Fig. 2

Phase diagrams of K-Mn oxide systems at E = 1.2 V. a pH-μ_K slice at 1/R = 0 indicating bulk stabilized phases. b 1/R-μ_K slice at pH = 2 illustrating the size-stabilized phases. c, d Pourbaix grand free-energies versus 1/R, showing the metastable energy landscape of crystal growth where [K⁺] = 10⁻⁶ M (c) and [K⁺] = 0.2 M (d). d is representative of pathways at both [K⁺] = 0.2 M and [K⁺] = 0.33 M due to their similar [K⁺]. The three reaction conditions are referred to as P_K = 0, P_K = 0.2, and P_K = 0.33, respectively, based on their [K⁺]. \(\mathrm{d}({{\Phi }}_{\delta - \mathrm{K}_{0.33}{\mathrm{MnO}_{\mathrm{2}}} - }{{\Phi }}_{\alpha - \mathrm{K}_{0.125}{{\mathrm{MnO}}_{\mathrm{2}}}})/{\mathrm{d}}\mu _{\mathrm{K}}= - 0.21\), which means that the driving force for the transformation from the δ to the α phase decreases at a larger [K⁺]. The red and blue arrows indicate the trend that the slope of \({{\Phi }}_{\delta - {\mathrm{K}}_{0.33}{\mathrm{Mn}}{{\rm O}_2}}\) and \({{\Phi }}_{\alpha - {\mathrm{K}}_{0.125}{{\mathrm{Mn}}{\rm O}_2}}\) changes with increasing [K⁺]