Corium lavas: structure and properties of molten UO2-ZrO2 under meltdown conditions

In the exceedingly rare event of nuclear reactor core meltdown, uranium dioxide fuel reacts with Zircaloy cladding to produce eutectic melts which can subsequently be oxidized by coolant/moderator water. Oxidized corium liquids in the xUO2·(100 − x)ZrO2 system were produced via laser melting of UO2-ZrO2 mixtures to temperatures in excess of 3000 K. Contamination was avoided by floating the droplets on a gas stream within an aerodynamic levitator and in-situ high-energy x-ray diffraction experiments allowed structural details to be elucidated. Molecular dynamics simulations well reproduced diffraction and density data, and show less compositional variation in thermal expansion and viscosity than suggested by existing measurements. As such, corium liquids maintain their highly penetrating nature irrespective of the amount of oxidized cladding dissolved in the molten fuel. Metal-oxygen coordination numbers vary with both composition and temperature. The former is due to mismatch in native values, nUO(x = 100) ≈ 7 and nZrO(x = 0) ≈ 6, and the requirement for oxygen site stabilization. The latter provides a thermal expansion mechanism.

Multiscale modelling of material processes is essential for both forensic investigation of historically rare fission reactor core meltdowns, as well as development and design of future power reactors with improved safety and performance [1][2][3] . At the smallest, atomic length scales, molecular dynamics based on semi-empirical interatomic potentials remains a popular and powerful tool [4][5][6][7][8] , allowing access to wider length and time scales as compared to full quantum mechanical treatments of the electronic structure. Experimental validation of such simulations is of paramount importance 1 , but data at the extreme temperatures reached during core meltdown is scarce owing to the inherent practical challenges. Recently measurements pertaining to the atomistic structure of molten fuel and fuel-bearing phases have become possible [9][10][11] . Herein we seek to apply such techniques to the molten corium phase, obtained during extensive fuel-clad interaction, and to exploit these measurements by evaluating promising interatomic potentials which, in turn, are used to predict key structural and bulk physical properties including thermal expansion and viscosity. Melt viscosity in particular is an important property governing the spreading, and thereby the cooling, of corium in core-catchers 12 . The potentials and properties derived are suitable for larger scale simulations, including solid-melt interfaces between core materials, or to study the incorporation and behaviour of fission products. Moreover the predicted physical properties can be used to inform higher level finite element codes as part of multiscale reactor models.
A severe accident scenario involving melting of the reactor core leads to the formation of corium lavas which are initially composed primarily of fuel and cladding materials. The most common such materials are based on UO 2 and zirconium respectively. Subsequent interaction with water present as coolant and/or neutron moderator leads to oxidation of U-Zr-O liquids and H 2 generation. In this paper, we study the structure and properties of oxidized molten corium in the UO 2 -ZrO 2 pseudobinary obtained by laser heating ~100 mg quantities of ceramic material during levitation on a gas jet 13,14 within a hermetically sealed chamber 9,10 , Fig. 1B. In-situ measurements using high-energy synchrotron x-rays yield the melt's structure factors, S(Q), which provide atomistic level structural information used to verify the efficacy of molecular dynamics (MD) models, themselves based on indicate negligible differences between measurements made in Ar and highly reducing Ar:5%H 2 gases (cf. 20 and 21 mol% UO 2 samples). It is important to bear in mind that at such high temperatures, and at low oxygen potentials, the formation of low valence states of uranium, initially U 3+ , can be significant. However, previous work on UO 2-x melts 9 revealed only subtle changes in the liquid diffraction patterns under different oxygen fugacities, from which a U 3+ content of ≲ 27% was estimated in Ar:5%H 2 . Since U 4+ is favored thermodynamically in Ar 9,17 (owing to ppm levels of oxygen) and, based on the previous work, can be maintained kinetically even at lower oxygen potentials, we neglect the presence of lower valence states in our analyses.
Simulated melt densities were also obtained within reasonably good agreement with existing measurements [18][19][20][21] , Fig. 3, despite considerable uncertainty in the experimental density of molten ZrO 2 . The experimental verification of the MD models by x-ray diffraction and comparison of their densities to the literature, as well as their basis in the well-tested Yakub potentials for UO 2 4,11 , encourages further interrogation of their statistical structure and properties.
Composition Dependent Coordination Numbers. Several interesting phenomena are revealed in Fig. 4A, which shows that the cation-oxygen coordination numbers, n ZrO and n UO , as well as their mean, n RO , depend on both composition and temperature (see also Fig. S4). with Q max = 11.77 Å −1 and without any Q-dependent window or modification function. The inset shows the full detector panel with masked regions in red (mainly beamstop). In both parts A and B, the blue points correspond to the nominally 30 mol% UO 2 sample measured in Ar:5%H 2 which overlay those of the sample of the same nominal composition measured in Ar, demonstrating that the data are essentially indistinguishable.
Scientific REPORTS | (2018) 8:2434 | DOI:10.1038/s41598-018-20817-z At high temperatures solid UO 2 -ZrO 2 binaries form a cubic solid solution, with all cations coordinated to 8 oxygen ions. Further increasing the temperature leads to melting, upon which n ZrO and n UO have both been shown to decrease in the pure endmember dioxides 11,15,18 , as observed for several other oxides 15,22 , but importantly the effect is larger for zirconium. As such, molten ZrO 2 has a different structure from molten UO 2 , akin to the difference in room temperature crystal structures, where monoclinic zirconia contains 7-fold Zr, while UO 2 maintains the cubic structure with 8-fold U. The insight provided by our models of the binary melts is that this mismatch in endmember or 'native' R-O coordination numbers gives rise to linear composition dependence of both n ZrO (x) and n UO (x). Since the slopes ∂n ZrO /∂x ≠ ∂n UO /∂x, interpolation of n RO (x) between endmembers, Fig. 4A, is non-linear, and specifically, quadratic. A similar effect is observed for the alkali cations in chemically more complex mixed alkali silicate glasses 23,24 . Further support for the existence of this effect is provided by comparison of the appropriately weighted g UO (r) and g ZrO (r) taken from simulations of the endmember dioxides, to the x-ray diffraction data for the 27 mol% UO 2 corium binary. Figures S5 and S6 show that use of the endmember cation-oxygen bond length distributions leads to poorer agreement with the experimental corium T(r) in the region of the first peak, than does the simulation of the binary itself (Fig. 1F). We propose that the variation in local structure with composition can be qualitatively understood in terms of the stabilization of the various oxygen environments present.
Oxygen site stabilization. The concept of the Pauling electrostatic bond strength (EBS) 25 has been shown to be helpful in understanding amorphous structures 26 . Within this approach the M-O bonds are assigned an EBS of s = Z M /n MO and then the stability of the anionic oxygen sites are assessed by the proximity of the EBS sum, where the sum is taken over all O-M bonds to the given O 2− anion. Figure 4B shows a schematic representation of a 3-fold oxygen site in a material built up of only 6-fold M 4+ cations, similar to molten zirconia. The oxygen must be on average 3-coordinated given the universal relationship n jk = (c k /c j ) n kj where c k and c j are atomic fractions of the k th and j th atomic species respectively, or n OM = (1/2) n MO for MO 2 stoichiometry ( Fig. 4A right hand ordinate). In this case, ζ O = 2 and the site stability is maximized (Fig. 4B). Figure 4C to F illustrate the effect of introducing a larger cation, M' , with larger native n M'O (100) = 8. We have chosen the value 8, which is larger than n UO (100) ≈ 7, in order to simplify the discussion by means of an integer native O-M' coordination of n OM' (100) = 4. Figure 4C considers the dilute case, where M'O 2 is doped into the MO 2 melt such that the most likely 'mixed cation' oxygen sites are 3-fold, with two M and a single M' . With native coordination numbers maintained, ζ O = 1.83 < 2 and the site is less stable than in the undoped melt. One way to stabilize the site would Clearly the EBS approach is highly simplified, not all bonds to a given cation are equal, and in the actual melts there are wide bond-length distributions, Figs 1F and 2B, as well as distributions of coordination environments, Fig. S8. The flexibility introduced through variable bond lengths allows oxygen sites to be stabilized without necessarily any coordination changes. For example, a given cation can simultaneously form longer bonds to 4-fold oxygen sites, and shorter ones to 3-fold oxygen sites. The magnitude of the coordination changes within the EBS approach are suggested by the coordination numbers given in Figs 4D,F, S7C, S7E, which are those required to yield ζ O = 2. We reason that bond length variability will significantly depress the magnitude of these coordination changes. Furthermore, if one iterates around Figs 4B-F or S7 there is positive feedback, i.e. the two cation coordination numbers simply diverge in a runaway process. The limiting mechanism preventing this is the existence of minimum and maximum stable coordination numbers, which are controlled by the form of the pair potentials. Finally, the effect is smaller in our MD simulations given the smaller difference in native coordination numbers (n ZrO (0) ≳ 6, n UO (100) ≈ 7) than illustrated. Nonetheless, similar non-ideal mixing behavior can be expected to occur in related (Ti,Zr,Hf)O 2 -(Th,U,Pu,…)O 2 melts, but likely becomes negligibly small in more ideal cases, where cation radii are more closely matched, such as molten (U,Pu)O 2 MOX fuels. In real Pu bearing melts, hypostoichiometry and the occurrence of reduced Pu 3+ would play an important role, potentially again giving rise to composition dependent structures. Fig. 4A is that n ZrO (x,T) and n UO (x,T) both decrease linearly with T, and this is plotted explicitly in Fig. S4, which reveals that the temperature coefficients are approximately independent of composition (Table S1). The values ∂n ZrO /∂T ≈ −3.2(1)×10 −4 K −1 and ∂n UO /∂T ≈ −4.9(2)×10 −4 K −1 are comparable to that calculated for titanium in molten TiO 2 , ∂n TiO /∂T = −3.0(1)×10 −4 K −122 , and it is evident that the magnitude of the effect increases with decreasing field strength, Ti 4+ → Zr 4+ → U 4+ , and increasing anharmonicity of the metal-oxygen pair potential. As with the thermal coordination number reduction, the temperature dependencies of the melt densities, Fig. 3,  29 shaded, and for a 62 mol% UO 2 corium melt from Abalin, et al. 31 . The latter authors report the kinematic viscosity, and we have therefore multiplied by the melt density (Asmolov,et al. 20 ) to obtain the shear viscosities shown. The same viscosity data are plotted by Sudreau and Cognet 12 with about 15% lower magnitude, presumably due to use of smaller densities, the provenance of which are not stated by those authors. Recent MD results from Kim et al 5 . are also shown, the shaded region corresponding to their range of interpolated corium viscosities (their Fig. 8c). The solid and chained curves for ZrO 2 are based on Teter and EAM potentials, respectively. Note that we have plotted our own fits to the digitized data from Kim et al. 's Fig. 8a and b because their equations 38 and 39 (and their Table 5) are erroneous and do not reproduce their graphical data.

Temperature Dependent Coordination Numbers and Thermal Expansion. Also revealed in
Scientific REPORTS | (2018) 8:2434 | DOI:10.1038/s41598-018-20817-z are also linear. This is suggestive of coordination change playing a key role as a thermal expansion mechanism. Positive correlation between coordination number and density is observed in many other settings, for example: the tetragonal and cubic phases of ZrO 2 (8-fold Zr) are more dense than the monoclinic phase (7-fold Zr); both density and coordination numbers tend to increase with increasing pressure 27,28 ; and materials like amorphous SiO 2 and GeO 2 , composed of open networks of 4-fold tetrahedral cations, have lower atom number densities, despite their smaller cations, compared to melts of Ti, Zr, U dioxides, or the (U,Zr) corium melts, all of which contain more highly coordinated cations 11,15,18,22 . The calculated thermal expansion coefficients (TECs) of the corium melts at 3070 K are shown in Fig. 5A,B. They show remarkably little dependence on the corium composition, and lie between higher measured values for the endmembers 19,20 and the lower value measured for a corium binary 20 . As such the models show no deep minimum in TEC as implied by the regular solution interpolation of Asmolov, et al. 20 . Such a deep minimum would require much larger structural variations with composition than observed by our diffraction measurements and MD simulations. Diffusivity and Viscosity. The self-diffusion coefficients calculated for the corium liquids are large, of order 10 −5 to 10 −4 cm 2 s −1 (Fig. 5C), similar to the Yakub, et al. 4 model for UO 2 upon which our potentials are based 11 . Since no measurements of D i in corium liquids have been made, we have used the Stokes-Einstein (SE) relation to estimate the liquid viscosities from the D O : B O O Figure 5D. The result for UO 2 is slightly smaller than the recommended 29 measurements made by Woodley 30 in the 3140 to 3300 K range, although the agreement is reasonable given the experimental uncertainties and assumption of the SE relation. As with the TECs, little variation in viscosity is observed across the range of corium compositions, and little increase with ZrO 2 content is observed, in contrast to that implied by the comparison of the corium measurement of Abalin, et al. 31 to that of Woodley 30 . Abalin, et al. explain this discrepancy as due to build up of pressure within Woodley's apparatus as T increases -leading to an apparent suppression of the activation energy for viscous flow. Note however that the difference between Abalin, et al. 31 and Woodley 30 (or our SE estimates) amounts to a factor ≲ 1.4 (1.8), which is small -smaller than the change in viscosity of liquid water between room temperature and its freezing point. Considering that liquid viscosity varies over at least 16 orders of magnitude, from 10 12 Pas for supercooled liquids near their glass transition, down to 10 −4 Pas for boiling or supercritical water, a factor of < 2 is of little practical importance. The less precise molten UO 2 viscosity measurements of Tsai and Olander 32 are in fact larger than Woodley's, and similar to Abalin et al. 's corium measurement.
The key observation is that the corium liquids have incredibly low viscosities of only a few mPas, approaching that of cold water. Combined with their high mass densities, these fluids are highly penetrating and capable of traveling large distances within a reactor vessel during meltdown, independent of the amount of cladding incorporated into the molten fuel. This fluidity can enhance the spreading and cooling of melts within suitable core-catcher designs.
It is noteworthy that similar viscosities are obtained using the cationic self-diffusion coefficients in the Stokes-Einstein relation only if the diffusion of [MO n ] coordination complexes are considered, such that the hydrodynamic radii are significantly larger than the ionic radii (Fig. S10). This has been explicitly validated in mixtures of molten halides [33][34][35] .
Arima 6 has modelled other transport properties of corium melts, such as the thermal conductivities, and shown that they also show little variation with composition. Note however that the potential parameterizations used by Arima 6 are based on those of Pedone, et al. 36 (ξ = 0.60), which typically require further refinement to yield good agreement with liquid diffraction measurements at experimental temperatures 15,22 . Kim, et al. 5 recently computed similar UO 2 and ZrO 2 melt viscosities to ours by means of the Green-Kubo stress autocorrelation approach (Fig. 5D). The authors used embedded atom method (EAM) many-body CRG potentials 7,8 , and additionally tested Teter pair potentials 37 for ZrO 2 . They found some dependence on the choice of ZrO 2 interatomic potential, but nonetheless inferred by compositional interpolation little variation in viscosity across the corium pseudobinary, in accord with our findings.
We note that EAM potentials, unlike simple pair potentials, allow for violation of the Cauchy relations between elastic tensor components, and have been shown 7 to yield an improved bulk modulus for UO 2 as compared to the Yakub model upon which our potentials are based. Direct comparisons of the EAM models 5 to diffraction data for the molten end-members (Figs S11 to S13) reveal good agreement for UO 2 , but poorer agreement for ZrO 2 . Furthermore, whilst the density of molten UO 2 appears well predicted by the EAM CRG potentials, that of ZrO 2 is rather large, as compared to the measurements of Kohara, et al. 18 and the function used by Asmolov, et al. 20 , see Fig. 3. The Teter potentials 37 tested by Kim, et al. 5 conversely yield a rather low melt density (very similar number density to UO 2 ) and we have shown previously 22 that these potentials require refinement to yield agreement with experimental densities and melt diffraction data.

Conclusions
High energy x-ray diffraction from levitated droplets of molten corium has been used to validate classical molecular dynamics models of the same liquids. These models were based upon the well-tested UO 2 interatomic potentials of Yakub et al. 4 subsequently refined to similarly measured x-ray structure factors and densities of the UO 2 and ZrO 2 molten endmembers. Modelled properties including thermal expansion, self-diffusion coefficients and melt viscosities show very little dependence on U/Zr ratio in the UO 2 -ZrO 2 pseudobinary, and confirm the extremely fluid and highly penetrating nature of these liquids. Corium liquids are therefore capable of traveling large distances within a reactor vessel during meltdown, independent of the amount of cladding incorporated Scientific REPORTS | (2018) 8:2434 | DOI:10.1038/s41598-018-20817-z into the molten fuel. Such high fluidity can be beneficial for spreading and cooling core melts within suitable core-catcher designs. Mismatch between the native metal-oxygen coordination numbers, n UO ≈ 7 in molten UO 2 and n ZrO ≈ 6 in molten ZrO 2 , results in their variation with composition. In particular, both n UO and n ZrO decrease with increasing UO 2 content, while their mean value increases. This behavior is explained on the basis of the stabilization of the various 3-and 4-fold oxygen sites present in the melts. Temperature dependent coordination numbers are also observed, and this is suggested to be a primary mechanism for liquid thermal expansion. Temperature derivatives ∂n UO /∂T ≈ −5 × 10 −4 K −1 and ∂n ZrO /∂T ≈ −3 × 10 −4 K −1 , imply an increasing temperature dependence with increasing cation coordination and anharmonicity of the pair potentials. Our experimental verification of interatomic potentials for molten corium suggests their suitability for larger scale simulations, including melt-solid interface interaction, incorporation and behaviour of fission products, and for informing higher level finite element codes as part of multiscale reactor models.

Methods
Diffraction experiments, using 100.23 keV x-rays, were performed at beamline 6-ID-D of the Advanced Photon Source, Argonne, Il, using an aerodynamic levitator, 400 W, 10.6 μm laser heating, and a hermetically sealed chamber, Fig. 1B,C, as described previously 9,10 . For a single sample formed in-situ, a sample to detector distance of 306.6 mm was used, whilst for later experiments on sintered ceramic beads, the distance was 837.7 mm, both of which were calibrated using a crystalline CeO 2 standard. In the latter measurements a second ZnSe laser window was installed to increase redundancy in the containment of radioactive samples. The argon gas between the two windows was held at a slight overpressure such that a sensor could detect breach of either window and cut power to the laser.
One corium sample was produced by in-situ fusion of individual UO 2 and ZrO 2 ceramic beads in the levitator nozzle via laser melting. Based on the masses of the initial beads (28 mg ZrO 2 and 69 mg UO 2 ), a composition of 52.9 mol% UO 2 was obtained, Table S2. However, due to the high vapor pressure of UO 2 above the melt, significant enrichment in ZrO 2 occurred during the in-situ synthesis of the bead. This compositional change was assumed to take place primarily during the initial stages of fusion, which was confirmed by comparison of the individual 1 s diffraction patterns collected during the final 27 s melting stage (Fig. S1). The diffraction pattern was observed to be invariant over the latter 16 s, and only these data were averaged together and analyzed. The composition of the recovered material was determined by inductively coupled plasma optical emission spectrometry (ICP-OES) to be 27(3) mol% UO 2 , and this composition was used for the analysis.
Further samples were made by milling together powders of −270 mesh (53 μm) depleted UO 2 and 1 μm ZrO 2 . The mixtures were pressed into a tungsten die to form a pellet that was then sintered at 1700 • C for 1 hr in an argon atmosphere using an induction furnace. The pellets were removed from the die and ground using a 400 grit silicon carbide polishing wheel to form roughly spherical samples 2.5-3.0 mm in diameter, suitable for levitation. Details of three samples successfully measured in the molten state are recorded in Table S2. ICP-OES analyses reveal final sample compositions of 20, 21 and 4 mol% UO 2 , compared to their nominal values of 30, 30 and 15 mol% UO 2 respectively, consistent with the high vapor pressure of UO 2 and estimates based on the mass losses. Measurements on nominal compositions of 50 and 75 mol% UO 2 were attempted but samples were only partially melted and are not reported here. The likely reasons for a lack of complete melting are the scattering of laser light by airborne particles of condensed volatilized UO 2 , along with the effective laser power loss associated with an additional laser window.
Either Ar or 5%H 2 :95%Ar levitation gases were used. Although the latter is highly reducing, the effects on diffraction patterns of molten UO 2−x have been shown to be small 9 , and are expected to be correspondingly smaller in dilutions with zirconia. This is confirmed by the strong similarity of the diffraction data for the two samples with the same nominal, 30 mol% UO 2 , compositions measured in different gases. As such, the UO 2 component is considered stoichiometric in our analyses. Future investigations into the effect of U 3+ content on melt structure and properties would be of great value, but extremely challenging experimentally, e.g. by in-situ U L III edge XANES. Modelling of hypostoichometric melts on the other hand poses a more tractable avenue. The fully corrected structure factors, S(Q) -1, are available as supplementary data online. Residual background intensity varying slowly with Q was removed using the top-hat convolution method (6.0 Å −1 top-hat width) and real-space intensity below 1.0 Å was supressed 38 , which is important for direct comparison to the MD simulated functions.
Atomistic models of corium melts were obtained by classical molecular dynamics (MD) simulation using the DL_POLY classic code 39 . Seven compositions were investigated, including the two endmembers, the ICP-OES measured final corium compositions of 4, 20 and 27 mol% UO 2 , and additionally 50 and 75 mol% UO 2 . As a starting point Morse potential parameterizations previously refined to measured x-ray structure factors of molten UO 2 at 3270 K 11 and molten ZrO 2 at 3170 K 15 were chosen. Those for UO 2 are based on the well-tested Yakub potentials 4 , but with simplified interatomic pair-potentials taking the form The second term is the short range Morse potential, parameterized by E jk , a jk and r 0jk , and the first term represents the Coulombic energy between ions of formal charge, Z j , in units of the electron charge, e, and 0 < ξ ≤ 1 is the ionicity parameter. Since the two starting models are incompatible due to their different ξ = 0.55 and 0.60, for UO 2 and ZrO 2 respectively, a new model was derived for liquid ZrO 2 using ξ = 0.55 and identical O-O interactions to the UO 2 model 11 . E ZrO and r 0ZrO were refined to the x-ray S(Q) and measured density 18,20 , Fig. 3, within the isothermal-isobaric, NPT ensemble. The starting point for all MD trajectories was the 4800 atom configuration used in a study of molten TiO 2 22 , which has the same stoichiometry, and similar atom number density (0.0726 Å −3 ) to the corium melts. The ZrO 2 refinement runs began with 5 ps of equilibration (forces capped at 10 3 k B T/Å), Scientific REPORTS | (2018) 8:2434 | DOI:10.1038/s41598-018-20817-z followed by a further 20 ps NPT trajectory, using a 1 fs timestep. A further 25 ps in the isothermal-isochoric NVT ensemble were used for calculation of the pair distribution functions, g jk (r) 40 , which were appropriately weighted, Fourier transformed and summed to yield the x-ray S(Q) (Figs S2, S3, 1,2). Ultimately the parameter E ZrO was increased, resulting in an increase in system density toward the measured values 18,20 , Fig. 3, and r 0ZrO was adjusted to maintain the U ZrO (r) potential minimum at the same distance as that of the reference potential 15 . Any further increase in E ZrO , and thereby the density, had a detrimental effect on the agreement between measured and simulated x-ray S(Q), Fig. S3, which may explain the low density of 0.068 Å −3 obtained by Skinner, et al. 15 during the original ξ = 0.60 refinement. The finalized potential parameters are given in Table S3. These were used for modelling of binary corium melts where U and Zr cations were decorated randomly onto the starting configuration 22 . For each composition the models were equilibrated at 3670 K for 50 ps followed by a further 50 ps (NPT) over which reported volume (density), mean-square-displacement (MSD) and coordination data were collected. The temperature was then reduced by 200 K and the procedure repeated (with just 25 ps equilibration) until T = 2670 K. Note that several of the lowest temperature points sampled correspond to supercooled liquids. This allows for interpolation of results through the melting points, as well as for isothermal comparisons across the full composition range. The liquidus temperature ranges from 3120 K (pure UO 2 ), down to a minimum of 2850 K at 40 mol% UO 2 41 . An additional 3170 K step was included for pure ZrO 2 to match the x-ray experimental temperature 15 . A further 50 ps (NVT) was simulated for collection of g jk (r) for comparison to experimental S(Q), and to calculate coordination number and bond-angle distributions using the RINGS code 42 . In comparisons of x-ray and MD simulated S(Q) and T(r), the MD simulation closest to the experimental temperature (Table S2) was chosen, and the average density term, T 0 (r) = 4πρ 0 r, added to the experimental T(r) = D(r) + T 0 (r), was based on the MD simulated atom number density, ρ 0 . Deviations from a statistical distribution of U and Zr cations were not detected, based on the observed U-U, Zr-Zr and Zr-U coordination numbers.
Note that any effects of charge transfer or disproportionation via 2U 4+ ⇄U 3+ + U 5+ have been neglected in our simulations. Gillan 43 estimates the free energy for this process in molten UO 2 at ΔF = 0.7 to 1.7 eV which yields between 4% and 18% of each of the [U 3+ ] = [U 5+ ] populations at the melting point. Over the 1000 K T range of our simulations, these become 2-6% and 15-20%. Clearly this is non-negligible from the point of view of electronic processes, whereby small polaron hopping provides a conduction mechanism. However, we expect the effect on structure and bulk properties such as viscosity to be much weaker. Consider the success of our and previous approaches 4-7 in correctly predicting a wide range of properties in the absence of charge transfer. Nonetheless future investigation into the effects of charge transfer is of interest. For example, models such as ours could be used as the basis for more accurate predictions of ΔF than those of Gillan 43 who used the Ornstein-Zernike equation with hypernetted-chain closure relation to estimate the melt structure.
Data availability. The fully corrected structure factors, S(Q) -1, are available as supplementary data online.
All other data are available upon request from the corresponding authors.