The critical role of hydrogen on the stability of oxy-hydroxyl defect clusters in uranium oxide † is © The Royal Society of

Despite considerable work applying ab initio techniques to model the role of defects on mechanical, structural and electronic properties of oxides, there has been little on the role of trapped hydrogen, despite it being virtually always present. We propose a framework for identifying reversible and irreversible hydrogen traps. We demonstrate that the thermodynamic stability of oxy-hydroxyl defects is de ﬁ ned by an interplay of formation and binding energies. This framework is applicable to all oxides and is essential for describing the solubility and di ﬀ usivity of hydrogen at the macroscopic level. For the most important actinide oxide in nuclear energy, uranium oxide, hydrogen signi ﬁ cantly impacts the stability of oxygen defect clusters, and with increased local hydrogen concentration it forms irreversible traps. Crucially, hydrogen stabilises isolated Willis clusters, named after their discoverer and originally reported in 1963, which all subsequent ab initio calculations have predicted to be unstable, but of course, none considered hydrogen.


Introduction
The presence of hydrogen in oxide materials is oen overlooked despite its signicant impact on materials properties, [1][2][3][4] including mechanical resistivity, 5 conductivity, 6 (photo-)catalysis, 7-9 gate materials, 10 insulator to metal transition 11 and chemo-and photo-chromism. 12 Perhaps one of the most compelling arguments, for investigating hydrogen defects, is that control over these materials properties would only be gained when a complete survey of the chemistry and atom-level structures of complex defects is compiled.
Hydrogen is of primary concern in metals and uranium is no exception, with its legacy from the rst generation of nuclear reactors. As a metal, uranium is susceptible to oxidation in the presence of water vapour and oxygen forming a thin uranium oxide coating layer that provides a protective physical barrier that quenches corrosion of the metal. In common with other metals, 5 the aqueous corrosion of uranium generates hydrogen species that permeate through the oxide layer [13][14][15] and react with the metal to form hydride species. 16 However, uranium hydride is pyrophoric and thus it is a major concern in the design of long-term storage facilities. To prevent formation of the hydride, trapping of hydrogen within the oxide coating layer is required, and this can be best achieved by identifying and controlling the defects that efficiently trap hydrogen.
In the case of uranium oxide, experimental data on solubility and diffusivity of hydrogen is limited, [17][18][19] and the nature of the diffusing hydrogen species is still undetermined, with both molecular 17 and atomic 18,19 hydrogen proposed. The latter, in particular, is essential for predicting any diffusivity and trapping. Evidence of hydrogen trapping was rst reported by Sherman and Olander,19 who saw the release of hydrogen from samples at high temperatures (800-1450 C). Wheeler suggested that this hydrogen may be in the form of strongly bound species (i.e. hydroxyl groups) that are effectively trapped. 17 There is also evidence that oxygen deciency increases hydrogen solubility compared to oxygen-rich conditions. 19 In the latter, the excess oxygen is incorporated as isolated oxygen interstitials at low oxygen partial pressure, which cluster at higher oxygen concentration. [20][21][22][23][24][25][26][27][28] This leads to a variety of potential hydrogen traps. A further level of complexity is then due to the microstructure of polycrystalline oxide materials, which has an impact on the diffusivity of hydrogen, [17][18][19] as it does for the diffusivity of oxygen in uorite oxides. [29][30][31][32] Glascott proposed a comprehensive mathematical model for hydriding initiation in uranium metal, which included hydrogen diffusivity in the oxide coating layer. 33,34 However, such macroscopic modelling is unable to provide insight into the structural features that interact directly with hydrogen in the material. This can only be realised through atomistic modelling that can identify the geometry of hydrogen trapping sites.
Recently, ab initio studies have begun to characterise the nature of hydrogen species, and have found that both hydroxyl groups and hydride species may form, with the latter identied as more favourable. 35,36 Other theoretical work has focussed on the adsorption, desorption and dissociation of water on uranium oxide surfaces, 37,38 which is one of the routes that hydrogen can be incorporated into the uranium oxide lattice and before reaching the metal. Whereas there is data suggesting the existence of hydrogen traps in oxygen-rich uranium oxide, there are no atom-level models to support this proposition. The present work applies ab initio modelling techniques to examine the dependence of hydrogen trapping on oxygen defect clusters under oxygen-rich conditions.

First principle calculations
A cubic unit cell comprised of 32 UO 2 units was generated by a 2 Â 2 Â 2 expansion of the uorite 4 UO 2 unit cell. All calculations were performed using the VASP code, 39 with PAW pseudopotentials and the GGA PBE functional. 40 The Dudarev 41 approach was used for the implementation of the onsite Hubbard U parameter, with U ¼ 4.5 eV and J ¼ 0.54 eV. 42 The planewave cut-off energy was set at 400 eV and a G-centred 2 Â 2 Â 2 Monkhorst-Pack k-point grid was used for all calculations, in line with previous work 22,25,43 and convergence criteria for the electronic and ionic relaxations were 1 Â 10 À6 eV per atom and 1 Â 10 À2 eV A À1 , respectively. All calculations were performed at constant pressure with collinear 1k antiferromagnetic ordering. This is considered an appropriate approximation for the experimentally observed 3k non-collinear antiferromagnetic ordering, 29,44 and is line with previous work. 22,25,26,28 VASP accounts for relativistic effects through scalar relativistic methods 45,46 and spin orbit coupling (SOC). In our previous work, 35 we examined the effect of SOC on hydrogen defects in UO 2 . The produced a consistent shi in the energies, which is in line with previous work for UO 2 . 47,48 SOC does not remove the tetragonal distortion caused by the antiferromagnetic ordering, which can only be removed using the experimentally observed 3k non-collinear ordering. 44 The effect of SOC has also been shown to be negligible on structure, relative stability and electronic properties of other actinide oxides. 49 For all these reasons and as the inclusion of SOC vastly increases the computational expense, it has not been included in here. All structural models were generated using the METADISE 50 code and all structural gures were created using VESTA. 51

UO 2+x structural models
The two smallest oxygen clusters proposed in uranium oxide are the split di-interstitial (I 2 x ) and the 2 : 2 : 2 Willis cluster (W 222 ), shown in Fig. 1. We use the nomenclature common in the literature for the split interstitial clusters (I 2 x ) 22 and have adapted this to also refer to the Willis clusters (W 222 ). Upon the incorporation of excess oxygen the UO 2 lattice is oxidised, such that for every additional oxygen, two uranium atoms are oxidised from U 4+ to U 5+ . The Kroger-Vink notation is commonly used in the literature for defects in solids 22,26 and we use this to describe the point defects used as reference states.
A I 2 x cluster (Fig. 1a) has a central oxygen vacancy ðV O Þ surrounded by three oxygen interstitials ðO 00 i Þ located approximately 1.6Å away in h111i directions. This causes the displacement of three lattice oxygen by approximately 0.3Å from their lattice sites, highlighted in blue in Fig. 1a. In the models presented here the excess oxygen changes the stoichiometry from UO 2.00 to UO 2.06 , and leads to the oxidation of four nearest neighbour uranium ions to the cluster from U 4+ to U 5+ . A I 2 x cluster is found to have a formation energy of À123 kJ mol À1 in agreement with previously reported literature values. 22,23 A W 222 (Fig. 1b) consists of two V O and four O 00 i each located approximately 1Å from an octahedral interstitial site; two O 00 i are found in h110i directions (O 0 ) and two in h111i directions (O 00 ). The W 222 cluster was the rst defect cluster proposed to rationalise the UO 2+x structure, 20 and was initially predicted to be stable using interatomic potential model calculations. 21 However, the advent of DFT has demonstrated single, isolated W 222 clusters are unstable with respect to I 2 x clusters. 23 Recently, DFT calculations have found that edge-sharing W 222 clusters are stable in UO 2+x , but only in a narrow composition range (0.125 < x < 0.25), 25 becoming unstable by x ¼ 0.33, 26 emphasising how sensitive UO 2 defect chemistry is to oxygen partial pressure.

UO 2+x H y structural models
Hydrogen was added to the UO 2.06 models of W 222 (Willis cluster, Fig. 1b) and I 2 x (split di-interstitial cluster, Fig. 1a) clusters and the local hydrogen concentration was varied from 117 to 467 mg H/g UO 2 , which is equivalent to y in UO 2+x H y varying from around 0.03 to 0.12. This is achieved by the addition of 1 to 4 hydrogen species to the oxygen clusters in simulation cell of U 32 O 62+2 . The addition of hydrogen to the oxygen clusters resulted in the formation of hydroxyl species within the cluster. As the defect clusters are made of oxygen and hydroxyl species these defect clusters will now collectively be referred to as oxy-hydroxyl clusters. The chosen congurations were selected to maximise the interactions between the oxygen and hydrogen species; details in the ESI. † The formation of each hydroxyl group within the oxygen defect split di-interstitial and Willis clusters (each containing 4U 5+ ) results in the reduction of a U 5+ to U 4+ . The concentration of U 5+ in each oxy-hydroxyl cluster is reported in ESI. † It is important to note that, hydrogen can interact with oxygen clusters in different ways, and as we discuss later this drastically impacts the stability of the oxy-hydroxyl cluster. The newly formed hydroxyl group can hydrogen bond to lattice oxygen ions (L) or defect oxygen ions belonging to the cluster (D). This makes only two different conformations for each local hydrogen concentration in a I 2 x ( L I 2 x and D I 2 x ), as it is composed of three equivalent O 00 i (Fig. 1a). The lower symmetry of the Willis cluster (arising from the two symmetrically inequivalent O 00 i ) is responsible for the greater number of possible distinct congurations; e.g. D W O 0 ;2O 00 222 denotes three hydrogen atoms, one at a O 0 interstitial site and one at each of the two O 00 sites, and all three species are hydrogen bonded oxygen species of the defect cluster (D).

Energetics of hydrogen in UO 2+x
In order to dene irreversible and reversible hydrogen traps it is necessary to introduce two energy terms, the formation and the binding energies of oxy-hydroxyl clusters.
The formation of the oxy-hydroxyl clusters is considered as the energy of dissolving hydrogen and oxygen gas into a UO 2 matrix according to eqn (1). In order to assess whether the oxy-hydroxyl clusters are more energetically favourable than isolated point defects, eqn (3) is considered, where the most stable isolated hydrogen defect is ½U U þ H 0 i comprising of a hole on the uranium as U 5+ ðU U Þ compensating the hydride ion ðH 0 i Þ. The charge state of the hydrogen interstitial is dependent on the location in the lattice (Fig. 2), where the hydride is more stable than the protonic form by 26 kJ mol À1 . The isolated oxygen interstitial compensated by two U 5+ is given by ½2U i . These interstitial defects come together to form the oxy-hydroxyl defect cluster [cluster], either based on the split di-interstitial, in the presence of hydrogen. In every case, hydrogen now forms protonic defects and so reduces the U 5+ associated with the hydride to U 4+ and reduces an additional U 5+ associated with the O 00 i , leading to the formation of the cluster in a simulation cell with a stoichiometry of UO 2.06 H y .
The preference for formation of oxy-hydroxyl clusters over isolated point defects (i.e. separated oxygen interstitial O 00 i and hydride) is expressed using the binding energy (E Bind ) of the oxyhydroxyl cluster per hydrogen atom in the simulation cell, n, where each of the defect energies are dened as the difference in energy between the defective cell and the equivalent stoichiometric cell containing the same number of UO 2 formula units, e.g. E cluster ¼ E UO 2.06 H y À E UO 2 . The most stable isolated interstitial species within UO 2 for hydrogen (i.e. the hydride anion) and oxygen (i.e. an interstitial oxygen). For an isolated hydrogen ðE H 0 i Þ this is a hydride (E Form ¼ 19 kJ mol À1 ), 35 and an O 00 i in stoichiometric UO 2 ðE O 00 i Þ for oxygen, with a formation energy calculated to be À119 kJ mol À1 in agreement with previous literature. 27 The reference energy for uranium oxide is the energy of stoichiometric UO 2 . A negative value of binding energy indicates an energetic preference for the interstitial species to cluster in an oxy-hydroxyl defect cluster rather than form isolated point defects within the UO 2 lattice, i.e. hydride and O 00 i . We follow the procedure of Minervini et al. 52 for calculating the binding energy.
The most stable oxy-hydroxyl UO 2+x H y clusters The structures of the most stable congurations at each local hydrogen clusters at each local hydrogen concentration, 117 to 467 mg H/g UO 2 , are presented in Fig. 3 along with their formation and binding energies. Data for all other congurations along with their energetics is presented in the ESI. † The most stable cluster is based on a I 2 x (split di-interstitial) geometry at local concentrations of 117 and 351 mg H/g UO 2 , which correspond to 1 and 3 hydrogen species added to the oxygen cluster (À146 and À95 kJ mol À1 /H). This results in the reduction of 1U 5+ to 1U 4+ and 3U 5+ to 3U 4+ within the UO 2 lattice respectively, as the oxidised UO 2+x including a split diinterstitial cluster contains 4U 5+ . Whereas the most stable W 222 (Willis) cluster is found at local concentrations of 234 mg H/g UO 2 and 467 mg H/g UO 2 , which correspond to 2 and 4 hydrogen species added to the oxygen cluster (À119 and À76 kJ mol À1 /H), this is balanced by the reduction of 2U 5+ at 234 mg H/g UO 2 and complete reduction of all U 5+ at 467 mg H/g UO 2 . In these clusters there is a strong interaction between the hydroxyl groups and lattice oxygen atoms. There seems to be an odd-even relationship for the most stable oxy-hydroxyl clusters. Oxygen clusters with odd numbers of interstitials oxygen atoms, the I 2 x , are stabilized by an odd number of hydrogen species, whereas oxygen clusters with even numbers of oxygen interstitials (W 222 ) are stabilized by an even number of hydrogen species.
Energetics of oxy-hydroxyl UO 2+x H y clusters as a function of cluster geometry Fig. 4 shows the formation (eqn (2)) and binding (eqn (4)) energies (in kJ mol À1 ) per hydrogen species as a function of local hydrogen concentration ([H] in mg H/g UO 2 ) for the most stable congurations of the oxy-hydroxyl clusters. We provide separate gures depending on the nature of the oxygen cluster either split di-interstitial (I 2 In each case addition of hydrogen to the oxygen defect cluster resulted in favourable formation energies, however these become less favourable with increasing hydrogen concentration and the effect is less pronounced for D W 222 (Fig. 4d). Additionally, oxy-hydroxyl clusters where a hydrogen bonding network is formed with lattice oxygen atoms ( Fig. 4a and c) are always favoured compared to those with a hydrogen bonding network formed within the oxygen of the defect cluster ( Fig. 4b and 3d). This shows a preference for the hydroxyl groups in the cluster to be arranged such that they hydrogen bond with oxygen of the lattice (L) rather than oxygen of the defect (D).
The preference for defect clusters over isolated point defects is expressed by the binding energy. This is generally unfavourable (i.e. isolated point defects dominate) at low hydrogen concentrations (117-234 mg H/g UO 2 ) and becomes increasingly favourable (i.e. defect clusters dominate) at higher concentrations (351-467 mg H/g UO 2 ). At the highest concentration (467 mg H/g UO 2 ) all clusters with favourable formation energies also have favourable binding energies (Fig. 4). There is no straightforward correlation between stability of the clusters and the total number of hydrogen bonds (Fig. 4). It appears that as the hydrogen concentration increases the number of hydrogen bonds per hydrogen decreases with the exception of L W 222 clusters (Fig. 4c). We have investigated hydrogen interaction with two types of oxygen defect cluster (split di-interstitial (I 2 x ) and 2 : 2 : 2 Willis (W 222 ) clusters) in uranium oxide. We found that the stability and geometry of these clusters radically change as a function of local hydrogen concentration. These oxygen defect clusters act as hydrogen traps and thus will macroscopically affect the material in terms of mechanical properties and hydrogen solubility and diffusion. This will signicantly impact long term storage and disposal of uranium metal as hydrogen diffusion through the protective oxide layer forms pyrophoric hydride at the interfaces.
We found that, remarkably, the presence of hydrogen stabilises the Willis cluster when an appropriate hydrogen bonding network is established, suggesting that trapped hydrogen may have been present when these clusters were originally reported in uranium oxide. 20 Although the Willis cluster is stabilized by hydrogen, it is only the most stable defect at 234 and 467 mg H/g UO 2 , with the split di-interstitial the most stable at 117 and 351 mg H/g UO 2 . This indicates that oxygen defect chemistry in UO 2 is not only strongly dependent on oxygen stoichiometry, 25,26 but also on hydrogen partial pressure. This opens key chemical challenges to determine whether other oxy-hydroxyl clusters will be directly stabilised, to what extent these clusters will appear in other materials, and whether the properties of oxide materials will be affected.
For all clusters, with one exception (ESI Table 4 †), addition of hydrogen resulted in the formation of hydroxyl groups, which are essential for the formation of a hydrogen bonding network capable of stabilising oxy-hydroxyl clusters. Although there is little experimental work on the interaction of hydrogen with UO 2 , hydroxyl groups were suggested as potential traps for hydrogen species by Wheeler 17 at concentrations of up to 10%. Sherman and Olander 19 also acknowledged the presence of hydrogen bound to a variety of defects (rather than simply dissolved as interstitial species) and proposed that oxygen vacancies may be a stable location for hydrogen defects. Indeed, our data shows that the Willis cluster can only be stabilized by forming a hydrogen bonding network that prevents an interstitial oxygen (O 00 ) from moving to the vacant oxygen site (Fig. 1b) and thus relaxing to a I 2 x cluster. In some of the defect congurations, generally the least stable ones, we observe hydrogen species positioned at oxygen vacancy sites but still bound to an oxygen interstitial, which is in line with the proposition put forward by Sherman and Olander. 19 Fig. 4 Energetics of oxy-hydroxyl clusters. Formation energy (red triangles) and binding energy (blue squares) per hydrogen, and total number of hydrogen bonds (black diamonds) as a function of local hydrogen concentration for the most stable oxy-hydroxyl defect clusters.
Based on the formation energies (which dene the thermodynamic stability) and binding energies (which dene the preference of having oxy-hydroxyl) we can identify, from a thermodynamic viewpoint, reversible and irreversible hydrogen traps in uranium oxide. This provides atom-level parameters that could be implemented in mathematical models to determine hydrogen transport in uranium oxide, 17,19,33,34 and each different hydrogen trap now may be considered explicitly. We suggest that oxy-hydroxyl clusters with favourable formation (i.e. negative) and binding energies (i.e. negative) correspond to irreversible traps, as they will form and will not dissociate into isolated defects. Clusters with a favourable formation energy and an unfavourable binding energy (0 < E Bind < 80 kJ mol À1 ) represent reversible hydrogen traps. These correspond to metastable intermediates that are reversible in nature, as the clusters can form but they are not the most stable defect congurations however they do not have sufficient energy for the hydroxyl bond to dissociate. 35 Whereas traps with a binding energy greater than the threshold of 80 kJ mol À1 are still metastable intermediates but are less likely to exist, as there is sufficient energy to convert the hydroxyl groups into hydride defects. This threshold for the binding energy of reversible traps (80 kJ mol À1 ) corresponds to the energy barrier to form an isolated hydride group from a stable isolated hydroxyl species. 35 Using our denition for the irreversible and reversible nature of hydrogen traps in UO 2 , we have made an assessment of the probability of these oxy-hydroxyl clusters to be present within the lattice. This helps rationalise the experimentally observed reversible and irreversible hydrogen traps in UO 2, by providing atom level insights into the nature of defect clusters within UO 2 . However, our calculations do not attempt to determine the absolute concentration of traps as this would require the calculation of free energy terms and is beyond the scope of this work, but represents a key area of future development. Furthermore, while the formation and binding energies allow us to dene the nature of traps as reversible or irreversible, simply being an irreversible trap does not guarantee a high concentration of that trap. As in order for the cluster to have a greater concentration than the isolated point defects, its binding energy should be lower than the formation energy of the individual components. 53 Thus, a trap can be irreversible in nature and have a low concentration. Within the cluster congurations studied in this work, the only case where the binding energy is greater than the formation energies of the individual point defects is the L W 2O 0 ;2O 00 222 cluster (Fig. 3d), which suggests that this might be the dominant irreversible trap for hydrogen in UO 2 .
We have generated thermodynamic distributions of irreversible and reversible hydrogen traps as a function of temperature, by calculating the probability of different clusters based on their formation energy. Eqn (5) displays the calculation of free energy, where the DG i is the free energy for the oxyhydroxyl cluster i, U is the degeneracy of the oxy-hydroxyl cluster (we report the degeneracy of a cluster in ESI †), E Form,i is the formation energy calculated in eqn (2) and we rescale them so that this value to zero for the most stable defect cluster, R is the gas constant and T is the temperature.
The probability is then given by While this approach does not explicitly include the effect of temperature on the defect clusters, as the comparison is between defects all in the solid phase the cohesive energy term will be dominant and entropy terms are unlikely to reverse the conclusions. This again represents key areas for future research, both for the explicit inclusion of temperature and congurational entropy on the stability of defect clusters.
The thermodynamic distribution of all the clusters at each hydrogen concentration is presented in Fig. 5. The calculated probabilities show the predicted nature of the oxy-hydroxyl clusters based on the oxygen defect geometry (i.e. Willis or split di-interstitial) and trap nature (i.e. reversible or irreversible). As the concentration of hydrogen increases, there is a clear shi from reversible trapping, which dominates at the lowest concentration 117 mg H/g UO 2 , to irreversible trapping, independent of cluster type (Fig. 5). Although we cannot compare our data directly with experimental de-trapping data, 17,19 as this accounts for kinetic parameters, it is intriguing that experimental data suggests that hydrogen dissolved in uranium oxide is not released by a diffusion controlled process. Examining the desorption curve for uranium oxide single crystal shows a rst desorption peak at approximately 1100 K and a second and more discernible desorption peak at approximately 1700 K, associated with energies of 80 and 210 kJ mol À1 respectively. 19 This precludes hydrogen diffusion via interstitial sites in the UO 2 lattice and supports the idea that hydrogen is bound to discrete sites, as we see in our ab initio model structures. These hydrogen traps will be characterised by a binding energy and a rate constant of release.
While we calculated the binding energies for oxy-hydroxyl defect clusters in uranium oxide, we have no information on the rate constants for hydrogen release, which depend on temperature. When considering a thermodynamic distribution that accounts for all hydrogen traps at all local hydrogen concentrations (Fig. 6), irreversible trapping (i.e. negative binding energy) appears at temperatures higher than 900 K and is slightly more likely to be based on the Willis geometry (1.3% at 1000 K and 5% at 2000 K) compared to the split di-interstitial (0.3% at 1000 K and 3.5% at 2000 K). Up to 10% of irreversible hydrogen traps appear at 2000 K, whereas the majority (up to 90%) of hydrogen traps will still metastable intermediates (i.e. reversible traps) based on the I 2 x cluster (Fig. 6). As the irreversible traps are stabilised by a more complex hydrogen bonding network, they are more stable at higher local hydrogen concentrations and thus their thermodynamic distribution increases with temperature, contrary to the metastable (reversible) traps that are mainly oxy-hydroxyl clusters with low local hydrogen concentration and a weaker hydrogen bonding network.

Conclusions
To conclude, we found that hydrogen strongly affects the structure and energetics of oxygen defect clusters in uranium oxide, as has been shown for many other oxides. [1][2][3][4][5] Our analysis denes a thermodynamic route to determine reversible (metastable) and irreversible hydrogen traps, and show that there are stable local oxygen-rich regions (i.e. oxygen defect clusters) that can trap and sustain high local hydrogen concentrations despite low overall concentration in the sample. Our ab initio approach allows for quantitative measurement of the population of hydrogen traps, which changes as a function of local hydrogen concentration and temperature. This shows that we need atom-level models and insight if we are to predict macroscopic properties such as the dynamics of hydrogen defects and microstructural changes under non-equilibrium conditions that can arise in service due to the presence of extreme environments (e.g. radiation, temperature). Thus this work demonstrates when investigating defect clusters in oxides that the inclusion of hydrogen as part of these studies is overdue.

Conflicts of interest
There are no conicts to declare.