Estimating Reaction Rate Constants from Impedance Spectra: Simulating the Multistep Oxygen Evolution Reaction

TL;DR

This work develops a method to extract potential-independent rate constants for the multistep OER at semiconductor photoanodes from electrochemical impedance spectroscopy data. It derives a microkinetic-based impedance model, introduces reduced-order representations valid at different potentials, and leverages a sample maximum likelihood estimator that jointly uses multiple frequencies and potentials with numerical stabilization. The approach is validated on simulated hematite EIS data, showing that two potentials are needed for unique identifiability and that the estimates recover the impedance with high fidelity. The framework provides a practical pathway to quantify OER kinetics from impedance measurements and informs design of more efficient photoelectrochemical cells, with extension to real data in a companion paper.

Abstract

The efficiency of water electrolysis in a photoelectrochemical cell is largely limited by the oxygen evolution reaction (OER) at its semiconductor photoanode. Reaction rate constants are key to investigating the slow kinetics of the multistep OER, as they indicate the rate-determining step. While these rate constants are usually calculated based on first-principles simulations, this research aims to estimate them from experimental electrochemical impedance spectroscopy (EIS) data. Starting from a microkinetic model for charge transfer at the semiconductor-electrolyte interface, an expression for the impedance as a function of the rate constants is derived. At lower potentials, the order of this impedance model is reduced, thus eliminating the rate constants corresponding to the last reaction steps. Moreover, it is shown that EIS data from at least two potentials needs to be combined in order to uniquely identify the rate constants of a particular reduced order model. Therefore, this work details a sample maximum likelihood estimator that integrates not only multiple frequencies, but also multiple potentials simultaneously. Measuring multiple periods of the current density and potential signals, allows this frequency domain estimator to take measurement uncertainty into account. In addition, due to the large numerical range of the rate constants, various scaling methods are implemented to achieve numerical stability. To find suitable initial values for the highly nonlinear optimization problem, different global estimation methods are compared. The complete estimation procedure of the rate constants is illustrated on simulated EIS data of a hematite photoanode.

Figures (8)

• Figure 1: Normalised equilibrium species coverages at different potentials for a hematite photoanode. The coloured regions correspond to different reduced model orders. The normalised coverage of O$_2^*$ only starts to increase at higher potentials.
• Figure 2: Bode plots of the fourth order model compared with the second order model at 1.5 V (top), and with the third order model at 1.6 V, 1.7 V and 1.8 V (bottom).
• Figure 3: Means and standard deviations over 40 realisations of the logarithmic cost in the estimated rate constants $\log_{10} V_\mathrm{SML}(\hat{\bf k})$, after 25 (blue), 50 (red) and 100 (yellow) iterations of the genetic algorithm, for varying crossover rates $P_c$ between 0.6 and 1. The black dotted line indicates the cost in the true rate constants $\log_{10} V_\mathrm{SML}({\bf k}_0)$. Simulation of the second order impedance model with 40 dB noise, $N_\mathrm{exc} = 90$ excited
• Figure 4: Means and standard deviations over 100 realisations of the logarithmic cost in the estimated rate constants $\log_{10} V_\mathrm{SML}(\hat{\bf k})$, over the iterations of the different global optimization algorithms: ES (blue), PSO (red) and GA (yellow). The black dotted line indicates the cost in the true rate constants $\log_{10} V_\mathrm{SML}({\bf k}_0)$.
• Figure 5: Poles (crosses) and zeros (circles) of the fourth order impedance model as a function of potential. Reduction of the model order eliminates the higher order pole-zero paths, marked with different colours. For example, the model of order $n=1$ only contains the blue path, the model of order $n = 2$ contains the blue and the red path, etc. Except for the poles of the first order model (dashed blue line), the poles and zeros of the reduced order models approximately coincide with those of the fourth model. The horizontal dashed lines indicate the potential boundaries of the different model orders (see Figure \ref{['fig:SpeciesCoverage']}). The blue region corresponds to the measurable frequency band [10 mHz, 200 Hz] considered in Figure \ref{['fig:BodePlotsReducedOrder']}.