Electrode covering effect on the electric response of a cell to external stimulus
Introduction
The electric response of an electrolytic cell to an external voltage is well described by the Poisson–Nernst–Planck model based on the equations of continuity for the positive and negative ions, and on the equation of Poisson for the actual electric potential across the sample. It has been described in details by Macdonald considering different types of boundary conditions [1]. In the simplest case the electrodes are assumed blocking. In this framework the ionic current densities vanish on the electrodes and in the external circuit the electric current is just a displacement current due to the time variation of the surface electric displacement. The case where the electrodes are not blocking and a dc current is possible has been described using of two different models. According to the first model [2], the conduction current in the external circuit is supposed proportional to the surface variation of the bulk density of charges with respect to the thermodynamic equilibrium value (infinite sample, in the absence of external electric field). In the second model [3] the conduction current is assumed proportional to the surface electric field. Several papers have been published on the effect of the partially blocking character of the electrodes on the electric response of a cell to a ac or dc external voltage [4], [5], [6], [7], [8], [9], [10]. All the papers consider the limiting electrodes homogeneous. It is well known that the transfer of charge from the cell to the external circuit can take place via the formation of an intermediate state related to the adsorption [11], [12] or formation of bubbles on the electrodes [13], responsible for a non-uniformity of the electrodes. This non-uniformity is related to the modification of the surface properties of the electrodes concerning the charge exchange with the external circuit. For instance, in the case of a photoelectrochemical cell for water splitting small bubbles are formed on the nucleation centers of the anode surface. The concentration and the size of the bubbles depend on the operational parameters of the cell, as current density, electrolyte flow conditions, pressure or temperature. As the bubbles adhere on a part of the electrode surface, this is no longer homogeneous and the electric response of the cell depends on the surface density of the bubbles, on their characteristic size and also on their residence time on the electrode surface [14], [15]. The problems related to the bubbles formation have been analyzed [16], [17], [18], [19], [20], [21] mainly from the mechanical point of view. A model taking into account the surface tension of the bubble with the electrodes and with the liquid, to determine the critical dimension of the bubble has been proposed [22]. Also the role of the flow of the liquid in contact with the electrode on the departure of the bubble [23], as well as of the influence of the Archimede buoyancy force on their motion have been investigated [23].
In our paper we analyze the role of the surface non-homogeneities on the electric response of the cell. We consider the case where the external voltage is periodic of amplitude and circular frequency and on one electrode there is a deposition of ions or formation on bubbles. The sample is assumed a slab of thickness d, limited by square electrode of area S, of side , with , in such a manner that the mathematical problem can be supposed one-dimensional. This means that all the physical quantities necessary for the description of the system, as the ionic bulk density, the ionic current density or the electric potential, depend only on the distance from the electrodes and on the time. In the absence of an applied voltage, the electrodes are supposed homogeneous. The presence of the ionic surface deposition or of the adhering bubbles on one of the electrodes is responsible for the surface modulation of the properties of the electrode describing the charge exchange between the bulk and the external circuit. Our description is based on the concept of equivalent electrode, defined as formed by two homogeneous parts, one not covered, with the surface area , and the other having a covering material, with the area , such that . This assumption implies that the transversal variations of the physical quantities characterizing the properties of the sample are very small with respect to the variations along the normal to the electrodes. In the slab approximations this hypothesis is usually verified when the bubble adhering or the material deposition are random mechanisms on the electrodes. In this approximation, the sample can be considered, from the electric point of view, as a parallel of two cells limited by homogeneous electrodes of surface area and .
Section snippets
The system
We consider a cell in the shape of a slab of thickness d and surface area S. The z-axis of the cartesian reference frame is normal to the limiting surfaces, coinciding with the electrodes and located at z = ±d/2. The dielectric constant of the medium is indicated by and supposed, in the considered frequency range, not dispersive. The bulk density of ions responsible for the observed dielectric dispersion of the cell is indicated by N, and the electrical charge by q. We assume that only the
Blocking and transparent electrodes
In the case when the electrodes are perfectly blocking, for the reactance presents a minimum and the resistance, in the series representation, has a change of curvature [27]. It is convenient to measure the circular frequency in units of , and express the impedance in units ofwhich has the dimension of a resistance and it is independent of the thickness of the sample. In the following we will consider the cases: (1) the two electrodes are blocking, subscript b, (2) the
Covering effect on the electric impedance
Let us consider now a cell limited by transparent electrodes and assume that due to the material deposition on an electrode the surface properties are changed from transparent to blocking. In the case of the bubbles adhering the surface modification is easily understood because in the region occupied by the bubbles no active sites for the reactions are present. In the other case where the exchange of charge is made via the formation of an intermediate state due to adsorption, the presence of
Covering effect on the complex dielectric constant
In this analysis the spectra of the real, , and imaginary, , parts of the dielectric constant of the cell are considered. Using the complex dielectric constant is possible to define a complex capacitance related to the impedance of the cell by , in absolute units, or byin dimensionless form. The relation between the real and imaginary parts of and the impedance of the cell isas it follows form Eq. (11). From Eq. (12), taking into account that
Conclusions
We have investigated the effect of the charge exchange properties of an electrode on the electric response of an electrolytic cell when a material is deposited on the electrode surface. The cases when the electrodes are perfectly transparent to the charges or when deposited material covered electrodes are blocking have been considered. In this framework the electrochemical cell is described as a parallel of two electric impedances, relevant to the uncovered and covered parts of the electrodes.
Conflict of interest
There is no conflict of interest.
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