A general theory of coupled ion–electron transfer (CIET) is presented, which unifies quantum kinetics of electron transfer (ET) with classical kinetics of bond-breaking ion transfer (IT) in a general framework of non-equilibrium thermodynamics [1]. In the limit of large reorganization energy, the theory predicts Marcus kinetics of “electron-coupled ion transfer” (ECIT). In the limit of large ion-transfer energies, the theory predicts generalized Butler–Volmer kinetics of “ion-coupled electron transfer” (ICET), where the charge transfer coefficient and exchange current are connected to microscopic properties of the electrode/electrolyte interface. In the ICET regime, the reductive and oxidative branches of Tafel’s law are predicted to hold over a wide but finite range of overpotentials, bounded by the ion-transfer energies for oxidation and reduction, respectively. The probability distribution of transferring electron energies in CIET smoothly interpolates between a shifted Gaussian distribution for ECIT (as in the Gerischer–Marcus theory of ET) to an asymmetric, fat-tailed Meixner distribution centered at the Fermi level for ICET. The latter may help interpret asymmetric line shapes in x-ray photo-electron spectroscopy (XPS) and Auger electron spectroscopy (AES) for oxidized metal surfaces in terms of shake-up relaxation of the ionized atom and its image polaron by ICET. In the limit of large overpotentials, the theory predicts a transition to inverted Marcus ECIT, leading to a universal reaction-limited current for metal electrodes, dominated by barrierless quantum transitions. Uniformly valid, closed-form asymptotic approximations are derived that smoothly transition between the limiting rate expressions for ICET and ECIT for metal electrodes, using simple but accurate mathematical functions. The theory is applied to lithium intercalation in lithium iron phosphate (LFP) and found to provide a consistent description of the observed current dependence on overpotential, temperature and concentration, learned from x-ray images [2]. CIET theory thus provides a critical bridge between quantum electrochemistry and electrochemical engineering.[1] M. Z. Bazant, Unified quantum theory of electrochemical kinetics by coupled ion-electron transfer, Faraday Discussions 246, 60-124 (2023).[2] H. Zhao, H. D. Deng, A. E. Cohen, J. Lim, Y. Li, D. Fraggedakis, B. Jiang, B. D. Storey, W. C. Chueh, R. D. Braatz, and M. Z. Bazant, Learning heterogeneous reaction kinetics from X-ray movies pixel-by-pixel, Nature 621, 289-294 (2023). Figure 1
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