# Kinetics of Ions in Post-Lithium Batteries
## 3 Post-Lithium Ionic Conductors  
### Sodium-Ion Batteries  
Sodium (Na) is being actively considered as an alternative ion for energy storage applications [44; 48; 89; 90; 91; 92; 93; 94; 95; 96; 97; 98; 99; 100; 101; 102; 103; 104; 105]. In previous studies, we considered a number of sodium-based materials including NaZr\({}_{2}\)(PO\({}_{4}\))\({}_{3}\), Na\({}_{3}\)Fe\({}_{2}\)(PO\({}_{4}\))\({}_{3}\), Na\({}_{3}\)V(PO\({}_{4}\))\({}_{2}\), NaNiO\({}_{2}\), NaFeO\({}_{2}\) and Na\({}_{2}\)MnSiO\({}_{4}\). Table 1 reports the activation energies of diffusion of promising Na of candidate Na-ion battery materials, which range from 0.26 to 0.81 eV [44; 94; 95; 96; 97; 94]. It is observed from Table 1 that NaZr\({}_{2}\)(PO\({}_{4}\))\({}_{3}\) (known as NZP- it is NASICON-type) has a very low activation energy of diffusion (0.26 eV) [47]. These static atomistic simulations are in agreement with previous molecular dynamic calculations by [PERSON] et al. [20] that calculated activation energy for Na diffusion of 0.23 eV.Figure 1 is a schematic representation of the long-range Na-ion diffusion and the migration energy profile for the Na pathway in NaZr\({}_{2}\)(PO\({}_{4}\))\({}_{3}\)[47]. DFT simulations indicate that NaZr\({}_{2}\)(PO\({}_{4}\))\({}_{3}\) is a wide-gap semiconductor, however, the exoergic incorporation of sodium leads to Na\({}^{+}\) ions and electrons. Thus, it is an electronically conductive/high-capacity material that can be used as an electrode [47]. \begin{table}
\begin{tabular}{c c c} \hline
**Material** & **Activation Energy/eV** & **Comments** \\ \hline NaZr\({}_{2}\)(PO\({}_{4}\))\({}_{3}\) & 0.26 & Ref. [47], vacancy mechanism \\ Na\({}_{3}\)Fe\({}_{2}\)(PO\({}_{4}\))\({}_{3}\) & 0.45 & Ref. [95], vacancy mechanism \\ Na\({}_{3}\)V(PO\({}_{4}\))\({}_{2}\) & 0.59 & Ref. [94], vacancy mechanism \\ NaNiO\({}_{2}\) & 0.67 & Ref. [96], ab-plane \\ NaFeO\({}_{2}\) & 0.65–0.67 & Ref. [97], vacancy mechanism \\ Na\({}_{2}\)MnSiO\({}_{4}\) & 0.81 & Ref. [41], vacancy mechanism \\ \hline \end{tabular}
\end{table} 
Table 1: The activation energies of diffusion of Na in candidate Na-ion battery materials. Figure 2 is a schematic representation of the crystal structure (space group C2/\(c\)) and the Na ion diffusion paths calculated in Na\({}_{3}\)Fe\({}_{2}\)(PO\({}_{4}\))\({}_{3}\). Here the green, purple, and light blue atoms represent the Na ion hopping trajectories [95]. Although there is no experimental work on the diffusion pathways in Na\({}_{3}\)Fe\({}_{2}\)(PO\({}_{4}\))\({}_{3}\) there are previous atomistic studies on related materials that show the same similar energetics and diffusion mechanisms [30; 31]. For example, in the classical atomistic simulation investigation by [PERSON] et al. [31] in Na\({}_{2}\)FeP\({}_{2}\)O\({}_{7}\), calculated three-dimensional long-range Na ion migration paths in different directions with activation energies for Na self-diffusion 0.33-0.49 eV. [PERSON] et al. [30] employed computational modeling to study Na ion diffusion in NaFePO\({}_{4}\) material. They predicted that the lowest energy path is in the [010] direction with an activation energy of only 0.30 eV [30]. Fe-based polyanion materials have reasonably low activation energy for Na-ion diffusion and this suggests that they can be considered for energy storage applications [30; 31; 95]. Figure 1: Schematic representation of (**a**) the long-range Na-ion diffusion and (**b**) the energy profile for the Na pathway required to migrate via the vacancy mechanism in NaZr\({}_{2}\)(PO\({}_{4}\))\({}_{3}\)[47]. Figure 2: Schematic representation of (**a**) the crystal structure (space group C2/\(c\)) and (**b**) the Na ion diffusion paths (along the _bc_-plane) calculated in Na\({}_{3}\)Fe\({}_{2}\)(PO\({}_{4}\))\({}_{3}\). Here, the green, purple, and light blue atoms represent the Na ion hopping trajectories [95]. Figure 3 represents the crystal structure (space group C2/_c_) and the Na ion diffusion paths along the _ab_-plane in monoclinic Na\({}_{3}\)V(PO\({}_{4}\))\({}_{2}\)[94].