of the electron density and is coherent with the fact that, as we showed in section 3.3, the higher the electron density at the saddle point of a He migration, the higher is the associated energy barrier. In conclusion, we think that our results for the 2D diffusion of helium in boron carbide are quite robust. Future developments might be devoted to the influence of intrinsic defects, in particular vacancies, on the mobility of helium.
5. Summary The results previously presented concerning the energy of the interstitial and substitutional helium defects lead to the first conclusion that icosahedra are not concerned at all in the insertion of helium, at least in thermodynamical equilibrium. Indeed creating a vacancy on one of the icosahedral sites is quite expensive (between 4.2 and 5.6 eV) and inserting a helium atom inside an icosahedron costs 7.23 eV. The migration of helium in boron carbide then only concerns the inter-icosahedra channels, namely the interstitial sites located around the C-B-C chains and the chains atomic sites. Concerning the mobility of He interstitials in boron carbide, the calculated migration paths and energy barriers suggests a 2D diffusion regime, along 〈111〉 planes, with an activation energy between 1.2 and 1.3 eV. 3D diffusion is expected to occur only at higher temperature, with an activation energy of 2.2 eV or slightly lower when the stoichiometry shifts to the boron-rich side. Our results are necessary input data for any kinetic model willing to describe He bubbles formation and their size and shape evolution during and after irradiation.
Acknowledgements Kevin Gillet is gratefully acknowledged for pointing out the instability of vacancies and substitutional He on equatorial sites. This work was granted access to the HPC resources of TGCC and CINES under the allocation 2016A0010906018 made by GENCI.
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