MD simulations of primary damage formation in L12 Ni3Al intermetallics

Journal of Nuclear Materials 522 (2019) 123e135

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MD simulations of primary damage formation in L12 Ni3Al intermetallics Roman Voskoboinikov National Research Centre Kurchatov Institute, Kurchatov Square, Moscow, 123182, Russia

h i g h l i g h t s
Collision cascades in Ni3Al were modelled over a wide temperature and PKA energy range.
The number of Frenkel pairs and anti-sites are found as a function of (EPKA , T).
Collision cascades initiated by Al PKAs generate more radiation damage.
Collision cascades initiated by Ni PKAs generate more disorder.
Preferred formation of Ni SIAs was observed under all temperatures and PKA energies.

a r t i c l e i n f o

a b s t r a c t

Article history: Received 17 October 2018 Received in revised form 4 May 2019 Accepted 6 May 2019 Available online 11 May 2019

Computer modelling by molecular dynamics (MD) was employed to study radiation damage created in collision cascades in L12 Ni3Al intermetallic compound. Either Al or Ni primary knock-on atoms (PKA) with PKA energy 5 keV  EPKA  20 keV were initiated in the intermetallic crystals at ambient temperatures 100 K  T  1200 K. At least 24 different cascades for each (EPKA , T, PKA type) set were simulated in order to mimic an isotropic spatial and random temporal distribution of PKAs. The total yield of nearly 1000 simulated cascades provides the largest yet reported sampling to get statistically reliable quantitative results. The obtained MD simulation data were analysed to get the number of Frenkel pairs, the fraction of point defects in point defect clusters, the fraction of Al and Ni vacancies, self-interstitial atoms (SIA) and anti-sites created in collision cascades as a function of (EPKA , T, PKA type). It was found that under identical simulation conditions, collision cascades in L12 Ni3Al initiated by Al PKAs generate more radiation damage, whereas collision cascades initiated by Ni PKAs produce more chemical disorder. Preferred formation of Ni SIAs in collision cascades in Ni3Al was observed under all ambient temperatures and PKA energies. © 2019 Elsevier B.V. All rights reserved.

Keywords: L12 Ni3Al intermetallics Primary damage Collision cascades Molecular dynamics Frenkel pairs Anti-sites

1. Introduction Alongside with the reliability, nuclear waste management, longer residence time of nuclear fuel and its extended burn-up, higher operating temperatures determine the sustainable economic and environmental perspectives of nuclear power. The increase of the operating temperature of future nuclear power plants requires refractory structural materials capable of withstanding simultaneous impact of external loading, chemically active/oxidation environment, high thermal flux and fast particle irradiation without exhibiting significant degradation of service properties over an extended time period.

E-mail address: [email protected]. https://doi.org/10.1016/j.jnucmat.2019.05.009 0022-3115/© 2019 Elsevier B.V. All rights reserved.

Laves phases and Ni3Al intermetallic precipitates are key strengthening components in a number of existing and developing austenitic and martensitic steels with high aluminium content and advanced Ni-based refractory alloys qualified for use in neutron irradiation environment. The high-temperature creep resistance of these materials is either determined or at least significantly improved by intermetallic precipitation strengthening, and degradation of their service properties due to radiation-induced decomposition and dissolution of Ni3Al precipitates could have harmful effects. Phase stability and order-disorder phase transformations in NieAl alloys exposed to irradiation have been extensively studied both experimentally [1e9] and theoretically [10e13] over several decades. In order to gain a deeper insight into ballistic effects, primary damage formation, disordering kinetics, amorphisation,

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etc. in intermetallic compounds subjected to fast particle irradiation in the nuclear stopping power regime, a number of atomicscale modelling studies have been conducted as well, see e.g. [14e32]. In the undertaken researches, primary attention was paid to radiation-induced disordering and dissolution of Ni3Al precipitates in the surrounding nickel matrix, whereas formation of radiation defects was very much overlooked. Moreover, narrow ambient temperature range, low projectile energies, small or unknown sampling, no analysis of projectile mass and stopping power effects and element partition in the implemented modelling programmes make problematic the application of the published quantitative results in a broader context of the development of new breeds of advanced radiation tolerant structural and functional materials with ordered crystal structure. A large programme of primary damage formation in collision cascades in Ni3Al over a wide range of temperature and primary knock-on atom energy is therefore carried out. 2. Problem set Formation of primary radiation defects in materials subjected to fast particle irradiation in the nuclear stopping power regime occurs during relaxation of collision cascades that are initiated by the recoil of primary knock-on atoms (PKA) with energy EPKA T 1 keV. The cascade process is characterised by the temporal and spatial scale of the order of z 2e20 ps and z 5e30 nm, respectively, which is beyond the resolution range of existing experimental equipment but it can be modelled by applying molecular dynamics (MD) simulation techniques. The stochastic nature of primary damage formation phenomena and broad scattering of the outcomes of MD simulations require representative sampling of collision cascades. In order to reach this objective, a series of collision cascades in L12 Ni3Al crystal initiated by PKAs with the same energy at the same ambient temperature T needs to be generated. The necessary size of the sampling set in this research is justified a posteriori by analysing the convergence of the number of residual radiation defects with raising the number of collision cascades in a series with the same (EPKA , T, PKA type). The undertaken study of primary damage formation in L12 Ni3Al exposed to fast particle irradiation is therefore reduced to an extensive programme of MD simulations of a series of collision cascades under a wide range of irradiation conditions (EPKA , T, PKA type) followed by a thorough analysis of residual radiation defects and statistical treatment and interpretation of the generated simulation data. 3. Simulation techniques Semi-empirical EAM-type [33] interatomic potential [34] is applied for evaluation of the interatomic forces in the simulated L12 Ni3Al crystal. At short distances the pair parts of AleAl, NieNi and AleNi potentials were modified by fitting to the ZBL universal repulsion potential [35,36] following the procedure described in [37]. The experimentally measured threshold displacement enNi ergies in aluminium EAl d ¼ 16±3 eV [38e41] and nickel Ed ¼ 23± 2 eV [42,43] were used as fitting parameters. The threshold displacement energies of the modified AleAl and NieNi interNi atomic potentials are 13 eV  EAl d  14 eV and 23 eV < E d  24 eV, respectively. The modified interatomic potentials were employed for evaluation of the threshold displacement energy in L12 Ni3Al. The calculated value 24 eV  EdNi3 Al  25 eV matches the experi3 Al mentally measured one ENi ¼ 24±1 eV [42]. The fitting procedure d does not affect the equilibrium lattice parameters, cohesive energies, vacancy formation energies, elastic constants, stacking fault energies and the free surface energy of the two elemental metals

and Ni3Al intermetallic compound. MD simulations of collision cascades in L12 Ni3Al were conducted at ambient crystal temperatures T ¼ 100, 300, 600, 900 and 1200 K. By applying the virial theorem [44,45], the equilibrium lattice parameters were found to fit the zero internal pressure for the selected simulation temperatures. The corresponding values are provided in Table 1. MD simulations were conducted in the NVE (microcanonical) statistical ensemble. MD cell has a cubic shape with f100g faces. Periodic boundary conditions were applied at all faces. Collision cascades were initiated by Al or Ni PKAs with energy EPKA ¼ 5, 10, 15 and 20 keV. In order to reduce the probability of channelling and focussing phenomena and imitate an isotropic spatial and random temporal distribution of PKAs, they were introduced at different crystal sites, along one of 〈123〉 crystallographic directions at different moments of time. A series of at least 24 collision cascades was simulated for each (EPKA , T, PKA type) set. The outcomes of SRIM2013 [36,46] calculations of the electronic Se and nuclear Sn stopping of Al and Ni recoils in Ni3Al target are provided in Table 2. The calculations show that in the considered PKA energy range, the electronic to nuclear stopping ratio of Ni Ni projectiles in Ni3Al intermetallics is SNi e =Sn z 0.03e0.05, i.e. at the ballistic phase of cascade evolution, the energy loss of Ni recoils to the electronic subsystem of Ni3Al target is marginal and therefore can be neglected. The upper-bound estimate for the total energy loss of a 20 keV Al Al Al projectile to the electronic subsystem of Ni3Al is SAl e =ðSn þ Se Þz Al 18% or 3.6 keV, since SAl =S goes down with the decrease of recoil e n energy, see Table 2. In fact, it is noticeably lower because PKA energy in collision cascades is transferred to both Ni and Al atoms, the atomic ratio of Ni and Al atoms in Ni3Al intermetallics is 3:1, and according to Table 2, the electronic loss of Ni recoils is negligible. Moreover, existing experimental studies, see e.g. [47] and references therein, have demonstrated that SRIM calculations heavily overestimate electronic stopping at low projectile energies (( 0.01 MeV/amu). It has been found [48] that the experimental ranges of heavy or medium heavy projectiles with low energies exceed SRIM calculations by as much as 40e50%. Elastic scattering due to thermal oscillations of target atoms is one more factor that reduces the Se =Sn ratio. Hence, the total energy loss to the electronic subsystem and therefore the overall impact of the electronic stopping to the residual number of radiation defects created in collision cascades in the bulk of Ni3Al intermetallics is anticipated at a level well below the influence of the accuracy of the employed interatomic potential, the current limits to the precision of experimental data and the dispersion of MD simulation results. Therefore, no electronic stopping has been included into consideration in the undertaken MD simulations. MD cell size was chosen in accordance with PKA energy, see Table 3. The number of atoms Nbox in MD cell was scaled by PKA energy as z 102 atoms/eV. L12 Ni3Al crystals were equilibrated at the selected simulation temperature for 1:5  104 MD steps before PKA initiation. No temperature control/damping was applied. Typical change of the effective Maxwell temperature during MD cell equilibration and at different stages of the simulation of a collision cascade initiated by Al PKA with EPKA ¼ 20 keV in L12 Ni3Al intermetallics at T ¼ 300 K is shown in Fig. 1. The injected PKA energy is not extracted from the simulated

Table 1 Temperature dependence of the equilibrium lattice parameter a in L12 Ni3Al. T, K

0

100

300

600

900

1200

a, Å

3.571

3.5739

3.5802

3.5901

3.6008

3.614

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Table 2 The results of SRIM2013 [46] calculations of the electronic stopping Se , nuclear stopping Sn (both in MeV/(mg,cm2)) and projected ranges RAl and RNi (in Å) of Al and Ni recoils, respectively, with energy E in Ni3Al target. E, keV

SAl e

SAl n

Al SAl e =Sn

RAl

SNi e

SNi n

Ni SNi e =Sn

RNi

RAl =RNi

1 3 5 10 15 20

0,04798 0,08310 0,10730 0,15170 0,18580 0,21460

0,5573 0,7686 0,8561 0,9434 0,9689 0,9724

0,08609 0,10812 0,12534 0,16080 0,19176 0,22069

15 32 46 79 110 142

0,02808 0,04863 0,06278 0,08878 0,10870 0,12560

1052 1628 1927 2331 2546 2680

0,02669 0,02987 0,03258 0,03809 0,04269 0,04687

13 24 33 51 68 84

1.15 1.33 1.39 1.55 1.62 1.69

Table 3 The number of atoms, Nbox , in MD cell at different PKA energies. EPKA , keV

5

10

15

20

Nbox

500000

1048576

1492992

2048000

system. The corresponding temperature increase after relaxation of a collision cascade region does not exceed z 40 in any of the simulations, see Fig. 1. At the initial stage of a collision cascade, a small fraction of aluminium and nickel atoms is moving with high velocity whereas the rest of the L12 Ni3Al crystal remains in the thermodynamic equilibrium at the chosen simulation temperature. Because the convergence of the velocity-Verlet algorithm [49] that is employed in this study depends on the time step t evaluated in accordance with the velocity of the fastest atom, direct integration of the equations of motion of all atoms in the MD cell would lead to a nonoptimal use of high performance computing (HPC) resources. In order to facilitate calculations and optimise utilisation of HPC facilities, the methodology proposed in [50] has been implemented. Following a carefully tailored selection criteria, the simulated crystal is divided into two variable subsets of “cold” and “hot” atoms that are treated separately. The equations of motion of “hot” atoms are integrated under the assumption of “frozen” “cold” atoms, whereas the ensemble of “cold” atoms is regularly adjusted in accordance with the evolution of the subset of “hot” atoms. The ensemble of “hot” atoms grows in expense of the “cold” ones until all atoms in the MD cell become “hot” and the implemented methodology is reduced to the conventional velocityVerlet algorithm. The selection criteria, energy conservation and the effectiveness of the approach based on [50] have been tested previously and employed for MD simulations of radiation damage

in copper [51], a-zirconium [52,53], aluminium [54e56], g-TiAl [57,58] and a2 -Ti3Al [59,60] intermetallic compounds and studying the interaction of collision cascades with screw and edge dislocations [61,62]. The Lindemann spheres [63], the Wigner-Seitz cell method [64], cluster analysis [51] and the local geometry approach [61,62] were applied for identification of point defects and anti-sites remained after relaxation of collision cascades. The threshold radius of 0.3a, where a is the equilibrium lattice parameter, was used in the Lindemann spheres, and the radius of the second coordination sphere was applied for point defect cluster identification in cluster analysis. 4. Results and discussion 4.1. The number of Frenkel pairs The number of Frenkel pairs created in collision cascades initiated by Al and Ni PKAs in L12 Ni3Al is shown in Figs. 2 and 3, respectively, as a function of (EPKA , T). The obtained results are accompanied by the outcomes of other studies [28,29,32]. When PKA energy is expressed in keV, the average number of Frenkel pairs fits the power law


 NFP ¼ AðTÞ  Em PKA ;

(1)

where m ¼ 0.87 and 0.85 for Al and Ni PKAs, respectively, and AðTÞ values are given in Table 4. A similar hNFP i fEm PKA scaling with m ¼ 0.83 and 0.81 for collision cascades initiated by Al and Ni PKAs, respectively, was observed in [32]. Also the change from sublinear to linear dependence of defect production on EPKA was detected in collision cascades initiated by Ni PKAs with EPKA > 10 keV there. The observed discrepancies with the current study can be caused by a

Fig. 1. The effective Maxwell temperature T, time step t, the number of Frenkel pairs NFP and the number of antisites NAS at different stages of the evolution of a typical collision cascade initiated by Al PKA with EPKA ¼ 20 keV in the bulk of L12 Ni3Al intermetallics at 300 K.

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Similarly to elemental metals [51e56] and other intermetallic compounds [57e60], NFP goes down with raising irradiation temperature. Temperature increase shortens replacement-collision sequences, and thermal oscillations of atoms enhance scattering of high energy recoils. Consequently, denser collision cascades are formed at elevated temperatures. In an infinite isotropic solid at temperature T, the temperature Tc in the centre of a heat pulse at the time tc is determined by [65] 3=2

Tc  Tftc

Fig. 2. The number of Frenkel pairs, NFP , generated in collision cascades initiated by Al PKAs in the bulk of L12 Ni3Al intermetallics as a function of (EPKA , T). Each scatter point shows NFP created in one cascade. Overlapping points are shown with offset. hNFP i is the average over a series of collision cascades with the same (EPKA , T). The standard error of the mean is shown by bars.

Fig. 3. The number of Frenkel pairs, NFP , produced in collision cascades initiated by Ni PKAs in the bulk of L12 Ni3Al intermetallics as a function of (EPKA , T). Each scatter point shows NFP formed in one cascade. Overlapping points are shown with offset. hNFP i is the average over a series of collision cascades with the same (EPKA , T). The standard error of the mean is shown by bars.

Table 4 AðTÞ values at different ambient temperatures and PKAs. T, K

100

300

600

900

1200

Al PKA Ni PKA

5.2 5.0

4.9 4.3

4.4 3.6

3.7 2.9

3.2 2.4

number of reasons including the employed EAM interatomic potentials, different fitting procedures to ZBL repulsion potential at short distances, the cell MD acceleration technique implemented in [32], overestimated contribution of the electronic stopping, and an unevident procedure that was applied in [32] to convert the ballistic energy into PKA energy.

:

Hence, slower cooling of collision cascades occurs at higher ambient temperatures. With the increase of irradiation temperature, a narrower spatial distribution of primary damage created in collision cascades, longer relaxation of the domain of displaced atoms and higher diffusivity result in facilitated recombination of radiation defects and therefore reduction of the number of residual point defects. hNFP i produced in collision cascades initiated by Al PKAs is higher than that created in collision cascades initiated by Ni PKAs under the same irradiation conditions. The difference varies from z 10% at low T up to z 40e50% at T ¼ 1200 K. Radiation defects created in collision cascades enhance diffusion of the constituent components in irradiated materials, and that can be experimentally measured. The experimental study of diffusion in irradiated model NieAl alloys with 8.5, 11.1 and 13.1 at.% Al demonstrated [66] that collision cascades initiated by Niþ irradiation are by a factor of 0.1 less effective in promoting diffusion than those produced by protons. And in general, atomic displacements from irradiation by light ions are more than one order of magnitude more efficient in promoting solute diffusion than those produced by heavy ion irradiation [66]. It is an indirect indication of a larger number of residual radiation defects remained after relaxation of collision cascades initiated by lighter ions, and that matches the outcomes of the conducted MD simulations. The average number of Frenkel pairs hNFP i, the median number of Frenkel pairs 〈Nmed FP 〉, the corresponding dispersion s and the standard error of the mean serr in collision cascades initiated by Al and Ni PKAs with PKA energy 5 keV  EPKA  20 keV in the bulk of L12 Ni3Al intermetallics at temperature 100 K  T  1200 K are provided in Table 5. According to SRIM calculations, projected ranges of 5e20 keV Al PKAs exceed those of Ni PKAs by z 40e70%, see Table 2. Consequently, Al and Ni PKAs produce collision cascades with essentially different shapes and sizes, see Fig. 4 for comparison. Fast Al projectiles create extended multiply connected domains of displaced atoms along the projectile trajectory, see Fig. 4a, while twice pffiffiffi heavier than Al but initially z 2 slower Ni projectiles generate dense simply connected regions of displaced atoms, Fig. 4b. As a result, during and after relaxation of collision cascades initiated by Al PKAs, radiation defects are scattered over a wide region, see Fig. 5a, and the probability of their recombination is therefore lower than that of lesser dispersed radiation defects created in collision cascades initiated by Ni PKAs, see Fig. 5b. Thus, on average, the residual number of defects in the former case turns out to be higher than that in the latter case, see also Table 5. High stopping power of Ni projectiles in L12 Ni3Al intermetallics leads to high local energy density release in collision cascades initiated by Ni PKAs. That triggers formation of shock waves that propagate primarily along 〈111〉 crystallographic directions, see Fig. 6a. It is an additional mechanism of energy transfer from a collision cascade area away to the bulk of the material that competes with primary damage formation and reduces the residual number of radiation defects created in collision cascades initiated by Ni PKAs. No noticeable shock wave formation was observed in

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Table 5 The average number of Frenkel pairs hNFP i, the median number of Frenkel pairs hN med FP i, the dispersion s and the standard error of the mean serr in collision cascades initiated by Al or Ni PKAs with PKA energy 5 keV  EPKA  20 keV in the bulk of L12 Ni3Al intermetallics at temperature 100 K  T  1200 K.

Fig. 4. Typical morphologies of collision cascades initiated by Al (a) and Ni (b) PKAs with PKA energy EPKA ¼ 20 keV in the bulk of L12 Ni3Al intermetallics at ambient temperature T ¼ 100 K. Blue, green, orange and yellow spheres denote Al displaced atoms, Ni displaced atoms, vacancies in the aluminium sublattice and vacancies in the nickel sublattice of L12 Ni3Al intermetallic compound, respectively. For interpretation of the reference to colour in this figure legend, the reader is referred to the electronic version of this paper. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

collision cascades initiated by Al PKAs in Ni3Al intermetallic compound. Recently it has been suggested [30] that the number of Frenkel pairs and the number of anti-sites created in a collision cascade in the bulk of L12 Ni3Al intermetallics are governed by the collision cascade shape. However, the present study shows that the collision cascade region morphology, radiation damage formation and spatial separation of residual radiation defects are all determined

by the stopping power and energy transfer from a fast projectile to target atoms. A higher stopping power of Ni projectiles in L12 Ni3Al intermetallics leads to the creation of dense cascades accompanied by the formation of shock waves, while a lower stopping power of Al projectiles leads to the creation of less regular domains of displaced atoms along the projectile trajectory. As a result, in the latter case the number of survived defects is higher, whereas in the former case the number of radiation defects is lower but they tend

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Fig. 5. Spatial distribution of residual point defects after relaxation of the collision cascades shown in Fig. 4. Colour coding similar to that in Fig. 4 is retained. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

Fig. 6. (a) Shock wave formation in a collision cascade initiated by the recoil of a 20 keV Ni PKA in the bulk of L12 Ni3Al intermetallics at temperature T ¼ 300 K. Blue, green, orange and yellow spheres denote Al displaced atoms, Ni displaced atoms, vacancies in Al sublattice and vacancies in Ni sublattice, respectively. For interpretation of the reference to colour in this figure legend, the reader is referred to the Web version of the paper. (b) Spatial distribution of residual point defects and their clusters after relaxation of the collision cascade shown in Fig. 6a. Formation of a vacancy cluster in the centre of the collision cascade is accompanied by the formation of three small 1/2〈110〉 dislocation loops on the periphery. 1/
N4 Ni-base superalloy [9]. (For interpretation of the 2〈110〉 dislocation loops have been observed experimentally in 0.5 mm L12 Ni3Al intermetallic precipitates in irradiated Rene references to colour in this figure legend, the reader is referred to the Web version of this article.)

to cluster more, see the examples in Figs. 5b and 6b and subsequent Section 4.2. A straightforward correlation between the number of Frenkel pairs and the number of anti-sites is discussed further in Section 4.3. According to Table 5, under considered simulation conditions, h NFP i and hN med FP i are equal or within the range of the corresponding serr , which is (i) a feature of a symmetric distribution of NFP around its mean, (ii) few “outlier” collision cascade events, if any, do not spoil the statistics, and therefore hNFP i is a good representative quantitative measure of primary damage formation in L12 Ni3Al intermetallics exposed to fast particle irradiation in the nuclear stopping power regime. The dispersion of NFP is high even in collision cascades simulated under the same (EPKA , T, PKA type) set. NFP scattering broadens with increasing EPKA and/or decreasing T, see Figs. 2 and 3 and Table 5. Because of the stochastic nature of primary damage

formation phenomena, good statistical sampling of MD simulations of collision cascades is necessary to make a quantitative valuation of residual radiation defects. In the undertaken study, sampling size is validated a posteriori using the dependence of hNFP i and hNmed FP i on the number n of the collision cascades in a series with the same (EPKA , T, PKA type) set of parameters. In the example in Fig. 7, the average hNFP i and the median hN med FP i in collision cascades in Ni3Al initiated by Al PKAs with PKA energy EPKA ¼ 20 keV converge to their “stationary” values within the simulated series of 24 cascades. The same assessment has been accomplished for collision cascades simulated under all other values of (EPKA , T, PKA type) parameters. 4.2. Fraction of vacancies and interstitial atoms in point defect clusters Ni3Al (or g0 ) precipitation strengthened Ni-base refractory alloys

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Fig. 7. The average number of Frenkel pairs, hNFP i, and the median number of Frenkel pairs, hN med FP i, as functions of the number n of 20 keV collision cascades in the sampling set. The standard error of the mean is shown by bars.

Fig. 8. The total number of SIAs, NSIA4 , in SIA clusters larger than 4 against the total number of vacancies Nvac3 in vacancy clusters larger than 3 in individual collision cascades initiated by Al and Ni PKAs in the bulk of L12 Ni3Al intermetallics. Straight line corresponds to NSIA4 ¼ Nvac3 .

X-750 and 718 are widely used for manufacturing of fasteners, coil, helical and flat springs, centering pins and spacers in cores of modern commercial light and heavy water nuclear power reactors [8,67e71]. Size distribution of Ni3Al fine precipitates in these and other similar commercial Ni-base alloys is within z 10e25 nm range. Therefore studying clustering of point defects, which is the main cause of radiation hardening of structural materials exposed to fast particle irradiation, in the case of L12 Ni3Al intermetallics is of minor practical importance. Nevertheless, there is at least one published experimental study [9] of radiation effects in a two
N4 with significant (z 60%) fraction of phase Ni-base alloy Rene L12 Ni3Al intermetallic phase. Alongside with PWA1480, SRR99,
N4 belongs to a family RR2000, CMSX2, CMSX3, CMSX6, etc., Rene of first generation superalloys that have been developed for investment casting of single-crystal turbine blades with extended creep rupture life [72,73]. There are no nuclear reactor concepts/ designs and/or known applied nuclear engineering projects that consider g =g0 two-phase Ni-base cast alloys as candidate reactor structural materials. However, studying formation of vacancy and self-interstitial atom (SIA) clusters in collision cascades in L12 Ni3Al intermetallics might be interesting from the point of view of gaining insight into radiation effects in solids with ordered crystal structure and development of new breeds of radiation resistant structural and functional materials. The total number of SIAs, NSIA4 ¼ SNSIA , in SIA clusters with NSIA  4 against the total number of vacancies, Nvac3 ¼ SNvac , in vacancy clusters with Nvac  3 in individual collision cascades in L12 Ni3Al intermetallics is shown in Fig. 8. In general, Nvac3  NSIA4 at low temperatures and Nvac3  NSIA4 at elevated temperatures are fulfilled in collision cascades initiated by both Al and Ni PKAs in L12 Ni3Al intermetallics. At low irradiation temperature, there is a large fraction of collision cascades with vacancy clusters that are not accompanied by SIA clusters. At elevated temperatures, a lot of simulation results in Fig. 8 are aligned along the axes, i.e. in contrast to other materials, no strong correlation in the nucleation of SIA and vacancy clusters in collision cascades in the bulk of L12 Ni3Al intermetallics is observed. Moreover, often if vacancy clusters are formed, SIA clusters are not formed and vice versa. The fraction of SIAs in SIA clusters, NSIA4 =NFP , against the fraction of vacancies in vacancy clusters, Nvac3 =NFP , in collision cascades in L12 Ni3Al intermetallics is shown in Fig. 9. The trend observed in Fig. 8 is expectedly retained and Nvac3 =NFP  NSIA4 = NFP

Fig. 9. The fraction NSIA4 =NFP of SIAs in SIA clusters larger than 4 against the fraction of vacancies Nvac3 =NFP in vacancy clusters larger than 3 in individual collision cascades initiated by Al and Ni PKAs in the bulk of L12 Ni3Al intermetallics. Straight line corresponds to NSIA4 =NFP ¼ Nvac3 =NFP .

at low temperatures and Nvac3 =NFP  NSIA4 =NFP at elevated temperatures but the highest fraction of point defects in point defect clusters is achieved in low energy collision cascades, while the largest number of point defects in point defect clusters is obviously obtained in high energy collision cascades. In order to analyse temperature and PKA energy dependence of Nvac3 =NFP and NSIA4 =NFP further, the raw data from individual cascades in Fig. 9 was averaged over MD simulations with the same (EPKA , T, PKA type) set of parameters, see Figs. 10 and 11. Due to a broader scattering and spatial separation of residual radiation defects created in collision cascades initiated by Al PKAs, the probability of their agglomeration into clusters is lower, and therefore both hNvac3 =NFP i and hNSIA4 =NFP i there are also lower than those in collision cascades initiated by Ni PKAs in the whole range of simulated PKA energies and temperatures. Similarly to other materials, thermal stability of vacancy clusters in L12 Ni3Al intermetallics decreases with raising ambient temperature. Consequently, hNvac3 =NFP i goes down with temperature

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Fig. 10. The fraction of vacancies hNvac3 =NFP i in vacancy clusters  3 averaged over a series of collision cascades with the same (EPKA , T, PKA type) set of parameters as a function of (EPKA , T). The standard error of the mean is shown by bars.

Fig. 12. The average number of NiAl and AlNi anti-sites created in collision cascades in Ni3Al as a function of (EPKA , T, PKA type). The standard error of the mean is shown by bars.

collision cascades, and the number of clustered SIAs also increases due to their high diffusivity. As a result, the average fraction hNSIA4 =NFP i raises as well, since it is averaged over all collision cascades in a series with the same (EPKA , T, PKA type) including those with NSIA4 ¼ 0. 4.3. The number of anti-sites

Fig. 11. The fraction of SIAs hNSIA4 =NFP i in SIA clusters  4 averaged over a series of collision cascades with the same (EPKA , T, PKA type) set of parameters as a function of (EPKA , T). The standard error of the mean is shown by bars.

increase, see Fig. 10. In contrast to hNvac3 =NFP i, the fraction of SIAs in SIA clusters gradually increases with raising temperature, see Fig. 11. The increase of hNSIA4 =NFP i is driven by diffusivity of SIAs, which is a thermally activated process. High mobility of SIAs facilitates their clustering and recombination of isolated SIAs and vacancies. Both phenomena raise hNSIA4 =NFP i at elevated temperatures. The difference in PKA energy dependence of hNvac3 =NFP i and hNSIA4 =NFP i is also determined by the difference in diffusivity of vacancies and SIAs. Vacancies are not very mobile even at high temperatures, and the increase of their number with raising EPKA in accordance with Eq. (1) does not change the ratio between the number of isolated and clustered vacancies. The number of residual point defects created in low energy collision cascades is not very high, and SIA clusters are not formed in all collision cascades initiated by 5 keV PKAs, see 5 keV cascade simulation results aligned along the horizontal axis in Figs. 8 and 9. Raising PKA energy increases the number of Frenkel pairs created in

In addition to Frenkel pairs, collision cascades in intermetallic materials generate anti-sites. The number of NiAl and AlNi antisites1 created in collision cascades in Ni3Al as a function of (EPKA , T, PKA type) averaged over series of collision cascades with the same simulation parameters is shown in Fig. 12. The corresponding average values, dispersion s and the standard error of the mean serr are provided in Table 6. The number NAlNi of AlNi anti-sites exceeds the number NNiAl of NiAl anti-sites, see Fig. 12. However, strictly speaking, that cannot be deduced from the comparison of the mean values because hNAlNi i and hNNiAl i are within the range of the corresponding serr , see Table 6. Therefore the difference DNAS ¼ NAlNi  NNiAl was calculated in individual simulated cascades and the average hDNAS i over a series of collision cascades with the same (EPKA , T, PKA type) was evaluated afterwards, see the outcomes of the calculations in Table 7. The imbalance between the number of AlNi and NiAl antisites is small comparing to both NAlNi and NNiAl but it is comparable with the number of Frenkel pairs created in collision cascades, see Table 5, and that plays a key role in the partition of residual radiation defects, see Section 4.4 for further details. The number of both NiAl and AlNi anti-sites created in collision cascades initiated by Ni PKAs exceeds the number of NiAl and AlNi anti-sites created in cascades initiated by Al PKAs under identical (EPKA , T), and the difference increases with raising ambient temperature T and PKA energy. It has already been established in Section 4.1 that in contrast to the number of anti-sites, the number of Frenkel pairs created in collision cascades initiated by Ni PKAs is lower than that in collision cascades initiated by Al PKAs under the

1 The atomic notation Mex, where Me ¼ Ni, Al or V (for vacancy), and the subscript x ¼ i, Al, Ni denotes an interstitial position, a position in the aluminium sublattice or a position in the nickel sublattice of L12 Ni3Al intermetallics, respectively, is employed in the present paper, see [74] for other generally accepted notations.

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Table 6 The average number of antisites hNNiAl i and hNAlNi i, the corresponding dispersion s and the standard error of the mean serr in collision cascades initiated by Al or Ni PKAs with PKA energy 5 keV  EPKA  20 keV in the bulk of L12 Ni3Al intermetallics at temperature 100 K  T  1200 K. Extra significant digits are retained in hNNiAl i and hNAlNi i since the difference hNNiAl i hNAlNi i might be of interest and its accuracy is higher than the accuracy of both hNNiAl i and hNAlNi i, see also Table 7.

same (EPKA , T) parameters. In order to clarify a correlation between the number of anti-sites and the number of Frenkel pairs created in collision cascades, the difference hNAS i NAS against the difference hNFP i NFP is demonstrated in Fig. 13. Here NAS and NFP are the number of anti-sites and the number of Frenkel pairs, respectively, created in individual collision cascades and hNAS i and hNFP i are the corresponding average values over a series of collision cascades with the same (EPKA , T, PKA type) set. Most data points in Fig. 13a reside in the second and forth quadrants of the Cartesian coordinate system, i.e. the increase of the number of anti-sites NAS in an individual collision cascade above the corresponding mean hNAS i is accompanied by the decrease of the number of Frenkel pairs NFP below the mean hNFP i and vice versa. It is also a robust confirmation that L12 ordered crystal structure and order-disorder phase transformation make a key contribution to the resistance of Ni3Al intermetallics against the formation of radiation defects. The raw data from MD simulations of collision cascades with the same (EPKA , T, PKA type) set in Fig. 13a is approximated by a linear fit with z 10e20% accuracy, see a few examples in Fig. 13b. The figure contains free families of curves that illustrate key factors affecting

the slope of the linear fit. The steepest slope occurs in collision cascades initiated by Al PKAs with PKA energy EPKA ¼ 5 keV in L12 Ni3Al intermetallics at T ¼ 100 K. Raising PKA energy from 5 keV to 10, 15 and 20 keV leads to gradual slope reduction because of NFP f Em PKA power law dependence with m < 1. Temperature dependence of the correlation between NFP and NAS is illustrated by the outcomes of MD simulations of 20 keV collision cascades. According to Figs. 2, 3 and 12 and Tables 5 and 6, NFP decreases with raising ambient temperature, whereas NAS increases with raising ambient temperature. Consequently, the slope of the linear fit in Fig. 13b also reduces with temperature increase. Under the same simulation conditions, Al PKAs initiate collision cascades that create more radiation damage, and Ni PKAs initiate collision cascades that create more chemical disorder, see Figs. 2, 3 and 12 for comparison. Hence, with the same set of parameters (EPKA , T), in the former case MD simulations of collision cascades are approximated by the linear fit with a steeper slope in Fig. 13b comparing to the linear approximation in the latter case. Experimental studies of irradiated two phase g =g0 model and commercial Ni-base refractory alloys estimated the number of

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Table 7 The difference hDNAS i ¼ hNAlNi NNiAl i between the number of AlNi anti-sites and the number of NiAl anti-sites evaluated in individual collision cascades and averaged over cascades with the same simulation parameters (EPKA , T, PKA type), the corresponding dispersion s and the standard error of the mean serr .

Fig. 13. (a) A correlation between the number of Frenkel pairs NFP and the number of anti-sites NAS created in individual cascades in L12 Ni3Al intermetallics exposed to fast particle irradiation. (b) Fig. 13a raw data from MD simulations of collision cascades initiated by Al and Ni PKAs with PKA energy EPKA ¼ 20 keV in L12 Ni3Al intermetallics at temperature ranging 100 K  T  1200 K and collision cascades initiated by Al PKA with PKA energy ranging 5 keV  EPKA  20 keV in L12 Ni3Al intermetallics at T ¼ 100 K are approximated by a linear fit.

replacements per displacement at a level varying from 10 [3] to 70 [6]. In the undertaken study, the number of anti-sites scales with the number of Frenkel pairs as

hNAS i  NAS ¼ 1=k , ðhNFP i  NFP Þ; where k varies from 7.9±0.5102 for collision cascades initiated by Al PKAs with PKA energy EPKA ¼ 5 keV in L12 Ni3Al at T ¼ 100 K to 1.4±0.5102 for collision cascades initiated by Ni PKAs with PKA energy EPKA ¼ 20 keV in L12 Ni3Al at T ¼ 1200 K. 4.4. Partition of point defects The difference between the number of AlNi and NiAl anti-sites

has been observed in all simulated collision cascades, see Section 4.3, and NAlNi  NNiAl is fulfilled, see Table 7. This could be due to either preferred formation of vacancies in the aluminium sublattice of L12 Ni3Al intermetallics or preferred formation of Ni SIAs or both. In order to reveal the mechanism(s) of radiation damage partition, the number of residual vacancies in the nickel sublattice, NVNi , against the number of vacancies in the aluminium sublattice, NVAl , remained after relaxation of collision cascades in Ni3Al, and the number of Ni SIAs, NNii , against the number of Al SIAs, NAli , are built and presented in Figs. 14 and 15, respectively. The NVAl =NVNi ratio in individual collision cascades in Fig. 14 scatters around the stoichiometric composition of Ni3Al intermetallics. Under identical (EPKA , T) simulation conditions, larger NVAl and NVNi values are achieved in collision cascades initiated by Al PKAs (open symbols in

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Fig. 14. The number of vacancies in the nickel sublattice NVNi against the number of vacancies in the aluminium sublattice NVAl created in individual collision cascades in Ni3Al. Each scatter point reflects the residual number of vacancies in the nickel and aluminium sublattices of Ni3Al intermetallics remained after relaxation of one cascade. The dash line shows the stoichiometric composition of Ni3Al.

Fig. 16. The mean fraction of vacancies in the aluminium and nickel sublattices of L12 Ni3Al intermetallic compound, hNVAl =NFP i and hNVNi =NFP i, respectively, and the mean fraction of Ali and Nii SIAs, hNAli =NFP i and hNNii =NFP i, respectively, averaged over collision cascades with the same (EPKA , T). The standard error of the mean is shown by bars.

Fig. 14). Raising PKA energy and/or reducing ambient temperature lead to the increase of both NVAl and NVNi but their ratio is retained close to the stoichiometric ratio of Al and Ni atoms in Ni3Al. Hence, there is no partition of vacancies in L12 Ni3Al intermetallics, and vacancies do not contribute to the observed imbalance in the number of NiAl and AlNi anti-sites. In contrast to vacancies, the relative number of Ali is well below the stoichiometric composition of Ni3Al, according to Fig. 15. No isolated aluminium SIAs were observed under all simulation conditions. Ali SIAs appear only as a part of SIA clusters, see the examples in Figs. 5b and 6b. Since the following expression is fulfilled, see, e.g. [14] or [15],

SIA, nickel SIA and AlNi anti-site in L12 Ni3Al, respectively, all isolated aluminium SIAs relax through the formation of nickel SIAs and AlNi anti-sites:

EfAl > EfNi þ EfAl ; i

i

Ni

where EfAl , EfNi and EfAl are the formation energies of aluminium i

i

Ni

Ali þ NiNi /Nii þ AlNi : To quantify partition of point defects created in collision cascades, the data from individual MD simulations presented in Figs. 14 and 15 was averaged over collision cascades with the same (EPKA , T). The outcomes of the undertaken calculations are presented in Fig. 16. On average, the ratio of vacancies in the aluminium and nickel sublattices of L12 Ni3Al intermetallic compound matches its stoichiometric composition within the range of a few per cent. The average fraction of Ali , hNAli =NFP i, does not exceed z 1e2%. Under considered simulation conditions, the average values hNVAl =NFP i, hNVNi =NFP i, hNAli =NFP i and hNNii =NFP i are independent of PKA energy and ambient crystal temperature. 5. Summary

Fig. 15. The number of nickel SIAs NNii against the number of aluminium SIAs NAli created in individual collision cascades in Ni3Al intermetallics. Each scatter point shows the residual number of nickel and aluminium SIAs remained after relaxation of one cascade.

MD simulations were carried out to study primary damage created in collision cascades in L12 Ni3Al intermetallic compound and generate a reference database of radiation defects there for future use. Either Al or Ni PKAs with energy 5 keV  EPKA  20 keV were initiated in the intermetallic crystals at temperature ranging from 100 K to 1200 K. At least 24 different cascades for each (EPKA , T, PKA type) set were simulated in order to mimic an isotropic spatial and random temporal distribution of PKAs. The total yield of nearly 1000 simulated cascades is the largest yet reported for this intermetallic material. The obtained MD simulation results were analysed to get the number of Frenkel pairs NFP as a function of (EPKA , T, PKA type). It is found that NFP has a symmetric distribution around its mean, and therefore the average hNFP i is simple but an adequate representative quantitative measure of primary damage formation in L12 Ni3Al intermetallics exposed to fast particle irradiation. It was observed that the average number of Frenkel pairs created in collision cascades initiated by Al PKAs is from z 10% (low EPKA ) up to z 40e50% (high EPKA ) higher than that produced in collision cascades initiated by Ni PKAs under identical simulation conditions. At the same time,

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the number of anti-sites produced in collision cascades initiated by Ni PKAs is higher than that produced in collision cascades initiated by Al PKAs. In other words, under the same (EPKA , T), lighter PKAs generate more radiation damage in collision cascades in L12 Ni3Al, and heavier PKAs produce more chemical disorder in L12 crystal structure of the intermetallic compound. A simple connection between the number of anti-sites NAS and the number of Frenkel pairs NFP created in collision cascades has been established. It was observed that in collision cascades with the same (EPKA , T, PKA type), the correlation between NAS and NFP is approximated by a linear function. Raising NAS leads to NFP reduction and vice versa. The increase of PKA energy, simulation temperature and stopping power of the projectile decreases the slope of the linear fit. It is suggested that radiation effects in Ni3Al intermetallics are governed by the stopping power of projectiles. Projectiles with low stopping power initiate collision cascades with irregular multiplyconnected domain of displaced atoms that is extended along the projectile trajectory and create radiation defects scattered over a broad area, which hinders recombination and formation of point defect clusters. Projectiles with high stopping power quickly lose their energy to the target and initiate dense potato-shaped collision cascades that produce radiation defects within the reach for both clustering and recombination. The mean fraction of vacancies in vacancy clusters, hNvac3 =NFP i, and the mean fraction of SIAs in SIA clusters, hNSIA4 =NFP i, were evaluated as a function of simulation parameters. Under the same (EPKA , T), both hNvac3 =NFP i and hNSIA4 =NFP i averaged over collision cascades initiated by Ni PKAs exceed those averaged over collision cascade initiated by Al PKAs. Raising ambient temperature reduces hNvac3 =NFP i and increases hNSIA4 =NFP i due to lower thermal stability of vacancy clusters and higher mobility of SIAs at elevated temperature. Raising PKA energy increases the number of Frenkel pairs created in collision cascades that is accompanied by the increase of hNSIA4 =NFP i but does not affect hNvac3 =NFP i. Low mobility of vacancies in L12 Ni3Al intermetallics prevents their agglomeration into clusters over the time frame of cascade relaxation, whereas high mobility of SIAs facilitates both clustering and recombination of isolated point defects and therefore increases hNSIA4 =NFP i with the increase of NFP . Partition of radiation defects in collision cascades in L12 Ni3Al intermetallic compound was considered. It was observed that the ratio between the number of vacancies in the aluminium and nickel sublattices of L12 Ni3Al intermetallics matches its stoichiometric composition under all simulated conditions, whereas the average fraction of Al SIAs does not exceed z 1e2%. Preferred formation of Ni SIAs in collision cascades in L12 Ni3Al is accompanied by the corresponding difference between the number of created AlNi and NiAl anti-sites. Declaration of competing interests The author declares no affiliations with or involvement in any organisation or entity with any financial or non-financial interest in the subject matter or materials discussed in this research paper. Data availability The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.

#1600. Simulation software, numerical methods and point defect identification and visualisation techniques were developed under partial support from the Russian Foundation for Basic Research, grant #17-03-01222a. MD simulations were carried out using HPC resources of the Federal Centre for Simulation and Data Processing for Mega-science Facilities at NRC Kurchatov Institute (ministry subvention under agreement RFMEFI62117X0016), http://ckp.nrcki. ru/. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.jnucmat.2019.05.009. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]

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Acknowledgement The research was supported by NRC Kurchatov Institute, project

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