Asymptotic Derivation of Equivalent Circuit Model for the Stack-Electrode Supercapacitors

Supercapacitors are promising electrochemical energy storage devices due to their prominent performance in rapid charging/discharging rates, long cycle life, stability, etc. Based on a recently proposed stack-electrode model, this work develops a matched asymptotic expansion method to derive a generalized equivalent circuit model for the description of the charging dynamics of supercapacitors with porous electrodes. Coupling leading-order solutions between every two stacks by continuity of ionic concentration and fluxes leads to an ODE system, which is a generalized equivalent circuit model for zeta potentials, with the potential-dependent nonlinear capacitance and resistance determined by physical parameters of electrolytes, e.g., specific counterion valences for asymmetric electrolytes. Linearized stability analysis on the ODE system after projection is developed to characterize the charging timescale. Validity of the derived asymptotic solutions is numerically verified. Further numerical investigations on the charging timescale demonstrate that the proposed generalized equivalent circuit model, as well as companion linearized stability analysis, can faithfully capture the charging dynamics of symmetric/asymmetric electrolytes in supercapacitors with porous electrodes.