On a possible physical origin of the constant phase element

Electrochemical impedance spectroscopy of La0.6Sr0.4Co0.2Fe0.8O3-δ nanofiber cathodes for intermediate temperature-solid oxide fuel cell applications: A case study for the ‘depressed’ or ‘fractal’ Gerischer element

“…Indeed, it has been demonstrated on theoretical grounds that a fractal geometry of the interface causes frequency dispersion; nevertheless, a correlation between the fractal dimension of the surface and the CPE exponent α has not been found [24]. Recently, Niya et al [25] have proposed that the anomalous diffusion theory can offer a theoretical demonstration of Eq. 1, and that the CPE power α is equal to the subdiffusion power, being related to the fractal dimension and the pore size distribution of the porous media.…”

A General Equivalent Electrical Circuit Model for the characterization of MXene/graphene oxide hybrid-fiber supercapacitors by electrochemical impedance spectroscopy – Impact of fiber length

“…with the CPE impedance being given by [4,40] 𝑍 𝐶𝑃𝐸 𝑐𝑡/𝑑𝑙 = 1 (iω) 𝛼 𝑐𝑡/𝑑𝑙 Q 𝑐𝑡/𝑑𝑙 (5) and 0 ≤ 𝛼 𝑐𝑡/𝑑𝑙 ≤ 1 is the CPE exponent and 𝑄 𝑐𝑡/𝑑𝑙 the CPE magnitude or apparent capacitance (in 𝐹. 𝑠 −𝛼 ). CPEs allow to enhance the fit quality of Nyquist plots showing a depressed semi-circle centered below the x-axis that cannot be properly simulated with classical capacitances.…”

“…Nevertheless, discussion remains about the physical meaning of CPEs, with explanations being often linked to the porous nature of the electrodes such as surface roughness [42,43], fractal dimension [44,45] or pore size distribution [46]. Niya et al [40] used anomalous diffusion theory to derive analytically the expression of the CPE impedance. They demonstrated that the CPE exponent 𝛼 is equal to the subdiffusion power and can give information about the impact of the porosity on ion diffusion.…”

Reliability of Poisson–Nernst–Planck Anomalous Models for Impedance Spectroscopy

“…Such scenarios may be attributed to unusual relaxation processes related to non-Debye relaxations, which may be handled in terms of a fractional kinetic equation. Furthermore, the asymptotic result for the impedance is related with the behavior exhibited by CPEs, which in turn may be related to differential operators of fractional order. , Thus, in order to cover a broad set of relevant experimental situations, we unify and extend, from the formal point of view, the previous boundary conditions to a fractional one with the possibility of describing a wide range of scenarios depending on the choice of the kernel κ­( t , ν) and the fractional differential operator. This extended approach is able to reproduce the behavior exhibited by the experimental data and permits one to consider the superposition of different surface phenomena according to the choice of the kernel.…”