Accepted Manuscript A defect model for UO2+x based on electrical conductivity and deviation from stoichiometry measurements Philippe Garcia, Elisabetta Pizzi, Boris Dorado, David Andersson, Jean-Paul Crocombette, Chantal Martial, Guido Baldinozzi, David Siméone, Serge Maillard, Guillaume Martin PII:
S0022-3115(16)31130-8
DOI:
10.1016/j.jnucmat.2017.03.033
Reference:
NUMA 50199
To appear in:
Journal of Nuclear Materials
Received Date: 16 November 2016 Revised Date:
27 February 2017
Accepted Date: 22 March 2017
Please cite this article as: P. Garcia, E. Pizzi, B. Dorado, D. Andersson, J.-P. Crocombette, C. Martial, G. Baldinozzi, D. Siméone, S. Maillard, G. Martin, A defect model for UO2+x based on electrical conductivity and deviation from stoichiometry measurements, Journal of Nuclear Materials (2017), doi: 10.1016/j.jnucmat.2017.03.033. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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A defect model for UO2+x based on electrical conductivity and deviation from stoichiometry measurements
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Philippe Garciaa,∗∗, Elisabetta Pizzia , Boris Doradob, David Anderssonc , Jean-Paul Crocombetted, Chantal Martiala , Guido Baldinozzif, David Sim´eonee , Serge Maillarda , Guillaume Martina a CEA,
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DEN, DEC, Centre de Cadarache, F-13108 Saint Paul Lez Durance, France b CEA, DAM, DIF, F-91297 Arpajon, France c Materials Science and Technology Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA d CEA, DEN, DMN, SRMP, Centre de Saclay, F-91191 Gif-sur-Yvette, France e CEA, DEN, DMN, SRMA, LRC-CARMEN, Centre de Saclay, F-91191 Gif-sur-Yvette, France f CNRS, SPMS, UMR 8580, LRC CARMEN, Ecole Centrale Paris, F-92295 Chˆ atenay-Malabry, France
Abstract
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Electrical conductivity of UO2+x shows a strong dependence upon oxygen partial pressure and temperature which may be interpreted in terms of prevailing point defects. A simulation of this property along with deviation from stoichiometry is carried out based on a model that takes into account the presence of impurities, oxygen interstitials, oxygen vacancies, holes, electrons and clusters of oxygen atoms. The equilibrium constants for each defect reaction are determined to reproduce the experimental data. An estimate of defect concentrations and their dependence upon oxygen partial pressure can then be determined. The simulations carried out for 8 different temperatures (973 – 1673 K) over a wide range of oxygen partial pressures are discussed and resulting defect equilibrium constants are plotted in an Arrhenius diagram. This provides an estimate of defect formation energies which may further be compared to other experimental data or ab-initio and empirical potential calculations. Keywords: uranium dioxide, point defects, oxygen cluster, transport properties, electrical conductivity, deviation from stoichiometry
∗ Corresponding
author +33 4 42 25 41 88, fax: +33 4 42 25 32 85 Email address:
[email protected] (Philippe Garcia)
∗∗ telephone:
Preprint submitted to Elsevier
March 27, 2017
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1. Introduction
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Uranium dioxide has been extensively studied for over 50 years ([1, 2]). Despite this, there still does not exist a comprehensive understanding of prevailing point or complex defects or how these defects control most of its physical properties. The reason is probably to be found in the complexity of the material which is stable over large deviations from stoichiometry. This comes about through modifications mainly in oxygen related defect concentrations ([3, 4]) and is made possible by the property uranium has of existing under different charge states (3+, 4+, 5+ and 6+). The nature and concentration of point defects may be obtained through structural and spectroscopic characterisations of the material, but experimental devices usually only enable post-annealing characterisations to be carried out when in-situ characterisations are required. It is encouraging that reliable first principle determinations of point defect formation and migration energies are now emerging ([5, 6]). These approaches rely upon improved descriptions of the strong correlations which exist between the uranium 5f electrons and can ideally complement structural characterisations of the material. Also very useful in setting up a complete picture of the nature of defects and their concentrations as a function of temperature, oxygen partial pressure and possibly impurity concentrations, is the determination of the material’s physical properties as a function of these parameters. The aim then is to develop a point defect model or theory capable of reproducing these properties. Many atomic transport properties of oxides have been analysed in this way (see for instance [7]) but the focus of this work is the electrical conductivity data obtained by Ruello et al. [8] and to a lesser degree the deviation from stoichiometry measurements selected by Perron [9]. Ruello’s work has been selected because it constitutes to our knowledge the most comprehensive contribution to the study of electrical properties of UO2+x . Perron’s work on the other hand goes back further but we checked that the data we used is entirely consistent with data more recently compiled by G`eneau et al. [8]. In the present study, we attempt to develop a point defect model, based on defect chemistry which reproduces these data. In relation to UO2 , one finds two categories of point defect models depending upon whether or not these models account for the fact that defects are charged. Amongst the latter category one finds the approaches of Lidiard [10], subsequently apllied by Matzke [11] and more recently by Crocombette [12], Freyss et al. [13] or Dorado et al. [14] to evaluate point defect concentrations as a function of deviation from stoichiometry. In the more recent studies, neutral defect formation or migration energies calculated from first principles are used to provide a theoretical approach to point defect concentration or even self-diffusion coefficient estimates. Although internally consistent because the defects modelled bear no charge, the main shortcoming of all these models lies in that they fail to capture the essential role of electronic defects which explain the material’s electrical properties. Also because charged defects are generally more stable than neutral defects [15], it is essential they be formally taken into account. It is relatively straightforward to do so [2, 16, 17] and the improved 2
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2. Modelling hypotheses
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treatment of strong correlations now available [5, 6, 18, 17] opens up the prospect of defect related quantities being calculated with reasonable accuracy. It is this latter approach which we report here. The first part of this work is devoted to laying out the principles and hypotheses that underpin the model. Particular attention is paid to building into the model our knowledge of the electrical properties of UO2 but also, following [2], of another important property which is the relationship between deviation from stoichiometry and oxygen partial pressure. We then go on to derive the model equations and how the model parameters are determined, initially from an asymptotic analysis of the available experimental data. Finally, we discuss the ability of the model to reproduce the electrical conductivity measurements of [8] between 973 K and 1673 K and to give an estimate of defect concentrations as a function of oxygen partial pressure. Particular attention is devoted to comparing the activation energies of the mass balance equations involving the defects to defect formation energies derived from first principles.
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The model developed is similar to others reported in the past (notably [2] and [16]) and is based on defect chemistry. Its underlying principles are the same as those of the model reported recently [17] and are recalled here for the sake of completeness. Thermodynamic equilibrium is expressed through the so-called mass-action laws. In this approach, defects are assumed to be far apart which means that the dilute limit approximation (i.e. low defect concentrations) applies. Also strong defect interactions which are known to occur in oxide systems are neglected. There exists no simple analytical expression of thermodynamic equibrium in the general case of high defect concentrations and models have to be developed specifically for a material [19]. These approaches are thought to be necessary in defect concentration regimes typically greater than 10−2 . Hence the equations described below are applicable in theory to low deviations from stoichiometry only but we apply them in this study to deviations from stoichiometry in excess of this value. We follow the assumption of many other authors ([8, 20]) that thermally activated electronic disorder is controlled by the disproportionation of two U4+ (5f 2 ) ions into one U5+ (5f 1 ) ion and one U3+ (5f 3 ) ion. U5+ ions may also be formed through a charge compensation mechanism on introduction of an oxygen interstitial or vacancy as suggested from recent electronic structure calculations [21]. Because UO2 is assumed to be an electronic conductor, the electrical conductivity of the material is expressed as the sum of an electron and a hole contribution. In the following, the electron excess cation defect (cation site on which an excess electron is localised) is noted in Kr¨ oger and Vink notation U′U and the electron deficient cation defect U◦U , so that the electrical conductivity is given by [22]: σ = µp × Ns,U (µrel [U′U ] + [U◦U ]) e
(1)
where e is the electron charge, µp is the hole mobility and µrel is the ratio of the electron to hole mobility. Ns,U is the cation site concentration. For an adiabatic, 3
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σ = Ns,U
µ0 × exp T
m − ∆H k T b
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small polaron hopping mechanism, the hole mobility may be expressed [23] as m µp = µT0 exp − ∆H so that: kb T (µrel [UU′ ] + [UU◦ ]) e
(2)
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where T represents temperature, µ0 is a constant and ∆Hm is the charge migration enthalpy. Although no direct measurement of µp is reported for UO2 , an estimate can be obtained from an analysis of the temperature dependence of electrical conductivity data at low oxygen partial pressures and at temperatures below the transition from an intrinsic behaviour. It is widely documented ([20, 24]) that in this case, UO2 is a p-type conductor and the hole concentration is determined by impurities, the valence of which is less than 4+. As a result, the Arrhenius dependence of σT provides an estimate of the hole mobility, i.e. on the one hand (∆Hm ), and the product of µ0 with the impurity concentration. Dudney [20] proposed a value for µ0 of 0.0554 m2 KV−1 s−1 (widely used) and Ruello [24] an activation energy of 0.26 eV for the study of a particular single crystal sample. This activation energy is determined unequivocally from an analysis of the low temperature data. On the other hand µ0 cannot, and only the product of µ0 with y (the extrinsic carrier concentration) may be determined directly from the data. We will see below that studying deviation from stoichiometry as a function of oxygen partial pressure provides a more appropriate estimate of µ0 . We also assume following [24] that the electron and hole mobilities are similar (i.e. µrel ∼ 1) although this inference is essentially indirect, i.e. no straightforward measurement of the electron mobility has ever, to our knowledge, been carried out for UO2 . The relevance of such a hypothesis will be looked at in the discussion section. As mentioned above, we use Kr¨oger-Vink notation to express thermodynamic equilibria between the various point defect species. The disproportionation equation provides the following relationship: ◦ ′ 2UX U ⇋ UU + UU
(3)
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4+ where UX ) U designates a uranium atom on an ordinary uranium lattice site (U with the superscript X indicating a neutral defect. Although negatively charged interstitials and conversely positively charged oxygen vacancies have been assumed for many years, it is only recently that the analysis of oxygen self-diffusion experiments of [21, 25] has shown that negative doubly charged oxygen interstitials actually exist. This work also highlighted the relevance for UO2 at least for small deviations from stoichiometry of the following defect equilibrium equation which describes the introduction in the material of an oxygen interstitial from the gas phase: ′′ 1 X ◦ 2UX U + O2 + Vi ⇋ Oi + 2UU 2
(4)
where ViX is a vacant interstitial site. This defect is not to be confused with a uranium or oxygen vacancy. Rather, it is an empty octohedral site in the 4
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fluorite structure and its modeling is required because we describe situations in which deviation from stoichiometry is substantial. As a result of this, the concentration of available interstitial sites normalised to the total number of cation sites may deviate substantilly from one with an effect on the expression of the various equilibrium constants (see equations 8, 9, 10). The model may thus account for the conservation of the total number of octohedral interstitial sites, the occupation of which does not exceed a value corresponding to the formation of U4 O9 , i.e. to a deviation from stoichiometry of roughly 0.25. The model also describes the sub-stoichiometric region of the phase diagram. However substantial (i.e. measurable) deviation from stoichiometry in the substoichiometric region is only observed at high temperature (greater than 1873 K, see for instance [26]) and very low oxygen partial pressure. This corresponds to conditions that in the main lie outside those of the electrical conductivity data used to validate the model. Following the majority of previous authors (e.g. [27] and [28]), we assume that sub-stoichiometry is due to the presence of oxygen vacancies. Assuming oxygen vacancies of charge +p constitute the predominant anion-type defect population, one would expect (see appendix A) that deviation from stoichiometry follows a power law dependence of oxygen partial pressure with an exponent of − 12 close to stoichiometry (i.e. when U′U ∼ U◦U ) ′′ and this irrespective of the charge of the point defect, so long as Oi are the interstitial species. As x decreases still further and electrons become the predominant electronic defect species, the slope of this property is expected to be 1 close to − 2(p+1) . The results of thermogravimetric experiments of Tentenbaum and Hunt [29], albeit at temperatures in excess of 2000 K, indicate an Log(x ) vs. Log(pO2 ) slope that changes from -0.5 to roughly -0.25 as sub-stoichiometry increases. This would suggest a singly charged oxygen vacancy at least at large deviations from stoichiometry. The results of Javed [26] on the other hand would tend to indicate characteristic slopes below -0.5. To complement this analysis, it is interesting to turn to recent charged first-principles defect calculations. Crocombette and coworkers [18] have shown that in sub-stoichiometric material, singly charged or even neutral oxygen vacancies could exist. The authors also show that close to stoichiometry, the differences in formation energies between singly and doubly charged vacancies are slight and that hyper-stoichiometry will favour doubly charged vacancies. We therefore surmise here the existence of doubly charged positive vacancies only. Our representation of the material could easily be adapted to account for differing charged states. Oxygen disorder on the anion sublattice is therefore assumed to result from the following Frenkel equilibrium: ′′
X ◦◦ OX O + Vi ⇋ Oi + VO
(5)
It has long been demonstrated for UO2 that in the hyperstoichiometric region of the phase diagram, excess oxygen atoms readily agglomerate to form so-called clusters. Based on neutron diffraction studies carried out on UO2.12 single crystals at 1100 K, Willis [4] showed the existence of complex defects which adopted a so-called 2:2:2 configuration. In this configuration, the actual 5
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interstitial sites are unoccupied, vacancies are also formed on normal lattice sites and there are two kinds of interstitial atoms noted O′ and O′′ (note that the superscripts here bear no reference to charge). The fact that occupation numbers for all three sites are roughly the same justifies reference to a (2:2:2) type defect, which we rename W in the remainder of this work. There remains to determine an appropriate charge for this defect. Again we turn to Ruello’s data [8] with which we aim to remain consistent. The author showed a square root dependence of electrical conductivity upon oxygen potential above 1273 K and at high oxygen potentials in a region where the material has an intrinsic behaviour (region 2 described in appendix B.1). Assuming n excess oxygen atoms make up the defect and that a single type of defect predominates, then it is quite straightforward to prove that the electrical conductivity is proportional to n oxygen partial pressure to the power 2(p+1) where -p is the apparent charge of the defect. Assuming, as suggested by Willis that n equals two, a charge of -1 for this type of cluster is compatible with the experimental results. Note also (see for instance [9]) that for deviations from stoichiometry between 5 × 10−3 and 10−1 the slope of the Log(x ) vs. Log(pO2 ) curve is also close to 0.5. This result is entirely consistent with the simple point defect model assuming (2:2:2) singly charged (negative) type clusters constitute the majority anionic defect population. Note also that the present model makes no assumption about the shape of the clusters of oxygen interstitials. Recent first principles calculations [5, 30] point to an arrangement of 2 interstitials different from the (2:2:2) Willis cluster. Furthermore this split di-interstitial configuration as it is called was found to be singly negatively charged in the substoichiometric regime [15]. Our model is consistent with any di-interstitial configuration since it makes no assumptions in regard to the details of atomic arrangements within the clusters. The mass balance equation describing the formation of such a cluster may be written as follows:
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′ ◦ X X O2 + 2OX O + 2Vi + UU ⇋ W + UU
(6)
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Note that this di-interstitial consumes two addition oxygen ions (incorporated from the gas phase), two empty interstitial sites (ViX ) and two normal oxygen sites (OX O ). It is quite probable that clusters of different composition exist in UO2+x and that their proportion varies depending upon oxygen potential and temperature but in the absence of straightforward experimental evidence, we choose here to consider only one type of cluster. Another interesting point raised by Willis concerns uranium vacancies. Their existence is also a means to accommodate deviation from stoichiometry as would suggest the increase with oxygen partial pressure of uranium self-diffusion coefficients [17]. Willis [4] also used neutron diffraction on stoichiometric UO2 and following oxidation to U4 O9 to address this issue. The number of uranium atoms in an average cell was found in both cases to be about 4. The error could be estimated as corresponding to a deviation from stoichiometry of 0.02 so well below the quantities of excess oxygen required for forming U4 O9 . From these considerations, we also assume here that uranium vacancies constitute a 6
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U◦U
+1
Hole, i.e. U5+ ion
U′U
-1
Electron, i.e. U3+ ion
OX O
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Oxygen atom on an ordinary anion lattice site
O2
0
Oxygen molecule present in the gas phase
-2
Doubly charged oxygen interstitial
0
Vacant interstitial site
+2
Doubly charged oxygen vacancy Singly charged di-oxygen cluster
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Uranium atom on an ordinary cation lattice site
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UX U
Effective charge 0
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O′′i ViX ◦◦ VO
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Table 1: Defects considered in the model
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minority defect for all deviations from stoichiometry. Table 1 summarises the nature of defects the model takes into account.
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3. Model equations
In the dilute limit approximation, the activity of each defect species is given by its corresponding site fraction. If ∆Gα designates the Gibbs free energy of defect formation equation (α), then thermodynamic equilibrium determines relationships between the activities, i.e. site fractions under our approximations. Site fractions are proportional to the defect concentrations normalized to the uranium site concentration. Thermodynamic equilibrium is therefore expressed as four relationships, each corresponding to the four chemical equlibria described in the previous section: [U◦ ] [U′ ] ∆Ge Ke = U 2U = exp − (7) kb T UX U
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2 ∆GOi [Oi′′ ] [U◦U ] exp − = 2 X √ kb T UX Vi pO2 U h ′i W [U◦U ] ∆GW = 2 2 = exp − kb T OX ViX UX O U pO2 h ′′ i ◦◦ Oi [VO ] ∆GAF KAF = X X = exp − kb T OO Vi
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(8)
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KOi =
(9)
(10)
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where the square brackets represent defect concentrations normalised to the uranium site concentration Ns,U and pO2 the equilibrium oxygen partial pressure. There are eight unknowns to this problem: in addition to the four defect equilibrium relationships, one guarantees electroneutrality and three additional equations express the constraints imposed by the crystalline structure: h ′i h ′′ i ◦◦ ] (11) y + 2 Oi + [U′U ] + W = [U◦U ] + 2 [VO
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where y represents the total charge due to all negatively charged substitutional impurities associated with aliovalent (bi or tri-valent) cations. If one assumes that all cation impurities are substitutional and trivalent then y represents their concentration. Through charge compensation effects, these impurities create electron holes which are responsible for the extrinsic behaviour observed at low oxygen partial pressures and temperatures. Additional constraints, which express the conservation of cation, anion and interstitial sites, are written as follows: y + [U′U ] + [U◦U ] + UX (12) U =1 h i X ′ ◦◦ ]=2 (13) OO + 2 W + [VO h ′′ i h ′i Oi + ViX + (2 + α) W = 1 (14)
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Equation (12) is an approximate formulation which implicitly assumes that all cation impurities are substitutional and trivalent. An empirical parameter α is introduced at this point that limits deviation from stoichiometry to a value of the order of 0.25 which is the composition of U4 O9 . At this phase limit, if the di-interstitial cluster constitutes the majority defect then deh population i ′ viation from stoichiometry is simply approximately 2 W which limits the cluster concentration to 0.125. At this point there are no interstitial sites readily available to accept additional interstitials and equation (14) yields a value of approximately 6. The physical reason why cluster concentrations are limited is probably to be found in the fact that as their concentration increases they are brought closer and closer to each other and Coulombic interactions are no longer negligible. Strong electrostatic repulsive forces appear between clusters that no doubt constitute a driving force for their ordering and results in the formation of a new U4 O9 phase as demonstrated recently [31]. This ordering of clusters can 8
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probably be associated with the levelling off of both deviation from stoichiometry and electrical conductivity at high oxygen partial pressures (see Fig. 1 for example). In Desgranges et al.’s study [31], it was actually demonstrated that it is the ordering of more complex cuboactahedral clusters which accompanies oxidation of UO2+x to U4 O9 . There is to our knowledge just a single neutron diffraction study which reports clusters in UO2+x (UO2.13 ) closer to a cuboctahedron type composition [32]: this indicates that it is likely that clusters of different compositions and charge may exist and co-exist. The actual details of how, why and indeed if the majority defect population in UO2+x changes from a (2:2:2) to a cuboctahedron configuration are unknown. The electrical conductivity is obtained from relationship (2). Also, an estimate of deviation from stoichiometry x may be calculated, assuming uranium vacancies to be negligible in comparison to oxygen defect populations: h ′i h ′′ i ◦◦ ]=x (15) Oi + 2 W − [VO 4. Results and sensitivity analysis
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In analysing Ruello’s work [8] with this model, our aim is to determine whether values of defect equilibrium constants may be determined that simultaneously reproduce electrical conductivity and deviation from stoichiometry data adequately. A mathematical model was therefore set up that solves equations (7) to (14) simultaneously for given values of the equilibrium constants, α and µ0 . Because of the number of physical constants involved, it was necessary to devise a method for providing a initial estimate of these constants based on an asymptotic model approach which is described in appendix B. We consider ranges of temperature and oxygen potential over which only a restricted set of defects predominate and all other defects may be neglected in the analysis. Low (i.e. 973 K, 1073 K, 1173 K) and high (i.e. 1473 K, 1573 K, 1673 K) temperature data are first analysed separately. Indeed, as indicated from Ruello et al.’s Seebeck coefficient measurements [8], at low temperature, under all oxygen partial pressures studied, there is no transition from a n to p type conduction. Hence the only defects are thought to be holes, oxygen interstitials, di-interstitials and negatively charged impurities. To solve this problem in terms of electrical condutivity, it is necessary to determine four constants: KW , KOi , y and µ0 . Three ranges of increasing oxygen potential are identified in which extrinsic immobile cationic defects, extrinsic immobile cationic defects and isolated oxygen interstitials and finally di-interstitial clusters constitute the majority negatively charged defect populations. A different asymptotic model for each of these three oxygen partial pressure regions provides three different equations whence comparison to the relevant electrical conductivity data yields estimates of three physical constants. The fourth equation is provided by deviation from stoichiometry data. At high temperature, electrical conductivity goes through a minimum and further assuming that electrons and holes have similar mobilities Ke may be
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KW
Ke
KAF
1.3 × 10−3
1.6 × 108
1.8 × 10−12
5.6 × 10−13
1073
3.2 × 10−4
7.9 × 106
3.1 × 10−11
5.2 × 10−12
1173
2.5 × 10−4
6.3 × 105
3.3 × 10−10
3.4 × 10−11
1273
1.1 × 10−4
7.7 × 104
2.4 × 10−9
1.6 × 10−10
1373
6.5 × 10−5
1.3 × 104
1.3 × 10−8
6.1 × 10−10
1473
4.1 × 10−5
2.7 × 103
6.3 × 10−8
1.7 × 10−9
1573
2.7 × 10−5
6.9 × 102
1.6 × 10−7
6.6 × 10−9
1673
1.9 × 10−5
2.1 × 102
7.1 × 10−7
1.1 × 10−8
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estimated. The oxygen potential value for which electrical conductivity is minimal may be regarded as the point at which the material is strictly stoichiometric and this further provides an estimate of KAF . Equilibrium constants, for which whenever possible the asymptotic analysis provides starting values, and the α parameter are determined with the complete model for a data set, i.e. at a given temperature, by minimising the difference between the experimental data points and the equivalent physical quantities determined from model calculations and application of equations (2) and (15). Greater emphasis is given to the electrical conductivity data. Table 2 provides a summary of model parameters determined from the asymptotic analysis or from extrapolation of these values at lower or higher temperatures, as appropriate. Table 3 indicates results obtained from fitting the data with the complete model. As can be seen from comparing Tables 2 and 3, values of the estimates or their extrapolation are in general very close to the physical constants derived upon considering the model in its complete form. Also indicated in Table 3 are the uncertainties associated with each of the model parameters based on an estimated relative error of ±5% for electrical conductivity data and ±5 10−3 postulated for deviation from stoichiometry data. These uncertainties are estimated from the minimum and maximum values of the constants that reproduce the experimental data within the uncertainties postulated for both these properties (deviation from stoichiometry and electrical conductivity).
Table 2: Model parameters determined from the asymptotic analysis presented in B or extrapolated values (underlined) with a hole mobility pre-exponential factor of 0.26 m2 KV−1 s−1 .
The main conclusions one may draw from this analysis are as follows: • It is the combined analysis of deviation from stoichiometry and electri10
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KOi
KW
Ke
973
1.3 × 10−3±0.4
1.6 × 108±0.4
2.0 × 10−10
1073
3.2 × 10−4±0.4
7.9 × 106±0.4
5.0 × 10−10
1173
2.0 × 10−4±0.4
6.3 × 105±0.4
1.0 × 10−9
1273
1.3 × 10−4±0.4
4.0 × 104±0.4
5.0 × 10−9±0.2
4.0 × 10−13
1373
7.9 × 10−5±0.4
1.8 × 104±0.4
1.6 × 10−8±0.2
1.0 × 10−12
1473
5.6 × 10−5±0.4
6.3 × 103±0.4
6.3 × 10−8±0.2
4.0 × 10−9±0.4
1573
2.4 × 10−5±0.4
1.6 × 103±0.4
1.7 × 10−7±0.2
1.0 × 10−8±0.4
1673
1.3 × 10−5±0.4
3.2 × 103±0.4
7.1 × 10−7±0.2
3.2 × 10−8±0.4
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KAF
5.0 × 10−15 4.0 × 10−14 1.3 × 10−13
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Table 3: Physical parameters obtained from minimising the difference between experimental data and results from the model in its complete form. A pre-exponential factor for hole mobility of 0.26 m2 KV−1 s−1 is assumed. Underlined values correspond to situations where only maximum values may be derived.
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cal conductivity data that enables us to determine all model parameters (equilibrium constants, α and µ0 ) unequivocally. A best-estimate value of 0.26 m2 KV−1 s−1 was derived from the data analysis for the hole mobility pre-exponential factor (see appendix B for details) whence a value for the concentration of negative cation charges is obtained (5.2 × 1024m−3 ) based on the low temperature extrinsic electrical conductivity data.
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• In principle deviation from stoichiometry measurements should be carried out on the same set of samples as those used to measure out electrical conductivity since as now widely reported, impurity content will modify these material properties. However, we only consider deviations from stoichiometry in excess of about 5 × 10−3 first because it constitutes the minimal error associated with this kind of measurement, second because it is only at low oxygen partial pressures that extrinsic behaviour, by definition different from one set of samples to the other, dominates. In the deviation from stoichiometry region we have considered as relevant for model benchmarking, most samples are expected to behave intrinsically unless purposefully doped. • Our analysis shows that KAF and Ke cannot be determined from the low temperature electrical conductivity data (i.e. 973 K, 1073 K, 1173 K). This is also the case for KW with the exception of the 1173 K data. However, KW may be estimated with reasonable accuracy in the low temperature 11
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range based on deviation from stoichiometry measurements. Conversely because of the presence of impurities and the limited impact of isolated oxygen interstitials upon deviation from stoichiometry, KOi may not be determined from deviation from stoichiometry measurements.
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Fig. 1 gives an example of both calculated and measured electrical conductivity and deviation from stoichiometry data at 1173 K and 1573 K. Also shown in Fig. 2 are the corresponding defect concentrations. As expected, at 1173 K, di-interstitial clusters only appear at the highest oxygen potential values. Electronic defects constitute the majority defect population at all temperatures and oxygen partial pressures. The equilibrium constants determined from the data analysis are plotted in an Arrhenius representation in Fig. 3a, b, c and d. Activation energies are reported for all four equilibrium constants in Table 4.
Formation energy (eV) experiment ab initio
empirical potentials
2.2 ± 0.1
1.7 [33]
3.7 [34]
O Frenkel pair
2.2 ± 0.1
3.3 [33]
4.8 [34]
Single oxygen interstitial
−0.8 ± 0.1
−0.5 / − 0.6 [17]
between -0.94 and -0.7 eV [34]
Di-interstitial
−2.3 ± 0.2
−1.9 [35]
-
0.7 − 0.9 [35]
0.7 [34]
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e-h pair
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Binding energy of di-interstitial
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Table 4: Comparison of first principles and empirical potential estimates of defect formation energies with results from the analysis of experimental data.
5. Discussion 5.1. Determination of the equilibrium constants As Fig. 3 indicates, the equilibrium constants are roughly aligned when plotted in an Arrhenius representation which gives credit to the activation energies derived. It is however worth mentioning that not all equilibrium constants may be determined at all temperatures. Indeed, an accurate determination of equilibrium constants can only be derived first if a defect involved in this equilibrium is actually present in substantial quantities under the conditions investigated and if the property is actually sensitive to the presence of that defect. Electrical 12
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conductivity is more directly sensitive to charge carrier concentrations and deviation from stoichiometry to disorder on the oxygen sublattice. In relation to the di-interstitial constant for instance, analysing deviation from stoichiometry data at high oxygen potentials with an asymptotic model, provides good estimates of the corresponding equilibrium constant because di-interstitials constitute the majority defect population and that property is sensitive to it. Another example is the mono-interstitial formation constant. Fig. 4 indicates the experimental data at 1173 K in comparison with model calculations assuming the presence of both di-interstitials and mono-interstitials on the one hand, and di-interstitials alone on the other.
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This figure illustrates that electrical conductivity measurements are sensitive, albeit in a small range of partial pressures, to the presence of oxygen mono-interstitials, which gives credit to any defect characteristics derived from this property analysis. Conversely, Fig.1 shows that because of the accuracy one may assume for thermogravimetric measurements, deviation from stoichiometry estimates are too insensitive in the region where mono-interstitials prevail for any meaningful data to be derived in relation to these defects. If improved accuracy were possible then it would be necessary to measure deviation from stoichiometry and electrical conductivity on identical sets of samples to draw 16
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combined, coherent conclusions. Along these lines, it is important to note that at the lowest temperatures, the Frenkel defect or the electron hole formation constants may be discounted altogether with no loss in terms of model representation (see Fig.5).
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Figure 5: Experimental data and corresponding asymptotic model results at 973 (a), and 1173K (b). Corresponding defect concentration values are also given at 973 K (c) and 1173 K (d).
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In other words it is impossible to determine Frenkel equilibrium constants from the lowest temperatures. The green points reported in Fig.3 represent maximum values for the formation constants (oxygen Frenkel and electron-hole formation) above which the model can no longer reproduce the experimental data adequately, irrespective of all other equilibrium constants. Regarding the Frenkel disorder equilibrium constant in particular, its influence is only notable at the highest temperatures and its determination relies on an estimate of the oxygen potential at which electrical conductivity is minimal (with the added hypothesis that electrons and holes have similar mobilities, see appendix B, section B.2). The electrical conductivity data shows that it is difficult to determine this partial pressure adequately. So it is expected that using electrical conductivity measurements (or indeed deviation from stoichiometry) will lead to a generally poor estimate of the Frenkel formation constant. 5.2. Nature and formation energy of defects Fig.2a shows that the crossover point from a regime where mono-interstitials predominate to a regime where di-interstitials become the majority defect pop17
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ulation occurs at 1173 K at an oxygen potential between 10−14 atm. and 10−13 atm. The corresponding deviation from stoichiometry lies between 10−4 and 10−3 which is very low indeed. Beyond that crossover point, the charge of the di-interstial defect does not follow the value expected from a purely ionic model. This is consistent with the work of Crocombette et al. [18] who have shown from first principle theoretical approaches that a decrease of the Fermi level (which corresponds to increasing hyper-stoichiometry) will favour singly charged and possibly even neutral oxygen mono-interstitials. Still more recently Crocombette’s defect cluster calculations [15] have shown that decreasing Fermi levels (i.e. increasing oxygen partial pressures) will favour both interstitial clustering and a decrease in the (absolute) charge state of the most stable anion cluster. For instance, the most stable di-interstitial (noted IX2) reported in [15] at high oxygen partial pressures has a charge of -1, as transpires from the present analysis. We can also quantitatively compare the results of our data analysis to such charged defect calculations. The results of Andersson et al. [33] are reported in Table 4. A good agreement is noted for both the electron-hole formation energy and the doubly charged mono-interstitial. One may also note the very close agreement with the band gap of 2.1 eV determined from optical methods by Schoenes [36]. A lack of accuracy in the Frenkel pair formation constant makes comparisons difficult between experiment and theory. We have also used results derived from Dorado’s work [35] to estimate the formation energy of diinterstitial clusters. This exercise is also extremely encouraging since it shows very close theoretical and experimental values (−2.3 eV vs. − 1.9 eV for the experimental and theoretical values respectively). Even closer agreement may be derived from comparing binding energies for the di-interstitial (i.e. the difference between twice the energy corresponding to the single interstitial and the energy corresponding the di-interstitial (approximately 0.7 eV in both experimental and theoretical approaches). The reason for this is possibly that calculation of such differences eliminates the need for an oxygen molecule reference state. It is worth comparing our results to empirical potential calculations. Over the past forty years, a very large number of studies have been reported relating to these simulation techniques applied to UO2 (see for instance [37, 34, 38, 39, 40, 41, 42, 43, 44, 45]). Originally, they involved an explicit account of the individual charge states of uranium ions which is key to understanding or modelling non stoichiometry ([37, 34, 41]). However, possibly because of the complexity behind developing such potentials when the system is made more complex (such as when radiation damage is modelled or when fission products are added), work has generally focused upon rigid ion models for which only exact stoichiometry may be described. The defect energies derived from the shell model approaches [37, 34] may be compared to the experimental data we report here, particularly when the corresponding defect equilibria involve electronic defects, such as is the case for oxidation reaction (4) or the electron hole equilibrium (3). When ionisation states for uranium ions different from +4 are not considered, which is the case of most empirical potential approaches, we can only compare our experimental data to defect equilibria involving simultaneous conservation of 18
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atoms and electronic defects. In this category we have the oxygen and uranium Frenkel equilibria and the Schottky equilibrium. The properties we are concerned with here restrict us for all studies but that of Catlow [37], Jackson et al. [34] and Yabub et al. [41], to the oxygen Frenkel equilibrium. Most of the empirical potential studies reported above place the anion Frenkel equilibrium energy at between 4 and 5 eV, i.e. about double the experimental value we report. One must however bear in mind the difficulties in determining estimates of the Frenkel pair equilibrium constants below 1473 K (see section 5.1). The work of Jackson et al. provides defect formation energy values which are in the main close to our experimental determinations as can be seen in Table 4, especially for the energy associated with oxidation reaction 4 (between -0.94 eV and -0.72 eV reported from empirical potential calculations for temperatures ranging between 773 K and 1273 K and -0.8 eV from the analysis of experimental data). Also the binding energy for a di-interstitial or cluster reported by Jackson is roughly 0.9 eV (compared to 0.7 eV in our study). These results suggest that both experimental and theoretical approaches are extremely encouraging. Also, interesting indications stem from the close inspection of Fig.5 which indicates deviation from stoichiometry as calculated by the model and corresponding experimental data at 973 K. As the oxygen partial pressure rises the slope of the Log(x ) vs. Log(pO2 ) curves changes from roughly 1 2 to 1 possibly indicating the emergence of a defect population of charge p and n containing n excess oxygen atoms such that 2(p+1) ∼ 1. Assuming a single negative charge would indicate that the cluster is made up of roughly 4 excess oxygen atoms (with respect to the original fluorite structure), consistent with the 5 excess oxygen cuboctahedron originally proposed by Bevan et al. [46]. This is also consistent with the quad-cluster structure suggested by Andersson et al. [5] containing 4 excess oxygen atoms. Finally, the fact that this defect equilibrium approach is capable of reproducing properties over a wide range of temperature and oxygen partial pressure would suggest that uranium defects constitute the minority defect population. This confirms the broadly accepted view that electronic defects are present at higher concentrations than defects on the anion sublattice which themselves are present at higher concentrations than defects on the cation sublattice. Also, as in the analysis of Ruello [24], in the work detailed in the previous sections we have assumed that the electron and hole mobilities are equivalent. This hypothesis is expected to have an impact upon the data analysis, hence the equilibrium constants derived; at high temperature mainly since it is at the higher temperatures (above 1473 K) that conduction undergoes a transition from p to n type regime as the oxygen partial pressure decreases. In fact, as seen in appendix B, section B.2, Ke and KAF are the constants mainly affected by the hypothesis of similar hole and electron mobilities. We have checked that a five fold modification of the electron relative mobility does not substantially modify values of KW or KOi , nor does it in general modify the activation energies of any of the equilibrium constants.
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6. Conclusions
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In this work, a point defect model has been developed for UO2+x based on point defect equilibria and a combined analysis of electrical conductivity and deviation from stoichiometry data. Although ideally, this requires that property measurements be carried out on samples from identical batches, our model very satisfactorily reproduces both these physical properties over a wide range of temperatures and oxygen partial pressures. In this model, defects on the cation sublattice other than electronic defects are neglected which is consistent with the generally held view in relation to UO2 that electronic disorder dominates over anion disorder, which itself is more significant than cation disorder. A rather large value for the preexponential factor of the hole mobility is deduced from this analysis (0.26 m2 KV−1 s−1 ), approximately 5 times greater than the value suggested by Dudney et al. [20]. Based on the temperature analysis, it appears that formation energies for mono-interstitials (−0.8 eV), di-interstitials (−2.3 eV), and the electron-hole formation energies (2.2 eV) can be determined with reasonable accuracy. It is less straightforward to determine the formation energy of the oxygen Frenkel pair from the data analysis, probably because under the conditions examined, oxygen vacancies constitute a minority defect population. The energies derived for the mono-interstitials, the electron-hole pair and the oxygen di-interstitial are shown to be close to values derived from first principles. In addition to this, the charge and apparent composition of di-interstitials is well in line with DFT+U calculations which predict a decrease in the charge per additional oxygen interstitial that makes up the cluster. Our results show that in relatively pure material, oxygen clustering is predominant at very low levels of hyperstoichiometry, corresponding to deviations from stoichiometry between ∼ 4 × 10−4 and ∼ 10−3 for temperatures in the range 973 K to 1673 K. Although extremely encouraging, these conclusions should be backed up by further investigations and in particular, direct mobility measurements of electrons and holes should be carried out.
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A. Asymptotic model in an intrinsic, substoichiometric regime
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In this appendix we derive simplified expressions for deviation from stoichiometry and electrical conductivity under certain limiting conditions. We assume the majority anion-type defect is the oxygen vacancy but treat the general case where the formal charge of the oxygen vacancy is +p. The electroneutrality equation may be expressed as: h ′′ i h ◦i 2 Oi + [UU′ ] = [UU◦ ] + p VOp (16) Frenkel equilibrium (i.e. equation 10) has to be re-written to take into account a charge state for the oxygen vacancy which may differ from +2: ◦
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electrical conductivity reflects the level of electronic disorder. The difficulty with this type of analysis lies in determining a set of model parameters that reproduces the experimental data over a wide range of oxygen partial pressures and temperatures adequately but also in demonstrating this set is unique. To do this we need to determine ranges of temperature and oxygen potential over which only a restricted set of defects predominate and all other defects may be neglected in the analysis. We first focus on the low temperature electrical conductivity data (i.e. 973 K, 1073 K, 1173 K) so that only negatively charged ′′ ′ aliovalent ions (y), h◦ , Oi and W defects need be considered.
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The region in which equation (24) is relevant is known as region 2. Note that a transition occurs at the point where isolated oxygen interstitials predominate ′′ over extrinsic electronic defects, i.e. when y ∼ 2[Oi ] and σ is equal to 2 σplat. , where σplat. is the electrical conductivity in the purely extrinsic regions. Fig.6 indicates that electrical conductivity data exists beyond this transition oxygen partial pressure only at 1173 K. A simplified model may be developed that accounts for the oxygen defects that are isolated only. This model uses equations (24) and (8) (with ViX ∼ UUX ∼ 1) the combination of which yields: p (25) [UU◦ ]3 − y [UU◦ ]2 = 2KOi pO2 Equation (25) may be rendered adimensional by dividing both sides of the equation by y 3 which yields: x3 − x2 = 2
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γ, i.e. yO3 i may be determined so that solution to equation (26) reproduces as σ many experimental data points ( σplat , where σplat is the electrical conductivity value in the extrinsic region) as possible. With increasing oxygen partial pressures and as extrinsic carriers and isolated oxygen interstitials become negligible, the appropriate electroneutrality equation becomes: h ′i ′′ W + 2 [Oi ] ≈ [UU◦ ] (27)
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which suggests Log(x) varies as 12 .Log(pO2 ) Note that under those assumptions, Log( σ) has the same dependence upon Log(pO 2 ). We focus on the highest temperature data we are concerned with here for which there are indications that the data cover all three regions. KW is estimated at approximately 5.6×105 from fitting equation (29) to Perron’s [9] deviation from stoichiometry data as shown in Fig. 7. A unique value of γ, estimated at γ ≈ 5 × 107 , is determined which provides an adequate fit of the simplified model (equation (26)) to experimental conductivity values corresponding to regions 1 and 2 (see Fig. 8). In order to determine pO2,trans at a given temperature from experiment, it is necessary that for some values of the oxygen potential defects W′ and Oi ′′ coexist, which appears to be the case at 1173 K. It was further assumed that the experimental data point obtained at this temperature and at the highest oxygen partial pressure corresponds to region 3 where clusters constitute the majority oxygen defect population. Electrical conductivity may therefore be approximated beyond this point by a straight line in a Log–Log representation with a slope of 0.5 which goes through this last data point. The intersection between this line and the optimised theoretical curve describing regions 1 and 2 provides an estimate of pO2,trans (10−13 , see Fig.8). KOi is then estimated from (28) at 1.7×10−4, y from equation (26) and the value of γ previously determined. The value of σ in the extrinsic region then enables (equation (23)) µ0 to be estimated. A numerical application yields values of 4.6 × 1024 m−3 and 0.29 m2 KV−1 s−1 for y and µ0 respectively. 23
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KOi 2.5 × 10−4 3.2 × 10−4 1.3 × 10−3
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At this stage, a model that simultaneously takes into account only negatively charged cation impurities, holes, oxygen interstitials and complex defects was applied to optimise the values of corresponding parameters µ0 , y, KOi and KW at 1173 K. Values of 0.26 ± 0.05m2KV−1 s−1 and 5.2 × 1024m−3 were determined for µ0 and y respectively. The value of µ0 appears to be approximately five times the value suggested by Dudney et al. [20] (0.0554m2KV−1 s−1 ). We then went on to reproduce the data at 973 K and 1073 K considering this last estimate of µ0 and using a simplified model, equation (26). The optimsed values of KOi and KW at all three lower temperatures are given in Table 5.
Table 5: Best estimate determination of equilibrium constants KOi and KW obtained using a simplified model with a value of µ0 of 0.26 m2 KV−1 s−1
The fact that the simplified model reproduces the experimental data adequately at all three temperatures is consistent with the statement that under those conditions, electrons and oxygen vacancies are negligible in comparison with holes and oxygen interstitials or clusters.
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B.2. Analysis of the high temperature data (1473 K, 1573 K, 1673 K) We follow Ruello [24]’s hypotheses in the sense that electrons and holes are assumed to have similar mobilities. As a result, a combination of equations 1 and 7 suggests that electrical conductivity data shows a minimum for values of [U◦U ] given by: p (30) [UU◦ ] ≈ [UU′ ] ≈ Ke
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Note that pO2,min may also be considered as the oxygen partial pressure at which exact stoichiometry is obtained. However, because of the shape of the electrical conductivity curves, there is probably a relatively large uncertainty in the determination of pO2,min and hence KAF . Table 2 indicates all values of KOi , KW , Ke and KAF either obtained directly from analysing the data with the asymptotic model or extrapolated at lower or higher temperatures. Values of µ0 and y are assumed as indicated in the previous section (0.26 m2 KV−1 s−1 and 5.2 × 1024 m−3 respectively). Note that if electrons and holes are assumed to have different mobilities, then the analytical expressions derived from the asymptotic model are modified as follows:
p 2 KO pO2,min KOi pO2,min i + (1 − µrel ) 3/2 √ 2µ2rel Ke2 4µrel Ke
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(35)
[1] A. B. Auskern and J. Belle, J. of Nucl. Mat. 3, 267 (1961). [2] F. A. Kr¨ oger, Z. Phys. Chem. 49, 178 (1966).
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[3] B. T. M. Willis, Acta Crist. A 34, 88 (1978).
[4] B. T. M. Willis, J. Chem. Soc. Faraday Trans. 83, 1073 (1987).
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[5] D. A. Andersson, T. Watanabe, C. Deo, and B. P. Uberuaga, Phys. Rev. B 80, 060101 (2009).
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[6] B. Dorado, G. Jomard, M. Freyss, and M. Bertolus, Phys. Rev. B 82, 035114 (2010). [7] R. Dieckmann, Solid State Ionics 12, 1 (1984).
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[8] P. Ruello, G. Petot-Ervas, and C. Petot, J. Am. Ceram. Soc. 88, 604 (2005). [9] P. O. Perron, Thermodynamics of non-stoichiometric uranium dioxide, Report AECL-3072 (1968).
[10] A. B. Lidiard, J. Nucl. Mater. 19, 106 (1966). [11] H. Matzke, J. Nucl. Mater. 30, 26 (1969). [12] J. P. Crocombette, F. Jollet, T. N. Le, and T. Petit, Phys. Rev. B 64, 104107 (2001). [13] M. Freyss, T. Petit, and J. P. Crocombette, J. Nucl. Mater. 347, 44 (2005). [14] B. Dorado, J. Durinck, P. Garcia, M. Freyss, and M. Bertolus, J. Nucl. Mater. 400, 103 (2010). 27
ACCEPTED MANUSCRIPT
RI PT
[15] J.-P. Crocombette, Phys. Rev. B 85, 144101 (2012). [16] K. Park and D. R. Olander, High Temp. Science 29, 203 (1990).
[17] B. Dorado, D. A. Andersson, C. R. Stanek, B. P. Uberuaga, G. Martin, M. Bertolus, M. Freyss, and P. Garcia, Phys. Rev. B 86, 035110 (2012).
[19] S. Ling, Phys. Rev. B 49, 864 (1994).
SC
[18] J.-P. Crocombette, D. Torumba, and A. Chartier, Phys. Rev. B 83, 184107 (2011).
M AN U
[20] N. J. Dudney, R. L. Coble, and H. L. Tuller, J. Am. Ceram. Soc. 64, 627 (1981). [21] B. Dorado, P. Garcia, G. Carlot, C. Davoisne, M. Fraczkiewicz, B. Pasquet, M. Freyss, C. Valot, G. Baldinozzi, D. Sim´eone, et al., Phys. Rev. B 83, 035126 (2011). [22] P. Kofstad, Nonstoichiometry, Diffusion, and Electrical Conductivity in Binary Oxides (1972). [23] J. M. Casado, J. H. Herding, and G. J. Hyland, J. Phys.: Condens. Matter 6, 4685 (1994).
D
[24] P. Ruello, Ph.D. thesis, Ecole Centrale Paris (2001).
TE
[25] P. Garcia, M. Fraczkiewicz, C. Davoisne, B. Pasquet, G. Carlot, D. Sim´eone, G. Baldinozzi, and C. Petot, J. Nucl. Mater. 400, 112 (2010). [26] N. A. Javed, J. Nucl. Mater. 46, 219 (1972).
EP
[27] G. M. Murch and R. Thorn, J. Nucl. Mater. 71, 219 (1978). [28] K. C. Kim and D. R. Olander, J. Nucl. Mater. 102, 192 (1981). [29] M. Tetenbaum and P. D. Hunt, J. Chem. Phys. 49, 4739 (1968).
AC C
[30] H. Y. Geng, Y. Chen, Y. Kaneta, M. Iwasawa, T. Ohnuma, and M. Kinoshita, Phys. Rev. B 77, 104120 (2008). [31] L. Desgranges, G. Baldinozzi, G. Rousseau, J.-C. Ni?pce, and G. Calvarin, Inorg. Chem. 48, 7585 (2009). [32] A. D. Murray and B. T. M. Willis, J. Solid State Chem. 84, 52 (1990). [33] D. A. Andersson, B. P. Uberuaga, P. V. Nerikar, C. Unal, and C. R. Stanek, Phys. Rev. B 84, 054105 (2011).
[34] R. A. Jackson, A. D. Murray, J. H. Harding, and C. R. A. Catlow, Philosophical Magazine A 53, 27 (1986). [35] B. Dorado, Ph.D. thesis, Universit´e de Marseille (2010). 28
ACCEPTED MANUSCRIPT
RI PT
[36] J. Schoenes, J. of Appl. Phys. 49, 1463 (1978). [37] C. R. A. Catlow, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 353, 533 (1977).
[38] N.-D. Morelon, D. Ghaleb, J.-M. Delaye, and L. V. Brutzel, Philosophical Magazine 83, 1533 (2003).
SC
[39] L. V. Brutzel, J.-M. Delaye, D. Ghaleb, and M. Rarivomanantsoa, Philosophical Magazine 83, 408 (2003).
[40] K. Govers, S. Lemehov, M. Hou, and M. Verwerft, Journal of Nuclear ˝ Materials 366, 161U177 (2007).
M AN U
[41] E. Yakub, C. Ronchi, and D. Staicu, Journal of Nuclear Materials 389, 119 (2009). [42] D. S. Aidhy, P. C. Millett, T. Desai, D. Wolf, and S. R. Phillpot, Phys. Rev. B 80, 104107 (2009). [43] G. Martin, P. Garcia, C. Sabathier, L. V. Brutzel, B. Dorado, F. Garrido, and S. Maillard, Physics Letters A 374, 3038 (2010). [44] F. Devynck, M. Iannuzzi, and M. Krack, Phys. Rev. B 85, 184103 (2012).
D
[45] M. Cooper, C. Stanek, J. Turnbull, B. Uberuaga, and D. Andersson, Journal of Nuclear Materials 481, 125 (2016).
AC C
EP
TE
[46] D. J. M. Bevan, I. Grey, and B. T. M. Willis, J. Solid State Chem. 61, 1 (1986).
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