Potential Step for Double-Layer Capacitances Obeying the Power Law

Abstract

Potential-step chronoamperometry was made at a platinum wire electrode in KCl aqueous solution at the aim of finding the behavior of the power law of the time or the constant phase element for the double-layer (DL) capacitances. The logarithmic current decays linearly with the time shorter than 0.1 ms, and then it obeys the power law in which it has a linear relation with the logarithmic time in the millisecond time domain. The transition from the exponential decay to the power law was expressed theoretically for the model of a series combination of the resistance and the DL capacitance. The expression predicts that the double logarithmic plots of the current–time provide a capacitance value at 1 s from the intercept, independent of the resistance. This prediction was demonstrated experimentally in KCl solutions of which concentrations ranged from 1 mM to 0.5 M. The capacitance can be evaluated simply by chronoamperometry on a 1 s time scale without considering any resistance effect. The capacitance values did not vary with the applied potential.

Similarity of Heterogeneous Kinetics to Delay of Double-Layer Capacitance Using Chronoamperometry

“…where C1s is the DLC per area at t = 1 s, and λ is the number close to 0.1, according to previous results [40]. It is not in the form of exp(−t/RsC) for a solution resistance, Rs and the DLC, C, because it largely deviates from an ideal capacitance due to frequency dispersion.…”

“…We describe the time dependence of DLC, which is not only reflected in heterogeneous kinetics but also helpful for subtracting the DLC components from the observed components. When voltage V is applied in a step form to DLC, the chronoamperometric current density at t > 0.3 ms can be calculated as follows [40]:…”

“…Each contribution is examined here in order to find which current contributes the most to c s . Chronoamperometric curves were calculated using Equations ( 1), ( 4) and ( 5) for our experimental values of c*, D [21], λ [40] and µ, where the gamma functions were evaluated with use of the approximate Equation (6.1.35) in [43]. They are shown in Figure 3A, and the Cottrell plots (I vs. t −1/2 ) are shown in Figure 3B.…”

“…Each contribution is examined here in order to find which current contributes the most to cs. Chronoamperometric curves were calculated using Equations ( 1), (4), and (5) for our experimental values of c*, D [21], λ [40] and µ, where the gamma functions were evaluated with use of the approximate equation ( 6 It is interesting to compare negative capacitance with Butler-Volmer kinetics, which is expressed in chronoamperometry as [47]: 1), ( 4) and ( 5) at c* = 1 mM, D = 0.7×10 −5 cm 2 s −1 , µ = 0.1, λ = 0.1, V = 0.2 V, C 1s = 30 µF cm −2 , C rx = 70 µF cm −2 and the electrode area 1.77 mm 2 .…”

Impedance Response of a Thin Film on an Electrode: Deciphering the Influence of the Double Layer Capacitance

“…Such an approximation is easy to verify a posteriori when data analysis is performed assuming that each of the contributions is a capacitance and that the order of magnitudes for the double layer capacitance is between 10 to 50 μF cm −2 for a wide range of electrolyte concentrations and over a large potential window [1] . However, both the electrochemical impedance of passive film [2–5] and double layer relaxation [6–8] usually exhibit a non‐ideal behavior, thus resulting in the use of constant‐phase elements (CPE) to account for the distribution of dielectric properties of the interface [8–10] as well as experimental factors (position of the reference electrode, adsorption of impurities) [11–13] …”