Atomistic simulations and experimental measurements of helium nano-bubbles in nickel

Journal of Nuclear Materials 495 (2017) 475e483

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Atomistic simulations and experimental measurements of helium nano-bubbles in nickel E. Torres a, *, C. Judge a, H. Rajakumar a, A. Korinek b, J. Pencer a, G. Bickel a a

Canadian Nuclear Laboratories, Chalk River Laboratories, Chalk River, ON K0J1J0, Canada Canadian Centre for Electron Microscopy, Brockhouse Institute for Materials Research, McMaster University, 1280 Main Street West, Hamilton, ON L8S4L8, Canada b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 15 June 2017 Received in revised form 30 August 2017 Accepted 30 August 2017 Available online 4 September 2017

Deterioration of nickel-based alloy components in fission reactors is associated with the generation of helium from neutron-induced nuclear reactions. In tests at the macroscopic scale, deleterious effects of helium on the mechanical properties of nickel alloys are observed upon the formation and growth of helium bubbles. In order to enhance the understanding of helium effects in nickel, the properties of helium bubbles can be investigated by atomistic scale simulations. In the present work, we have studied helium bubbles in pure nickel, with diameters from 1.0 to 5.0 nm, using molecular dynamics (MD) simulations. The properties of nano-sized helium bubbles as a function of the helium-to-vacancy ratio are calculated at 600 K. The conditions for helium bubbles in mechanical equilibrium with the nickel matrix are determined. The results from simulations are found to be in good agreement with experimental data. Crown Copyright © 2017 Published by Elsevier B.V. All rights reserved.

Keywords: DFT Molecular dynamics EELS TEM Helium bubbles Irradiation effects Nickel embrittlement

1. Introduction Helium is chemically inert and essentially insoluble in nickel. Its production, through neutron capture and transmutation, and retention in nickel has been observed to cause degradation in material performance. In fission reactors, a significant amount of helium is created from nickel in a two-step reaction 58 Niðn; gÞ and 59 Niðn; aÞ. The production and trapping of helium generated in irradiated structural alloys is also expected to affect components in advanced fission reactors, such as the Canadian SCWR [1], and molten salt reactors [2]. Because of its low solubility in nickel-based alloys, the aggregation of helium into bubbles eventually leads to material damage, as manifested in the deterioration of the mechanical properties of the alloys. An extensive review of helium effects in alloys can be found in Ref. [3] and references therein. The importance of helium effects in metals was recognized early on, e.g., the review by Ullmaier [4], but the observed embrittlement of irradiated metals caused by the formation of helium bubbles continues to be an

* Corresponding author. E-mail address: [email protected] (E. Torres). http://dx.doi.org/10.1016/j.jnucmat.2017.08.044 0022-3115/Crown Copyright © 2017 Published by Elsevier B.V. All rights reserved.

active area of research [3,5]. An understanding of the fundamental mechanisms of helium-embrittlement and the conditions in which material degradation occurs is important in the development of innovative solutions to the helium degradation effects in nickel alloys. Considerable experimental effort has been devoted to understand the behavior of helium in austenitic steels and nickel alloys [3,6]. In experiments, nanoscopic helium bubbles can be directly observed and recorded using electron energy loss spectroscopy (EELS) and scanning transmission electron microscopy (STEM) [3,6,7]. Subsequently, associated properties of helium bubbles can be characterized from the measured bubble diameters and EELS maps of helium. In atomistic simulation studies, to investigate more complex helium effects, an accurate description of the properties of helium atom impurities and small clusters in the metal lattice is required. Basic properties, such as the solubility, the barrier for migration, trapping sites, and the structure of clusters, are determined by the behavior of helium atoms at different sites in perfect and imperfect lattices. While properties and formation of helium clusters in nickel have been investigated using atomistic simulations [8e10], there are no satisfactory atomistic scale studies of helium bubbles in nickel reported so far. The majority of previous atomistic models of

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helium in nickel have relied on early electronic structure calculations [11], used with a few developed helium-nickel interatomic potentials [12e14]. However, the interatomic potentials used in these studies were later shown to be inaccurate, based on comparisons with recent density functional theory (DFT) studies [10,15,16]. In more recent theoretical investigations, significant efforts have been devoted to develop a reliable He-Ni interatomic potential [10,17]. Despite the fact that recently reported interatomic potentials better represent some of the energetics of helium impurities in nickel, there are still discrepancies with fundamental details of helium-nickel interactions obtained from DFT electronic structure calculations. The description of the bubble-metal interface in atomistic simulations mainly depends on the helium and nickel interactions. In addition, the properties at the interior of helium bubbles are in part determined by the structure at the bubble interface. Consequently, inaccuracies in the currently available He-Ni potentials make them unsuitable for the modeling of helium bubbles in nickel. In this work, a molecular dynamics study of nano-size helium bubbles in nickel is presented. An improved He-Ni interatomic potential based on a recently proposed potential function [10] is derived. The revised He-Ni potential accurately reproduces the formation energies of single helium defects in nickel as determined from DFT calculations. The improved interatomic potential is used in MD simulations of helium nano-bubbles in nickel. A systematic characterization of size-dependent properties of bubbles with diameters from 1.0 to 5.0 nm is performed. Helium bubbles in mechanical equilibrium conditions are determined from the analysis of the helium-to-vacancy ratio. The properties of bubbles under equilibrium conditions calculated from MD simulations are compared to experimentally determined data. The reported experimental and simulation results are found to be in good agreement.

were left free to relax in total energy and structure optimizations. A calculation of the reference state energy for a helium atom in the gas phase was performed in a cubic supercell of 20 Å in each Cartesian direction. This supercell size was found sufficient to eliminate self-interactions, therefore ensuring substantial accuracy. The calculation only included the G-point, while spin-polarization was not considered. Molecular dynamics simulations were carried out with the large-scale atomic/molecular massively parallel simulator (LAMMPS) code [22] including the GPU accelerator package [23]. Simulations were performed with periodic boundary conditions imposed on all Cartesian directions. The Ni-Ni interactions were described using the potential of Bonny et al. [24]. This potential was chosen because it accurately reproduces point defects in nickel. The He-He interactions were modeled using the Beck potential [25]. This potential reproduces the second virial coefficient for helium, and is therefore adequate for the study of helium bubbles in this work. The He-Ni interactions are described using the interatomic potential developed here. The interstitial and substitutional sites in fcc nickel are indicated in Fig. 1. The formation energies for a single He atom as an interstitial (Eint ), at the octahedral (oct), tetrahedral (tet) or crowdion (crd) interstitial sites, is given by

2. Computational methods

Esub=f ¼ EðNi þ Hen VÞ  ½ðN  1Þ=N EðNiÞ  nEðHeÞ;

First principles total energy calculations were performed using DFT, within the generalized-gradient approximation (GGA), as implemented in the PWscf electronic-structure code distributed with the Quantum ESPRESSO integrated suite [18]. The exchange and correlation energy contributions were described by the Perdew-Burke-Ernzerhof (PBE) functional [19]. The electronic wavefunction was expanded in a plane-wave basis set determined by the kinetic-energy cut-off of 950 eV. The Brillouin zone integration was performed on an 888 Monkhorst-Pack k-point mesh [20]. A first-order Methfessel-Paxton electron smearing scheme with parameter s ¼ 0:2 eV was found suitable for structure optimizations. Spin polarization was considered for pure nickel and with helium impurities. Total energy results were converged up to an error threshold of 0.01 meV per atom. Parameters used in DFT calculations were determined from the convergence of defect formation energies within 1.0 meV of accuracy. Zero-point energy contributions were not included in calculations of the defect formation energies. However, a contribution of  0:01 eV has been estimated [16]. Density functional theory calculations of defect free bulk nickel, and including solute helium atoms, were performed using a 333 periodic supercell of the conventional fcc unit cell of nickel. The defect free nickel supercell contains a total of 108 atoms. Nickel atoms were described with a scalar relativistic ultrasoft pseudopotential, including a non-linear core correction, while helium atoms were described by a non-relativistic ultrasoft pseudopotential [21]. The initial configurations of the magnetic moments on nickel atoms were provided to start the calculations in the ferromagnetic (fm) ordered state. However, the magnetic moments

where EðNi þ Hen VÞ is the total energy of the nickel supercell with n helium atoms in a nickel vacancy (V). For the particular case of a substitutional helium atom n ¼ 1. The parameters of the He-Ni interatomic potential were determined based on a two-step MD optimization and validation procedure, performed using supercells with dimensions of 333 and 666 lattice units, respectively. The validation was performed using DFT results as the reference data. The formation energies of interstitial and substitutional helium defects were determined using the Polak-Ribìere formulation of the conjugate gradient method, as implemented in the cfg module in LAMMPS. A force tolerance of 1:0  106 eV/Å was used in all energy optimizations. Molecular dynamics simulation of helium bubbles were performed in a cubic nickel supercell of 404040 lattice units,

Eint ¼ EðNi þ Heint Þ  EðNiÞ  EðHeÞ;

(1)

where EðNi þ Heint Þ is the total energy of bulk nickel with an interstitial helium atom impurity, EðNiÞ is the energy of perfect bulk nickel with a total of N atoms and EðHeÞ is the energy of an isolated helium atom. The formation energy of a helium atom as a substitutional (Esub ) impurity or a helium cluster (Ef ) in a single nickel vacancy is approximated as

(2)

Fig. 1. Substitutional (1) and interstitial octahedral (2), tetrahedral (3), and crowdion (4) helium impurity sites are indicated with solid black circles. Nickel atoms in the conventional fcc unit cell are represent by gray spheres. Gray colored walls indicate the octahedral (left panel) and tetrahedral (right panel) volumes.

E. Torres et al. / Journal of Nuclear Materials 495 (2017) 475e483

including a total of 256000 nickel atoms. At the temperature of T ¼ 0 K the supercell has 14.1 nm in each Cartesian direction. After equilibration at 600 K, the supercell increased to an average size of 14.3 nm in each coordinate direction. The calculated lattice parameter of nickel, at 600 K and external pressure of zero bar, is 3.572 Å. The supercell size was found appropriate to incorporate helium bubbles in the nanometer scale. Helium bubbles were constructed inside the nickel matrix using a two-step procedure. In the first step, a spherical shaped vacancy cluster is generated by removing nickel atoms localized within a given radius. The center of the sphere was positioned on a nickel atom in the middle of the supercell. In the second step, helium atoms are placed in random positions inside the spheroidal void volume. The numerically created random helium atom distribution may therefore incorporate atoms in close contact. In order to remove high force components inside a newly generated helium bubble, a preliminary structure relaxation was carried out using the conjugate gradient method. The energy and force tolerance of 1:0  1014 (unitless) and 1:0  1014 eV/Å were applied, respectively. The temperature, volume and the pressure, of the constructed nickel-helium systems, were initially equilibrated during 10 ps of simulation time using a Langevin thermostat and Berendsen barostat in conjunction with the microcanonical (NVE) ensemble. After switching to the Nose-Hoover thermostat-barostat (NPT) ensemble, the temperature, pressure and volume of the preequilibrated system were conserved. This clearly indicates that a significant reduction of the total equilibration time is obtained by pre-equilibrating the system. The total simulation time of 200 ps within the NPT ensemble, using a time-step of 0.2 fs, was found sufficient to obtain steady state results of helium bubble properties. The time-average pressure in a helium bubble was approximated as


 1 P ¼  sxx þ syy þ syy ; 3

(3)

with saa the diagonal components of the stress tensor [26] calculated in LAMMPS using the following expression

2 3 NU X 1 X 1 4 sab ¼ F 5r þ mi vi 5vi 5; U i 2 jsi ij ij

(4)

where U is the Voronoi [27] bubble volume confining NU helium atoms, Fij is the force on atom i due to atom j, rij is the position vector of atom j relative to atom i, mi is the mass of atom i, vi is the velocity of atom i and the 5 symbol denotes the tensor product. The change in supercell volume DV was calculated by

DV ¼ VNiþHe ðTÞ  VNi ðTÞ;

477

300 kV, equipped with a CEOS image and probe corrector (CEOS GmbH, Heidelberg, Germany). The microscope is equipped with a Gatan Quantum GIF (Gatan Inc., Pleasanton, CA). The dwell time for each pixel was 0.01 s. The GIF entrance aperture was 5 mm. The collection semi-angle was approximately 40 mradians, and the convergence semi-angle was 19 mradians. The pixel size in the object plane was set to 0.15 nm. The probe size of the microscope is < 0:1 nm. The thickness of the focused ion beam (FIB) section was approximately 20 nm. The resulting data was analyzed using Gatan Digital Micrograph 4.0 software. The helium is quantified by extracting the 1s/2p transition edge at  22 eV. Note that pressurized bubbles shift the 1s/2p transition to higher energies, so signal extraction is done specifically for each bubble to quantify the atomic helium [30e35]. A second order log polynomial fit with a hydrogenic cross section for helium was used to isolate Helium signals from the plasmon and background signals. The obtained EELS maps showed residual helium content outside of physical bubbles. Using this, locally, as a background estimate for helium, the atomic densities from individual bubbles were further corrected by extracting the atomic helium density from the bulk material. The atomic densities were extracted from the areal densities using an EELS thickness map, the material, and accounted bubble sizes (assuming spherical bubbles).

4. Results and discussion The ground state bulk properties of nickel, as determined from DFT/PBE calculations, are the lattice constant a0 ¼ 3:52 Å, cohesive energy Ece ¼ 4:84 eV and magnetic moment m ¼ 0:63 mB . These values are in good agreement with DFT studies by other research groups [15,16,36], and experimentally measured values [37]. Formation energies from DFT and MD calculations of single helium atoms in nickel, as interstitial impurities and substitutional defects, are summarized in Table 1. Formation energies (c), (d) and (e) in Table 1, were calculated from molecular dynamics simulations using the Morse [12], piece-wise [17] and Morse-2G potentials [10], respectively. A detailed comparison of previously reported He-Ni interatomic potentials can be found in Ref. [10]. In addition, formation energies listed in Table 1(f) were obtained with MD simulations using the He-Ni interatomic potential developed in this work (vide infra). From the comparison of the results provided in Table 1, shortcomings in MD simulations to correctly describe energies of single helium atoms in nickel, as obtained with DFT, are revealed. In particular, molecular dynamics simulations, based on previously reported He-Ni interatomic potentials, fail to reproduce the relative stability of interstitial energies as obtained in DFT calculations. For

(5)

where VNi is the supercell volume with defect free bulk nickel and VNiþHe is for a nickel supercell with a helium bubble, and T is the temperature of system.

3. Experimental methods Inconel X-750, irradiated up to 80 dpa in a high thermal flux at 300e330  C and 25000 appm helium were characterized with Electron Energy-Loss Spectroscopy (EELS). The details of the neutron irradiation experiments can be found in Ref. [28]. This material has been shown to have a very high density of bubbles within the grain interior ranging in size from 1 to 10 nm [29]. EELS elemental maps of helium bubbles were acquired using an FEI Titan3 TEM (FEI Company, Eindhoven, Netherlands), operated at

Table 1 Formation energies for substitutionally (Esub ) and interstitially, tetrahedral (Etet ), octahedral (Eoct ) and crowdion (Ecrw ), sited helium atom in nickel. Energies are in eV. Method

Esub

Etet

Eoct

Ecrw

DFTa DFTb MDc MDd MDe MDf

3.318 3.185 2.722 2.908 3.733 3.607

4.577 4.460 4.520 4.499 4.501 4.483

4.735 4.589 3.928 4.653 4.616 4.684

4.769g 4.651 4.612 4.613 4.572 4.725

a b c d e f g

DFT/PBE Ref. [10]. DFT/PW91 Ref. [16]. Using the Morse potential Ref. [12]. Using Zhang-HeNi potential Ref. [17]. Using the Morse-2G potential Ref. [10]. Using the Morse-3G potential in this work. Value calculated using DFT/PBE in this work.

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instance, the Morse potential, derived from the early results by Melius et al [11] and used in MD investigations [9,12,38], incorrectly gives the octahedral rather than tetrahedral site as the minimum energy interstitial location for helium. More recently reported HeNi interatomic potentials [10,17] reproduce interstitial energies in better agreement with DFT. However, as shown in Table 1, the formation energy of a helium interstitial atom in a crowdion site is underestimated, and is predicted to be more stable than the octahedral site, in clear disagreement with DFT results. An accurate description of He-Ni interactions is essential for a realistic description of the helium-nickel interface in MD simulations. Thus, the observed discrepancy between molecular dynamics models and DFT have to be addressed. In order to accurately describe formation energies of helium atom impurities in nickel, we opted to develop a new He-Ni interatomic potential based on our recently proposed pair potential function [10]. The general expression of the pair potential function has the following form: m h i X 2 EðrÞ ¼ D0 e2aðrr0 Þ  2eaðrr0 Þ  Ai eBi r ;

(6)

i

where the first term is a Morse potential, the second term is a linear combination of Gaussian functions, and m determines the number of Gaussian functions. This potential function, referred hereafter as Morse-mG, is both continuous and differentiable everywhere. We modified the LAMMPS source code to explicitly include the above potential function. The Morse-mG potential function was shown to effectively capture the medium-range interactions while still correctly describing the near- and long-range asymptotic behaviors, therefore preserving the transferability of the potential to different atomic configurations. The parameter optimization was performed using DFT reference data, and the details of the procedure are outlined in Ref. [10]. It was necessary to extend m ¼ 2 to m ¼ 3 in order to achieve accurate results. Therefore, the He-Ni interatomic potential developed in this work is referred as Morse-3G. The parameters, defining the Morse-3G potential, obtained by the optimization routine, are listed in Table 2. At the cut-off distance, the potential reaches 4106 eV. In molecular dynamics, the computational cost substantially increases with the cut-off radius of the potential. As such, it is very advantageous to use interatomic potentials with small cutoff radii in MD simulations. Table 1(f) summarizes MD formation energies of helium impurities calculated using the Morse-3G potential. The energies obtained with the Morse-3G potential (see Table 1(f)) are in very good agreement with DFT results, and therefore, the Morse-3G potential is shown to be suitable for simulations of helium bubbles in nickel. In Fig. 2, the potential energy surfaces describing He-Ni interactions as described by the Morse-3G potential, along with the Morse-2G potential [10] and the early Morse interatomic potential by Xia et. al [12], are superimposed for comparison. The three potentials in Fig. 2 show the same long range asymptotic behavior. The Morse-3G and Morse-2G behave similarly in the range of

Table 2 Parameters for the Morse and Gaussian (Gi ) functions in the Morse-3G He-Ni interatomic pair potentials. rc is the pairwise cut-off radius. Morse

G1 G2 G3

D0 ðeVÞ 0.000276

a 1.386787

AðeVÞ

B

20.893039 7.606549 49.713764

1.122149 0.604257 2.085666

r0 ð
AÞ 4.113218

rc ð
AÞ 4.11 AÞ rc ð
4.11 4.11 4.11

Fig. 2. Potential energy surfaces of He-Ni interatomic potentials. (open circles) The optimized Morse-3G developed in this work. (open diamonds) The Morse-2G from Ref. [10]. (open squares) The Morse from Ref. [12].

medium energies, while the Morse-2G and Morse potentials differ considerably in the short range interactions. The behavior of the Morse-2G potential in the short range was found to be the source for the discrepancy with results from DFT calculations. This issue was addressed by including an additional Gaussian. (G3 in Table 2) function in the Morse-2G potential to allow fine-tuning of short range interaction energies. As shown in Table 1(f), use of the resulting interatomic potential eliminated the discrepancies between DFT and MD formation energies. Formation energies of helium clusters from n ¼ 1 to 13 atoms, calculated with MD/Morse-3G and DFT using a nickel supercell of 333 units, are listed in Table 3. The energies obtained using the Morse-2G and DFT results with a 222 supercell from Ref. [10] are also included for comparison. The significant differences in formation energies in DFT results for n  6 were found to arise from spurious supercell size effects. In particular, calculations performed in a 222 supercell exhibit a large reduction of the magnetic moments of the nickel atoms as the cluster size increases, and eventually becoming zero for clusters with n ¼ 10 or larger size. The decrease of the magnetic moment is not significant in

Table 3 Formation energy (Ef ) for the most stable helium clusters Hen from n ¼ 1 to 13 helium atoms. Energies are in eV. n

MDa

DFTb

MDc

DFTd

1 2 3 4 5 6 7 8 9 10 11 12 13

3.73 7.21 10.53 13.77 17.27 20.60 24.25 27.75 31.35 34.85 38.19 41.52 44.70

3.36 6.38 9.40 12.35 15.72 18.88 22.58 26.33 30.08 34.94 38.93 42.59 46.39

3.61 7.08 10.36 13.77 17.01 20.34 24.00 27.41 31.06 34.57 37.92 41.16 44.34

3.32 6.30 9.24 12.04 15.18 18.20 21.46 24.71 28.03 31.69 35.05 38.03 41.16

a b c d

Morse-2G DFT using Morse-3G DFT using

He-Ni potential [10]. a 222 supercell [10]. He-Ni potential. a 333 supercell.

E. Torres et al. / Journal of Nuclear Materials 495 (2017) 475e483

calculations using a 333 supercell. Therefore, the DFT calculations using the larger supercell significantly reduce size effects and provide more accurate reference data. Comparison of the results presented in Table 3 clearly show that the Morse-3G potential better reproduces the formation energies of helium clusters as determined from DFT calculations, and it is therefore suitable for larger scale simulations. Molecular dynamics simulations of helium bubbles at 600 K were performed using the Morse-3G potential. Fig. 3 shows a cross section of the nickel supercell including a 5.0 nm diameter helium bubble. In Fig. 3, the distance between two nearest neighboring helium bubbles images is  10:0 nm. Therefore, the size of the supercell is adequate for simulations of the helium bubble sizes of interest in this work. We investigate helium bubbles with size and temperature conditions within a range observed in experiments [5,7]. However, in order to keep the bubble size appropriate within the simulated supercell, only bubbles with diameters between 1.0 and 5.0 nm are considered. The number of nickel atoms (VNi ) removed only depends on the required bubble diameter, Db . Table 4 summarizes the bubble diameters considered in this work along with the respective number of nickel atoms removed in each case. The number of nickel vacancies, in the performed MD simulations, is strictly determined by the initial bubble size. Under this condition, the characteristics and properties of the bubbles primarily depend on the number of helium atoms trapped in them. Therefore, the behavior of the nickel-helium system as a function of the helium-to-vacancy ratio (He/VNi ) was initially examined. We investigated the effect on the supercell and the bubble properties for values of He/VNi in the range from 0.2 to 1.2. In general, in the investigated He/VNi range, simulation time and temperature, we did not observe He atom emissions from the bubble into the nickel lattice, or the formation of nickel self-interstitial defects. Therefore, the structure and the properties of the helium bubbles were stable under the simulation conditions. The change in supercell volume, bubble pressure and helium density were characterized in terms of He/VNi to analyze their relationship. In Fig. 4, the results for the investigated bubble diameters are shown. As can be seen from Fig. 4(a), for each helium bubble size there is a particular He/VNi value, between 0.4 and 0.6, for which no significant change of the supercell volume is observed. Consequently, helium bubbles with He/VNi within this range do not produce a significant stress field in the nickel lattice. This finding indicates

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Table 4 Number of nickel atoms (VNi ) deleted as function of the sphere diameter (Db ). Db (nm) VNi

1.0 55

1.5 177

2.0 381

3.0 1289

4.0 3055

5.0 5979

Fig. 4. (a) Volume change of the supercell, (b) bubble pressure and (c) helium density as a function of the helium-to-vacancy ratio (He/VNi ).

Fig. 3. A sliced cross section of the model system of a Db ¼ 5.0 nm helium bubble in nickel. Helium and nickel atoms are colored in blue and gray, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

that material swelling is evidence of the formation of large helium bubbles with high He/VNi values, and therefore they may be in overpressurized conditions. In fact, the existence of overpressurized bubbles have been reported in experimental findings [39]. When no stress field is produced in the supercell, the helium bubble is said to be in mechanical equilibrium with the nickel lattice and behave as a neutral object in it. Therefore, by monitoring the change in volume with the He/VNi ratio, the helium concentration at which the bubbles is in mechanical equilibrium conditions can be determined by MD simulations. In Fig. 4(b), the pressure in the bubbles is plotted for various bubble sizes as a function of the He/VNi ratio. In general, the pressure increases with the increase of He/VNi . Fig. 4(b) shows that the bubble pressure depends on both the ratio He/VNi and the bubble size. The highest pressure is characteristic of small bubbles, independent of He/VNi . This behavior was observed in experiments by

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Trinkaus [6]. The high compressibility of small bubbles results in negligible changes of the total supercell volume as a response to the variation of the He/VNi ratio. The large packing of helium atoms in small bubbles suggests that a significant amount helium can be stored in the nickel lattice without a significant change in volume, as has been suggested from experiments [40]. The helium densities obtained from simulations are plotted as a function of the He/VNi ratio, in Fig. 4(c), for various bubbles sizes. The differences in helium densities are relatively large for small He/ VNi values, and become less dependent on the bubble diameter as the He/VNi ratio increases. This is indicated by the convergent trend in Fig. 4(c). The small difference in the density curves for Db ¼ 4.0 and 5.0 nm suggests a weak effect of the helium-metal interface at the interior of large bubbles. In experiments, the presence of bubbles with high helium densities may indicate the presence of overpressurized bubbles. Stress fields, originating from overpressurized bubbles, may significantly impact the mechanical performance of the host metal. As has been explained above, in order to assess the conditions at which helium bubbles are in mechanical equilibrium, we searched for the number of helium atoms (NHe ) at which the supercell volume with a helium bubble is identical to that of bulk single crystal fcc nickel. In practice, the equilibrium conditions were determined within a volume difference of (0:002%. The NHe values, and the associated He/VNi ratios, for helium bubbles in mechanical equilibrium are listed in Table 5. We found that the He/VNi ratio increases as the bubble diameter decreases. The calculated pressure and helium density for bubbles in equilibrium as a function of the diameter are presented in Fig. 5, both quantities were found to monotonically decrease with the increase of the bubble diameter. In the discussion above, the helium densities were calculated assuming a homogeneous distribution of helium atoms inside the bubble volumes. However, variations in the spatial distribution of helium atoms may arise from size effects and interactions at the nickel metal interface. In order to investigate these potential effects, we have calculated the atomic and areal helium density profiles across the bubbles. The atomic and areal helium densities, shown in Fig. 6, were obtained from the average of 24 independently simulated helium bubbles. For each bubble a total of 1000 snapshots were stored during 200 ps simulation time. The error bars of helium densities from simulations are very small and therefore are not visible, indicating a high confidence and reproducibility in the obtained results. Significant differences in atomic helium densities can be directly observed in Fig. 6, therefore, indicating a substantial effect of the bubble size on the helium density variation. Nevertheless, the calculated atomic densities are in good agreement with experimental results at low temperature ( 773 K) reported in Ref. [7]. The atomic helium densities of various bubble radii, from the center of the bubble outward, are presented in Fig. 6(top). A significant impact of the bubble size on the atomic helium densities can be observed and becomes more prominent as the bubble diameter decreases. A characteristic local maximum density peak at the bubble-metal interface, in the range approximately 0.1e0.3 nm from zero, shows discernible size effects. Moreover, the strongly non-uniform distribution of He atoms in small bubbles (Db  2:0 nm) indicates a transition from helium in gas phase to the

Table 5 Determined number of helium atoms (NHe ) and helium-to-vacancy ratio (He/VNi ), as a function of the diameter (Db ), for bubbles in mechanical equilibrium. Db (nm) NHe He/VNi

1.0 33 0.60

1.5 100 0.56

2.0 204 0.54

3.0 635 0.49

4.0 1416 0.46

5.0 2640 0.44

Fig. 5. Pressure (top) and helium density (bottom) for bubbles in mechanical equilibrium.

Fig. 6. Atomic helium density (top) and areal helium density (bottom) from molecular dynamics simulations of bubbles in mechanical equilibrium.

conformation of helium clusters as the bubble size decreases. The density at the interior of large bubbles (not including the bubblemetal interface) is virtually independent of the radius, indicating that inside large bubbles helium atoms are homogeneously distributed. The size dependence of the density peak at the bubblemetal interface indicates that He atom behavior in the outermost region of the bubble is dominated by the He-Ni interactions. Therefore, the structure of the bubble-metal interface is

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determined by the He-Ni interatomic potential. As such, the physical validity of the He-Ni potential plays an important role in correctly describing helium bubbles. The calculated areal densities are plotted in Fig. 6(bottom). The effects of the bubble size can be also observed in the areal densities. Therefore, it is expected that spatial variations in helium density can be observed experimentally. The helium atomic and areal densities directly obtained from EELS experimental data are shown in Fig. 7. Each density curve in Fig. 7 was obtained from the statistical analysis of 24 different helium bubble EELS maps. The relatively small error bars in Fig. 7 provide a good indication of the accuracy of the measurements. Fig. 8 shows the distribution helium bubbles in a Fresnel contrast image at a focus of 300 nm in bright field using a FEI Titan3 TEM operated at 300 kV. The sample was irradiated at 300 + C to  80 dpa and 25000 appm helium, and prepared with a FIB to a nanoscale lamellae. Interestingly, the experimental atomic helium density curves in Fig. 7(top) exhibit a helium density peak characteristic of the helium-nickel interface, also observed in results from simulations. However, the helium density peak is not clearly visible for the 3.0 nm bubble. Furthermore, the atomic helium densities for the 2.0 nm bubble clearly show large oscillations in comparison to larger bubbles. This behavior is also observed in simulations and supports the identification of a plausible phase transition from helium gas to cluster structures as the bubble diameter decreases. The atomic densities at the interior of the investigated bubbles are in the range from 10 to 30 He/nm3. Areal densities plots, presented in Fig. 7(bottom), show a steady increase from the bubble-nickel interface to the interior until it reaches a nearly constant value at the center, following a similar trend with the bubble size as observed in simulations. Fig. 9 shows the experimental and simulated areal density of a 5 nm helium bubble for comparison. The reported theoretical and experimental results for helium bubbles in nickel are in good qualitative agreement. Nevertheless, discrepancies between the helium densities from simulations and

Fig. 7. Atomic helium density (top) and areal helium density (bottom) from EELS experiments at room temperature.

481

Fig. 8. Size distribution of helium bubbles in nickel observed in a Fresnel contrast image. Helium bubbles appear as bright areas enclosed by nearly circular dark boundaries.

Fig. 9. Experimental (top) and simulated (bottom) areal helium densities for a 5 nm bubble. The color gradient from blue to red correspond to helium densities from low to high. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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experiments can be also observed. Features in the experimentally determined areal density, not observable in the simulated density as seen in Fig. 9, indicate an influence of the bubble environmental conditions in the irradiated samples that is not represented in the simulations. The source of the differences is still a matter of investigation. Differences in metal composition (e.g. pure nickel in the simulations versus X-750 in the experiments), or the presence of hydrogen could contribute to the observed differences [5]. Inclusion of these details could improve agreement between theoretical predictions and experimental observations in irradiated nickel-based alloys. 5. Summary and conclusions Molecular dynamics simulations based on previously reported He-Ni interatomic potentials cannot reproduce the relative stability of helium defects in nickel. We have parameterized a helium-nickel interatomic potential, the Morse-3G potential, to be used in conjunction with the nickel-nickel potential by Bonny et al., and have demonstrated the applicability to the modeling of interstitial and substitutional helium defects in nickel. The MD results using the Morse-3G are in good agreement with DFT results, thus providing an alternative to the available He-Ni potentials. The Morse-3G potential was employed in MD simulations to study nano-sized helium bubbles in nickel. We have investigated the effect of the helium-to-vacancy ratio on the properties of the bubbles. It was demonstrated that the helium-to-vacancy ratio plays a decisive role in modulating the bubbles properties. Overpressurized bubbles, with high helium-to-vacancy ratio, lead to the generation of substantial stress fields and may result in material swelling. The conditions for helium bubbles in mechanical equilibrium with the nickel lattice were determined. An increase in the packing fraction of helium atoms and pressure was observed as the bubble diameter decreases. As the bubble size decreases, the helium atoms show a transition from helium in gas phase to a more ordered cluster-like structures. The calculated helium densities in the bubbles in mechanical equilibrium are found in good agreement with experimental findings. The dependence of the bubble structure with the size and the bubble size distribution may impact the mechanical properties of nickel. Therefore, the presented computational results provide an important foundation for theoretical and experimental studies of size and distribution of helium bubbles in Ni-based alloys. Acknowledgments The authors acknowledge the Government of Canada for providing funding for this work through Atomic Energy of Canada's Federal Science and Technology Program. References [1] D. Guzonas, R. Novotny, Supercritical water-cooled reactor materials Summary of research and open issues, Prog. Nucl. Energy 77 (2014) 361e372. [2] J. Serp, M. Allibert, O. Benes, S. Delpech, O. Feynberg, V. Ghetta, D. Heuer, D. Holcomb, V. Ignatiev, J.L. Kloosterman, L. Luzzi, E. Merle-Lucotte, J. Uhlír, R. Yoshioka, D. Zhimin, The molten salt reactor (MSR) in generation IV: overview and perspectives, Prog. Nucl. Energy 77 (2014) 308e319. [3] Y. Dai, G.R. Odette, T. Yamamoto, 1.06-The effects of helium in irradiated structural alloys, in: R.J.M. Konings (Ed.), Comprehensive Nuclear Materials, Elsevier, Oxford, 2012, pp. 141e193. [4] H. Ullmaier, The influence of helium on the bulk properties of fusion reactor structural materials, Nucl. Fusion 24 (8) (1984) 1039. [5] C.D. Judge, N. Gauquelin, L. Walters, M. Wright, J.I. Cole, J. Madden, G.A. Botton, M. Griffiths, Intergranular fracture in irradiated Inconel X-750 containing very high concentrations of helium and hydrogen, J. Nucl. Mater 457 (2015) 165e172. [6] H. Trinkaus, Energetics and formation kinetics of helium bubbles in metals, Radiat. Eff. 78 (1e4) (1983) 189e211.

[7] W. Qiang-Li, H. Kesternich, D. Schroeder, H. Schwahn, Ullmaier, Gas densities in helium bubbles in nickel measured by small angle neutron scattering, Acta Metall. Mater 38 (12) (1990) 2383e2392. [8] J. Adams, W. Wolfer, Formation energies of helium-void complexes in nickel, J. Nucl. Mater 166 (3) (1989) 235e242. [9] H. Gong, C. Wang, W. Zhang, J. Xu, P. Huai, H. Deng, W. Hu, The energy and stability of helium-related cluster in nickel: a study of molecular dynamics simulation, Nucl. Instr. Meth. Phys. Res. B 368 (2016) 75e80. [10] E. Torres, J. Pencer, D.D. Radford, Density functional theory-based derivation of an interatomic pair potential for helium impurities in nickel, J. Nucl. Mater 479 (2016) 240e248. [11] C.F. Melius, C.L. Bisson, W.D. Wilson, Quantum-chemical and lattice-defect hybrid approach to the calculation of defects in metals, Phys. Rev. B 18 (4) (1978) 1647e1657. [12] J. Xia, W. Hu, J. Yang, B. Ao, X. Wang, A comparative study of helium atom diffusion via an interstitial mechanism in nickel and palladium, Phys. Status Solidi B 243 (3) (2006) 579e583. [13] M.I. Baskes, C.F. Melius, Pair potentials for fcc metals, Phys. Rev. B 20 (8) (1979) 3197e3204. [14] A. Adams, W. Wolfer, On the diffusion mechanisms of helium in nickel, J. Nucl. Mater 158 (1988) 25e29. [15] X.T. Zu, L. Yang, F. Gao, S.M. Peng, H.L. Heinisch, X.G. Long, R.J. Kurtz, Properties of helium defects in bcc and fcc metals investigated with density functional theory, Phys. Rev. B 80 (5) (2009) 054104. [16] D.J. Hepburn, D. Ferguson, S. Gardner, G.J. Ackland, First-principles study of helium, carbon, and nitrogen in austenite, dilute austenitic iron alloys, and nickel, Phys. Rev. B 88 (2) (2013) 024115. [17] W. Zhang, C. Wang, H. Gong, Z. Xu, C. Ren, J. Dai, P. Huai, Z. Zhu, H. Deng, W. Hu, Development of a pair potential for NiHe, J. Nucl. Mater 472 (2016) 105e109. [18] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G.L. Chiarotti, M. Cococcioni, I. Dabo, A. Dal Corso, S. de Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A.P. Seitsonen, A. Smogunov, P. Umari, R.M. Wentzcovitch, Quantum espresso: a modular and open-source software project for quantum simulations of materials, J. Phys. Condens. Matter 21 (39) (2009) 395502 (19pp). [19] J.P. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77 (18) (1996) 3865e3868. [20] H.J. Monkhorst, J.D. Pack, Special points for brillouin-zone integrations, Phys. Rev. B 13 (12) (1976) 5188e5192. [21] We used the pseudopotentials ni.pbe-n-rrkjus_psl.0.1.upf and he.pbe-mt_ fhi.upf from the quantum espresso pseudo-potential database, http://www. quantum-espresso.org. [22] S. Plimpton, Fast parallel algorithms for short-range molecular dynamics, J. Comp. Phys. 117 (1) (1995) 1e19. [23] W.M. Brown, S.J.P.P. Wang, A.N. Tharrington, Implementing molecular dynamics on hybrid high performance computers - short range forces, Comp. Phys. Comm. 182 (2011) 898e911. [24] G. Bonny, N. Castin, D. Terentyev, Interatomic potential for studying ageing under irradiation in stainless steels: the FeNiCr model alloy, Model. Simul. Mater. Sci. Eng. 21 (8) (2013) 085004. [25] D. Beck, A new interatomic potential function for helium, Mol. Phys. 14 (4) (1968) 311e315. [26] A.P. Thompson, S.J. Plimpton, W. Mattson, General formulation of pressure and stress tensor for arbitrary many-body interaction potentials under periodic boundary conditions, J. Chem. Phys. 131 (15) (2009) 154107. [27] C.B. Barber, D.P. Dobkin, H. Huhdanpaa, The quickhull algorithm for convex hulls, ACM Trans. Math. Softw. 22 (4) (1996) 469e483. [28] M. Griffiths, G. Bickel, S. Donohue, P. Feenstra, C. Judge, L. Walters, M. Wright, Degradation of ni-alloy components in candu reactor cores, in: 16th Int. Conference on Environmental Degradation of Materials in Nuclear Power Systemsewater Reactors, 2013. Asheville, North Carolina, USA. [29] C.D. Judge, H. Rajakumar, A. Korinek, G. Botton, J.I. Cole, J.W. Madden, J.H. Jackson, P.D. Freyer, L.A. Giannuzzi, M. Griffiths, High Resolution Transmission Electron Microscopy of Irradiation Damage in Inconel X-750, 2017. Portland, USA. €Ger, R. Manzke, H. Trinkaus, G. Crecelius, R. Zeller, J. Fink, H.L. Bay, [30] W. Ja Density and pressure of helium in small bubbles in metals, J. Nucl. Mater 111 (1982) 674e680. [31] W. J€ aGer, R. Manzke, H. Trinkaus, R. Zeller, J. Fink, G. Crecelius, The density and pressure of helium in bubbles in metals, Radiat. Eff. 78 (1e4) (1983) 315e325. €Ger, R. L€ [32] W. Ja aSser, T. Schober, G.J. Thomas, Formation of helium bubbles and dislocation loops in tritium-charged vanadium, Radiat. Eff. 78 (1e4) (1983) 165e176. chard, M. Walls, M. Kociak, J.P. Chevalier, J. Henry, D. Gorse, Study by [33] S. Fre EELS of helium bubbles in a martensitic steel, J. Nucl. Mater 393 (1) (2009) 102e107. [34] R.F. Egerton, Electron Energy-loss Spectroscopy in the Electron Microscope, Springer Science & Business Media, 2011. [35] C.D. Judge, The Effects of Irradiation on Inconel X-750, Thesis, Department of Materials Science and Engineering, McMaster University, 2015.  Andrieu, D. Monceau, First-principles nickel database: entable, E. [36] D. Conne ergetics of impurities and defects, Comput. Mater. Sci. 101 (2015) 77e87.

E. Torres et al. / Journal of Nuclear Materials 495 (2017) 475e483 [37] C. Kittel, Introduction to Solid State Physics, Wiley, Hoboken, NJ, 2005. [38] C. Wang, C. Ren, W. Zhang, H. Gong, P. Huai, Z. Zhu, H. Deng, W. Hu, A molecular dynamics study of helium diffusion and clustering in fcc nickel, Comput. Mater. Sci. 107 (2015) 54e57. [39] G. Amarendra, B. Viswanathan, A. Bharathi, K.P. Gopinathan, Nucleation and

483

growth of helium bubbles in nickel studied by positron-annihilation spectroscopy, Phys. Rev. B 45 (1992) 10231e10241. [40] D. Preininger, D. Kaletta, The growth of gas-bubbles by coalescence in solids during continuous gas generation, J. Nucl. Mater 117 (1983) 239e243.