Computational Materials Science 147 (2018) 168–175

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Cell Molecular Dynamics for Cascade (CMDC): Molecular dynamics simulation of cascades for realistic ion energies q Jean-Paul Crocombette DEN-Service de Recherches de Métallurgie Physique, CEA, Université Paris-Saclay, F-91191 Gif sur Yvette, France

a r t i c l e

i n f o

Article history: Received 11 September 2017 Received in revised form 2 February 2018 Accepted 3 February 2018 Available online 13 February 2018 Keywords: Cascade Molecular dynamics

a b s t r a c t The CMDC code especially designed to accelerate molecular dynamics simulations of cascade is presented. Using on-the-fly addition and removal of cells during the simulation, it allows the description of the atomic displacements induced by a cascade. The principles of the code are explained and some details are given about the set-up of the parameters of the simulations. Some comparisons with standard molecular dynamics for results and calculation times are provided. A case study on ion irradiation in UO2 thin samples is then presented. It provides insight on how to actually calculate the primary damage induced by an ion irradiation. It also shows that the irradiation damage in thin sample can be different from what it would be in thick material especially in terms of subcascade statistics. Ó 2018 Elsevier B.V. All rights reserved.

1. Introduction High velocity neutrons or ions travelling in a material lose part of their energy by ballistic collisions with some of the atoms of the material [1]. Such collisions transfer high kinetic energies to these atoms, designated as Primary Knock-on Atoms (PKA). The PKAs in turn create displacement cascades which are made of the series of subsequent atomic collisions and displacements induced by the PKAs. Displacement cascades form the so-called primary damage of irradiation. The tool of choice to model the primary damage is Molecular Dynamics (MD). The principle of such calculations is rather simple and can be summarized as follows. Atoms are put in a big box according to the atomic structure of the material under study. A fitted empirical potential accounts for their interactions. Initial atomic positions and velocities are pre-determined by thermodynamic equilibration. The MD simulation is then initiated by giving to one of these atoms a large initial velocity which corresponds to the PKA initial (kinetic) energy. Then the atomic movements of all the atoms in the box are followed by MD. It is also possible to model the cascade induced by an external ion entering the material through a surface. MD simulations enable one to distinguish three phases of a cascade, see Fig. 1: – the ballistic phase corresponding to the series of two-body atomic collisions. It lasts for about 1 tenth of a pico-second q

This article belongs to the special issue: Radiation Effects E-mail address: [email protected]

https://doi.org/10.1016/j.commatsci.2018.02.008 0927-0256/Ó 2018 Elsevier B.V. All rights reserved.

and results in the displacement of atoms far from their original sites; many defects are created during this phase; – the thermal phase, created by the thermal and pressure wave of the cascade, which lasts for a few pico-seconds and results in disordering and melting of part of the material and the subsequent cooling down of the disordered zone which partly restores the crystalline structure leaving defects in the material; – and finally the diffusive phase which takes place at much longer time scales and consists basically in thermally activated diffusion processes of the defects created by the cascade. This last phase is inaccessible to standard MD simulations, though advanced simulation methods exist to deal with it [2]. MD simulations of cascades is thus rather straightforward, although a few special tricks must be applied to the simulations (for a detailed description see [3]). First the inter-atomic potential must include a specific short range term [4] which describes the high interaction energies experienced in a cascade. Second to speed-up the calculations, it is customary to use a variable time step in the simulations. Indeed at the beginning of the simulations, due to the very high velocity of the PKA, the MD time step must be very small (10 attoseconds or less). At the end of the ballistic phase atomic velocities tend to decrease to reach more regular values during the thermal phase. The time step of the simulation is then automatically adapted by step by step increase up to standard MD values around 1 fs [5]. Moreover to avoid a large increase of temperature at the end of the simulation, one uses some sort of temperature control at the external borders of the box, which are supposed far from where the cascade will take place. This

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Fig. 1. Sketch of the evolution of the number of defects in a cascade over time. The domains of the ballistic, thermal and ballistic phases are indicated.

temperature control effectively pumps out the heat introduced by the cascade. Finally, the effect of electronic loses can be taken into account, either roughly modelled by the application of a slowing term on fast moving atoms [6], or more accurately through a description of the electron dynamics in the so-called twotemperature model described elsewhere in this volume [7]. These standard MD simulations of cascades offer an accurate description of the ballistic processes of the cascade. However they suffer from some problems. Among them the fact that the inelastic or electronic loss are not well described is the subject of intensive research described elsewhere in this issue [5]. Another problem, on which we should concentrate in this paper, is the fact that these calculations are computationally intensive. An high energy cascade calculation requires in the order of 105 to 106 CPU hours. Indeed one must follow the atomic movements of all atoms in a box the size of which must be related to the kinetic energy of the PKA: the larger the PKA energy, the bigger the box must be and the more CPU the simulation will take. One of the critical choice of MD simulations of cascade is thus the size of the MD box which has to be done before-hand, prior to the actual MD simulations. One should take a box containing all the space the cascade may go into. But choosing a too large box would result in CPU time spent uselessly. At the opposite, with a too small box one may end-up with the cascade crossing the borders of the box, creating spurious interactions of the cascade either with the temperature controlled zone or with itself due to the periodic boundary conditions. One of the nasty little secrets of MD simulations of cascade is the fact that, more than rarely, simulations are put to garbage because the box was chosen too small or the cascade had an unusual spread. One then has to choose a box larger than what will eventually be necessary to accommodate the cascade. Even for an optimal choice of its size, many parts of the MD box are just present in case the cascade would go there which it will eventually not do. The energy of the PKA is also limited by the maximum size of box one can deal with using the CPU power at hand. Following the improvements of computers, the maximum PKA energies have naturally increased over the years but they remain limited. The maximum PKA energies, as of today, are of the orders of a few hundreds of keV. With such large energies it is not possible to do multiple cascade simulations and the statistics of the results are very limited. The limitation of the energy of the PKA is problematic. For neutrons irradiation, the problem is however limited as the cascade created, in a reactor, by collisions of thermal neutrons with atoms are of not too large energies. Moreover they are well disconnected from each other because of the large free flight path of the neutrons between two collision events. It is therefore possible to consider the irradiation damage induced by thermal neutrons as a

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collection of cascades of medium energies (in the order of 10100keVs). These cascades can be modelled by regular MD. Nevertheless, fusion environment are more problematic as 10 MeV PKAs are possible. Such high energy cascades cannot be modelled by regular MD. For ion irradiations, the problem is more acute. Indeed the free flight path of an ion between subsequent collisions of the ions with atoms of the material is nanometric. The possible decomposition of the damage into well separated cascade is thus much less obvious than for neutron irradiation. To describe properly the damage induced by an ion, one would need to model the complete trajectory of the ion with all its collisions and subsequent displacements as a single cascade simulation. This is not feasible with regular MD as soon as the ion energy exceeds a few hundred keVs. One final problem induced by the limitations on box size is the fact that MD simulations are focused on compact cascades. First, chaneling events which are possible for cascades in crystalline materials can spread the damage beyond the MD box. Second, in alloys (e.g. UO2), cascades are always calculated for the heavier component PKAs (uranium) and not the lighter ones (e.g. oxygen). Naturally this choice is partly justified by the facts that at equivalent energy heavy PKAs will experience a larger proportion of ballistic loses and thus induce more atomic displacements than their lighter counterparts. However it is also due to the fact that cascades created by light PKA are much less compact and are therefore much more CPU demanding to be modelled by standard MD. To alleviate these problems of box size, the Cell Molecular Dynamics for Cascade (CMDC) code [8] was developed recently. The idea of the CMDC code is to have a tool which would allow dealing, within the MD framework, with large PKA energies without the trouble of preliminary box setting. The ultimate goal of the development is to obtain the same accuracy as standard MD simulations for a much lower computational cost. With such a tool one can consider large energies corresponding to actual ion experiments and to make multiple calculations even at very large energies which allows to do proper statistics on the primary state of damage in terms of number of defects created, subcascade division, etc. In the next section the principles of CMDC are exposed and comparison is made with standard MD simulations. Examples of CMDC calculations in the case of uranium dioxide are given in the following section.

2. Cell molecular dynamics for cascade 2.1. Principles As in regular MD the main parameter of a CMDC cascade simulation is the empirical potential describing the interactions between atoms. This potential must include a short-range term, usually of the ZBL type [4]. The main difference between standard MD and CMDC simulations lies in the building of the box. A crystal has a periodically repeated structure. It can then easily be divided in cells with the same atomic structure. The on-the-fly addition or removal of such cells is at the core of CMDC. This idea relates CMDC to MD codes devoted to the determination of the depth profile of implanted ions: the MD-Range [9] and REED (Rare Events Enhanced Domain following) [10] codes share the principle of constantly adding and removing portions of the material to follow the trajectory of the implanted ion. As both codes are concerned with the determination of dopant profiles, they adjust the box relative to the position of the ion. CMDC inherits the principle of constantly changing the simulation box. But with CMDC the aim is to describe all of the cascade and not only the PKA trajectory. Therefore the PKA is not treated differently than other atoms. Cells are included in the simulation where and when necessary to

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describe the cascade, based on local criteria described in the following. The basic cell is chosen as the smallest supercell of the crystalline unit cell whose dimensions are larger than the cut-off of the empirical potential of inter-atomic interactions. For metals interacting through short-range interactions the basic cell contains about twenty atoms while for ionic compounds interacting at larger range the basic cell may contain a few hundreds of atoms. In active cells MD atomic movements are followed by a regular MD algorithm. When the cascade spreads out more and more cells are added to the simulation. These cells remain active up to the time when cascade is over in their area. They are then removed from the MD simulations (they become inactive) and the positions of atoms they contain are frozen and stored in memory. In a cascade, the atoms will deviate from their regular motion when they start to interact with atoms participating in the cascade, i.e. fast moving atoms. In cell language that is translated as: when a fast moving atom enters a cell, all the neighboring cells that were still inactive are turned on and become active. The actual turning on of the neighboring cell is driven by thresholds on velocities which be discussed below. When any of these thresholds is superseded, the neighboring cells are turned on. At that time the atoms in the turned-on cells are given some initial velocity and random displacement corresponding to the equilibrium temperature assigned to the material. The MD algorithm (in practice velocity Verlet) is then applied in the active cells. Symmetrically, cells are turned off, when the cascade is locally over. Thresholds of the same nature as for the waking up of cells are used to determine which cells can be turned off. At this time the positions of the atoms inside the cell are frozen. The method is schematically exposed in Fig. 2. In this figure the fast moving atoms have travelled along the dotted lines and are presently at their ends. To enter into some details, one must distinguish various types of cells: – Active cells where the cascade is actually taking place, i.e. the ones where the thresholds have been exceeded. – Active cells which are neighbors of the formers. The number of shells of neighboring active cells can be chosen in the simulation. By default only one shell of neighboring cell is activated so that in regular calculations, the simulation starts with 3 ⁄ 3 ⁄ 3 = 27 active cells. With this set-up, all atoms in cells participating to the cascade are interacting with moving atoms. It is however possible to add more shells of active cells up to second or third neighbors (with respectively 125 or 216 active cells at the beginning). This option could be activated to allow more space for the thermal wave to spread around the trajectories of the fast moving atoms.

– An external layer of ghost cells with frozen atoms which exist to allow force calculations on the peripheral cells. There is usually only one shell of external ghost cells but for some potentialstwo shells of ghost shells are necessary to properly calculate the forces in the outer active cells. This is the case for EAM potentials because the force on an atom depends on the positions of atoms situated at two times the cut-off distance. The turning on and off of cells is automated (see below). Beyond the variation of the number of active cells, a space variable time step is implemented in CMDC to further speed-up the calculations. During the simulation each active cell has its own period for the calculation of the forces on the atoms it contains. This period is based on the maximum velocity of atoms in its surrounding cells and itself. It goes from 1 atto-second near the path of the PKA to the standard value of 1 femtoscond. The global time step of the simulation is defined as the minimum value of the period of force calculations of all the cells. Every global time step the positions of all atoms are updated (in active cells), so that cells are always synchronized. But the forces are updated more or less frequently, according to the local period of force calculation of each cell. The last ingredient in CMDC simulations is the inclusion of a slowing down term applied on moving atoms. This term is parametrized according to the electronic losses as extracted from the SRIM code [11]. It has a double use. First, as in many standard MD calculations, it models in an approximate way the electronic losses experienced by moving atoms. The inclusion of electronic slowing down then ensures a correct stopping power (with on average the correct proportion of electronic and ballistic losses). This is especially important at high kinetic energies for which electronic losses are the major source of energy loss. A large difference may then appear between the initial kinetic energy of the PKA and the socalled ‘‘ballistic energy” which is the part of the energy lost in ballistic collisions. Second, it constantly pumps out part of the kinetic energy of the atoms. The total kinetic energy of the atoms in the simulation therefore decreases which ensures that the simulation will indeed come to an end. 2.2. On/off turning of cells The turning on of cells proceeds as follows. All cells flagged as participating to the cascade are surrounded by active cells. Thus, when the cascade spreads out, it will enter a cell which is already active. This will be detected as an increase of a quantity over the corresponding threshold in a peripheral active cell. This cell then

Fig. 2. Schematics of CMDC code.

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becomes part of the cascade. Those of its surrounding cells which were not active are turned on and become new surrounding active cells. The corresponding outer ghost cells are also added at that time. Conversely a cell is turned off when all test quantities in the cell and all its neighbors become lower than the corresponding turning off thresholds. The number of active cells thus increases as the cascade develops and slowly decreases as atoms come to rest. There comes a point where no cell is active anymore. The cascade is then over. Unlike regular MD simulations, in CMDC, neither the simulated time nor the number of time steps are predetermined. After completion of the cascade all cells are reactivated and the atomic positions are quenched by an energy minimization algorithm. Such a procedure stabilizes atomic positions. It makes regular atoms return to their perfect crystalline positions-which eases subsequent defect detection, and suppresses mechanically unstable defects, assuring for instance the recombination of very close interstitial-vacancy pairs. Note that equivalent final minimizations are commonly performed in standard MD cascade simulations. It is however especially important in CMDC as cells may be frozen as soon as their thermal agitation becomes less than the threshold which may forbid the relaxation of unstable defects. The final structure can then be analyzed in terms of atomic displacements and created defects. The update procedure (determining which cells are active) described above is performed every iteration counted according to the global time step of the simulation. At every global iteration, the local time steps of each cell are also modified if necessary. One could imagine various tests to determine which cells are to be flagged as active or not in the cascade. In the basic version of CMDC, the actual turning on of the neighboring cell is driven by thermal and kinetic effects. Thresholds are defined on the maximal kinetic energy of the atoms in the cell and on its temperature (i.e. the average kinetic energy of the atoms of the central cell). This allows a good description of the ballistic phase of the cascade as well as its induced thermal wave. An important part in the preparation of CMDC calculations is the determination of thermal thresholds for turning on and turning off of cells. Compared to usual cascade simulations, these thresholds are additional parameters which have no correspondence in standard MD. This choice is somewhat tricky and remains partly artisanal. Some guidelines may however be provided. Threshold should be larger than values reached by thermal fluctuations. Indeed thresholds below these fluctuations would lead to the spontaneous turning on of cells even in the absence of any cascade. At the opposite too large threshold would lead to spatial and temporal confinement of the cascade. Such calculations can be performed if one only wants to describe the ballistic phase of the cascade with CMDC, though other simulations tools, such as the ones based on Binary Collision Approximations (BCA) are better designed for that purpose. If one wants to describe properly the thermal phase one should then choose not too large values for the thresholds. Fortunately, we have noted that when low or medium energy cascades are considered, there is an interval in between the minimum and maximum values where the geometry of cascades and the number of created defects is statistically invariant with thresholds. When standard MD simulations were available we observed that threshold in this range lead to results which are close to what one obtains with standard MD simulations (see below examples of comparisons between standard MD and CMDC cascade calculations). Thresholds should then be chosen somewhere in this interval. Finally, the turning off thresholds for each quantity must be chosen lower than the ones corresponding for turning on, as one wants a cell detected as participating in the cascade to be included in the simulation for some time. It is also possible to define a minimum turned-on time for active cells. This

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proves especially important in specific situations exemplified below by the UO2 case. 2.3. Comparisons with standard MD In this section, we shall briefly present comparisons of the predictions of CMDC to the ones of standard MD in terms of number of created defects before discussing the speed-up of CMDC calculations compared to standard ones. A first comparison of the results of CMDC vs standard MD simulations has been already published in the case of iron [8] and will be only briefly summarized here. We considered PKAs for various energies ranging from a few eV up to 1 MeV. For each energy, 100 calculations were performed. We compared the prediction of CMDC (with thermal thresholds only) and MD on the number of created defects at each energy. The creations of defects predicted by CMDC and standard MD are in very good agreement. CMDC reproduces accurately the deviation of the number of created defect from the linear NRT law which was first predicted by standard MD (see Fig. 3). Another detailed comparison has been done for uranium PKAs in UO2. In Fig. 4 the number of defects created by uranium PKAs in UO2 calculated by standard MD [12] and by CMDC are compared for PKA energies up to 80 keV. Two sets of CMDC runs are shown. The first one corresponds to CMDC calculations performed with simple thermal thresholds for turning off of cells. In these calculations, the kinetic energy thresholds are such that the all the cells are turned off after less than 0.5 ps. One can see that for these CMDC runs the number of defects is consistently larger than the one obtained with MD. To get a better agreement, we had to perform additional calculations with a different turning off criterion. Beyond the thermal criterion, each cell, once turned-on, remains active for at least 5 ps. One can see that the results of these runs are extremely close to the standard MD numbers. Forcing the cells to remain active for a specific time naturally increases the computational time. It was necessary in the UO2 case to do such ‘‘long runs” to allow metastable interstitial vacancy pairs to recombine in the first picoseconds following the end of the cascade. Note that this metastability effect is particularly important in UO2. In this material it is indeed known that close Frenkel recombinations in UO2 exhibit quite a complex energy landscape with low energy barriers for vacancy-interstitial distances of a few nearest neighbors [13]. The need to do long runs depends on the material. It

Fig. 3. Defect production in iron. Comparison of the defect surviving ratio compared to NRT predicted by CMDC (black circles) and standard MD (blue losanges). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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Fig. 4. Comparison of defect creation between CMDC and standard MD for U PKA in UO2. Number of created vacancies as a function of ballistic energy. Black and blue lines are CMDC short and extended runs respectively (see text). Red points are standard MD calculations form the literature [12] using the same potential. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

can be tested with CMDC on medium energy cascade calculations. For instance, it has not been observed for metals and alloys (e. g. iron [8] and Ni3Al [14]) where long runs decrease the number of defects by less than 10%. CMDC was conceived to speed-up the MD simulations of cascades. One observes that there is indeed a large speed-up compared to MD. One should first note that the measured CPU time of cascade simulations using CMDC increases much less steeply with the PKA energy. This can be explained as follows. The length of the PKA track naturally increases with its energy. One observes that the distance between the initial and final positions of the PKA evolves sub-linearly with the PKA energy with a power between 0.7 and 0.9 depending on the material. In CMDC the number of active cells is roughly proportional to the length of the PKA track At the opposite in MD, as one does not know what this track will be, one has to take a cubic box, the edge of which is larger than the expected track length. This results in a cubic variation of number of atoms with track length. Moreover not all the cells are active at the same time in CMDC calculations. Considering, as an example, one particular but typical 1.8 MeV cascade in iron (see Fig. 5), one obtains that the maximum number of atoms in active cells at any time of the simulation is 4 105; the total number of atoms that were in active cells at some point in the simulations is 1.5 106;

while the number of atoms needed to fill a cubic box containing the cascade is 2 109. This last number is the absolute minimum of atoms that would be needed with standard MD to accommodate the cascade. Because of the blind guess one has to do with such calculations, the actual number of atoms needed to model such a cascade with standard MD is quite larger. One then expect CMDC to be at least four orders of magnitude faster than MD for such cascades. Moreover, in CMDC like in standard MD, the main part of the CPU time is spent in the double loop of force calculation. The inclusion of a space variable period of force calculation then allows for an additional speed-up as forces are not calculated or all atoms at every iteration. In the present cascade, one can estimate that the space variable time step induce an additional factor 10 on the calculation speed-up. Quantitative comparisons of CPU times have been performed for iron. We obtained that the calculations time of CMDC cascades goes from about 1 h for 100 keV PKAs to about 100 h for 1 MeV. The estimated speed-up compared to standard MD goes from about 1000 for 100 keV to more than 100 000 for 1 MeV PKAs. The ability of CMDC to deal with large energies allowed us to tackle realistic ion energies as exemplified in the next section on UO2. 3. Case study: ion irradiation on UO2 thin sample This part deals with the modeling of cascade created by ions in UO2. It offers the opportunity to give some details on how to proceed in practice to simulate the cascade produced by irradiations with external ions of realistic energies. The simulations presented in this section were performed in connection with the experimental work of Dr. Claire Onofri [15]. Part of this work was the comparison of two different ion irradiations of uranium dioxide. UO2 samples were 100 nm thick polycrystalline samples. They were irradiated either with 390 keV Xe ions or with 4 MeV Au ions. Such conditions were chosen to estimate the effect of ion implantation on the response of the material. Indeed SRIM calculations predicted that high energy Au ions go all the way through the thin samples and are therefore not implanted in the material. At the opposite lower energy Xe ions are expected to end their trajectory in the material which results in their implantation in the sample. Analyzing the irradiated materials, one could then expect to detect the effect of rare gas ion implantation on the evolution of uranium dioxide under irradiation. SRIM calculations also predicted that, for these ions and energies, equal ion doses (in ions/cm2) would lead to equivalent damage expressed in terms of displacement per atoms (dpa). Comparisons between the two irradiations were therefore carried out at the same doses. The results of this experimental study are not discussed here and the reader is refered to Dr. Onofri thesis [15] and related papers [16,17] to access these results. 3.1. Methodological details

Fig. 5. 1.8 MeV cascade track in iron. Yellow and blue spheres are vacancies and interstitials respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

We made CMDC simulations of these two experiments to compare the primary damage induced by these two irradiations. We present in some details, the specifics of the methodology as it applies to other modeling of experimental ion irradiations. CMDC is able to deal with external ion irradiation. In such situations, the ion projectile comes from the outside of the material and enters it through a surface of specific or random direction. One can also deal with thin materials, in which case, one must also include a back surface to allow the possibility of ion escape the material at the back of the sample. To reproduce irradiations of large grain polycrystals one simply considers a random crystallographic direction for the surface of the material.

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The procedure mentioned above for thin sample calculations is directly applicable for metals or metallic alloys. However it often does not work for ionic materials. Indeed the surfaces of such compounds are very energetic and atoms close to the surface usually experience large rearrangements. For instance, when, using MD and ionic empirical potentials, one makes a simple cut of an UO2 crystal through a random direction, the atomic positions are very unstable. The surface then largely heats up and some atoms are even ejected from the surface because of their high potential energy. It is then impossible to stabilize these surfaces with MD simulations. This is not an artefact from simulations. Indeed, in reality, such surfaces, when they exist, experience large and complex reconstructions and noticeable deviations from stoichiometry. For CMDC calculations, the situation is even worse as the created surface cells spontaneously heat up which leads to the continual addition of other cells even in the absence of any cascade. To address this problem we performed pseudo-surface calculations considering bulk cascades with random PKA directions. But the atoms situated at the back, along the PKA direction, of its initial position and further than the depth of the sample in the same direction were kept frozen. The presence of such frozen atoms stabilizes the atoms situated in the 100 nm thick slice of the material. Atoms which go either way of the slice are immediately stopped. With such a scheme one reproduces the effect of the limited depth of the samples but one loses the surface effects, such as forward and backward sputtering. As in any MD simulations, empirical potentials must be provided to describe the inter-atomic interactions. In the present case no empirical potential exist for U and O interactions with Au in UO2. To circumvent this problem we chose to use simplified potentials for the interactions of the atoms of the material with the external ions. Universal interatomic potentials [4] were thus used for Xe and Au interactions with U and O. Using such simple pair potentials for the interactions of external ions with atoms of the material does not limit the accuracy of the calculations. Indeed these potentials are valid down to a few tens of eV and they are anyway always included in potentials at short range in MD simulations of cascade. This means that the energies and momentum of the interatomic collisions with the external ion down to energies below the displacement energies of the atoms will be identical for these crude potentials to what they would be with carefully crafted potentials. The ballistic phase will therefore be well reproduced. At the opposite, accurate potentials must be used for the atoms of the material to properly describe the thermal and healing phase of the cascade. This is the case in the present simulations for UO2. 3.2. Results Using the general CMDC methodology with the specific features described above, we reproduced the ion irradiation of UO2 thin samples with either Au 4 MeV or Xe 390 keV ion. 25 cascades were performed for each ion. Fig. 6 presents the number of dpa created in the depth of the UO2 for the two irradiations. In SRIM calculations the number of dpa is calculated with the number atoms displaced from their initial sites irrespective of their final destination. In CMDC, in accordance with standard MD, dpa is calculated with the number of vacancy defects at the end of the simulation. In both simulations these quantities are divided by the number of atoms in the portion of material under consideration to arrive at a ‘‘per atom” quantity. One observes that the damage is increasing with depth for Xe irradiations while for Au irradiation it increases after 40 nm. This is in agreement with SRIM calculations. However, a quite noticeable difference appears in the amount of dpa for the two irradiation conditions. For the same 5  1014 ions/cm2 dose, the maximum damage for Xe irradiations is close to 2 dpa while it barely exceeds 1 dpa for Au ions. This is in disagreement with

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Fig. 6. dpa production for Xe 390 keV and Au 4 MeV ions in 100 nm UO2 thin sample as a function of depth for a dose of 5
1014 ions/cm2.

SRIM results which predicted the same 4 dpa maximum damage for both irradiation conditions at this dose. The fact that the number of defects predicted by CMDC is lower than with SRIM is not surprising. Indeed the deviation between SRIM (or more generally BCA) dpa and CMDC (or more generally MD) dpa is a well-known fact. BCA codes, such as SRIM, tend to overestimate the amount of created defects compared to more precise MD simulations [18, 19, 14] and experiments [20]. Nevertheless the fact that the ratio of defect production predicted by SRIM and MD simulations is different in the two irradiations is less expected. The two Xe and Au irradiations were initially chosen to be of the same dose so that the amount of damage would be identical based on SRIM predictions. Unfortunately, the present CMDC calculations made subsequently, show that there is in fact a factor 2 of difference in the damage created by the two ions at the same dose. Such a difference in defect production efficiency (comparing CMDC to SRIM) is due to different healing of defects during the thermal phase of the cascades. Such an effect cannot be modelled by SRIM or other BCA type calculations and can only be reproduced by MD simulations. Another difference between the two irradiations appears when one considers the creation of subcascades by the ion irradiations. The relationship between PKA energy, defect and subcascade creation in ordered alloys has been discussed previously [14]. We showed that the commonly assumed relationship between linear defect production as a function of ballistic energy at high PKA energy and the division in subcascades is not valid for alloys. Our definition of subcascades is based on the analysis of remaining defects at the end of the cascade: one defines defects belonging to a subcascade as defects connected (e.g. closer than 1.5 nm) to other defects connected to at least a given number (e.g. 8) of other defects. Such a definition allows discarding the linear tracks of defects along the high velocity ion paths which do not form subcascades. The size distributions of subcascades in various irradiation conditions are presented in Fig. 7. First one can note that both types of irradiation create subcascades of various sizes which may contain up to thousands of defects. Such subcascades are rare compared to the smallest ones but their contribution to the total defect production is far from negligible. It is worth stressing that such large subcascades cannot be produced by lower energy projectiles such as the 32 keV U PKAs shown in blue in left panel of Fig. 7. The assumption that one may reproduce the defect creation under ion irradiation by accumulating (within MD) low energy uranium PKAs proves therefore not valid.

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Fig. 7. Distribution of subcascades created by various cascades in UO2. Comparison of Xe 390 keV (red) and Au 4 MeV (black) ion irradiation in 100 nm thin sample to bulk U 32 keV (left panel in blue) and Au 4 MeV (green right panel). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Second comparing Xe and Au irradiations in UO2 thin samples one observes that Xe irradiation tend to create larger subcascades than the Au irradiation. The spatial distributions of the defects created by the two irradiations are therefore different. It may seem surprising that 4 MeV Au irradiation creates fewer large subcascades than 390 keV Xe irradiations. Indeed one would expect heavy projectiles with large energy to create more of larger subcascades compared to lower energy, lighter ions. This unexpected feature actually proves to be a thin sample effect. This is shown by the subcascade distribution of 4 MeV Au PKA in bulk uranium dioxide (right panel of Fig. 7). When one compares the subcascade distribution of 4 MeV Au ions in bulk vs thin sample UO2, one clearly sees first the Au PKAs create much more subcascades in bulk than in thin sample. Naturally, as Au ions are not implanted and go through the thin sample, the ballistic energy actually lost in the sample is lower than what it would be in bulk UO2. However there is about five times more subcascades in Au bulk irradiations than in the thin sample while only 50% more ballistic energy is lost in bulk compared to in the film. Second the distribution of subcascade size is notably different in the two situations. Bulk Au irradiation tends to create large subcascades. As expected these subcascades are larger than the ones created by Xe irradiations. Irradiating thick UO2 samples with 4 MeV Au would therefore result in larger subcascades than with 390 keV Xe. One major result of the present calculations is therefore that irradiations performed on thin samples may lead to quite different damage than in thick material. Not only is the number of created defects different but the microstructure of these defects, exemplified here in terms of subcascades sizes in also quite different.

4. Conclusion The CMDC code has been presented. It allows simulating cascade induced by high energy ions or PKAs which are out of reach of MD simulations, with minimal loss of accuracy compared to the standard MD calculations. Comparisons with results from the literature showed that CMDC is able to reproduce the defect production predicted by regular MD calculations. The last section of the paper dealing with ion irradiation of UO2 thin samples gives an example of a study which can be performed solely with CMDC. It showed that the microstructural damage induced by cascades can be different in bulk materials and in thin samples.

CMDC is not intended to replace standard MD simulation of cascades but to calculate many high energy events with almost the same precision as in standard MD at an extremely lower price in terms of CPU time. It requires a little bit more tuning than usual MD simulations, especially for the determination of the parameters for turning on and off of cells. While the comparison with cascades obtained with regular MD when they are available is always a good idea, it is not mandatory to have such results to do CMDC calculations. Indeed the parameters of the simulations can be determined using CMDC only. The CMDC code is still a work in progress. Two directions of improvement can be mentioned. First, CMDC presently fails to reproduce the clustering of interstitials which is observed at the end of cascades in metallic materials. CMDC predicts much less clustering than regular MD simulations. It has been shown [21] that these clusters are induced by the interaction of pressure waves created within the cascade. In order to be able to reproduce such pressure waves, we are currently working on extending the criteria for turning on and off of cells to enable the detection of perturbations in pressure close to the cascade track. Second, the description of electronic loss remains rather basic in CMDC. One could contemplate to implement in CMDC a two-temperature description of the electronic loss [7]. References [1] G.S. Was, Fundamentals of radiation materials science: metals and alloys, Springer, Berlin New York, 2007, xxii, 827 p.. [2] A.F. Voter, F. Montalenti, T.C. Germann, Extending the time scale in atomistic simulation of materials, Annu. Rev. Mater. Res. 32 (2002) 321–346. [3] J.P. Crocombette, Cascade Modeling, in: S. Yip (Ed.), Handbook of Materials Modeling, Springer, Berlin, 2005, p. 987. [4] J.P. Biersack, J.F. Ziegler, Refined universal potentials in atomic collisions, Nucl. Instr. Meth. 194 (1982) 93–100. [5] N.A. Marks, M. Robinson, Variable timestep algorithm for molecular dynamics simulation of non-equilibrium processes, Nucl. Instrum. Meth. Phys. Res. Sect. B-Beam Interact. Mater. Atoms 352 (2015) 3–8. [6] A.E. Sand, K. Nordlund, On the lower energy limit of electronic stopping in simulated collision cascades in Ni, Pd and Pt, J. Nucl. Mater. 456 (2015) 99– 105. [7] D.M. Duffy, Two-temperature model, electron-phonon coupling, Computat. Mater. Sci., 2017 (this issue). [8] J.-P. Crocombette, T. Jourdan, Cell Molecular Dynamics for Cascades (CMDC): a new tool for cascade simulation, Nucl. Instrum. Methods Phys. Res., Sect. B 352 (2015) 9–13. [9] K. Nordlund, Molecular dynamics simulation of ion ranges in the 10–100 keV energy range, Comput. Mater. Sci. 3 (1995) 448–456. [10] K.M. Beardmore, N. GrÃßnbech-Jensen, Efficient molecular dynamics scheme for the calculation of dopant profiles due to ion implantation, Phys. Rev. E 57 (1998) 7278–7287.

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