Frequency Dependence of Effective Capacitance Cec for Polyaniline Membrane-Based pH Sensor and its Extension to the Gouy–Chapman–Stern Model

This study proposed an effective capacitance (Cec) for bare and conducting polymer-covered electrodes using electrochemical impedance spectroscopy (EIS). Bare electrodes show three regimes: potential-dependent Helmholtz capacitance, Gouy–Chapman–Stern diffusion capacitance (1 MHz–10 Hz), and complex low-frequency responses, deviating from semi-infinite Warburg diffusion (1 Hz–10 mHz). Polyaniline (PANI) and poly(3,4-ethylenedioxythiophene) PEDOT-based electrodes exhibit larger potential-dependent diffusion pseudocapacitance (1 MHz–10 Hz) and the absence of a Warburg tail or a nearly horizontal low-frequency slope at 0.01–0.026 (1 Hz–10 mHz). A high-frequency Cec of a PANI membrane correlates with bulk electrolyte concentration, while bare electrodes are less affected and dominated by Helmholtz capacitance. The equivalent circuit of the time-dependent EIS impedance spectrum for bare electrodes and PANI and PEDOT-based electrodes shows parallel capacitor behavior in combination with high-frequency capacitance (1 MHz–10 Hz) and a low-frequency response (1 Hz–10 mHz). The mathematical simulation of effective capacitance Cec with respect to time period t (f−1) follows two time constants (τ = RC), representing double-layer capacitance or pseudocapacitance (τ1) and complex low-frequency responses or Warburg diffusion (τ2) for bare electrodes and conducting polymer-based electrodes, respectively. This simulation analysis also elucidates the frequency dependence of the Warburg characteristic frequency (ω) and the extension of the double-layer capacitance diffuse distance LD for H+ with GC electrodes to approximately 7.37–15.15 μm over a time interval of ca. 1 s (t = f−1). The diffusion coefficient Di of K+ ion transfer through a PEDOT solid contact from 1 mC (0.1 µm) to 10 mC (1 µm) is in the range of 0.57–12 × 10−10 cm2·s−1, following a power law with an exponent of 1.75 with respect to the polymerization time of PEDOT, which is inconsistent with Fick’s law.