Frequency dependence of the phenomenological parameters describing adsorption processes in supercapacitors
Introduction
The electric response of an electrolytic cell is usually described by means of a continuous model based on the equation of continuity for the ions present in the medium, and on the equation of Poisson for the actual potential distribution across the cell, known as the Poisson-Nernst-Planck (PNP) model [1]. These equations have to be solved taking into account the boundary conditions imposed on the limiting electrodes. We have recently investigated the voltammogram of electric double layer capacitors and found that boundary conditions based on the adsorption phenomenon of ions on the electrodes describe well our experimental data [2]. However, in our previous work, from the fit procedure, it was clear that the parameters describing the adsorption phenomenon on the electrodes are frequency dependent. At a first sight this experimental result looks rather strange because the adsorption phenomenon is related to an adsorption energy [3], which is supposed independent of the frequency of the external field responsible for the ion accumulation close to the electrodes. The goal of the present paper is to investigate the influence of a random distribution of the potential energy responsible for the adsorption phenomenon on the effective adsorption coefficient and desorption time. In our analysis we consider the case in which the kinetic equation describing the adsorption phenomenon is of Langmuir type [4]. It is characterized, in the ideal case of homogeneous surface, by frequency independent adsorption coefficient κ and desorption time τ. To investigate the frequency dependence of the effective adsorption coefficient and desorption time, the Lissajous figures relevant to the electric current across the cell and to the applied potential, for a well defined frequency, are investigated for different types of electrodes. To theoretically relate the experimental data with the proposed model, by means of the PNP model we determine the electrical impedance of the cell, and compare the current-voltage characteristics evaluated in this framework with our experimental data. Since the theoretical expression of the impedance depends on the adsorption parameters, from the best fit of the Lissajous figures we determine the effective adsorption coefficient, κ, and relaxation time, τ, versus the angular frequency of the applied voltage, ω.
According to our experimental data, the effective κ and τ strongly depend on ω. To interpret our results, we extend the model of adsorption assuming a distribution of relaxation times. Other models have been recently proposed to investigate the influence of the adsorption phenomenon on the response of an electrochemical cell, based on fractional diffusion or introducing a kernel in the integro-differential equation representing the boundary conditions, which is not only related to simple relaxation mechanisms [5,6]. The analysis presented in Refs. [5,6] discusses in details the possible mechanisms responsible for the adsorption phenomenon. Nevertheless, the proposed theoretical model is rather complicated, and the physical meaning of the different contributions to the boundary conditions is not obvious. The analysis presented by us follows, somehow, the one proposed for the conduction in disordered solids, based on the concept of diffusion time distribution [7]. Our model is simple, and the number of phenomenological parameters small. However, as it is shown, its theoretical predictions for the frequency dependencies of κ and τ well compare with the experimental data.
Our paper is organized as follows. In Sect. 2 the Langmuir model is presented. Starting from the kinetic equation for the adsorption on the limiting surface, we show how it is possible to rewrite it in the integral form used to describe the time evolution for a generic linear system. Using this equation and assuming a distribution of relaxation times, expressions for the effective adsorption coefficient and desorption time are deduced. In Sect. 3 the electrodes’ preparation and the experimental set up used to study the electrical response of the cells are reported. Section 4 is devoted to the evaluation of the effective adsorption parameters, based on the analysis of Lissajous figures. In Section 5 the frequency dependence of the effective adsorption parameters is presented, and discussed in terms of the model proposed. Section 6 is devoted to the Conclusions.
Section snippets
Adsorption phenomenon and kinetic equation
Adsorption phenomena are often investigated by means of the Langmuir isotherm [4].where κ and τ are the adsorption coefficient and the relaxation time, respectively. In Eq. (1) is the bulk density of particles just in front of the surface, and the surface density of particles already adsorbed. According to the kinetic equation (1) the rapidity of adsorption at time t depends on the bulk and surface density at the same time t. The main characteristic of Eq. (1) is
Experiment
Electrochemical cells were tested in EL-CELL® AQU version for aqueous electrolyte and STD for aprotic electrolyte. AQU cells are specifically designed to remove any contribution coming from equipotential surfaces that may react, i.e. the cell is hosted in a polymeric environment made of PolyEther Ether Ketone (PEEK)and contacted through gold probes on electrodes’ back sides. STD ones are composed by stainless steel 316 L package which is inert in aprotic environment and stable with common
Evaluation of the adsorption parameters
To test the model proposed in Sect.2, we have investigated the electric response of a cell, in the shape of a slab, to a monochromatic external difference of potential , of amplitude and circular frequency ω. In order to be in the linear limit, the amplitude of the applied voltage was mV. The Lissajous figures representing the current across the cell versus the applied difference of potential are of elliptic shape, confirming the linear response of the cell to the
Results
From the best fit of the Lissajous figures, we have determined the frequency dependence of the effective parameters entering the Langmuir kinetic equation. For different ion concentrations, NaCl 1 M and, respectively, 0.1 M, in Fig. 4 we report the frequency dependence of κ (a) and τ (c) for the cell with aqueous electrolyte whereas in Fig. 4 (b) and (d) the dependence of the same parameters for the cell with organic electrolyte, for the GF cells. In our best fit procedure we determine not only
Conclusions
We have shown that a random distribution of relaxation times could be responsible for a frequency dependence of the effective adsorption coefficient and desorption time. The presented model allows to derive mathematical expressions for the effective parameters entering into the kinetic equation of the adsorption of Langmuir type. Our analysis is based on a generalization of the theory proposed for linear systems, where the response function involves a homogeneous distribution of relaxation
Acknowledgements
Many thanks are due to Marco Fontana (IIT-Torino) for the FESEM images, and to Mara Serrapede (IIT-Torino) for very useful discussions. This work was supported by the MEPhI Academic Excellence Project (agreement with the Ministry of Education and Science of the Russian Federation of August 27, 2013, project no. 02. a03.21.0005).
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