Theoretical study of xenon adsorption on UO 2 surfaces

During in-pile operation, ﬁ ssion gases such as xenon are produced and precipitates in nuclear fuel as nano-bubbles, modifying the thermomechanical properties of UO 2 fuel matrix. This work aims at precising the nature and the scope of the xenon — UO 2 matrix interaction. To this end, we present a theoretical study of xenon adsorption on UO 2 surfaces, based on Monte Carlo simulation and semi-empirical interatomic potential. We determine xenon adsorption isotherms on previously relaxed ( 111 ) and ( 100 ) UO 2 surfaces, and we compare them to the incorporation ones inside an empty box in order to isolate interface effects. A speci ﬁ c attention is given to the microstructure of xenon in these systems. Finally, an analysis of the mechanical properties ( pressure and stress pro ﬁ les near by the surface ) is achieved in order to investigate the sensitivity to surface orientation and to get pertinent quantities that will supply micromechanical models at higher scale.


Introduction
Uranium dioxide UO 2 is the standard fuel in nuclear pressurized water reactors (PWR). During the reactor operation, the fuel pellets undergo thermal and mechanical stresses. For this reason it is very important to understand the thermomechanical properties of this system both in normal operation conditions and accidental situations (from 300 to 2000 K). During uranium fission reactions, rare gases such as xenon are produced within the fuel pellets. Due to their low solubility, these gases are generally found in bubble shape, from nanometer to micrometer diameters. Due to their impact on the overall behaviour of the fuel rod, fission gases behaviour in oxide fuel has been the focus of attention for decades from both experimental [1][2][3] and theoretical [4][5][6][7][8] points of view. The internal pressure of the bubbles, particularly in the event of a rapid increase in temperature, is suspected of generating internal stress in the fuel. Evaluating this pressure is a key matter especially for nuclear safety aspects [9]. Usually, the pressure inside the bubbles is determined using either the perfect gas or the van der Waals equations of state. Unfortunately, these expressions are totally unsuited in the temperature and pressure ranges encountered in nuclear fuel intragranular bubbles. Indeed, experimental results indicate that the xenon pressure order of magnitude is of a few GPa in these systems [1,2]. That is the reason why alternative equations of state have been suggested [7,8], but they are still not taking into account the Xe-UO 2 interactions at the gas-matrix interface, which implies neglecting confinement effects although they are known to be very significant [5,6]. Similarly, the pressure inside the fuel matrix is determined through the Laplace's law and using average bubble radius and UO 2 surface tension, without taking into account the Xe-UO 2 interactions.
The present work aims to overpass these limitations by means of atomistic simulations. Indeed, computer simulations have been considered as a new research tool for a better understanding of processes occurring at the atomic scale, but also for processes that could not be observed directly by experimental means. A previous work [10] focused on xenon incorporation in UO 2 nanocavities by mean of GCMC approaches, revealed that xenon adsorption isotherms exhibit an abrupt transition from a dilute phase towards a dense one as a function of increasing chemical potential. Whatever the temperature, the corresponding internal gas pressure values are in agreement with most recent atomistic studies, but definitely prove that mesoscopic models based on gas state behaviour are totally unsuited and must be changed. In order to have a global physical view of the situation, the next step is now to investigate also, by similar approaches, the pressure inside the matrix.
At the atomic scale, intragranular bubbles take the shape of an octahedron, presenting mainly (111) and (100) facets [3]. Given the complexity of this octahedron stability study, we have simplified the problem by decoupling the various effects. We have previously [11] studied the relaxation process of these surfaces and characterized the resulting UO 2 surface microstructure, as well as the energetic and stress profile in its vicinity. In all cases, it was found that whatever the quantity under consideration is (relaxation, energy, stress), the presence of the surface only affects the outermost layer for the (111) direction, but a more extended region for the (100) one. This result will have important consequences on micromechanical models that should take into account these two different behaviours, for instance by introducing different effective surface thicknesses. The next step of this study is now to extend it to the case of xenon-UO 2 system and to answer the question: what is the stress propagation inside the xenon-UO 2 system and does this propagation depends on the UO 2 surface orientation?
For this purpose, we determine here the adsorption isotherms of xenon on previously relaxed surfaces, and we compare them to the incorporation ones inside an empty box in order to isolate gas-matrix interface effects. A specific attention is given to the microstructure of xenon in these systems. Finally, an analysis of the mechanical properties (pressure and stress profiles near by the surface) is achieved in order to get the pertinent quantities that will supply micromechanical models at higher scale.

Methods
2.1. Simulation systems 2.1.1. UO 2 surface modelling As previously described [6,10,11], the system modelled here is xenon atoms in interaction with a network of UO 2 , generated by infinitely duplicating the elementary cell in the three space directions. The UO 2 elementary cell is fluorite type and its lattice constant is a=0.547nm as determined experimentally at 296 K [12].
Such structure cannot simply be reduced to a stacking of atomic planes along any direction. Thus, in the (111) direction, the structure can be viewed as a stacking of 'triple layers', made up of O-U-O successive atomic planes. The situation is simpler in the (100) direction along which the structure is an alternate stacking of oxygen and uranium planes.
In practice, we use as a starting point a perfect fluorite structure cell of 2592 atoms (6×6×6 supercells), corresponding to a UO 2 volume of about 35 nm 3 , and we extend Lz to = = Lz a 12 6.564 nm to mimic a surface perpendicular to the Z direction.

Pure xenon
With respect to the pure xenon system (in the absence of UO 2 surfaces), which is used as a reference, we have considered an empty box with fixed dimension Lx=Ly=Lz=3.272 nm duplicated in the three space directions by periodic boundary conditions. The size of this system was chosen such as the cell parameter exceeds 2r Xe−Xe (r Xe−Xe =1.5 nm is the range of xenon-xenon interactions) and that the total cell volume is similar to the free volume considered for the two previously mentioned surface situations. It should be noticed that the minimal density variation accessible by our study is bounded by the simulation cell size: . Therefore, the very low density systems (for highly negative chemical potentials) will not be relevant.

Monte carlo thermostatistic tools
We have separated our study in two successive steps: the first one is to relax the matrix surface in absence of xenon, described in [11], and the second one is to adsorb xenon on the relaxed surface. We have checked the validity of our assumption not to relax the matrix under xenon incorporation by adding in some high density cases post-incorporation relaxation studies, without any changing in the main conclusions. All the simulations are performed by Monte Carlo simulations, in canonical (NVT) ensemble for relaxation steps, and in grand canonical (μ,V,T) ensemble for the incorporation study.
In the canonical ensemble, the only proposed events between a configuration i with energy E i and a configuration i+1 with energy E i+1 are the atom displacements, which are accepted with the probability P i,i+1 : with k the Boltzmann constant and T the temperature.
In the grand canonical ensemble two other types of events are proposed, the addition or the removal of one xenon atoms, with the respective probabilities + P i i add , 1 and + P i i rem , 1 [13]: where N i is the number of atoms in the configuration i, V the system volume, L = p h kTm 2 the De Broglie wavelength, m the xenon atomic mass, h the Plank constant and μ the fixed chemical potential.
In practice, the type of event is chosen randomly, according to the following distribution: 40% for moves, 30% for additions and 30% for removals. Finally, the displacement vector is randomly chosen too, in a distance range from 0 to 5% of the Xe-Xe equilibrium distance. The Grand Canonical Monte Carlo (GCMC) approach allows us to determine isothermal xenon incorporation curve. The thermodynamic equilibrium is assumed to be reached when the statistical number of created particles equals that of the removed ones, which generally requires more than 3×10 8 proposed steps. Additional Canonical Monte Carlo steps are then achieved to average the physicals quantities of interest (the number of particles N Xe , the xenon pressure P, the system energy E...). Typically, 10 000 configurations each 3000 steps are used, to avoid correlation between consecutive steps.

Energetic model
Semi-empirical approaches are better adapted here than ab initio ones since the present work involves the study of thousands atom systems. In addition, the physical properties of interest, such as mechanical or thermal properties, do not require a precise calculation of the electronic structure.

UO interaction model
There exist a variety of UO 2 effective energetic models in the literature [14][15][16][17]. We choose here the Basak potential [18] which accounts for the interactions involved in UO 2 and correctly reproduces macroscopic quantities such as the elastic constants, especially at high temperature and for structures far from the perfect crystal [12,19]. In addition, this potential reproduces fairly well surface properties, and in particular surface energies for main orientations [20]. Within this potential the total energy E tot is written analytically as a function of the atomic positions, as a sum of an electrostatic term (E Coulomb ), a Buckingham-Hill (E Buck ) and a Morse one where E Coulomb is given by: where q i and q j are the particle charges for i, j = O, U (with q 0 =−1.2 and q U =2.4 [18]), and r ij the interatomic distances. This term is computed using an usual Ewald summation.
The Buckingham Hill component has the following analytical form: and the Morse component this one: The parameters A ij , ρ ij , C ij , D ij , β ij and * r ij used for the previous equations, based on the Basak potential for U-O interactions [21], Grimes one for Xe-UO interactions [16] and modified Brearley potential for Xe-Xe interactions [22,23], are summarised in table 1.

Stress profile determination
The symmetric per-atom stress tensor for each i atom is computed through the LAMMPS code following the formula [24]: where a and b take on values (x, y, z), r i a is the i-atom position, f ij is the force resulting from the pairwise interaction (except Coulombic contribution) between the i and j atoms; f ij c b is the force resulting from the short range Coulombic interaction between the i and j atoms. K space is the contribution from long-range Coulombic interactions, calculated using the method described in [25] for the Ewald method. In the stress expression, the force term expressed in (8) needs to be divided by the per-atom volume to have unit of stress. However, it is not easy to unambiguously define an individual atomic volume, and we have chosen to keep here this stress * volume formulation, which leads to use eV unit.
As previously done [11], the off-diagonal S xy , S yz and S xz components of the tensor are found negligible, consistently with no residual shear stress induced by the presence of the surface, therefore they are not represented here. We have also checked that, due to the crystallographic symmetries of the structure, the S xx and S yy components are equal, and we will use the following notation = = // S S S xx yy and = S S zz . These per-atom values are averaged over an hundred configurations generated by LAMMPS Molecular Dynamic at 300 K neglecting the thermal expansion, and they are then averaged in layers perpendicular to the surface to obtain stress profiles.

Temperature effect
The physical incorporation of xenon inside the previously described systems has been achieved within GCMC, for different chemical potentials and temperatures. The values of explored chemical potentials have been chosen in the characteristic range of the xenon state inside the fuel, namely [−2 eV; 10 eV], taking into account the energy of a xenon atom inside the UO 2 matrix in interstitial (∼20 eV) or vacancy (∼5-10 eV) positions [6,26,27]. The temperature range has been chosen to be consistent with the in-pile conditions, and extends from 300 to 2000 K. Figure 1 represents the corresponding incorporation isotherms with or without UO 2 surfaces. We recover the previous qualitative results of isothermal adsorption in nanobubbles [10]: all curves exhibit the same general behaviour with a first zero or almost zero part followed by a more or less abrupt jump to a new plateau at a critical chemical potential μ c . This behaviour is characteristic of a phase transition from dilute to condensed system. Both the values of the critical chemical potential μ c and that of the saturation density ρ s at the plateau, depend on the temperature:  • at 300 K the xenon saturation density ρ s corresponds to about 6.5 g.cm −3 , which is around twice ρ eq =3.6 g.cm −3 , the metallic xenon phase density [28]; • the higher the temperature, the lower μ c is and the higher ρ s is. For example, ρ s increases by about 40 % between 300 and 2000 K. Furthermore, temperature increase smoothes the phase transition. This temperature dependence of ρ s value is far from being intuitive, and should be related to thermal fluctuations as previously discussed [10].
Based on the comparison of the three system isotherms, the impact of the UO 2 surfaces on the number of adsorbed xenon atoms seems to be negligible in all the temperature range and in almost the whole range of chemical potentials (m < 4 eV).

Xenon structuring 3.2.1. Pure xenon
In order to clarify this qualitative description of a diluted/dense phase transition, we have characterized the different obtained microstructures. In all the temperature range the structuring of xenon is very comparable, so that we limit ourselves here to the presentation of the results obtained at 1200 K. In the figure 2, we can observe a very dilute phase at low chemical potential and, on the contrary, a dense phase, which in addition seems to have a crystalline order at high chemical potential.
In order to characterize the nature of this crystalline order, we have analysed the local atomic environments by polyhedral template matching, using the Ovito [29] software. The result is shown in figure 3 in which the evolution of the different atomic populations as a function of the chemical potential is represented for the three isotherms (pure Xe, on UO 2 (100) or (111) surfaces). It can be observed that the condensed phase of xenon is constructed by co-existing close-packed (FCC/HCP) and more open (BCC) phases, which is consistent with the data from the literature [30,31], although it is difficult to conclude as to a real thermodynamic coexistence between the different structures, or the prevalence of one over the others.
In what concerns the xenon structuring, we can note from the previous figure that it occurs at lower chemical potentials in presence of a surface than in an empty box, and in the former case for a slightly higher chemical potential on a (100) surface than on a (111) surface. Therefore, the UO 2 surface presence seems to facilitate the structuring of the xenon, the (111) surface appearing more efficient in this way, even if it does not impact the number of Xe adsorbed atoms as stated above. We will see in the next section that this will have consequences on mechanical properties.
Xenon BCC phase can be noticed, which could suggest that by preventing the development of this phase, HCP / FCC / BCC competition is no longer as important and no longer interferes with the ordering of the xenon system which oscillates between the different crystal orders.

UO 2 (111) surface effect on Xe adsorption
In order to clarify the UO 2 surface impact on xenon structuring, we carried out a more detailed structural analysis of systems at 1200 K. In the figure 4, which illustrates the distribution of the atoms as a function of their position perpendicularly to the surface for the configuration obtained for μ=4 eV, we notice that xenon atoms form planes, parallel to the (111) surface and constituted of about the same number of xenon atoms as in the first ). This plane structuring is observed for μ2 eV. More precisely, we observe a succession of planes parallel to the (111) surface, regularly distributed at an average distance of 2.9 to 3.0 Å for μ varying between 1 and 4 eV, as illustrated in figure 4. For lower chemical potential, we have also observed the presence of some (one for μ=0 eV, 3 for μ=1 eV)) xenon planes in direct vicinity of UO 2 surface, which confirms that the surface presence helps the plane structuring of the xenon. This structuring is consistent with the results of Geng et al [32]. As soon as Xe slab structuring is achieved, the distance between this first Xe slab and the last O one is of the some order of magnitude than the Xe-Xe interplane distance and decrease from 3.02 Å (μ=1 eV) to 2.87 Å (μ=4 eV).
Further analysis shows that, when the xenon density permits, the first xenon planes (at the interface) have exactly the same number of atoms as the uranium and oxygen planes (θ=1). From the point of view of structuring, the xenon atoms belonging to these planes assume exactly the same positions as the neighbouring uranium or oxygen atoms (see figure 5), thus indicating a pseudomorphism between the UO 2 planes and those of the adsorbed xenon, which therefore form a hexagonal plane (111). The same pseudomorphism was observed at 300 K for a chemical potential between 2 and 4 eV, at 600 K for a chemical potential between 1 and 4 eV and at 1200 K for a chemical potential of 2 eV. These conditions correspond to xenon density range of 5.6-6.7 g.cm −3 (higher than ρ eq =3.6 g.cm −3 .) In these cases, the mean lateral distance between the first nearest neighbours of the Xe d Xe−Xe =3.81 Å is very close to that of the uranium d U−U and the oxygen d O−O of the same plane. This value is less than the distance between the first nearest neighbours of xenon in the FCC and BCC systems at equilibrium at 0 K, which is 4.3 and 4.24 Å, respectively. The xenon is therefore compressed laterally, but less than in the direction perpendicular to the surface (3 Å at 1200 K and μ=2 eV). A tetragonalization of the xenon structure is thus observed, which tends to compress it even more at the interface (2.8 Å at 1200 K and μ=2 eV).
The observed pseudomorphism can be seen as the signature of effective lateral interactions (as meditated by the substrate) between xenon atoms, much weaker than the interactions of the latter with the UO 2 substrate atoms. This effect obviously depends on the chemical potential, since an increase of the later induces an increase of the density, which leads to the incorporation of xenon atoms both in and between the planes described above, and then strongly disturbs the scheduling.

UO 2 (100) surface effect on Xe adsorption
The figure 6 illustrates the distribution of the atoms as a function of their position perpendicularly to the surface for the configuration obtained for μ=4 eV. As previously on the (111) surface, we observe a succession of planes parallel to the (100) surface, regularly distributed at an average distance of 2.9 to 3.2 Å for μ varying between 0 and 4 eV. The most striking feature is that, although well defined, this layers are thicker than in the previous case, and are significantly more close-packed than the underlying O plane (q~4 3, i.e. almost the density of a (111) layer).
Moreover, the analysis of the xenon planes shows that, in no case, we obtain planes identical in number and/ or position to those of the oxygen or uranium ones at the surface.
All these results seem to reveal that, whatever the orientation of the UO 2 substrate, the xenon grows as a stacking of (111) planes.

UO 2 strain
We have also studied the evolution of the UO 2 layer structure as a function of the xenon chemical potential. We did not observe any additional restructuring of the UO 2 surfaces in the presence of Xe, whatever the chemical potential used, which is consistent with the results of [11]: the restructuring of the UO 2 surfaces is essentially driven by the Coulomb forces and little affected by the presence of other species.
However, the presence of Xe leads to a pressurization of the UO 2 surface and therefore, a priori, to a contraction of the latter in the direction perpendicular to the surface. In order to clarify these results, and in particular the effect of strains related to the presence of Xe on the UO 2 slabs, we were particularly interested in the evolution of the U-U and O-O interplane distances as a function of the chemical potential.
Let us first consider the (111) orientation. We observe that the interlayer distance decreases as μ increases, and in the same amount for all the atomic layers. More precisely, we can fit linearly this μ dependency of the interlayer distance d s=111 as: The found values of d s 0 and a s are tabulated in table 2.    We can also notice, as in [11], that the dilation of the outermost (OUO) trilayer ( ). We can notice that the slope for the surface interlayer distance decrease is about twice as large as in the bulk. The surface planes seem to accommodate the xenon countermeasures more than the bulk ones, which are very weakly impacted.
Finally, we can compare the effect of surface orientation on the interlayer distance sensitivity to the xenon presence. The main result is that the xenon impact on the UO 2 bulk is lower for the (100) than for the (111) orientation, contrary to its impact on the UO 2 surface layer. These results seem to indicate that the surface (100), more disordered than the (111), dampens the stress induced by the xenon presence.

Local stress analysis
In order to clarify previous results, it is now necessary to analyse the stress levels in the xenon and the bulk, which are fundamental parameters for higher scale micromechanical models. As a reminder, the constraint analysis in the pure UO 2 surface was described in a previous work [11], which in particular highlights the importance of the electrostatic contribution and a behaviour very different between the apolar (111) and polar (100) surfaces.

(111) surface
The // S and S ⊥ profiles as a function of the xenon chemical potential at 1200 K are plotted on the figure 7 (O atoms on the left and U on the right). We compare the evolution of these profiles between the different layers of atoms from 'C1', the outermost layer, to the 'Bulk', i.e. the layer in the middle of the slab and exhibiting the same behaviours as an infinite perfect solid layer. First of all, we can observe that whatever the xenon chemical potential, the stress profile is rapidly damped into the bulk, only the surface layer being found to undergo a different stress from the bulk.
From this graph, we notice that the curves 'C2', 'C3' and 'Bulk' are superimposed. This confirms that only the outer part of the first triplane (OUO) is impacted by the presence of xenon. Moreover, the difference between the stress in the surface plane and that in the plane of the volume leads to the measurement of the surface stress in the presence of the xenon and therefore to its evolution with the chemical potential.

Oxygen atoms
For the oxygen atoms, we notice that the parallel stress // S decreases almost linearly with the increase of the chemical potential: the oxygen atoms are all the more compressed as the xenon potential increases. The decrease of this component between the two extreme chemical potentials (−1.6 and 5 eV ) is about 0.85 eV. This variation is almost identical for the different layers, which suggests that the effect of increasing the density of the xenon system has an homogeneous effect on the parallel stress. The deviation of this stress tensor component between the external surface 'C1' and that of the volume 'Bulk' leads to a surface constraint D/ / S 2.2 eV at 1200 K independent of xenon chemical potential, and hence of xenon density.
The perpendicular stress S ⊥ also decreases almost linearly with the increase of the chemical potential. This decrease is directly attributable to the increase in the density of xenon on the surface, which induces an increase in the pressure exerted on the surface of UO 2 . In contrast to S // , the difference between this component at the external surface stress and that in the volume (ΔS ⊥ ), depends on the chemical potential of the xenon: it decreases by 0.9 eV in the absence of xenon to 0.5 eV for μ=5 eV.
Thus, the first xenon effect is to compress the outermost oxygen planes as its density increases.

Uranium atoms
For uranium atoms, we notice that parallel stress S // in the different layers is almost constant with the chemical potential. This is due to the fact that uranium atoms remain almost immobile during the relaxation of the surface (after adsorption) and that they do not feel the presence of the xenon atoms, which is screened by the external oxygen layer. This observation is consistent with the fact that the parallel component of the stress tensor ΔS // on the surface remains identical to its value (0.8 eV ) without xenon. The perpendicular stress of the uranium atoms decreases in a similar way (with a slope twice as great) to that of the oxygen atoms with the chemical potential (the xenon decreases the stress in tension). As for the perpendicular component of the surface stress, it remains almost constant and equal to its value (;−0.8 eV ) without xenon.

Xenon atoms
The stress profiles evolution as a function of the chemical potential for the different xenon atom layers is plotted in figure 8. We can see that the two components are almost identical, indicating an isotropy of the state of stress in the xenon layer.
This stress becomes more and more negative when the chemical potential increases in association with higher compression. The comparison between these profiles for the different layers shows that these quantities are independent of the distance to the surface of UO 2 (profiles are almost superimposed). It can therefore be deduced that there is no need to define a 'surface stress' in the xenon layer in the sense that we have done for the UO 2 substrate.

The whole system
In the figure 9, we plot the per atom total constraint ( ) ( ) Tr S 3 averaged by 'effective UO 2 layer', as a function of its azimuthal position at 1200 K for various values of the chemical potential (-1.6 and 5 eV ). The effective layer for orientation (111) refers in the UO 2 substrate to (OUO) triplanes and the xenon layer to atomic planes parallel to the surface.
We can see on this profile that the role of the interface is only sensitive to the first (OUO) triplane, which is less compressed than those of the volume. The full layer is in compression, especially at high chemical potential. This compression that results from the presence of the xenon on a layer of confined UO 2 will be useful in the perspective of implementation of our results in micromechanical codes.

(100) surface
Similarly to the (111) surface, the evolution of the perpendicular S ⊥ and parallel S // stress tensor components with the xenon chemical potential at 1200 K is plotted in figure 10, for the O (left) and U (right) atoms and for the different atom layers of the (100) surface.
It can be seen in this figure that, unlike the (111) surface, it is the stress state in the first two atomic planes of both U and O that differs significantly from that of volume and remains non-negligible in the third plane. Thus, as in our analysis of the (100) surface without xenon [11], we find that the first two planes of uranium and oxygen are perturbed by the presence of the surface, and remain so, even in the presence of xenon.

Oxygen atoms
For the oxygen atoms, similarly to the (111) surface, the parallel stress component S // decreases as the chemical potential increases. The decrease of this component between extreme chemical potentials (−1 and 5 eV ) or the surface layer (C1), about 0.85 eV, is higher than that of (111) surface (about 0.65 eV in this range). We can also notice that the bulk stress component S // (Bulk) remains almost constant in this range of chemical potential. The perpendicular stress component S ⊥ also decreases with the chemical potential increase, but here, this linear decrease seems to affect all the oxygen layers until the bulk.

Uranium atoms
Concerning the uranium atoms, contrary to the (111) surface, the S // (C1) component decrease with the xenon chemical potential increase is high (about 1.1 eV) whereas it remains almost null for the internal layers. The surface stress S ⊥ also decreases according to the chemical potential from 5 to 4.2 eV. As for oxygen, the stresses in the second and third plane of uranium vary in the same order of magnitude.

Xenon atoms
In figure 8 (right), we plot the stress evolution as a function of the chemical potential for the different xenon atom layers. From the first observation, we can notice that a very similar behaviour to the one of the (111) surface is observed, namely that the two parallel and perpendicular components are quasi-identical, which is the signature of an isotropic state of stress in the xenon layer. Moreover, the orders of magnitude are the same for the two surfaces, which indicates that the stress in the xenon layer does not depend on the orientation of the substrate of UO 2 on which it is adsorbed. We can see on this profile that the role of the interface is only sensitive to the two first layers of UO 2 , which are less compressed than those of the volume. The full layer is in compressive stress, especially at high chemical potential.

Discussion
As illustrated in figure 12, we can summarize all the previous results by a quite different behaviour of the UO 2 surface as a function of its orientation: in the (111) surface, only the first surface layer is affected by the presence of the xenon, and the per atom stress evolves linearly with the xenon one. For the (100) surface, the xenon stress impact is converged only above the fourth layer, and the stress that is viewed by the bulk layers is lower than in the (111) case. The (100) surface leads to a more efficient damping, which can be connected and imputed to a greater accommodation of the surface due to its initial disorder.

Conclusion
We have presented here a theoretical study of xenon incorporation in UO 2 (111) and (100) surfaces, by mean of GCMC approaches based on semi-empirical potentials. The corresponding results allowed us to highlight the following points.
During Xenon incorporation in an empty box or xenon adsorption on UO 2 surfaces, we observed a phase transition from a diluted to dense phase. The adsorption isotherms depend on the temperature of the system and the chemical potential of the xenon but are independent of the surface presence. The dense phase of xenon has a complex structure with the coexistence of BCC and FCC phases, and adopts a (111) stacking perpendicular to the UO 2 surface, whatever its orientation ((111) or (100)). This highly compressed layered structure of xenon develops from a pseudomorphic interface in the case of UO 2 (111) and a disordered interface buffer layer in the case UO 2 (100).
The mechanical analysis of the surfaces after xenon adsorption revealed a decrease in the stress of oxygen and uranium atoms induced by the presence of xenon on the UO 2 surfaces. This decrease is all the more important as the density of the xenon atom increases. It can also be noted that the difference between the stresses of the surface layer and the bulk increases with the chemical potential in the case of the (100) surface but remains constant in the case of the (111) surface, which seems to indicate that the surfaces (100) make it possible to better mitigate the xenon pressurization and to limit the stresses in the matrix, most probably due to its structural disorder. It would obviously be interesting to compare the results that we have just summarized with experimental data and/or other theoretical works. Unfortunately, the only reported experimental observations concern bubbles included in matrices and not surfaces. Moreover, at least to our knowledge, there are no other calculations performed on systems studied here, either in DFT or using semi-empirical potentials. These results should now be introduced in advanced micromechanical models, which should take into account the different behaviours associated to each surface orientation, for instance using various effective surface thickness and surface tension values.