Anisotropic Thermal Conductivity in Uranium Dioxide
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ARTICLE Received 9 May 2014 | Accepted 27 Jun 2014 | Published 1 Aug 2014 DOI: 10.1038/ncomms5551 Anisotropic thermal conductivity in uranium dioxide K. Gofryk1,w,S.Du2,w, C.R. Stanek3, J.C. Lashley1, X.-Y. Liu3, R.K. Schulze1, J.L. Smith1, D.J. Safarik1, D.D. Byler3, K.J. McClellan3, B.P. Uberuaga3, B.L. Scott4 & D.A. Andersson3 The thermal conductivity of uranium dioxide has been studied for over half a century, as uranium dioxide is the fuel used in a majority of operating nuclear reactors and thermal conductivity controls the conversion of heat produced by fission events to electricity. Because uranium dioxide is a cubic compound and thermal conductivity is a second-rank tensor, it has always been assumed to be isotropic. We report thermal conductivity measurements on oriented uranium dioxide single crystals that show anisotropy from 4 K to above 300 K. Our results indicate that phonon-spin scattering is important for understanding the general thermal conductivity behaviour, and also explains the anisotropy by coupling to the applied temperature gradient and breaking cubic symmetry. 1 Materials Technology-Metallurgy, Materials Science and Technology Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA. 2 Physics and Chemistry of Materials, Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA. 3 Materials Science in Radiation & Dynamical Extremes, Materials Science and Technology Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA. 4 Materials Physics and Applications Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA. w Present address: Fuel Modeling and Simulation Department, Idaho National Laboratory, P.O. Box 1625, Idaho Falls, Idaho 83415, USA (K.G.); Present address: Division of Functional Materials and Nanodevices, Ningbo Institute of Materials Technology and Engineering, Chinese Academy of Sciences, Ningbo, Zhejiang 315201, China (S.D.). Correspondence and requests for materials should be addressed to C.R.S. (email: [email protected]) or to D.A.A. (email: [email protected]). NATURE COMMUNICATIONS | 5:4551 | DOI: 10.1038/ncomms5551 | www.nature.com/naturecommunications 1 & 2014 Macmillan Publishers Limited. All rights reserved. ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms5551 oday nearly 20% of the world’s electricity is generated by originated from molecular dynamics (MD) simulations using 6–8 nuclear energy from UO2 fuel. The thermal conductivity of the direct method , which calculates thermal conductivity by TUO2 governs the conversion of heat produced from fission measuring the temperature gradient in a rectangular cuboid events into electricity and it is an important parameter in reactor with heat added and removed at a hot and cold plate, respectively. design and safety. Scatter among thermal conductivity (k) data The simulations predict large differences for the effective from previous experimental studies on UO2 single crystals cannot thermal conductivity in the o1004 o1104 and o1114 easily be explained by sample impurities or inherent measure- crystallographic directions at 300 K, see Fig. 1a. In an attempt to ment errors. (see ref. 1 and references therein). We hypothesize eliminate boundary condition artifacts related to scattering by the that much of this scatter is due to hidden anisotropy, overlooked hot and cold plates, the thermal conductivity is obtained for in previous experiments that did not specify single crystal infinite simulationÀÁ cell length by extrapolation7,8. Figure 1a shows À 1 À 1 orientation. The crystal symmetry of materials dictates the that k ¼ fLz (Lz is the length of the simulation cell) does symmetry of the response to applied potential gradients. not obey the expected linear relationship for UO2 in the o1004 Thermal conductivity is a second-rank tensor (that is, k ¼ kij) direction, and is instead quadratic. The extrapolated values at that governs the flow of energy under an applied temperature 300 K in the inset of Fig. 1a (based on the quadratic fits) show gradient. For materials with cubic point and space group symmetry that the anisotropy between o1114 and o1004 is B15%. This it is isotropic. Thermal conductivity is almost exclusively result is noteworthy as the quadratic term indicates that the dominated by phonons in insulators and semiconductors. At intrinsic material properties are affected, whereas a linear term very low temperatures, where ballistic phonon transport may would correspond to a mere change in the phonon scattering occur, exceptions to the isotropy of cubic materials have been length. The underlying physics is illustrated by the projected MD reported, for example, ref. 2. This can be explained by the thermal trajectory of an oxygen ion, see Fig. 1b. This simulation was conductivity being a fourth rank tensor (k ¼ kijlm) related to the performed in the same rectangular cuboid cell as the thermal elastic stiffness tensor when phonon scattering is minimal3. conductivity simulations, but without a temperature gradient. Here we show by experiments on single crystals that Although the average position coincides with the cubic UO2 UO2, which crystallizes in the cubic fluorite structure ðÞFm3m , fluorite structure, the shape of the distribution is an ellipsoid exhibits anisotropic thermal conductivity at temperatures at rather than the expected sphere. The ellipsoidal character is still which ballistic mechanisms are not possible. The anisotropy is present during short time intervals in a cubic simulation cell, rather due to the applied temperature gradient interacting with however it averages to a sphere since the ellipsoid orientation the electric moments on uranium ions thus affecting the phonon- along the crystallographic axes is uncorrelated in space and time. spin interactions to slightly break cubic symmetry. This effect is Apparently, the trajectory is sensitive to small perturbations, such caused by the magnetic properties of UO2 and the degenerate as the dimensionality of the simulation cell or the applied bands giving rise to Jahn–Teller distortions. This result is temperature gradient. These perturbations break the underlying unexpected since the crystal structure for both the paramagnetic cubic symmetry, and lead to a preferential ellipsoid orientation and the antiferromagnetic phases of UO2 are cubic and there is along one of the three o1004 axes. This orientation bias gives no clear evidence of non-cubic distortions in diffraction rise to anisotropy in the physical properties. experiments4. These results should also apply to the inter- pretation of other magnetic strongly correlated materials 5 exhibiting Jahn–Teller physics . Measurements of anisotropic UO2 thermal conductivity. These MD results inspired a detailed experimental study of the UO2 Results thermal conductivity as a function of temperature with a thermal Molecular dynamics simulations of UO2 thermal conductivity. gradient applied in different crystallographic directions on Our consideration of anisotropic UO2 thermal conductivity oriented single crystal bars. We find that the overall shape of the ab61.8 –1 –1 ∞=20.6 Wm K 0.10 1.00 〈111〉 Exp., Calc. 0.95 〈110〉 Exp., Calc. 61.6 300 K 〈100〉 Exp., Calc. –1 ) –1 0.08 0.90 61.4 0.85 (mKW 300 [001] (Å) –1 0.06 T (K) –1 –1 ∞=24 Wm K 61.2 〈 〉 0.04 100 〈111〉 61.0 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 1.0 1.2 1.4 1.6 1.8 –1 –1 [100] (Å) Lz (nm ) ÀÁ À 1 À 1 Figure 1 | Anisotropic UO2 thermal conductivity from MD simulations. (a) The thermal conductivity (k) variation (k ) as a function of the length Lz of the simulation cell for the temperature gradient applied in the o1004 and o1114 lattice directions at 300 K. The lines represent linear and quadratic 7,8 fits to the data used for extrapolation to infinite length (kN) of the simulation cells . The inset compares the experimentally measured thermal conductivity values (Exp.) in the o1004 o1104 and o1114 crystallographic directions with the MD simulations (Calc.). The length of the MD simulation cells were B30 nm for these data points. The values have been normalized to the highest thermal conductivity value at 300 K (k300). (b) Projected trajectory of one oxygen ion during a 500 ps MD run in the rectangular cuboid simulation cell at 300 K. The MD simulation was performed in the NVE ensemble without an applied thermal gradient. The horizontal axis is the [100] and the vertical axis the [001] direction of the fluorite lattice. The elongated direction of the simulation cell is along the vertical axis. The ellipsoid character of the distribution is emphasized by the dashed red line as compared with the white line representing the spherical distribution. 2 NATURE COMMUNICATIONS | 5:4551 | DOI: 10.1038/ncomms5551 | www.nature.com/naturecommunications & 2014 Macmillan Publishers Limited. All rights reserved. NATURE COMMUNICATIONS | DOI: 10.1038/ncomms5551 ARTICLE thermal conductivity is atypical for insulating crystals without same energy range as the occupied phonons, otherwise scattering defects, see Fig. 2. For all directions, pronounced double-peak is negligible10,16–18. There are no excitation levels in the triplet behaviour in the thermal conductivity is observed with maxima ground state of uranium ions above TN that can interact with occurring at B10 and 220 K and a minimum at the Ne´el tem- phonons19,20. However, inelastic neutron scattering experiments 1,9 perature TN ¼ 30.8 K, as reported previously . This behaviour have identified a dynamic Jahn–Teller distortion in o1004 has been observed in other materials with antiferromagnetic lattice directions that splits the ground state into three singlets transitions9–13, which is due to resonant scattering of phonons with dispersive peaks of magnetic character between 3 and through excitations of the spins on the magnetic ions in the 10 meV19,20. These excitations are in the appropriate energy paramagnetic phase and absence of this scattering mechanism range for interaction with phonons and give rise to significant 9–13 below TN due to spin ordering .