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Effect of Radiation Defects on Thermo–Mechanical Properties of UO_{2} Investigated by Molecular Dynamics Method

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## Abstract

**:**

_{2}within 600–1500 K has been studied using the molecular dynamics method. Two types of point defects have been investigated in the present work: Frenkel pairs and antisites with concentrations of 0 to 5%. The results indicate that these point defects reduce the thermal expansion coefficient (α) at all studied temperatures. The elastic modulus at finite temperatures decreases linearly with the increase in concentration of Frenkel defects and antisites. The extent of reduction (R) in elastic modulus due to two different defects follows the trend R

_{f}> R

_{a}for all studied defect concentrations. All these results indicate that Frenkel pairs and antisite defects could degrade the performance of UO

_{2}and should be seriously considered for estimation of radiation damage in nuclear fuels used in nuclear reactors.

## 1. Introduction

_{2}) is widely used as a nuclear fuel in the nuclear industry for various nuclear power reactors [1]. Thus, the safe operation of a nuclear reactor correlates strongly with the stability of UO

_{2}. However, under extreme conditions, different radiation defects (e.g., vacancies, interstitials, and voids) would be created within the nuclear fuel due to irradiation. These defects would lead to severe degradation of the physical, thermal, and mechanical properties of the nuclear fuels [2,3,4]. For example, irradiation-induced fission products and vacancies can produce bubbles and voids, causing swelling and fragmentation which thus deteriorates the performance of fuels [5]. Therefore, to investigate the effect of radiation-induced defects on the thermo-mechanical properties of uranium dioxide is essential.

_{2}. For example, Hobson et al. analyzed porous UO

_{2}with porosity levels of 4.11 to 8.58% and observed the relationship of reductions in thermal conductivity to the temperature [6]. An experimental study on the effect of soluble fission products on thermal conductivity was also performed, which found that at lower temperatures the thermal conductivity would decrease with an increase in fission product concentration; however, at higher temperatures the concentration of fission products has only a slight influence on thermal transport [7].

_{2}and reported that thermal conductivity decreased with the increasing concentration of Schottky defects [10]. Furthermore, the thermal transport of ThO

_{2}, as an alternative to conventional uranium nuclear fuel, was also investigated extensively. For example, Park et al. [11] investigated the effect of vacancies and substitutional defects on the thermal transport of ThO

_{2}by employing reverse non-equilibrium molecular dynamics (NEMD). The authors reported that the effect of thorium vacancy defects on the thermal transport of ThO

_{2}is even more detrimental than that of oxygen vacancy defects. In addition, compared to vacancy defects, substitutional defects in ThO

_{2}slightly affect the thermal transport [11]. To investigate the effect of irradiation-induced fission products on the thermal conductivity of thorium dioxide, Rahman et al. [5] examined the effect of Xe and Kr with impurity concentrations of 0 to 5% on the thermal conductivity of ThO

_{2}with the molecular dynamics method, and found that Xe and Kr resulted in significant reductions in the thermal conductivity of ThO

_{2}.

_{2}matrix containing different concentrations of porosity and observed that the elastic modulus decreased with an increase in porosity concentration [13]. Rahman et al. examined the effect of fission products (Xe and Kr) and porosity on mechanical properties of ThO

_{2}within 300–1500 K using molecular dynamics simulations. By comparing the effect of fission products and porosity, the authors reported that the fission products resulted in a stronger reduction in elastic modulus than the porosity [14].

_{2}have been studied by different groups, to our best knowledge no investigation has been performed about the effect of Frenkel defects and antisites on the thermo-mechanical properties of irradiated UO

_{2}. Considering its importance, in this work, the influences of Frenkel defects and antisites on the thermal expansion coefficient and elastic modulus of uranium dioxide are investigated extensively via molecular dynamics simulations. The thermal expansion coefficient of perfect and damaged systems is evaluated from changes in lattice parameters. Three independent elastic constants are calculated for each system, which are used to estimate the elastic modulus. The reduction in the elastic modulus induced by Frenkel defects and antisites is also calculated as a function of concentrations of defects in the system. A comparison is finally made between the effects of Frenkel defects and antisite defects to provide more understanding about the structure and property changes of UO

_{2}after irradiation. In the following sections, the computational method is first presented. The results and discussion are provided in Section 3. The conclusion is made in the last section.

## 2. Computational Method

_{2}, the Coulomb interaction is further included with the original pair. The computational box used in this work is a 10 × 10 × 12 extension of fluorite (CaF

_{2}) unit cells containing 14,400 atoms. The lattice parameter for the computational box is the equilibrated lattice parameter at the investigated temperature. The periodic boundary condition is applied in all directions.

_{2}is close to two [19], which is in agreement with the results presented by Devanathan et al. [20] and Van Brutzel et al. [21]. In order to create this structure of uranium dioxide with Frenkel defects and maintain the neutral charge of the system, uranium and oxygen atoms are removed from the system by keeping 1:2 ratio. The same amount of uranium and oxygen atoms are then randomly inserted at the octahedral interstitial positions of the face-centered cubic (fcc) cation sublattice [22]. Different from Frenkel defects, oxygen-antisites are created by substituting O atoms with U atoms. Similarly, uranium-antisites are created by substituting U atoms with O atoms. In order to maintain the stoichiometry of the defected system, the number of O-antisites is equal to that of U-antisites. In order to investigate the effect of defect concentration, UO

_{2}structures with 1%, 2%, and 5% Frenkel defects and 1%, 2%, and 5% antisite defects are built for further simulations. It should be noted that in the present work the concentration is defined as the value before MD relaxation at given temperatures. The main reason is that the relaxations at different temperatures could result in different concentration values after full relaxation. In order to avoid this misunderstanding during the investigation of concentration effects in this work, the concentration value before MD relaxation is used. For each defect concentration the statistical results are made based on 3 samples by randomly creating Frenkel or antisite defects.

_{2}after irradiation. The timestep of 1 fs is used for all simulation processes.

_{2}is FCC and thus the thermal expansion coefficient is isotropic.

_{11}, C

_{12}, and C

_{44}) need to be calculated. These elastic constants can be calculated by applying elementary strain in six directions and measuring the changes in the six stress components. In this work, the strain to induce the deformation of the simulation box was set to be 10

^{−5}. Based on the dependence of the stress on the strain, these constants are calculated as described in [24]. Based on these three constants, the bulk modulus (B), shear modulus (G), and Young’s modulus (Y) can be calculated. The bulk modulus is calculated with the following equations [14].

_{11}+ 2C

_{12})/3

_{V}) using the Voigt method and the shear modulus (G

_{R}) using the Reuss method need to be obtained, which can be determined using the following Equations (3) and (4), respectively.

_{V}= (C

_{11}− C

_{12}+ 3C

_{44})/5

_{R}= (5(C

_{11}− C

_{12})C

_{44})/(4C

_{44}+ 3(C

_{11}− C

_{12}))

_{V}and G

_{R}.

_{V}+ G

_{R})/2

## 3. Results and Discussion

#### 3.1. Effect of Defects on Lattice Parameter and Thermal Expansion Coefficient

_{2}as the function of temperature is plotted in Figure 1. The error bars in the figure correspond to the standard deviation calculated among the five different statistical lattice constants calculated at the given temperature. For comparison, the results from the VASP calculation by Wang et al. [26] and from the experimental measurement by Taylor et al. [27], Yamashita et al. [28], and Momin et al. [29] are also included in Figure 1. It is also clear that L increases linearly with increases in temperature from 0 K to 1500 K as investigated in this work. From these results, the results from the present MD agree better with the experimental value than those from VASP calculations. Thus, the MD method and the related empirical potential could be used for the present purpose for further simulations.

_{2}on defect concentration at different temperatures is provided for Frenkel defects (dash) and antisite defects (solid). From Figure 2, it is clear that the lattice parameter of the system increases with an increase in Frenkel defect concentration from 0 to 5% at all investigated temperatures in the present work, although the increases are limited around 1.0%.

_{2}, which reported that the L of the system increased linearly with increases in porosity concentration. The reason for the above difference may be mainly from the property of anisotropic effects induced by antisite defects and interstitials in Frenkel pairs, which is different from the isotropic vacancy or porosity. It should also be noted here that in this work, only defect concentrations up to 5% are considered. If higher concentrations were considered, the lattice constant may change accordingly.

_{2}as a function of temperature. The uncertainty of the thermal expansion coefficient at different temperatures has also been calculated by changing the defect distribution but keeping the same concentration as stated in the computational method section. As shown in Figure 3, the uncertainty is limited for the three cases investigated in the present work. For comparison, the experimental results [27], MD derived values [30], and first principles data [31] for pure UO

_{2}are included in the figure.

_{2}. Different from the effects of Frenkel defects, Figure 3b clearly indicates that for the concentrations of antisites investigated in the present work, the thermal expansion coefficient of the system increases with an increase in temperature. When the temperature is 600 K, the thermal expansion coefficient decreases with an increase in antisite defect concentration. When the temperature is 900 K or 1200 K, the thermal expansion coefficient has similar values for systems containing antisite defects less than 2%, but decreases around 25–30% when the antisite defect concentration is 5%. When temperature is 1500 K, the thermal expansion coefficient has the same value for systems containing 1% to 5% antisite defects, which is lower than that of the perfect system. Comparing the results shown in Figure 3a,b, it could be found that antisite defects have stronger effects than Frenkel defects on the thermal expansion coefficient of UO

_{2}.

#### 3.2. Elastic Modulus of UO_{2}

_{2}are initially calculated from three independent elastic constants (C

_{11}, C

_{12}, and C

_{44}) using Equations (2), (5) and (6). Figure 4 depicts the dependence of the bulk modulus of perfect UO

_{2}on temperature calculations of systems. For comparison, the experimental results from Belle et al. [33], the MD derived bulk modulus from Basak et al. [23], and ab initio data calculated by Wang et al. [26] are also included in this figure. It is clear that the present study has similar results to those obtained by previous MD calculations and experiments but lower than those from the VASP calculation. Figure 4 also indicates that the bulk modulus of perfect UO

_{2}derived in this study decreases with an increase in temperature, which has been confirmed in the previous study by Dorado et al. [34].

_{2}at different temperatures as a function of Frenkel defect concentration (dash) and antisite defect concentration (solid). Figure 5 shows that both a Frenkel defect and an antisite defect could considerably decrease the bulk modulus of UO

_{2}, showing a linear decreasing dependence on temperature from 600 to 1500 K. With an increase in defect concentration, the elastic modulus also decreases accordingly, as shown by the figure. The extent of reduction in the elastic modulus for the system containing defects becomes smaller with increasing temperature and defect concentration. In addition, Figure 5 demonstrates that Frenkel defects increase the bulk modulus to a larger extent compared to that induced by antisite defects.

_{2}. Firstly, for the concentration of defects investigated in this work, the shear modulus decreases with an increase in temperature. The extent of reduction in the shear modulus decreases with increases in temperature from 600 to 1500 K. Similar to the effects on the bulk modulus shown in Figure 5, it can be seen from Figure 6 that the increase of defect concentration could significantly reduce the shear modulus of UO

_{2}. However, the extent of reduction in G resulting from Frenkel defects is larger than that observed for antisite defects. For example, for 5% Frenkel and antisite defects there is a maximum of 20% and 17% reduction in G at all temperatures, respectively.

_{2}containing different concentrations of Frenkel (dash) and antisite defects (solid) as the function of the temperature is plotted in Figure 7. From this figure, it is clear that Young’s modulus of uranium dioxide decreases linearly with the increase in temperature for Frenkel and antisite defects within the given concentrations. This result is similar to that reported by Jelea [13] et al. who observed that Young’s modulus for damaged UO

_{2}with different percentages of porosity linearly decreases with increases in temperature. Similar to the results of the bulk modulus and the shear modulus, Young’s modulus also decreases with increasing temperature and concentration of defects. The relative change in Y due to Frenkel defects is larger than that observed for antisite defects. Within the given temperatures, a maximum of 20% reduction in Y is observed for UO

_{2}systems containing 1%, 2%, and 5% Frenkel defects. In contrast, for antisite defects with the same concentrations, there is a maximum of 16% reduction in Y at all temperatures.

#### 3.3. Reduction of Elastic Modulus of UO_{2} by Frenkel Defects and Antisites

_{p}− M)/M

_{p}

_{p}and M represent the elastic modulus of a perfect and defective UO

_{2}, respectively.

_{f}> R

_{a}for all studied defect concentrations.

## 4. Conclusions

_{2}via the molecular dynamics method in the temperature range of 600 to 1500 K. The results indicate that both Frenkel defects and antisite defects reduce the thermal expansion coefficient. However, the reduction in the thermal expansion coefficient due to antisite defects is larger than that observed for Frenkel defects. For the elastic modulus, the calculated bulk, shear, and Young’s modulus of the pure UO

_{2}are in agreement with the experimental values. Furthermore, the present results indicate that both Frenkel defects and antisite defects reduce the elastic modulus at all temperatures. The degree of reduction in the elastic modulus increases with increasing concentrations of defect. In addition, the percentage of reduction in the elastic modulus due to Frenkel and antisite defects follows the trend R

_{f}> R

_{a}at all studied defect concentrations. All these calculated values can be used to predict the performance of UO

_{2}under irradiation used in the nuclear reactor environment.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Dependence of lattice constant of UO

_{2}on defect concentration at different temperatures for Frenkel defects (dash) and antisite defects (solid).

**Figure 3.**Thermal expansion coefficient for pure UO

_{2}and for defective UO

_{2}with different percentages of Frenkel defect (

**a**), and antisites (

**b**), as a function of temperature. For comparison, the experiment studies by Taylor et al. [27], the MD results by Cooper et al. [30] and the first-principles data by Yun et al. [31] are included in the figure.

**Figure 5.**Variation of the bulk modulus of UO

_{2}containing different concentrations of Frenkel (dash) and antisite defects (solid) versus temperature. The fitted lines are also included in the figure.

**Figure 6.**Variation of the shear modulus of UO

_{2}containing different concentrations of Frenkel (dash) and antisite (solid) defects versus temperature. The fitted lines are also included in the figure.

**Figure 7.**Variation of Young’s modulus of UO

_{2}containing different concentrations of Frenkel (dash) and antisite (solid) defects versus temperature. The fitted lines are also included in the figure.

**Figure 8.**R

_{B}as a function of the concentration of Frenkel defects (

**a**) and antisites (

**b**) at different temperatures.

**Figure 9.**R

_{G}as a function of the concentration of Frenkel defects (

**a**) and antisites (

**b**) at different temperatures.

**Figure 10.**R

_{Y}as a function of the concentration of Frenkel defects (

**a**) and antisites (

**b**) for different temperatures.

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**MDPI and ACS Style**

Wang, Z.; Yu, M.; Yang, C.; Long, X.; Gao, N.; Yao, Z.; Dong, L.; Wang, X.
Effect of Radiation Defects on Thermo–Mechanical Properties of UO_{2} Investigated by Molecular Dynamics Method. *Metals* **2022**, *12*, 761.
https://doi.org/10.3390/met12050761

**AMA Style**

Wang Z, Yu M, Yang C, Long X, Gao N, Yao Z, Dong L, Wang X.
Effect of Radiation Defects on Thermo–Mechanical Properties of UO_{2} Investigated by Molecular Dynamics Method. *Metals*. 2022; 12(5):761.
https://doi.org/10.3390/met12050761

**Chicago/Turabian Style**

Wang, Ziqiang, Miaosen Yu, Chen Yang, Xuehao Long, Ning Gao, Zhongwen Yao, Limin Dong, and Xuelin Wang.
2022. "Effect of Radiation Defects on Thermo–Mechanical Properties of UO_{2} Investigated by Molecular Dynamics Method" *Metals* 12, no. 5: 761.
https://doi.org/10.3390/met12050761