- Email: [email protected]

Contents lists available at ScienceDirect

Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci

Thermophysical properties of (UxAm1−x)O2 MOX fuel a,⁎

a

b

T b

Ericmoore Jossou , Linu Malakkal , Jayangani Ranasingh , Barbara Szpunar , Jerzy Szpunar a b

a

Department of Mechanical Engineering, College of Engineering, University of Saskatchewan, 57 Campus Drive, Saskatoon S7N 5A9, Saskatchewan, Canada Department of Physics and Engineering Physics, College of Art and Science, University of Saskatchewan, 116 Science Place, Saskatoon S7N 5E2, Saskatchewan, Canada

A R T I C LE I N FO

A B S T R A C T

Keywords: Uranium dioxide Nuclear fuel Molecular dynamics MOX Thermal conductivity

The eﬀect of the addition of americium on the thermophysical properties of uranium dioxide (UO2) has been systematically studied by molecular dynamics (MD) simulation technique in the whole concentration range of americium and the temperature range from 300 K to 3200 K. The predicted thermophysical properties for (UxAm1−x)O2 solid solutions agree well with the available experimental data. The lattice parameters decreased with the increase in americium concentration and obey Vegard's law up to 2000 K. There is no signiﬁcant change in the enthalpy, heat capacity, lattice expansion, and thermal conductivity as we increased the concentration of americium. Overall, a series of empirical models are derived for the thermophysical properties of (UxAm1−x)O2 MOX fuel based on the MD data.

1. Introduction

where An = Ce, U, Th, Pu, Np, Cm, and Am. We decided to use the same potential to study the thermophysical properties of (UxAm1−x)O2 MOX fuel (where x = 0.00, 0.25, 0.50, 0.75, and 1.00) from 300 K to 3200 K with emphasis on the calculation of lattice constants, linear coeﬃcient of thermal expansion, enthalpy, heat capacity at constant pressure, elastic constants, and the thermal conductivity. Furthermore, the eﬀect of porosity on the thermal conductivity is reported given that ~5% porosity is often present as a result of the pores formed during sintering [11]. The focus on thermal expansion is due to the importance of fuel pellet–cladding interaction, which has been studied for UO2, ThO2, NpO2, and MOX fuels [3,7,12–14]. Thermal conductivity plays a major role in eﬃcient heat dissipation and might even delay the onset of accidents when properly tuned. There is a signiﬁcant body of work to determine thermal conductivity by experiments and theoretical calculations. For example, Li et al. [15] used a partial ionic model of BornMayer-Huggins interatomic potential to predict rather small degradation in thermal conductivity due to mix cation lattice of (U0.7xPu0.3Amx)O2 which has been observed independently by Cooper et al. in (PuxTh1−x)O2 and (UxTh1−x)O2 [4]. Ma et al. [1] showed that oxygen vacancies have a signiﬁcant eﬀect on the thermal conductivity of (U0.75Pu0.25)O2−y. They noted that when oxygen vacancies exist, the inﬂuence of plutonium concentration on the thermal conductivity is very small such that it can be ignored. The elastic properties in fuel materials play a signiﬁcant role in the mechanical properties of fuel during irradiation and directly determines the phonon group velocities, which are important parameters that govern the thermal properties of

Uranium dioxide (UO2) is the standard fuel in today's Light Water Reactors. It is usually subjected to extreme conditions such as high radiation dosage and temperature. This leads to the generation of minor actinides (MAs) oxides such as AmO2, CmO2, and NpO2 during reactor operation. Due to the large yield, long half-life, and high radioactivity of the MAs, it is crucial to develop technologies for their reuse, just like it was done for PuO2 and ThO2 oxides. In comparison to (UxPu1−x)O2, (UxTh1−x)O2, and (UxNp1−x)O2 MOX, there are limited studies on the thermophysical properties of (UxAm1−x)O2 fuel, and therefore, no reliable literature is available on (UxAm1−x)O2 fuel [1–5]. One of the promising techniques for the reuse of MAs is to add MA elements into the MOX fuel, which can serve as fuel for fast breeder reactors and/or transmutation reactors [6]. The use of MA-based MOX fuel is dependent on a thorough understanding of its thermophysical properties. These properties are also needed for prediction of in-service fuel behavior when used in the parametrization of performance codes. In recent time, Bellino et al. [7] determined the thermophysical properties of (NpxPu1−x)O2 MOX fuel using molecular dynamics while Cooper et al. [4] calculated the thermal conductivity of (UxTh1−x)O2 and (UxPu1−x)O2. They reported that the thermal conductivity of MOX fuel is independent of solid solution composition at high temperature. However, there is minimal work on thermophysical properties of MAbased MOX fuel. Given the success of the empirical potential developed by Cooper et al. [3,4,7–10] (hereafter referred to as the CRG potential) in reproducing the thermophysical properties of actinide oxides, AnO2

⁎

Corresponding author. E-mail address: [email protected] (E. Jossou).

https://doi.org/10.1016/j.commatsci.2019.109324 Received 28 July 2019; Received in revised form 24 September 2019; Accepted 28 September 2019 0927-0256/ © 2019 Elsevier B.V. All rights reserved.

Computational Materials Science 172 (2020) 109324

E. Jossou, et al.

where n is the number of moles. The ﬁrst derivative of the enthalpy was calculated by ﬁtting a straight line to the enthalpy at a given temperature T and the data at T ± 25 K on either side. The elastic constants were computed using the 10 × 10 × 10 supercells after checking size dependence using larger system sizes (15 × 15 × 15 and 20 × 20 × 20 ) with no signiﬁcant change in the results obtained. These simulations are carried out in an isothermal condition such that a thermostat is applied to prevent temperature ﬂuctuation. Since actinide dioxides systems have cubic structure, there are three independent elastic constants (C11, C12, and C44) calculated which are further used to compute the bulk modulus (B), shear modulus (G) and Young’s modulus (E) using Hills-Voigt-Reuss approach [23–25].

materials. Recently, Rahman et al. [16] compared the thermomechanical properties of (UxTh1−x)O2, (UxPu1−x)O2, and (PuxTh1−x) O2 for x = 0.3 and 0.5 using MD simulation in the temperature range of 300 K–1500 K. They reported the eﬀect of porosity and ﬁssion gas on the degradation of elastic properties of ThO2 [17]. 2. Computational method 2.1. Lattice expansion, heat capacity, and elastic constant calculation Molecular Dynamics simulations are carried out using the Largescale Atomic/Molecular Massively Parallel Simulator (LAMMPS) code [18]. The interatomic forces in the AnO2 are described by the manybody potential developed by Cooper et al. [9]. In this interatomic model, the potential energy, Ei , of an atom i with respect to all other atoms has two components which include a pairwise (BuckinghamMorse type) and a many-body embedded atom method (EAM) contribution [3]. The detailed formalism of the interatomic interaction is well described in Refs. [19,20]. A 10 × 10 × 10 supercell is used for all MD simulations and is considered suﬃcient to obtained converged thermal properties given rigorous supercell size eﬀects testing at 300 K carried out by Malakkal et al. on ThO2 using the same interatomic potential [21]. The structures of the mixed oxides are created by randomly distributing uranium and americium ions on the 4a Wyckoﬀ sites in the supercell. In reality there is likely a short range order and maybe other factors that move the system from being truly random. Such randomness in the structure could be generated by special quasirandom structures (SQS) approach which mimics, as closely as possible, the most relevant, nearest neighbor pair and multisite correlation functions of an inﬁnite random solid solution within a ﬁnite supercell [22]. However, recent work by Ghosh et al. [5] on the thermophysical properties of (Th, U)O2 and (Th, Pu)O2 mixed oxides system using both the randomized and SQS structures produced same results. This gave us conﬁdence to use the randomly generated structures for the simulations. In the calculation of the lattice expansion and heat capacity, the system was ﬁrst equilibrated using an isothermal-isobaric (NPT) ensemble with a Nosé-Hoover barostat and thermostat with a damping time of 0.5 ps and 0.1 ps respectively at zero external pressure. Equilibration is achieved when there is convergence in total energy and volume of the system. These structures are equilibrated for 25 ps for temperatures between 300 K and 3200 K at 25 K intervals with the lattice parameter and enthalpy calculated by averaging the data from the ﬁnal 5 ps of the simulation. Subsequently, the linear thermal expansion coeﬃcient (αp ) at constant pressure and linear thermal expansion (LTE) in percentage is then calculated using Eqs. (1a) and (1b):

2.2. Thermal conductivity calculation Equilibrium Molecular Dynamics (EMD) approach, also referred to as the Green-Kubo method, is used in the calculation of the phononassisted thermal conductivity (kph ) of the end member and MOX fuel. The kph of a material is related to the equilibrium ﬂuctuations of the heat current vector, S through the ﬂuctuation–dissipation theorem. For instance, the kph in the x direction is given by Eq. (3) [26,27]:

kph, x =

∞ 1 ∫ 〈Sx (t ) Sx (0) 〉 dt , KB VT 2 0

(3)

where KB is the Boltzmann constant, t is time, T is temperature, 〈Sx (t ) Sx (0) 〉 is the ensemble average of Sx in the x-direction, usually called the heat current autocorrelation function (HCACF). Sx denotes the heat current in the x-direction, which can be computed by Eq. (4) [28,29] as follows:

Sx =

∑ Ei vi,x + i

1 2

∑ ∑ (Fij (vi + vj) rij,x i

j>i

(4)

where Ei and vi are the total energy and velocity of atom i, respectively, Fij is the force exerted by atom j on atom i and rij is the instantaneous and equilibrium interatomic separation vectors between atoms i and j. A simulation cell of size 10 × 10 × 10 was equilibrated using the NPT ensemble for 4,000,000 time steps followed by equilibration under a constant volume and energy ensemble (NVE) to allow the volume to expand for 4,000,000 time steps at the temperatures of interest. To achieve energy conservation, a time step of 1.0 fs was utilized during the simulations. To avoid statistical noise during heat ﬂux collection, NVE was run for an additional 2,000,000 time steps. The HCACFs were evaluated from the heat current time series. 3. Results and discussion 3.1. Lattice parameters

1 ∂L αp = ⎛ ⎞ L ⎝ ∂T ⎠ p L (T ) − L (298K ) ⎞ LTE(%) = ⎛ × 100 L (298K ) ⎝ ⎠ ⎜

The average lattice parameters of pure UO2, AmO2, and (UxAm1−x) O2 MOX fuel as a function of temperature are shown in Fig. 1 alongside experimental data for UO2 [30,31] and AmO2 [32]. Clearly, from Fig. 1, the calculated lattice parameters agree well with the experimental values, which is not surprising given that the potential was ﬁtted to these results. However, Vegard's law is valid up to ~2400 K, such that the maximum deviation of the calculated results from the Vegard's law is ~0.08%. The subtle dip at 2400 K in Fig. 1 corresponds to the onset of superionic transition usually caused by high temperature oxygen disorder in AnO2 oxide fuels. Similar behavior has been observed in (ThxPu1−x) O2 [33], (NpxPu1−x)O2 [7], (UxTh1−x)O2 [34] and (UxPu1−x)O2 [3]. Furthermore, the lattice parameter of (UxAm1−x)O2 (x = 0, 0.25, 0.50, 0.75, and 1.00) decreases slightly with an increase in the Am content, as shown in Fig. 2. The diﬀerence in the covalent radius of U (1.96 Å) and Am (1.80 Å) atoms cause strain in the lattice in order to accommodate Am substitution into the U cation sublattice. Based on Figs. 1 and 2, we concluded that the lattice parameter of (UxAm1−x)O2 decreases with

(1a)

⎟

(1b)

where L is the lattice parameter at room temperature and T is the temperature. The ﬁrst derivative of the lattice parameter with respect to temperature (slope of the lattice-temperature graph) is represented as ∂L , while the details of its numerical derivation from the MD simula∂T tions is well described in Ref. [3]. It is important to note that for each temperature of interest, the points within T ± 25 K on either side is considered to reduce numerical ﬂuctuations. The speciﬁc heat capacity at constant pressure (Cp) is estimated by taking the ﬁrst derivative of the enthalpy increment with respect to temperature at constant pressure using the following relationship:

Cp =

1 ⎛ ∂H ⎞ n ⎝ ∂T ⎠ p

(2) 2

Computational Materials Science 172 (2020) 109324

E. Jossou, et al.

O2 [3] and (UxTh1−x)O2 [34] MOX fuels. We further compared our results with experimental data by Halden et al. [36], Grønvold [37], Christensen [38] for pure UO2 and experimental data reported by Prieur et al. [39] in the case of AmO2 as shown in Fig. 3b-c. There is an excellent agreement between experiment and MD data presented in Fig. 3b-c except for data by Halden et al., where there is an over estimation in the experimental LTE at temperature > 1750 K for pure UO2. This eﬀect can be observed more clearly in a plot of the linear thermal expansion coeﬃcient (α) as a function of temperature (Fig. 4a). Speciﬁcally, the superionic transition temperature is ~2600 ± 25 K for UO2, which is in agreement with a previously calculated value of 2600 K using the same CRG potential [3] and experimentally determined value of 2670 K [40]. The transition peak temperatures are estimated to be 2575, 2550, 2525, and 2525 K for (U0.75Am0.25)O2, (U0.5Am0.5)O2, (U0.25Am0.75)O2 and AmO2, respectively. This implies that the (UxAm1−x)O2 solid solution tend to experience disorderliness in the oxygen sublattice as the concentration of Am is increased in the MOX. This eﬀect is much less noticeable in (UxTh1−x)O2 as observed by Cooper et al. due to the higher superionic transition temperature in comparison to UO2 while such disorderliness is more prominent in (UxPu1−x)O2 compared to (UxAm1−x)O2 [3]. We further compared our results with the ﬁtting model derived from experimental data by Martin [41] for pure UO2 and experimental data reported by Fahley et al. [42] and Sobolev [43] in the case of pure AmO2 as shown in Fig. 4b–c. The average αUO2 is estimated here to be 12.14 × 10−5K−1 in the temperature range of 300–2000 K which is comparable to the result by Martin [41] while average α AmO2 is 12.22 × 10−5K−1 which is in agreement with experimental data reported by Fahley et al. [42] and theoretical prediction by Sobolev [43]. It is well known that the thermal expansion of solids is determined by the anharmonicity of the interatomic interactions in such material. Therefore, the lower calculated value of MD simulation for AmO2 is most likely due to the lower anharmonicity. Furthermore, it is clear that the αUO2 is not signiﬁcantly aﬀected by the addition of Am (~3% increase in α of (UxAm1−x)O2 MOX fuel) which is similar to the result of the substitution of U with Pu by Cooper et al. [3]. The slight increase in α (Ux Am1 − x ) O2 as we increase the concentration of Am is due to the relatively small diﬀerence in αUO2 and α AmO2 , respectively. Further comparison of the simulated data with Vegard’s law [44] shows (Fig. 5) a deviation of –0.2% to 4% for the temperature range of 300 K–2000 K. At higher temperatures; especially at the superionic transition point, there is a huge deviation from Vegard’s law reaching a maximum and minimum of + 25% and −25% respectively depending on the concentration of americium. Previous simulations of (UxPu1−x)O2 [3] and (NpxPu1−x)O2 [7] support Vegard’s Law below 2000 K. Therefore, the LTE and α in the temperature range of 300 K–2000 K data were ﬁtted as a function of temperature using a third order polynomial function described in Eq. (6):

Fig. 1. The lattice parameter of (UxAm1−x)O2 MOX fuel dependence on the temperature.

Fig. 2. The lattice parameter of (UxAm1−x)O2 MOX fuel dependence on the americium concentration.

increase in Am concentration. Therefore, the lattice parameter a as a function of temperature T in Kelvin and Am concentration x is given by Eq. (5):

a (T , x ) = a0 + a1 T 3 + a2 T 2 + a3 T + mx

(5)

The coeﬃcients of Eq. (5) are determined by the least-squares ﬁtting technique with a relative coeﬃcient of 0.992. The error margin of the Equation for estimation of the lattice parameter is ± 0.0001 Å. The ﬁtting parameters derived from Eq. (5) are reported in Table 1.

LTE (α ) = l 0 (α 0) + l1 (α1 ) T + l2 (α2 ) T 2 + l3 (α3 ) T 3

3.2. Thermal expansion

The ﬁtting parameters l provided in Table 2.

In Fig. 3a the linear thermal expansion (LTE) is plotted as a function of temperature with no signiﬁcant change as we increase the concentration of Am in UO2 until the onset of superionic transition as underscored by the divergence in the plots starting at temperature of 2400 K. Similar behavior was observed in (NpxPu1−x)O2 [7], (UxPu1−x)

a0

a1 (10−12)

a2 (10−8)

a3 (10−5)

m

(UxAm1−x)O2 (UxPu1−x)O2 [35]

5.39 5.45

4.46 1.90

−1.03 0.28

7.57 5.16

0.11 −0.07

(α0), l

1

(α1), l

2

(α2) and l

3

(α3) are

3.3. Enthalpy and heat capacity We have calculated the enthalpy increment (ΔH) of (UxAm1−x)O2 over a temperature range 300–3200 K. In addition, UO2 and AmO2 endmembers were also computed and compared with literature data, with an excellent agreement, as shown in Fig. 6a–c. The ΔH increases linearly with temperature below 2400 K while temperature >2400 K shows a ‘bump’ similar to LTE plot signifying the supersonic transition behavior due to anion sublattice pre-melting (Inset in Fig. 6a). Fig. 6b shows a comparison of calculated enthalpy increase in UO2 in this study with experimental data by Vălu et al. [46] and Konings et al. [47] where there is about ~3% underestimation in

Table 1 Parameters for a third-degree polynomial that ﬁts the change in the lattice parameters as a function of temperature and americium concentration. MOX

0

(6)

3

Computational Materials Science 172 (2020) 109324

E. Jossou, et al.

Fig. 3. a–c. Linear thermal expansion (LTE) as a function of temperature for (UxAm1−x)O2 with comparison of pure UO2 with experimental data from Halden et al. [36], Grønvold [37], Christensen [38] and AmO2 with data from Prieur et al. [39].

Fig. 4. a–c. Linear thermal expansion coeﬃcient (α) data as a function of temperature for (UxAm1−x)O2 with a comparison of pure UO2 with experimental data from Martin [41] and AmO2 with data from Fahley et al. [42] and Sobolev [43].

our calculated data especially in the temperature range of 300–1400 K. We observed excellent agreement between calculated ΔHs for AmO2 and correlation derived from experimental data by Nishi et al. [45], while the nonstoichiometric AmO2 and MOX data from Vălu et al. [46] lies slightly below and above the calculated results respectively. The speciﬁc heat capacity of UO2, AmO2, and (UxAm1−x)O2 are shown in Fig. 7a–c, along with experimental values. Although the heat capacity of pure UO2 is underestimated in comparison to the results of Vălu et al. [46] at high temperature while pure AmO2 is appreciably underestimated. Generally, the MD results are in reasonable agreement with previous data by Carbajo et al. [48], and Thiriet et al. [49]. Our results show that the heat capacity curve of AmO2 is close to that of UO2. This means that blending UO2 with AmO2 would not require any change in cladding or system design to accommodate a change in the thermophysical properties. At temperatures above 2400 K, the Cp shows the characteristic peak (see Fig. 7a) attributed to the superionic transition of the ﬂuorite-type structure [47]. This transition is due to the formation of oxygen disorder, but its mechanism is not well understood. The melting points (Tm ) of the end member oxides is related to the superionic transition peak temperature (Tα )by Eq. (7) [48].

Fig. 5. The deviation of α from Vegard’s Law as a function of temperature is shown for three compositions of the (UxAm1−x)O2 solid solution.

Tm =

Tα 0.85

(7)

The predicted melting points of UO2 and AmO2 are 3059 and 4

Computational Materials Science 172 (2020) 109324

E. Jossou, et al.

Table 2 Fitting parameters of LTE and α are given by LTE (α) = l0 (α0) + l

1

(α1)T + l2 (α2)T2 + l3 (α3)T3.

Fuel

l 0 (10−3 )

l1 (10−6 )

l2 (10−10 )

l3 (10−13 )

α 0 (10−6 )

α1 (10−9 )

α2 (10−13 )

α3 (10−16 )

AmO2 (This work) Am-MOX [39] UO2 (This work) UO2 [41]

−2.92 −3.88 −2.90 −2.66

9.50 13.95 9.49 9.80

6.50 50.11 5.35 2.71

2.86 20.75 3.28 4.39

9.40 – 9.54 –

1.81 – 1.03 –

1.09 – 7.96 –

2.35 – 0.71 –

elastic constant values (C11, C12, and C44) and elastic moduli are plotted as a function of americium concentration in Fig. 9a–b and ﬁtted to a linear function. The temperature dependence of the bulk, Young’s and shear moduli are calculated for pure for UO2, AmO2 and (UxAm1−x)O2 solid solution as a function of americium concentration as shown in Fig. 10a–c. The values of the elastic moduli decreases with increase in temperature. A similar trend is observed in previous studies in actinide oxides [16]. Softening of the elastic modulus as temperature increase is due to the increase in thermal vibration of atoms. Based on the assumption that the elastic moduli are linear in the temperature range of 300–2000 K, the elastic moduli are ﬁtted to Eq. (12):

2824 K, respectively. In comparison with the international nuclear safety, center recommended data of 3120 ± 30 K for UO2, the predicted melting point obtained from the MD approximation is found to be underestimated whereas the predicted melting temperature is overestimated for AmO2 in comparison to the experimentally measured value of 2773 K [49]. In Fig. 8 the deviation from Kopp-Neumann law [51] is plotted UxAm1−x)O2 for x = 0.25, 0.5 and 0.75. There is deviation of −2% to 2% in the temperature range of 300 K–2000 K. However, beyond 2000 K, the heat capacity no longer obey KoppNeumann law due to the disorderliness in the oxygen sublattice of the fuel with a deviation of –18% to 18%. This has implication for hightemperature oﬀ normal reactor conditions such as Loss of coolant accident scenarios and suggests the need to blend MOX with thermal conductivity and ﬁssion gas retention enhancers such as molybdenum, Al2O3, and Cr2O3 [52]. The ΔH of both AmO2 and UO2 were ﬁtted using the least-square approach into the Maier–Kelly empirically derived temperature dependent model with a constraint (Eqs. (8) and (9)), such that ΔH = 0 at a temperature of 300 K [53]. In a similar approach, an empirical model was derived by imposing a boundary condition for C p, UO2 = 76.00 and C p, AmO2 = 75.78kJmol−1 at T = 300 K respectively. Eqs. (10) and (11) represent the high-temperature heat capacities of AmO2 and UO2 obtained in this study.

ΔHUO2 = −26.5 + 8.2273 × 10−2T + 2.11939 × 105T −2

ΔH AmO2 = −26.1 + 8.2173 ×

10−2T

+ 1.3113 ×

105T −2

Y (T ) = εY T + Yo

(12)

where, Y (B, E and G) and Yo (Bo , Eo and Go ) denotes the elastic modulus at ﬁnite temperature and 0 K respectively while εY is the corresponding temperature coeﬃcient. εY (εB , εE , and εG ) is a measure of the rate of change in elastic modulus with temperature. Fig. 11 is a plot of the temperature coeﬃcients of B, E, and G as a function of americium concentration. From Fig. 11, the temperature coeﬃcient increase with an increase in americium concentration while for B, E, and G, the value of ɛ follow the trend εE > εB > εG . It is worth noting that the rate of change of εG as you move from UO2 to the AmO2 end is very small.

(8) (9)

C p, UO2 = 73 + 7.5 × 10−3T + 6.9993 × 10 4T −2

(10)

C p, AmO2 = 74 + 7.0 × 10−3T + 1.8513 × 10 4T −2

(11)

3.5. Thermal conductivity It is important to determine a converged value of the HCACF integral, which is proportional to the thermal conductivity based on Eq. (3). A normalized HCACF for UO2 and AmO2 at 300 and 2100 K respectively is shown in Fig. 12a–b. The HCACF is well converged after 7000 ps for both temperatures considered here. Therefore, we speciﬁed the converged value of the HCACF integral by averaging its value between 7000 and 10000 ps as presented in the shaded region of the insets

3.4. Elastic constants The elastic constant are 0 K serves as input for calculating the temperature coeﬃcient of the elastic modulus. The three independent

Fig. 6. a–c. The enthalpy increments of the UO2, AmO2 and (UxAm1−x)O2 calculated in this study together with enthalpy equations of AmO2, AmO1.5 obtained by Nishi et al. [45], AM10 (U0.9123Am0.0877O2−y) and AM20 (U0.8105Am0.1895O2−y) intermediate compositions measured by Vălu et al. [46] and UO2 assessed by Konings et al. [47]. 5

Computational Materials Science 172 (2020) 109324

E. Jossou, et al.

Fig. 7. a–c. Heat capacity of the UO2, AmO2 and (UxAm1−x)O2 calculated as a function of temperature together with experimental results for UO2 [46,48] and AmO2 [45,49,50] end members.

the experimental data are corrected to 100% theoretical density (TD) using Maxwell-Eucken equation:

k100% =

k 95% (1 + βp) 1−p

(13)

whereβ = 0.05 and p is the porosity such that for 95% density pellets, p = 0.5 as recommended by the IAEA [54]. k100% and k 95% are the thermal conductivity at 100 and 95% TD respectively. Our results (Fig. 13) are overestimated in comparison with experimental data. The overestimation is due to the presence of strong spin-phonon scattering component in UO2 and AmO2, respectively. However, spin-phonon scattering is not captured by the CRG potential used in these MD simulations [55]. The results of thermal conductivity that we obtained using the EMD method are higher than the values determined by Maxwell et al. using the same method [56]. This might be because the authors used a smaller supercell size, which is half the one deployed in this study. Rigorous size dependence test is important for calculating converged thermal properties using the EMD technique. There is an insigniﬁcant diﬀerence when compared to the result of Cooper et al., which is based on the NEMD technique [4]. Fig. 14a–b compares calculated thermal conductivity of (UxAm1−x) O2 (where x = 0, 0.25, 0.5, 0.75, 1.0) as a function of temperature. In Fig. 14a, we report a sharp reduction in the thermal conductivity, particularly at low temperatures, due to a reduction in the phonon mean free path because of the scattering of the phonons in a non-uniform cation sublattice. Similarly, Fig. 14b shows that although AmO2 has a slightly higher thermal conductivity than UO2, there is a marginal

Fig. 8. The deviation of speciﬁc heat capacity from Kopp-Neumann Law as a function of temperature is shown for three compositions of the (UxAm1−x)O2 solid solution.

in Fig. 12a–b. Thermal conductivity for stoichiometric UO2 and AmO2 was calculated for validation purpose. For comparisons with the MD results,

Fig. 9. a–b. Elastic constants and modulus at 0 K as function of americium concentration for (UxAm1−x)O2 (where x = 0, 0.25, 0.5, 0.75, 1.0). 6

Computational Materials Science 172 (2020) 109324

E. Jossou, et al.

Fig. 10. a–c. Elastic constants and modulus as function of temperature and americium concentration for (UxAm1−x)O2 (where x = 0, 0.25, 0.5, 0.75, 1.0).

distance between diﬀerent cation sites that limit the phonon mean free path [4]. To further validate the above assertion, the calculated thermal conductivities of (UxAm1−x)O2 fuels are ﬁtted to the defect-phonon and ph–ph scattering relation adapted for MOX fuels by Adachi et al. [61] based on the original Klemens-Callaway theory for thermal conductivity of disordered alloys [62]:

k=

1 xWUO2 + (1 − x ) WAmO2 + x (1 − x ) CU − Am

(14)

whereWUO2 = aUO2 + bUO2 T , WAmO2 = a AmO2 + b AmO2 T are the thermal resistivity of the end member and CU − Am is due to the random distribution of U and Am atoms on a given sublattice. However, Eq. (14) is reduced to Eq. (15) when x = 0 or x = 1. This is essentially a rational ﬁt for the end member oxides.

k=

1 a + bT

(15)

where a and b are the defect-phonon scattering and ph–ph scattering contributions respectively. The value of a and b constants estimated by rationally ﬁtting the thermal resistivity versus temperature data (see Table 3). The eﬀect of the disordered cation sublattice in (UxAm1−x)O2 is comparable to (UxNp1−x)O2 [63] while it is 5 and 20 order of magnitude reduced in (UxPu1−x)O2 and (UxTh1−x)O2 [4] respectively as demonstrated by CU − Am term when the MD data is ﬁtted to Eq. (14) as shown in Table 3. The values of the constant, a for (UxAm1−x)O2 derived from the simulations increase with increasing AmO2 concentration, consistent with a reduction of thermal conductivity values compared to those of pure UO2 and AmO2, due to increasing defect-phonon scattering. Furthermore, our calculated values show no systematic trend for constant b as we increased the concentration of Am while the values of ph–ph scattering constant b are nearly constant within a window of ~4% deviation. This observation is not surprising due to the fact that the end member which constitute the MOX fuel has similar ﬂuorite crystal structures. Therefore, the ph–ph scattering is not expected to change signiﬁcantly with the concentration of Am within the

Fig. 11. Temperature coeﬃcient of elastic modulus as function of americium concentration for (UxAm1−x)O2 (where x = 0, 0.25, 0.5, 0.75, 1.0).

reduction at low temperatures due to substitutional uranium additions. However, at higher temperatures, as phonon–phonon (ph–ph) interactions become the limiting factor in determining the phonon mean free path, the proportionate eﬀect of the non-uniform cation sublattice is insigniﬁcant. The sharp reduction of thermal conductivity in U0.5Am0.5O2 can be attributed to the diﬀerence in the lattice parameter of the end member oxides and the mass diﬀerence of the actinide atom involved. For instance, the diﬀerence in the lattice parameters of AmO2 and UO2 is ~0.0939 Å. This is responsible for the lattice strain around the cation sublattice in UxAm1−xO2. This becomes signiﬁcant when x = 0.5 since the substitutional defects become saturated. Furthermore, the diﬀerence in masses of U238 and Am243 also contribute to the change in thermal conductivity as the concentration of Am equals U. Similar trend has been observed in (U0.50Th0.50)O2, and this has been ascribed to the 7

Computational Materials Science 172 (2020) 109324

E. Jossou, et al.

Fig. 12. Normalized HCACF as a function of correlation time in (a) UO2 (b) AmO2 calculated at 300 and 2100 K, respectively.

Fig. 13. . Shows the bulk values as a function of temperature for unirradiated UO2 and AmO2 with comparison to the experimental thermal conductivity of Wiesenack et al. [57], Saoudi et al. [58], Uchida et al. [59]and MD data from Cooper et al. [4], Maxwell et al. [56] and Bakker et al. [60].

Fig. 14. The eﬀect of the addition of (a) U to AmO2 and (b) Am to UO2 for a range of (UxAm1−x)O2 (where x = 0, 0.25, 0.5, 0.75, 1.0), as predicted using the equilibrium molecular method. Eq. (14) has been ﬁtted to the data points from MD and is shown using solid lines.

atoms. Additionally, the MD results indicate a large decrease in thermal conductivity at low temperatures, and the lowest thermal conductivity occurs at about the x = 0.5. This implies that the distance between diﬀerent cation sites is smaller and has a larger eﬀect on limiting the phonon mean free path. This trend is consistent with previous studies where MOX structures are generated randomly or by SQS approach [4,13,15,17,63].

temperature range of this study. Similarly, the calculated values for (UxAm1−x)O2 are higher compared to that of UO2 and AmO2, which is consistent with thermal conductivity of MOX being lower than those of the end members UO2 and AmO2 across the temperature range considered in this work. Fig. 15 shows the variation of the thermal conductivity for (UxAm1−x)O2 solid solution as a function of the americium content for temperature ranging from 300 K to 2100 K. It is interesting to ﬁnd that the thermal conductivity of (UxAm1−x)O2 is lower than both pure UO2 and AmO2, which may be caused by the phonon-lattice strain interaction which is due to the size and mass diﬀerences between U and Am

4. Conclusions MD simulations were performed to calculate the thermal expansion, 8

Computational Materials Science 172 (2020) 109324

E. Jossou, et al.

CRediT authorship contribution statement

Table 3 A comparison of constants a and b of the equations W = a + b T for (UxAm1−x) O2 MOX fuel (where x = 0, 0.25, 0.5, 0.75, 1.0) derived from MD simulations and from experimental measurements. The value of CU − Am is 1.13 × 10−2 mKW−1. Fuel

Work was done

a (× 10−2 ) mKW−1

b (× 10−4 ) mW−1

UO2

MD, This work MD Experiment Experiment (U0.75Am0.25)O2 (U0.5Am0.5)O2 (U0.25Am0.75)O2 MD, this work Experiment Experiment

2.781 1.896 6.240 6.548 3.052 3.171 3.329 5.015 10.060 3.894

2.387 1.923 2.399 2.353 2.227 2.324 2.396 2.397 1.664 2.559

MOX

AmO2

Ericmoore Jossou: Data curation, Investigation, Formal analysis, Methodology, Validation, Visualization, Writing - original draft. Linu Malakkal: Validation, Writing - review & editing. Jayangani Ranasinghe: Validation, Writing – review & editing. Barbara Szpunar: Conceptualization, Investigation, Funding acquisition, Project administration, Resources, Software, Supervision, Writing - review & editing. Jerzy Szpunar: Project administration, Resources, Software, Supervision, Writing - review & editing. Acknowledgements The authors gratefully acknowledge Compute Canada (WestGrid) and University of Saskatchewan’s Research Cluster (Plato) for a generous amount of CPU time. We also acknowledge the ﬁnancial support of the Canadian National Science and Engineering Research Council (NSERC) and the University of Saskatchewan’s International Dean’s Scholarship. Thanks to Dr. Rahman for a fruitful discussion. Data availability The raw/processed data required to reproduce these ﬁndings cannot be shared at this time as the data also forms part of an ongoing study. References [1] J. Ma, J. Zheng, M. Wan, J. Du, J. Yang, G. Jiang, Molecular dynamical study of physical properties of (U0.75Pu0.25)O2–x, J. Nucl. Mater. 452 (2014) 230–234. [2] S. Nichenko, D. Staicu, Molecular dynamics study of the mixed oxide fuel thermal conductivity, J. Nucl. Mater. 439 (2013) 93–98. [3] M.W.D. Cooper, S.T. Murphy, M.J.D. Rushton, R.W. Grimes, Thermophysical properties and oxygen transport in the (UxPu1−x)O2 lattice, J. Nucl. Mater. 461 (2015) 206–214. [4] M.W.D. Cooper, S.C. Middleburgh, R.W. Grimes, Modelling the thermal conductivity of (UxTh1−x)O2 and (UxPu1−x)O2, J. Nucl. Mater. 466 (2015) 29–35. [5] P.S. Ghosh, N. Kuganathan, C.O.T. Galvin, A. Arya, G.K. Dey, B.K. Dutta, R.W. Grimes, Melting behavior of (Th, U)O2 and (Th, Pu)O2 mixed oxides, J. Nucl. Mater. 479 (2016) 112–122. [6] T. Wakabayashi, Transmutation characteristics of MA and LLFP in a fast reactor, Prog. Nucl. Energy 40 (2002) 457–463. [7] P.A. Bellino, H.O. Mosca, S. Jaroszewicz, Evaluation of thermophysical properties of (Np, Pu)O2 using molecular dynamics simulations, J. Alloy. Compd. 695 (2017) 944–951. [8] M.J. Qin, M.W.D. Cooper, E.Y. Kuo, M.J.D. Rushton, R.W. Grimes, G.R. Lumpkin, S.C. Middleburgh, Thermal conductivity and energetic recoils in UO2 using a manybody potential model, J. Phys.: Condens. Matter 26 (2014) 495401. [9] M.W.D. Cooper, M.J.D. Rushton, R.W. Grimes, A many-body potential approach to modeling the thermomechanical properties of actinide oxides, J. Phys.: Condens. Matter 26 (2014) 105401. [10] M.W.D. Cooper, C.R. Stanek, J.A. Turnbull, B.P. Uberuaga, D.A. Andersson, Simulation of radiation driven ﬁssion gas diﬀusion in UO2, ThO2 and PuO2, J. Nucl. Mater. 481 (2016) 125–133. [11] S. Nichenko, D. Staicu, Thermal conductivity of porous UO2: molecular dynamics study, J. Nucl. Mater. 454 (2014) 315–322. [12] T. Matsumoto, T. Arima, Y. Inagaki, K. Idemitsu, M. Kato, K. Morimoto, M. Ogasawara, Thermal conductivity measurement of (Pu1−xAmx)O2 (x=0.028, 0.072), J. Alloy. Compd. 629 (2015) 92–97. [13] J.-J. Ma, J.-G. Du, M.-J. Wan, G. Jiang, Molecular dynamics study on thermal properties of ThO2 doped with U and Pu in high-temperature range, J. Alloy. Compd. 627 (2015) 476–482. [14] W. Li, J. Ma, J. Du, G. Jiang, Molecular dynamics study of thermal conductivities of (U0.7−xPu0.3Amx)O2, J. Nucl. Mater. 480 (2016) 47–51. [15] H. Xiao, C. Long, X. Tian, H. Chen, Eﬀect of thorium addition on the thermophysical properties of uranium dioxide: atomistic simulations, Mater. Des. 96 (2016) 335–340. [16] M.J. Rahman, B. Szpunar, J.A. Szpunar, Comparison of thermomechanical properties of (UxTh1x)O2, (UxPu1x)O2 and (PuxTh1x)O2 systems, J. Nucl. Mater. 513 (2019) 8–15. [17] M.J. Rahman, B. Szpunar, J.A. Szpunar, Eﬀect of ﬁssion generated defects and porosity on thermo-mechanical properties of thorium dioxide, J. Nucl. Mater. 510 (2018) 19–26. [18] S. Plimpton, Fast parallel algorithms for short-range molecular dynamics, J. Comput. Phys. 117 (1995) 1–19. [19] M.S. Daw, M.I. Baskes, Embedded-atom method: derivation and application to impurities, surfaces, and other defects in metals, Phys. Rev. B. 29 (1984) 6443–6453.

Fig. 15. Thermal conductivities calculated by MD simulations vary with different Am content for (UxAm1−x)O2 MOX fuel (where x = 0, 0.25, 0.5, 0.75, 1.0).

elastic constants, and thermal conductivity of UO2, AmO2, and (UxAm1−x)O2 MOX fuels. The main conclusions are as follows: I. With increasing AmO2 concentration, the lattice parameters of (UxAm1-x)O2 decrease due to the smaller ionic radius of Am+4 compared to that of U+4. The lattice parameters variation with composition also obeys Vegard's law. II. The characteristic peaks of the superionic transition were observed in both the linear thermal expansion coeﬃcient and in the speciﬁc heat for temperatures above 2400 K. The melting points of the end member was deduced from the peak of the superionic transition using an empirical model. The result is consistent with experimental values of the end member of the (UxAm1−x)O2 MOX fuel. III. The elastic constants and elastic moduli were calculated as a function of temperature from 300 to 3200 K. The elastic constants and temperature coeﬃcient of the elastic moduli were found to obey Vegard's law for the diﬀerent americium concentrations. IV. The calculated values of thermal conductivities agree with the literature data. With the temperature increasing, the thermal conductivity obviously decreases. V. The parameters a and b derived from the thermal resistivity variation data of (UxAm1−x)O2 MOX fuel is consistent with the classical phonon transport model for oxide systems. Overall, this work provides a more complete reference data for the thermophysical properties of Am-containing MOX fuels.

9

Computational Materials Science 172 (2020) 109324

E. Jossou, et al.

Inorg. Nucl. Chem. Lett. 10 (1974) 459–465. [43] V. Sobolev, J. Nucl. Mater. (2009) 45–51, https://doi.org/10.1016/j.jnucmat.2009. 01.005. [44] L. Vegard, Die Konstitution der Mischkristalle und die Raumfüllung der Atome, Zeitschrift Für Phys. 5 (1921) 17–26. [45] T. Nishi, A. Itoh, K. Ichise, Y. Arai, Heat capacities and thermal conductivities of AmO2 and AmO1.5, 414 (2011) 109–113. [46] O.S. Vălu, D. Staicu, O. Beneš, R.J.M. Konings, P. Lajarge, Heat capacity, thermal conductivity and thermal diﬀusivity of uranium–americium mixed oxides, J. Alloy. Compd. 614 (2014) 144–150. [47] R.J.M. Konings, Thermochemical and thermophysical properties of curium and its oxides, J. Nucl. Mater. 298 (2001) 255–268. [48] J.J. Carbajo, G.L. Yoder, S.G. Popov, V.K. Ivanov, A review of the thermophysical properties of MOX and UO2 fuels, J. Nucl. Mater. 299 (2001) 181–198. [49] C. Thiriet, R.J.M. Konings, Chemical thermodynamic representation of Am O2–x, J. Nucl. Mater. 320 (2003) 292–298. [50] S.W. Proceedings, U. Kingdom, Advanced Reactors with Innovative Fuels, (n.d.). [51] K. Hermann, G. Thomas III, Investigations of the speciﬁc heat of solid bodies, Philos. Trans. R. Soc. London 155 (1865) 71–202. [52] J. Arborelius, K. Backman, L. Hallstadius, M. Limbäck, J. Nilsson, B. Rebensdorﬀ, G. Zhou, K. Kitano, R. Löfström, G. Rönnberg, Advanced doped UO2 pellets in LWR applications, J. Nucl. Sci. Technol. 43 (2006) 967–976. [53] C.G. Maier, K.K. Kelley, An equation for the representation of high-temperature heat content data, J. Am. Chem. Soc. 54 (1932) 3243–3246. [54] I. Atomic, E. Agency, Thermal conductivity of uranium, Dioxide (1966). [55] X.-Y. Liu, M.W.D. Cooper, K.J. McClellan, J.C. Lashley, D.D. Byler, B.D.C. Bell, R.W. Grimes, C.R. Stanek, D.A. Andersson, Molecular dynamics simulation of thermal transport in UO2 containing uranium, oxygen, and ﬁssion-product defects, Phys. Rev. Appl 6 (2016) 44015. [56] C.I. Maxwell, J. Pencer, Molecular dynamics modelling of the thermal conductivity of oﬀ-stoichiometric UO2 ± x and (UyPu1−y)O2 ± x using equilibrium molecular dynamics, Ann. Nucl. Energy 131 (2019) 317–324. [57] W. Wiesenack, Review of Halden reactor project high burnup fuel data that can be used in safety analyses, Nucl. Eng. Des. 172 (1997) 83–92. [58] M. Saoudi, D. Staicu, J. Mouris, A. Bergeron, H. Hamilton, M. Naji, D. Freis, M. Cologna, Thermal diﬀusivity and conductivity of thorium-uranium mixed oxides, J. Nucl. Mater. 500 (2018) 381–388. [59] T. Uchida, T. Arima, K. Idemitsu, Y. Inagaki, Thermal conductivities of americium dioxide and sesquioxide by molecular dynamics simulations, Comput. Mater. Sci. 45 (2009) 229–234. [60] K. Bakker, R.J.M. Konings, On the thermal conductivity of inert-matrix fuels containing americium oxide, J. Nucl. Mater. 254 (1998) 129–134. [61] S. Adachi, J. App. Phys. 102 (2007) 063502. [62] J. Callaway, H.C. von Baeyer, Eﬀect of point imperfections on lattice thermal conductivity, Phys. Rev. 120 (1960) 1149–1154. [63] P.S. Ghosh, A. Arya, N. Kuganathan, R.W. Grimes, Thermal and diﬀusional properties of (Th, Np) O2 and (U, Np) O2 mixed oxides, J. Nucl. Mater. 521 (2019) 89–98.

[20] M.S. Daw, M.I. Baskes, Semiempirical, quantum mechanical calculation of hydrogen embrittlement in metals, Phys. Rev. Lett. 50 (1983) 1285–1288. [21] L. Malakkal, A. Prasad, E. Jossou, J. Ranasinghe, B. Szpunar, L. Bichler, J. Szpunar, Thermal conductivity of bulk and porous ThO2: atomistic and experimental study, J. Alloy. Compd. 798 (2019) 507–516. [22] J. von Pezold, A. Dick, M. Friák, J. Neugebauer, Generation and performance of special quasirandom structures for studying the elastic properties of random alloys: application to Al-Ti, Phys. Rev. B 81 (2010) 94203. [23] W. Voigt, B.G. Teubner (Ed.), Lehrbuch der Kriystallphysik (1928), p. 962 (Leipzig, Germany). [24] A. Reuss, Z. Angew. Math. Mech. e J. Appl. Math. Mech./Zeitschrift fr Angewandte Math. und Mech. 9 (1929) 49. [25] R. Hill, Proc. Phys. Soc. Sect. A 65 (1952) 349. [26] R. Kubo, Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems, J. Phys. Soc. Japan. 12 (1957) 570–586. [27] M.S. Green, M.R. Processes, Markoﬀ Random Processes and the Statistical Mechanics of Time-Dependent Phenomena, 1281 (2000). [28] Z. Liang, A. Jain, A.J.H. Mcgaughey, P. Keblinski, Molecular simulations and lattice dynamics determination of Stillinger-Weber GaN thermal conductivity, 125104 (2015) 1–5. [29] M. Kaviany, I. Introduction, Phonon Transport in Molecular Dynamics Simulations : Formulation and Thermal Conductivity Prediction, 39 (2006) 169–255. [30] J.K. Fink, Thermophysical properties of uranium dioxide, J. Nucl. Mater. 279 (2000) 1–18. [31] Thermophysical properties of materials for water cooled reactors, (1997). [32] P. Martin, D.J. Cooke, R. Cywinski, A molecular dynamics study of the thermal properties of thorium oxide, J. Appl. Phys. 112 (2012) 73507. [33] C.O.T. Galvin, M.W.D. Cooper, M.J.D. Rushton, R.W. Grimes, Thermophysical properties and oxygen transport in (Thx, Pu1−x)O2, Sci. Rep. 6 (2016) 36024. [34] M.W.D. Cooper, S.T. Murphy, C.M. Paul, M.J.D. Rushton, R.W. Grimes, Thermophysical and anion diﬀusion properties of (Ux,Th1 − x)O2 Proc. R. Soc. A 470: 20140427. [35] H. Balboa, L. Van Brutzel, A. Chartier, Y. Le, Assessment of empirical potential for MOX nuclear fuels and thermomechanical properties, J. Nucl. Mater. 495 (2017) 67–77. [36] M. Pukari, M. Takano, T. Nishi, Sintering and characterization of (Pu, Zr)N, J. Nucl. Mater. 444 (2014) 421–427. [37] F. Grønvold, J. Inorg. Nucl. Chem. 1 (1955) 357. [38] J.A. Christensen, J. Am. Ceram. Sot. 46 (1963) 607. [39] D. Prieur, R.C. Belin, D. Manara, D. Staicu, J. Richaud, J. Vigier, A.C. Scheinost, J. Somers, P. Martin, Linear thermal expansion, thermal diﬀusivity and melting temperature of Am-MOX and Np-MOX, J. Alloy. Compd. 637 (2015) 326–331. [40] J.P. Hiernaut, G.J. Hyland, C. Ronchi, Premelting transition in uranium dioxide, Int. J. Thermophys. 14 (1993) 259–283. [41] D.G. Martin, The thermal expansion of solid UO2 and (U, Pu) mixed oxides — a review and recommendations, J. Nucl. Mater. 152 (1988) 94–101. [42] J.A. Fahey, R.P. Turcotte, T.D. Chikalla, Thermal expansion of the actinide dioxides,

10