Swelling Mechanism of Lattice with the Ingrowth of the Defects in UO2

Swelling of uranium dioxide with ingrowth of defects by irradiation is not fully understood. Experimental and theoretical groups have attempted to explain this phenomenon with various complex theories. In this study, experimental lattice expansion and super saturation of the lattice were well reproduced by molecular dynamics simulation method. From the resemblance with the experimental data, here it is manifested that only oxygen frenkel pairs were created in the fission induced lattice while alpha particle radiation causes both oxygen and considerable amount of uranium defects. Moreover, in this work, defects are divided into two sub-groups as obstruction and distortion and it is shown that obstruction type frenkel pairs merely responsible for the lattice swelling for both fission and alpha particle radiation. Evidently relative lattice expansion varies linearly with the obstruction type of survived uranium defects. Additionally, at high concentrations, some of the obstruction type uranium frenkel pairs forming double or triple structures with oxygens in their octahedral cages which increase the slope of the linear dependence.

nucleus produced in alpha decay) and neutrons are the reasons for irradiation damage.
Moreover electrons, X-rays and gamma rays would enhance damage but generally ignored (17). When an 2 UO crystal exposed to radiation, frenkel pair (FP) defects are created in the direction of the radiation path. FPs are lattice vacancies and the atoms which jumped from the lattice site to an interstitial position by the enforcement of irradiation. Furthermore, if the implemented dose is increased, complete amorphization of the crystalline may occur at ambient condition (17). Experimentally, Nakae et al. (18)(19)(20) and Weber (21,22) have studied the fission and α-particle dose effect and temperature dependence of lattice parameter, lattice strain and their recovery behavior of irradiated 2 UO .
There have been radiation damage molecular dynamics simulation studies in which defect production and clustering by energetic uranium recoils in 2 UO have been investigated (23). In addition, Aidhy et.al. (24) have explored the kinetic evolution of irradiation-induced point defects in 2 UO by molecular dynamics simulation at 1000K. They have observed that if the 3 defects are present in only one sublattice, the FPs recombine during equilibration, whereas if defects are present in both sublattice they form clusters and conclude that the cation sublattice is primarily responsible for the radiation tolerance or intolerance of material. However, to our knowledge, computer simulations of the lattice swelling with the defect ingrowth and the lattice recovery with temperature in defected 2 UO have not been considered so far.
In the present study, molecular dynamics simulation calculations were carried out for the supercell of 2 UO . The defected samples were prepared according to the number of defects which correspond to the suggested experimental dose. Two different type of partially ionized rigid ion potentials, existed in literature (1,5), were used for the interionic interactions. The ingrowth of defect versus swelling of the lattice is investigated and the results are compared with the experimental data.

Procedure
Crystalline uranium dioxide with four uranium and eight oxygen ions in its unit cell has the fluoride type structure. Every uranium ion is at the center of surrounding cube and coordinated to eight oxygen anions. Oxygen is surrounded by four Uranium ions. MD cell is constructed by setting 500 cations and 1000 anions with array 5x5x5 supercells in five mutually orthogonal directions. The calculations have been carried out by the MD code Moldy (25). Long-range coulomb interactions are accounted with Ewald's summation (26).
The positions and velocities of the ions are calculated by Beeman's algorithm which is predictor-corrector type, with the time step Δt=1.0 fs. The system has been simulated at constant pressure and temperature (NPT) ensemble at 300K by applying the Nose-Hoover thermostat and Parinello-Rahman constant stress method. Equilibrium run was performed for 10 ps and then the data were accumulated over the following 40 ps. 4

Sample Preperation
The irradiated samples of 5x5x5 supercell with the different defect concentrations have been prepared by randomly replacing an ion from its lattice site to an interstitial position on the layers. A representative sample of defected supercell is given in Figure 1. Here after, through this article, these ions are called initial frenkel pair (IFP) defects. After the equilibration, they are called frenkel pair (FP) defects. To take into account the annihilation effect, IFP defects have been created such that one layer is defected at every two layers, so nearby vacancies and interstitials do not terminate each other directly. Each defected layer has approximately the same number of IFPs. The simulation procedure for the defected and perfect supercell boxes is the same. Samples of supercell boxes of uranium dioxide with IFPs are prepared for several different defect concentrations, based on the experimental dose (17). Supercell boxes are built up with oxygen IFPs or uranium IFPs but not both together in the same sample in order to correlate the sublattlice effect with irradiation type. The interionic interactions have been modeled by using two different types of rigid ion pair potentials parameterized by Yakub et.   Figure 3 that is used to determine the maximum and the minimum values of r. The distortion and obstruction type defects (see Figure 2) can be, respectively, considered as displacements of the ions into the channels and occupation of the central positions of the channel by the ions. Obstruction type ions have a constant number of ions surrounding it but distortion type ions are much more mobile. It is observed in Figure   3 that the distinct pre-peak just before the principal peak reflects the distribution of the obstruction type of FPs.  . These calculated numbers of defects are consistent with the visual observation of the VMD snapshot given in Figure 2.

Swelling of the lattice with ingrowth of defects
The difference between the lattice expansion with fission damage and alpha particle damage is about a few orders of magnitude (17) where the fission damage is less effective than the alpha particles. The relative lattice expansion o a a  has been calculated at 300K for the IFP defect numbers varying up to 40 for 5x5x5 supercell system. Comparing the results with experimental relative lattice expansion have shown that the calculations carried out with the defected sample separately with Oxygen IFPs and Uranium IFPs resembles the experimental results with fission and alpha particles, respectively. Therefore, it is more convenient to present the results in two separate sections.

Defected Supercell with Uranium Frenkel Pairs
The aim of this part is to correlate relative change of lattice parameter of alpha irradiated 2 UO with MD simulation of defected 2 UO supercell with uranium FPs and to interpret the results.
This section is separated into two subsections because fraction of defects is an order of magnitude higher than oxygen IFPs part. MD simulation cell is built for both 5x5x5 and 8x8x8 unit cells in order to observe the influence of defects images which is caused by periodic boundary conditions. Results for 5x5x5 supercell and 8x8x8 supercell with Uranium IFPs are compared so that they could be confirmed.

5x5x5 Supercell
The number of uranium IFPs has been varied up to 30 which were enough to observe the  with the obstruction type of oxygen defects and the distortion type of uranium defects which have, respectively, almost no contributions to the total defect up to about ~15 and ~10 uranium IFPs. Therefore, these are not included in Figure   14 7. Until the saturation level, the lattice expansion has an exponential dependence to the number of distortion type of oxygen (see Figure 7(a)), distortion+obstruction types of oxygen (see Figure 7(c)), distortion+obstruction types of oxygen+uranium (see Figure 8(a)) and initial uranium FP (see Figure 8(b)). The maximum lattice expansion observed is about 1.4% for the Yakub potential and 0.5% for the Günay potential which are respectively correspond to volume change of 4.2% and 1.5%. For comparison, the experimental lattice expansion with alpha dose by Weber (21) is also shown as inset in Figure 8( suggested that such concentration of defects signals isolated defects and negligible clustering.
Both potentials give almost the same number of obstruction type defects as Weber had found.
It should be also noted that lattice expansion in Figure 7(b) varies linearly with the obstruction type of survived uranium defects which is interesting. Evidently, these uranium defects are the reason for lattice expansion which is coordinated with six uranium ions. According to Eq. 2, the tangent of Figure 7(b) gives volume increment per obstruction type uranium frenkel pair,

8x8x8 Supercell
Here similar procedure is used as the 5x5x5 supercell. Uranium IFP defects are created up to  Obstruction type oxygen defects are relatively much less than distortion type oxygen defects and can only be observed above 40 IFP uranium ions.  In the previous chapter, it is also observed in Figure 6, obstruction type oxygen FPs appear above 15 uranium IFPs. As we mentioned in the previous section obstruction type oxygen defects are only a very small fraction of oxygen defects. From the slope of Yakub potential graph in Figure 10, (18). These outcomes give the idea that small amount of oxygen defects penetrate into the octahedral cages of obstruction type uranium defects. It could be observed that some octahedral cages are hosting one or sometimes two oxygen atoms and these oxygens are vibrating at this interstitial site as an obstruction type defect. Not all of the obstruction type oxygens involve in this incident. This event, which can also be observed visually, result an increase in the slope in Figure 10 above 30 uranium FPs. In Figure 12, when obstruction type oxygens appear above 40 up to 80 uranium IFPs, swelling of the lattice is exponentially decaying with an increasing form, saturated to an upper value.
Here it might be speculated that oxygen and uranium ions, which existed in the octahedral 21 cage as a couple, at some point collapse results a decrease or recombine results an increase in the lattice so that the high scattering of the data above 40 in Figure 12 is observed. When obstruction type uranium ions are created and obstruction type oxygens combine more and more with obstruction type uranium ions, swelling of the lattice reaches a saturation point until no space left for the obstruction defects. These saturation points are % 4 . For the uranium IFP sample, the simulations results are similar to the experimental data obtained in the study of defect production and annealing with the high energy 4 He ions (28).
Because of the large vibrations and the size of oxygen ions, distortion type of defects may also give rise to an enhanced scattering yield even that they are not exactly occupying the 22 center of the channel. Lattice expansion clearly varies linearly with the obstruction type of survived uranium defects while the rest have the exponential dependence. Six coordinated (U-U) uranium interstitials are the reason for lattice expansion. Although there are considerable differences between the saturation value of lattice expansions from experiment and our simulation, the estimated obstruction type of defect concentration is almost the same as experimental value. Saturation value of lattice expansion for both fission and alpha particle irradiation exist because interstitial ion volume grow to an upper limit, but at some point volume is too big for the obstruction type ion to have a stable point.
Next step of this study will concern with evolution of the lattice recovery with the temperature for the supercell with oxygen and uranium FPs. This will enable us to understand the effect of the sublattice on the recovery procedure.