Prussian Blue (PB) has attracted much attention as a cathode active material with large cages for the next-generation batteries. Not only conventional Li-ion (Li⁺), utilization of Na-ion (Na⁺) and K-ion (K⁺) for the inserted ions has been intensively investigated so far. However, the microscopic diffusion mechanisms of the alkali ions in PB remain still unclear. This study aims to clarify the stable occupation sites and diffusion mechanisms of Li⁺, Na⁺, and K⁺ in A-PB by using A₄Fe₄[Fe₄(CN)₆] model systems (A = Li, Na, K). We used density functional theory (DFT) based calculations with Nudged Elastic Band (NEB) method and molecular dynamics (MD) sampling, as well as a DFT-surrogate technique to quickly search for stable ion occupations in ionic compounds [1].

In the PB framework, body center (BC), face center (FC), off-center FC (off-FC), and transport hub (TH) sites [2] were considered for the A⁺ ion sites (Fig. a). With the DFT-surrogate technique followed by DFT geometry optimization, we found that Li⁺ and Na⁺ occupy the off-FC sites, close to the CN^– anions, in their most stable configurations, while stable K-PB prefers the BC sites for K⁺. This can be explained with the ionic radius and acidity. For the choice of four out of the eight cages, the configuration with the furthest distance between each A⁺ pair is chosen probably due to the Coulomb repulsion among A⁺ ions.

Then, we carried out DFT-MD analysis of A⁺-ion self-diffusion via NVT ensemble with T = 300-1000K. Focusing on the 700 K results, we found that (i) Li⁺ has the self-diffusion coefficient (D*) in the order of 10^–4 cm²/s, and Na⁺ self-diffusion coefficient in the order of 10^–6 cm²/s is much lower than Li⁺, while K⁺ has almost D* ≈ 0. The complementary NEB calculations indicate that the K⁺ diffusion barrier is 1200 meV, too high to diffuse, while that for Na⁺ is just 74 meV, rather low in contrast to its low D*.  One possibility will be that the repulsion between A⁺-ions prevent the hopping to configurations with A⁺ ions in the neighboring cages. This will be discussed more in the presentation.

Visualizations of A⁺-ion trajectory density in PB unit cell are shown in Fig. b-d. It is suggested that Li⁺, and Na⁺ ions take three-dimensional pathways, possibly involving the TH sites connecting to the stable FC, and off-FC ones, respectively (see Fig. b, c). For the K-FB system, K⁺ ions hinge around BC (Fig. d) due to the barrier described above. These findings offer fundamental insights into mono-cation size effects on self-diffusivities within PB crystal bulk.

Fig.  (a) Single cage of Prussian Blue (PB) crystal and possible occupation sites. Trajectory densities calculated at 700 K set-up for (b) Li, (c) Na, and (d) K ions.