In this chapter, we reviewed numerical method based on the FE space discretiza-tion and standard time discretizadiscretiza-tion for the DAREs. To that end, we follow the description on the FE found for example in the book by Brenner and Scott [38] or
Ern and Guermond [97]. In two dimensions, we have applied the combination of the FE discretization with the time integrators such that LI, Impl, ETD, EXPR and ROCK2 method to a variety of linear and non-linear DAREs. However, the main goal of this chapter is to describe the time integrators methods that will be used in this thesis.
One dimensional discontinuous Galerkin method for Cyclic
Voltammetry models
Contents
3.1 Introduction to cycle voltammetry . . . 34 3.2 DG for electron transfer only model . . . 37 3.3 DG for electro catalytic model . . . 72 3.4 Summary . . . 87 The biggest challenge in the study of the cyclic voltammetry models is to find the analytical solution of their governing equations. So the electro-chemists rely on the software package or finite difference methods to solve numerically these equations [33, 36, 34, 35]. But this approach is strongly inefficient while inverting the cyclic voltammetry models by fitting its signal response. Therefore in this chapter, we proposed a more efficient numerical method, to solve the governing equations of the cyclic voltammetry models. In order to do so, we first give a brief review on cyclic voltammetry in Section 3.1, then we investigate the numerical resolution, using DG space discretization, of two cyclic voltammetry problems respectively in Section 3.2 and Section 3.3.
The contribution here is the investigation of the cyclic voltammetry, based on the
novel combination of the DG space discretization and standard time discretization method. The DG method relies on the Legendre polynomials while the standard time discretization method used are Impl and ETD method. The DG analysis presented here, can be used for any diffusion and second order non linear reaction term: it has not been applied in this context before. Contrary to expectations, while combine with the DG spatial discretization, the standard implicit time integrator out performs the methods such as the ETD method and adaptive time stepping method ode15s. We also see that the DG method performs better than the standard FE method.
3.1 Introduction to cycle voltammetry
Cyclic voltammetry (CV) is a technique used to study electrochemical reaction mech-anisms that give rise to electroanalytical current signals. It has been frequently used by electrochemists for a variety of purposes, including chemical and biochemical sensing (e.g. glucose sensors [187], gas detectors [3, 200], pH meters [224]), tech-nological applications (e.g. electroplating [224], electrochromic displays [28, 192]), energy storage (e.g. solar cells [223], batteries [146]), imaging, synthesis, which underpin much of modern biology and nanotechnology [140, 158, 138]. There are several good texts that investigate the theory and practice of CV in depth, see for example [22, 163].
3.1.1 Cycle voltammetry experiment
CV involves applying a voltage to an electrode immersed in an electrolyte solution, and seeing how the system responds. Let us consider for example the electron transfer only process at the electrode represent as follows
Q−*)−kf
kb
Qn++ ne−, (3.1)
where n is the number of electrons transferred per molecule, the rate constants kb
and kf are given by the Buttle-Volmer kinetics Equation [66]. We have
kb = k0exp
(1 − α)nF
RT(E − E0)
, kf = k0exp
(−α)nF
RT(E − E0)
, (3.2)
where E(V ) is the potential applied to the electrode, F is Faraday constant, T (◦K) is the Kelvin temperature, R is the universal gas constant, α ∈ [0, 1] is the charge transfer coefficient, k0(ms−1) is the the value of the rate constants at the formal potential E0(V ). These rate constants describe how the flux of electrons in the electrode solution interface depends on the applied potential.
In a typical CV experiment, the potential is swept linearly with time from some starting potential, E1, where species Q is stable to some other potential, E2, at which electron transfer between species Q and the electrode is rapid, and species Q+ is formed. The potential is then swept back to E1, causing electron transfer in the opposite direction and the reformation of Q. The rate of change of the potential from the initial potential, E1 to the so called vertex potential, E2, and back again is called the scan rate(ν in V s−1) [66]. This potential waveform is shown in Fig 3.1.
Figure 3.1: The waveform of the potential applied during a typical cyclic voltam-metry experiment. In this case the initial potential, E1 = −10V , and the vertex potential, E2 = 10V , and the scan rate, ν = 0.1V.s−1.
On the forward sweep, the potential E, is given at any time t by E = E1+ νt.
At the time t = tswitch, the potential reach the reverse potential E2 and change the direction. On the reverse sweep, the potential E, is given at any time t > tswitch by E = E2−ν(t−tswitch) or equivalently E = 2E2−E1−νt, since the time the potential change the direction tswitch is (E2 − E1)/ν. The process can then be repeated in a periodic, or cyclic manner. Therefore, according to (3.2), the rate constants kb and kf are function of the time t (i.e. kb := kb(t) and kf := kf(t)).
Throughout this process the current, I, (proportional to the rate of electron transfer) is recorded. We plot in Fig 3.2, the current-potential curve (or voltam-mogram) where Ipc and Ipa are called the peak cathodic and peak anodic current.
The peak currents Ipc and Ipa are respectively associated the peak potential Epc and Epa. Note that in Fig 3.2, as the potential is scanned in the positive direction, the current rises to a peak and then decays. The current depends on two steps in the overall process, the movement of electroactive material to the surface and the electron transfer reaction.
Figure 3.2: Voltammogram produced by the application of the potential waveform.
We plot in this figure, a typical cyclic voltammogram where Ipc and Ipa show the peak cathodic and peak anodic current respectively associated the peak potential Epc and Epa
This technique is extremely useful experimentally as the resulting peak shaped
signal provides a direct fingerprint of the features of the reduction and oxidation processes. Analysis of the position and shape of the peaks can give important in-formation about the nature of the electrochemical process taking place and about the chemical species themselves. The modelling of the CV experiment requires the definition of the electrical perturbation applied as well as the system under study, in term of mass transport, boundary conditions and heterogeneous or homogeneous chemical reaction. Therefore the mathematical problems faced in CV involve the resolution of a system of PDEs by means of analytical, semi-analytical or numerical methods. The solution of the problems are the concentration profiles of the species present in the chemical reaction, from which the voltammogram is deducted. Never-theless it is not always feasible to use the analytical methods due to the complexity of the problems. The numerical methods offer a very accurate approximation to the true solution.
Unfortunately, simulation is usually obscure for non-theoreticians who often have to rely on software packages such as pdepe of MATLAB, which uses the FE method, described in [205], for the space discretization and ode15s algorithm, described in [201, 202], as time discretization. This will allow us to introduce the DG method in a way that allows any researcher or student to develop their own research and teaching tools for the study of voltammetry.