CHEM-E4175 - Fundamental Electrochemistry, 09.01.2019-18.02.2019

We have previously stated that impedance is a frequency-dependent quantity and therefore the most convenient and reasonable way to determine it is in the frequency domain. The conversion from a time variable to a frequency variable is done using Laplace transformation, and because of this, we start with a couple of important remarks concerning Laplace transforms [1]. The Laplace transform is a linear operator:

The symbol \( \mathcal{L} \) represents the Laplace transform, a and b are constants, and f(t) and g(t) arbitrary functions of time; \( \overline{F}(s) \) and \( \overline{G}(s) \) are transforms of f(t) and g(t). The bar over the function highlights the frequency domain and s is the Laplace variable (frequency).

The transform being linear means that non-linear factors, such as product k_f c^s cannot be converted (k_f is a potential dependent rate constant and c^s surface concentration of the reacting species). Non-linear factors therefore have to be linearized near measurement potential, and because of this, the magnitude of the input signal has to be small enough for a valid linearization. A few examples of the most common linearizations are shown in the following subchapters. Impedance of any given system can be determined following the steps below:

The third step deserves a couple of remarks. The Laplace transform of a sinusoidal signal takes the form \( 1/(s^2+\omega^2) \), so multiplication with a binomial \( (s^2+\omega^2) \) removes the denominator. All other signals without such a denominator approach zero at the limit \( s=j\omega \). Actually, there is no need to apply Equation (8.15) in practice because all terms without \( \overline{E}(s) \)or \( \overline{I}(s) \) can be eliminated. The following examples will clarify the method.

8.3.1 Linearization of current-voltage curves and faradaic impedance

Current-overpotential equation (6.27) was derived in Chapter 6:

Let us linearize this equation at the point of equilibrium \( (i=0, \eta=0, c_{\text{R}}^s=c_{\text{R,0}}, c_{\text{O}}^s=c_{\text{O,0}}) \):

Equation (8.17) can be written directly in the Laplace domain as follows:

Now we need to determine the surface concentrations  \( \overline{c}_{\text{R}}^s(s) \) and  \( \overline{c}_{\text{O}}^s(s) \) from the transfer equations, which we already did in Chapter 7. Substituting Equations (7.7) and (7.8) to Equation (8.18) gives

Dividing Equation (8.19) by \( \overline{i}(s) \) and replacing s with \( j\omega \), impedance can easily be found:

Note that because i stands for current density, we needed to divide the expression by the area A. The first term of Equation (8.20) represents charge transfer resistance, R_ct. The rate constant of the reaction, k⁰, can be determined by substituting the expression for the exchange current density in Equation (8.20). The latter term of the equation is known as the Warburg element, which describes diffusion. Because \( j^{1/2}=(1-j)/\sqrt{2} \), it is evident that the real and imaginary parts of the Warburg element are equal. The faradaic impedance of the electrode reaction, Z_f, is therefore

where \( \sigma \) is the Warburg element

The characteristic feature of Warburg impedance is an increasing straight line at an angle of 45 degrees. When faradaic impedance and electrode capacitance are in parallel and the solution resistance is added in front of the whole circuit, an impedance plot shown in Figure 8.9 is obtained. The whole circuit is called the Randles circuit or Randles equivalent circuit, shown in Figure 8.11.

Figure 8.9 shows how the values for R_s and R_ct can easily be obtained. The semicircle and the straight line are usually overlapping, so only the left-hand side (high frequencies) of the semicircle is visible. Despite this, it is easy to determine the numerical values using modern software; we will come back to the fitting of circuit components later in this Chapter. 

            If the solution resistance and capacitance have been eliminated from the experimental data, the residual faradaic impedance can be divided into real and imaginary parts:

Plotting the real and imaginary parts as a function of \( 1/\sqrt{\omega} \), two straight lines are obtained. Of these two, the real part intersects the vertical axis at R_ct and the imaginary part at the origin. Both of the lines have a gradient of s. The plot is sometimes called the Randles plot.

If the impedance measurement is conducted off the equilibrium point, the measurement arrangement has to be such that a real stationary state can be reached (see comments at the beginning of Chapter 7.2). If the experiment is fast enough (< 1 min), we can compromise this requirement because the conditions during the experiment are nearly constant. Linearization is easier straight from the Butler-Volmer equation

which will, after linearization and using the Laplace transform, take the form

Quantities \( \Delta\overline{c}_{\text{R,O}}^s(s) \) represent the deviation from the surface concentration caused by the AC input and are determined by the constant current at each data point, see Equation (7.55); \( \Delta\overline{i}(s) \) and \( \Delta\overline{E}(s) \) are the deviations from constant current and voltage, i.e. amplitudes of the AC signal.

Deviations from the surface concentrations are obtained as usual:

We substitute Equation (8.26) in Equation (8.25) and place the current and voltage terms on different sides of the equation:

After little rearrangement, impedance is expressed as

where parameter \( \displaystyle\lambda=\frac{k_f}{\sqrt{D_{\text{R}}}}+\frac{k_b}{\sqrt{D_{\text{O}}}} \).

Surface concentrations of the stationary state \( c_{\text{R}}^s \)   and \( c_{\text{O}}^s \) can naturally be determined from Equation (6.18), but if we want to see how the charge transfer resistance and Warburg element change as a function of potential, we will first sum up the surface concentrations using the general equations (7.7) and (7.8):

The surface concentrations can be determined using Equation (8.24), taking into account that \( k_f=k^0\theta^{1-\alpha} \) and \( k_b=k^0\theta^{-\alpha} \); \( \theta \) is defined in Equation (7.5).

Approximation, which eliminates the final term of the numerator, is based on practical experience: There is no point in running impedance measurements with high DC current densities because of the formation of convective currents. The term i/Fk⁰ is no more than 10^-5 mol cm^-3 in magnitude. It is possible to derive the results (8.30) and (8.31) directly from the assumption that the surface concentrations follow the Nernst equation; in fact, this assumption is typically made in AC voltammetry.

We can now write the charge transfer resistance and the potential dependency of Warburg element explicitly:

Figure 8.12. Potential dependency of charge transfer resistance (left) and Warburg element (right) without their constants. \( \xi \) = 1 and n = 1; \( \theta \) = exp[(nF)/(RT)(E - E^0’)].

The value of the Warburg element becomes two orders of magnitude greater than that of the charge transfer resistance when the system is polarized, i.e. mass transfer to the electrode will be a limiting factor as shown in Figure 8.12. This is another reason why impedance measurements should not be conducted with high overpotential values.