Diffusion impedance of electroactive materials, electrolytic solutions and porous electrodes: Warburg impedance and beyond

The diffusion coefficient is a key property of materials. Electrochemical impedance spectroscopy (EIS) is a routine tool to determine the diffusion coefficient. Albeit being versatile for varied electrochemical systems and powerful in distinguishing multiple processes in a wide frequency spectrum, the EIS method usually needs a physical model in data analysis; misuse of models leads researchers to provide unwarranted interpretation of EIS data. Regarding diffusion, the simple and elegant formula developed by Warburg has been serving as the canonical model for more than a century. The classical Warburg model has very strict assumptions, however, it is used in a wide range of scenarios where assumptions may not be satisfied. It is the main purpose of the present article to define the boundary of applicability of the Warburg model and develop alternative models for cases beyond the boundary. In so doing, the Warburg model is revisited and its limitations and assumptions are scrutinized. Afterwards, new impedance models for more complicated and realistic scenarios are developed. The present article features: (1) generalization of the boundary condition when treating diffusion in bounded space and geometrical variants; (2) diffusion impedance in porous electrodes and fractals; (3) the effect of electrostatic interactions and coupling between diffusion and migration on the diffusion impedance in electrolytic solutions; (4) introduction of homotopy perturbation method to treat the convective diffusion; (5) physical interpretations of diffusion impedance behaviors.