Model-free capacitance analysis of electrodes with a 2D+1D dispersion of time constants

A constant phase element (CPE) is often used in electrochemical impedance spectroscopy to describe the non-ideal impedance of solid/electrolyte interfaces. The generalization of its mathematical description leads to the concept of a variable or generalized phase element αgpe, which, in association with Brug's expression, can be used to determine the interfacial capacitance of metallic electrodes, whose time-constant distribution is restricted to a bidimensional plane (C2D). In this work, a heuristic approach is used to extend the generalization principle even further, in order to arrive at an expression of capacitance, Cgpe, that is valid for electrodes with time-constant dispersions that also spread into the electrode bulk, causing a 2D+1D time-constant distribution. As a proof of concept, the proposed method is applied in the analysis of the electrochemical impedance of n-doped hydrogen-terminated Si(100) electrode in 100 mM Na2SO4, in a potential range that extends from accumulation to depletion. The single crystalline structure and polished surface of the electrode grants a quasi-ideal capacitive behavior to the system that allows comparison of the proposed analysis with the traditional representation of effective capacitance Ceff. Both methods yield similar results in the depletion region, where the impedance response is predominantly defined by the ideal behavior of the silicon space charge region. In accumulation, however, when impedance is influenced by the somewhat larger distribution of relaxation times of the electric double layer, Ceff shows frequency dispersion, whereas Cgpe does not. A dimensional analysis of the functional elements in Cgpe reveals the underlying reason for this intriguing behavior and also shows that C2D is, in fact, just an approximate expression of the more general Cgpe. It is also shown that the generalized capacitance representation allows for a clear detection of the onset of size border effects, and these appear at a higher frequency than what is observed in the Ceff representation. [Display omitted]