CHEM-E4106 - Electrochemistry D, Lecture, 9.1.2023-21.2.2023

The analysis of all transient methods starts from Fick’s second law, Equation (3.54) and the boundary condition (3.48). We only consider a trace ion in a one-dimensional, semi-infinite space x ≥ 0. The Fick’s 2^nd law is first converted into the Laplace domain:

where the upper bar denotes a variable in the Laplace domain; $c_i^b$ is the initial or the bulk concentration of ion i, and s is the Laplace variable*. The general solution of this ordinary differential equation is

7.1.1 Potential step

In the potential step method, the electrode potential is stepped at t = 0 to a new value and then kept constant. If the reaction is reversible, the surface concentrations of the redox couple obey the Nernst equation:

Let us assume that we initially only have reduced species R present and the potential step is made in the anodic direction, i > 0, whereby c_R at the electrode surface is lowered due to its oxidation. Hence

because $c_o^b=0$. Using Equation (7.6) current density $\bar i_s$ can be solved:

If the potential is stepped to such high anodic potential $\xi \theta \gg1$ that the surface concentration of R immediately drops to zero, we see directly from Equation (7.7) that

Equation (7.11) is known as the Cottrell equation, which states that, along with Equation (7.10), current under diffusion control decays in a manner proportional to t^−1/2. These equations bear the peculiarity that at time t = 0 current should be infinite. This naturally never happens because a small solution resistance R_s always remains between the electrode and the Luggin capillary (see Chapter 4). Current at t = 0 is therefore equal to $\Delta E/R_s$ where $\Delta E$ is the height of the potential step. Electrodes also always have double layer capacitance, C_dl, and charging this capacitance makes current decay exponentially, $i_c=(\Delta E/R_s)e^{-t/(R_sC_{dl})}$. Total current is the sum of this charging current, i_c, and diffusion current, Equation (7.10) or (7.11). Charging time is, however, very short on solid electrodes, only a few microseconds, and usually we do not notice it at all.

The current from Equation (7.10) can be written in terms of the Cottrell current, Equation (7.11):

When current $i(t)=\frac{1}{2}i_d(t)$, the final term of this equation vanishes and we have reached the half-wave potential E_1/2:

If an electrode reaction has a limited rate, the Nernst equation (7.5) cannot be used as the boundary condition, but Equation (6.12) is used instead. In the Laplace domain it reads

where k_ox and k_red are the rate constants of the oxidation and reduction reactions, respectively, as discussed in the previous chapter. Inserting the surface concentrations, Equations (7.7) and (7.8), into this equation, current density takes on the following form:

Equation (7.19) is of the form $\frac{1}{\sqrt{s}(\sqrt{s}+\lambda)}$ for which the inverse transform is found tabulated:

At low values of x the function can be linearized, and current density is given by:

Plotting current density as a function of $\sqrt{t}$ and extrapolating to zero, the intercept at the current axis gives the value of k_ox. In principle, this could be seen directly from current at t = 0, but this value may be blurred due to the charging current or the limited response time of the potentiostat.

Potential steps are mainly used to determine electrochemical rate constants, as indicated above. In systems in which an electrode reaction is accompanied by a homogeneous reaction, a second step to the reverse potential can be made. The response to the second step now depends on the rate of the homogeneous reactions, which can be utilized in the analysis; these problems are beyond the scope of this book.

7.1.2 Linear potential scan or cyclic voltammetry

Linear potential scan or cyclic voltammetry is probably the most commonly applied electrochemical method. Although its calculus is more complicated than that of a potential step above, and although the accurate determination of kinetic parameters is prone to errors from various sources, it is an excellent semi-quantitative method. With reasonable training and experience, it is easy to deduce what kind of phenomena will prevail in the system. Estimates of the electrode reaction rate and the number of electrons exchanged are readily available, and the presence of an adsorption step or a following homogeneous reaction can easily be detected, for example. More specific reaction mechanisms, such as disproportionation also have their own very characteristic responses. The analysis of an electrochemical system is  therefore commonly initiated with potential scans at varying scan rates, after which a more specific method can be chosen that applies to that particular problem.

As the name tells, the electrode potential is scanned linearly as a function of time. If the scan is reversed immediately from the end value back to the initial potential, the procedure is called cyclic voltammetry. The scan rate is usually kept the same in the forward and backward scans, although this is not mandatory; the theory of cyclic voltammetry has been developed for constant scan rates. The potential function in cyclic voltammetry can formally be written as

Because the quantity $\theta$ in Equation (7.5) is no more constant, the Nernst equation cannot be transformed to the Laplace domain (Equation (7.6)). Instead, the boundary condition is written as

In the above equations, u is the dummy time variable. Inserting Equations (7.24) and (7.25) into Equation (7.23) we find that 

$\xi$ is defined earlier. The problem is reduced to the solution of the convolution integral on the left-hand side where the current function resides. In addition, the integral has a singularity at the upper integration limit. Equation (7.26) is written in a dimensionless form for the numeric solution. The time variable u is replaced with a dimensionless potential variable $\theta= \sigma u$, leading to $du=d \tau/ \sigma$. The current function i(u) is formally replaced with a new function $g( \tau)$. The convolution integral is now in the form of

This is further divided by $nF\sqrt{\pi D_{\text{O}}}c_{\text{O}}^b$, resulting in the dimensionless current function $\chi( \tau)$:

where $C_r=c_{\text{O}}^b/c_{\text{R}}^b$. The real physical current (density) is expressed by the dimensionless current (density) as

1.            The peak current density, i_p, is

         Equation (7.33) is known as the Randles-Sevcík equation. It is most often used to determine the diffusion coefficient of a trace ion by plotting peak currents as the function of $\sqrt{v}$. Alternatively, if the diffusion coefficient is known, the concentration can be calculated.

2.      The potential of the peak current, E_p, is related to the half-wave potential, E_1/2, with

         E_p/2 is the potential at which current is half of the peak current; finding this potential from a graph is easier than E_p because the current peak may be rather flat.

3.      The separation between the current peaks in forward and reverse directions, $\Delta E_p$, is 59/n mV at 25 ºC; this is the diagnostic criterion for the reversibility of an electrochemical reaction.

4.      The mid-point of the potentials of the peak currents is ~E_1/2.

If an electrode reaction has a limited rate, the Nernst equation is replaced by the kinetic equation (6.12) into which the surface concentrations, Equations (7.24) and (7.25) are added. Potential is now included in the rate constants k_ox and k_red, Equation (6.10). The numeric solution is rather cumbersome, so we will only present the results. The sluggishness of an electrode reaction is seen as an increase of $\Delta E_p$ as the function of the scan rate and as a decrease of the peak currents. Table 7.1 shows n$\Delta E_p$ as the function of the dimensionless parameter $\psi$.

$\sigma$ and $k^0$ are defined in Chapter 6, and $\xi$ and $\sigma$ are as above.

As Table 7.1 shows, the peak separation $\Delta E_p$ is practically the same as in the reversible case when $\psi$ > 20. A typical value of a diffusion coefficient is of the order of 10^-5 cm² s^-1, and since the scan rate cannot be increased without limits – for technical reasons 1 V s⁻¹ often is the upper limit – the upper limit of k⁰ that can be determined with cyclic voltammetry is of the order of 0.5-1 cm s⁻¹. The accuracy of the analysis based on Table 7.1 is not very good at higher values of $\psi$: When $\psi$ decreases from 20 to 5, $n\Delta E_p$ increases only 4 mV, which is the level of measurement accuracy due to the flatness of the current peaks. Yet cyclic voltammetry is an excellent preliminary method because if we see, for example, that the peak separation increases with the scan rate, we know that the electrode reaction is under kinetic control and we get an estimate for the value of k⁰. Other techniques such as potential step or impedance spectroscopy can then be used for more accurate determination.

The left-hand panel of Figure 7.5 shows a few simulated voltammograms with varying values of parameter $\psi$, keeping the scan limits and $\alpha$ constant. The peak separation $\Delta E_p$ does not quite correspond to Table 7.1 because the smaller $\psi$ is, the more sensitive is $\Delta E_p$ to the switching potential where the scan direction is reversed.

In the right panel, parameter $\psi$ is kept constant and the value of the charge transfer coefficient $\alpha$ is varied. As can be seen, the effect of $\alpha$ on the shape of the voltammograms is rather small. Increasing $\alpha$ increases the oxidation current peak and decreases the reduction current peak. Marcus theory showed that a is not a constant but varies slightly as the function of potential. The value 0.5 is therefore most commonly assumed to be $\alpha$, which is quite an accurate assumption around the half-wave potential.

If the peak separation $\Delta E_p$ is less than 59/n mV, the electrode reaction is probably linked to an adsorption step. If adsorption is the rate-limiting step, the oxidation and reduction current peaks can be found at the same potential and are directly proportional to the scan rate, not to the square root of the scan rate like under diffusion control.

Electrode capacitance C_dl also contributes to the measured current. The capacitive (charging) current, i_C, is

The exponential term of the above equation decays very fast. At solid electrodes, the contribution of the capacitive current is significant only at very high scan rates while at immiscible liquid-liquid interfaces it can be quite high. We derived above that the faradaic current is proportional to the square root of the scan rate. The total current i_T is therefore

where A is a constant. Plotting $i_{\text{T}}/\sqrt{v}$ as a function of $\sqrt{v}$ at each potential, a straight line is obtained with the slope of C_dl. Figure 7.7 shows just such an analysis of the water-1,2-dichloroethane (DCE) interface.