Ab Initio Atomistic Thermodynamics of Water Reacting with Uranium ...

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Ab Initio Atomistic Thermodynamics of Water Reacting with Uranium Dioxide Surfaces P. Maldonado,*,† L. Z. Evins,‡ and P. M. Oppeneer† †

Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden Swedish Nuclear Fuel and Waste Management Co., Blekholmstorget 30, SE-10124 Stockholm, Sweden

‡

ABSTRACT: Using first-principles simulations, we study the temperature- and pressure-dependent adsorption reaction of water on the flat (111) and (211) and (221) stepped surfaces of uranium dioxide. Our calculations are based on the density functional theory (DFT) corrected for on-site Coulomb interactions (DFT+U) for describing the chemical interaction of water with UO2, in combination with ab initio molecular dynamics simulations to capture the temperature dependence of the reaction. We compute the pressure−temperature phase diagrams and establish the thermodynamic boundaries which govern the feasibility of water adsorption at these surfaces. Effects of water coverage on the surface adsorption reaction have been taken into account. We find that the dissociative adsorption reaction of water on stepped surfaces can be analyzed as two separated reactions, the dissociative water adsorption on the step edge and the water adsorption on the terrace. The most stable water adsorption upon modification of the water partial pressure and temperature is adsorption on the (211) step edge, followed by adsorption on the (221) step edge and being the least favorable for the (111) surface. We conclude that these UO2 surfaces will always react with water at room temperature and atmospheric pressure, leading to water dissociation and a modification of the step morphology.

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reducing;4 however, it has been shown that oxidizing conditions enhance the dissolution.5 In a reducing repository environment, which is expected, for example, in the Swedish KBS-3 system,6 oxidative corrosion could still be affecting the spent fuel due to the radiolysis of water, which produces oxidants that can attack the UO2 surface. However, in the KBS-3 system, oxidative dissolution is expected to be suppressed by the so-called hydrogen effect, which has seen to be effective while there is anoxic corrosion of metallic iron.7 Thus, nonoxidative dissolution of UO2 must be considered as an important process by which the radionuclides contained within the spent fuel matrix will be released to the water. The first step in fully understanding this process is to investigate the energetics of water reactions with UO2 surfaces. Uranium dioxide has been the object of extensive studies during the last decades, both experimentally4,7−13 and computationally.14−23 The complexity of the material, involving

INTRODUCTION The treatment of spent nuclear fuel is currently the issue of an intense debate. Different strategies have been proposed to solve this issue. Nuclear transmutation1 and nuclear reprocessing2 are promising strategies that however do not represent a complete solution to the problem, since they also produce nuclear waste. In addition, these routes are only compatible with a commitment to nuclear power for the foreseeable future. If nuclear power is to be phased out and replaced with other energy sources, direct disposal of the spent nuclear fuel in deep geological repositories3 emerges as the most promising solution for effective long-term isolation of the spent fuel. However, its implementation requires a fundamental understanding of fuel corrosion processes in order to present a safety case based on a scientifically sound estimation of the environmental impact of the planned repository. Uranium dioxide UO2 is the most commonly used fuel in the nuclear reactors operating today, leading to a spent nuclear fuel consisting of mainly UO2with only a small fraction of highly radiotoxic long-lived actinides and fission products. Uranium dioxide has a very low solubility in water if conditions are © 2014 American Chemical Society

Received: February 18, 2014 Revised: April 1, 2014 Published: April 1, 2014 8491

dx.doi.org/10.1021/jp501715m | J. Phys. Chem. C 2014, 118, 8491−8500

The Journal of Physical Chemistry C

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approximation (L(S)DA) as the exchange-correlation functional. To improve the treatment of the highly correlated system, we have used density functional theory including an additional Hubbard term (DFT+U).43−45 Within the L(S)DA +U approach, we have used the Liechtenstein et al. formulation,44 where Hubbard and exchange parameters, U and J, respectively, are introduced to account for the correlation between the uranium 5f electrons; this helps to remove the selfinteraction error and improves the description of correlation effects. We have chosen a Hubbard U value of 4.5 eV and an exchange parameter J value of 0.54 eV which has been shown to provide good results.22 To deal with the problem of the generation of metastable states when using the DFT+U methodology, we have used the occupation matrix control (OMC) method proposed by Dorado et al.27 This method consists of the direct control of the strongly correlated electron occupation matrices. We have observed that, regardless of what the initial nondiagonal occupation matrices as proposed by Dorado et al. are, the final state will only depend on the sign of one of the two independent variables of the occupation matrix. Therefore, we only need to check two initial occupation matrices to get the lowest-energy state of the simulated cell. The electron−ionic-core interaction on the valence electrons in the systems has been represented by the projectoraugmented wave potentials (PAWs).46 For U and O atoms, the (6s, 7s, 6p, 6d, 5f) and (2s, 2p) states were treated as valence states. A plane-wave basis with an energy cutoff of 520 eV was used to expand the electronic wave functions. The Brillouin zone integrations were performed on a special k-point mesh generated by the Monkhorst−Pack scheme47 (6 × 6 × 6 in the bulk). The sampling of the Brillouin zone for the surfaces was performed with a 3 × 3 × 1 Monkhorst−Pack kpoint mesh, and a Gaussian smearing of SIGMA = 0.1 eV was used. The electronic minimization algorithm used for static total-energy calculations was a combined blocked Davidson +RMM-DIIS algorithm. The optimized surface geometries were obtained by minimizing the Hellmann−Feynman forces until the total forces on each ion converged to better than 0.02 eV/ Å. To assess the vibrational modes, we imposed the criterion that the total forces should be less than 0.0004 eV/Å on the nonfrozen ions. We use the collinear 1k antiferromagnetic order in which the spins of uranium atoms change spin along the Oz axis, as an approximation of the noncollinear 3k antiferromagnetic order found as the experimental ground state.48 Although below a Néel temperature of 30 K, UO2 exhibits a static Jahn−Teller distortion of the oxygen lattice, to avoid a complication arising when modeling the surfaces in this work, we only consider the fluorite structure. Within the fluorite structure, two different optimized geometries that provide similar results have been obtained, i.e., the fcc and tetragonal geometries. Although we have found that the systems adopt a tetragonal lattice in the ground state, the energetic difference with respect to the fcc lattice is only of a few meV, as has also been shown previously.49 For this reason, when constructing the surfaces, we have only considered the fcc geometry. The surfaces are also built by assuming an antiferromagnetic order (see below). The bulk occupation matrices have been used as initial occupation matrices for surface calculations. The spin− orbit interaction has not been included because it was previously shown to have a very small influence on the structural properties, phonons, or surface energies of UO2.15,21,50,51

strongly correlated electrons and magnetic properties along with the failure of standard approximations to describe those materials, evidences the necessity to get a better understanding of the surface properties and surface reactivity with atomic and molecular complexes. Previous works have shown that to achieve a reliable description of the strongly correlated 5f-electrons a DFT description supplemented with on-site Coulomb correlations is essential.23,24 However, the use of the DFT+U formalism may create numerous local-energy minima, or metastable states, which restricts the knowledge of the system. To circumvent this problem, different approaches have been proposed, i.e., the ramping method,25 the quasi-annealing approach,26 or the occupation matrix control method (OMC).27 In spite of the controversy about the efficiency and accuracy of the different approaches to obtain the ground state, the OMC approach has been proven to provide a good route to obtaining the ground state at low computational costs. The understanding of the surface reactivity requires a fundamental knowledge of the electronic surfaces, surface geometry, and surface stability. A large number of works have been recently dedicated to improve the understanding of those quantities for both anhydrous and oxygen rich environments in fluorite materials.21,28−36 Thus, it has been shown that the (111) surface is the most stable surface, and that the surface geometry plays a very important role, leading to the prediction of the relative surface stability order.36 Since water is an essential part of corrosion and dissolution reactions,37−39 it is crucial to understand the interaction of water with the solid surface. Despite the extensive experimental investigations dedicated to spent nuclear fuel dissolution, it is only recently that a few theoretical works14,16,19,21 have addressed the characterization of the uranium dioxide−water interaction. Although they have used similar methodologies, the provided adsorption energy results differ from one work to another. Moreover, only an analysis of the adsorption on the (111) surface has been carried out, although this is the less reactive surface and therefore the less important one in dissolution processes. To the best of our knowledge, only Alexandrov et al.40 addressed the study of more suitable surfaces for dissolution, though only for ThO2 and CeO2. It is also important to mention that in the previous works there is a lack of simulations taking into account thermodynamic conditions, which could explain the feasibility of the water adsorption upon modification of temperature and water partial pressure. Therefore, an encompassing study to clarify the magnitude of the adsorption energies, along with a study of the interaction of water with stepped surfaces, taking into account the environmental conditions of pressure and temperature is still needed. With the aim to better understand and clarify adsorption of water onto different surfaces, we study the adsorption energy and dissociative energy of water molecules onto different surfaces and aim to understand the mechanisms of UO2 dissolution and how surface structure and morphology affect the water−solid reactions. This is investigated here for different degrees of water coverage on the surfaces as well as for varying pressures and temperatures.

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METHODOLOGY Computational Methods. The calculations were performed with the Vienna Ab-initio Simulation Package (VASP version 5.2.12)41,42 with the local spin-polarized density 8492



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To validate our calculations, we compared the equilibrium V and the bulk moduli of the bulk fcc crystal with the experimental values. The values found for the lattice constant, 5.449 Å (5.47 Å), and the bulk moduli, 218 GPa (207 GPa), are in good agreement with the experimental values given in parentheses. These values are in slightly better agreement with the experimental results than those obtained when using GGA +U.50 Each surface has been modeled as periodically repeated slabs containing 3, 7, and 12 atomic planes for the (111), (211), and (221) surfaces, respectively, with a vacuum gap bigger than >15 (we have vacuum thickness convergence). To account for the antiferromagnetic character of UO2 and to examine a number of surface coverages for one-sided water adsorption, we have used 2 × 2 surface unit cells for the (111) and (211) surfaces and a 1 × 2 supercell for the (221) surface. Ferromagnetic and Antiferromagnetic Order in the OMC Approximation. The close energies between the ferromagnetic and antiferromagnetic configurations52−54 suggests the option to simplify the modeling of surfaces by resorting to the ferromagnetic configuration.21 However, we find two reasons to work with the AFM ordering. First, we obtain a better description of the ground state by avoiding the use of one unnecessary approximation. The second reason is related to the OMC approximation used in the present work and its description of the FM ordered state. We have found that the OMC scheme fails to provide the lowest state when working with certain supercell choices. In Table 1, we show the different AFM and FM energies

magnetic symmetries of the surfaces when using the FM configuration, we have used solely AFM ordered surfaces. Ab Initio Thermodynamics of Water Adsorption. When computing the water adsorption on different surfaces, there are many reactive sites to be considered, in addition to many different orientations of the water molecule. As an efficient way of considering the many configurations, and to have a control of the system at different temperatures, we have used ab initio molecular dynamics (AIMD) simulations40 with the NVT ensemble. The Berendsen and Nosé thermostats55−58 were used to increase the temperature and to thermalize the system, respectively. Different initial atomic configurations of water molecules were adopted as the initial configuration in the AIMD simulations, in which the temperature was increased to 300 K with a Berendsen thermostat with a time step of 1 fs for each molecular dynamics step. At 300 K, the configuration space is probed using a NVT ensemble, where the lowestenergy configuration along the AIMD simulations is chosen. From this one, a DFT+U calculation where all the ions are allowed to relax is performed. The AIMD simulations were performed at the Γ-point of the mesh k-points grid, with the cutoff energy reduced to 250 eV. To study the interactions of the water molecules with the surfaces and possible adsorption or dissociative adsorption processes, a convergence study of all the quantities has been carried out. To analyze the thermodynamic effects of the different water reactions on UO2 surfaces when exposed to water environments, it is necessary to take into account the temperature and the water partial pressure. This is achieved by using ab initio atomistic thermodynamics.33,59−61 This approach assumes that the system is in thermodynamic equilibrium with a gas phase environment, which is treated as a reservoir, therefore exchanging particles with the system without changing its chemical potential. It is based on the calculation of the Gibbs free energy of the system, G(T, P), and its modification in a given reaction. The DFT+U total energies, EDFT+U, are related to the internal energy, W(S, V). At 0 K and in a vacuum with constant composition, it is equivalent to the Gibbs free energy, G(0, 0), which is in turn equivalent to the Helmholtz free energy at zero temperature F(0, V). Therefore, the total Gibbs free energy of a system, assuming negligible rotational and translational contributions and with electronic excitation energies much greater than kBT, can be written as

Table 1. Calculated AFM and FM Total Energies for Different Simulation Cells along with the Employed Magnetic and Dynamic Configurations

2UO2 4UO2 8UO2 16UO2

magnetic order

energy (eV)

magnetic symmetry

dynamic symmetry

AFM FM AFM FM AFM FM AFM FM

−63.807 989 −63.798 947 −127.616 32 −123.651 73 −255.248 73 −247.316 77 −510.468 62 −510.396 32

D4h D4h D4h Oh D4h Oh D4h D4h

D4h D4h Oh Oh Oh Oh D4h D4h

GSystem(T , P) = EDFT + U + Gvib(T , P)

(1)

where Gvib comes from the contribution of vibrational motion. As the pV contribution to the Gibbs free energy can be considered negligible (several orders of magnitude smaller than the other considered terms), the above equation can be rewritten as

computed for different supercells along with the point symmetry of the magnetic and dynamic configurations to which the system is restricted for each simulation cell. We observed that when we work with 4 or 8 formula units in the simulation cell, in the FM case, the simulation gets trapped in a metastable state (italicized values in Table 1), regardless of any initial occupation matrix, whereas for the other two cases shown (2 and 16 formula units in the simulation cell) the agreement between the FM and AFM energies is acceptable, which is a sign that the ground state has been reached. The reason for the disagreement for the former structures is the difference of magnetic structure which gives rise to different point symmetries D4h and Oh, and a drift of the electronic occupation to wrong states. When the symmetries are turned off, the states have a larger Hilbert space available and recover the AFM occupation matrices, correspondingly correcting the energies. Hence, to avoid possible problems related to the

GSystem(T , P) = EDFT + U + F vib(T )

(2)

where Fvib is the vibrational free energy within the harmonic approximation for n fundamental modes (with frequencies of ωi) and can be expressed as n

F vib(T ) =

⎡

⎛ ⎛ −ℏωi ⎞⎞⎤ 1 ℏωi + kBT ln⎜⎜1 − exp⎜ ⎟⎟⎟⎥ ⎢⎣ 2 ⎝ kBT ⎠⎠⎥⎦ ⎝

∑⎢ i=1

(3)

For the specific case of water reactions on UO2 UO2 + H 2O ⇋ H 2O/UO2 8493



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the change of the Gibbs free energy or adsorption Gibbs free energy, Gads, can be written as Gads(T , P) = G H 2O/UO2(T , P) − G UO2(T , P) − G H2O(T , P) +U +U DFT + U = E HDFT − E UO − E HDFT 2O/UO2 2 2O vib ( T ) − μ H O (T , P ) + FHvib2O/UO2(T ) − FUO 2 2

vib (T ) = Eads + FHvib2O/UO2(T ) − FUO 2

− ΔμH O(T , P) 2

(4)

where ΔμH2O is the difference between the experimental chemical potential (μH2O) and the Gibbs free energy of water at zero temperature and pressure (EHDFT+U ), which includes both 2O vibrational and rotational effects,62 and Eads is the value of the binding energy of the water molecule to the surface per simulation cell, i.e., +U DFT + U +U Eads = E HDFT − E UO − E HDFT 2O/UO2 2 2O

Figure 1. Side view of the two distinct adsorption structures of water adsorption onto (111) UO2: (a) dissociative adsorption; (b) molecular adsorption. The numbers represent the bond lengths in Å. Gray, red, and pink balls represent U, O, and H atoms, respectively.

Despite the structural differences between the configurations, the adsorption energy for the molecular adsorption is −1.10 eV, very similar to the energy of dissociative adsorption. We carried out a study using the nudged elastic band method (NEB) to determine the barrier energy, and the result shows that the processes are virtually barrierless. It is worth pointing out how, and why, the water adsorption energies calculated here differ from previously published theoretical results,16,21 where the adsorption energy calculated for 1 monolayer (ML) coverage is roughly 4 times smaller (from −0.22 to −0.29 eV). As we will see in the following, although the coverage dependence has a direct effect on the adsorption energy, it is still small compared to the absolute value of the adsorption energies and it cannot be the reason for the large difference in theoretical absorption energies. This significant difference is probably due to the fact that we are using the OMC scheme in combination with the DFT+U approach, which guarantees obtaining the lowest-energy state for the given uranium environment for every calculation we carried out, unlike previous works. Thus, the possibility to fall into different metastable states for adsorbate-free surfaces and adsorbent surfaces, in combination with the large difference in energy between the ground and metastable states, leads to unreliable adsorption energies when one does apply a procedure to isolate the ground state. Only Boettger et al.14 using a scalar (fully) relativistic approach provide similar adsorption energies, −1.16 eV (−1.08 eV), although their results lack the on-site Coulomb correlation, which precisely is the reason that they avoid wrong energy minima. However, the on-site Coulomb interaction has been shown to be essential for the study of other properties of UO2, as mentioned previously. Hence, our findings once more emphasize how important it is to use a procedure to obtain the true ground state, to study both total energies and reaction processes. We have further tested dispersion corrections (DFT-D2)64 to study H 2 adsorption on UO2 surfaces, finding a negligible contribution to the adsorption energy. Therefore, a negligible contribution of the long-range dispersion forces when applied to water adsorption on UO2 is expected. Having determined the adsorption energy values, we can now examine the relevance of thermodynamic effects on the previous adsorption reactions, which would reveal the feasibility

(5)

Equation 4 can therefore finally be rewritten as vib (T ) Gads(T , P) = Eads + FHvib2O/UO2(T ) − FUO 2

− ΔHH2O(T , P 0) + T ΔSH2O(T , P 0) − kBT ln(PH2O/P 0)

(6)

where PH2O is the water partial pressure, and P0 is the reference pressure of the water vapor (1 atm in our case), and in addition ideal gas behavior of H2O is assumed. ΔH (ΔS) is the enthalpy (entropy) at 0 K minus that at temperature T and P0. We have ignored the effect of the dipole perpendicular to the surface in single-sided adsorption, since this has been shown by Skomurski et al.16 to be only about 0.01 eV.

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RESULTS Water Adsorption onto (111) UO2. When studying the water adsorption onto (111) UO2, the use of AIMD simulations at 300 K directs all the initial water configurations to only two final adsorbed structures, i.e., dissociative adsorption and molecular adsorption. The former case is the final configuration found in most of the AIMD simulations and presents a splitting of the water molecule into one OH group and one H atom, as illustrated in Figure 1a (plotted with VESTA63). After an ionic relaxation using the DFT+U, the OH group is oriented with the H atom pointing upward and the O atom sits on top of one of the surface U atoms at a distance of 2.23 Å, while the bond length between the O and the H is 0.97 Å. The second H, located at a distance of 1.45 Å from the O atom, is bonded to one of the surface oxygens with a bond length of 1.05 Å. The adsorption energy, eq 5, of this reaction is −1.12 eV. Conversely, when a molecular adsorption is found as the final structure, after the DFT+U relaxation, the water molecule is always found with the hydrogen bonds parallel to the surface and with the oxygen atom just atop one of the surface U atoms (Figure 1b). Unlike the case of dissociative adsorption, the two hydrogen atoms are bonded to the water oxygen at a distance of 1 Å. With respect to the surface, the distance between the water oxygen and the U atoms is about 2.48 Å, larger than that obtained for the dissociative case. 8494



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illustrates, we have probed the capacity of the AIMD simulations as an exceptional tool to give physical insight into the reaction processes, through providing useful and reliable final configurations. We have also assessed the effects of water coverage on the adsorption energy and Gibbs free energy, by analyzing coverages from one to four H2O molecules adsorbed on the stoichiometric UO2 (111) slab, which correspond to 0.25, 0.50, 0.75, and 1.0 ML. Similar to the above-discussed case of 0.25 ML coverage, different initial configurations have been studied throughout with the help of AIMD simulations. Notably, as the water coverage increases, a mixed adsorption geometry characterized by a combination of molecular and dissociated water molecules is found, regardless of the initial configuration. Similar results were found by Alexandrov et al. when studying the water adsorption on (111) CeO2.40 The final geometries are illustrated in Figure 3, for the four different cases of water

of a water molecule to adsorb on the (111) surface. In doing this, we have only taken into account, in eq 6, the vibrational modes of the water molecule adsorbed on the surface as well as those of the oxygen atoms at the top layer of the surface, while leaving the rest of the substrate fixed when using eq 3. To perform this investigation, we assume that the vibrational modes belonging to the frozen atoms cancel out with the vibrational modes of the frozen atoms in the adsorbent-free surface. Here it is important to remember that the condition of a stable adsorption reaction requires a negative Gibbs free energy (exothermic reaction), and therefore, the stability of the reaction is limited by the temperature T of the system and the pressure of the gas phase with respect to the standard state (PH2O/P0). These two variables define the phase diagram boundaries determined by eq 6. In Figure 2, we show the

Figure 3. Side view of the computed water adsorption reactions for four different water coverages on the (111) UO2 surface.

Figure 2. Ab initio computed phase diagram of water molecule adsorption on the (111) UO2 surface. The blue line indicates the range of water partial pressures at which the water molecule desorbs at room temperature.

coverage ranging from 0.25 to 1.0 ML. The final orientations of the water molecules on the surface correspond to the already studied dissociative adsorption and water adsorption orientations (see Figure 1). The adsorption energies per water molecule for the different coverages studied are illustrated in Table 2. It can be seen that the adsorption energy decreases

calculated pressure−temperature phase diagram, highlighting with a shaded area the region of temperature and water partial pressure where the dissociative adsorption is feasible. Since the phase diagrams for dissociative adsorption and molecular adsorption are almost the same, we only show the one for dissociative adsorption. As can be seen in Figure 2, at low vapor partial pressure, the interaction between water molecules and the surface will be small, and therefore, the adsorption will only occur at low temperatures, with the molecules becoming desorbed with increasing temperature. By increasing the partial pressure, we obtain an increase of the boundary temperature at which the reaction is still exothermic. A blue vertical line in Figure 2 indicates the partial pressure (0.17 atm) at 300 K below which the adsorption is not feasible if the partial pressure is decreased. Hence, in spite of the dependence of the dissociative adsorption on pressure and temperature, the high negative value of the adsorption energy, Eads, allows the reaction to take place at room temperature for a large range of water partial pressures. Therefore, we have found boundaries for the two most probable processes of water adsorption on the (111) surface, i.e., molecular adsorption or dissociative adsorption, within a range of temperatures and water partial pressures. The water dissociative adsorption produces a hydroxylated surface (i.e., a surface with hydroxyl groups attached) which is highly relevant for investigations of the UO2 dissolution mechanisms. The dissociative reaction is notably found to be feasible on the (111) UO2 surface for a large range of accessible temperatures and water partial pressures. As this analysis

Table 2. Calculated Adsorption Energies per Water Molecule (Eads) on the (111) UO2 Surfaces, as a Function of the Water Coverage coverage (ML) Eads (eV)

0.25 −1.12/−1.10

0.50 −1.23

0.75 −1.13

1.0 −1.09

with increasing water coverage until a minimum value is reached (−1.23 eV) for a coverage of 0.5 ML. The reason, already pointed out by Alexandrov et al.,40 is the formation of efficient H-bonds with the surface that strengthens stability. Higher values of the water coverage provide slightly lower adsorption energies (yet of the same size as the adsorption energy for 0.25 ML), probably due to the more important effect of the repulsive interaction between the different water molecules. The presence of adsorbed water molecules influences not only the adsorption geometry of a new adsorption occurring but also its adsorption energy. We have also carried out a study of the (111) surface of CeO2. We find the adsorption energy vs coverage behavior is similar, with the lowest adsorption energy at 0.66 ML coverage. In addition, we have analyzed the Gibbs free energy as a function of the water coverage to study the feasibility of the reaction occurring at different temperatures and water pressures. In Figure 4, the calculated phase diagram for the 8495



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step edge (see Figure 5). This adsorption reaction is found to produce a hydroxylated surface. After the corresponding DFT

Figure 4. Ab initio computed phase diagram of water molecule adsorption onto the (111) UO2 surface, depending on the water coverage. The differently colored areas indicate where water adsorption is feasible for the different coverages as indicated in the inset. See text for further explanation.

different coverage reactions is presented, indicating the different boundaries at which the water adsorptions are feasible (the darker the color, the lower the coverage at which the adsorption reaction is feasible). Although the curves have a similar qualitative behavior, we observe that the adsorption at 1 ML coverage is less stable upon modification of the temperature and water partial pressure than the adsorption at other coverages. On the other hand, at low/medium temperatures (until about 1175 K), the most stable water coverage configuration is 0.50 ML, changing to 0.25 ML for higher temperatures, despite its higher adsorption energy value per water molecule (see Table 2). As an example of the importance of the water coverage on a water adsorption reaction, the temperature to desorb the water molecules from the surface increases by about 120 K at a water partial pressure of 1 atm, when considering 1.0 or 0.50 ML coverage. As a matter of comparison, the adsorption values published previously16,21 have been adopted in our approach to study the thermodynamic effects for such values. Using the vibrational energies presented here, we cannot obtain, for these values,16,21 a stable water adsorption down to 0 K. This stands in marked contrast to our results in Figure 4. Clearly, an expected feasibility or, conversely, improbability of water reacting with UO2 surfaces has far reaching consequences for estimating the long-term dissolution stability of oxide fuel material in geological repositories. Our findings particularly highlight the need to include and analyze the thermodynamics of the reaction. Water Adsorption on Stepped UO2 Surfaces. In the preceding, we have already observed the thermodynamic and water coverage effects on the water adsorption at the (111) surface and exemplified its importance as a precursor of dissolution processes. However, the surface morphology of the nuclear fuel material continuously changes due to dissolution reactions. Therefore, to predict dissolution reactions, it is essential to consider the water adsorption onto different morphologies. To do so, we have here chosen the (211) and (221) surfaces, which, in according with the structural picture drawn by Maldonado et al.,36 can be seen as surfaces made of a terrace oriented in the {111} plane and with steps oriented in [100] and [110] directions, respectively. Our AIMD simulations, performed for one water molecule on the (211) surface, predict the most stable adsorption geometry to be that where the water molecule adsorbs at the

Figure 5. Top: side view of the one-molecule water adsorption onto the step edge of the (211) UO2 surface (bond lengths in the inset are given in Å). Bottom: computed phase diagram of water molecule adsorption on the edge of the (211) surface. The shadowed area represents the region where the reaction is exothermic.

+U ionic relaxation, we find that the OH group is adsorbed directly onto the step edge, while the dissociated hydrogen, after the attraction to one of the superficial oxygen atoms at the edge, forms another OH group next to the OH water group. The bond lengths between the displaced oxygen (water oxygen) and the superficial U atoms are 2.25 Å (2.30 Å) and 2.46 Å (2.42 Å), and its distance to the H atom is 0.99 Å (0.98 Å). Therefore, the adsorption of one water molecule triggers a modification of the surface morphology, exposing one of the U atoms, and making the surface more suitable for dissociative reactions. The computed adsorption energy, Eads, is −2.39 eV, which is lower than the value obtained in the adsorption study at the flat (111) surface. As a consequence, it is expected that improved adsorption boundaries can be obtained when the thermodynamics is taken into account. Similarly to the case of the adsorption on the (111) surface, we consider the vibrational modes of the water molecule and the vibrational modes of the oxygen atoms in the top layer of the surface. In Figure 5, we show the calculated phase diagram of this reaction, indicating with the shadowed area the region of temperatures and pressures where the adsorption reaction is feasible. By comparing it with the phase diagram of the dissociative adsorption on the (111) surface (Figure 2), it can be seen that this reaction is more stable with respect to modification of the temperature and pressure (to about 700 K at atmospheric pressures, and stable at any pressure at room temperature). This is mainly due to the stronger adsorption. It can furthermore be expected that water coverage along with the different available adsorption sites (terrace and edge) will influence the Gibbs free energy, and therefore the feasibility of the reaction. We have therefore carried out an analysis of the 8496



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only adsorbent-free surfaces are possible. This picture supports the idea of two independent reaction processes for water adsorption on the (211) surface, namely, adsorption on the step edge (more stable against modification of T and PH2O) and adsorption on the terrace. Similarly, the adsorption on the (221) surface is computed to happen at the step edge through a dissociative adsorption (see Figure 7). The OH group sits atop one of the U atoms of the

adsorption reaction for coverages of two and three water molecules. In Figure 6, we present the final adsorption

Figure 6. Top: side view of the three-molecule water adsorption on the (211) UO2 surface. Bottom: the computed pressure−temperature phase diagram of the reaction. The shadowed areas indicate the regions where the reaction is feasible, only at the step (yellow) or at the step and the terrace (brown).

configuration after the AIMD simulation and the DFT+U relaxation, for the case where three water molecules are adsorbed. We observe that there is dissociative adsorption of two water molecules at the step edge, while the third one sits atop one of the U atoms of the (111) terrace. The hydrogen atoms from the dissociative adsorption are located atop the oxygen atoms at the edge. This configuration leads to a computed adsorption energy per water molecule of −1.86 eV. Note that this value differs more than 0.50 eV with respect to the case of only one water molecule; the reason lies in the different adsorption energies of the different sites. Thus, this average quantity is obtained as a combination of two contributions, easily distinguishable and of different magnitude, viz., the adsorption energy coming from the dissociative adsorption on the edge (around −2.40 eV) and the adsorption energy coming from the adsorption on the terrace (around −1.10 eV). For the same reason, when we assess the adsorption energy of two water molecules on the step edge, we have found a value of −2.30 eV, only 0.09 eV different from the case with only one water molecule. This small difference can be explained because of the different atomic distributions that we find when there is one or two water molecules that adsorb. Hence, we can conclude that the adsorption reaction occurs as a combination of two processes that can be analyzed separately, a dissociative adsorption of water molecules on the step edges and an adsorption of water molecules on the (111) terrace. This second process follows the same adsorption behavior as the above studied case of adsorption onto the (111) surface. The dynamics of this reaction can be illustrated by a phase diagram. In Figure 6, we show the different range of water partial pressure and temperature at which the adsorption on the terrace and the step (red-brown region) and the adsorption only on the step (yellow region) are feasible. Below these areas,

Figure 7. Top: side view of the one-molecule water adsorption reaction at the step edge of the (221) UO2 surface (bond lengths in the inset are represented in Å). Bottom: calculated phase diagram of water molecule adsorption on the edge of the (211) surface. The shadowed area represents the region where the reaction is exothermic.

step oriented in the [110] direction, and at a distance of 2.20 Å, while the dissociated hydrogen bonds to one of the neighbor oxygen atoms at a distance of 1.75 Å from the OH oxygen. The bond orientation of the H atoms with respect to the oxygen atoms is perpendicular to the [110] direction of the step with bond lengths of 0.97 and 1.00 Å for the OH hydrogen and superficial O, respectively. The adsorption energy is −1.40 eV, lower than the adsorption on the (111) surface but higher than the same on the (211) surface. As we observed before, this value determines the stability of the reaction under the effect of temperature and water partial pressure. The adsorption of one water molecule on the edge of the (221) surface is more stable upon modification of temperatures and water partial pressures than the adsorption on the (111) surface (see Figure 7). On the other hand, the adsorption reaction is less stable when comparing with adsorption on the edge of the (211) surface. To analyze the influence of the adsorption site and of water coverages, we have studied the adsorption Gibbs free energy for an adsorption reaction with two and three water molecules. 8497



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The final distribution of three water molecules adsorbing onto the (221) surface is similar to that onto the (211) surface; i.e., two water molecules adsorb at the step edge and one water molecule absorbs on the (111) terrace (see Figure 8). However,

where kad is a temperature-dependent constant. As a consequence of the non-multistep dependence of the model, it fails to describe the adsorption rates of stepped surfaces. We have already pointed out the reason for this failure, which lies in the different adsorption energies between the step and the terrace of the stepped surfaces. Hence, the reaction rate of the these surfaces should be computed as the reaction rates of two different processes, adsorption on the terrace and adsorption on the steps. The reaction rate of these two processes will only differ in the value of kad. In accord with the classification of surfaces into families reported by Maldonado et al.,36 the results presented here for the (211) and (221) surfaces can be generalized for any surface within the same family. Consequently, water molecules will adsorb more favorably on the step edges than on the terraces. This methodology can be readily extended to calculate reaction rates of chemical reactions occurring on the surface and those which cause dissolution, where reactions at the step and the terrace are expected to have different dynamics of the reaction. Thereby, it provides insight not only into the reaction rates but also microscopic insight regarding the sites where the reaction mainly happens.

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CONCLUSIONS Through the use of ab initio DFT+U theory in combination with atomistic thermodynamic simulations, we have found that water adsorption onto the (111) UO2 surface happens through two processes, dissociative adsorption and molecular adsorption, of which the first one is found to be energetically more favorable. Thermodynamic considerations reveal that the energetically most favorable water coverage is either 0.5 or 0.25 ML depending on the temperature and water partial pressure. The 1 ML coverage adsorption is the least favorable water coverage. The dissociative adsorption of water on the (111) surface is found to be feasible for a large range of temperatures and water partial pressures, which indicates that this is an important step in the mechanistic description of the dissolution process. The adsorption reactions of water on the more reactive (211) and (221) surfaces have also been studied. We find that, when only one water molecule is present, it adsorbs dissociatively at the edge for both surfaces, with a somewhat stronger adsorption occurring at the (211) surface. When this happens, the adsorption is accompanied by a modification of the step morphology. This can be regarded as the beginning of dissolution. We have found that the adsorption energy differs appreciably between the step edges and the terraces, favoring the adsorption at the edges. Hence, the reaction processes with water vapor can be seen as a combination of two adsorption reactions: the reaction of the water molecule at the (111) terrace and the water reaction at the edges. We have found that at room temperature and atmospheric pressure water reacts with UO2 surfaces, leading to dissociation and modification of the step morphology. The most stable water adsorption upon modification of the water partial pressure and temperature is adsorption on the (211) step edge, followed by adsorption on the (221) step edge, and being the least favorable for the (111) surface. Dissociative adsorption of water at a step edge is predicted to always occur at atmospheric pressure, for the (221) step edge for temperatures up to some 500 K and for the (211) step edge for temperatures up to 1175 K. These results establish that UO2 in nuclear fuel material will always react with water under equilibrium

Figure 8. Top: side view of the three-molecule water adsorption on the (221) surface of UO2. Bottom: the computed phase diagram of the water-surface reaction. The shadowed areas indicate the regions where the reaction is feasible, only at the step (yellow) or at the step and the terrace (brown).

for this stepped surface, one of the two water molecules adsorbs dissociatively at the edge, while the other one adsorbs in its molecular form. The adsorption of the remaining water molecule on the (111) terrace is a dissociative adsorption. The adsorption energy per water molecule increases with respect to the case of only one water molecule, from Eads = −1.40 eV to Eads = −1.20 and −1.22 eV, for two and three water molecules, respectively. The reason, as explained for the adsorption on the (211) surface, is the different adsorption energy values on the terrace and on the edge. The computed phase diagram (Figure 8) shows, as for the adsorption on the (211) surface, the range of water partial pressures and temperatures at which the two adsorption processes occur. A behavior similar to the adsorption onto the (211) surface is found; water adsorption on the step edge is more stable than adsorption on the terrace upon modification of the temperature and water partial pressure. Since the adsorption reaction on the various surfaces is exothermic, the kinetics of the reaction at a fixed temperature can be simplified by the Langmuir model, where the rate of adsorption, rad, is a function of the partial pressures (PH2O) and the concentration of vacant surface sites ([S]), and follows from

rad = kadPH2O[S] 8498



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conditions, leading to water dissociation and initiation of the dissolution of the fuel material. An important advantage of the methodology presented in this work is that it allows the determination of the range of temperature and water partial pressure at which an adsorption reaction is feasible. Furthermore, it provides detailed insight of the kinetics of water reaction on stepped surfaces, and it also reveals the influence of different morphologies and water coverages on the stability of the reaction. These features together make a powerful tool for ab initio investigations of corrosion reactions and of molecule−surface reactions in general.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The research leading to these results has received funding from the European Atomic Energy Community’s Seventh Framework Programme (FP7) under Grant Agreement No. 269903 (REDUPP). We acknowledge computer time received through the Swedish National Infrastructure for Computing (SNIC). We thank Boris Dorado, Marjorie Bertolus, and Michel Freyss for the use of OMC in VASP.

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