The electric characteristics of electric-double layer capacitors (EDLCs) are determined by their capacitance which is usually measured in the time domain from constant-current charging/discharging and cyclic voltammetry tests, and from the frequency domain using nonlinear least-squares fitting of spectral impedance. The time-voltage and current-voltage profiles from the first two techniques are commonly treated by assuming ideal S_sC behavior in spite of the nonlinear response of the device, which in turn provides inaccurate values for its characteristic metrics. In this paper we revisit the calculation of capacitance, power and energy of EDLCs from the time domain constant-current step response and linear voltage waveform, under the assumption that the device behaves as an equivalent fractional-order circuit consisting of a resistance R_s in series with a constant phase element (CPE(Q, α), with Q being a pseudocapacitance and α a dispersion coefficient). In particular, we show with the derived (R_s, Q, α)-based expressions, that the corresponding nonlinear effects in voltage-time and current-voltage can be encompassed through nonlinear terms function of the coefficient α, which is not possible with the classical R_sC model. We validate our formulae with the experimental measurements of different EDLCs.

Black phosphorus (BP) is a two-dimensional material that has gained attention for various applications due to its exceptional physical and chemical properties such as tunable bandgap and high charge carrier mobility. However, the critical step towards the application of BP is still contingent on successfully exfoliating the bulk material into few-layer phosphorene nanosheets. Herein, we show a one-step, facile, and environmentally friendly method of bipolar electrochemical exfoliation-and-deposition of BP nanosheets on a positive feeding electrode. The exfoliated-and-deposited material is high-quality phosphorene with an orthorhombic crystal structure, and it adheres onto the substrate with a fractal and structured morphology. The electrochemical performance of the material was evaluated in a symmetric two-electrode configuration soaked in aqueous electrolyte for capacitive energy storage application. The results revealed fractional-order capacitive behaviour in the low-to-medium frequency, and a high stability and reversibility for at least 40 000 cycles. The power performance is outstanding with 0.35 mW cm⁻² at a constant current rate of 0.5 mA cm⁻².