Cyclic voltammetry (CV) is a useful analytical tool for electrochemical engineers. It is often the first test performed to characterize a new electrolyte or an electrode. You will need to be familiar with the basic experiment and the interpretation of the data in order to obtain quantitative information. We begin at a potential that is well below the equilibrium potential of the reaction of interest. The potential is ramped up linearly to some maximum potential (called the switching potential or peak potential) at a constant rate, ν [V·s⁻¹], called the sweep rate. The sweep in the positive direction is called the anodic scan. At the peak potential, the sweep rate is reversed (−ν) and the potential is returned to its original value. The negative scan is termed the cathodic scan. This triangular wave may be repeated many times as shown in Figure 6.9.

Imagine that we have a generic redox reaction:

Let’s begin by assuming that we have kinetics at the electrode that can be described by the Butler–Volmer expression from Chapter 3:

(3.17)

η_s is, of course, the surface overpotential defined as

(3.15b)

ϕ₁ is the potential of the working electrode and ϕ₂ is the potential of the solution measured with a reference electrode located just outside the double layer. U is the equilibrium potential defined against that same reference electrode.

If we simply use Equation 3.17 to calculate the current when the surface overpotential undergoes a triangle wave as depicted in Figure 6.9, the response is the solid black line shown in Figure 6.10. Note that the current is identical on the anodic and cathodic sweeps and only depends on the overpotential. In other words, there is only one value of the current for each value of the overpotential, and the current does not depend on the direction that the voltage is changing. This response was calculated with what we learned in Chapter 3 for steady-state conditions under kinetic control. However, for the common sweep rates used in CV, too much of the physics of the problem have been neglected. First of all, since the voltage changes constantly during the experiment, the double layer is also changing by either being charged or discharged; consequently, there is current associated with the constant charging or discharging of the electrical double layer. An equivalent circuit representation of this situation was shown in Figure 6.6. The faradaic current is given by the BV equation. The non-faradaic current associated with the double-layer charging is

(6.11)

where C_DL is the double-layer capacitance of the electrode [F·m⁻²]. We also make the simplifying assumption that C_DL does not depend on potential. With this assumption, Equation 6.11 tells us that the current associated with the double-layer charging is proportional to the sweep rate. The double-layer current is shown as the solid gray line in Figure 6.10 for the forward and reverse voltage sweeps. It is constant for a given sweep rate and direction, but changes sign when the sweep direction changes from positive to negative. The total current is the sum of the faradaic and double-layer current:

(6.12)

The total current response for the CV that includes both the faradaic and charging current is shown as the dashed lines in Figure 6.10. The figure showing the relationship between potential and current is called a cyclic voltammogram, although it doesn’t yet have the prototypical shape.

Another important phenomenon that needs to be included is diffusion of products and reactants. So far we have assumed that both species (O and R) are present uniformly at the bulk values in the solution. Recall from Chapter 3 that the exchange-current density depends on the concentration of reactants. Therefore, a changing concentration implies that the exchange-current density, i_o, is not constant. Also, as we have seen from Chapter 4, mass transfer of species to and from an electrode can limit the rate of reaction. The equilibrium potential of the cell can change with composition too—think back to the Nernst equation from Chapter 2.

Before we go through the mathematical analysis of this situation, let’s reason through what might happen during the CV when mass transfer is considered. Imagine that our bulk solution has an equal amount of R and O. If the current follows the trajectory as shown in Figure 6.9 for the anodic sweep, we would quickly deplete R from the solution near the electrode. Molecular diffusion would allow transport of R from the bulk to the electrode surface, but there would be a finite maximum rate of replenishment; that is, a limiting current. Above this point, increases in potential would not result in an increase in the current. At the same time, we would be building an excess of O at the electrode surface. When the peak potential in the triangular wave is reached, the potential sweep is reversed. At the time of reversal, there is little R present and a large amount of O at the electrode surface. Instantaneously, the current associated with the double layer reverses sign (since  changes sign), and the faradaic current also starts to decrease as the overpotential diminishes. When the potential becomes low enough that the cathodic reaction is favored, O is consumed. This cathodic reaction continues until a limiting current is reached again. The current voltage response for this process is shown in Figure 6.11 for a typical voltammogram with fast or reversible electrode kinetics.

Because of changing concentration profiles, the shape of the voltammogram is different for the initial sweep and subsequent sweeps. Multiple sweeps are common experimentally, and sweeping is most often done experimentally from the open-circuit potential. The simulation results shown in this section follow multiple sweeps, where a quasi-stationary condition has been reached.

The steps for calculating this relationship between the current and voltage are analogous to those worked previously, and assume the presence of supporting electrolyte. For one-dimensional transport in the absence of bulk flow, Equation 4.25 once again becomes the well-known diffusion equation:

As before, we can use Faraday’s law to relate the current to the flux of species at the electrode surface by, for example, setting  (see Equation 4.17), where N_O is the flux of species O and is positive when O moves away from the surface. Since we are specifying the potential of the WE, this potential combined with the BV equation is one of our boundary conditions. Another boundary condition expresses the fact that the concentrations far from the surface of the WE are equal to the bulk values. Finally, we add an initial condition, which also sets the composition of each species to its bulk value. Both analytical and numerical solutions of the diffusion equation are possible. The results shown here were obtained numerically for multiple voltage sweeps. Note that many of the analytical relationships in the literature, including popular textbooks, apply strictly to a single linear voltage sweep (LSV) rather than to multiple cycles and involve additional simplifying assumptions. Care should be taken to ensure that the assumptions made in deriving a relationship are consistent with your experimental system.

Whether solving the equations analytically or numerically, it is important to express the exchange-current density as a function of concentration. It is that concentration dependence that keeps the current from increasing with potential once the mass-transfer limit is reached. Using Equation 3.20 for the exchange-current density with the reference concentration taken as the bulk concentration and γ₁ = γ₂ = 0.5 yields

(6.13)

An additional simplification is possible for reversible systems where kinetic rates are sufficiently rapid that the potential does not move significantly from its equilibrium value. In such cases, the ratio of c_O and c_R at the surface is directly related to the potential through the Nernst equation. This simplification is used in many of the analytical solutions available and is explored as part of a numerical solution in Problem 6.24.

A typical cyclic voltammogram is shown in Figure 6.11 for reversible kinetics. There are two peak currents and two peak potentials—These values are the basis for the primary interpretation of the voltammogram. For reversible kinetics at 25 °C, the distance between the two peaks is

(6.14)

as long as the sweep extends ∼300 mV above and below the equilibrium potential in either direction. A smaller sweep will lead to a slightly greater distance between the peaks (e.g., 0.059/n for a sweep that extends 120 mV). It follows from Equation 6.14 that for reversible kinetics, the number of electrons transferred can be determined from the distance between peaks. Conversely, if n is known, the peak separation can be used to infer whether or not the reaction is reversible.

The concentrations of O and R during CV just before the peak current are displayed in Figure 6.12. Here the concentration of the reduced species is low and the concentration of the oxidized species is high. The penetration depth of this concentration difference depends on the square root of time. Thus, the slower the scan rate, the deeper the penetration, and the lower the limiting current.

The height of the peaks due to the faradaic current is proportional to , whereas the double-layer charging current is proportional to ν, the scan rate. Figure 6.13 shows the effect of scan rate, ν, on the voltammogram with the double-layer charging removed. The results in Figure 6.13 are for reversible (fast) kinetics. Under such conditions, the peak position is constant and the peak height is a function of the scan rate.

When examining a cyclic voltammogram, the waveforms are frequently described in terms of how facile the electrochemical reactions are: reversible, quasi-reversible, and irreversible kinetics. We’ve already seen examples of reversible kinetics (see Figures 6.11 and 6.13). Figure 6.14 provides three CVs for reactions with more and more sluggish kinetics; that is, decreasing exchange-current density, i_o. The peak currents decrease as the kinetics become slower. Also, in contrast to the reversible case, the peak potentials are not constant for finite kinetics. For irreversible or quasi-reversible kinetics, as the kinetics decrease the peak potentials move farther and farther apart. Although not shown in Figure 6.14, as the scan rate increases, the peak separation increases further—also different from the reversible case.

An electrochemically reversible reaction has a high exchange-current density. As a result, the surface overpotential is small and the potential of the electrode is close to its equilibrium potential as approximated by the Nernst equation, which is the primary condition for reversibility. Quasi-reversible reactions have slower kinetics where the potential differs significantly from the Nernst value. Here we refer to irreversible reactions as reactions for which the reverse reaction (cathodic or anodic) does not take place to any appreciable extent. Unfortunately, these terms are not particularly precise and are not used consistently in the literature. For example, reactions with very slow kinetics are often referred to as electrochemically irreversible reactions, even though they are chemically reversible (the reverse reaction does take place). The behavior is summarized in Table 6.3.

Table 6.3 Impact of the Kinetics of the Reaction on the Shape of Cyclic Voltammogram

Reversible	Quasi-reversible	Irreversible
Peak currents increase with scan rate	Peak currents increase with scan rate	Peak current increases with scan rate
Peak position and separation are constant, given by Equation 6.14	Peak separation increases with scan rate	Only one peak current observed since reverse reaction is not significant
	Voltammograms are more drawn out and exhibit a larger separation

We close this section with one final note. In Equation 6.1 we saw that the applied potential includes the influence of concentration and ohmic drop in solution, due to the fact that the reference electrode is not located just outside the double layer. Consequently, the CV data, as measured, will include these influences and may need to be corrected. Ways for handing these effects during measurement are considered in Section 6.9.

ILLUSTRATION 6.2

Please identify each of the following figures as reversible, quasi-reversible, or irreversible (the answer is below each figure).

The peaks are symmetrical and separated by about 60 mV. This reaction is reversible. (Adapted from Phys. Chem. Chem. Phys., 15, 15098 (2013).)

Reaction of 2-nitropropane.

The oxidation and reduction peaks are separated by more than 0.5 V. Here both peaks are drawn out rather than sharp. This is a quasi-reversible system. (Adapted from J. Chem. Ed., 60, 290 (1983).)

1.5 M LiTFSI in a mixture of ethylene carbonate and diethyl carbonate. There is an oxidation current at high potentials, but no reverse reaction. This system would be described as irreversible. In fact, the reaction occurring is the oxidation of the solvent. This test establishes the potential limits for the stability of the solvent. (Adapted from Phys. Chem. Chem. Phys., 15, 7713 (2013).)

Platinum oxidation

Although the peak potentials are not widely separated, the peaks are extremely drawn out. Furthermore, the peak potentials shift slightly with scan rate This reaction is quasi-reversible. (Adapted from Phys. Chem. Chem. Phys., 16, 5301 (2014).)