Combined ab initio and empirical model of the thermal conductivity of uranium, uranium-zirconium, and uranium-molybdenum

In this work we developed a practical and general modeling approach for thermal conductivity of metals and metal alloys that integrates ab initio and semiempirical physics-based models to maximize the strengths of both techniques. The approach supports creation of highly accurate, mechanistic, and extensible thermal conductivity modeling of alloys. The model was demonstrated on α-U and U-rich U-Zr and U-Mo alloys, which are potential fuels for advanced nuclear reactors. The safe use of U-based fuels requires quantitative understanding of thermal transport characteristics of the fuel. The model incorporated both phonon and electron contributions, displayed good agreement with experimental data over a wide temperature range, and provided insight into the different physical factors that govern the thermal conductivity under different temperatures. This model is general enough to incorporate more complex effects like additional alloying species, defects, transmutation products, and noble gas bubbles to predict the behavior of complex metallic alloys like U-alloy fuel systems under burnup.

Figure 1

The calculated phonon dispersion curves along [100], [010], and [001] directions for α-U. The dots represent Crummett et al.'s experimental data [49] obtained from inelastic neutron scattering.

Figure 2

Plot of the phonon contribution to the α-U thermal conductivity in the temperature range of 43 to 933 K. The different curves correspond to the phonon contribution to the thermal conductivity along different crystallographic directions and the black curve indicates the average over the different directions. Only the phonon-phonon scattering is included in these values.

Figure 3

Plot of the phonon scattering contributions to the α-U thermal conductivity in the temperature range of 43 to 933 K. The different curves correspond to the different phonon scattering contributions to the thermal conductivity and the black curve indicates the phonon thermal conductivity obtained from Eq. (2).

Figure 4

Plot of $\left(\frac{\sigma }{\tau }\right)$ (electrical conductivity divided by relaxation time) of electrons in α-U over the temperature range of 43 to 933 K. The curves with different colors indicate the value of $\left(\frac{\sigma }{\tau }\right)$ along different crystallographic directions and the black curve indicates the average over the different directions.

Figure 5

Plot of fitted resistivity model [using Eq. (17), solid curves] of α-U resistivity from 43 to 933 K, compared with single crystal α-U experimental resistivity data (symbols). The curves with different colors represent the fitted resistivity along different crystallographic directions, and the shaded areas represent the error ranges. The experimental data for the [010] and [001] directions were obtained from Pascal et al. [36] and Raetsky [35], and the [100] data were from Raetsky [35].

Figure 6

Calculation results of α-U resistivity from 43 to 933 K (curves), compared with experimental data (symbols). The filled circles represent experimental data of single crystal α-U from Refs. [35, 36], and the open circles represent experimental data of polycrystalline α-U from Refs. [18, 50, 51, 52]. The solid black curve $\left({\rho }_{\mathrm{total}}\right)$ is the calculated resistivity of α-U, and the dashed orange, purple, and blue curves, labeled ${\rho }_{\mathrm{e}\text{–}\mathrm{ph}}$, ${\rho }_{\mathrm{e}\text{–}\mathrm{e}}$, and ${\rho }_0$, respectively, are the electron-phonon, electron-electron scattering contributions, and residual resistivity due to electron scattering with point defects.

Figure 7

Plot of our thermal conductivity model from 43 to 933 K (solid curves), compared with polycrystalline α-U experimental thermal conductivity data (symbols). The curves with different colors represent the calculated thermal conductivity along different crystallographic directions. The experimental thermal conductivity data were obtained from Refs. [10, 11, 12, 13, 50, 53, 54].

Figure 8

Calculation results of α-U thermal conductivity from 43 to 933 K (curves), compared with polycrystalline α-U experimental thermal conductivity data (symbols). The solid black curve $\left({\kappa }_{\mathrm{total}}\right)$ is the calculated thermal conductivity of α-U, and the dashed orange and purple curves, labeled ${\kappa }_{\mathrm{ph}}$ and ${\kappa }_{\mathrm{e}}$, respectively, are the phonon contribution and electronic contribution to the thermal conductivity, respectively. The experimental thermal conductivity data were obtained from Refs. [10, 11, 12, 13, 50, 53, 54].

Figure 9

Calculation results of U-rich U-Zr alloys thermal conductivity from 300 to 933 K (solid curves), compared with U-Zr experimental thermal conductivity data (symbols). The dashed lines are values based on empirical models developed by Kim et al. [3]. The curves and symbols with different colors represent the U-Zr thermal conductivities in different Zr at %. The experimental thermal conductivity data were obtained from Refs. [10, 11, 13, 56, 58].

Figure 10

Calculation results of U-rich U-Mo alloys thermal conductivity from 300 to 933 K (solid curves), compared with U-Mo experimental thermal conductivity data (symbols). The dashed lines are values based on empirical models developed by Kim et al. [4]. The curves and symbols with different colors represent the U-Mo thermal conductivities in different Mo at %. The experimental thermal conductivity data were obtained from Refs. [10, 11, 13, 57, 59, 60, 61, 62, 63].

Figure 11

Calculation results of U-rich U-Zr alloys thermal conductivity from 300 to 933 K (solid curves), compared with U-Zr experimental thermal conductivity data (symbols). The black dot is the data point to which the model was fitted.

Figure 12

Calculation results of α-U thermal conductivity from 43 to 933 K (curves), compared with polycrystalline α-U experimental thermal conductivity data (symbols). The solid curve is obtained by fitting to the single crystal resistivity data from 43 to 300 K, while the dashed curve is fitted to all available single crystal resistivity data from 43 to 933 K.