Fig. 5: Schematic of inner-sphere energetics.

a Simplified schematic of the free energy, G, for the water dissociation and hydroxide solvation reaction during the alkaline HER (\(2{{{{\rm{H}}}}}_{2}{{{\rm{O}}}}+2{{{{\rm{e}}}}}^{-}\to {{{{\rm{H}}}}}_{2}+2{{{{\rm{OH}}}}}^{-}\)), where the water molecule and electron are at the initial state and a metal hydride, MH^, and solvated hydroxide, OH^-_solv at the final state. Black lines represent the energetics at electrochemical equilibrium and the blue line under applied bias, where F is Faraday’s constant and η the overpotential. b While G is reduced under bias, the free enthalpy, H, of the final state can in fact increase or decrease under bias (blue lines). For the conditions studied in Figs. 1–4, the bias leads to energetically reversible build-up of excess charge, which induces an electrostatic potential drop, \(\varDelta {\phi }_{{ex}.{charge}}\), and associated electric fields. These can impact the activation entropy in the solvent, \(\varDelta {S}_{{sol}}\). In contrast, Butler-Volmer theory assumes that the bias reduces ΔH by\(\varDelta {\phi }_{{neutral}}\) and that \(\varDelta {S}_{{sol}}\) is constant with bias. In this case, the electrochemical bias simply manifests itself as an electrostatic potential difference, \(\varDelta {\phi }_{{neutral}.}\) because the chemical potential differences are constant. Here, charge at the interface is either constant with bias or does not impact the kinetics. c With constant excess charge, σ, (and constant electric fields) the activation entropy, \(\varDelta {S}_{{sol}}\), stays constant and \(\varDelta {\phi }_{{neutral}}\) directly changes the enthalpy of the solvated ion, similar to outer-sphere reactions. Note, in these cases \(\varDelta {\phi }_{{neutral}}\) is the driving force and not an electric field. d Bias dependent build-up of excess charge at the interface leads to an electrostatic potential drop and electric fields that can increase the entropy of the solvation transition state, and, thus, the activation entropy, \(\varDelta {S}_{{sol}}\), and drive the reaction forward, i.e. reduce ΔG reaction in panel a, despite an increasing ΔH in panel b. In this case, the reaction is driven by entropic changes, in contrast to the enthalpic changes in panel c.