The intercalation phase diagram of Mg in V$_2$O$_5$ from first principles

We have investigated Mg intercalation into orthorhombic V$_2$O$_5$, one of only three cathodes known to reversibly intercalate Mg ions. By calculating the ground state Mg$_x$V$_2$O$_5$ configurations and by developing a cluster expansion for the configurational disorder in $\delta$-V$_2$O$_5$, a full temperature-composition phase diagram is derived. Our calculations indicate an equilibrium phase separating behavior between fully demagnesiated $\alpha$-V$_2$O$_5$ and fully magnesiated $\delta$-V$_2$O$_5$, but also motivate the existence of potentially metastable solid solution transformation paths in both phases. We find significantly better mobility for Mg in the $\delta$ polymorph suggesting that better performance can be achieved by cycling Mg in the $\delta$ phase.