The Electrochemical Impedance Spectroscopy Response of Marcus-Hush-Chidsey Heterogeneous Kinetics
Authors: Knudsen Kristian B. , McCloskey Bryan D. 
Published: February 2020

Over the past decade researchers have attributed an additional physical process to the power loss at fast current rates to the reorganization of the solvation shell when a solvated ion is reduced or oxidized. The heterogeneous electron transfer to the solvated molecule either attracts or repels the constituents of the inner solvation shell and can lead to a significant energy loss. In fact, a recent theoretical study by Smith and Bazant illustrate through multiphase theory of an intercalating porous Li-ion electrode that the loss of the lithium ion’s solvation shell, when intercalated, contributes to a significant overpotential (50-100 mV) at fast charging rates1.
Kinetic theories that describe heterogeneous charge-transfer where electrons are transferred from an electrode to a reactant in solution without adsorption and breaking of bonds has been intensively studied over the past century. Today three main theories are used to describe aprotic electron transfer: Butler-Volmer (BV), Marcus-Hush (MH), and Marcus-Hush-Chidsey (MHC). The Butler-Volmer theory2,3 is widely recognized and applied to electrochemical data to explain the evolution of rate constants as a function of overpotential. However, the BV theory provides little physical insight into understanding the limitations of the electron transfer whereas the Marcus-Hush theory has been able to predict the evolution of homogeneous rate constants as function of activation energy decades before they were experimentally confirmed - especially for the so-called inverted region4–9. The Marcus-Hush theory includes the reorganization energy (λ), the parameter causing the inverted region, and accounts for the energy change in the solvation shell of the redox active molecule. The Marcus-Hush-Chidsey theory is based on the original work by Marcus and Hush and was combined with the Fermi-Dirac distribution describing the probability of electrons around the Fermi level in a conducting electrode10,11. MHC theory therefore provides a microscopic description of the heterogeneous rate constants that, among other, captures the reorganization energy during an electron transfer and depending on the reorganization energy forces the rate constants to significant deviate from the classical BV theory as shown in the figure.
However, extracting kinetic contributions from any electrochemical system through voltammetry has previously proven to be difficult, as fast kinetics and mass-transport often complicate matters. Electrochemical impedance spectroscopy (EIS) is a technique that relies on a small periodic ac amplitude around a dc input. The ac input can be measured over a wide frequency range allowing for the careful separation of kinetic-, double-layer, and mass-transport contribution as each process often occurs at different time scales. Kinetic contributions are referred to as charge-transfer resistances (RCT), double-layer as capacitors and their derivatives, while mass-transport have a wide range of terms.
Herein, we present formulas for the charge-transfer resistance that is governed by infinite or finite MHC kinetics and compare these to the classical BV representations. This allows for the numerical simulation and fitting of the EIS response for any electrochemical system that is governed by MHC kinetics. In addition, we have also solved the RCT governed by the simplified MHC terms developed by Zeng et al.12, which simplify the otherwise complex Fermi-Dirac integrals. Consequently, this opens the possibility of predicting and determining the reorganization energy by EIS.

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Figure 1