Polyanionic Lattice Modifications Leading to High-Entropy Sodium Ion Conductors: Mathematical Solution of Accessible...

Sodium zirconium double phosphate NaZr₂ (PO₄ )₃ can be used as a starting point for investigations of high-entropy materials. Apart from the frequently used approach of partial substitution with four or more different transition metal cations, this class of materials also allows multiple substitutions of the phosphate groups. Herein modifications of the polyanionic lattice are considered and high-entropy compositions are numerically determined with up to eight elements on the central tetrahedral lattice site of the so-called NaSICON structure. For this study, the chemical formula was fixed as Na₃ Zr₂ (EO₄ )₃ with E=B, Al, Si, P, As, Sb, S, Se and Te. The number of compositions increases exponentially with the increasing number of elements involved and with decreasing equal step size for each element. The maximum number of 237258 compositions is found for Na₃ Zr₂ ([B,Al,Si,P,As,Sb,S,Se]O₄ )₃ with a step size of 0.1 mol/formula unit. Of this compositional landscape, 143744 compositions fulfil the definitions of high-entropy materials. The highest entropy factor of ΔS_config /R=-2.0405 is attributed to the compositions Na₃ Zr₂ (B_0.5 Al_0.6 Si_0.4 P_0.3 As_0.3 Sb_0.3 S_0.3 Se_0.3 )O₁₂ and Na₃ Zr₂ (B_0.6 Al_0.5 Si_0.4 P_0.3 As_0.3 Sb_0.3 S_0.3 Se_0.3 )O₁₂ .