Frequency Response of the Graphene Electrode with Finite Conductivity in Electrochemical Systems

We investigate the frequency-dependent electrochemical behavior of a finite-sized graphene disk electrode (GE) in contact with an electrolyte under a three-dimensional cylindrical configuration. The study is performed by modeling the GE with a finite conductivity derived from its intrinsic quantum-mechanical properties to derive an analytical expression for its quantum surface impedance and then examine the charge relaxation dynamics in both the frequency and time domains. The analysis reveals a transition from Warburg-type impedance in the high-frequency regime to RC-circuit behavior at low frequencies, governed by the quantum capacitance and conductivity of graphene. Next, to interface the GE with the electrolyte, the three-dimensional problem is reduced to a one-dimensional problem by introducing the surface averaging of the variables using the application of the Hankel transform. This enables us to generalize the impedance modeling of our recent work [J. Electroanal. Chem. 2023, 946, 117711] where GE was idealized as a uniformly charged capacitor. The results are discussed under the influence of ion concentration, electrode radius, and external circuitry both in the regime where the electrochemical impedance spectroscopy (EIS) and the electrochemical capacitive spectroscopy (ECS) are applicable and beyond.