# Thermodynamics of Liquid Uranium from Atomistic and Ab Initio Modeling
## 5 Discussion  
Figure 9 shows the atomic energy of uranium metal through the sequences of the heating (1-2), melting (2-3), heating (3-4), cooling (4-5), solidification (5-6), cooling (6-7), heating (7-8), remelting (8-9), and final heating (9-10) stages. The liquid paths, 3-4, 4-5, and 9-10, are fully reversable, which means that the atomic structure of liquid uranium is temperature-independent (there are no phase transitions). The calculated melting tempera  
Figure 8: The self-diffusion coefficient for liquid uranium. The results of the previous ab initio calculations are taken from Ref. [10]. The experimental data are taken from Ref. [49]. ture, \(T_{2}\) = 1445 K, is shown by the chain line. As we discussed before, the heat of melting at \(T_{2}\) is \(\Delta H_{M}\approx\) 9.07 kJ/mole and the heat of solidification at 800 K is \(\Delta H_{S}\approx\) 7.06 kJ/mole. After solidification, 5-6, cooling, 6-7, and heating, 7-8, uranium metal remels, 8-9, at the temperature, \(T_{rm}\approx\) 1400 K, which is smaller than the calculated melting temperature, \(T_{2}\), and the remelting enthalpy \(\Delta H_{rM}\approx\) 7.84 kJ/mole is about 13.6% smaller than the calculated melting enthalpy \(\Delta H_{M}\approx\) 9.07 kJ/mole at \(T_{2}\) = 1445 K.  
In order to investigate the asymmetry between superheating (sh), \(T_{sh}\) = 1875 K, and supercooling (sc), \(T_{sc}\approx\) 800 K, temperatures in solid-liquid transitions in uranium metal, we employed the formalism developed by [PERSON] et al. [50] and [PERSON] and [PERSON] [51] which is based on the classical nucleation theory of [PERSON] [52]. The authors [50,51] introduced the degree of superheating, \(\Theta_{sh}\) = \((T_{\rm sh}-T_{m})/T_{m}\), and degree of supercooling, \(\Theta_{sc}\) = \((T_{m}-T_{\rm sc})/T_{m}\). According to Ref. [51], \(0\leq\Theta_{sh}\leq 1/3\) or \(T_{\rm sh}\leq 4/3\)\(T_{m}\); and \(0\leq\Theta_{sc}\leq 2/3\) or \(T_{\rm sc}\geq 1/3\)\(T_{m}\). Taking into account the calculated values of \(T_{sc}\), \(T_{m}\) = \(T_{2}\), and \(T_{sh}\) for uranium metal, \(\Theta_{sh}\approx\) 0.2976 and \(\Theta_{sc}\approx\) 0.4464. Thus, in the case of the EAM [4], uranium metal can be supercooled \(\Theta_{sc}/\Theta_{sh}\approx\) 1.5 times more than it can be overheated. The classical nucleation theory predicts the relation [50,51]:  
\[{\Theta_{sh}}^{2}(1+\Theta_{sh})={\Theta_{sc}}^{2}(1-\Theta_{sc}). \tag{6}\]  
In the ideal case, \(\Theta_{sh}\) = 1/3 and \(\Theta_{sc}\) = 2/3, and \(\Theta_{sh}\)\({}^{2}(1+\Theta_{sh})\) = \(\Theta_{sc}\)\({}^{2}(1-\Theta_{sc})\) = 4/27 \(\approx\) 0.14815. According to our calculations, for \(\Theta_{sh}\approx\) 0.2976, \({\Theta_{sh}}^{2}(1+\Theta_{sh})\approx\) 0.11492, and for \(\Theta_{sc}\approx\) 0.4464, \({\Theta_{sc}}^{2}(1-\Theta_{sc})\approx\) 0.11031. So, even our \(\Theta_{sh}\) and \(\Theta_{sc}\) are different from the ideal case of 1/3 and 2/3, respectively, and Relation (6) holds an accuracy of 4%. By using the classical nucleation theory of [PERSON] [52], [PERSON] et al. [50,53], [PERSON] [54], and [PERSON] et al. [55], we derived an equation that gives the melting temperature, \(T_{m}\), from the superheating, \(T_{sh}\), and supercooling, \(T_{sc}\), temperatures:  
\[T_{m}=T_{sh}+T_{sc}-\sqrt{T_{sh}T_{sc}} \tag{7}\]  
By substituting \(T_{sh}\) = 1875 K and \(T_{sc}\) = 800 K into Equation (7), one obtains \(T_{m}\approx\) 1450 K, which is five degrees above the melting temperate \(T_{2}\) = 1445 K calculated independently by the coexistence simulation of the solid and liquid phases discussed in Section 4. The heating and cooling curves in Figure 9 reveal a significant hysteresis effect which manifests itself as the superheating and supercooling phenomena. One should notice that when the supercooled liquid uranium solidifies, it has slightly higher energy than the ideal  
Figure 9: Atomic energy of uranium for heating and cooling cycles. solid.