Step Change in Potential or Current for a Semi-Infinite Planar Electrode in a Stagnant Electrolyte

As shown in Figure 6.3, we consider a planar WE located at x = 0. The CE is located sufficiently far from the working electrode so that it has no effect on the results. The reference electrode is assumed to be positioned just outside of the double layer. As before, the potential of interest is the potential of the WE relative to the RE. The electrolyte is assumed to be stagnant so that the bulk fluid velocity is zero.

Cottrell Equation

In this section we consider a potential step experiment where the potential is changed instantaneously at the start of the experiment, and the current is measured as it varies with time. This technique is referred to as chrono-amperometry, where “chrono” refers to time and “amperometry” to the measurement of current. We restrict our analysis to a case where the overpotential of an electrode is changed from zero to some large positive or negative value, a magnitude so large that a limiting current is reached.

The case that we examined in Illustration 4.2 is essentially the same as that considered here. For the one-dimensional, planar geometry with no convection, Equation 4.25 reduces to

(6.2)

which is also known as Fick’s second law. The initial and boundary conditions for this problem are

These conditions assume that the reaction is limited by the transport of a reactant to the surface. One could also consider the case where the reaction is limited by transport away from the surface owing to, for example, a saturated layer on the surface, with similar results. The solution is well known and can be expressed in terms of the similarity variable η, where . The concentration profile in terms of η is

where the error function is defined as

The concentration profiles under these conditions were shown in Figure 4.3. For electrochemical systems, we are generally interested in the current or potential response. If we consider that the diffusing species undergoes an electron-transfer reaction at the surface, the flux of that species to the surface can be related to the current by Faraday’s law:

(4.17)

Differentiating the error-function solution above and substituting yields

(6.3)

Here we see that the current depends inversely on the square root of time. This relationship is known as the Cottrell equation. Regardless of which electrochemical system is being analyzed, you should always be mindful of how the current changes with time. This dependence on the reciprocal of the square root of time is indicative of mass-transfer control. Also note that with this semi-infinite geometry, a steady-state solution does not exist.

What can we learn by using the potential step method as an analytical technique? A plot of the current versus the reciprocal of  results in a straight line, with the current density approaching zero at long times. From the slope of the line, either the diffusivity, D_i, the number of electrons transferred, n, or concentration, c_i^∞, can be calculated if the other quantities are known. We’ll see later in this chapter that there are other, perhaps better, approaches to determining these quantities. Nonetheless, it is worth taking a minute to examine a couple of our assumptions. First, we assumed that the CE was far enough away to be treated as infinity. Just how far is far enough? The length over which concentration changes occur is proportional to , and can be seen in Figure 4.3. If the distance between electrodes is at least an order of magnitude larger, the supposition is good. Second, we assumed the electrolyte was stagnant. Often it is difficult to achieve a stationary electrolyte with this configuration. Small vibrations in the laboratory can affect the results. Additionally, the flow of current will cause some heat generation and temperature differences, as well as concentration differences. Both result in density differences and free convection as described in Chapter 4. The net result is that the flow is not zero; and worse yet, the fluid velocity is not well defined. These challenges can be mitigated if the time of measurement is short.

If the overpotential is not sufficiently large so that the reaction is mass-transfer controlled, there is no simple analytical solution. Thus, this experiment depends on achieving a limiting current. Finally, we must consider the possibility of side reactions occurring. The overpotential cannot be so large as to promote other reactions, such as the electrolysis of the solvent.

Sand Equation

Whereas the above analysis describes a controlled potential technique, we can easily imagine a controlled current method. In this experiment, a step change in current is made for the same semi-infinite, planar system. Generally the equipment for controlled current experiments is cheaper, but unfortunately the analysis is not as straightforward as with a controlled potential method.

Our starting point is again the convective−diffusion equation (Equation 4.25) and, since the geometry and assumptions are the same, it reduces to Equation 6.2. The boundary conditions are different for this galvanostatic experiment. In this case, the current I(t) is prescribed at the surface; therefore, the flux of any species involved in the reaction is known at all times. Here we will only consider a step change to a constant value of current. Figure 6.4 shows the concentration profiles at several different times. Note that the slope at the electrode surface is constant—directly proportional to the current density. At longer times, the concentration of limiting reactant at the surface decreases, and the effects of the step change in current propagate farther away from the electrode surface. The time at which the surface concentration drops to zero is known as the transition time, τ, which is given by

(6.4)

This relationship is known as the Sand equation. For a semi-infinite case, the current cannot be sustained indefinitely because eventually the concentration of the reactant approaches zero at the transition point. As this reactant concentration decreases, the overpotential at the electrode increases to maintain constant current. At some point, the overpotential becomes very large and a second reaction, for example, electrolysis or corrosion, occurs. This behavior is shown in Figure 6.5, which plots the overpotential as a function of time. This overpotential remains relatively constant until the concentration of reactant approaches zero. At that point, the exchange-current density becomes small and the overpotential increases sharply.Similar to the Cottrell equation, this analysis allows one to determine either n, the bulk concentration, or the diffusivity, assuming the other two quantities are known. Like the step change in potential experiment, it can be difficult to prevent unwanted convection, leading to differences between the experimental results and the analysis. Note that, in addition to the utility of this specific analytical technique, it is important to understand the behavior exhibited in Figure 6.5, which is another indication of a system that is mass-transfer limited.

The system is likely mass-transfer limited if

1. at constant potential, the current decreases inversely with the square root of time;
2. at constant current, the overpotential increases dramatically to sustain the current;
3. at steady-state, a change in potential does not result in a change in current.