Small amplitude potential step

From an experimental point of view, it may be worth mentioning that the artificial membrane is reasonably stable, though only for a short time. It is therefore also a clear example of an object that should be studied by means of a quick method. The (small-amplitude) potential step method has been applied successfully [113], as has the FFT method described in Sect. 2.5.5. With the latter, the analysis in terms of an equivalent circuit like in Fig. 29 was demonstrated quite spectacularly [114]. 

The traditional way is to measure the impedance curve, Z(co), point-after-point, i.e., by measuring the response to each individual sinusoidal perturbation with a frequency, to. Recently, nonconventional approaches to measure the impedance function, Z(a>), have been developed based on the simultaneous imposition of a set of various sinusoidal harmonics, or noise, or a small-amplitude potential step etc, with subsequent Fourier- and Laplace transform data analysis. The self-consistency of the measured spectra is tested with the use of the Kramers-Kronig transformations [iii, iv] whose violation testifies in favor of a non-steady state character of the studied system (e.g., in corrosion). An alternative development is in the area of impedance spectroscopy for nonstationary systems in which the properties of the system change with time. 

Let us now consider potential step chronoamperometry in some detail. We consider a surface-deposited polymer film of uniform thickness L to which a small amplitude potential step is applied. This ensures that only a small change in the polymer oxidation state is effected. Structural changes will be minimal. Initially at time t = 0 before applying the step the redox center concentration in the layer is uniform and has the value c, = where denotes the total redox center concentration in the layer. After applying the small amplitude step, the redox centre concentration at jc = 0 (which defines the support electrode/film interface) is given by Cf (see Fig. 1.46). Let c x, t) denote the concentration of redox centers in the film as a function of distance x and time t. The boundary value problem can be stated as follows... 

The mathematics outlined in this section are fairly well applicable to a technique named differential pulse polarography [21, 48, 50]. The potential is programmed as a sequence of normal potential steps each with a small amplitude potential pulse superimposed [see Fig. 14(a)]. The problem of separating the small amplitude current is met by sampling the current at two different moments in one period (i) just before the application of the small amplitude pulse, i.e. at t t0 and (ii) at a moment tm — t0 + tp within the duration of the pulse. The difference of the two samples is recorded as a function of E. It is not difficult to... 

The bottom line is to avoid large-amplitude perturbations it is much better instead to apply a small-amplitude potential or current step to the film. In such a circumstance only a minor perturbation in polymer structure is effected. Tis comment applies in particular to analyzing electronically conducting polymers, but it should also be kept in mind when examining redox polymer films. Hence a large number of small steps is much preferred to a single large one. 

As an alternative to potential step experiments, current steps have also been used.163,166,167 Again, small-amplitude experiments are preferable,163 and a migration model should be used for data analysis.167... 

Differential Pulse Voltammetry. As illustrated in Figure 38, in the Differential Pulse Voltammetry (DPV) the perturbation of the potential with time consists in superimposing small constant-amplitude potential pulses (10 < AEpUise <100 mV) upon a staircase waveform of steps of constant height but smaller than the previous pulses (1 < AEbase < 5 mV). 

First of all, the mathematical background will be developed for the case of a simple electrode reaction O + n e = R. In this treatment, contrasts like potential versus current perturbation, large amplitude versus small amplitude, and single step versus periodical perturbation are emphasized. While discussing these principles, the most common methods derived from them will be briefly mentioned. On the other hand, it will be shown that, by virtue of the method of Laplace transformation, these methods have much in common and contain, in principle, the same information if the detected cell response is of the same order. 

It may be emphasized here that such an elaboration is possible for any small amplitude perturbation technique. It is only necessary to explicitize either the first-order current or the first-order interfacial potential, corresponding to the type of perturbation, to be able to derive expressions for 7q, 1 and AEl. So, the treatment is also useful to estimate the error due to second-order non-linearity in the step methods. However, a separate measurement of the second-order effect can only be done with (sinusoidal) a.c. perturbation. In Table 5, the explicit expressions for SF pertaining to the four methods mentioned in Sect. 2.4.1 are given in such a way that the connection between them is clearly shown. 

All the previous techniques described in this chapter used large perturbations of the system for recording the transient response of the system. It is the case, for instance, with potential sweeps (CV) and potential or current step (PITT and GITT). Another way to characterize an electrochemical system is to perturb the system initially at the steady state by the use of an alternative signal of small amplitude this method is used in EIS. 

Another methodology consists in applying small amplitude perturbations where the steps or sweeps are sufficiently small to yield a Unear current-potential relationship (Figure 11.15). This approach is at the heart of electrochemical impedance methods. 

A crucial step between any proposed potential and the experimentally-observed properties of a system is the characterisation of the nuclear motion. For clusters this step is particularly demanding because the concepts of equilibrium geometry and small amplitude motion, which have proved so fruitful in the analysis of the spectra of chemically-bound molecules, are almost always invalid. This means that methods based solely on perturbation theory are also unlikely to be successful unless a new and reliable zeroth order model for such systems can be developed. 

It is worth noting that diffusional (random walk or stochastic) phenomena manifest themselves in terms of a square root relationship, such as t - (potential-step), v (potential-sweep), or (small-amplitude ac perturbation). 

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