Mathematical Model Based on Staircase Structure for Impedance Analysis of Non-Ideal and Non-Uniform Processes in Porous Electrodes

Electrochemical impedance spectroscopy analysis for porous electrodes in energy conversion and storage, described as resistance and capacitance, is widely used to understand their electrochemical processes. These processes include: (i) electronic resistance inside the electrode ( R e ), (ii) ionic resistance in the electrolyte bulk and inside the porous electrode ( R ion ), (iii) electrical double layer formation ( C dl ) and charge transfer resistance at the electrode electrolyte interface ( R ct ) , and (iv) diffusion for charge compensation. As one of the mathematical models in electrochemical impedance spectroscopy for porous electrodes, a transmission line model is known, in which the pore structure is described as a cylinder geometry. 1 We have proposed an analytical method that combines the transmission line model with electrochemical impedance spectroscopy using a symmetric cell, 2 and have investigated resistance separation in lithium-ion batteries and its effect on battery performances. 3-5 Due to mathematical constraints, the transmission line model is limited to a representation of an ideal, uniform process model. Here we propose a mathematical model based on a network model consisting of a staircase structure for electrochemical impedance spectroscopy at porous electrodes that can describe not only ideal and uniform processes but also non-ideal and non-uniform processes (Fig. 1), and aims to help elucidate their borderline behavior. 6 The proposed staircase model ( Z SCM ) gives a series equivalent circuit consisting of Z 0 , which is a series circuit of electrode/electrolyte interface impedance ( Z int0 ) and electronic resistance ( R e0 ), and electrolyte resistance ( R ion0 ) as an initial step without pores ( Z SCM0 = R ion(0) + Z 0 ). The one-step model ( Z SCM1 ) is then given a series equivalent circuit consisting of the parallel circuit of Z SCM0 and Z 1 and the electrolyte resistance R ion(1) in series ( Z SCM1 = R ion(1) + ( Z SCM0 −1 +Z 1 −1 ) −1 ). The staircase model is calculated by incorporating the 1 prior step (Fig. 2). Thus, the model with n steps ( Z SCM(n) ) is composed of a series equivalent circuit of the parallel circuit of Z SCM(n−1) and Z n and the electrolyte resistance R ion(n) ( Z SCM(n) = R ion(n) + ( Z SCM(n−1) −1 + Z n −1 ) −1 ). Fig. 3 shows Nyquist plots computed for each input parameter at 50 steps for non-Faradic and Faradic processes in the staircase model. The results are in good agreement with the profiles calculated for the transmission electric model. 2 The presentation will further show examples of the calculated results using the staircase model for non-ideal and inhomogeneous processes and discuss the analysis of impedance behavior using real electrodes reflecting non-uniform processes. References R. de Levie, Electrochim. Acta , 9 , 1231-1245 (1964). N. Ogihara, S. Kawauchi, C. Okuda, Y. Itou, Y. Takeuchi and Y. Ukyo, J. Electrochem. Soc. , 159 , A1034-A1039 (2012). N. Ogihara, Y. Itou, T. Sasaki and Y. Takeuchi, J. Phys. Chem. C , 119 , 4612-4619 (2015). N. Ogihara, Y. Itou and S. Kawauchi, J. Phys. Chem. Lett. , 10 , 5013-5018 (2019). Y. Itou, N. Ogihara and S. Kawauchi, J. Phys. Chem. C , 124 , 5559-5564 (2020). N. Ogihara and Y. Itou, Phys. Chem. Chem. Phys. , 24 , 21863-21871 (2022). Figure 1