Accepted Manuscript

Molecular dynamical study of physical properties of (U0.75Pu0.25)O2-x

J.J. Ma, J.J. Zheng, M.J. Wan, J.G. Du, J.W. Yang, G. Jiang

PII: S0022-3115(14)00275-X

DOI: http://dx.doi.org/10.1016/j.jnucmat.2014.05.008

Reference: NUMA 48133

To appear in: Journal of Nuclear Materials

Received Date: 29 October 2013

Accepted Date: 3 May 2014

Please cite this article as: J.J. Ma, J.J. Zheng, M.J. Wan, J.G. Du, J.W. Yang, G. Jiang, Molecular dynamical study

of physical properties of (U0.75Pu0.25)O2-x , Journal of Nuclear Materials (2014), doi: http://dx.doi.org/10.1016/

j.jnucmat.2014.05.008

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Molecular dynamical study of physical properties of(U0.75Pu0.25)O2−x

J.J.Maa, J.J.Zhengb, M.J.Wana, J.G.Duc, J.W.Yanga, G.Jianga,∗

aInstitute of Atomic and Molecular Physics Sichuan University, Chengdu 610065, ChinabInstitude of Theoretical Physics and Department of Physics, Shanxi University, Taiyuan

030006, ChinacCollege of Physical Science and Technology, Sichuan University, Chengdu 610064, China

Abstract

Physical properties of mixed-oxide fuel (U0.75Pu0.25)O2−x (x =0.0, 0.02, 0.06,

0.1, 0.15, 0.2, 0.25) have been investigated by the molecular dynamic (MD)

simulation in the temperature range 300-3000K. The lattice parameter, lin-

ear thermal expansion coefficient, compressibility, bulk modulus and thermal

conductivity are systematically investigated and analyzed by comparison with

experiments and previous calculations. The calculated results of physical prop-

erty are in good agreement with the experimental values and literature data.

The oxygen vacancies have a significant effect on thermal properties of MOX.

As oxygen vacancies increase, the bulk modulus gradually tend to be linear rela-

tionship with temperature and the thermal conductivity decreases very clearly,

also, the temperature dependence weakens. In addition, we found that the in-

fluence of plutonium concentration for thermal conductivity is so small that it

can be ignored when oxygen vacancies are existent.

1. Introduction

Uranium-plutonium mixed oxide fuel (MOX) is regarded as an important

candidate in the energy resources due to the effective utilization rate of nuclear

∗Corresponding author. Tel./fax: +86 28 85408810.Email address: gjiang@scu.edu.cn (G.Jiang)

Preprint submitted to Journal of Nuclear Materials May 12, 2014

fuel and significant function to protect the environment in the process of dis-

posing nuclear waste. At the moment, the MOX-fuel have been loaded into5

the commercial light water reactor (LWRs. (U, Pu)O2 contained high Pu con-

centrations (15 − 30%) will be widely used in the fast breeder reactors (FBR)

or transmutation rectors in future [1, 2, 3, 4, 5, 6]. In order to operate and

develop the nuclear reactor system safely, it is crucial to forecast the physical

properties of MOX-fuel [7], especially, the understanding of thermal expansion10

and thermal conductivity which is avail to design installation of nuclear fuel.

These properties of MOX have been studied extensively in the past, such as

thermal conductivity.

Gibby et al. [8] comprehensively investigated the thermal conductivity of

(U,Pu)O2 containing up to 30% PuO2 and established the basic approaches15

to measure the characterizations of nuclear fuel. It was found that the ther-

mal conductivity of (U,Pu)O2 had a small but systematical decrease with in-

creasing of Pu content. Philipponneau [4] developed an equation for FR-MOX

fuel by reviewing the previous results and studying the thermal conductivity

of (U,Pu)O2−x mixed oxide fuel. He recommended to neglect the effect of the20

amount of plutonium on thermal conductivity in high Pu concentration range

and observed that the decrease of conductivity is due to a decrease of the O/M

(i.e. O/(U+Pu)) ratio. Duriez et al. [2] preformed an experimental measure-

ment of the thermal conductivity of low Pu content (U1−yPuy)O2−x mixed

oxides in the ranges 0.03 ≤y≤ 0.15 and 1.95 ≤x≤ 2.00. Through the compari-25

son with high plutonium content FBR fuel, they proposed that the effect of the

O/M radio should be taken into account when predicting thermal conductivity

of mixed oxide fuels.

Due to fluctuations of O/M ratio in experiments, these results are not ac-

curate to describe the effect of oxygen-to-metal radio. With the development30

of computer simulation techniques, molecular dynamics (MD) can be a useful

technique to obtain the information of oxide fuels in recent years. The phys-

ical properties of UO2 [9, 10, 11], PuO2 [12], UO2−x [13], PuO2−x [7, 14],

and (U,Pu)O2 [15] have been evaluated over a wide temperature, above room

2

temperature, the calculated results are almost in agree with the experimental35

values. Arima et al. [1] have performed MD simulations of hypostoichiometric

MOX-fuel U0.8Pu0.2O2−x (x = 0.0 − 0.06) solid solutions in the temperature

range from 300 to 2000K. Their MD simulation results indicate that the oxy-

gen deficiency x has a larger effect on the thermal conductivity. However, their

simulation just in a small temperature (300 − 2000K) and oxygen deficiency40

(0.0− 0.06) range, there is still no reliable conclusion about the effect of oxygen

vacancy on the physical properties of MOX. In present work, the MD simula-

tion of mixed oxide fuel, (U0.75Pu0.25)O2−x (x=0.0, 0.02,0.06,0.1,0.15,0.20,0.25),

has been performed to evaluate the lattice parameter, linear thermal expan-

sion, compressibility, bulk modulus and thermal conductivity in temperature45

range 300− 3000K. In order to ensure the influence of high Pu content on ther-

mal conductivity of MOX, we further investigated the thermal conductivity of

U1−xPuxO2 (x = 0.15, 0.2, 025, 0.3) and compared the results with UO2 which

had been system systematically reported by Watanabe et al.[16, 17].

This paper is organized as follows: In Sec. II and III, we present a detail50

description of the MD simulation details and potential function, respectively.

The results are presented in Sec. IV. Here, we show our results and compare

with the available experimental and theoretical studies. Finally, we give our

main conclusions of physical properties of (U0.75Pu0.25)O2−x in Sec. V.

2. Simulation details55

The defects of U0.75Pu0.25O2 may be existed in oxygen vacancy or metal

clearance. To investigate physical properties between different ratios of the

defects, we selected defects in oxygen vacancy as initial model, so that the

same structure was kept in the case of different ratio. The MD calculation

for U0.75Pu0.25O2 was performed for a system of 6 × 6 × 6 supercell which in-60

cludes 2592 ions (864 cations and 1728 anions) as initial structure arranged in

a CaF2 type crystal structure. Fig. 1 presents the structure of unit cell of

U0.75Pu0.25O2−x. Based on the prefect structure of CaF2, the (U4+ ,Pu4+)

3

Figure 1: The unit cell of (U0.75Pu0.25)O2−x.

and O2− ions were arranged at each regular site of the CaF2 structure. One

atom of oxygen was taken out as deficiency which is showed by hollow circular65

in the picture. For the other MOX compositions included oxygen deficiency,

we take U0.75Pu0.25O2−x where x = 0.02 as example, first we remove small

amounts of oxygen ions as deficiency randomly in the initial structure, then

select the two +4 ions replaced with Pu3+ or U3+ near each oxygen vacancy

to preserve charge neutrality. The structure for each of the other composi-70

tion (x = 0.06, 0.1, 0.15, 0.2, 0.25) can be obtained in the same way, the only

difference is the number of oxygen vacancy which is decided by the O/M ra-

tio. In the present study, calculations are well performed by the Large-scale

Atomic/Molecular Massively Parallel Simulator (LAMMPS) program [18] using

the molecular dynamics (MD) technique.75

The calculations were carried out in the temperature range from 300 to

3000K, and in the pressure range from 0.1MPa to 1.5GPa. They were con-

trolled by the Nose/Hoover temperature thermostat and Nose/Hoover [19, 20]

4

pressure barostat, respectively. These techniques in the MD simulation were

performed under standard constant pressure-temperature (NPT) and constant80

volume-temperature (NVT) ensembles. Also, the calculations employed 3D pe-

riodic boundaries and performed 10000 steps equilibration run at expected tem-

perature and pressure. Although the number of steps is small, the equilibrium

of system is achieved as judged from the changes in the temperature (2.5%),

internal energy (0.05%), and volume (0.3%).85

For the simulation of thermal conductivity, direct method and the Green-

Kubo method are applied most commonly. Schelling et al [21] systematically

explored the strengths and weaknesses of two methods through a model of crys-

talline silicon. The result of comparison is that both methods were in reasonable

agreement with the experimental value and proved that either method can be90

applied to compute bulk thermal conductivity in perfect crystalline solids. In

addition, they gave an analysis of finite-size effects, which indicated that the

effects are much more severe in the direct method due to the presence of real

interfaces at heat source and sink. Based on the findings of their study, the

Green-Kubo approach is adopted for simulation of thermal conductivity in the95

present study. In order to sufficiently converge the current-current autocorre-

lation function, the program needs very long simulation times (5 × 105 steps).

During the simulation, the supercell shape was kept to cubic structure even for

the system including oxygen defect.

3. Potential functions100

In this work, we applied Born-Mayer-Huggins (BMH) potential with the

partially ionic model (PIM) to each ion pair in simulated crystals. This potential

function is given by [10]

UPIM =zizje

2

rij+ f0(bi + bj)exp

ai + aj − rij

ai + aj− cicj

r2ij

. (1)

Where rij is the distance between i and j atoms, f0 is the adjustable para-

meter which equals to 4.1860, ai , bi and ci were determined by experimental105

5

Table 1: Potential parameters for the BMH type with PIM.

Ion Zi ai bi ci

(67.5%) (A) (A) (eV0.5A3) Sources

O2− -1.35 1.847 0.166 1294 Ref.[22, 23]

U4+ 2.70 1.318 0.0306 0.0 Ref.[10]

Pu4+ 2.70 1.272 0.0325 0.0 Ref.[10]

U3+ 2.025 1.318 0.0306 0.0 Ref.[24]

Pu3+ 2.025 1.2217 0.0128 0.0 Ref.[1]

data. zi and zj are the effective partial electronic charges on the ith and jth

ions, and the ionic bonding of 67.5% is assumed for (U0.75Pu0.25)O2−x system.

The first term on the right side of Eq. (1) represents long-range Coulomb inter-

action, and the Coulomb interaction energy is considered by Ewalds simulation.

Other terms represent short-range interactions: the second term is the repulsive110

potential between ionic cores; the last one which is called Morse-type potential

[25] is the attractive part of the Van der Waals interaction. The parameters (a,

b and c) are listed in Table 1.

In present work, We use Pu3+ potential for smaller value of x (i.e. x =115

0.02, 0.06, 0.1), and added U3+ ions to (U0.75Pu 0.25)O2−x system when x =

0.15, 0.2, 0.25. In order to investigate the influence of the ratios of trivalent

ions (U3+/ Pu3+) on the simulation results, we compare the lattice constants

at x = 0.02 and x = 0.25 under the different U3+/Pu3+ ratios. The lat-

tice parameters as a function of temperature are shown in Fig. 2, in which120

symbols represent the MD results and lines are the data listed in literature.

It is also confirmed that changes in lattice constants (0.08%) are extremely

small under different U3+/Pu3+ ratio. The results demonstrate that the influ-

ence of the ratios (U3+/ Pu3+) on the lattice constants of (U0.75Pu0.25)O2−x

(x = 0.02, 0.06, 0.10, 0.15, 0.20.0.25) can be ignored.125

6

Figure 2: The lattice constants of (U0.75Pu0.25)O2−x (x = 0.02, 0.25) as a function of tem-

perature under different U3+/Pu3+ ratio, along with Arima’s [1] results of (U1−yPuy)O2(y =

0.2, 0.3).

7

Figure 3: Lattice constants calculated by MD simulations as a function of temperature for

(U0.75Pu0.25)O2−x(x =0.0, 0.02, 0.06, 0.1, 0.15, 0.2, 0.25).

4. Results and discussion

4.1. Lattice parameters

To our knowledge, the temperature and oxygen deficiency ranges are quite

limited due to experimental conditions [8, 2, 26], but the high-temperature and

high oxygen deficiency lattice constants can be obtained from the MD calcula-130

tion. The variation of the lattice parameters of U0.75Pu0.25O2−x(x =0.0, 0.02,

0.06, 0.1, 0.15, 0.2, 0.25) with temperature calculated by the MD method are

shown in Fig. 3, along with Arima’s [1] theoretical results for U0.8Pu0.2O2−x

(x = 0.02 and 0.06), as well as Kuroski’s [15] result for U0.8Pu0.2O2. The

same trend exist among their results and the MD simulated lattice parameters135

of U0.75Pu0.25O2 in the temperature range from 300 to 2300K. The calculated

lattice constants of U0.75Pu0.25O2−x(x = 0.0, 0.02, 0.06) agree well with both

Arima and Kuroski data of U0.8Pu0.2O2−x(x = 0.0, 0.02, 0.06), which indicates

that the change (5%) of Pu concentration rarely affect lattice parameters of

8

MOX. Fig. 3 shows an increase in lattice constant of U0.75Pu0.25O2−x with140

increasing both oxygen vacancy x and temperature. Unfortunately, no avail-

able experimental or theoretical data to compare with our results in the high-

temperature and high oxygen deficiency conditions. The phenomenon of lattice

expansion is mainly cased by the presence of vacancies, the larger size of the

+3 ions, and the reduced strength of Coulombic interactions arising from the145

replacement of +4 ions by +3 ions, which had been reported by Watanabe [17]

in their previous work.

4.2. Linear thermal expansion coefficient

The linear thermal expansion coefficient (αlin) of (U0.75Pu0.25) O2−x can

be evaluated from the variation of the lattice parameters with temperature as150

follow:

αlin = 1L(T0)

(∂(L)∂T )

P, (2)

where L could be either unit cell length or the MD cell length. In the case

of (U0.75Pu0.25)O2−x, there is little experimental data for the linear thermal

expansion coefficient.

The averaged thermal expansion coefficients (temperature range 300-1270K)155

of present MD calculation when x=0.02 and 0.06 are plotted in Fig. 4, and

compared with previous determinations of (U1−yPuy)O2−x(y ∼ 19.1− 25%) by

Leblanc et al. [27], Lorenzelli et al. [28], Roth et al. [29], Gibby et al.[30] and

Arima et al. [1] The results are in good agreement with the literature data, and

consistent with Arima’s findings surprisingly.160

The variation of linear thermal expansion coefficient with temperature is

shown in Fig. 5, along with the results of (U1−yPuy)O2 (y = 0.2, 0.25) from

Martin [31], Skavdahl et al [32]. and Gibby [30]. The tendency is consistent

with previous research results, which the linear thermal expansion coefficient

increases with increasing temperature. It is a known fact that the interatomic165

spacing between the atoms is a function of temperature, and the atoms will

9

Figure 4: Calculated results of thermal expansion coefficient for (U0.75Pu0.25)O2−x as a

function of O/M ratio.

10

Figure 5: Calculated results of linear thermal expansion coefficient for (U0.75Pu0.25)O2−x as

a function of temperature.

vibrate and move further apart resulting from the increase of temperature. The

linear thermal expansion coefficients of (U0.75Pu0.25)O2−x increase with tem-

perature. The impact of the concentration of oxygen vacancy is negligible when

the temperature falls below 1300K, but that became obvious with the increas-170

ing temperature, especially x = 0.25. These results indicate that the values of

oxygen vacancy have a larger effect on the linear thermal expansion coefficient

in high temperature range.

4.3. Compressibility and Bulk modulus

The isothermal compressibility can be expressed as follows:175

β = − 1V (P0)

(∂(V )∂P )T , (3)

where P and V stand for pressure and volume (lattice), respectively. At the all

temperature range from 300 to 3000K, the compressibility was evaluated by the

11

Figure 6: Compressibility of (U0.75Pu0.25)O2−x as a function of temperature.

change of lattice volume with varying pressure from 0.1MPa to 1.5GPa under the

NPT ensemble. The variation of isothermal compressibility of (U0.75Pu0.25)O2−x

with temperature, along with Yamada’s [9] MD result of UO2 are shown in Fig.180

6. Yamada’s result show rapid increase beyond 2200K, where have a Bredig

transition due to the instability in oxygen sublattice. However, unstable oxy-

gen sublattice does not guarantee a drastic rise [33], because cation sublattice

might well remain stable above 2000K. We notice that our isothermal compress-

ibility of (U0.75Pu0.25)O2−x slightly increases with increasing temperature. In185

addition, the results suggest that the compressibility increases with x values of

oxygen deficiency, but the rate of increase with temperature is almost same at

different values of x.

The bulk modulus is defined as the reciprocal of the compressibility. In

this work, we present comparisons between our MD simulated values of bulk190

modulus of (U0.75Pu0.25)O2−x and the Arima’s results of (U0.8Pu0.2)O2−x (x =

0.02, 0.06) which are shown in Fig. 7. Although in the case of slightly different

12

Figure 7: Calculated results of bulk modulus for (U0.75Pu0.25) O2−x as a function of O/M

ratio, together with literature data.

13

ratios of Pu content, our calculated results for x = 0.02 and 0.06 are in excellent

agreement with Arima’s theoretical results. The phenomenon demonstrates

that the effect of plutonium concentration appears to be relatively small to the195

bulk modulus. The bulk modulus decreases with increasing temperatures and x

values of oxygen deficiency. When x = 0.02, 0.06, the temperature dependence of

the bulk modulus can be treated in two temperature regions: a gradual decrease

at low temperatures (below 1000K) and a rapid decrease at high temperatures.

However, when oxygen deficiency x > 0.2 , the bulk modulus decreases linearly200

with increasing temperature.

4.4. Thermal conductivity

The thermal conductivity was calculated by the Green-Kubo approach [34]

in equilibrium Molecular Dynamics (EMD) system. The Green-Kubo formulas

which relate the ensemble average of the auto-correlation function of the heat205

flux is usually given by:

κ =V

3KBT 2

∫ ∞

0

J(t)J(0)dt, (4)

where KB is the Boltzmanns constant, V is the simulated cell volume ,T is

the absolute temperature, and J(t) is the heat flux for the simulated cell [35].

The heat flux can be calculated from the fluctuations of per-atom potential

and kinetic energies, and per-atom stress tensor in a steady-state equilibrated

simulation. It is usually given by:

J =1V

∑i

eiVi +12

∑i<j

(fij (Vi + Vj))Xij

, (5)

where Vi is the velocity of atom i , fij andXij are the atomic force and distance

between i andj , respectively. ei is the instantaneous excess energy of atom i

which is given by:

ei =

12miv

2i +

12

∑i=j

u (rij)

− eav, (6)

14

Figure 8: Thermal conductivity for (U0.75Pu0.25)O2−x as a function of temperature.

where eav is the average energy of the system, mi is the mass of atom i, u is

the atomic potential between atom i and j , respectively.

In this work, we evaluated the thermal conductivity of (U0.75Pu0.25) O2−x210

(x = 0.0, 0.02, 0.06, 0.1, 0.15, 0.2, 0.25) by MD calculation in the temperature

range between 300K and 3000K. The thermal conductivity as a function of tem-

Table 2: Plutonium concentration dependence of thermal conductivitya of U1−yPuyO2.

Temperature UO2 U1−yPuyO2

(K) Busker Yamada y = 0.15 y = 0.2 y = 0.25 y = 0.3500 7.52 9.25 9.340 9.282 9.671 9.3781000 3.33 3.42 4.239 3.561 3.457 3.4891500 2.28 2.45 1.982 1.780 2.147 1.9362000 1.31 1.40 1.298 1.361 1.544 1.501

aThermal conductivities are in Wm−1K−1.

15

perature is shown in Fig. 8, together with data listed in literatures [1, 4, 5, 36].

Although only the lattice contribution to the thermal conductivity can be eval-

uated in the present case, the results of calculated thermal conductivity are con-215

sistent with the experiment data and theoretical values in the low temperature

and low oxygen deficiency (x = 0.02, 0.06) ranges. From the reports of Arima

et al. [10], we have known that the contribution of lattice defects (i.e. oxygen

vacancies) to thermal conductivity is large at low temperatures, whereas the

Umklapp process (phonon∼phonon interactions) dominates lowering the ther-220

mal conductivity at high temperatures. It is obvious that our calculated thermal

conductivities clearly decrease with temperature increasing and show the same

tendency with the results of Arima et al. [1], Ionue [5] and Mihaila et al. [36].

Philipponeau’s [4] recommendation consequences seem to give somewhat lower

values in the temperature range 500−1700K, this phenomenon may result from225

chemical instability of MOX which was pointed out by Inoue [5]. Furthermore,

with increasing x values of oxygen deficiency, the thermal conductivity decreases

obviously, and the degree of the temperature dependence also decrease. In or-

der to investigate the influence of Pu concentration, we further calculated the

thermal conductivity of U1−yPuyO2 (y = 0.15, 0.2, 0.25, 0.3) and compared with230

Watanabe’s [16] values of UO2 using Busker and Yamada potentials, which are

listed in table. 2. The deviation of thermal conductivity made by the different y

is not obvious, which indicate that the influence of Pu concentration is so small

that it can be ignored compared with impact of oxygen defect. Therefore, the

oxygen vacancy plays a more important role than the difference of plutonium235

concentration in evaluating the thermal conductivity of MOX fuel.

5. Conclusions

In summary, the lattice parameter, linear thermal expansion coefficient, com-

pressibility, bulk modulus and thermal conductivity of (U0.75Pu0.25)O2−x (x =

0.0, 0.02, 0.06, 0.1, 0.15, 0.2, 0.25) have been investigated through MD simula-240

tions using the BMH potential with the PIM in the temperature range between

16

300 and 3000K. The temperature dependences of calculated lattice parameter,

linear thermal expansion coefficient and compressibility for U0.75Pu0.25O2−x

well agree with the experimental observations and theoretical data in the low

temperature and low oxygen deficiency ranges. The calculated values of bulk245

modulus at x = 0.02, 0.06 are in good agreement with Arima’s theoretical data,

and gradually tend to be linear relationship with temperature when oxygen

deficiency x > 0.2. We also find that the oxygen vacancies have a significant

effect on thermal properties of MOX, especially thermal conductivity. As oxy-

gen vacancies increase, the thermal conductivity decreases very clearly, and the250

temperature dependence weakens. When oxygen vacancies are existent, the in-

fluence of plutonium concentration for thermal conductivity is so small that

it can be ignored. In addition, we provide a more complete data of physical

properties of MOX by MD simulation.

6. Acknowledgments255

The author J.G.Du acknowledges the funding supporting from National Nat-

ural Science Foundation of China (NO.11204193).

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