Suppressed thermal conductivity in hyperstoichiometric uranium dioxide controlled by phonon lifetimes Free

Uranium dioxide (UO₂) is the most widely used nuclear fuel, and its thermal properties are important to reactor safety and performance.¹ UO₂ is able to accommodate a variable stoichiometry, depending on temperature and oxygen pressure.^2,3 For example, UO₂ can be oxidized by water vapor in the rare event of a cladding breach in a reactor, which consequently raises the ratio of oxygen to uranium (O/U = 2 + x).² The excess oxygen atoms in hyperstoichiometric uranium dioxide (x > 0) were identified as interstitial defects that form partially ordered clusters.^4–6 In addition, the O/U ratio is one of the most important parameters governing fuel safety, because it significantly affects the thermal properties such as the melting point⁷ and thermal conductivity.^8–12 For example, Manara et al.⁷ found that the melting point of UO_2+x (x = 0–0.21) decreases as the O/U ratio increases. White and Nelson¹¹ reported that UO_2+x pellets exhibit a decrease in thermal conductivity with increasing temperature and follow 1/T dependence due to the anharmonic Umklapp phonon–phonon scattering at 363–1673 K, and the thermal conductivity of UO_2+x also decreases as x increases, indicating the importance of phonon-defect scattering.

Although there are many studies of the thermal properties of UO_2+x at high temperatures (300–1700 K),^7–13 there is little information on the thermal conductivity and phonon properties of UO_2+x at low temperatures (below 300 K). The effects of excess oxygen atoms (x) on the thermal conductivity of UO_2+x at low temperatures, especially around Néel temperature T_N = 30.8 K, are of critical importance in benchmarking theoretical models. In this study, the thermal conductivity of UO₂ and UO_2+x single crystals is reported in the temperature range of 2–300 K. We find that UO_2+x has a much smaller thermal conductivity than UO₂ at all temperatures except near the Néel temperature, where it is unaffected by the excess oxygen. The phonon density of states (PDOS) and scattering function measured by inelastic neutron scattering (INS) demonstrate that the suppressed thermal conductivity in UO_2+x originates with a decrease in phonon lifetimes with the addition of excess oxygen.

The depleted UO₂ crystals were sintered to 92% of the theoretical density at 1377 K and cooled down under Ar-6% H₂ flow in order to achieve a nominal oxygen stoichiometry of 2.00.¹⁴ The hyperstoichiometric UO₂ crystals were obtained by adjusting the partial pressure of oxygen and controlling the oxygen activity at 1273 K in a thermogravimetric analyzer. The final oxygen stoichiometry of the UO_2+x crystals was calculated from the sample weight change relative to the stoichiometric UO₂ reference data. In this method, the typical error for calculating the O:U is ±0.002.¹¹ Thermal conductivity measurements on UO₂ and UO_2+x single crystals with the size of ∼1 $×$ 1 $×$ 3 mm³ were performed in a DynaCool-9 quantum design measurement system. The measurements were performed in the continuous heating mode of the thermal transport Option (TTO) option using a pulse-power steady-state method. The crystals were measured from 2 to 300 K at 0.25 K/min.

The single crystals used in the thermal conductivity measurements are too small for direct INS measurement on phonon dispersion and especially phonon linewidths. Hence, to characterize the phonon properties, instead INS measurements on ∼10 g depleted UO₂ and UO_2.08 powders at 77 and 295 K were performed using the wide angular range chopper spectrometer (ARCS) at the Spallation Neutron Source (SNS) of Oak Ridge National Laboratory.¹⁵ The setup of the spectrometer was identical to previously reported PDOS measurements for UO₂.^14,16 An incident neutron energy of E_i = 120 meV was used, which is high enough to capture the phonon cutoff (maximum phonon energy) around 80 meV and allows for summing over enough zones in momentum space to obtain PDOS. Scattering introduced by the sample can and the cryostat were corrected by subtracting the corresponding spectra from a duplicate, empty sample can measurement. The corrected scattered neutron intensities, I(Φ, t), were converted to the scattering function S(Q, E), where Q is the momentum transfer magnitude and E is the energy transfer. Other experimental details can be found elsewhere^14,16 and are not repeated here. The neutron weighted PDOS g^NW(E) was then obtained by integrating S(Q, E) over Q values ranging from 4 to 10 Å⁻¹ for both UO₂ and UO_2.08, then correcting for multiphonon scattering (using an iterative procedure),¹⁷ the Debye–Waller factor, and thermal population factor, and subtracting the elastic peak. Note that the Q integration range (4–10 Å⁻¹) was intentionally selected to create a square box in S(Q, E) to eliminate effects of uneven Q averaging on the Debye–Waller factor correction while still including a wide enough angle range to obtain a sufficient zone average.

The measured neutron weighted PDOS can be expressed as

where g_i, M_i, and σ_i are the partial PDOS, the atomic mass, and the corresponding neutron scattering cross section¹⁸ of element i (i = U or O), respectively. In order to extract the neutron unweighted PDOS, we take advantage of the fact that nearly all phonon modes below 25 meV are from uranium, and nearly all phonon modes above 25 meV are from oxygen.^14,19,20

The specific heat at constant pressure, C_P, can be written as

where C_V is the specific heat at constant volume, α is the linear thermal expansion, B is the bulk modulus, v is the molar volume, and T is the temperature.²¹ The phonon contribution to the specific heat at constant volume $CVph$ can be obtained directly from the neutron unweighted PDOS using

where k_B and $gT0(E)$ are the Boltzmann constant and neutron unweighted PDOS at temperature T₀, respectively.¹⁶ 

The measured thermal conductivity of UO_2+x single crystals as a function of temperature is shown in Fig. 1(a). The thermal conductivity of UO_2+x for all values of measured x exhibits a peak at ∼10 K and a broad maximum at ∼220 K, and a minimum occurs at the Néel temperature T_N = 30.8 K, consistent with previous observations.^22–24 In addition, other than in the vicinity of T_N, the thermal conductivity of UO_2+x is significantly suppressed compared to UO₂ and decreases with increasing x [see Fig. 1(b)]. The decreasing thermal conductivity with increasing x is similar to that observed at high temperatures, where it has been attributed to phonon-defect scattering.^8–12 More specifically, the thermal conductivity of UO_2.08 [2.68 W/(m K)] is 67% of the value for UO₂ at 77 K [4.46 W/(m K)]. However, the thermal conductivity of UO_2+x is essentially independent of the x at T_N = 30.8 K [∼1 W/(m K)], and this is likely due to magnetic critical fluctuations that become the dominant factor in controlling the thermal conductivity right at the magnetic transition,²⁵ see Fig. 1(b).

To gain deeper insight into the suppressed thermal conductivity in UO_2+x, we first measured a neutron diffraction pattern of UO_2.08 in Fig. S1 of the supplementary material. It shows that UO_2.08 adopts a mix phase of UO₂ and U₄O₉, consistent with the reported phase diagram with 0 < x  < 0.25 at low temperatures.^13,26,27 We then measured their neutron weighted PDOS of UO₂ and UO_2.08 at 77 and 295 K, respectively, as shown in Figs. 2(a) and 2(b). Well-defined zone-boundary phonon peaks are observed at energies of 12, 21, 33, 56, and 72 meV for both UO₂ and UO_2.08 at 77 and 295 K, respectively. With reference to the phonon dispersion of UO₂,^14,28 the uranium-dominated transverse acoustic (TA) and the longitudinal acoustic (LA) zone-boundary phonon energies correspond to the 12 and 21 meV peaks. These peaks show a trivial difference between UO₂ and UO_2.08. The oxygen-dominated transverse optical (TO1 and TO2) and the longitudinal optical LO₂ zone-boundary phonon energies correspond to the remaining 33, 56, and 72 meV peaks, respectively. In contrast, these peaks in UO_2.08 are less intense and slightly broader than that in UO₂. In addition, there are some nonnegligible intensities above phonon cutoff (78 meV, maximum phonon energy in UO₂), which are consistent with nonlinear propagating modes (NPMs) identified in a previous study.²⁹ Most importantly, all of these peak positions in UO_2.08 show trivial shifts compared to UO₂, indicating similar phonon group velocities in these two materials. The calculated specific heat capacities (C_P) of UO₂ and UO_2.08 using the neutron unweighted 77 and 295 K PDOS based on Eqs. (2) and (3) are shown in Figs. 2(c) and 2(d). Our calculated C_P of UO₂ shows good agreement with the experimental C_P,^16,30 except in the vicinity of T_N where magnetic contributions are large.^31–33 Moreover, C_P of UO₂ and UO_2.08 are almost the same (less than 2% difference) over the entire temperature range of 2–300 K. Note that their $CVph$ also show trivial difference as shown in Fig. S2 of the supplementary material.

Thermal conductivity (k) of single crystals in the phonon gas model is quantitatively described by

where $CVph$⁠, V_g, and τ are the phonon heat capacity, the average phonon group velocity, and the average phonon lifetimes, respectively. With the similar $CVph$ and V_g in UO₂ and UO_2.08, the lower thermal conductivity in UO_2.08 must result from smaller average phonon lifetimes than UO₂. For example, given that the thermal conductivity of UO_2.08 is about 67% of the value of UO₂ at 77 K, the average phonon lifetimes in UO_2.08 are expected to be about 82% of the value of UO₂. To validate this, we plot the measured scattering function S(Q, E) as a function of E and Q at 77 and 295 K in Figs. 3(a)–3(d). Interestingly, all S(Q, E) contour plots exhibit near-sinusoidal shaped features in the energy range of 20–55 meV and flat features at 56 and 72 meV, corresponding to LO1, TO2, and LO2 phonon branches based on its known phonon dispersion.²⁸ Moreover, the LO1 phonon branches in UO_2.08 are noticeably broader than UO₂ at both temperatures, which indicate smaller phonon lifetimes in UO_2.08. (Note that in the previous work, the LO1 phonon branch was found to contribute the largest amount (>30%) to the total thermal conductivity in UO₂.²⁸)

Selected energy cuts of the measured scattering function at Q = [3.75, 3.8] and Q = [5.256, 5.306] for 77 K are shown in Figs. 3(e) and 3(f). The phonon energies of TO2 and LO2 modes in UO₂ are very close to UO_2.08, consistent with the PDOS (Fig. 2). The linewidths of the TO2 and LO2 modes in UO₂ [red symbols, lines, and labels in Figs. 3(e) and 3(f)] are comparable to previously measured linewidths (2–3 meV) in a UO₂ single crystal¹⁷—indicating that the powder averaged linewidths observed here are a reasonable representation of the direction resolved widths. (The preservation of these modes with powder averaging is a result of the high symmetry of these modes in the fluorite structure.¹⁹) Comparing samples, the UO_2.08 linewidths [blue symbols, lines, and labels in Figs. 3(e) and 3(f)] for the TO2 modes are about two to four times broader than UO₂, and the LO2 modes are about 1 to 1.3 times broader. These results are consistent with an overall smaller average phonon lifetime in UO_2.08. The phonon lifetimes (an inverse of linewidths) of the TO2 and LO2 branches in UO_2.08 and UO₂ are summarized in Table I. In brief, given the similar $CVph$ and V_g in UO₂ and UO_2+x (Figs. 2 and 3) and the observed smaller phonon lifetimes for select modes (Fig. 3 and Table I), we deduce that it must be the phonon lifetimes that control the composition dependence of the thermal conductivity of UO_2+x (Fig. 1).

In summary, we performed thermal conductivity and INS measurements on UO₂ and UO_2+x at low temperatures (2–300 K). It is found that the thermal conductivities of all UO_2+x single crystals exhibit a double-peak behavior, where maxima occur at 10 and 220 K and a minimum occurs at the Néel temperature T_N = 30.8 K, consistent with previous studies on UO₂. Except in the vicinity of T_N, the thermal conductivity of UO_2+x is significantly suppressed compared to UO₂ and decreases with increasing x. At T_N = 30.8 K, the thermal conductivity of UO_2+x is essentially independent of the x. Further INS measurements of the PDOS and scattering function demonstrate that the heat capacity and phonon group velocities of UO₂ and UO_2+x are similar, and thus, the suppressed thermal conductivity is attributed to shorter phonon lifetimes in UO_2+x. This conclusion is supported by observed shorter lifetimes for select phonon modes in the powder averaged scattering function.

See the supplementary material for the measured neutron diffraction patterns and heat capacity (⁠$CVph$⁠) of UO₂ and UO_2.08.

H.M., M.S.B., K.G., and M.E.M. were supported by the Center for Thermal Energy Transport under Irradiation, an Energy Frontier Research Center funded by the U.S. Department of Energy (DOE), Office of Science, United States, Office of Basic Energy Sciences. D.J.A. and K.G. acknowledge support from the Advanced Fuel Campaign program (INL). Portions of this research used resources at the Spallation Neutron Source, a U.S. DOE Office of Science User Facility operated by the Oak Ridge National Laboratory. We are grateful to David A. Andersson and Christopher R. Stanek for providing UO_2+x crystals for the thermal conductivity measurements. This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).