BASIC POTENTIAL STEP METHODS
5.1 OVERVIEW OF STEP EXPERIMENTS .1 Types of Techniques
С Н AfcT E R
5
BASIC POTENTIAL
5.1 Overview of Step Experiments 157
Function
generator Potentiostat
E controlled
Figure 5.1.1
Experimental arrangement for controlled-potential i(t) measured experiments.
alternatively as an active element whose job is to force through the working electrode whatever current is required to achieve the desired potential at any time. Since the cur- rent and the potential are related functionally, that current is unique. Chemically, it is the flow of electrons needed to support the active electrochemical processes at rates consistent with the potential. Thus the response from the potentiostat (the current) actu- ally is the experimental observable. For an introduction to the design of such apparatus, see Chapter 15.
Figure 5.1.2a is a diagram of the waveform applied in a basic potential step experi- ment. Let us consider its effect on the interface between a solid electrode and an unstirred solution containing an electroactive species. As an example, take anthracene in deoxy- genated dimethylformamide (DMF). We know that there generally is a potential region where faradaic processes do not occur; let E\ be in this region. On the other hand, we can also find a more negative potential at which the kinetics for reduction of anthracene be- come so rapid that no anthracene can coexist with the electrode, and its surface concentra- tion goes nearly to zero. Consider E2 to be in this "mass-transfer-limited" region. What is the response of the system to the step perturbation?
First, the electrode must reduce the nearby anthracene to the stable anion radical:
An + e —> An* (5.1.1)
This event requires a very large current, because it occurs instantly. Current flows subse- quently to maintain the fully reduced condition at the electrode surface. The initial reduc- tion has created a concentration gradient that in turn produces a continuing flux of anthracene to the electrode surface. Since this arriving material cannot coexist with the electrode at E2, it must be eliminated by reduction. The flux of anthracene, hence the cur-
j _
-ел
t3 > t2 > f 1 > 0
0 t 0 x 0 t (a) (b) (c)
Figure 5.1.2 (a) Waveform for a step experiment in which species О is electroinactive at E\, but is reduced at a diffusion-limited rate at Ei- (b) Concentration profiles for various times into the experiment, (c) Current flow vs. time.
158 Chapter 5. Basic Potential Step Methods
rent as well, is proportional to the concentration gradient at the electrode surface. Note, however, that the continued anthracene flux causes the zone of anthracene depletion to thicken; thus the slope of the concentration profile at the surface declines with time, and so does the current. Both of these effects are depicted in Figures 5.1.2b and 5.1.2c. This kind of experiment is called chronoamperometry, because current is recorded as a func- tion of time.
Suppose we now consider a series of step experiments in the anthracene solution dis- cussed earlier. Between each experiment the solution is stirred, so that the initial condi- tions are always the same. Similarly, the initial potential (before the step) is chosen to be at a constant value where no faradaic processes occur. The change from experiment to ex- periment is in the step potential, as depicted in Figure 5.1.3a. Suppose, further, that exper- iment 1 involves a step to a potential at which anthracene is not yet electroactive; that experiments 2 and 3 involve potentials where anthracene is reduced, but not so effectively that its surface concentration is zero; and that experiments 4 and 5 have step potentials in the mass-transfer-limited region. Obviously experiment 1 yields no faradaic current, and experiments 4 and 5 yield the same current obtained in the chronoamperometric case above. In both 4 and 5, the surface concentration is zero; hence anthracene arrives as fast as diffusion can bring it, and the current is limited by this factor. Once the electrode po- tential becomes so extreme that this condition applies, the potential no longer influences the electrolytic current. In experiments 2 and 3 the story is different because the reduction process is not so dominant that some anthracene cannot coexist with the electrode. Still, its concentration is less than the bulk value, so anthracene does diffuse to the surface where it must be eliminated by reduction. Since the difference between the bulk and sur- face concentrations is smaller than in the mass-transfer-limited case, less material arrives at the surface per unit time, and the currents for corresponding times are smaller than in experiments 4 and 5. Nonetheless, the depletion effect still applies, which means that the current still decays with time.
Now suppose we sample the current at some fixed time т into each of these step experiments; then we can plot the sampled current, /(т), vs. the potential to which the step takes place. As shown in Figures 5.1.3b and 5.1.3c, the current-potential curve has a wave shape much like that encountered in earlier considerations of steady-state voltammetry under convective conditions (Section 1.4.2). This kind of experiment is called sampled-current voltammetry, several forms of which are in common practice.
The simplest, usually operating exactly as described above, is called normal pulse voltammetry. In this chapter, we will consider sampled-current voltammetry in a gen- eral way, with the aim of establishing concepts that apply across a broad range of par-
E
\ 4 5
(a) (b) (c)
Figure 5.1.3 Sampled-current voltammetry. (a) Step waveforms applied in a series of experiments, (b) Current-time curves observed in response to the steps, (c) Sampled-current voltammogram.
5.1 Overview of Step Experiments 159
t С
\ A +
)
т
(b)
Figure 5.1.4 Double potential step chronoamperometry.
(a) Typical waveform.
(b) Current response.
ticular methods. Chapter 7 covers the details of many forms of voltammetry based on step waveforms, including normal pulse voltammetry and its historical predecessors and successors.
Now consider the effect of the potential program displayed in Figure 5.1.4 a. The for- ward step, that is, the transition from E\ to E2 at t = 0, is exactly the chronoamperometric experiment that we discussed above. For a period r, it causes a buildup of the reduction product (e.g., anthracene anion radical) in the region near the electrode. However, in the second phase of the experiment, after t = т, the potential returns to E\9 where only the ox- idized form (e.g., anthracene) is stable at the electrode. The anion radical cannot coexist there; hence a large anodic current flows as it begins to reoxidize, then the current de- clines in magnitude (Figure 5.1.4/?) as the depletion effect sets in.
This experiment, called double potential step chronoamperometry, is our first exam- ple of a reversal technique. Such methods comprise a large class of approaches, all featur- ing an initial generation of an electrolytic product, then a reversal of electrolysis so that the first product is examined electrolytically in a direct fashion. Reversal methods make up a powerful arsenal for studies of complex electrode reactions, and we will have much to say about them.
5.1.2 Detection
The usual observables in controlled-potential experiments are currents as functions of time or potential. In some experiments, it is useful to record the integral of the current versus time. Since the integral is the amount of charge passed, these methods are coulo- metric approaches. The most prominent examples are chronocoulometry and double po- tential step chronocoulometry, which are the integral analogs of the corresponding chronoamperometric approaches. Figure 5.1.5 is a display of the coulometric response to the double-step program of Figure 5.1.4a. One can easily see the linkage, through the in- tegral, between Figures 5.1.4b and 5.1.5. Charge that is injected by reduction in the for- ward step is withdrawn by oxidation in the reversal.
Of course, one could also record the derivative of the current vs. time or potential, but derivative techniques are rarely used because they intrinsically enhance noise on the sig- nal (Chapter 15).
Several more sophisticated detection modes involving convolution (or semi- integration), semidifferentiation, or other transformations of the current function also find useful applications. Since they tend to rest on fairly subtle mathematics, we defer discussions of them until Section 6.7.
160 Chapter 5. Basic Potential Step Methods
Q
Figure 5.1.5 Response curve for double potential step chronocoulometry. Step t waveform is similar to that in Figure 5.1 Aa.
5.1.3 Applicable Current-Potential Characteristics
With only a qualitative understanding of the experiments described in Section 5.1.1, we saw that we could predict the general shapes of the responses. However, we are ulti- mately interested in obtaining quantitative information about electrode processes from these current-time or current-potential curves, and doing so requires the creation of a the- ory that can predict, quantitatively, the response functions in terms of the experimental parameters of time, potential, concentration, mass-transfer coefficients, kinetic parame- ters, and so on. In general, a controlled-potential experiment carried out for the electrode reaction
kf
(5.1.2) can be treated by invoking the current-potential characteristic:
i = FAk° [Co(0, t)e-^-E°r) ~ CR(0, , ) e a - a № *0' ) ] (5.1.3) in conjunction with Fick's laws, which can give the time-dependent surface concentra- tions CQ(0, t) and CR(0, f). This approach is nearly always difficult, and it sometimes fails to yield closed-form solutions. The problem is even more difficult when a multistep mechanism applies (see Section 3.5). One is often forced to numerical solutions or ap- proximations.
The usual alternative in science is to design experiments so that simpler theory can be used. Several special cases are easily identified:
(a) Large-Amplitude Potential Step
If the potential is stepped to the mass-transfer controlled region, the concentration of the electroactive species is nearly zero at the electrode surface, and the current is totally con- trolled by mass transfer and, perhaps, by the kinetics of reactions in solution away from the electrode. Electrode kinetics no longer influence the current, hence the general i-E characteristic is not needed at all. For this case, / is independent of E. In Sections 5.2 and 5.3, we will be concentrating on this situation.
(b) Small-Amplitude Potential Changes
If a perturbation in potential is small in size and both redox forms of a couple are present (so that an equilibrium potential exists), then current and potential are linked by a lin- earized i-r] relation. For the one-step, one-electron reaction (5.1.2), it is (3.4.12),