Molecular dynamics simulation of displacement cascades in U–Mo alloys

Nuclear Instruments and Methods in Physics Research B 321 (2014) 24–29

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Molecular dynamics simulation of displacement cascades in U–Mo alloys Xiao-Feng Tian a,⇑, Hong-Xing Xiao b, Rui Tang b, Chun-Hai Lu a a b

The College of Nuclear Technology and Automation Engineering, Chengdu University of Technology, Chengdu, China The National Key Laboratory of Nuclear Fuel and Materials, Nuclear Power Institute of China, Chengdu, China

a r t i c l e

i n f o

Article history: Received 4 November 2013 Received in revised form 10 December 2013

Keywords: U–Mo alloys Molecular simulation Cascade Radiation damage Defect

a b s t r a c t Molecular dynamics simulations are employed to investigate the displacement cascades in U–Mo alloys. The cascade process is analyzed in detail. The effects of initial directions of primary knock-on atom (PKA), Mo content and PKA energies on the final damage state are evaluated. The results suggest that the direction of the PKA has no effect on the final primary damage state. A high content of Mo will raise the number of defects and the probability of Mo replacement. Most of the sizes of defects cluster are no larger than three and the probabilities of producing larger interstitial and vacancy clusters are increased with higher PKA energy. The fractions of Mo interstitial in clusters with size larger than three and isolated Mo interstitials is low, while more than half the total Mo interstitials are contained in dumb-bells. Finally, it is found that the number of U–U dumb-bells is the highest and the number of Mo–Mo dumb-bells is the lowest in both alloys. The number of Mo–Mo dumb-bells seems to be independent of Mo content but the numbers of U–U and U–Mo dumb-bells decline with the increase of Mo content in alloys. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction The goal of the Reduced Enrichment for Research and Test Reactors (RERTR) Program initiated by the U.S. Department of Energy in 1978 is to convert research reactors from high enriched uranium (HEU) fuel to low enriched uranium (LEU). Over the last 30 years more than 40 research reactors have been converted from HEU to LEU fuels. These research and test reactors require the development of new LEU fuels. LEU fuels are considered to be about 20% enriched that requires increase of the fuel densities to compensate for the resultant loss of power [1–4]. Up to now, many different uranium alloys have been tested and U–Mo alloys have proved to be the most prominent candidates due to the stable swelling behavior under irradiation and high uranium density [5–7]. At high temperature, Mo exhibits a high solubility in the range of 0–35 at.% in the body-centered cubic (bcc) c-U phase, while partial decomposition of the c-phase can occur below 833 K and the equilibrium state corresponds to a mixture of orthorhombic a-phase and body centered tetragonal c phase (U2Mo intermetallics) [5]. However, it has been experimentally proved that a metastable c-state can be retained up to room temperature by rapid cooling from the c-phase of bcc c-uranium with proper molybdenum content [8]. For instance the single-phase c-alloys U-4 at.% Mo by centrifugal atomization method [3], U-13 at.% Mo by splat cooling method [9], U-15 at.% Mo under ultrafast cooling (from the melting ⇑ Corresponding author. Tel./fax: +86 28 85405234. E-mail address: [email protected] (X.-F. Tian). 0168-583X/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nimb.2013.12.013

temperature to room temperature) [8], U-16.5 at.% Mo under normal furnace cooling conditions [5], have been reported. In order to extend the lifetime and improve performance of the developed fuels for future reactors, we need a fundamental understanding of their behavior under irradiation. At lower energies, energetic recoils produced by fission dissipate most of their energy through ballistic collisions between atoms [10]. The ballistic shocks due to the energetic recoil of a daughter nucleus lead to numerous atomic displacements and radiation damage, which plays an important role in the processes of swelling, dislocations, crack and possible phase transitions of the fuels [11,12]. Furthermore, radiation cascades in fuels can result in degradation of thermomechanical properties, retention of fission products, formation of defect clusters and fission bubbles. Although a number of irradiation experiments have been performed to determine the basic irradiation behavior of U–Mo fuel [13,14], the processes of displacement cascades are not fully understood. Especially, an understanding of the numbers and types of defects present in the primary damage state, as well as the effect of the recoil energy on damage production, is lacking. Molecular dynamics (MD) method was demonstrated to be perfectly suitable for reproducing displacement cascades in solid [15–19]. We aim in this study to model and understand the collision cascades and associated primary damage in U–Mo alloys by molecular dynamics simulation. The rest of the paper is structured as follows: in Section 2, we will describe the methodology employed to perform the simulations. Our results are presented and discussed in Section 3. Lastly, conclusions are drawn in Section 4.

X.-F. Tian et al. / Nuclear Instruments and Methods in Physics Research B 321 (2014) 24–29

2. Computational procedure The interatomic potential used in this work was developed recently by Smirnova et al. [20]. This embedded atom method-like potential was fitted using a force-matching technique and a dataset of ab initio atomic forces. The test results suggest that this new EAM potential is suitable to study the atomistic processes of defect evolution taking place in the U–Mo nuclear fuel. The code LAMMPS [21] was employed to perform the simulations. The MD runs were performed with a system containing 64  64  64 conventional body-centered cubic unit cells including 524288 atoms under periodic boundary conditions (PBC). To evaluate the effect of molybdenum content on primary damage, the alloys U-8 at.% Mo and U-21.6 at.% Mo were considered and Mo atoms were randomly distributed in the alloy. Prior to simulating the cascade event, the simulation block was equilibrated at 300 K in an NPT ensemble to ensure that the system has adequate time to sample a suitable local equilibrium structures. Temperature and pressure of the system were corrected by Nose/Hoover thermostat [22] and Nose/Hoover barostat [23,24] respectively. The equilibrium lattice parameter of U-8 at.% Mo and U-21.6 at.% Mo were respectively 3.50 and 3.41 Å at the end of our NPT simulations. In the cascade simulations, energetic U recoils with kinetic energies ranged from 0.1 up to 10 keV were simulated with the constant NVE ensemble. The initial direction of the primary knock-on atom (PKA) was chosen randomly and ten cascade simulations were performed for each set of recoil energy to provide a statistical approach to the modelling results. A variable time step in the 0.01–1 fs range, which depends on the maximum kinetic energy in the system, was used for all simulations. Each simulation was run 30 ps, which is sufficient to allow the cascade to anneal and return to thermal equilibrium. A thermostat is applied at the box sides by rescaling the velocities of the atoms in a 3 Å layer width around the simulation box. This thermostat can extract the excess of the kinetic energy which arise from the cascade and minimize the interaction of heat wave with periodic images of itself.

3. Results and discussion 3.1. The evolution of the cascade After a PKA is introduced in alloys, many atoms will leave their original lattice site due to binary collision between atoms. However, the atoms leaving their origin lattice site by less than 1 Å are considered to still vibrate around their origin positions. Fig. 1 shows a typical snapshots of atoms displaced by more than 1 Å from their initial position during displacement cascade with 5 keV kinetic energy PKA taking place in U-8 at.% Mo alloy. We define these atoms as displaced atoms in following paper. Within 0.3 ps of PKA introduction, a limited number of atoms are displaced and release their kinetic energy by binary collisions. Then a ballistic phase is presented during which many sub-cascades are created. In this stage, some atoms can migrate to a long range before undergoing another collision. After this collision occur, a new displacement cascade, which is called sub-cascade, is created. Fig. 1(a) and (b) shows the ballistic phase at two different moments. Subsequently, no new sub-cascades are observed. The number of the displaced atoms increases although a limited number of atoms return to their initial lattice site (which can be seen by comparing Fig. 1(b) with Fig. 1(c)). This stage is called thermal spike and last until about 1.6 ps when the number of displaced atoms reaches a peak, as shown in Fig. 1(c). In next stage, significant recovery of the crystalline lattice appears and the whole system cools down under the influence of the thermostat applied in the

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outer boundary layer. Fig. 1(d) shows the snapshot of the displaced atoms at 25.6 ps. During the displacement cascade, the displaced atoms may migrate to either interstitial site or other lattice site. If the displaced atoms are located at interstitial site, the association of a vacancy and an interstitial is defined as the Frenkel pair. In other case, the displaced atoms locate at crystalline positions create replacements rather than Frenkel pairs. Typical time evolutions of displaced atoms, Frenkel pairs and replacements are given in Fig. 2. Similar evolutions are observed for U and Mo atoms in the U-8 at.% Mo alloy. Within 0.3 ps after PKA is introduced, no U and Mo replacements are observed. In the following stage, the numbers of replacements increase gradually to constant values at 14.6 ps and 8.6 ps for U and Mo respectively. After the initial knock-on event occurred at t = 0, the numbers of the displaced atoms and Frenkel pairs increase with time and reach peak values synchronously at about 1.6 ps for both U and Mo. Then the numbers of the displaced atoms decrease resulting from the recovery of the crystalline lattice. The evolutions of the Frenkel pair are affected by decrease of displaced atoms and increase of the replacements in this stage. Thus, the numbers of Frenkel pairs decrease significantly with time after thermal spike. Finally, the numbers of the Frenkel pairs are almost invariable after 14.6 and 8.6 ps for U and Mo respectively. 3.2. Direction influence The influence of the initial recoils directions versus the number of displaced atoms and Frenkel pairs created in U-8 at.% Mo alloy was also investigated, as shown in Table 1. To evaluate the influence of PKA direction, three typical PKA directions, [1 0 0], [1 1 0] and [1 1 1], were chosen and ten cascade simulations were performed for each direction with 5 keV PKA. No significant effect of initial PKA direction on the number of the displaced atoms and Frenkel pairs has been found. Averaging the total 30 PKA simulations, we note that the fractions of Mo displaced atoms and Mo Frenkel pairs among total displaced atoms and total Frenkel pairs are 10.7% and 15.4% respectively. Both are higher than the content of Mo in alloy. This result indicates that the defects of molybdenum are more easily produced than uranium defects in U-8 at.% Mo alloy. Fig. 3 shows typical diffusion distance distributions of U and Mo atoms displaced over distances smaller than 10 Å at two different stages. In our PKA simulations in U-8 at.% Mo, the maximum of diffusion distance for displaced U and Mo atoms is 9.6 and 9.4 Å respectively. In the stage of thermal spike, the numbers of U and Mo atoms decreases significantly with the increase of diffusion distance. After the system cools down, there are several favorite positions for the displaced atoms. The peak around 3 Å corresponds to atoms which have migrated to their first nearest and second nearest neighbor lattice site, while the highest values appear at the first nearest lattice site for both U and Mo. Small fraction of the displaced atoms have moved to third and forth nearest neighbor lattice site, which corresponds to the peaks around 5 and 6 Å respectively. 3.3. PKA energy influence We now pay our attention to the effect of initial PKA energy on the final damage. The number of defects will increase with higher PKA energy due to a higher temperature in the core of the cascade and more collision sequences [25]. In the process of cascades, the creation of an interstitial is accompanied by the creation of a vacancy, which results in formation of a Frenkel pair. The number of total Frenkel pairs produced by 0.1, 0.5, 1, 2, 5 and 10 keV PKA is plotted in Fig. 4. We note that higher content of Mo in c-U will

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X.-F. Tian et al. / Nuclear Instruments and Methods in Physics Research B 321 (2014) 24–29

(a) 0.3ps

(b) 0.7ps

(c) 1.6ps

(d) 25.6ps

200

2200 2000 1800 1600 1400 1200 1000 800 600 400 200 0

180

dispalced atom Replacement Frenkel pair

160 140

dispalced atom Replacement Frenkel pair

120

Number

Number

Fig. 1. Snapshots of the U (green) and Mo (blue) atoms displaced by more than 1 Å during the displacement cascade initiated at energy of 5 keV along 1 1 1 directions. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

(a)

100 80 60

(b)

40 20 0

0 2 4 6 8 10 12 14 16 18 20 22 24 26

0 2 4 6 8 10 12 14 16 18 20 22 24 26

Time (ps)

Time (ps)

Fig. 2. Time evolution of the number of displaced atoms, Frenkel pairs and replacement atoms created: (a) uranium atoms; (b) molybdenum atoms.

Table 1 The number of atoms displaced and Frenkel pairs created by 5 keV PKA in U-8 at.% Mo for three crystallographic directions. Direction

U displaced

Mo displaced

U Frenkel

Mo Frenkel

100 110 111 Average

1009 (89.4%) 976 (89.2%) 1047 (89.4%) 1011 (89.3%)

120 118 125 121

145 141 158 148

27 26 27 27

(10.6%) (10.8%) (10.6%) (10.7%)

(84.2%) (84.7%) (85.6%) (84.6%)

(15.8%) (15.3%) (14.4%) (15.4%)

increase the number of defects, which agrees with the conclusion obtained in Table 1 that Mo defect is more easily created that U defects. Additionally, the numbers of Frenkel pairs in our simulations

are higher than in general alloys and metals produced by cascades. This may attributed to that the interatomic potential used in this work underestimate the threshold displacement energy for U–Mo alloys. The evolutions of the number of Frenkel pairs with the initial energy of the PKA can be fitted to a power law Nf / Em. It is reported that the value of m is weakly dependent on the material and temperature [26]. In our simulations, the exponent is equal to 0.83 for U-8 at.% Mo and to 0.86 for U-21.6 at.% Mo, both are close to the values for other alloys and metals [25–27]. The replacement fraction of U and Mo atoms is reported in Fig. 5. With lower PKA energies, the volume of the cascade core and the temperature during collisions is limited. The excess of

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X.-F. Tian et al. / Nuclear Instruments and Methods in Physics Research B 321 (2014) 24–29 30

1.6 ps 25.6 ps

80

60

40

(a)

20

1.6 ps 25.6 ps

25

Number of atoms

Number of atoms

100

0

20 15 10

(b)

5 0

1

2

3

4

5

6

7

8

9

10

1

2

3

Distance [A]

4

5

6

7

8

9

10

Distance [A]

Total number of surviving Frenkel pairs

Fig. 3. Distribution of the atoms as a function of the distance traveled starting from their initial position: (a) uranium atoms; (b) molybdenum atoms.

perature for each sub-cascade; hence the replacement fraction is independent of initial PKA energies. With initial PKA energy higher than 2 keV, the increase of Mo content has no influence on the replacement fraction of U atoms but can boost slightly the replacement fraction of Mo, as shown in Fig. 5. In our simulations, the replacement fraction of Mo is about 81% in U-8 at.% Mo and the value is about 88% in U-21.6 at.% Mo at high PKA energy, while the replacement fraction of U is about 84% in both alloys. Defects produced during cascades easily gather into defect cluster. The distribution of cluster size was examined and the results were plotted in Fig. 6. The second nearest neighbor distance was

100

10

U-8 at.% Mo 0.83 NFP=45.5E U-21.6 at.% Mo 0.86 NFP=54.3E 0.1

1

10

MD cascade energy (keV) 1.0

U-8 at.% Mo U-21.6 at.% Mo Partitioning probabilities

Fig. 4. Total number of Frenkel pairs as a function of the cascade energy by MD simulations in U-8 at.% Mo and U-21.6 at.% Mo alloys.

1.0

Replacemnent/diffused

0.9 0.8 0.7 0.6

U in U-8 at.% Mo Mo in U-8 at.% Mo U in U-21.6 at.% Mo Mo in U-21.6 at.% Mo

0.5 0.4

(a)

0.9 0.8 0.7 0.6 0.5 0.4 0.3

0 5 10 inter 15 20 5 stitia ls cl 2 30 5 3 uste r siz 40 45 e 50

0.2 0.1 0.0 10K 5K 2K y rg 1K ne 0.5K A e PK 0.1K

0.3 0

1

2

3

4

5

6

7

8

9

10

11

1.0

Energy (KeV)

the kinetic energy which arise from the cascade can dissipate in short time. So, a large fraction of displaced atoms in interstitial positions have not enough energy or time to combine with vacancy before the system cools down and low replacement fraction for both U and Mo atoms is obtained. With the increase of PKA energy, the volume of cascade core, which is the volume affected by the associated high temperature, will increase. The system need longer time to dissipate the kinetic energy. So, more displaced atoms have chance to migrate into crystalline site and higher replacement fraction is observed. When the PKA energy is higher than 2 keV, sub-cascades will be created in ballistic phase. There is no more increase of the volume of cascade core and time spent at high tem-

Partitioning probabilities

Fig. 5. Replacement fractions of U and Mo atoms as a function of the PKA initial energies in U-8 at.% Mo and U-21.6 at.% Mo alloys.

U-8 at.% Mo U-21.6 at.% Mo

0 5 10 5 vaca 1 20 5 ncy clus2 30 35 ter s 40 45 ize 50

(b)

0.9 0.8 0.7 0.6 0.5 0.4 0.3

0.2 0.1 0.0 10K 5K

2K 1K y erg 0.5K en A 0.1K K P

Fig. 6. Size distribution of defect clusters versus PKA energies: (a) interstitial cluster; (b) vacancy clusters.

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X.-F. Tian et al. / Nuclear Instruments and Methods in Physics Research B 321 (2014) 24–29 0.40

Mo in all interstitials isolated Mo interstitials Mo in dumbbells Mo in large cluster (>3) Mo in cluster

(a)

0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00

Fraction of Mo interstitial

Fraction of Mo interstitial

0.40

0.35

Mo in all interstitials isolated Mo interstitials Mo in dumbbells Mo in large cluster (>3) Mo in cluster

(b)

0.30 0.25 0.20 0.15 0.10 0.05 0.00

0

1

2

3

4

5

6

7

8

9

10 11

0

1

2

3

Energy (KeV)

4

5

6

7

8

9

10 11

Energy (KeV)

Fig. 7. The fraction of Mo in all interstitial defects, isolated interstitials, interstitials clusters, dumbbells and large interstitial clusters (size is larger than three): (a) U-8 at.% Mo; (b) U-21.6 at.% Mo alloys.

3.4. Composition influence Concerning the chemical nature of interstitials clusters, Fig. 7 shows the fraction of Mo in all interstitial defects, isolated interstitials, interstitials clusters, dumbbells and large interstitial clusters (size is larger than three). At low PKA energy for U-8 at.% Mo, the fraction of Mo interstitials in Fig. 7(a) is significantly higher than that at high PKA energy. This is probably due to the fact that the averaged number of Mo interstitials produced at low PKA energy and low Mo content is rather small, resulting in poor statistics of the results. In the alloy with higher Mo content, the results will be improved, as can be found in Fig. 7(b) for U-21.6 at.% Mo. It is notable that the content of Mo in alloys has no obvious effect on the proportion of Mo interstitials among the total interstitial atoms at high PKA energies. Averaging the total 30 cascade simulations with PKA energies higher than 2 keV, the fraction of Mo in total interstitials is 12.5% and 13.2% in U-8 at.% Mo and U-21.6 at.% Mo respectively. The reason is that the replacement fraction of Mo will increase although the fraction of displaced Mo atoms is significantly higher in U-21.6 at.% Mo compared U-8 at.% Mo. Besides, Fig. 7 reveals that the fractions of Mo interstitial in cluster with size lager than three and isolated Mo interstitial is low, while more than half the total Mo interstitials are contained in dumbbells. Fig. 8 gives us the number of U–U, U–Mo, Mo–Mo dumbbells versus PKA energies. Independently of the recoil energy and Mo content, we found that the number of U–U dumbbells is the highest and the number of Mo–Mo dumbbells is the lowest by looking at the chemical composition of the produced dumbbells. At all PKA

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U-U U-8 at.% Mo U-Mo U-8 at.% Mo Mo-Mo U-8 at.% Mo U-U U-21.6 at.% Mo U-Mo U-21.6 at.% Mo Mo-Mo U-21.6 at.% Mo

40

Number of dumbbells

chosen to be the cut-off radius for vacancy cluster connectivity and the third nearest neighbor distance was chosen for interstitial clusters in this work. The trend of distribution of cluster size is similar in both alloys. We get that the size of most interstitial and vacancy clusters in both alloys is not lager than three. The probabilities of producing lager interstitial and vacancy clusters are increased with higher PKA energy. The largest interstitial cluster observed in our simulations involves 45 interstitials in U-8 at.% Mo and 48 interstitials in U-21.6 at.% Mo, while the size of the largest vacancy cluster are 47 and 48 respectively. All of the largest defect clusters are created at 10 keV. Fig. 6(a) shows that the fraction of isolated interstitials decreases with the increase of PKA energy, which indicates that the fraction of interstitials contained in clusters is lager at higher PKA energy. For example, the fraction of surviving interstitials in clusters produced in U-8 at.% Mo and U-21.6 at.% Mo increased respectively from 29% and 19% at 0.1 keV PKA energy to 56% and 60% at 10 keV PKA energy. In comparison, The PKA energy seems to have a weaker influence on the size distribution of vacancy clusters, as can be seen in Fig. 6(b).

35 30 25 20 15 10 5 0

0

1

2

3

4

5 6 7 Energy (KeV)

8

9

10

11

Fig. 8. Number of dumbbells versus PKA energy in U–Mo alloys.

energies considered in our work, higher Mo content in alloy has nearly no effect on the number of Mo–Mo dumbbells but reduce the number of U–U and U–Mo dumbbells. In both alloys, the number of dumbbells of three different types calculated in our work finally follows nearly linear energy dependence when PKA energy is higher than 2 keV.

4. Conclusions The displacement cascade in U–Mo alloys, with recoil energies ranging from 0.1 to 10 keV, was studied by molecular dynamics simulations based on recently proposed interatomic potential. The PKA energies The alloys U-8 at.% Mo and U-21.6 at.% Mo were chosen to evaluate the effect of Mo content on the final damage. The results show that there is no significant influence of the initial recoils directions on the number of final displaced atoms and Frenkel pairs. At the end of cascades, there are several favorite positions for the displaced atoms. Most of displaced atoms have migrated to their nearest and second near neighbor lattice site and small fraction has moved to third and forth neighbor lattice site. The number of defects and replacement fraction will rise with the increase of PKA energy. At high PKA energies, the replacement fraction of Mo in U-8 at.% Mo and U-21.6 at.% Mo is 81% and 88% respectively. The replacement fraction of U is 84% in both alloys. The study of spatial distribution of displaced atoms shows that most of the replacement occur at nearest and second near neighbor lattice site of displaced atoms.

X.-F. Tian et al. / Nuclear Instruments and Methods in Physics Research B 321 (2014) 24–29

The size of most interstitial and vacancy clusters produced by cascades in both alloys is not lager than three. The probabilities of creating lager interstitial and vacancy clusters are increased with higher PKA energy. The content of Mo in alloys has no obvious effect on the proportion of Mo interstitials among the total interstitial atoms at high PKA energies. The fractions of Mo interstitial in cluster with size lager than three and isolated Mo interstitial is low, while more than half the total Mo interstitials are contained in dumbbells. It is found that the number of U–U dumbbells is the highest and the number of Mo–Mo dumbbells is the lowest in both alloys. The number of Mo–Mo dumbbells seems to be independent of Mo content but the numbers of U–U and U–Mo dumbbells decline with the increase of Mo content in alloys. Acknowledgment This work was supported by the National Natural Science Foundation of China (Grant Nos. 91226203 and 11205146). Reference [1] J.H. Kittel, B.R.T. Frost, J.P. Mustelier, K.Q. Bagley, G.C. Crittenden, J. VanDievoet, J. Nucl. Mater. 204 (1993) 1. [2] J.L. Snelgrove, G.L. Hofman, M.K. Meyer, C.L. Trybus, T.C. Wiencek, Nucl. Eng. Des. 178 (1997) 119. [3] K.-H. Kim, D.-B. Lee, C.-K. Kim, G.E. Hofman, K.-W. Paik, J. Nucl. Mater. 245 (1997) 179. [4] M.K. Meyer, G.L. Hofman, S.L. Hayes, C.R. Clark, T.C. Wiencek, J.L. Snelgrove, R.V. Strain, K.-H. Kim, J. Nucl. Mater. 304 (2002) 221.

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