The MD simulation of thermal properties of plutonium dioxide

Physica B 407 (2012) 4595–4599

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The MD simulation of thermal properties of plutonium dioxide Wan Mingjie a, Zhang Li a, Du Jiguang a, Huang Duohui a,b, Wang Lili c, Jiang Gang a,n a b c

Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, China Academic Affairs Office, YiBin University, YiBin 644007, China Institute of Calculation, CAEP, Mianyang 621900, China

a r t i c l e i n f o

abstract

Article history: Received 26 July 2012 Received in revised form 7 August 2012 Accepted 8 August 2012 Available online 16 August 2012

The thermodynamic properties of PuO2 have been investigated between 300 and 3000 K by molecular dynamics (MD) simulation with empirical interaction potential. The properties include melting point, lattice parameter variation, enthalpy and heat capacity. The melting point of two-phase simulation (TPS) is in agreement with the experimental value, and it gives a much lower value than one-phase simulation (OPS). The lattice parameter and heat capacity at high temperatures are 1 expressed as aðTÞ ¼ 5:38178 þ 4:38  105 T þ 6:5525  109 T 2 þ 0:9362  1012 T 3 and C P ðKJUmol U 6 1 2 474:5=T 2 474:5=T =ðT ðe 1Þ Þ þ 9:337  10 T, respectively. True linear thermal expansion K Þ ¼ 18648:8e coefficient (TLTEC) a is about 8.89  10  6 K  1 at 300 K. Our simulation results are in good agreement with experimental and other theoretical data. & 2012 Elsevier B.V. All rights reserved.

Keywords: Plutonium dioxide Melting point Linear thermal expansion coefficient (TLTEC) Heat capacity

1. Introduction In the energy resources, uranium and plutonium have the most important role to play in the future. The actinide oxides are the dominant fuel of nuclear reactors [1], and plutonium dioxide (PuO2) is a key compound of MOX fuel. The melting point, thermal expansion and heat capacity of PuO2 for high temperature are of interest to understand and predict the reactor fuel performance. It provides great challenges for theoretical and experimental studies of PuO2. Due to the high temperature condition and radioactive toxicities, it is difficult to experimentally evaluate the melting point and thermal properties of actinide dioxide with accuracy. Take the melting point of PuO2 for example; since the 1960s, several investigators have reported the melting point of PuO2. The melting point ranged from 2511 to 3017 K [2–9]. Adamson et al. [2] and Gue´neau et al. [3] obtained values of 2660 and 2701735 K, respectively. In 2010, Bruycker et al. [4] reassessed the melting point of PuO2. In the work, the peak laser power varied between 630 and 720 W, and the value is evaluated to be 3017728 K. It is at least 300 K higher than other experimental data. MD simulations [10–14] are performed for evaluating the thermal properties of PuO2. However, most of the previous studies were performed in lower temperatures limited to 1500 K. The thermodynamic properties are also important at or near melting point. In this paper, a major goal of this report is to

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Corresponding author. Tel./fax: þ 86 28 85408810. E-mail address: [email protected] (J. Gang).

0921-4526/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physb.2012.08.010

investigate the melting point, thermal expansion, enthalpy and heat capacity of PuO2 by MD simulation at temperatures varied between 300 and 3000 K.

2. Interatomic potential functions In the present work, all calculations have been performed by using the Born–Mayer–Huggins (BMH) potential with the partially ionic model (PIM) [11]. The PIM potential function is written as [11]   zz a þa r c Fðrij Þ ¼ i j þ f 0 ðbi þ bj Þexp i j ij  6ij ð1Þ r ij bi þbj r ij where rij is the distance between i and j atoms, f0 is the adjustable parameter which equals 4.186 J A˚  1 mol  1, ai, bi and cij were determined based on experimental data, and zi is the effective charge of type i. It has been used so far to evaluate the thermodynamic properties of actinide oxides. The ionic bounding of the PuO2 system is assumed as 67.5%. The first on the right side of Eq. (1) describes the long-range Coulomb interaction; the Coulombic interaction energy is considered by Ewald’s simulation. Other terms produce the short-range interactions— the second one is the repulsive potential between ionic cores and the last one is the attractive part of the Van der Waals interaction. Potential parameters for Pu–Pu, Pu–O and O–O pairs are obtained from Ref. [11], and the potential energy as a function of the distance between ions (including Pu–Pu, Pu–O and O–O pairs) was described in Fig. 1.

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Fig. 1. Potential energies for the interaction atoms. (a) Pu–Pu; (b) Pu–O; and (c) O–O. Potential functions were proposed by Arima et al.

3. Computational detail The calculations below used the molecular dynamics (MD) technique and the LAMMPS program [15] was employed for all simulations. The program was developed at Sandia National Laboratories. Two types of MD simulation were tested to evaluate the melting point of PuO2; one was one-phase simulation (OPS) and the other one was two-phase simulation (TPS). The simulations were performed using a Nose/Hoover temperature thermostat and Nose/Hoover [16,17] pressure barostat with a time step of 1 fs. These techniques in the simulations can be performed under a normal pressure and temperature (NPT) ensemble. Total simulation times varied between 100 ps and 500 ps. The calculations employed 3D periodic boundaries. For OPS, perfect PuO2 crystalline was simulated using a 6  6  6 supercell of fluorite structure, and in the initial state solid was prepared under 300 K; for TPS, solid and liquid phases coexisted in the initial state, listed in Fig. 2(a), the two phases were equilibrated at 300 and 4500 K, and the ˚ For all calculations, we varied distance between two boxes was 3 A. the temperature at intervals of 20 K near the melting point, and the error of the melting point is estimated to be 710 K.

4. Result and discussion 4.1. Melting point Melting point calculations of OPS and TPS of PuO2 in the NPT ensemble are performed using Arima et al. potentials. To determine the melting point, we simulated the PuO2 system at

different temperatures and observed the change of the density of PuO2 crystalline. It is well known that the saltation of density has occurred at melting point. Densities of PuO2 as a function of the temperatures of OPS and TPS are shown in Figs. 3 and 4, respectively. In both OPS and TPS, it is clearly seen that all calculated densities decrease gradually with increase in the temperature, and drop steeply at melting point. Our OPS gives a value of 4450710 K; the value is much higher than the experimental values [2–9]. Then, we performed TPS according to the following conditions: the system was divided into two boxes, one was equilibrated at 4500 K which was in a liquid state, and the other was in a solid state under 300 K. Each cell size was 6  6  6 unit cells. This situation was different from the experimental one. From Fig. 4, we can see that the melting point of TPS is 3320710 K; the melting point is much lower than that obtained from OPS. In our study, the value is 620 K higher than that in Adamson et al. [2], 660 K higher than that in Gue´neau et al. [3] and 300 K higher than that in Bruycker et al. [4]. It is comparable to data of Bruycker. In 2009, Arima et al. [18] evaluated the melting point of UO2 using this potential function, and the value was 3675725 K, which was 550 K higher than INSC recommended data. So, the melting point is overvalued using PIM potential functions. On the other hand, for TPS with 6  6  6  2 unit cells, the tendency of structure of PuO2 crystalline due to increase in temperature is described in Fig. 2. Fig. 2(a) is the initial structure, and Fig. 2(b)–(d) are relaxed 1000 ps at 2500 , 3300 and 3320 K, respectively. The radial distribution function (RDF) of Pu–O pair is shown in Fig. 5, where we can clearly see that g(r) is almost the same between 3320 and 4000 K, and the crystalline melted at 3320 K. We also obtain the melting point of PuO2 as 3320710 K from RDF.

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Fig. 4. Relationship between PuO2 density and temperature for TPS. MD calculations were performed for 6  6  6  2 unit cell.

Fig. 2. Configuration of supercell of TPS with 6  6  6  2 unit cell. (a) In the initial structure, left and right sides in the supercell were equilibrated at 300 and 4500 K, respectively; (b) relaxation after 1000 ps at 2500 K; (c) relaxation after 1000 ps at 3300 K; (d) relaxation after 1000 ps at 3320 K. These pictures were drawn by a 3D visualization program: VMD.

Fig. 5. Radial distribution function (RDF) of Pu–O pairs at 2500, 3300, 3320 and 4000 K.

4.2. Thermodynamic properties The thermodynamic properties of solid PuO2 were performed in NPT ensemble using Arima’s potential. In this paper, we have investigated the thermal expansion, enthalpy and heat capacity of plutonium dioxide.

Fig. 3. Relationship between density and temperature of PuO2 for OPS. MD calculations were performed for 6  6  6 unit cell.

Comparing the two methods, the initial supercell of TPS contains the disorder phase in liquid state which plays a role as a trigger for melting. For this reason, the result of TPS is much better than that of OPS.

4.2.1. Thermal expansion The lattice constants for PuO2 estimated by MD simulation as a function of temperature between 300 and 3000 K are shown in Fig. 6. A lot of potentials have been employed to estimate the lattice constant. We make a comparison between the results of our study and those of the previous ones. From Fig. 6, it can be seen that the calculated results are in good agreement with the experimental data and other theoretical data at lower temperatures. Our results are in quite satisfactory agreement with the data by Martin [19] up to 2000 K. It underestimates Martin’s results when the temperature is higher than 2000 K. The value estimated by Kurosaki’s potential [12] overestimates Martin’s and Yamashitai et al.’s [14] results below 2000 K.

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4.2.2. Enthalpy and heat capacity Then, we calculate the enthalpy variation with temperature at constant pressure. In this paper, the enthalpy data are fitted using a nonlinear least-squares method by the equation [24] H0T H0300 ðKJUmol

1

h i Þ ¼ C 1 y ðey=T 1Þ1 ðey=300 1Þ1 þ C 2 ðT 2 3002 Þ

ð4Þ

where, T is in K, and y is Einstein temperature. The fitted parameters are listed in Table 3, and the curve of enthalpy variation with temperature is shown in Fig. 7. At lower temperatures, our simulation results of enthalpy variation are in agreement with Fink’s [24]and Oetting and Bixby’s [25], and at higher temperatures, the results are smaller than Fink’s [24] and Ogard’s [26]. We give the expression of enthalpy variation of PuO2 at high temperature as H0T H0300 ðKJUmol

1

Fig. 6. Comparison of the present simulated and the previous values of lattice constant versus temperature. The temperature varied from 300 to 3000 K.

Þ ¼ 39:3ðe474:5=T 1Þ1 10:59 þ 4:6685  106 T 2

The heat capacity Cp is given by   @H Cp ¼ @T P

ð5Þ

It can be calculated by differentiation of Eqs. (4) and (5). These yield Cp ¼

C 1 C 22 eC 2 =T 2

T ðeC 2 =T 1Þ2

þ2C 3 T

ð6Þ

The heat capacity equals 70.3 K  1 at 300 K, which is a little bigger than Fink’s 66.24 K  1. Cp of PuO2 is shown in Fig. 8. We can see that the calculated heat capacity variations are close to the analytical data by Fink [24]. The analytical expression of heat capacity is given by 1

C p ðKJUmol

UK1 Þ ¼ 18648:8e474:5=T =ðT 2 ðe474:5=T 1Þ2 Þ þ 9:337  106 T

5. Conclusions

Fig. 7. Enthalpy of PuO2 as a function of temperature. The temperature varied from 300 to 3000 K.

True linear thermal expansion coefficient (TLTEC) is calculated from 300 to 3000 K, and it can be expressed as   1 dLðTÞ aðTÞ ¼ ð2Þ L300 dT P where, L300 is the lattice constant at 300 K and L(T) is the lattice constant at temperature T. The variation in TLTEC with temperature is shown in Fig. 7. Measured lattice parameters are fitted as a function of temperature, which is given by [14] 2

LðTÞ ¼ a0 þ a1 T þa2 T þ a3 T

3

ð3Þ

The fitted L(T) results of our paper and Martin are listed in Table 1, and the calculated a(T) are listed in Table 2. The lattice parameter of PuO2 at high temperature can be described as aðTÞ ¼ 5:38178þ 4:38  105 T þ 6:5525  109 T 2 þ 0:9362  1012 T 3

From Table 2, we can see that our TLTEC values are 8.89 and 11.78  10  6 K  1 at 300 and 1200 K, respectively. The results are in agreement with experimental data [20–23] and other theoretical values [2,14].

MD simulations have been employed to evaluate the melting point and some properties of PuO2 using the interatomic potential developed by Amira. For evaluating the melting point, two types of simulation are applied in the present work, which are one-phase simulation (OPS) and two-phase simulation (TPS). The results show that TPS gives much lower melting point than OPS, and the melting point of TPS is in agreement with experimental value, which results in the disordered liquid phase helping to melt the PuO2 crystal. In addition, for thermodynamic properties, thermal expansions and heat capacity of PuO2 are investigated between 300 and 3000 K.

(1) The lattice parameter is expressed as aðTÞ ¼ 5:38178 þ 4:38  105 T þ 6:5525  109 T 2 þ 0:9362  1012 T 3

True linear thermal expansion coefficient (TLTEC) is about 9.2  10  6 K  1 in our study at 300 K, which is a little higher than experimental data.

(2) The heat capacity is expressed as Cp ðKJ  mol

1

 K1 Þ ¼ 18648:8e474:5=T =ðT 2 ðe474:5=T 1Þ2 Þ þ 9:337  106 T

The calculated heat capacity variations are close to the analytical data by Fink.

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Table 1 The fitted parameters of lattice constants for plutonium dioxide.

This work Yamashita et al. [14] a

˚ a0 (A)

a1 (10  5) (A˚  K  1)

a2 (10  9) (A˚  K  2)

a3 (10  12) (A˚  K  3)

˚ L300 (A)

5.38178 5.38147

4.38 4.452

6.5525 7.184

0.9362 0.1995

5.3955 5.3951a

The values were obtained at 293 K.

Table 2 True linear expansion coefficients at 300 and 1200 K for plutonium dioxide.

300 K (  10  6) 1200 K (  10  6) a b c

This worka

Yamashita et al.b[14]

Marplesa[22]

Taylorb[21]

TPRCb[20]

Fahey et al.b[23]

Chu et al.c[10]

8.89 11.78

9.04 11.61

8.4

8.84 12.27

8.1 12.0

8.4 12.14

8.3–8.5

The values were obtained at 300 K. The values were obtained at 293 K. The value was obtained at 298 K.

Grant no. 426030106: 12-YY03-7. Our thanks are also due to Dr. Tian Xiaofeng for his help with using LAMMPS.

Table 3 The fitted parameters of heat capacity for plutonium dioxide.

This paper Fink [24]

C1 (10  2) (KJ  mol  1  K  1)

y (102) (K)

C2(10  6) (KJ  mol  1  K  2)

8.283 8.7394

4.7449 5.8741

4.6685 3.9780

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[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

Fig. 8. Heat capacity variations of solid PuO2.

Acknowledgments This work is supported by the Equipment Pre-Research Project of the National of China Academic of Engineering Physics under

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