The model potential used in this study calculated the critical temperature as 2,600 K and the ratio of the critical temperature to the melting point (*T*_{m} ≅ 3,200 K) is estimated as *T*_{c}/*T*_{m} ≅ 0.8125 which is within the suggested range of the PbF_{2}-type superionic conductor [19]. The temperature dependence of oxygen diffusion coefficients *D*_{o} obtained from the slope of the time-dependent mean square displacements at each temperature is presented in Figures 1 and 2. The self-diffusion coefficient of ions is calculated by Einstein relation, where ${r}_{i\mathrm{\alpha}}\left(t\right)$ is the position of the *i*th ion at time *t*,
$\u3008{\left|{r}_{i\mathrm{\alpha}}\left(t\right)-{r}_{i\mathrm{\alpha}}\left(0\right)\right|}^{2}\u3009=6Dt$(2)

Diffusion path of the ions are changed by the nearby molecules, where the path resembles a random walk. This problem was investigated by Albert Einstein and acquired an equation for Brownian motion.

Figures 1 and 2 show the diffusive behavior of oxygen ions in the perfect crystal (U_{2048}O_{4096}), the defective systems like oxygen vacancy, uranium vacancy, oxygen interstitial, uranium interstitial, oxygen Frenkel defect, uranium Frenkel defect and Schottky-type defect. Three separate regions can be considered from the graphs: extrinsic diffusion is assisted by irradiation-induced defects that dominate diffusion from the thermal vibrations at lower temperatures (1,200 < *T*_{e} < 1,750), intrinsic diffusion operates at higher temperatures (1,800 < *T*_{i} < 2,600), where the diffusion is formed by the thermal energy results, dynamic Frenkel defects and in the third region, superionic diffusion which occurs at temperatures between the Bredig transition and the melting (2,600 < *T*_{s} < 3,200). In Figures 1 and 2, oxygen self-diffusion coefficients are presented at temperatures from 1,000 to 3,200 K. These graphs are plotted as ln(*D*) versus *T*^{–1} in order to display Arrhenius law:
$D=A\times K\times \left(1-K\right)\times exp\left(\frac{-H}{kT}\right)$(3)where *k* is the Boltzmann constant, *A* is the pre-exponential factor, *K* is the concentration of defects and the *H* is the activation energy of defect. For our supercell box, concentration of defects can be expressed as the ratio of number of defects to the total number of anions or cations.

In the extrinsic region, in Figure 1, oxygen diffusion is more intense in the oxygen-defected systems compared to the perfect crystal and the Frenkel-defected supercell. Also it has value comparable with the interstitial oxygen and Schottky in this region. Concentration of oxygen vacancy is 0.0244 % in the MD supercell. At low temperatures, the oxygen diffusion of Schottky type has the lowest value within the defective systems even though it has the highest defect concentration of 0.0488 %. In the case of Li_{2}O, this has been interpreted as occurrence of the clustering of vacancies by Oda [2], namely the clustering of O vacancies with a U vacancy in U_{2047}O_{4094} reduces the oxygen diffusion. All of the defective and crystal systems, except oxygen vacancy, are approaching each other in the intrinsic region as temperature increases. The discrepancies between *D*_{o} of all defective and non-defective systems are completely dissolved in the superionic region and comparable to that of liquid state which is typical of a PbF_{2}-type superionics except the oxygen vacancy defect. The similar temperature dependence of the diffusion behavior of the defective and non-defective systems is also observed for Li_{2}O [2].

Figure 1: Oxygen diffusion calculations of oxygen vacancy, oxygen interstitial, oxygen Frenkel defect, Schottky defect and crystal supercells by MD simulations. Data are compared with experimental values.

In Figure 2, almost all systems behave the same and approach to each other as the temperature is increased, like intrinsic and superionic regions. For the extrinsic part, diffusion of all types of defective systems is higher than the crystal system. Temperatures below 1,200 K are not taken into account for the extrinsic range because too much vibration of the mean square displacement data presents a huge amount of uncertainty in the slope and consequently in the diffusion data. These vibrations mainly occur because MD box is exposed to constant pressure ensemble.

Figure 2: Oxygen diffusion calculations of uranium vacancy, uranium interstitial, uranium Frenkel defect, Schottky defect and crystal supercells by MD simulations. Data are compared with experimental values.

The overall temperature dependence of oxygen diffusion is modeled [2] and expressed as
${D}_{\text{Oxy}}={D}_{\text{radiation}\text{\hspace{0.17em}}\text{defects}}+{D}_{\text{thermally induced defects}}+{D}_{\text{dynamic Frenkel defects}}$(4)

Radiation defects are emerged in crystals by irradiation of fission products, α-particles, α-recoil atoms and neutrons. These extrinsic sources enhanced the diffusion of oxygen ions in the extrinsic region. Thermally induced defects were created with the increase of temperature in the extrinsic region. Dynamic Frenkel defects also appear with the increase of temperature to the critical point (~2,600 K) in the intrinsic region. This type of defect eventually causes collapse of oxygen sublattice at the critical point.

Table 1: *Z*_{U}=2.2208 and *Z*_{O}=–1.1104. The unit of lengths and energies are Å and eV, respectively.

## Irradiation-induced defects in the extrinsic region

Radiation may affect the material and changes the crystal structure, conductivity, lattice parameter and stoichiometry as the defects are created. Radiation-induced defects are extrinsic-type defects and in this temperature range diffusion coefficient is individually evaluated from the Arrhenius plot in Figure 1 and summed in order to obtain radiation-induced diffusion as follows:
${D}_{\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\text{\hspace{0.17em}}\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{s}}={D}_{\mathrm{i}\mathrm{\_}\mathrm{o}\mathrm{x}\mathrm{y}}+{D}_{\mathrm{v}\mathrm{\_}\mathrm{o}\mathrm{x}\mathrm{y}}+{D}_{\mathrm{f}\mathrm{\_}\mathrm{o}\mathrm{x}\mathrm{y}}+{D}_{\mathrm{s}}$(5)Here “i, v, f and s” denote interstitial-, vacancy-, Frenkel- and Schottky-type defects. Diffusion coefficients of each type can be summed because at low defect concentrations it is presumed that defect mechanisms operate independently. Low concentration of defects results in the absence of defect clusters.

Matzke [20] compared the properties of irradiated UO_{2} samples and demonstrated that different sources have different impacts on the sample. To elucidate this idea experimental data could be observed, for example, lattice expansion of crystal due to fission [21] is almost 10 times smaller than α-irradiated [22] UO_{2} sample. This huge inequality emerges from the different types of sublattice defects, fusion particles mainly create oxygen-type defects; however, α-particles create both oxygen- and uranium-type defects. This difference controls the physical properties like lattice parameter, conductivity or diffusion coefficient. For this reason, in order to evaluate diffusion induced by neutrons eq. (5) is used and for the α-particles eq. (6) is used which are given as follows:
${D}_{\text{radiation\hspace{0.17em}defects}}={D}_{\text{i\_oxy}}+{D}_{\text{v\_oxy}}+{D}_{\text{f\_oxy}}+{D}_{\text{s}}+{D}_{\text{i\_ur}}+{D}_{\text{v\_ur}}+{D}_{\text{f\_ur}}$(6)Additional components are added to eq. (5) to consider uranium defects. All these terms of the sum for eqs (5) and (6) are individually evaluated and given in .

Table 2: *A* is the pre-exponential factor; *H* is the activation energy from the Arrhenius law.

Kim and Olander [9] obtained experimental data points of UO_{2-x} between 1,473 and 1,873 K for various oxygen vacancy concentrations such as *x*=0.005, 0.01, 0.03, 0.05 and determined the value of activation energy 0.50736 eV and pre-exponential factor as 4.4×10^{–4} cm^{2}/s for concentrations *x*/2. Low *x* values are considered as stoichiometric UO_{2}. As the value of *x* is increased, oxygen vacancies dominate the diffusion coefficient values. In this study, both experimental and MD simulation data points divided into two parts as intrinsic and extrinsic regions: below and above 1,750 K. As a result, experimental data of Kim and Olander must be reconsidered and experimental constants are reobtained from the fit of Arrhenius plot as given in and . The activation energy of oxygen vacancy resembles the data of Kim and Olander [9]. Moreover, activation energy of crystal structure for MD is in good agreement with stoichiometric data of Marin and Contamin [10] in . Note that concentrations for crystal and Frenkel defect are taken as 1.

Above 1,700 K in Figure 1, both MD and experimental data demonstrate a transition region (~1,700–1,800 K) and follows an increase for the diffusion data. Figure 1 demonstrates that slope of all the data change in the intrinsic range. Murch [23] summarizes this event and concludes that as the temperature increases, a second regime appears and the slope becomes steeper in the second range. Intrinsic region is evaluated separately and following data found in .

Table 3: *A* is the pre-exponential factor; *H* is the activation energy from the Arrhenius law.

Vacancy data is placed in to compare with experimental value of Kim and Olander for intrinsic region. Again, activation energy for both experimental data and the MD vacancy simulation data are in good agreement. In addition to this, crystal data will be used to calculate dynamic Frenkel defects in the following sections.

## Thermally induced defects in the extrinsic region

In the extrinsic region, both Schottky and Frenkel defects may be generated. In this region, this event occurs by thermal effects independent from a source of radiation. Thermally induced defects in the extrinsic region are neither a radiation-induced defect in the extrinsic region nor intrinsic range dynamic Frenkel defect. As a thermally induced defect, both Schottky and Frenkel defects may be taken into account but Schottky defect formation energy is higher than the Frenkel defect. For this reason, Schottky defect has limited effect and only Frenkel-type defects are taken into account. Following equation is used [2]:
$D={A}_{\text{i}}\times K\times \mathrm{exp}\left(-{H}_{\text{i}}/kT\right)+{A}_{\text{v}}\times K\times \mathrm{exp}\left(-{H}_{\text{v}}/kT\right)$(7)
$K=\sqrt{1/2}\times \mathrm{exp}\left(-{E}_{\text{f}}/2kT\right)$(8)

Here *K* is the concentration of Frenkel pairs and *E*_{f} is the formation energy of Frenkel pairs. In order to calculate the formation energy of oxygen Frenkel pairs, Mott-Littleton [24] approach is implemented. Gulp program is used to obtain results [25]. The radii of the spheres are considered as 9 and 20 Å for Mott-Littleton approach. Radii of the region 1 and region 2a are increased until satisfactory convergence is achieved. 2×2×2 supercell of UO_{2} is used and oxygen vacancy and interstitial ions are placed not too close to each other to avoid direct annihilation. For this reason, a uranium ion resides between two defected sites.

Arima [3] have also calculated the formation energy of Frenkel defects for Yakub potential and found a value between 5.1 and 5.3 eV. Activation energies of oxygen from crystals with vacancy and interstitial ions at and Frenkel defect formation energy from (*E*_{f}= 5.9 eV) are used to calculate the diffusion of thermally induced defects.

Table 4: Formation energies of oxygen Frenkel pairs for various potentials from an MD study paper [7], this work and experimental data are listed.

## Dynamic Frenkel defects in the intrinsic region

As the temperature increases to the critical point oxygen ions become more mobile. At about 1,800–2,600 K, this type of Frenkel pairs is more effective than the other type mechanisms. Dynamic Frenkel pairs leave their sites gradually and contribute to total diffusion until 0.8 *T*_{m} [1]. Above this temperature to the melting point oxygen sublattice totally melts and behaves like a liquid where the uranium sublattice still remains like solid. Here, stoichiometric UO_{2} data of MD simulation in is taken into account to calculate diffusion coefficient.

## Overall contribution to oxygen diffusion of irradiated UO_{2}

There are several reasons for oxygen diffusion in irradiated UO_{2} fuel at different temperatures from room temperature to the melting point. These contributions consist of parts from thermal effects and radiation in the extrinsic region, in addition to this dynamic Frenkel defects in the intrinsic region. It should be noted that these contributions could be directly summed [28, 29]. It is presumed that defect mechanisms operate independently because of low-defect concentration, which means the absence of defect clusters and also nearby defects do not annihilate each other. Also these contributions have their independent effects change with the temperature. In order to observe these contributions to the total oxygen diffusion, 100×*D*_{x}/*D*_{Oxy} value is calculated for each part of the sum. *D*_{Oxy} value is given in eq. (4).

In Figure 3, at low temperatures radiation defects dominate the diffusion mechanism which means under reactor conditions, main reason for oxygen diffusion in UO_{2} is fast neutrons, α-particles or recoil atoms. Radiation defect contribution (both neutron and α-particles) to diffusion dominates the overall diffusion up to ~1,200–2,000 K. Above this critical point, contribution from the dynamic Frenkel defects is observed. The effect of thermally induced defects is limited for all the temperature ranges and it has no influence on the overall diffusion and the contribution is almost zero for the irradiated UO_{2}. When the irradiation source changed from neutrons to α-particles the contribution of each component is changed as shown in Figure 3. It results in a considerable amount of change as indicated with the arrows. α-Irradiation-induced defects dominate the overall oxygen diffusion for a few hundred Kelvin more than the neutron irradiation. Thermally induced defects normally should be dominant at low temperatures, but results in very small oxygen diffusion. Main reason for this in UO_{2} crystal is the high value of formation energy of oxygen Frenkel pairs that results in very low concentration of these defects created with thermal effects and consequently very small diffusion coefficient of oxygen.

Figure 3: This figure shows the percentage of contribution to the total oxygen diffusion. If radiation source is changed from neutron to α-particles, contributions from each fragments change in the direction of the arrows.

Radiation-induced defects consists of parts as oxygen vacancy, oxygen interstitial, oxygen Frenkel and Schottky defects. Their individual contributions to radiation-induced oxygen diffusion are observed in Figure 4. Interstitial defects are more effective about 1,000 K but for higher temperatures oxygen vacancy and Frenkel defects are more operative.

Figure 4: Contributions from oxygen vacancy, interstitial-, Frenkel- and Schottky-type defects to irradiation-induced oxygen diffusion. Here irradiation source is considered as neutrons.

In Figure 5, all types of defects contribute to the irradiation-induced oxygen diffusion. At about 1,000 K interstitial-type oxygen and about 1,800 K oxygen vacancy contribute the most to the diffusion in the extrinsic region.

Figure 5: Contributions from oxygen and uranium vacancy, interstitial-, Frenkel- and Schottky-type defects to irradiation-induced oxygen diffusion. Here, irradiation source is considered as α-particles.

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