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Journal of Nuclear Materials

journal homepage: www.elsevier.com/locate/jnucmat

Thermal conductivity of porous UO2: Molecular Dynamics study

S. Nichenko a, D. Staicub'*

a Isotope and Elemental Analysis, Department Nuclear Energy and Safety, Paul Scherrer Institut, 5232 Villigen PSI, Switzerland bEuropean Commission, Joint Research Centre, Institute for Transuranium Elements, P.O. Box 2340, D-76125 Karlsruhe, Germany

ARTICLE INFO ABSTRACT

Classical Molecular Dynamics (MD) was used to investigate the effect of nanometric size pores on the thermal conductivity of irradiated UO2. The Green-Kubo approach was used for the thermal conductivity calculation.

The effects of pores size, porosity and pores separation were simulated. A comparison with existing theoretical models is presented and an analytical model adapted to irradiated fuel is obtained. The results demonstrate that, for realistic bubbles size and concentrations, the impact on the fuel thermal conductivity is higher than predicted by the correlations used to quantify the impact of porosity: the impact of 0.3 vol.% of nanometric pores is of the same order of magnitude as that of 4.5 vol.% of micrometric pores.

© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND

license (http://creativecommons.org/licenses/by-nc-nd/3XI/).

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Article history: Received 7 May 2014 Accepted 5 August 2014 Available online 27 August 2014

1. Introduction

Gas pores are created in irradiated nuclear fuels as a result of the formation and low solubility of the fission gases. The size of the pores ranges from few angstroms to micrometers. The overall impact of pores on the thermal conductivity is usually included in the porosity effect which is evaluated with formulas obtained applying the Fourier law to porous materials. The total porosity ranges from about 5% to 15% depending on burn-up and radial position in the fuel pellet. The higher values of the porosity correspond to the high burn-up structure observed at the pellet rim at high burn-ups. However, the nanometric size pore (nanopore) volume fraction is never explicitly taken into account in the correlations available for the thermal conductivity of irradiated fuels. Depending on how the porosity is determined, the nanometric pore volume fraction is included or not. If the porosity is determined with the application of optical micrography of the irradiated fuel, the nanop-ores are not visible and their volume fraction is neglected. When the porosity is determined from the difference between the measured density and the density for the matrix of irradiated fuel, the nanopore volume fraction is included in the porosity.

The Fourier law is applied assuming that the size of the pores is large compared to the mean free path of the phonons. A direct consequence of this assumption is that the porosity effect on the thermal conductivity does depend on pores shape and spatial arrangement, but does not depend on pores size. This assumption

* Corresponding author. E-mail addresses: Sergii.Nichenko@psi.ch (S. Nichenko), Dragos.Staicu@ec. europa.eu (D. Staicu).

is not verified in the case of nanopores with high concentration. Because of their high concentration (about 1017 cm-3), and very small size (a few nm), nanopores could be a parameter strongly affecting the thermal conductivity by scattering phonons. However, no quantitative assessment of this impact in irradiated nuclear fuels is available in the literature. In order to assess the real impact of nanopores, a realistic calculation is required, based on real size and concentrations.

2. Pores in irradiated fuels

Pores with micrometric size are observed in nuclear fuels. The fresh fuel pore size distribution curve can have one or two modes, with sizes between 1 and 10 im. About 5 vol.% porosity is already present in unirradiated fuel as a result of the admixture of pores former before sintering.

Fission gas pores are observed in irradiated fuels within the grains and along the grain boundaries. Both populations are expected to impact the fuel thermal conductivity. The irradiation conditions and the burn-up are described as having a limited influence on the size and density of the intragranular pores [1]. If irradiation temperature increases from 900 to 1400 °C the pores radius increases from about 0.5 nm to about 1 nm, while the pores concentration decreases from 1018cm-3 to 6 ■ 1017 cm-3 [2], representing respectively 0.052 and 0.25 vol.%. Irradiation temperature and burn-up both influence the pores size and pores concentration. A second population of pores, with a radius of 5-10 nm and a density of 1015 cm-3 appears at high burn-ups or high irradiation temperatures [3,4]. The pores size distribution changes from monomodal to bimodal due to pores growth and coalescence.

http://dx.doi.org/10.1016/jonucmat.2014.08.009 0022-3115/® 2014 The Authors. Published by Elsevier B.V.

This is an open access article under the CC BY-NC-ND license (http://creativec0mm0ns.0rg/licenses/by-nc-nd/3.0/).

This analysis shows that the relevant nanopores radius is between 0.5 and 1 nm with a volume fraction below 1%.

3. Impact of porosity and bubbles on the thermal conductivity: Review

The impact of porosity on the thermal conductivity of a material is quantified by solving the Fourier heat transport equation. The material is considered to be heterogeneous, i.e. composed by a matrix containing inclusions. The effective or equivalent thermal conductivity of the porous material is defined assuming that the thermal conductivity of both constituents, as well as the geometry, is known. A large number of analytical solutions is available, corresponding to various pore shapes, spatial arrangements and porosity range. Except in some ideal cases, the analytical solutions are obtained with approximations, for instance neglecting the interactions between the pores. The equation of Maxwell [5] can, for instance, be used for spherical non interacting pores, i.e. with a low porosity.

Already in 1973 Ondracek and Schutz [6] stated that the thermal conductivity of solid material depends not only on the pores volume fraction, but also on their shape, relative size, distribution and orientation. And these parameters should be taken into account for correct estimation of thermal conductivity of a porous material. Numerical solutions of the Fourier law can be obtained by the finite elements technique. The advantage of this approach is, however, often limited as the real three dimensional geometry is not known, or can not be used because of calculation size limitations. The impact of porosity on the thermal conductivity of nuclear fuels is usually quantified by a formula resulting from the equation of Maxwell with, in some cases, a correction for the radiative heat transfer in the pores, for instance the equation recommended by Brandt and Neuer [7].

Usually, the impact of the nanopores on the thermal conductivity is not explicitly taken into account, but is included in the porosity effect, applying the Fourier law to spherical pores. However, the Fourier law cannot be applied for pores of a nanometric size, because the assumption that the pores size is large compared to the mean free path of the phonons is not verified. In the case of nanopores, the thermal conductivity not only depends on the porosity but also on the pores size.

A number of theoretical works have been done on nanoporous materials with the attempt to investigate an influence of not only porosity on thermal conductivity but also of pores size and distribution [8-10]. These works showed that the thermal conductivity of a porous material does depend not only on the porosity but also on the average distance between the pores. It was also demonstrated experimentally [11] that the thermal conductivity of nanoporous Si is strongly dependent on the pore size at a given porosity.

Alvarez et al. [8] applied a phonon hydrodynamic approach for the investigation of nanoporous silicon thermal conductivity and obtained an equation for the thermal conductivity depending on the porosity and a so called Knudsen number, which is the ratio of the phonon mean-free path to characteristic size of the system. This formula was applied for large volume fractions of nanopores (from 40 to 90 vol.%). The radius of the pores was taken for the characteristic size. Applying the expressions developed by Alvarez et al. [8] demonstrates that the thermal conductivity of nanopor-ous silicon is lower for higher nanopore volume fraction and also for smaller pore size. This behavior is related to the phonon ballistic effects [8].

Sellitto et al. [12] used the same approach as Alvarez et al. [8] to investigate the influence of porosity and of pore size on the thermal conductivity degradation in porous silicon for porosities from 30% to 90%. Three different geometrical arrangements of the pores

(simple cubic, body-centered cubic and random) have been considered. It has been shown that, for any given value of the porosity, for decreasing pore size the effective thermal conductivity decreases whatever the geometrical arrangement of the pores is. Sellitto et al. [12] has also showed that the body-centered cubic distribution leads to the highest decrease in the effective thermal conductivity among the considered geometrical arrangements.

Lee and Grossman [9] performed a Molecular Dynamics study of the nanoporous Si thermal conductivity. In this work the dependence of the thermal conductivity on pores radius and pores separation was investigated for porosities from 7% to 38%. Lee and Grossman [9] concluded that the Si thermal conductivity depends on both the nanopores size and separation. In their work they showed that the thermal conductivity of nanoporous Si decreases with a decrease of pores size at a given porosity. The decrease of the thermal conductivity was related to the reduction of the pho-non mean-free path caused by the increased phonon scattering at the pores surface. Furthermore, the decrease of the thermal conductivity is more pronounced at low porosities (below 20%).

Coquil et al. [13] performed non-equilibrium Molecular Dynamics simulations of amorphous nanoporous silica. The porosity varied between 10% and 35%. In this work Coquil et al. [13] claimed that thermal conductivity of nanoporous silica does not depend on the pores radius and only on the porosity of the material. This claim is in contradiction to the statements of the above authors [6,8-11]. The presented results on the thermal conductivity of nanoporous silica are in good agreement with the coherent potential approximation [13] but are higher than the experimental results [14]. This discrepancy in the results was assumed to be related to the presence of "necks" connecting the pores in a real amorphous mesoporous silica.

In order to asses different models in this work, the thermal conductivity of UO2 with a nanopore volume fraction of 0.32 vol.% was calculated at 500 K with the formulas of Alvarez et al. [8] and Sellitto et al. [12] (see Fig. 1), considering pores radii from 1 to 100 A. The value of Fink [15] was used for the thermal conductivity of dense UO2 (6.55 Wm-1 K-1) and the pores were attributed zero thermal conductivity. For large pores (100 A radius), the effective thermal conductivity is equal to the prediction obtained with the formula of Maxwell [5]. The predicted impact of the nanopores is much higher if the pore radius is 5 A: 5.37 Wm-1 K-1 for Alvarez et al. [8] and 6.40 W m-1 K-1 for Sellitto et al. [12] (pore concentration of 6 ■ 1015 cm-3) compared to 10A: 6.16Wm-1K-1 for Alvarez et al. [8] and 6.45 Wm-1 K-1 for Sellitto et al. [12] (pores concentration of 7.6 ■ 1014cm-3). In the case of the 10 A pores, the difference between the Alvarez et al. [8] and Maxwell [5]

Pores radius, A

Fig. 1. Thermal conductivity at 500 K of UO2 containing 0.32 vol.% of nanopores.

thermal conductivity values is of 5% and between the Sellitto et al. [12] and Maxwell [5] is of 0.6%, which is sufficient in order to justify further investigations. Furthermore, the applicability of the Alvarez and Sellitto equations to the low porosities relevant to nuclear fuels needs to be checked.

4. Theoretical background

In the present study classical Molecular Dynamics was used with a force field model based on the empirical pair potential developed by Arima et al. [16-18]. This force field model was chosen due to its successful application to UO2 and its extension to mixed oxide fuels [16-18]. In particular this model shows correct description of the lattice constant as well as its temperature dependence (thermal expansion) in the whole temperature range of interest [16,19,20]. The model leads to thermal conductivity values in good agreement with literature data [19-22] in a wide temperature range, with however an overestimation at low temperatures [23,24].

For the Coulomb interaction, a finite-range pair potential proposed by Wolf et al. [25,26] was used. In our previous work [19] we proposed to use, for the UO2 system, the truncation radius for this potential equal to 9.329 A. Our tests have shown that this value of the truncation radius leads to an accurate calculation of the Coulomb interaction and to numerically correct results [19].

For the calculation of the thermal conductivity in equilibrium Molecular Dynamics (EMD) the Green-Kubo approach [27] was used. This method predicts the thermal conductivity from the analysis of the time-correlation of equilibrium heat flux fluctuations.

All the simulations were done for the NPT ensemble with periodic boundary conditions and the time-step of 0.5 fs. The detailed description of the modeling procedures and used approach is presented in our previous work [19]. In [20] we describe the sensibility of the Green-Kubo approach for the calculation of the thermal conductivity of nuclear fuels and propose an algorithm for the improvement of the accuracy of the obtained values.

5. Results

Porous UO2 was modeled using the MD simulation in the temperature range from 500 up to 1500 K. The literature review has shown that nanopores with a radius of 5 to 10 A and a volume fraction of 0.052 and 0.25% are relevant for nuclear fuels. These two parameters can be used to calculate the distance separating two adjacent nanopores. This separation distance is a parameter relevant for the assessment of the thermal conductivity degradation as it can be directly compared to the mean free path of the phonons.

For the purpose of the investigation of the UO2 thermal conductivity dependence on both the porosity and pore separation, several different configurations of the main cell were simulated. Porosities from 0.32% to 3.3% were considered in order to determine the influence of this parameter over the extended range and to enable the comparison with analytical models. The size of the main cell varied from 6 x 6 x 6 unit cells (2592 ions) up to 20 x 20 x 20 unit cells (96,000 ions). In each of the main cells only one pore was introduced. As a result of the periodic boundary conditions, a system with simple cubic distribution of pores was simulated.

For the investigation of the pore separation and size influence on the thermal conductivity at constant porosity, two different cases were considered. In the first case a main cell of 6 x 6 x 6 (2592 ions) with a single pore of 6 A radius was simulated. In the second case a main cell of 10 x 10 x 10 (12,000 ions) with a single pore of 10 A radius was simulated. In both cases the porosity was kept at a constant level of 3.3%, but the minimum separation between the pores in the first case is 20.8 A and 34.7 A in the second case.

For the investigation of the porosity influence on the thermal conductivity three other cases for the main cell sizes of 12 x 12 x 12 (20,736 ions), 15 x 15 x 15 (40,500 ions) and 20 x 20 x 20 (96,000 ions) were considered. In all the cases a single pore of 10 A radius was simulated. Thus, including the 10 x 10 x 10 case, four cases with the same pore size but different porosities were simulated. A summary of the simulated cases in presented in Table 1.

The MD calculation was performed for the isothermal-isobaric (NPT) statistical ensemble to simulate the lattice at constant temperature and pressure, using the Berendsen thermostat and baro-stat [28]. These conditions correspond to the conditions of a real experiment, but the temperature regulation requires a careful analysis as it may impact the thermal conductivity results [29]. To control pressure and temperature, a weak coupling to an external bath is simulated.

The results of the thermal conductivity calculations are presented in Fig. 2. Due to the presence of the pores, the thermal conductivity of the porous UO2 is lower than for pure crystalline UO2. The impact of the nanopores is stronger than predicted with the Maxwell equation. For instance, for a volume fraction of 3.3% of nanopores of 6 A radius the thermal conductivity at 500 K is reduced by 44%, while the formula of Maxwell predicts a reduction by 4.8%. The impact of pore size decreases as temperature increases, due to the decrease of the phonons mean free path: at 1000 K the thermal conductivity is reduced by 16%, compared to 4.8% obtained with the Maxwell formula.

For the convenience ofthe analysis a separate graph at constant porosity for the above described cases (A and B) is presented in Fig. 3. Fig. 3 shows that thermal conductivity for the case A (main cell of 6 x 6 x 6) is lower than for the case B (main cell 10 x 10 x 10), even though the porosity is the same in both cases. Besides that, in the case B the radius of the pore is bigger. The fact that in case A thermal conductivity is lower can be explained by the increased phonon scattering on the pores surface due to the decreased average distance between the pores, that is an increased concentration of the pores which behave as phonon scattering centers. For the same porosity the average pore separation depends on the average pore size. Thus, our results agree with the behavior observed by Alvarez et al. [8] and Sellitto et al. [12] but are in contradiction with the claim that the thermal conductivity of porous materials depends only on the porosity but not on the pores size stated by Coquil et al. [13].

In Fig. 4 the comparison of the obtained conductivity results for the cases with different porosity but the same pore size (B, C, D and E) is presented. Fig. 4 demonstrates that thermal conductivity of the porous material is strongly dependent on the porosity. But, as it was described above, the thermal conductivity of the porous material depends not only on the porosity but also strongly depends on the pore separation as it can be seen from Figs. 3 and 4.

Figs. 3 and 4 show that thermal conductivity of the porous material strongly depends on the temperature. The thermal conductivity decrease is less pronounced for the higher temperatures. In general, the thermal conductivity of the porous material is approaching the thermal conductivity of the pure material with increasing temperature. This effect is explained by the decreased phonon mean-free path related to the increased phonon-phonon scattering.

6. Development of an analytical model

6.1. Comparison to literature data

The presented results on the MD calculation of the thermal conductivity of porous UO2 demonstrate the influence of not only the

Table 1

Simulated cases.

Case Main cell Radius (A) Porosity (%) Separation (A) Concentration, 1014, cm~3

A 6 x 6 x 6 6 3.3 20.8 364.92

B 10 x 10 x 10 10 3.3 34.7 78.82

C 12 x 12 x 12 10 1.5 45.6 35.83

D 15 x 15 x 15 10 0.86 62.0 20.54

E 20 x 20 x 20 10 0.325 89.4 7.64

■ Pure UO

\\ \\ -- U02 3.3 vol.% porosity, Maxwell eq.

\\ case porosity, radius, separation

0 A 3.3 %, 6A, 20.8A

< □ B 3.3 %, 10A, 34.7A

0 c 1.5%, 10A, 45.6A

\_ < D 0.86 %, 10A, 62.0A

□ 0 E 0.325 % 10A, 89.4A

0 0 vNJ

1000 Temperature, K

Fig. 2. Thermal conductivity of porous UO2.

\ ■ Pure U02

\ case \ 0 A \ □ B porosity, 3.3 %, 3.3 %, radius, separation 6A, 20.8A 10Â, 34.7Â

□ □ \ 0 n 0 □ ° °

0............ i ■

1000 Temperature, K

Fig. 3. Thermal conductivity of porous UO2 at constant porosity but different separation.

Pure UO,

case porosity, radius, separation = B 3.3%, 10A, 34.7Â o C 1.5%, 10Â, 45.6Â < D 0.86%, 10Â, 62.OA > E 0.325%, 10A, 89.4Â

1000 Temperature, K

Fig. 4. Thermal conductivity of porous UO2 for the same pore size but different porosity.

presence of nanopores on the thermal conductivity degradation but also the pores size and separation. In this work the applicability of the equations of Alvarez et al. [8] and Sellitto et al. [12] derived using the phonon-hydrodynamic approach is tested.

The equation for the thermal conductivity of porous silicon presented by Alvarez et al. [8] (Eq. (1)) has been developed for a randomly distributed array of spheres in the silicon matrix. In this equation a so called Knudsen number, Kn = l/d, is used, that incorporates nonlocal effects dependent on the ratio of the mean free path, l, to the characteristic size of the system, d. In this case the characteristic size of the system is the pore radius.

keff =-

f/+18/ï^1 +72V/

In this equation f (/ ) represents the Maxwell term which describes the dependence of the thermal conductivity on the porosity of the material, /, derived using the Fourier law, f (/) = (1 - /)3 [8]. A(Kn) is a numerical function of the Knudsen number, Kn, proposed by Millikan [30] and has the form A(Kn) = 0.864 + 0.290exp(-0.625d/l). The second term in the denominator describes the role of the pores size in the thermal conductivity degradation. The results of the calculation using Eq. (1) together with the obtained MD results are presented in Fig. 5.

Even though the equation presented by Alvarez et al. [8] (Eq. (1)) reproduces the trend of the obtained results, it overestimates the influence of the nanopores size on the thermal conductivity leading to a much higher degradation.

Sellitto et al. [12] investigated the influence of porosity, pore size and also pore arrangement on the thermal conductivity of the porous silicon. The first term in the denominator of the derived equation Eq. (2), responsible for the effect of porosity, is the same as in the equation presented by Alvarez et al. [8].

keff =

m+18/Ch

In this equation Ci is the Cunningham correction factor (Eq. (3), where r is the bubbles radius) which is used in fluid dynamics to account for the non-continuum effects when calculating the drag on small particles. The expression for the second correction factor

■ Pure UO

case porosity, radius, separation

o A 3.3 %, 6Â, 20.8Ä

\ □ B 3.3 %, 10Â, 34.7A

tv o C 1.5% 10A, 45.6A

< D 0.86 », 10A, 62.0A

□ □ "O^ > E 0.325 %, 10Â, 89.4A

o 0--O. «

0--0 0-0--0- ^o-o-ig

Alvarez (2010), RD

1000 Temperature, K

Fig. 5. Comparison of the thermal conductivity of porous UO2 obtained by MD (thick symbols) and calculated with Eq. (1) [8] (lines with the same symbols) for a random distribution of the pores (RD).

C2 (Eq. (4)) depends on the pore arrangement in the material matrix.

C1 = 1 + 2r (1.257 + 0.4 ■ e-11r/l)

Since in our simulation a simple cubic distribution of the pores in the UO2 matrix was modeled, we are using the expression of the C2 for the simple cubic distribution presented by Sellitto et al. [12].

C2 = 1 - 1.76 ■ + /

The calculated values of the thermal conductivity, obtained using Eq. (2), together with the obtained MD results are presented in Fig. 6. This figure shows that Eq. (2) proposed by Sellitto et al. [12] underestimates the influence of the porosity on the thermal conductivity decrease.

The term (1 + pj V/) presented in the equation of Alvarez et al. [8] (Eq. (1)) is the same as the term presented in the work of Sellitto et al. [12] for the C2 coefficient, and corresponds to the random distribution of the pores in the material. But in our case we have a simple cubic distribution in the pores, which is represented in Eq. (2) by the C2 coefficient described by Eq. (4). This can be the reason why Eq. (1) presented by Alvarez et al. [8] overestimates the influence of the porosity compared to our MD results.

To test this assumption we replaced the (1 V/) term in Eq. (1) with the C2 for the simple cubic distribution of the pores represented by Eq. (4). The resulting equation is Eq. (5). The calculated values of the thermal conductivity of porous UO2 together with the obtained MD results are presented in Fig. 7. This figure shows that the derived Eq. (5) leads to the overestimation of the porosity influence on the thermal conductivity drop for porous UO2 and this overestimation is even more pronounced that in case of equation presented by Alvarez et al. [8] (Eq. (1)) for the random distribution.

keff =

f (/) + 2 V 1+A(Kn)

(1 - 1.76 ■ 3/ +

Thus, the above discussion has shown the necessity of a development of a new model.

6.2. Improved model

A fitting procedure was used to derive a new equation Eq. (6) that better describes the influence of porosity and pores size on the UO2 thermal conductivity. This equation corresponds to a simple cubic distribution of the pores in UO2. The influence of the porosity and pores size on the thermal conductivity degradation was optimized through the adjustment of the fitting coefficients according to the obtained MD results.

Pure UO.

case porosity, radius, separation

3.3 %, 6A, 3.3 %, 10A, 1.5%, 10A, 0.86%, 10A, 0.325 %, 10A,

20.8Â 34.7Â 45.6Â 62.0Â 89.4A

Alvarez (2010), SCD

1000 Temperature, K

Fig. 7. Comparison of the thermal conductivity of porous UO2 obtained by MD (thick symbols) and calculated with Eq. (5) (lines with the same symbols) for simple cubic distribution of the pores (SCD).

The calculated values of the thermal conductivity with Eq. (6) together with the thermal conductivity values obtained with the application of MD simulation are presented in Fig. 8.

keff =

ff(/ + 3 . 43 ■ /0 ■ 7 ■ Kn0■ 9 ■( - e-

The presented equation correctly describes the influence of the porosity on the thermal conductivity degradation of the porous UO2 with the consideration of the pores size and pores separation for the simple cubic distribution of the pores.

7. Application to irradiated fuel

The impact of the nanopores present in irradiated fuels on the thermal conductivity is investigated in representative cases in this section. Two irradiated fuels were selected, with 4.5 and 6.9 vol.% total porosity, corresponding to burn-ups of about 34 and 52 GWdt-1 [31]. The difference between the two irradiated fuels with different total porosity values is mainly due to the micromet-ric pores, while the nanometric bubble populations are similar [31]. The size distribution of the volume fraction (Fig. 9) is represented by the sum of two log-normal distributions, corresponding to the nanometric and micrometric pores (Fig. 4 in [31]). The contribution to porosity, considering pores of increasing radius, is shown in Fig. 10, for the irradiated fuels with 4.5 and 6.9 total porosity vol.%.

A preliminary analysis has shown that the effective thermal conductivity formulas (Eqs. (1), (5) and (6)) obtained for

Fig. 6. Comparison of the thermal conductivity of porous UO2 obtained by MD (thick symbols) and calculated with Eq. (2) [12] (lines with the same symbols).

Fig. 8. Comparison of the thermal conductivity of porous UO2 obtained by MD (symbols) and calculated with Eq. (6) (lines with symbols).

nanometric pores cannot be extended to pores of micrometric size, because the function f (/) = (1 - /)3 overestimates the impact of micrometric pores. The effective thermal conductivity of the porous fuel was therefore calculated by separating the contributions of the nanopores and micrometric pores. The effect of the nanop-ores was investigated by considering the equations of Alvarez adapted to a simple cubic distribution (Eq. (5)), and the equation obtained in this work using MD results (Eq. (6)). The effect of the micrometric pores was estimated with the equation of Maxwell-Eucken [32]. The threshold in bubbles radius used to distinguish between nano and micrometric pores is 100 A. For this value of the radius, the bubble size is much larger than the phonons mean free path and the bubble size does not impact the effective thermal conductivity anymore, while the volume fraction of pores with size between 0 and 100 A is of 0.7 vol.%, i.e. represent only a small fraction of the total porosity. The influence of this threshold value on the effective thermal conductivity was quantified by considering also a value of 30 A, corresponding to 0.3 vol.% of nanopores. The effective thermal conductivity for a distribution of nanopores of different sizes was calculated using an iterative process by applying the formulas (Eqs. (5) and (6)) to a first medium constituted by bulk UO2 (6.55 W m-1 K-1) containing the smallest pores. This effective medium was then considered as containing the next population of pores. The impact of this procedure on the calculated effective thermal conductivity was assessed by also calculating the effect corresponding to nanopores of only one radius representing the whole volume fraction of nanopores. The radius of these pores corresponds to the average radius of the nanopores weighted by their relative concentrations. For the fuels considered, this corresponds to nanopores with a radius of 0.76 nm and a volume fraction of 0.7%. The effect of the pores with radius higher than 100 A was estimated with the equation of Maxwell-Eucken [32] (Eq. (7)), taking into account the thermal conductivity of the UO2 matrix (km, which already includes the effect of the nanopores), of the pores (ki, set to zero) and their respective volume fractions (vi). In fact, the effective thermal conductivity obtained by applying the equation of Maxwell-Eucken only depends on the total porosity, and not on their size or sizes distribution, and the result is identical to the prediction of the classical Maxwell equation [5]. This result is due to the hypothesis that pores are spherical and not interacting, which is a realistic approximation as the porosity is low.

keii 1 + 2£n=1*,

ki-km 1 v 1 ki+2km

1 -£ n=1 V

ki-km 1 V ki+2km

£ o Q_

10° Iff8 10"7 10"6

Pores radius, m

Fig. 9. Pores size distributions for irradiated UO2 fuels with total porosity fractions of 4.5 and 6.9 vol.%.

0.070.06-& 0.05-q. 0.04-

B 0.03E ' 3 0.020.01 -0.00-

-4.5 vol. % total porosity - 6.9 vol. % total porosity

10"® 10~7 Pores radius, m

Fig. 10. Porosity considering pores of increasing radius, for irradiated fuels with 4.5 and 6.9 total porosity vol.%.

In order to assess the importance of the choice of an appropriate model for the investigation of the impact of nanopores, three effective thermal conductivity calculations were done. For the first calculation, the thermal conductivity decrease when pores of increasing size are taken into account was calculated with the formulas of Maxwell applied for all pores sizes (label Maxwell/Maxwell in Tables 2 and 3). For the second calculation, a combination of the formulas of Alvarez and Maxwell was used (label Alvarez/ Maxwell in Tables 2 and 3). For the third calculation, a combination of the formula obtained in this work (Eq. (6)) with equation of Maxwell was used (label Eq. (6)/Maxwell in Tables 2 and 3). The predicted thermal conductivity degradation taking progressively into account nanopores up to the radius of 100 A (Fig. 11) is of 6.48 Wm-1 K-1with the model of Maxwell, 6.16 Wm-1 K-1with the model of Alvarez, and 5.83 W m-1 K-1with Eq. (6). If the nanopores effect is considered using the weighted average radius, and the total volume fraction of 0.7%, the thermal conductivity is of 5.62 W m-1 K-1 with the model of Alvarez and 5.77 W m-1 K-1 with Eq. (6). This stronger impact can be explained by the higher concentration of bubbles simultaneously taken into account in the calculation, and the smaller distance between the bubbles, in comparison with the approach where the bubbles are introduced progressively. With both approaches, the results obtained with Eq. (6) are very close (5.83 versus 5.77 W m-1 K-1) and the calculation is therefore considered as reliable.

If the predicted thermal conductivity degradation is calculated taking progressively into account nanopores up to the radius of 30 A (0.32 vol.%), the value obtained with Eq. (6) is of 5.93 W m-1 K-1. If the nanopores effect is considered using the weighted average radius (0.73 nm) with the total volume fraction of 0.32%, the thermal conductivity obtained with Eq. (6) is of 6.07 Wm-1 K-1, value which shows that the calculation method does not strongly impact the result.

The effective conductivities (keff ), and their ratio to the bulk thermal conductivity (kef /km) were calculated (Table 2) and very similar results were obtained with the two calculation assumptions (thresholds of 30 or 100 A for the nanopores formula application). The impact of nanopores is found to be significantly higher than predicted with the Fourier law. For the irradiated fuel with 4.5 vol.% total porosity, at 500 K the nanopores introduce a supplementary conductivity degradation of 8%, which are to be added to the 7% degradation due to the porosity when evaluated with the Fourier law.

For the irradiated fuel with 6.9 vol.% total porosity, at 500 K the nanopores introduce a supplementary conductivity degradation of 8%, which are to be added to the 10% degradation due to the volume fraction when evaluated with the Fourier law. Compared to

Table 2

Effective conductivity (Wm-1 K-1) at 500 K (kef) for a total porosity of 4.5% (km is the bulk conductivity).

T (K) Conductivity with nanometric pores Conductivity with all porosity (keff) kf /km

Threshold Threshold Threshold

100 A 30 A 100 A 30 A 100 A 30 A

Maxwell/Maxwell 6.48 6.52 6.12 6.12 0.93 0.93

Alvarez/Maxwell 6.16 6.24 5.8 5.87 0.89 0.90

Eq. (6)/Maxwell 5.83 5.93 5.51 5.57 0.84 0.85

Table 3

Effective conductivity (Wm-1 K-1) at 500 K (keg) for a total porosity of 6.9% (km is the bulk conductivity).

T (K) Conductivity with nanometric pores Conductivity with all porosity (keg) keg =km

Threshold Threshold Threshold

100 A 30 A 100 A 30 A 100 A 30 A

Maxwell/Maxwell 6.48 6.52 5.89 5.89 0.90 0.90

Alvarez/Maxwell 6.16 6.24 5.62 5.66 0.86 0.86

Eq. (6)/Maxwell 5.83 5.93 5.31 5.36 0.81 0.82

VE 6-4 -

■S 6.0-

■o "

g 5.8-

| 5.6-

£ 5.4-

10"8 10"8 10"7 10"6 10"5 Pores radius, m

Fig. 11. Thermal conductivity decrease when bubbles of increasing size are taken into account (with a radius threshold of 30 A for the nanopores size).

their impact at 500 K, at 1000 K, the effect of the nanopores is reduced by a factor 3.7, inducing a conductivity degradation of about 2%.

The effect is significant and justifies a revision of the irradiated fuel conductivity models. The final values predicted by the thermal conductivity correlations will not be affected, when these are adjusted to experimental results, but the distribution of the burn-up effect between the different mechanisms (porosity, fission products, radiation damage,...), could be reviewed. A possibility is to include this effect in the radiation damage contribution, as the concentration of nanopores, as well as the concentration of radiation damage, reach saturation values with the increase of burn-up. The impact of increased irradiation temperature may require additional modeling, as both effects are not similarly affected: the radiation damage concentration decreases, while the bubble size increases by coalescence, both phenomena leading to a recovery in the thermal conductivity if the bubbles volume fraction remains constant as their size increases. This latter phenomena requires additional modeling by coupling the calculation to a fission gas behavior model, as an increase in temperature leads to supplementary fission gas precipitation.

8. Conclusions

The impact of porosity on the effective thermal conductivity of irradiated UO2 fuel, as predicted with the Fourier law, is

under-evaluated because of the inappropriate treatment of the contribution of the nanometric pores. The impact of nanometric pores on the thermal conductivity was investigated by MD, the results were compared to formulas available in the literature and a correlation adapted to the pore size and porosity relevant for irradiated fuels was proposed. This correlation takes into account the phonons mean free path, which is strongly impacted by the presence of nanometric pores with high concentrations. Calculations considering realistic size distributions and porosity were done for irradiated UO2 fuel, combining the correlation obtained for nano-metric pores, where the pores size has a strong effect, with the formula of Maxwell applied for pores size larger than 100 A, where the pores size has no impact. The main approximation of this model is the iterative introduction of nanometric pores of increasing sizes, in order to take into account the real sizes distribution. The results have shown that the conductivity degradation due to the nanometric size pores is of the same order of magnitude as the degradation due to the macroscopic pores volume fraction, while it represents only about 10% of the volume fraction of voids. This result demonstrates the necessity of a separate modeling of the effect of the nanometric pores in the irradiated fuel thermal conductivity correlations, as their impact is much larger than predicted from their volume fraction. The thermal conductivity values predicted by the correlations based on experimental results will not change, but an improved understanding of the phenomena responsible for the conductivity degradation with burn-up will be achieved, with a better quantification of the contribution of the different mechanisms.

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