A model for electrode effects based on adsorption theory
Introduction
Electrode effects have been observed and theoretically analyzed by several authors [1], [2], [3]. Starting from the pioneeristic work of Kellogg [4] it has been observed that at a certain value of voltage between two electrodes in an aqueous electrolyte, there is significant deviation from the standard electrolytic regime: the gas formed at the electrodes, for a critical value of the potential, coalesces, forming a unique continuous gas, which acts as an envelope around the electrodes, giving a sudden drop down in the current at the electrodes, normally followed by a luminous discharge (glow discharge plasma [5], [6]). This process called also electrolytic plasma process has been widely used for practical purpose as surface treatment process for generating oxide coatings on metals, in a similar way to anodizing, but in this case the resulting plasma modifies the structure of the oxide layer [7], [8].
To our knowledge an analytic model describing all this process is still missing. Good reviews on the electrochemical discharges, relevant to its discovery and recent applications, have been published by Gupta et al. [9] and by Wuthrich and Bleuler [10]. In a previous paper Wuthrich and Bleuler proposed also an interesting model for the electrode effects based on the percolation theory [11]. In the same way, our group recently used a percolation approach in order to study the time variation of the photoelectric process during bubbles generation [12]. The goal of the present work is to develop a model for the same effect based on the adsorption phenomenon. The adsorption phenomenon is supposed to be well described by a kinetic equation similar to Langmuir's isotherm, valid in the limit of small adsorption [13]. We assume that the electrical response of the working cell can be described by an electrical circuit formed by a bulk resistance in series with a surface layer characterized by a resistance and a capacitance [14]. Due to the adsorption phenomenon, the surface resistance and capacitance are the parallel of the covered and uncovered parts [15]. In this framework we show that the time dependence of the current contains two characteristics time: one short, related to the charging of the surface layer, and one long related to the kinetics of the coverage. The predictions of the model are validated with experimental data concerning the time dependence of the current and the current-voltage characteristics of a photo-electrolyzer, which is constituted by a BiVO4 photoanode and a Pt cathode assembled in a home-designed cell. In addition, a normalized current-voltage curve is compared with literature data based on another electrolytic applications, in order to assess the versatility of the developed model.
This paper is organized as follows. In Sect. 2 the electrical response of a simple circuit is considered, to show the divergence of the electric current in an ideal electrochemical cell modeled as a parallel formed by a resistance and a capacitance only. The presence of a bulk resistance on the time dependence of the current is discussed in Sect. 3, whereas the analysis of an electric circuit simulating a real cell is reported in Sect. 4. The influence of the adsorption on the covering effect is considered in Sect. 5, where the kinetic equation used in the analysis is introduced. In that section a simple expression for the time dependence of the covering ratio, valid when the bulk resistance of the cell is negligible with respect to that of the naked interface layer is deduced. The comparison of the theoretical predictions with the experimental data, along with the best obtained fit, is presented in Sect. 6. The final Sect. 7 is devoted to the conclusions.
Section snippets
Electrical response of a simple circuit
Let us consider a simple cell whose electrical response to an external excitation can be simulated by an electrical resistance Rs in parallel with a capacitance Cs. When the circuit is submitted to a voltage V0(t), due to an external power supply, the electric current in the circuit is i(t) = iR(t) + iC(t). In this relation iR(t) is the conduction current in Rs and iC(t) the displacement current in Cs (see Fig. 1a). By means of simple considerations we get iR(t) = V0(t)/Rs and iC = CsdV0(t)/dt, and the
Real Cell
In a real cell, in the presence of the electrode effect, the electrode is partially covered by bubbles coming from the bulk. In this case, the electrical response of the cell can be simulated by the circuit shown in Fig. 1b: the series of a bulk resistance Rb and a surface layer whose resistance, Rs, and capacitance, Cs, are given bywhere the subscripts c and nc refer to the covered and uncovered part of the electrode. According to (3) the effective resistance and
Analysis of the electric circuit simulating a real cell
From the discussion reported above for what concerns the electric response the cell can be simulated by the series of Rb with the parallel (Rs, Cs). When the cell is subjected to an external voltage V0(t) we havewhere, as above, iR(t) is the conduction current across Rs. Furthermore, since Rs and Cs are in parallel we have RsiR(t) = Q/Cs, where Q is the electrical charge on Cs. From this condition we get Q = RsCsiR. The total current in the circuit is then i(t) = iR + dQ/dt, and
Kinetic equation for the covering effect
In a working electrotytic cell the coverage is related to the electric charges that during the deposition produce bubbles of gas on the surface. These bubbles change the properties regulating the exchange of charge from the bulk and the external circuit, and hence the electrical properties of the interface bulk-electrode [15]. We assume that the kinetic of the formation of bubble on the electrode is described bywhere β = Sc/S is the coverage. According to Eq. (16) the bubble
Description of the experimental set-up and measurements
The electrical response of an electrochemical cell for the water photo-electrolysis reaction (see Fig. 3a) was characterized, in order to use the experimental data for the validation of the here proposed model. In a photo-electrolyzer, water is splitted into its components, i.e. oxygen and hydrogen, mainly using solar energy. A compact home-designed case similar to the one reported in our previous works [21], [22] was used for the experiments, in order to minimize the distance between the
Conclusions
We have proposed a model for the electrode effects based on the adsorption mechanism. It has been assumed that the electric response of a working electrochemical cell is well described by a simple circuit formed by a bulk resistance in series with a surface layer characterized by a resistance in parallel with a condenser due to the properties of the liquid-electrode interphase. During the formation of the bubbles on the electrode, the electrical properties of the interface change. In
Acknowledgment
Many thanks are due to N. Penazzi for useful discussion and to Mauro Raimondo for performing FESEM measurements. The financial support from the European Commission on the 7th Framework Program (NMP-2012 Project Eco2CO2 nr.309701 and FCH-JU Call 2011-1 Project ArtipHyction nr.303435) is gratefully acknowledged.
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