Intercalated Cation Disorder in Prussian Blue Analogues: First-Principles and Grand Canonical Analyses

Prussian Blue analogues (PBAs) are a promising class of intercalation materials for electrochemical energy storage and water treatment, and intercalated cation disorder within them results from partial cation occupation within their vacant interstitial sites. In this study, we investigate the influence of intercalated cation ordering on the electrochemical equilibrium of PBAs. First-principles calculations were performed for supercell configurations with two distinct arrangements of cations, vacancy-pair configurations, and vacancy-pair free (VPF) configurations (VPFCs), to analyze the energetic and electronic structure differences between them. VPFCs were found to be thermodynamically favored at 300 K. The energetic cost of vacancy-pair formation was also attributed to charge sharing between cations and surrounding atoms. Using these findings, a vacancy-pair free (VPF) model was then constructed based on grand canonical ensemble (GCE) theory to predict the composition-dependent equilibrium potential for intercalation into sodium nickel hexacyanoferrate Na_1+xNiFe(CN)₆. Close agreement with experiment indicates the critical role of cation ordering in PBAs at equilibrium. In this GCE analysis, a hybrid Markov method was used to sample VPFCs at various x, showing two distinct patterns of cation/vacancy ordering. Finally, the VPF model was compared to a noninteracting solution (NIS). The failure of the NIS to predict potential at a low degree of intercalation further confirms that cation/vacancy ordering is essential to electrochemical equilibria in PBAs.