We study the effect of Zn alloying on the hydrogen desorption properties of Mg(BH$_4$)$_2$ using $it{ab initio}$ simulations. In particular, we investigate formation/reaction enthalpies/entropies for a number of compounds and reactions at a wide range of temperatures and Zn concentrations in Mg$_{1-x}$Zn$_x$(BH$_4$)$_2$. Our results show that the thermodynamic stability of the resulting material can be significantly lowered through Zn alloying. We find that e.g. the solid solution Mg$_{2/3}$Zn$_{1/3}$(BH$_4$)$_2$ has a reaction enthalpy for the complete hydrogen desorption of only 25.3 kJ/mol H$_2$$-$a lowering of 15 kJ/mol H$_2$ compared to the pure phase and a corresponding lowering in critical temperature of 123 K. In addition, we find that the enthalpy of mixing is rather small and show that the decrease in reaction enthalpy with Zn concentration is approximately linear.
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