Frequency Transition from Diffusion to Capacitive Response in the Blocked-Diffusion Warburg Impedance for EIS Analysis in Modern Batteries

The use of the Blocked-diffusion Warburg (BDW) impedance within electrochemical impedance spectroscopy (EIS) measurements can unveil diffusion properties of the electroactive material of modern batteries at different states-of-charge. The impedance response of the BDW comprises a diffusion response of charge carriers through a short-diffusion distance (e.g. the solid-phase in electroactive material of battery electrodes) and a capacitive response due to accumulation of charge carriers in a blocked-interface (e.g. impermeable current collector of a thin film electrode). This study has developed a mathematical expression based on the Newton-Raphson iteration method to calculate the frequency and time constant during the transition from diffusion to capacitive response in the BDW impedance. The mathematical procedure to calculate the frequency during the diffusion-capacitive transition response in the BDW has been written in a script in Matlab® software and is applied to BDW impedance responses reported in previous studies and extracted from EIS measurements in Li-ion and NiMH batteries. This study demonstrates that the time constant during the transition from diffusion to capacitive response in the BDW differs from the characteristic time constant commonly represented in the BDW mathematical expression. The characteristic time constant represented in the BDW mathematical expression is related to the rate of accumulation of charge carriers in the blocked-interface of the electrode. On the other hand, the time constant during the transition from diffusion to capacitance responses in the BDW impedance can be related to diffusion properties in solid-phase particles with heterogeneous size distribution in the electroactive material of modern battery electrodes.