Lattice thermal conductivity of UO2 using ab-initio and classical moleculardynamics

Hyoungchul Kim,1,2 Moo Hwan Kim,3 and Massoud Kaviany1,3,a)

1Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA2High-Temperature Energy Materials Research Center, Korea Institute of Science and Technology,Seoul 136–791, Republic of Korea3Division of Advanced Nuclear Engineering, Pohang University of Science and Technology, Pohang 790-784,South Korea

(Received 2 December 2013; accepted 16 March 2014; published online 26 March 2014)

We applied the non-equilibrium ab-initio molecular dynamics and predict the lattice thermal

conductivity of the pristine uranium dioxide for up to 2000 K. We also use the equilibrium classical

molecular dynamics and heat-current autocorrelation decay theory to decompose the lattice

thermal conductivity into acoustic and optical components. The predicted optical phonon transport

is temperature independent and small, while the acoustic component follows the Slack relation and

is in good agreement with the limited single-crystal experimental results. Considering the phonon

grain-boundary and pore scatterings, the effective lattice thermal conductivity is reduced, and we

show it is in general agreement with the sintered-powder experimental results. The charge and

photon thermal conductivities are also addressed, and we find small roles for electron, surface

polaron, and photon in the defect-free structures and for temperatures below 1500 K. VC 2014 AIPPublishing LLC. [http://dx.doi.org/10.1063/1.4869669]

I. INTRODUCTION

Uranium dioxide (UO2) is a stable fluorite structure

(CaF2 type, space group Fm3m) containing 4 U4þ and 8 O2�

ions in a conventional cell (Fig. 1). This crystalline solid is

one of the most common nuclear-fission fuel materials, with

high melting temperature (Tm� 3120 K), radiation stability,

and chemical compatibility. However, it has low thermal

conductivity (j), which directly influences its thermal stabil-

ity (e.g., local heating and swelling of the fuel pellet) and the

operating temperature.1,2 So, accurate thermal conductivity

data and its possible improvements are important.

To explain the heat conduction mechanisms in UO2, vari-

ous theoretical,3–5 computational,6–9 and experimental9–23

studies have been performed over the last five decades. These

include the effects of the crystalline structure,9,10,20–23

non-stoichiometry,5,16 porosity,12,18 and irradiation.11,19 The

results show that large grain and stoichiometry are favorable

for the heat conduction, while defects (e.g., pores remaining

after powder sintering, and radiation-caused voids, impurities,

and fission products) hinder the transport by the heat carriers.

Also, it has been suggested that charge carriers play a role

only at the high temperatures, while phonons are the domi-

nant carriers at low and moderate temperatures.1,4,5,10,20,22,24

This phonon behavior (and lattice dynamics) of UO2 is clearly

explained by the recent atomic-level simulations based on the

density-functional theory (DFT), including the prediction of

significant anharmonicity25 and mechanical/thermodynamic/

vibrational properties.26,27 In the high-temperature regime,

many report that the thermal conductivity of UO2 is domi-

nated by the charge-carrier transport.1,4,5,10,20,22,24 Despite the

semiconductor features of the UO2 crystal (bandgap energy

DEe;g � 2.0 eV),3,28–30 the classical band theory is inadequate

in describing its electrical properties.1,3,4,28,31–35 To

explain the high-temperature behavior of its thermal conduc-

tivity, the small-polaron hopping mechanism has been

suggested.1,4,28,31,32,35–37 However, so far due to the absence

of the first-principles based results, the fundamental under-

standing of the UO2 thermal conductivity mechanisms

(including the conditions for charged heat carriers) has been

lacking.

Here, we present a comprehensive analysis of the lattice

thermal conductivity, jL, of UO2 crystals employing the mo-

lecular dynamics (MD) (including ab-initio and classical

simulations) and analytic methods. We predict the

temperature-dependent lattice thermal conductivity of UO2

and compare our results with the available experimental

results for single crystals. We also use the heat-current auto-

correlation function (HCACF) decay theory to predict the

optical-phonon transport contribution. The effects of the po-

rosity and grain boundary scatterings are modeled using the

effective lattice thermal conductivity, hjLi. We combine

these to predict the total lattice thermal conductivity and

compare with the available experimental results of the

sintered-powder UO2 samples. Simple analyses on the elec-

tronic and radiative contributions are also addressed in

Appendices A to C.

II. MD SIMULATIONS

A. Non-equilibrium ab-initio MD

We calculate the electronic structures and the lattice dy-

namics of UO2 using the DFT method implemented in the

Vienna ab-initio Simulation Package (VASP).38 The

Perdew-Burke-Ernzerhof parameterization of the generalized

gradient approximation (GGA) for the exchange-correlateda)Electronic mail: kaviany@umich.edu

0021-8979/2014/115(12)/123510/8/$30.00 VC 2014 AIP Publishing LLC115, 123510-1

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functional,39 and the projector augmented wave method for

modeling the core electrons (energy cutoff ¼ 400 eV),40,41

are used. The standard DFT calculations fail to simulate the

strong correlations among the U 5 f electrons, as reported in

several studies.27,29,42–46 Here, we use one of the alterna-

tives, i.e., the GGAþ U method, to predict the electronic and

phonon behaviors of UO2 more accurately. We apply the

same value for the effective Hubbard parameter (U¼ 4.5 eV

and J¼ 0.51 eV) as that used in the literature.42–46 We have

reproduced the previous results (e.g., band structure and

electronic density-of-states)42–46 and verified the suitability

of the GGAþ U method.

To predict the lattice thermal conductivity, we use the

non-equilibrium ab-initio MD (NEAIMD) with the modified

VASP code based on the energy exchange method47,48 as

reported in the literature.49,50 The lattice thermal conductiv-

ity is obtained from this NEAIMD formalism as the ratio of

the imposed heat flux to the resulting temperature gradi-

ent,49,50 i.e.,

jL ¼ �½QðtÞ=A�ðdT=dxÞ�1; (1)

where the overbar designates the time average, Q(t) is the

heat flow rate, t is time, T is temperature, and A is the cross-

sectional area of the simulation cell. The heat flux is imposed

by dividing the simulation cell into twelve sections of equal

width, and exchanging kinetic energy (velocity swapping)

between the hot and cold sections. Because the exchange of

the kinetic energy results in a non-Newtonian dynamics in

the hot and cold sections, only the linear portion of the tem-

perature gradient is considered in calculating the temperature

gradient.

Three different UO2 supercells, 4� 2� 2 (total 192

atoms), 6� 2� 2 (288 atoms), and 8� 2� 2 (384 atoms),

are used to account for the effects of finite system size on the

NEAIMD simulations. To consider the thermal expansion

with temperature changes, we prepare supercells using the

experimental thermal expansion coefficient and lattice pa-

rameter, i.e., a(T)¼ a8(9.9734� 10�1þ 9.802� 10�6T �2.705� 10�10 T2þ 4.391� 10�13T3), with a�¼ 5.455 A at

T¼ 300 K.51 The Fermi-Dirac smearing factor (kB T, where

kB is the Boltzmann constant) for each temperature is also

applied to ensure reliable thermal-disordered atomic coordi-

nates. The Brillouin zone is sampled at only the C point.

After constant-temperature simulations with the Nos�e

thermostat for 1 ps (0.5 fs time steps), we collect atomic tra-

jectories for 22 ps (1 fs time steps).

B. Equilibrium classical MD

In order to analyse/scrutinize the NEAIMD results and

to clarify the validity of the available empirical potential

models, the lattice thermal conductivity is also calculated

using the equilibrium classical MD (ECMD) techniques and

the Green-Kubo HCACF decay,24,52,53 i.e.,

jL ¼V

kBT2

ð10

hqðtÞ � qð0Þi�

3dt; (2)

where V is the system volume and hqðtÞ � qð0Þi� is the

HCACF (superscript � indicates equilibrium). The heat cur-

rent vector ðqÞ is

q ¼ 1

V

d

dt

Xi

Eiri ¼1

V

Xi

Eiui þ1

2

Xi;j

ðFij � uiÞrij

" #; (3)

where Ei, ri, and ui are the energy, position, and velocity

vectors of particle (atom) i, and rij, and Fij are the intera-

tomic separation and force vectors (between atoms i and j).After examining the computational size effect on the ECMD

results, averages are obtained over the three directions for a

system consisting of 6� 6� 6 conventional unit cells (2592

atoms). The Verlet leapfrog algorithm with the Nos�e-Hoover

thermostat and the Berendsen barostat are used in the NpTensemble for 200 ps and then in the NVE ensemble for 100

ps. Then, the 3000 ps raw data are obtained for the calcula-

tion of q and the resultant HCACFs are then directly inte-

grated and the lattice thermal conductivity is determined as

the average value in the stable regime of the integral.

III. RESULTS AND DISCUSSION

In this section, using ab-initio and classical MD (CMD)

simulations of Sec. II, the UO2 lattice thermal conductivity

(including the effective value with the grain boundary and

porosity) is examined. To verify and assess our findings on

the phonon contribution, as dominant below T¼ 1500 K, the

electronic and radiative contributions are also investigated in

Appendices A to C.

A. Lattice thermal conductivity

As described in Sec. II A, the NEAIMD predicted lattice

thermal conductivity is computed as the ratio of the applied

heat flux to the resulting temperature gradient. The tempera-

ture distribution along the direction of the imposed tempera-

ture is computed using the average temperature at locations

marked by dividing the supercell in that direction into twelve

segments [Fig. 2(a)]. Using three different simulation-cell

sizes, as shown in Fig. 2(b), we verify the expected size

effect and extrapolate the lattice thermal conductivity for the

infinite structure, through the linear extrapolation of their

reciprocal relation. Due to the extensive computational

requirement of these actinide oxides NEAIMD simulations,

FIG. 1. Crystal structure of the pristine UO2 using (a) the primitive and (b)

the conventional cell (blue is U and red is O atom). The lattice parameter a�

at 300 K is also indicated.

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we had to limit the cell volumes (192, 288, 384 atoms). The

uncertainties in the data used in extrapolation of the lattice

thermal conductivity from the limited data sets are shown

with error bars in Fig. 2(b).

Figure 3(a) shows the temperature dependence of this

predicted lattice thermal conductivity for the bulk, pristine

UO2, and comparison with the available experimental

results13 and the analytical Slack relation (�T�1),24,54,55 i.e.,

jL;SðTÞ ¼3:1� 104hMiV1=3

�T3

D;1Thc2

GiN2=3; (4)

where hMi is the average atomic weight, N is the number of

atoms, V� is the average volume per atom, TD;1 is the

high-temperature Debye temperature, and hcGi is the

temperature-average Gr€uneisen parameter. The predicted

results are in good agreement with the Slack relation and yet

higher than the single-crystal experimental results. This is

expected due to the underestimation of the long-range pho-

non interactions within the limited size of the NEAIMD

supercell. Applying fitting to the power-allometric equation

ðjL ¼ aTbÞ, we find that these NEAIMD results show nearly

the T�1 temperature dependence (a¼ 9315.7 and b¼�1.07)

of the Slack relation.

We also compare the NEAIMD predictions with the

CMD (equilibrium and non-equilibrium methods) results (for

various empirical interatomic potential models)6–9,23,58–60,64 in

Figs. 3(b) and 3(c). We note that the two MD methods for pre-

dicting the lattice thermal conductivity of UO2 have different

advantages and drawbacks. Generally, the equilibrium method

offers size-independent and reliable results with long simula-

tion time (at least 1 ns) using �10,000 atoms, whereas the

non-equilibrium method depends strongly on the sample size

(length and number of atom in each section) and the imposed

heat flux. To reduce artifacts from the equilibrium and

non-equilibrium methods, results from both CMD methods are

shown in Figs. 3(b) and 3(c). Generally, non-equilibrium clas-

sical MD (NECMD) results are more scattered due to the sen-

sitivity to the simulation cell size.

To assess the phonon transport in UO2 using the avail-

able empirical potential models, the phonon dispersions

FIG. 2. (a) Variations of temperature along the imposed-temperatures direction (divided into twelve segments) using the normalized cell distance (x/l), where lis the cell length. (b) The inverse of the lattice thermal conductivity with respect to the inverse of computational supercell size l, found from the NEAIMD sim-

ulations, for three different temperatures.

FIG. 3. Variations of the lattice thermal conductivity of UO2 as a function

of temperature. (a) The NEAIMD predicted results, the available experimen-

tal results,28 the minimum conductivity,56,57 and the Slack relation. Similar

to (a) for the NEAIMD results and the (b) ECMD and (c) NECMD results

with various interatomic potential models.6–9,23,58–60,64 Using the Debye

temperature range, 377 K (Ref. 61) to 383 K (Ref. 62), and the Gr€uneisen pa-

rameter range, 1.55 (Ref. 26) to 2.0 (Refs. 21 and 63), the results of the

Slack relation are shown with gray-shaded band.

123510-3 Kim, Kim, and Kaviany J. Appl. Phys. 115, 123510 (2014)

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predicted by these models are shown in Fig. 4. The DFT

method directly calculates the force field using VASP38 and

PHONON66 codes. The phonon dispersion curves are pre-

dicted using fits to the interatomic force-constant tensors of

the Hellmann-Feynman (HF) forces. The total energy and

the HF forces are found starting from the fully relaxed con-

figuration, with the criterion that the initial ionic forces are

less than 10�5 eV/A. The ionic displacement of 0.03 A for

each atom is sampled along the three directions. In the em-

pirical potential model, the GULP (general utility lattice pro-

gram)67 code is used for the prediction of the phonon

properties. Compared with the experiments65 and the DFT

results, each empirical potential model has distinct features

as follows. (a) The predicted acoustic modes are similar (or

slightly underestimated) to the experiments and the DFT

results, whereas the optical phonon frequencies are not (or

severely overestimated). (b) The Arima model predicts large

phonon bandgap, not consistent with the experiments and

DFT results. (c) The DFT phonon dispersion shows the best

agreement with the experiments. Considering the lattice ther-

mal conductivity of CMD predictions, the Yamada intera-

tomic potentials results are in closest agreement with the

NEAIMD results and the Slack relation. This supports the

use of this empirical potential model for the investigation of

the lattice thermal conductivity using the ECMD, which also

allows for decomposition and extraction of the

optical-phonon component.

Using the Yamada interatomic potential model, we

decompose the ECMD results for the lattice thermal conduc-

tivity of UO2 into the acoustic and optical compo-

nents,24,52,54 jL ¼ jL;A þ jL;O, where A and O denote

acoustic and optical. First, the optical component of HCACF

is filtered by the fast Fourier transform, low-pass filter with a

cutoff frequency of 6 THz (obtained from the upper value of

the acoustic-phonon dispersion). Then, the acoustic compo-

nent is obtained by subtracting the optical component from

the total lattice thermal conductivity. The decomposed

results are shown in Fig. 5, and we note again that the acous-

tic contribution has the expected strong T�1 temperature de-

pendence (mainly representing the long-range acoustic

transport) and dominates the lattice thermal conductivity,

while as expected the optical component is relatively small

and temperature independent. This is consistent with the

phonon transport in the bulk, crystalline solids.24,54

Recently,68 the optical phonon transport in UO2 was

measured and found not to be negligible and also tempera-

ture dependent. The difference can be associated with a

more pronounced anharmonicity than has been included in

our analyses. However, our results are in line with the gen-

eral agreement regarding the insignificance of the

optical-phonon contribution.25

B. Effective thermal conductivity

To consider the effects of the grain boundary and the po-

rosity in the heat conduction of (usually sintered powder) UO2

bulk, the relaxation time approximation24,69 and the effective

medium theory70–72 are used. Starting with the Slack relation

for the pristine UO2 dominated by the phonon-phonon scatte-

ring, we consider the additional grain-boundary scattering for

the polycrystalline UO2. The dominant phonon-phonon relax-

ation time ðsp�pÞ is found from

sp�p ¼3jL;S

qcvu2p;g;A

; (5)

where q is density and up;g;A is the average sound speed.

This gives sp�pðT ¼ 300 KÞ ¼ 1:69 ps using the pristine UO2

properties [i.e., cv(T¼ 300 K)¼ 227 J/kg-K, q(T¼ 300 K) ¼11.0 g/cm3, up,g,T¼ 5750 m/s, and up,g,L¼ 3275 m/s]. Using

the Matthiessen rule, the overall relaxation time with inclu-

sion of the grain-boundary scattering sp�b is

1

sp¼X

i

1

sp;i� 1

sp�pþ 1

sp�b: (6)

The polycrystalline scattering (i.e., grain-size effect) is based

on the diffuse boundary reflection and transmission, i.e., the

Casimir boundary scattering model.24,69 With simplifying

assumptions (phonon mean-free-path equal to grain size, and

using the average phonon speed), we have

FIG. 4. Calculated phonon dispersions along high-symmetry directions in

the FCC Brillouin zone. The results are obtained from various empirical

potential models6–9,23,64 of UO2 and from the DFT. The available experi-

mental results65 are also shown.

FIG. 5. Variations of the decomposed, acoustic, and optical components of

the lattice thermal conductivity of the pristine UO2 crystal, with respect to

temperature. The ECMD results use the Yamada empirical potentials.9

123510-4 Kim, Kim, and Kaviany J. Appl. Phys. 115, 123510 (2014)

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1=sp�b ¼ up;g;A=dg, where dg is the grain diameter. Here, we

consider the half-mixed grain structure of dg¼ 0.01 and

0.1 lm, and the total relaxation time of the grain-boundary

scattering is obtained from 1=sp�b ¼ 1=sp�bðdg ¼ 0:01 lmÞþ 1=sp�bðdg ¼ 0:1 lmÞ. This gives sp�b ¼ 2:37 ps.

To consider the effect of pore structure in the heat con-

duction of UO2, the effective medium theory is applied.

Assuming negligible conduction through the pores (i.e.,

jf ; jL, where jf is the pore fluid conductivity) and

two-dimensional, periodic porous solids with continuous

pores, we derive the simple relation for the effective thermal

conductivity of porous media,70–72 hjLi ¼ jLð1� e1=2Þ,where e is the porosity.

Combining these two scattering effects, the phonon

grain-boundary and the pores (e¼ 0.05), the resultant effec-

tive lattice thermal conductivity is obtained, and shown in

Fig. 6(a). For these grains, the high-density sintered-powder

specimen (5% porosity), the grain-boundary scattering domi-

nates at low temperatures. As temperature increases, the

reverse trend is seen in Fig. 6(a). Figure 6(b) shows compari-

son between our predicted effective lattice thermal conduc-

tivity and the available experiments for sintered-powder bulk

UO2.1,13,22,73,74 The effective lattice thermal conductivity

results for the polycrystalline and porous UO2 are in good

agreement with the experiments. In Appendices A to C, we

estimate the contributions from electrons, small polarons,

and photons.

IV. CONCLUSIONS

Thermal energy transport in UO2 is of fundamental and

practical interests and here we approached the heat transport

mechanisms using the non-equilibrium ab-initio and equilib-

rium classical MD calculations and predict the lattice ther-

mal conductivity and its acoustic and optical components.

Comparison between the first-principles predictions and

sintered-powder experiments has been historically challeng-

ing. Our predicted results show the lattice thermal conductiv-

ity is dominated by the long-range acoustic phonon

transport, i.e., the Slack relation, and it is reduced by the

presence of the pores and by the grain-boundary scattering,

which when included compare well with the experiments.

We also predict the charge (including surface polarons) and

photon contributions to the thermal transport in UO2 and find

they are small in the defect-free structures and for tempera-

tures below 1500 K.

ACKNOWLEDGMENTS

This research was supported by the University of

Michigan Center for Solar and Thermal Energy Conversion,

an Energy Frontier Research Center funded by the U.S.

Department of Energy, Office of Science, Office of Basic

Energy Science under Award No. DE-SC-0000957, and the

WCU at Pohang University of Science and Technology by

National Research Foundation of Korea under Award No.

R3130005. We are thankful to Dubo Han for help with the

ECMD simulations.

APPENDIX A: CONDITIONS FOR POLARONFORMATION

The polaron hopping model can be examined by the

charge and bond analyses of the DFT results.75–77 Despite

the ionic bonds and the polarization effect of the UO2 struc-

ture, in our DFT results for the bulk, pristine crystalline

structure we do not observe any polaron formation, over the

temperature range of 0 to 2000 K. Since the pristine UO2

structure is a highly stable, homogeneous symmetric struc-

ture, we find no charge localization (or trapping) and struc-

tural distortion even at high temperatures. To explore such

charge localization using the DFT, we use the stoichiometric

and net-neutral UO2 slab models consisting of 72 atoms with

the vacuum gap of 10 A to minimize the interactions between

the slabs78 with a simple termination of (111) surface (hav-

ing the lowest surface energy79,80). For further validation of

the charge associated with each atom in the simulation cell,

here the Bader analysis81 of the charge-density grid is used

with the DFT charge density.

Using these simulations, we investigate the effective

charges (qe) of U and O atoms as functions of temperature,

and the results are shown in Fig. 7(a). Due to the strong

FIG. 6. (a) Variations of the predicted effective lattice thermal conductivity of UO2 with respect to temperature, including the porosity and grain-boundary

scattering effects. The results are from NEAIMD along with the grain-boundary scattering and porosity models. (b) Temperature dependence of the predicted

effective lattice thermal conductivity of UO2. Available experimental results13,22,73,74 and fit to them1,22 are also shown.

123510-5 Kim, Kim, and Kaviany J. Appl. Phys. 115, 123510 (2014)

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stability and high symmetry of the fluorite structure, we do

not expect the effective charge changes to be large.

Surprisingly, the calculated effective charge of the surface U

atoms is significantly reduced and scattered compared to the

O atoms, for T> 500 K. As proposed in Refs. 33, 37, and 82,

the polaron formation arise due to the charge localization of

the U ions, 2U4þ ! U3þ þ U5þ. Based on our predicted

results, the presence of the surface (or defects in general) in

UO2 allows for the formation of charge localization at high

temperatures. Another evidence for the existence of small

polarons is the lattice distortion caused by its polarization

field. This can be shown with the iso-charge-density contours

(with positive or negative values) from the differential

charge density, qþ1ðrÞ � q0ðrÞ, where qþ1ðrÞ is the charge

density of the supercell with the polaron and q0ðrÞ is the

charge density without any injected excess electron. In Fig.

7(b), the polarized and distorted bond lengths of the sur-

rounding O and U atoms, induced by the small polaron, are

evident. The average U-O bond length around the U4þ site

(2.34 A), on a pristine structure, decreases to 2.30 A for U5þ,

while for the U3þ site it increases to 2.53 A. Based on these

results, we identify that charge localized U atoms signifi-

cantly attract/repel adjacent ions (i.e., distort its lattice struc-

ture), and then the lattice distortions are achieved.

APPENDIX B: CHARGE THERMAL CONDUCTIVITY

The polaron electrical conductivity ðre;IÞ of UO2 with

open surface (pore) is modeled by31

re;I ¼re;�;I

T2xð1� 2xÞexp � Ea

kBT

� �; (B1)

where re;�;I is the pre-exponential factor and x is the defect

concentration. Here, for Ea, we use the activation energy of a

small polaron migration (Ea¼ 0.21 eV (Ref. 34)). From the

results of the extrinsic regime given in the literature,31 and

using re;I ¼ ð1:2� 106=TÞ2xð1� 2xÞexp½�0:21=ðkBTÞ�, we

find x¼ 0.01.

For the intrinsic electrical conduction model, we use

re;II ¼ neecle, where ne is carrier density and le is the carrier

mobility. The carrier density based on the quantum-mechanical

treatment of the conduction electrons in nondegenerate state

is28

ne ¼2ð2pme;ekBTÞ3=2

h3p

exp �DEe;g

2kBT

� �; (B2)

where me,e is the effective mass and hp is the Planck

constant. This relation is then approximated to the simple

form re;II ¼ re;�;IIexp½�DEe;g=ð2kBTÞ�, where re;�;II is the

pre-exponential factor. Based on the average results of the

previous report,28 we used re;�;II ¼ 3:569� 103 S=cm.

FIG. 7. (a) Temperature dependence of the effective charge of UO2 atoms in

the (111) plane. The left and right panels are for the pore surface and the

bulk structure, respectively. The error range is plotted with the shaded area

of each element. The calculated effective charge of U and O atoms at

T¼ 0 K are also shown with the dashed lines ðqe;U ¼ 2:40 and

qe;O ¼ �1:20 ecÞ. (b) The iso-charge-density difference surfaces of U5þ

(left) and U3þ (right) compared to U4þ. The cut-off bond lengths of U-O

used are 2.33 and 2.29 A, respectively. The iso-charge-density surfaces are

set as 0.0137 (left) and 0.0583 (right) ec-Bohr�3. Yellow and cyan colors

mean positive and negative charges.

FIG. 8. Calculated temperature dependence of (a) electrical conductivity

and (b) electronic thermal conductivity of UO2 with pores. The contributions

are from small polaron and from the intrinsic semiconductor conduction

electrons, activated with different energies. The available experimental

results28,31 are also shown.

123510-6 Kim, Kim, and Kaviany J. Appl. Phys. 115, 123510 (2014)

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Figure 8(a) shows the variations of the electrical conductiv-

ity as a function of the temperature, along with the experi-

mental results for the single crystal UO2.28

The variation of re with the inverse of temperature is

shown in Fig. 8(a) and the trends corresponding to the two

activation energies of 0.21 (for T< 2000 K) and 0.9 eV (for

T> 2000 K) are evident. Note that the small polaron conduc-

tion matches the intrinsic conduction at around 2000 K. This

is consistent with the available experiments.28,31 Applying

the Wiedemann-Franz law,24 re ¼ NL;�reT, where NL;� is

the Lorenz number, the electronic thermal conductivity is

obtained and its variation with respect to the temperature is

shown in Fig. 8(b). Similarly the polaron contribution domi-

nates for T< 2000 K, while the intrinsic conduction domi-

nates for T> 2000 K.

APPENDIX C: PHOTON THERMAL CONDUCTIVITY

The radiative thermal conductivity ðjphÞ of UO2 can be

expressed using the kinetics of photon emission and trans-

port. Here, we consider the role of photon contribution to

heat transfer in UO2 at high temperature. Under optically

thick approximation,24 the radiative thermal conductivity is

jph ¼16n2rSBT3

3rph; (C1)

where n is the index of refraction (2.3 for UO2), rSB is the

Stefan-Boltzmann constant (5.67� 10�8 W/m2-K4), and rph

is absorption coefficient. Here, we use a Rosseland average

absorption coefficient of 5� 103 1/m (the absorption coeffi-

cient given in the literature84 ranges from 0.6� 103 to

5.7� 103 1/m). Using these, at 1000 K and 2000 K, we have

jph ¼ 0:32 and 2.56 W/m-K, respectively. Considering the

porosity (e¼ 0.05), the effective radiative thermal conductiv-

ity is given by Schotte model83

hjphi ¼1� e

1=jL þ 1=jphþ ejph: (C2)

The variation of hjphi as a function of temperature is

shown in Fig. 9, and it is compared with the effective lattice

thermal conductivity. The radiative transport does play a role

above 1500 K, but negligible below 1500 K.

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