# Thermodynamics of Liquid Uranium from Atomistic and Ab Initio Modeling
## 4 Results  
Figure 1 shows the measured temperature of the coexistence simulation of the solid and liquid phases as a function of MD steps performed with a time step of 1 fs. In this case, we used the 20 \(\times\) 20 \(\times\) 100-BCC unit cell simulation box. As was mentioned in Section 2, the simulation box is divided into two equal parts. During the first 20,000 time steps, the upper part of the simulation box was heated up to \(T_{1}\) = 2500 K where it melts, and the lower part of the simulation box was kept at the initial temperature, \(T_{0}\) = 1300 K. During the second 20,000 time steps, the upper box of the simulation cell was cooled down to the initial temperature, \(T_{0}\) = 1300 K. Then, during the remaining 1,960,000 time steps of the simulation (1.96 ns), both parts of the simulation box were equilibrated together via an _NPH_ ensemble. The measured temperature of the whole box was about \(T_{2}\approx\) 1445 K, which we consider to be the melting temperature of uranium metal modeled by the EAM [4]. The calculated melting temperature of the EAM [4] uranium is very close to the calculated melting temperature of \(T_{3}\) = 1455 K defined by the EAM [3]. The original paper by [PERSON] et al. [4] did not specify the melting temperature of uranium metal modeled by the suggested EAM. [PERSON] et al. [39] used a similar methodology to estimate the melting temperature of uranium metal described by EAM potential [4]. According to the calculations by [PERSON] et al. [39], uranium melts at \(T_{4}\) = 1524 K, which is about 80 K above the results of the present calculations, \(T_{2}\approx\) 1445 K, and 116 K above the experimental point, \(Tm\) = 1408 K.  
As was mentioned in the Introduction, the measured temperature was recorded every 1000 MD time steps. This output result is shown in Figure 1 as a black line with a significant noise. In this plot, we also show the value of the measured temperature averaged every 1 ps (the red line), which shows significantly smoother behavior. Figure 2 shows the temperature dependence of the total energy of \(\gamma\)-U and liquid uranium derived from the MD simulations. In this case, we used the 20 \(\times\) 20 \(\times\) 20-BCC unit cells simulation box. Initially, the simulation box was equilibrated at \(T\) = 1400 K for 1,000,000 time steps and was then heated up to \(T\) = 2000 K for 100,000 time steps. Even though the previous calculations show that BCC uranium melts at \(T_{2}\approx\) 1445 K, the simulation box remained solid approximately up to \(T_{sh}\) =1875 K where the total energy jump signaled the melting. Then, we equilibrated the simulation box at \(T\) = 2000 K for 1,000,000 time steps and started to cool it down to \(T\) = 1400 K with the step of \(\Delta T\) = 100 K, the equilibration time at each temperature for 1 ns, and the cooling time for each \(\Delta T\) for 1 ns. Finally, opposite to Figure 1 when both the output \"snapshot\" and the average temperature are plotted, Figure 2 shows only the average energy output over the interval of 10,000 time steps. The melting enthalpy, (\(\Delta H_{M}\)), was calculated from the energy difference between the liquid and crystalline system at the calculated melting point, \(T_{2}\) = 1445 K. The required energies are obtained from the data shown in Figure 2. Our calculations reveal \(\Delta H_{M}\) = 94.02 meV/atom \(\approx\) 6.91 mRy/atom \(\approx\) 9.07 kJ/mole, which is close to the experimental value of 9.14 kJ/mole [40]. Figure 1: The measured temperature of the coexistence simulation of the solid and liquid phases as a function of MD steps. Figure 2: The atomic energy of uranium during heating, melting, and cooling as a function of temperature. Figure 3 shows the calculated density of liquid uranium as a function of temperature, together with the experimental data of Fokin [41].