- Email: [email protected]
Establishing the isotropy of displacement cascades in UO2 through molecular dynamics simulation
Nuclear Instruments and Methods in Physics Research B 268 (2010) 2915–2917
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Establishing the isotropy of displacement cascades in UO2 through molecular dynamics simulation Clare L. Bishop *, Robin W. Grimes, David C. Parfitt Department of Materials, Imperial College London, London SW7 2BP, UK
a r t i c l e
i n f o
Article history: Received 23 September 2009 Received in revised form 22 March 2010 Available online 8 May 2010 Keywords: Uranium dioxide Displacement cascades Molecular dynamics Radiation damage
a b s t r a c t Simulation of displacement cascades is a valuable approach in furthering our understanding of how the physical properties of nuclear fuel evolve. Molecular dynamics simulations of displacement cascades in uranium dioxide have been performed at three different primary knock-on atom energies. Various properties of the cascade (such as the spatial extent and total number of defects) are monitored as the cascade progresses. Both the statistical variation of these properties and the dependence on the crystallographic direction of the primary knock-on atom are investigated in order to determine the isotropy of these events. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction The enormous irradiation that uranium dioxide fuel is subjected to within a reactor has important consequences in terms of fuel performance and the mechanical properties of this material [1]. For example, defect clusters formed in the wake of fission tracks may act as nucleation sites for intra granular fission gas bubbles which degrade the thermal conductivity of the fuel [2–4]. Furthermore, the strain fields arising from highly defective regions may result in an increase in the dislocation density [5,6]. To aid in the successful characterisation of the radiation damage, we must establish if there is any anisotropy associated with the direction of the primary knock-on atom (PKA) which initiates the cascade. Higher energy displacement cascades in metals have been shown to be isotropic with respect to initial PKA direction but the more directional bonding and long range coulombic interactions within a ceramic might cause deviations from this behaviour [7,8]. However, previous simulations in the ceramic MgO have shown that anisotropy arising from the initial direction of the PKA may be negated by a low concentration distribution of dopant species [9,10]. Also, the utmost care in how the molecular dynamics simulations are implemented in order to investigate these issues must be taken. In particular, statistical effects must be accounted for when we use initial configurations with temperatures above 0 K. A comprehensive review of uranium dioxide pair potentials has been performed by Govers et al. [11,12], in which the potentials * Corresponding author. Tel.: +44 20 7594 6765; fax: +44 20 7594 6729. E-mail address: [email protected] (C.L. Bishop). 0168-583X/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2010.05.007
employed in this study are characterised in terms of both their bulk and defect properties. More recently a comparison of the behaviour of these potentials, when implemented in displacement cascade simulations, has been undertaken in which the potentials have been grouped according to anion mobility and the ionic conducting temperature [13]. The Morelon potential [14] was found to have the lowest average displacement threshold energy of the potentials studied. In this work, we utilise both the potentials of Grimes and Catlow [15] and that of Morelon et al. [14], the latter of which demonstrates a much greater anion mobility due to its partial charge parameterisation. 2. Methodology Molecular dynamics simulations were performed using the highly parallelised DL_POLY 3.09 code [16] with simulation cells containing 324,000 ions. A variable timestep algorithm was used to maximise the efficiency of the simulations. The temperature of the equilibrated starting configurations was 300 K. A pseudo Langevin thermostat of thickness 3 Å was embedded in the simulation cell walls to control the temperature of only the ions at the extremities of the relatively large simulation cell. As such, the conditions were sufficient to both contain the defective region within the non-thermostatted region and to ensure that the cascade did not interact with its periodic images. The cascade was initiated by scaling the velocity of a uranium PKA ion to 0.4 keV, 1 keV or 10 keV in a given crystallographic direction using the Grimes potential [15] and 10 keV in a given crystallographic direction using the Morelon potential [14]. In order to investigate the isotropy of the PKA direction, cascade simulations were performed in 24
2916
C.L. Bishop et al. / Nuclear Instruments and Methods in Physics Research B 268 (2010) 2915–2917
different crystallographic directions. For each direction, 10 simulations were performed, each from a different starting configuration. This study therefore reports the results from 960 separate displacement cascade simulations. Long range coulombic interactions were treated using an Ewald summation. The potentials that describe the short range inter-ionic interactions were that of Grimes and Catlow [15] and Morelon et al. [14]. The former was parameterised as a shell model to account for polarisation effects in uranium dioxide. However, it has been found that at high energies the shell–core interactions can become unstable and so, for the purposes of this work, the shells have been omitted. The Morelon potential has been widely utilised for cascade simulations [14,17–19] and has been parameterised such that it is successful in describing defect migration in uranium dioxide. Devanathan et al. have categorised five uranium dioxide potentials in respect to cascade simulations using a PKA of 1 keV [13]. It was found that the Morelon potential [14] displayed the greatest disruption to the fluorite lattice due to the high oxygen diffusion coefficient predicted via this potential. 3. Results The time evolutions of various properties of the resulting radiation damage was monitored for each of the cascade events. In particular, the number of Frenkel defects and the spatial distribution of the damage were calculated as a function of time for the initial uranium PKA energies of 0.4 keV, 1 keV and 10 keV. After the cascade initiation, the number of defects and the extent of the radiation damage were seen to increase during the first 0.35 ps, 0.45 ps and 0.6 ps, respectively. The amount of damage was then seen to decline rapidly due to defect recombination events occurring within the damaged region which stabilised by 1.3 ps, 1.5 ps and 2.4 ps, respectively. Fig. 1 shows the number of oxygen and uranium defects at the peak of the radiation damage for cascades performed using the Grimes potential [15]. The error bars in this figure represent the number of defects within a standard deviation from the mean (averaged over 10 simulations of differing starting configurations). The variation in the top panel (PKA of 0.4 keV) appears to indicate some anisotropy with respect to the crystallographic direction. For example, in the low index [1 1 1] direction there is little variation in
the number of defects formed during the 10 simulations, which appears to be lower than other PKA directions. However, on increasing the energy of the cascade, the direction of the initial uranium PKA trajectory becomes less significant. For the 10 keV PKA simulations, the larger variation in the number of defects formed between differing starting configurations has a much greater effect than any anisotropy resulting from the initial PKA direction. Simulations using the Morelon potential [14] with an initial PKA energy of 10 keV also demonstrate overall isotropy with respect to crystallographic direction. The importance of repeating the simulations using many initial starting configurations is evident in Fig. 2, which shows the result for a PKA of 10 keV using the Grimes potential [15]. Here we report the number of defects that persist to the end of the molecular dynamics simulation (at approximately 10 ps), when the radiation damage has stabilised and the majority of the recombination events have occurred. The points in the top panel show the number of defects based on one starting configuration for each crystallographic direction. On the basis of these results we might erroneously conclude that there is a strong anisotropy with reference to the initial direction. However, on repeating the simulations with 10 initial starting configurations (Fig. 2, second panel) we see that there is significant variation within each set of initial PKA directions. The individual points show the results for each starting configuration in a given direction while the error bars in these graphs represent the range given by one standard deviation from the mean for the 10 different initial configurations (Fig. 2, second panel). By averaging we see that the spread in the number of defects is actually due to the statistical variation resulting from the different initial configurations and should not be attributed to the initial crystallographic direction of the PKA. The spatial distribution of cascade damage can be measured by calculating the mean distance of the defects from the centre of the defective region (the centre is identified by calculating the average coordinates of the all the defects present). Fig. 3 shows the mean distance former at increasingly higher initial uranium PKA energies, using the Grimes potential [15]. We see that it appears to be largely independent of the initial PKA direction and that any
200
120 90
10 simulations
200
300
200
Ndefects
160
1 keV
Ndefects
120 80
60
120
10 keV
80
6000 4000
579 458 357 344 335 259 257 234 225 159 144 139 136 134 128 124 123 122 115 111 068 015 012 011 Fig. 1. The total number of defects (both interstitials and vacancies) for a given crystallographic direction at the peak of the radiation damage. Each error bar represents the mean and standard deviation of ten separate simulations using different starting configurations. The top, middle and bottom panels show the number of defects for a uranium PKA energy of 0.4 keV, 1 keV and 10 keV, respectively. As the energy of the PKA increases, the events become more isotropic with respect to the initial crystallographic direction.
579 458 357 344 335 259 257 234 225 159 144 139 136 134 128 124 123 122 115 111 068 015 012 011
Ndefects
1 simulation
0.4 keV
Ndefects
150
Ndefects
160
Fig. 2. The number of defects persisting to the end of the simulation as a function of the initial PKA direction utilising the Grimes potential. Considerable variation in the number of resulting defects is seen which means that it is essential to repeat the simulations using several initial configurations. The top panel shows the results of one molecular dynamics simulation. The second panel shows the results of four simulations for each PKA direction. The error bars represent the range of the number of defects one standard deviation from the mean. Finally, the third panel shows the results of 10 simulations for each PKA direction and the error bars represent the range one standard deviation from the mean of these simulations. The significant variation in the number of resulting defects mean that the distribution appears to be isotropic.
C.L. Bishop et al. / Nuclear Instruments and Methods in Physics Research B 268 (2010) 2915–2917
0.4 keV
r (Å)
12 8 4
1 keV
r (Å)
12 8 4
r (Å)
10 keV
30
20
579 458 357 344 335 259 257 234 225 159 144 139 136 134 128 124 123 122 115 111 068 015 012 011 Fig. 3. The mean distance of the defects from the centre of the defective region (defined as the average coordinates of all the defects present) at the end of the simulation. The spatial distribution of the damage is compared for 24 crystallographic directions. Each error bar represents the mean and standard deviation of 10 separate simulations using different starting configurations. The top, middle and bottom panels show the mean distance of the defects from the centre for a initial uranium PKA energy of 0.4 keV, 1 keV and 10 keV, respectively. The size of the defective region appears largely isotropic at all initial uranium PKA energies.
2917
defects (not shown here) which is another measure of the spatial extent of the cascade damage. It is of interest to differentiate between the type of defects formed as a result of the high energy event. Fig. 4 compares the number of uranium defects formed using both the Morelon et al. [14] and Grimes and Catlow [15] potential models for uranium dioxide. We see that the Morelon potential almost always gives rise to a greater number of uranium Frenkel pairs. However, on comparing the percentage of uranium defects between the two potentials (lower panel) we see that this is consistent, ranging between 15% and 27%. In other words, the increased disruption to the uranium lattice with the Morelon potential is due to the greater number of both anion and cation Frenkel defects formed presumably as a consequence of utilising the partial charge model. 4. Conclusions Molecular dynamics simulations of displacement cascades have been performed in UO2, over a range of crystallographic directions, taking into account the statistical variation in the number of defects formed and spatial extent of the damage. On increasing the energy of the PKA from 0.4 keV to 10 keV the damaged region becomes more isotropic with respect to the initial PKA direction.
NU Frenkel pairs
Acknowledgements This research was supported by the European Commission through the FP7 F-BRIDGE Project (Contract No. 211690). All the calculations were performed on the Imperial College High Performance Computing Service.
30 20 10
% (uranium defects)
References 0 25 20 15
579 458 357 344 335 259 257 234 225 159 144 139 136 134 128 124 123 122 115 111 068 015 012 011 Fig. 4. The number of uranium defects as a function of crystallographic direction of the PKA velocity (10 keV) at the end of the simulation. The results for the Morelon potential [14] is shown in black and the results of the Grimes potential [15] are shown in red. Both potentials appear to be isotropic with respect to crystallographic direction. The greater number of uranium defects in the top panel shown for the Morelon potential [14] is due to the greater ion mobility of this potential. In both cases, the uranium defects represent a similar proportion of the total number of defects formed (bottom panel).
hint of a directional dependence has certainly disappeared by 10 keV. Again, we observe considerable variation in the resulting damage between the 10 simulations for each of the directions, with the mean distance varying between 16 Å and 42 Å. A similar trend is observed for the maximum distance between the most extreme
[1] D.R. Olander, Fundamental Aspects of Nuclear Reactor Fuel Elements, California University, Dept. of Nuclear Engineering, Berkeley, USA, 1976. [2] J. Turnbull, J. Nucl. Mater. 38 (2) (1971) 203–212. [3] D. Olander, D. Wongsawaeng, J. Nucl. Mater. 354 (2006) 94–109. [4] M.S. Veshchunov, J. Nucl. Mater. 277 (1) (2000) 67–81. [5] M. Amaya, J. Nakamura, T. Fuketa, J. Nucl. Mater. 392 (3) (2009) 439–446. [6] J. Jonnet, P.V. Uffelen, T. Wiss, D. Staicu, B. Rmy, J. Rest, Nucl. Instrum. Meth. B 266 (2008) 3008–3012. [7] R.E. Stoller, A.F. Calder, J. Nucl. Mater. 283 (2000) 746–752. [8] M. Hou, A. Souidi, C.S. Becquart, C. Domain, L. Malerba, J. Nucl. Mater. 382 (2–3) (2008) 103–111. [9] A.R. Cleave, R.W. Grimes, R. Smith, B.P. Uberuaga, K.E. Sickafus, Nucl. Instrum. Meth. B 250 (2006) 28–35. [10] B.P. Uberuaga, R. Smith, A.R. Cleave, G. Henkelman, R.W. Grimes, A.F. Voter, K.E. Sickafus, Phys. Rev. B 71 (10) (2005) 104102. [11] K. Govers, S. Lemehov, M. Hou, M. Verwerft, J. Nucl. Mater. 366 (1–2) (2007) 161–177. [12] K. Govers, S. Lemehov, M. Hou, M. Verwerft, J. Nucl. Mater. 376 (1) (2008) 66–77. [13] R. Devanathan, J.G. Yu, W.J. Weber, J. Chem. Phys. 130 (17) (2009) 174502. [14] N.D. Morelon, D. Ghaleb, J.M. Delaye, L. Van Brutzel, Philos. Mag. 83 (13) (2003) 1533–1550. [15] R. Grimes, C. Catlow, Phil. Trans. Royal Soc. A 335 (1639) (1991) 609–634. [16] W. Smith, Mol. Simulat. 32 (12–13) (2006) 933–1121. [17] L. Van Brutzel, M. Rarivomanantsoa, J. Nucl. Mater. 358 (2–3) (2006) 209–216. [18] L. Van Brutzel, M. Rarivomanantsoa, D. Ghaleb, J. Nucl. Mater. 354 (2006) 28– 35. [19] L. Van Brutzel, E. Vincent-Aublant, J. Nucl. Mater. 377 (3) (2008) 522–527.
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Establishing the isotropy of displacement cascades in UO2 through molecular dynamics simulation Clare L. Bishop *, Robin W. Grimes, David C. Parfitt Department of Materials, Imperial College London, London SW7 2BP, UK
a r t i c l e
i n f o
Article history: Received 23 September 2009 Received in revised form 22 March 2010 Available online 8 May 2010 Keywords: Uranium dioxide Displacement cascades Molecular dynamics Radiation damage
a b s t r a c t Simulation of displacement cascades is a valuable approach in furthering our understanding of how the physical properties of nuclear fuel evolve. Molecular dynamics simulations of displacement cascades in uranium dioxide have been performed at three different primary knock-on atom energies. Various properties of the cascade (such as the spatial extent and total number of defects) are monitored as the cascade progresses. Both the statistical variation of these properties and the dependence on the crystallographic direction of the primary knock-on atom are investigated in order to determine the isotropy of these events. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction The enormous irradiation that uranium dioxide fuel is subjected to within a reactor has important consequences in terms of fuel performance and the mechanical properties of this material [1]. For example, defect clusters formed in the wake of fission tracks may act as nucleation sites for intra granular fission gas bubbles which degrade the thermal conductivity of the fuel [2–4]. Furthermore, the strain fields arising from highly defective regions may result in an increase in the dislocation density [5,6]. To aid in the successful characterisation of the radiation damage, we must establish if there is any anisotropy associated with the direction of the primary knock-on atom (PKA) which initiates the cascade. Higher energy displacement cascades in metals have been shown to be isotropic with respect to initial PKA direction but the more directional bonding and long range coulombic interactions within a ceramic might cause deviations from this behaviour [7,8]. However, previous simulations in the ceramic MgO have shown that anisotropy arising from the initial direction of the PKA may be negated by a low concentration distribution of dopant species [9,10]. Also, the utmost care in how the molecular dynamics simulations are implemented in order to investigate these issues must be taken. In particular, statistical effects must be accounted for when we use initial configurations with temperatures above 0 K. A comprehensive review of uranium dioxide pair potentials has been performed by Govers et al. [11,12], in which the potentials * Corresponding author. Tel.: +44 20 7594 6765; fax: +44 20 7594 6729. E-mail address: [email protected] (C.L. Bishop). 0168-583X/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2010.05.007
employed in this study are characterised in terms of both their bulk and defect properties. More recently a comparison of the behaviour of these potentials, when implemented in displacement cascade simulations, has been undertaken in which the potentials have been grouped according to anion mobility and the ionic conducting temperature [13]. The Morelon potential [14] was found to have the lowest average displacement threshold energy of the potentials studied. In this work, we utilise both the potentials of Grimes and Catlow [15] and that of Morelon et al. [14], the latter of which demonstrates a much greater anion mobility due to its partial charge parameterisation. 2. Methodology Molecular dynamics simulations were performed using the highly parallelised DL_POLY 3.09 code [16] with simulation cells containing 324,000 ions. A variable timestep algorithm was used to maximise the efficiency of the simulations. The temperature of the equilibrated starting configurations was 300 K. A pseudo Langevin thermostat of thickness 3 Å was embedded in the simulation cell walls to control the temperature of only the ions at the extremities of the relatively large simulation cell. As such, the conditions were sufficient to both contain the defective region within the non-thermostatted region and to ensure that the cascade did not interact with its periodic images. The cascade was initiated by scaling the velocity of a uranium PKA ion to 0.4 keV, 1 keV or 10 keV in a given crystallographic direction using the Grimes potential [15] and 10 keV in a given crystallographic direction using the Morelon potential [14]. In order to investigate the isotropy of the PKA direction, cascade simulations were performed in 24
2916
C.L. Bishop et al. / Nuclear Instruments and Methods in Physics Research B 268 (2010) 2915–2917
different crystallographic directions. For each direction, 10 simulations were performed, each from a different starting configuration. This study therefore reports the results from 960 separate displacement cascade simulations. Long range coulombic interactions were treated using an Ewald summation. The potentials that describe the short range inter-ionic interactions were that of Grimes and Catlow [15] and Morelon et al. [14]. The former was parameterised as a shell model to account for polarisation effects in uranium dioxide. However, it has been found that at high energies the shell–core interactions can become unstable and so, for the purposes of this work, the shells have been omitted. The Morelon potential has been widely utilised for cascade simulations [14,17–19] and has been parameterised such that it is successful in describing defect migration in uranium dioxide. Devanathan et al. have categorised five uranium dioxide potentials in respect to cascade simulations using a PKA of 1 keV [13]. It was found that the Morelon potential [14] displayed the greatest disruption to the fluorite lattice due to the high oxygen diffusion coefficient predicted via this potential. 3. Results The time evolutions of various properties of the resulting radiation damage was monitored for each of the cascade events. In particular, the number of Frenkel defects and the spatial distribution of the damage were calculated as a function of time for the initial uranium PKA energies of 0.4 keV, 1 keV and 10 keV. After the cascade initiation, the number of defects and the extent of the radiation damage were seen to increase during the first 0.35 ps, 0.45 ps and 0.6 ps, respectively. The amount of damage was then seen to decline rapidly due to defect recombination events occurring within the damaged region which stabilised by 1.3 ps, 1.5 ps and 2.4 ps, respectively. Fig. 1 shows the number of oxygen and uranium defects at the peak of the radiation damage for cascades performed using the Grimes potential [15]. The error bars in this figure represent the number of defects within a standard deviation from the mean (averaged over 10 simulations of differing starting configurations). The variation in the top panel (PKA of 0.4 keV) appears to indicate some anisotropy with respect to the crystallographic direction. For example, in the low index [1 1 1] direction there is little variation in
the number of defects formed during the 10 simulations, which appears to be lower than other PKA directions. However, on increasing the energy of the cascade, the direction of the initial uranium PKA trajectory becomes less significant. For the 10 keV PKA simulations, the larger variation in the number of defects formed between differing starting configurations has a much greater effect than any anisotropy resulting from the initial PKA direction. Simulations using the Morelon potential [14] with an initial PKA energy of 10 keV also demonstrate overall isotropy with respect to crystallographic direction. The importance of repeating the simulations using many initial starting configurations is evident in Fig. 2, which shows the result for a PKA of 10 keV using the Grimes potential [15]. Here we report the number of defects that persist to the end of the molecular dynamics simulation (at approximately 10 ps), when the radiation damage has stabilised and the majority of the recombination events have occurred. The points in the top panel show the number of defects based on one starting configuration for each crystallographic direction. On the basis of these results we might erroneously conclude that there is a strong anisotropy with reference to the initial direction. However, on repeating the simulations with 10 initial starting configurations (Fig. 2, second panel) we see that there is significant variation within each set of initial PKA directions. The individual points show the results for each starting configuration in a given direction while the error bars in these graphs represent the range given by one standard deviation from the mean for the 10 different initial configurations (Fig. 2, second panel). By averaging we see that the spread in the number of defects is actually due to the statistical variation resulting from the different initial configurations and should not be attributed to the initial crystallographic direction of the PKA. The spatial distribution of cascade damage can be measured by calculating the mean distance of the defects from the centre of the defective region (the centre is identified by calculating the average coordinates of the all the defects present). Fig. 3 shows the mean distance former at increasingly higher initial uranium PKA energies, using the Grimes potential [15]. We see that it appears to be largely independent of the initial PKA direction and that any
200
120 90
10 simulations
200
300
200
Ndefects
160
1 keV
Ndefects
120 80
60
120
10 keV
80
6000 4000
579 458 357 344 335 259 257 234 225 159 144 139 136 134 128 124 123 122 115 111 068 015 012 011 Fig. 1. The total number of defects (both interstitials and vacancies) for a given crystallographic direction at the peak of the radiation damage. Each error bar represents the mean and standard deviation of ten separate simulations using different starting configurations. The top, middle and bottom panels show the number of defects for a uranium PKA energy of 0.4 keV, 1 keV and 10 keV, respectively. As the energy of the PKA increases, the events become more isotropic with respect to the initial crystallographic direction.
579 458 357 344 335 259 257 234 225 159 144 139 136 134 128 124 123 122 115 111 068 015 012 011
Ndefects
1 simulation
0.4 keV
Ndefects
150
Ndefects
160
Fig. 2. The number of defects persisting to the end of the simulation as a function of the initial PKA direction utilising the Grimes potential. Considerable variation in the number of resulting defects is seen which means that it is essential to repeat the simulations using several initial configurations. The top panel shows the results of one molecular dynamics simulation. The second panel shows the results of four simulations for each PKA direction. The error bars represent the range of the number of defects one standard deviation from the mean. Finally, the third panel shows the results of 10 simulations for each PKA direction and the error bars represent the range one standard deviation from the mean of these simulations. The significant variation in the number of resulting defects mean that the distribution appears to be isotropic.
C.L. Bishop et al. / Nuclear Instruments and Methods in Physics Research B 268 (2010) 2915–2917
0.4 keV
r (Å)
12 8 4
1 keV
r (Å)
12 8 4
r (Å)
10 keV
30
20
579 458 357 344 335 259 257 234 225 159 144 139 136 134 128 124 123 122 115 111 068 015 012 011 Fig. 3. The mean distance of the defects from the centre of the defective region (defined as the average coordinates of all the defects present) at the end of the simulation. The spatial distribution of the damage is compared for 24 crystallographic directions. Each error bar represents the mean and standard deviation of 10 separate simulations using different starting configurations. The top, middle and bottom panels show the mean distance of the defects from the centre for a initial uranium PKA energy of 0.4 keV, 1 keV and 10 keV, respectively. The size of the defective region appears largely isotropic at all initial uranium PKA energies.
2917
defects (not shown here) which is another measure of the spatial extent of the cascade damage. It is of interest to differentiate between the type of defects formed as a result of the high energy event. Fig. 4 compares the number of uranium defects formed using both the Morelon et al. [14] and Grimes and Catlow [15] potential models for uranium dioxide. We see that the Morelon potential almost always gives rise to a greater number of uranium Frenkel pairs. However, on comparing the percentage of uranium defects between the two potentials (lower panel) we see that this is consistent, ranging between 15% and 27%. In other words, the increased disruption to the uranium lattice with the Morelon potential is due to the greater number of both anion and cation Frenkel defects formed presumably as a consequence of utilising the partial charge model. 4. Conclusions Molecular dynamics simulations of displacement cascades have been performed in UO2, over a range of crystallographic directions, taking into account the statistical variation in the number of defects formed and spatial extent of the damage. On increasing the energy of the PKA from 0.4 keV to 10 keV the damaged region becomes more isotropic with respect to the initial PKA direction.
NU Frenkel pairs
Acknowledgements This research was supported by the European Commission through the FP7 F-BRIDGE Project (Contract No. 211690). All the calculations were performed on the Imperial College High Performance Computing Service.
30 20 10
% (uranium defects)
References 0 25 20 15
579 458 357 344 335 259 257 234 225 159 144 139 136 134 128 124 123 122 115 111 068 015 012 011 Fig. 4. The number of uranium defects as a function of crystallographic direction of the PKA velocity (10 keV) at the end of the simulation. The results for the Morelon potential [14] is shown in black and the results of the Grimes potential [15] are shown in red. Both potentials appear to be isotropic with respect to crystallographic direction. The greater number of uranium defects in the top panel shown for the Morelon potential [14] is due to the greater ion mobility of this potential. In both cases, the uranium defects represent a similar proportion of the total number of defects formed (bottom panel).
hint of a directional dependence has certainly disappeared by 10 keV. Again, we observe considerable variation in the resulting damage between the 10 simulations for each of the directions, with the mean distance varying between 16 Å and 42 Å. A similar trend is observed for the maximum distance between the most extreme
[1] D.R. Olander, Fundamental Aspects of Nuclear Reactor Fuel Elements, California University, Dept. of Nuclear Engineering, Berkeley, USA, 1976. [2] J. Turnbull, J. Nucl. Mater. 38 (2) (1971) 203–212. [3] D. Olander, D. Wongsawaeng, J. Nucl. Mater. 354 (2006) 94–109. [4] M.S. Veshchunov, J. Nucl. Mater. 277 (1) (2000) 67–81. [5] M. Amaya, J. Nakamura, T. Fuketa, J. Nucl. Mater. 392 (3) (2009) 439–446. [6] J. Jonnet, P.V. Uffelen, T. Wiss, D. Staicu, B. Rmy, J. Rest, Nucl. Instrum. Meth. B 266 (2008) 3008–3012. [7] R.E. Stoller, A.F. Calder, J. Nucl. Mater. 283 (2000) 746–752. [8] M. Hou, A. Souidi, C.S. Becquart, C. Domain, L. Malerba, J. Nucl. Mater. 382 (2–3) (2008) 103–111. [9] A.R. Cleave, R.W. Grimes, R. Smith, B.P. Uberuaga, K.E. Sickafus, Nucl. Instrum. Meth. B 250 (2006) 28–35. [10] B.P. Uberuaga, R. Smith, A.R. Cleave, G. Henkelman, R.W. Grimes, A.F. Voter, K.E. Sickafus, Phys. Rev. B 71 (10) (2005) 104102. [11] K. Govers, S. Lemehov, M. Hou, M. Verwerft, J. Nucl. Mater. 366 (1–2) (2007) 161–177. [12] K. Govers, S. Lemehov, M. Hou, M. Verwerft, J. Nucl. Mater. 376 (1) (2008) 66–77. [13] R. Devanathan, J.G. Yu, W.J. Weber, J. Chem. Phys. 130 (17) (2009) 174502. [14] N.D. Morelon, D. Ghaleb, J.M. Delaye, L. Van Brutzel, Philos. Mag. 83 (13) (2003) 1533–1550. [15] R. Grimes, C. Catlow, Phil. Trans. Royal Soc. A 335 (1639) (1991) 609–634. [16] W. Smith, Mol. Simulat. 32 (12–13) (2006) 933–1121. [17] L. Van Brutzel, M. Rarivomanantsoa, J. Nucl. Mater. 358 (2–3) (2006) 209–216. [18] L. Van Brutzel, M. Rarivomanantsoa, D. Ghaleb, J. Nucl. Mater. 354 (2006) 28– 35. [19] L. Van Brutzel, E. Vincent-Aublant, J. Nucl. Mater. 377 (3) (2008) 522–527.