The values of lattice parameter, bulk modulus, elastic constants and cohesive energy are given in and compared with the available experimental data and ab initio results. The bulk modulus and elastic constants were calculated from the utilization of known Birch–Murnaghan equation of state [19, 20], which is in good agreement with experiment and ab initio results [21–25]. Contrary to the previous MD results [11–15], in the present paper the lattice parameter is underestimated about 1.3 % and 3 % compared to the ab initio calculations and experimental data, respectively [22, 23]. However, bulk modulus obtained with the previous potentials has been overestimated and varied between 200 and 239 GPa [11–15] when compared with the experimental result 178 GPa [21]. In the present study, the amount of deviation of bulk modulus from the experimental study is about 5 %, whereas other studies are between 12 % and 34 %. The main reason for these results is the weight of observables while finding the potential parameters. Groups place great emphasis on the lattice constant while developing potential parameters. In this study, we distribute the weight constant unevenly on lattice constant, bulk modulus, elastic constants and energy as we are performing fitting procedure. Temperature dependence of energy is also taken into account in order to mimic the fast-ion phase transition. Discrepancy between the calculated value and experimental data of *C*_{12} are presented in . This surely can be modified by changing the potential parameters but in that the indication of the phase transition can be lost.

Table 2: Comparison of calculated results with MD simulation, experimental and ab initio data.

$L/{L}_{0}$ is calculated, where ${L}_{0}$ is the lattice constant at 300 K, as a function of temperature together with the other simulation results [11–15] and the experimental data [22], which are available up to 2,000 K are presented in Figure 1. Figure 2 shows the linear thermal expansion $\Delta L/{L}_{0}$. All potentials produce a similar thermal expansion of the lattice between 300 and 2,000 K; discrepancy between the simulation and the experiment increases as the temperature increased. At about 2,100 K, there is a sign of a change in trend of the evolution of the data with temperature. As recommended by Fink [7], transition to the superionic phase may be expected at temperature between 2,160 and 2,370 K.

Figure 1: Lattice parameter evolution with temperature. Experimental data are from TPRC data series [22].

Figure 2: Evolution of relative linear thermal expansion of present results, MD data and experimental data with temperature.

The temperature dependence of the pair correlation functions ${g}_{ij}\left(r\right)$ given in Figure 3 was also considered in our calculations aiming to have quantitative measure of the spatial local order–disorder of the atomic structure of PuO_{2}.

Figure 3: Pair correlation functions of $\mathrm{P}\mathrm{u}{\mathrm{O}}_{2}$ at three different temperatures.

The radial distribution function is defined as follows:
$g(r)=\frac{\mathrm{\Delta}n(r)}{\mathrm{\rho}4\mathrm{\pi}{r}^{2}\mathrm{\Delta}r}$(2)

$\mathrm{\Delta}n(r)$ is the average number of atoms at a distance between $r$ and $r+\mathrm{\Delta}r$. $\mathrm{\rho}$ is the atomic density and $\mathrm{\Delta}r$ is the width of the shell. The peaks of ${g}_{ij}\left(r\right)$ of all three pairs of ions become lower and broader due to the larger thermal vibrations of ions in their lattice sites as the temperature increased up to 2,000 K. Right after 2,000 K, ${g}_{oo}\left(r\right)$ has very small oscillations and overlap of the principal peaks after the first peak, indicating a considerable degree of disorder of the oxygen sublattice while the plutonium ions are still located in the lattice sites with larger vibrations. This liquid-like behavior of the oxygen sublattice at solid phase at about 2,100 K is interpreted as the onset of the transition to the superionic phase of PuO_{2}.

For a species of *N* particle, the mean square displacement (MSD) is calculated as
$\u3008{\left|r(t)-r(0)\right|}^{2}\u3009=\frac{1}{N{N}_{t}}\sum _{n=1}^{N}\sum _{{t}_{0}}^{{N}_{t}}{\left|{r}_{n}(t+{t}_{0})-{r}_{n}({t}_{0})\right|}^{2}$(3)

where ${r}_{n}(t)$ is the position of particle *n* at time *t*. The MSD of oxygen ions at different temperatures is shown in Figure 4. The linear change of MSD with time indicates that the oxygen ions become more diffusive at about 2,100 K, while the MSD of Pu ions remains constant. Figure 5 illustrates the diffusion coefficient of both Pu and O ions in Arrhenius diagram (log *D* versus 1/*T*) that are calculated from the slope of the MSD data using Einstein relation
${D}_{i}=\underset{t\to \mathrm{\infty}}{lim}\frac{1}{6t}\u3008{\left|{r}_{i}\left(t\right)-{r}_{i}\left(0\right)\right|}^{2}\u3009$(4)

Figure 4: Mean square displacement versus time at different temperatures.

Figure 5: Evolution of diffusion coefficient with temperature for both plutonium and oxygen ions.

It is clearly evident from the anomalous increase of diffusion coefficient of oxygen ions at ~2,100 K that there is an onset of Bredig transition to the superionic phase. While the oxygen diffusion continuously increases beyond 2,100 K, the plutonium ions show almost no diffusive behavior. As experimental data for oxygen diffusion coefficient for PuO_{2} is only available up to 1,400 K [26], we are not able to compare self-diffusion coefficient. As the temperature increases, Pu ions also show diffusive behavior after 3,000 K and *D*_{Pu} is less than *D*_{O} about an order of 10.

The enthalpy change $\mathrm{\Delta}H={H}_{T}-{H}_{298}$ is presented in Figure 6 and compared with the experimental data taken from Fink [7] and MD simulation with shell model potential by Chu. Evidently, there is a discontinuity in calculated $\mathrm{\Delta}H$ at about 2,100 K and very good agreement with experimental data before and after this temperature. This justifies the potential parameters and validates the constructed model of PuO_{2}.

Figure 6: Enthalpy change of $\mathrm{P}\mathrm{u}{\mathrm{O}}_{2}$ with temperature.

The existence of the thermally activated transition into superionic phase can be confirmed by a *λ*-peak in the heat capacity at constant pressure ${C}_{p}\left(T\right)$. Many fluorite-type ionic crystals exhibit such a transition at about $0.8{T}_{m}$ [27]. The heat capacity was evaluated from the variation of the internal energy $E\left(T\right)$ with temperature at constant pressure and presented in Figure 7:
${C}_{p}\left(T\right)={\left(\frac{\mathrm{\partial}E}{\mathrm{\partial}T}\right)}_{p}$(4)

Figure 7: Temperature dependence of heat capacity for $\mathrm{P}\mathrm{u}{\mathrm{O}}_{2}$ for the present study at 100 K intervals and comparison with experimental data. The inset shows the results for the different box sizes at 50 K intervals.

A very weak increase in *C*_{p} up to 1,750 K is interpreted as increase in the anharmonicity of the lattice vibrations. Further increase in temperature results with the additional increase in calculated *C*_{p} and a *λ* shape peak is clearly produced at the critical temperature *T*_{c} = 2,055 K, which is close to the expected value of the phase transition temperature in fluorite type of ionic crystals [7]. In order to make sure that it is a $\mathrm{\lambda}$-peak transition in specific heat, we have increased the number of data by varying the temperature at 50 K intervals in NPT ensemble for MD simulation calculations. These calculations have been carried out for three different supercell boxes: 6 × 6 × 6 (864 Pu-1728 O), 7 × 7 × 7 (1372 Pu-2744 O) and 8 × 8 × 8 (2048 Pu-4096 O) to show the consistency of the calculations in terms of number of atoms. Results for heat capacity at constant pressure around the transition region are shown in Figure 7 as inset. It is clearly evident that the system undergoes a thermally activated *λ*-type Bredig transition to the superionic phase. Moreover, the linear increase of MSD of oxygen ions with time at about 2,100 K while that of plutonium remains constant presented in Figure 4 and anomalous increase of diffusion coefficient of oxygen ions at around the same temperature presented in Figure 5, that are due to the premelting of oxygen sublattice, also support the indication of lambda transition in heat capacity.

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