The grain-size effect on thermal conductivity of uranium dioxide
K. Shrestha, T. Yao, J. Lian, D. Antonio, M. Sessim, M. R. Tonks, K. Gofryk
TThe grain-size effect on thermal conductivity of uranium dioxide
K. Shrestha , T. Yao , J. Lian , D. Antonio , M. Sessim , M. Tonks , and K. Gofryk Idaho National Laboratory, Idaho Falls, Idaho, 83402, USA Department of Mechanical, Aerospace, and Nuclear Engineering,Rensselaer Polytechnic Institute, Troy, NY 12180, USA Department Department of Materials Science and Engineering, University of Florida,Gainesville, FL 32611, USA
We have investigated the grain boundary scattering effect on the thermal transport behavior of uraniumdioxide (UO ). The polycrystalline samples having different grain-sizes (0.125, 1.8, and 7.2 µ m) have beenprepared by spark plasma sintering technique and characterized by x-ray powder diffraction (XRD), scanningelectron microscope (SEM), and Raman spectroscopy. The thermal transport properties (the thermal conduc-tivity and thermoelectric power) have been measured in the temperature range 2-300 K and the results wereanalyzed in terms of various physical parameters contributing to the thermal conductivity in these materialsin relation to grain-size. We show that thermal conductivity decreases systematically with lowering grain-sizein the temperatures below 30 K, where the boundary scattering dominates the thermal transport. At highertemperatures more scattering processes are involved in the heat transport in these materials, making theanalysis difficult. We determined the grain boundary Kapitza resistance that would result in the observedincrease in thermal conductivity with grain size, and compared the value with Kapitza resistances calculatedfor UO using molecular dynamics from the literature. INTRODUCTION
Uranium dioxide is one of the most studied actinidematerials as it is used as the primary fuel in the commer-cial nuclear reactors.
There are around 500 active nu-clear reactors, producing more than 15% of the total elec-tricity worldwide. In a reactor, the heat energy producedfrom the nuclear fission events inside the fuel pellets istransformed into electricity. Thus, the heat transportmechanism, i.e. thermal conductivity of the fuel materialis an important parameter for fuel performance, regard-ing its efficiency and safety. A nuclear reactor operatesat extreme environments that can include high temper-ature, high pressure, and high irradiation. As a result,a fuel pellet undergoes severe structural changes underirradiation conditions, including grain subdivision, fis-sion gas bubbles growth and redistribution and extendeddefects accumulations.
Thermal properties of the fuelmaterial are greatly affected by these changes which ul-timately affect the performance of a reactor. Numeroustheoretical and experimental studies (see Refs. 6–9 andreferences therein) have been carried out to understandhow these microstructure changes affect thermal trans-port properties of UO .UO is a Mott-Hubbard insulator with an energy gapof ∼ It crystalizes in cubic, CaF type of struc-ture and orders antiferromagnetically at the N´eel temper-ature, T N = 30.5 K. In an insulator, the lattice vi-brations (phonons) responsible for the heat transport arescattered by different scattering centers, such as defects,grain boundaries, phonon-phonon, etc. Depending uponthe temperature range, different scattering mechanismsdominate at different temperature regimes.
For in- a) Electronic mail: email: [email protected] stance, umklapp phonon-phonon scattering dominatesthe thermal conductivity at high-temperature, while thepoint-defect and boundary scattering govern the heattransport at intermediate and low temperatures, respec-tively. At low temperatures where the phonon mean freepath is comparable to the grain-size, the grain boundaryscattering mechanism is the main factor limiting the ther-mal conductivity. The effect of grain-size on the thermalconductivity has been investigated at low temperaturesin other types of materials, such as semiconductors, ther-moelectrics, nanomaterials, and thin films.
In thecase of UO , most of the studies on thermal propertiesare focussed in the high temperature range (where nu-clear reactors operate) to better understand the fuel per-formance and reactor design. However, in order tobetter understand mechanisms that govern heat trans-port in this important technological material, and to ac-curately model this compound at all relevant tempera-tures, the effects of various scattering mechanism mustbe quantified.Here, we have carried out systematic studies on thegrain-size effect on thermal conductivity of UO by per-forming measurements at low temperatures to study dif-ferent scattering mechanisms, focussing on grain bound-ary scattering. The UO samples (having grain-sizes0.125, 1.8, and 7.2 µ m) have been synthesized by SparkPlasma Sintering technique and characterized by XRD,SEM, and Raman methods. We show that the grainboundary scattering parameters vary systematically withthe grain-size below 30 K. Such a behavior is not observedat higher temperatures where other scattering processesstart to dominate. The thermal conductivity data areanalyzed using the Callaway model and the variation ofdifferent parameters with the grain-size are discussed. Inaddition, the grain boundary scattering has been assessedin these materials using molecular dynamic simulationsat higher temperatures. a r X i v : . [ c ond - m a t . m t r l - s c i ] O c t FIG. 1. (Color online) (a), (b), and (c) shows the microstructure features of sintered UO fuel pellets with different grain-sizeof 0.125 µ m, 1.8 µ m, and 7.2 µ m, respectively. (d) XRD spectra show that the sintered pellets have UO x structure with ‘x’values calculated by peak positions as shown in (e) for the high angle section. Superimposing feature of Raman spectra (f)indicates similar degree of interaction between defects and UO crystal structure in the sintered pellets. EXPERIMENTAL
Polycrystalline UO fuel pellets with three differentgrain-sizes (their physical properties are summarized inTable I) were sintered by spark plasma sintering fromvarious batches of powder prepared from UO . pow-der purchased from International Bio-analytical Indus-tries Inc., USA. Detailed information on powder samplescan be found in Ref. [25 and 26]. Generally, the pelletswith a grain-size of 7.2 µ m were sintered directly from theas purchased UO . powder at 1600 o C for 5 mins undera pressure of 40 MPa. The pellets with a grain-size of 1.8 µ m were sintered from nano-crystalline UO . powder at1300 o C for 30 mins under a pressure of 40 MPa. Due tothe graphite die used in those two sintering routes, thepellets were in-situ reduced to hypo-stoichiometric. Thepellets with a grain-size of 0.125 µ m were sintered at 700 o C for 5 mins under a pressure of 500 MPa in WC die.These sintered pellets were hyper-stoichiometric and apost-sintering annealing was conducted in a tube furnacein 4% H /Ar gas atmosphere in order to reduce oxygen.The furnace was purged by 4 hrs gas flow at a rate of 200ml/min, then the reducing was conducted at 600 o C for24 hours at a gas flow rate of 50 ml/min. The sinteredpellets are carefully stored in an oxygen controlled envi-ronment with momentary exposure to air for microstruc- ture and phase characterization. The bulk density of thepellets was measured by an immersing method using DIwater as the media, calculated based on weight differencein air and water, against a theoretical value of 10.97 g/ccfor UO . Microstructure characterization was conductedusing a Carl Zeiss Supra 55 (Jana, Germany) field emis-sion SEM. Grain-size was determined using a rectangularintercept method following an ASTM E122-88 standard(1992). The average size is given by: D = (cid:115) Aπ (cid:0) N i + N (cid:1) (1)where A is the area of an arbitrary drawn rectangle, N i and N are the numbers of grains in the rectangleand on the boundary of the rectangle, respectively. Atleast two hundred grains were analyzed for each pellet.The grain-size uncertainties are standard deviations ofthe measured grain-size for the same pellet from differ-ent locations.X-ray diffraction (XRD) spectra of the sintered pel-lets were collected by a Panalytical X (cid:48) Pert XRD system(Westborough, MA, USA) using Cu K α ( λ = 1.5406 ˚ A )irradiation at room temperature. Before each run, the X-ray beam was aligned with a direct beam method througha 0.2 Cu beam attenuator. Sample height was aligned TABLE I. Physical properties of polycrystalline UO samples.Sample ID UO (0.125 µ m) UO (1.8 µ m) UO (7.2 µ m)Grain size ( µ m) 0.125 ± ± ± ± ± ± with respect to the X-ray beam using the bisect method.A scanning step of 0.013 o with 2 seconds per step wasused. The O/U ratio was determined from the follow-ing empirical equation: a = 5.4705 − x , where‘ a ’ is the derived lattice parameter and ‘ x ’ is the stoi-chiometry deviation of UO x from stoichiometric UO .To estimate the O/U ratio, peaks in the region of 55 −
90 were used as input and the calculated stoichiometriesare statistically summarized. Micro-Raman spectra werecollected at room temperature using a Renishaw Micro-Raman spectrometer excited by a green argon laser (514nm). A typical spectrum was acquired with an expo-sure time of 10 seconds and 3 accumulations with a laserpower of 20 mW. An extended scanning region from 200to 1500 cm − was chosen since it contains the featuredpeaks for UO . For each pellet, multiple locations werechecked so that the spectrum is representative.The thermal conductivity and Seebeck coefficient mea-surements of UO samples were carried out in a Physi-cal Properties Measurement System DynaCool-9 PPMS(Quantum Design) using the thermal transport (TTO)option and Pulse power method. The measurements werecarried out using the continuous mode by slowly varyingthe temperature (0.2 K min − ). Typical dimensions ofsamples were ∼ × × . I. RESULTS AND DISCUSSION
Figs.1a and c show the dense microstructure with var-ious grain-sizes, as summarized in Table I. All pellets arefully densified with measured density higher than 95 %TD. The XRD spectrum in Fig. 1d shows that the sin-tered pellets are single phase UO . Detailed spectra atthe high angle area (Fig. 1e) shows well-separated K α and K α peaks for the (331) and (420) planes. The graylines added sit on the exact two-theta angles for 0.125 µ msamples. The peaks for 1.8 µ m shifted slightly to lowerangles, while the ones for 7.2 µ m have a larger degree ofpeak shifting, indicating slight changes in the lattice pa-rameter with stoichiometry. However, the superimposingfeatures of the Raman spectra (Fig. 1f) shows the chem-ical bonding in those three-different grain-sized samplesare very similar, indicating comparable localized defectinteraction with the crystal structure of UO .Figure 2 shows the thermal conductivity of UO poly-crystalline samples in the temperature range, 2-300 K.For comparison, we have also included the temperaturedependence of the thermal conductivity of UO single-crystal. The thermal conductivity of all samples shows similar temperature dependence as that of UO sam-ples in the previous reports. All κ ( T ) curves con-sist a broad maximum at T ∼
220 K and a minimumat the N´eel temperature, T N = 30.5 K. In addition,there is a well-defined peak at T ∼
10 K. Previous stud-ies have revealed only a small difference in the thermalconductivity between single-crystal and polycrystallineUO . Above magnetic ordering, the thermal conduc-tivity of uranium dioxide single crystal is limited by 3-phonon Umklapp scattering processes together with res-onant scattering.
These mechanisms are associatedwith a short mean free path, which may imply that κ [ W / m K ] Temperature [K] UO corrected - 7.2 µ m UO corrected - 1.8 µ m UO corrected - 0.125 µ m UO measured - 7.2 µ m UO measured - 1.8 µ m UO measured - 0.125 µ m UO measured - single crystal T N U O ( . µ m ) U O ( . µ m ) U O ( . µ m ) U O po l yc r ys t a l U O s i ng l e c r ys t a l S [ µ V / K ] Temperature [K]
FIG. 2. (Color online) Temperature dependence of the ther-mal conductivity of the UO polycrystalline samples havingdifferent grain-sizes. The dotted lines show the measuredthermal conductivity of UO polycrystals while the symbolsrepresent the corrected values by taking into account the den-sity difference (see the text). The solid orange line shows theUO single crystal results (data taken from the Ref. 28). In-set: Seebeck coefficient of UO samples measured at roomtemperature. The thermoelectric data for stoichiometric UO polycrystalline sample is taken from Ref. 31. D ( K - s - ) grain size ( µ m) (a) (b) (c) (d) κ m a x ( W / m k ) grain size ( µ m) B ( s - ) grain size ( µ m) κ [ W / m K ] Temperature [K] UO corrected - 7.2 µ m UO corrected - 1.8 µ m UO corrected - 0.125 µ m UO measured - single crystal FIG. 3. (Color online) (a) The low temperature thermal conductivity of UO samples. The solid lines represent the least-squarefits of the Callaway model to the experimental data (see the text); the grain-size dependence of κ max (b), parameters D (c), B (d). The corresponding values for the UO single-crystal are displayed as a dotted orange horizontal line in the relevantgraphs. The red dotted lines are guides to the eye. grain boundaries have a relatively small effect on thephonon conduction at this temperature range (see Refs.15, 28, and 30).The measured thermal conductivities are all signifi-cantly lower than the single crystal value, even for the7.2 µ m grain size sample for which grain boundary scat-tering should be very low, as can be seen in the Figure 2.These κ ( T ) values may be also compared to the one ob-tained for single crystal UO material. Several possi-ble scenarios besides grain boundary scattering may con-tribute to the reduction of the thermal conductivity inthe polycrystalline samples as compared to a single crys-tal. The main source of the reduction comes from the factthat the polycrystalline UO samples have slightly lowerbulk densities ( ∼ In order to rescale our measured thermalconductivity values to 100% density we have used thephenomenological expression proposed by Brandt andNeuer; κ = κ p / (1 − αp ), where α = 2 . − . t . Theparameter p stands for a porosity factor, t = T(K)/1000,and κ ,p is the thermal conductivity of fully dense ( p = 0)and porous UO , respectively. In Figure 2 we also showthe corrected thermal conductivity of our polycrystallinesamples. After the density correction, the values of ther-mal conductivity are similar or slightly smaller than thatof the single crystal. Another source that could impactthe thermal conductivity might be related to the fact thatthe polycrystalline UO samples were synthesized usingnatural uranium isotope, whereas the UO single crys-tal consists of depleted uranium. Natural uranium con-tains slightly more fissile uranium U-235, about 0.72%,then the depleted uranium, 0.2%. This small percentagechange of U-235 atoms, however, should have a negligibleeffect on the thermal conductivity value. Lastly, it hasbeen shown that the oxygen off-stoichiometry, i.e. UO ± x has quite large impact on the thermal conductivity of ura-nium dioxide. Theoretical and experimental studies haveshown that both hyper- and hypo-stoichiometry lowerthe thermal conductivity of UO , and its values de-creases as much as 30% for UO . , as compared to sto- ichiometric UO at room temperature. In general, in order to precisely determine the oxy-gen content in UO an x-ray diffraction is widely used. Here, we have also adapted thermopower measure-ments to probe the oxygen stoichiometry in this mate-rial. Depending upon the oxygen content, two typesof charge carriers can exist in UO . In hyper-stoichiometric samples (UO x ) the positive hole-likecarriers would lead to positive Seebeck effect, whereas,in hypo-stoichiometric UO − x the negative electron-likecarriers will result in negative Seebeck effect. The in-set to Figure 2 shows the Seebeck coefficient ( S ) of theUO samples measured at T = 300 K. The Seebeck coef-ficient of the UO single crystal is positive with the value S ∼ µV /K , which is also close to S value for the poly-crystalline UO . The S value, however, changes for thedifferent grain-size UO samples and even changes signfor the samples having grain-size 1.8 and 7.2 µ m. Thenegative sign of the Seebeck coefficient might suggest thepresence of hypo-stoichiometric UO in 1.8 and 7.2 µ msamples. These results are consistent with the measuredstoichiometries shown in Table I.The presence of lower densities in the polycrystallinesamples, as compared to single crystals, will reduce thethermal conductivity in these materials, especially athigh temperatures. In addition, the variation of See-beck coefficient and XRD measurements suggest thatvery small oxygen off-stoichiometry might be present andplay a role in lowering of the thermal conductivity in theUO samples. If so, this implies that separating differentscattering mechanisms (especially grain boundary scat-tering) and sources of thermal resistance in UO aboveroom temperature is a challenging task. Grain boundaryscattering dominates the thermal resistance of a mate-rial in the low-temperature regime when the grain-sizeis comparable or smaller than the mean free path ofphonons. The grain boundaries behave then as scat-tering centers for phonons, which ultimately reduce thethermal conductivity values and governs the thermal con-ductivity in the low-temperature regime.
Figure 3a
TABLE II. Comparison between measured thermal conductivity values with effective thermal conductivities calculated usingEq.5 with the new fit value for the Kaptiza resistance R fit and the value from molecular dynamics simulations from the literature R MD . Though R fit is significantly larger than R MD , the calculated thermal conductivities only differ for the smallest grainsize. d ( µ m) Thermal Conductivity (W/mK)measured corrected Eq.5 with R fit Eq.5 with R MD single crystal 7.6 - - -0.125 6.1 6.7 6.7 6.91.8 6.8 7.5 7.5 7.57.2 6.5 7.2 7.6 7.6 shows the blown-up region of the thermal conductivitycurves of the UO samples shown in Fig.2 in the rangebelow 30 K. As can been seen, in this temperature rangethe thermal conductivity decreases systematically withlowering the grain-size, as expected. The variation ofthermal conductivity value κ peak measured at the peakposition near T =10 K (see the arrows in Fig.3a) is shownin Fig. 3b. It is observed that κ peak increases system-atically with the grain-size as expected for grain bound-ary scattering and approaches towards the single crystalvalue (shown by the horizontal line) as the grain-size isincreased. This dependence of thermal conductivity onthe grain-size observed in UO is consistent with grainboundary scattering being a main source of the thermalresistance. In order to get more information of how differ-ent scattering processes affect the thermal conductiv-ity in uranium dioxide, we have used the Callawaymodel to analyze the experimental data obtained.This phenomenological model has been previously usedto successfully describe the lattice thermal conductiv-ity in different materials. This approach takesinto account scattering by different scattering mecha-nisms such as grain boundaries, point defects, or/andUmklapp phonon − phonon processes. At low tempera-tures (below ∼
30 K), the thermal conductivity of insu-lators is mainly dominated by the boundary and pointdefects.
Therefore, in order to analyze our low-temperature thermal conductivity data of UO , we havetaken into account only the grain-boundary ( B ) andpoint-defect ( D ) scattering contributions. Within theframework of this model, the thermal conductivity canbe expressed as, κ ( T ) = k B π v (cid:18) k B T (cid:126) (cid:19) (cid:90) Θ D /T τ ph x e x ( e x − dx, (2)where v and τ ph represent the mean velocity of soundand the phonon relaxation time, respectively. The pa-rameter x stands for (cid:126) ω/k B T . The parameters (cid:126) , Θ D ,and k B are the reduced Planck’s constant, the Debyetemperature, and Boltzmann constant, respectively. Themean sound velocity was determined using the formula v = k B Θ D / (cid:126) √ π n , where n is the number of atoms perunit volume. Taking Θ D = 395 K , v is estimated tobe 3,171 m s − for UO . The relaxation time is takenas the sum of inverse relaxation times of the scatteringprocesses, i.e. τ − ph = τ − D + τ − B . The particular inverserelaxation times are given by the following expressions: τ − D = Dx T = D (cid:18) (cid:126) ωk B (cid:19) (3)and τ − B = B, (4)where D and B are the fitting parameters. The B value is large for the lower grain-size sample and it de-creases while increasing the grain-size as expected forgrain boundary effect. The solid lines in Fig. 3a represent fits of the Call-away model to the low temperature data of UO . Thevariations of the obtained parameters with grain-size areshown in Figs. 3c and d. The results for UO singlecrystal (no grain boundary scattering) has been shownas a dashed orange horizontal line in the correspondinggraphs. The grain boundary effect, described by the pa-rameter B , is higher in the smaller grain-size sample, andit decreases towards the value for the UO single-crystalas the grain-size is increased. The parameter D , relatedto defect scattering, is comparable to each other (see Fig.2c) suggesting the presence of similar number of defectscattering centers in the measured samples.An alternative means of quantifying the impact ofgrain boundary scattering on the thermal conductivityis the grain boundary (GB) Kapitza resistance (see Refs.50–52 and references therein). The GB Kapitza resis-tance R can be calculated using the following equation, R = d/κ eff − d/κ sc , where κ sc is the single crystal ther-mal conductivity and κ eff is the effective thermal con-ductivity of a polycrystal of grain size d . By solving forthe effective thermal conductivity, we obtain the equa-tion: κ eff = d κ sc Rκ sc + d (5)Molecular dynamics simulations have been used to cal-culate the Kapitza resistance in various UO GB’s (seeTable 1 in Ref. 53). The largest Kapitza resistance foundwas R MD = 1.69 × − m K/W at a temperature of 300K. The data obtained in this work provides an excellentmeans of calculating the Kapitza resistance from experi-mental data by determining the value for R that results inan effective thermal conductivity using Eq.5 that is clos-est to the corrected values for the three different grainsizes. Using this approach, R fit = 2 . × − m K/Wat 300 K; the effective thermal conductivity values usingthis value with the grain sizes from the three samples areshown in Table II and have a maximum error (comparedto the corrected values) of 5.7%. If the uncertainty in thegrain size measurements reported in Table I are consid-ered, the standard deviation of R at 300 K is found to be0 . × − m K/W. The R fit value from the experimentsis 4.7 standard deviations larger than the value from themolecular dynamics simulations, indicating that the dif-ference can not be explained with just experimental error.It is not surprising that the experimental value is largerthan the molecular dynamics value, since the simulationsassume a perfectly stoichiometric grain boundary with noimpurities and will thus have less scattering. However,the molecular dynamics value was close enough to theexperiential value that it did not add significantly moreerror in the calculated effective thermal conductivities,as shown on Table II. CONCLUSION
To summarize, we have synthesized UO samples hav-ing different grain size (0.125, 1.8, and 7.2 µ m) and in-vestigated the grain-size effect on thermal properties inthis material. The samples have been characterized byx-ray powder diffraction (XRD), scanning electron mi-croscope (SEM), and Raman spectroscopy. By perform-ing low-temperature thermal conductivity measurementswe have studied the grain-boundary scattering related tograin-size and its impact on the thermal conductivity inthese materials. Although the operating temperaturesin nuclear reactors are high ( ∼ at 300 K, and the value was signifi-cantly larger than a value from the literature obtainedusing molecular dynamics simulations. The knowledgeof the details of the grain boundary scattering mecha-nisms in UO will be useful for researchers working onmodeling and simulations of this nuclear fuel. The ap-proach presented here would be also useful to study ther-mal transport in other applied materials, especially ther-moelectrics. ACKNOWLEDGEMENTS
The work was supported by the DOE’s NE NuclearEnergy University Programs (NEUP), US Department ofEnergy’s Early Career Research Program, and AdvancedFuel Campaign. K. L. Murty and I. Charit, An Introduction to Nuclear Materials:Fundamentals and Applications, Wiley- VCH, 2013. W. C. Patterson, Nuclear Power (second edition), PenguinBooks,1983. O. Runnalls, Uranium Dioxide - a Promising Nuclear Fuel(Atomic Energy of Canada Limited, 1958). R. J. M. Konings, T. Wiss, and O. Bene, Nat. Mater. 14, 47(2015). E. Jin, C. Liu, and H. He, Int. Conf. Nuc. Eng. 1 (2016). S. Yamasaki, T. Arima, K. Idemitsu, and Y. Inagaki, Int. J.Thermophys. 28, 661 (2007). S. Motoyama, Y. Ichikawa, and Y. Hiwatari, Phys. Rev. B 60,292 (1999). T. Watanabe, S. B. Sinnott, J. S. Tulenko, R. W. Grimes, P. K.Schelling, and S. R. Phillpot, J. Nuc. Mater. 375, 388 (2008). P. Ruello, G. Petot-Ervas, C. Petot, and L. Desgranges, J. Am.Ceram. Soc. 88, 604 (2005). Y. Q. An, A. J. Taylor, S. D. Conradson, S. A. Trugman, T.Durakiewicz, and G. Rodriguez, Phys. Rev. Lett. 106, 207402(2011). J. Schoenes, Phys. Rep. 63, 301 (1980). T. Meek, B. von Roedern, P. Clem, and R. H. Jr., Mater. Lett.59, 1085 (2005). W. M. Jones, J. Gordon, and E. A. Long, J. Chem. Phys. 20, 695(1952). M. J. M. Leask, L. E. J. Roberts, A. J. Walter, and W. P. Wolf,J. Chem. Soc. 0, 4788 (1963). T. M. Tritt, Thermal Conductivity Theory, Properties, and Ap-plications, Vol. (Springer,2012). C. Kittel, Introduction to Solid State Physics, Vol. (Pushp PrintService, Delhi, 2006). D. Cahill, W. Ford, K. Goodson, G. Mahan, A. Majumdar, H.Maris, R. Merlin, and S. Phillpot, Appl. Phys. Rev. 93, 793(2003). S. Yoon, O.-J. Kwon, S. Ahn, J.-Y. Kim, H. Koo, S.-H. Bae,J.-Y. Cho, J.-S. Kim, and C. Park, J. Elec. Mat. 42, 3390 (2013). J. Yang, G. P. Meisner, D. T. Morelli, and C. Uher, Phys. Rev.B 63, 014410 (2000). D. Spiteri, J. Anaya, and M. Kuball, J. Apl. Phys. 119, 085102(2016). J. C. Thompsonand B. A. Younglove, J. Phys. Chem. Solids 20,146 (1961). X. Wang, Y. Yang, and L. Zhu, J. Appl. Phys. 110, 024312 (2011). T. Godfrey, W. Fulkerson, T. Kollie, J. Moore, and D. McElroy,J. Am. Ceram. Soc. 48, 298 (1965). T. Uchida, T. Sunaoshi, M. Kato, and K. Konashi, Prog. Nuc.Sci. and Tech. 2, 598 (2011). T. Yao, S. M. Scott, G. Xin, and J. Lian, J. Nucl. Mater. 469,251 (2016). T. Yao, S. M. Scott, G. Xin, B. Gong, and J. Lian, J. Am. Cera.Soc. (in press). K. Teske, H. Ullmann, and D. Rettig, J. Nucl. Mater. 116, 260(1983). K. Gofryk, S. Du, C. Stanek, J. Lashley, X. Liu, R. Schulze,J. Smith, D. Safarik, D. Byler, K. McClellan, B. Uberuaga, B.Scott, and D. Andersson, Nat. Commun. 5, 4551 (2014). V. Haase, H. Keller-Rudek, L. Manes, B. Schulz, G. Schu-macher, D. Vollath, and H. Zimmermann, Gmelin Handbook ofInorganic Chemistry, Vol. (Springer Verlag, 1986). J. P. Moore and D. L. McElroy, J. Am. Ceram. Soc. 54, 40 (1971). R. D. Coninck and J. Devreese, Phys. Stat. Sol. 32, 823 (1969). Thermal Conductivity of Uranium Dioxide, Technical ReportsSeries No. 59, IAEA, Vienna, 1966. J. K. Fink, J. Nucl. Mater. 279, 1 (2000). R. Brandt,G. Neuer, J. Non-Equilib. Thermodyn. 1, 3 (1976). G. A. Slack, Phys. Rev. 105, 829 (1957). X.-Y. Liu, M. D. Cooper, K. McClellan, J. Lashley, D. Byler,B. Bell, R. Grimes, C. R. Stanek, and D. A. Andersson, Sum-mary report on UO thermal conductivity model refinement andassessment studies., Vol. (Report LA-UR-16-28212, Los AlamosNational Laboratory, Los Alamos, NM, 2017). S. Nichenko and D. Staicu, J. Nucl. Mater. 433, 297 (2013). Thermodynamic and Transport Properties of Uranium Dioxideand Related Phases, Technical Reports Series No. 39, IAEA, Vi-enna, 1965. P. G. Lucuta, Hj. Matzke, and R. A.Verrall, J. Nucl. Mater. 223,51 (1995). P. G. Lucuta, Hj. Matzke, and I. J. Hastings, J. Nucl. Mater.232, 166 (1996). K. Gofryk, private communications J. Rubin, K. Chidester, and M. Thompson, O/M Ratio Mea-surement in Pure and Mixed Oxide Fuels - Where are We Now?,Vol. (Report LA-UR-00-5805, Los Alamos National Laboratory,NM). S. Iida, Jap. J. Appl. Phys. 4, 833 (1965). J. Callaway, Phys. Rev. 113, 1046 (1959). B. K. Agrawal and G. Verma, Phys. Rev. 126, 24 (1962). A. M. Toxen, Phys. Rev. 122, 450 (1961). K. Gofryk, D. Kaczorowski, T. Plackowski, J. Mucha, A. Leithe-Jasper, W. Schnelle, and Y. Grin, Phys. Rev. B 75, 224426(2007). J. Yang, D. T. Morelli, G. P. Meisner, W. Chen, J. Dyck, and C.Uher, Phys. Rev. B 65, 094115 (2002). G. Dolling, R. Cowley, and A. Woods, J. Phys. 43, 1397 (1965). G. L. Pollack, Rev. Mod. Phys. 41, 48 (1969) D. H. Hurley, M. Khafizov, and S. L. Shinde, J. Appl. Phys. 109,083504 (2011) M. Khafizov I. W Park, A. Chernatynskiy, L. He, J. Lin, J. J.Moore, D. Swank, T. Lillo, S. R. Phillpot, A. El-Azab, D. H.Hurley, J. Am. Ceram. Soc. 97, 562 (2014)53