Recall from Chapter 3 that the surface of the electrode is usually charged, and that the structure of the interface is described as a double layer. The charge associated with the double layer depends on the electrode potential, and current must flow to alter this charge as either more or less charge is stored at the interface when the potential is changed. In the above analysis, we neglected this charging current since it was presumably not important for the physical situation and timescales of interest. In contrast, here we explore the current associated with double-layer charging in some detail and identify its effect on our analysis. We start by creating a simple equivalent circuit model of the electrode. While there are numerous equivalent circuits developed to describe electrochemical systems, we will limit ourselves to the simplified Randles circuit shown in Figure 6.6. This equivalent circuit is an idealized electrical circuit that mimics the electrical response of the actual electrode. In general, there may or may not be a close physical connection between the electrical components of the equivalent circuit and the actual physical processes occurring at the electrode. Here, we consider a simplified system where the physical connection is apparent and important. The first element is R_Ω, which represents the resistance to current flow in the electrolyte. We next see that there are two paths for current to flow to or from the electrode. The first is associated with the charging of the double layer. This current does not involve electron transfer at the surface and is termed non-faradaic or a charging current density, i_c. The second path represents the faradaic current density, i_f; which is the current due to an electron-transfer reaction at the surface. The faradaic current density is generally described with the Butler–Volmer equation discussed in Chapter 3. The sum of these two is the measured current density, i_m:

(6.5)

Devices using the charge stored in the double layer are treated in detail in Chapter 11 on electrochemical double-layer capacitors. Here we simply want to describe the double layer as a capacitor in parallel with an element representing the resistance to charge transfer, R_f. This equivalent circuit concept will allow us to understand the response of the systems considered here and, more specifically, how the charging current can affect our analysis of electrochemical systems.

The behavior of an ideal resistor is familiar. We can describe the relationship between current and potential with Ohm’s law, namely, that the voltage across the resistor is directly proportional to the current flow (ΔV = IR_Ω). The ideal capacitor behaves differently, which can be expressed mathematically as

(6.6)

At steady state, dV/dt = 0. Therefore, the current is zero and the capacitor has a fixed charge that depends on the potential across the capacitor, Q = CV. Conversely, for a step change in potential or current, the capacitor first behaves as short circuit, providing no initial resistance to the flow of current. In response to a change in current between the WE and CE in Figure 6.4, at first all of the current would flow through the capacitor. Because of the resistance of the electrolyte (R_Ω) in series with the capacitor, this current is finite even for an ideal capacitor. As the charge builds up on the capacitor, the potential across the capacitor increases, as does the resistance to the flow of current through the capacitor. As a result, current begins to flow through the parallel path representing the faradaic reaction. Once the capacitor is fully charged, no current flows through the capacitor, and all of the current is faradaic.

How long does it take to charge the capacitor that represents the double layer? To address this question, let’s consider the response of a capacitor in series with a resistor to a step change in potential. This situation is similar to that just described, although simplified further to only consider the current flow through the capacitor as shown in Figure 6.7. The resistance in series with the capacitor comes from the ohmic resistance of the electrolyte.

As seen in the definition of capacitance, the charge, Q, is proportional to the potential across the capacitor. In practice, we cannot move the charges instantaneously; that is, there is some resistance to current flow. Assume that at time equal to zero, the potential applied to the circuit is changed to V₀. Current will flow in response to the change in voltage, and the charge on the capacitor will vary. After a time, the capacitor will be fully charged, and the current will be zero. We can define Q₀ to be the charge equal to CV₀, which is the charge at the final steady state. The potential across the capacitor is simply Q/C, where Q varies with time, and the potential drop across the resistor in series is IR_Ω. We can use Kirchhoff’s voltage law to get

(6.7)

where dQ/dt is the current. Equation 6.7 can be integrated to give the fractional charge on the capacitor:

(6.8)

The quantity R_ΩC has units of seconds  and is a characteristic time representing the time to charge or discharge the capacitor; it is called the charging time. Since the capacitance is the proportionality constant between charge and voltage, Equation 6.8 can also be written as

(6.9)

The current is obtained by differentiation of Equation 6.8:

(6.10)

For a step change in potential, the current decreases exponentially to zero with a time constant τ = R_ΩC. A large resistance implies that the current is small and, therefore, it takes longer to charge the capacitor. Similarly, as the capacitance increases, it will also take longer to charge or discharge the capacitor.

We can gain insight into the timescales that correspond to different physical processes in our electrochemical system by comparing the time constant for capacitor charging with the characteristic time for diffusion:

where C_DL is the specific capacitance of the double layer (capacitance per area). If τ_c is much smaller than τ_D, then the current is not affected by the charging process except at very short times following a step change in current or potential. Let’s pick some typical values (Table 6.2), and estimate these characteristic times. The resistance in series with the capacitor is taken to be the resistance of the electrolyte between the WE and CE.

Table 6.2 Physical Parameters That Affect the Charging and Diffusion Times

Quantity	Description	Value
L	Distance between CE and WE	0.001 m
Di	Diffusion coefficient	10−9 m2·s−1
CDL	Electrode capacitance per unit area	0.2 F·m−2
κ	Electrical conductivity of solution	10 S·m−1

The diffusion time is 10³ s, while the charging time is only 20 μs. This difference tells us that often we don’t need to worry about the charging time. There are some cases where this time constant for charging is important as discussed in subsequent sections. Also, charging times can be considerably longer for a porous electrode where the surface area and, therefore, the capacitance is high. The time constant for double-layer charging of porous electrodes is addressed further in Chapter 11, which examines electrochemical double-layer capacitors.

We have assumed that double-layer charging can be represented by a capacitor in series with the solution resistance, and defined a time constant, , that is equal to the product of the resistance and capacitance. At a time equal to , the system reaches 63.2% of the steady-state value (see Equation 6.9). It actually takes 3 to reach 95% of steady-state value, which is a reasonable approximation of completion.

It turns out that the resistor and capacitor in series is a simplification that provides a reasonable estimate of the double-layer charging time. In reality, the capacitor is in parallel with the faradaic resistance (see Figure 6.6), which influences the charging time and the extent to which the double layer must be charged since, in the presence of current, not all of the voltage drop needs to be across the capacitor. This issue is explored in more detail in Problem 6.4. A different time constant, , results that is smaller than that presented above, but approaches the same value as the faradaic resistance becomes large. Consequently, the above treatment overestimates the double-layer charging time and therefore provides a conservative estimate.

Current Interruption

Figure 6.8 shows the voltage response for a step change of current (current step is not shown). At a time of 20 s, the current is increased and held constant. As shown, the potential also changes, but not instantaneously. In fact, multiple time constants are evident. There is an immediate step change in potential associated with the ohmic resistance in the electrolyte. Next we see a relatively rapid increase in potential, shown in more detail in the inset, which is associated with charging of the double layer. Finally, the potential continues to increase. This further increase can be ascribed to diffusion and changes in concentration. Recall from Chapter 4 that there are usually concentration changes associated with the flow of current. Here, the reactant concentration becomes lower, decreasing the exchange-current density; thus, a higher overpotential is required to keep the current constant. Generally, the double-layer charging is fast compared to the time for diffusion. Charging or discharging of the double layer takes place fairly quickly, while concentration changes take significantly longer.

When the cell potential is recorded experimentally for a step change in current, one needs to consider the sampling rate of the measurement. The smooth, continuous curves seen in Figure 6.8 may not always be obtained. This particular challenge is explored in Illustration 6.1.

A comment regarding the counter electrode is appropriate here. Electroanalytical methods are almost invariably focused on the behavior of one electrode. Therefore, it is important that the other electrode, the counter electrode, does not interfere with the measurement. The counter electrode also has a double-layer capacitance that is in series with that of the working electrode. In order to avoid interference from the counter electrode, we ensure that its capacitance is much larger than that of the working electrode, since it is the smaller capacitor in series that controls the dynamic behavior (Why?). The easiest way to increase the size of the counter electrode capacitance is to increase its active area. Hence, we see a second reason for the counter electrode to be significantly larger than the working electrode in electroanalytical experiments.

ILLUSTRATION 6.1

During operation of a low temperature H₂-O₂ fuel cell at steady state, it is desired to periodically measure the resistance of the cell. Current interruption is proposed as a means of accomplishing the measurement. It is common to report the resistance in [Ω·m²] for electrochemical systems. We used the same symbol, R_Ω for both resistance quantities. Here, the current density is used in place of the current:

At 0.4 A·m⁻², the potential of the cell is 0.7 V. The sampling rate is every 3 ms, resulting in the data shown in part (a) of the following figure for current interruption (step to zero current) at t = 0. The estimated resistance is (0.98–0.7)/0.4 = 0.70 Ω·m². This value seems large, and your colleague suggests increasing the sampling rate. When the sampling rate is increased to once every 50 μs, the data in part (b) of the following figure are obtained. Clearly, the potential of the cell changes rapidly during the first few milliseconds following interruption of the current. The potential is then extrapolated back to time zero in order to estimate its value at the instant just after the current is interrupted. This value is 0.73 V:

Why was the first calculation so far off? When the current is interrupted, there are reductions in ohmic losses, kinetic losses, and mass-transfer losses. Although the ohmic loss changes instantaneously when the current is interrupted, other losses have a time constant associated with them. In particular, it takes time for the charge in the double layer to reach a new steady state; this change in the double layer is the main reason for the delayed response of the potential at short times. The sampling time must be short enough to distinguish the ohmic losses from losses associated with charging or discharging of the double layer.