We investigate the role of the diffuse layer of the ionic cloud on impedance spectroscopy measurements. The analysis is performed assuming that the ions have equal mobility, the electrodes are perfectly blocking and adsorption phenomenon can be neglected. We find that the dielectric permittivity, in the limit of high frequency ω, tends to the dielectric permittivity of the pure liquid as ω^−3/2. The relationship between the detected equivalent permittivity and conductivity of the cell with the real and imaginary part of the complex dielectric constant is discussed. We show also that the presence of the ions is responsible for a distribution of relaxation times. An application to nematic liquid crystals is presented.

The analogy between a transmission line and an electrolytic cell is not physically grounded.

Many fundamental physical problems are modeled using differential equations, describing time- and space-dependent variables from conservation laws. Practical problems, such as surface morphology, particle interactions, and memory effects, reveal the limitations of traditional tools. Fractional calculus is a valuable tool for these issues, with applications ranging from membrane diffusion to electrical response of complex fluids, particularly electrolytic cells like liquid crystal cells. This paper presents the main fractional tools to formulate a diffusive model regarding time-fractional derivatives and modify the continuity equations stating the conservation laws. We explore two possible ways to introduce time-fractional derivatives to extend the continuity equations to the field of arbitrary-order derivatives. This investigation is essential, because while the mathematical description of neutral particle diffusion has been extensively covered by various authors, a comprehensive treatment of the problem for electrically charged particles remains in its early stages. For this reason, after presenting the appropriate mathematical tools based on fractional calculus, we demonstrate that generalizing the diffusion equation leads to a generalized definition of the displacement current. This modification has strong implications in defining the electrical impedance of electrolytic cells but, more importantly, in the formulation of the Maxwell equations in material systems.

The ionic distribution induced by an external field is investigated by means of the Poisson-Nernst-Planck model, by taking into account the non-blocking properties of the limiting electrodes. Three types of models proposed for the description of real electrodes are considered. The first two assume an ionic current on the electrodes proportional to the variation of the bulk density of ions and to the surface electric field, respectively. The third model assumes that the sample is limited by perfectly blocking electrodes with a true resistance in parallel to the cell. Here we show that the first two models are equivalent, in the sense that it is possible to find a phenomenological parameter by means of which the predictions of the two models, for what concerns the spectra of the real and imaginary parts of the impedance of the cell, are the same. On the contrary, the third model is equivalent to the others only if the conduction current across the electrodes is small with respect to the displacement current.

We analyze in which experimental conditions the concept of electrical impedance is useful for an electrolytic cell. The analysis is performed by solving numerically the differential equations governing the phenomenon of the redistribution of the ions in the presence of an external electric field and comparing the results with the ones obtained by solving the linear approximation of these equations. The control parameter in our study is the amplitude of the applied voltage, assumed a simple harmonic function of the time. We show that the bulk distribution of ions close to the electrodes differs from the one obtained by means of the linear analysis already for small amplitudes of the applied voltage. Nevertheless, the concept of electrical impedance remains valid. For larger amplitudes, the current in the circuit is no longer harmonic at the same frequency of the applied voltage. Therefore the concept of electrical impedance is no longer meaningful.