A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 77
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It also offers access to a wider range oftime scales than can be achieved by any of the pulse polarographic techniques. An extensive review by Osteryoung and Q'Dea (50) provides many details beyond the introduction given here.(a) Experimental Concept and PracticeSquare wave voltammetry is normally carried out at a stationary electrode; such as anHMDE, and involves the waveform and measurement scheme shown in Figure 7.3.13. Asin other forms of pulse voltammetry, the electrode is taken through a series of measurement cycles; however there is no renewal of the diffusion layer between cycles. In contrast to NPV, RPV, and DPV, square wave voltammetry has no true polarographicmode.10 The waveform can be viewed as a special case of that used for DPV (Figure7.3.9), in which the preelectrolysis period and the pulse are of equal duration, and thepulse is opposite from the scan direction.
However, the interpretation of results is facilitated by considering the waveform as consisting of a staircase scan, each tread of which issuperimposed by a symmetrical double pulse, one in the forward direction and one in the10Sometimes the term square wave polarography is applied to the application of SWV at a slowly growingmercury drop issuing from a DME; however this practice is distant from the conventional meaning ofpolarography, which is built upon the dropping action and periodic renewal of the electrode during theexperiment.294Chapter 7. Polarography and Pulse Voltammetry- Cycle 1 -- Cycle 2 -- Cycle 3 -ForwardNsampleАЕ„АЕ^ReversesampleTimeFigure 7.3.13 Waveform and measurement scheme for square wave voltammetry. Shown in boldis the actual potential waveform applied to the working electrode.
The light intervening linesindicate the underlying staircase onto which the square wave can be regarded as having beensuperimposed. In each cycle, a forward current sample is taken at the time indicated by the soliddot, and a reverse current sample is taken at the time marked by the shaded dot.reverse. Over many cycles, the waveform is a bipolar square wave superimposed on thestaircase, and this view gives rise to the name of the method.11Figure 7.3.13 helps to define the principal parameters.
The square wave is characterized by a pulse height, Д£ р , measured with respect to the corresponding tread of the staircase, and a pulse width £p. Alternatively, the pulse width can be expressed in terms of thesquare wave frequency, / = 1/2fp. The staircase shifts by AES at the beginning of eachcycle; thus the scan rate v = AEs/2t^ = fkEs. The scan begins at an initial potential, Evwhich can be applied for an arbitrary time to initialize the system as desired.Current samples are taken twice per cycle, at the end of each pulse.
The forward current sample, /f, arises from the first pulse per cycle, which is in the direction of the staircasescan. The reverse current sample, iT, is taken at the end of the second pulse, which is in theopposite direction. A difference current Д/ is calculated as /f - iT. There is diagnostic valuein the forward and reverse currents; hence they are preserved separately. Consequently, theresult of a single SWV run is three voltammograms showing forward, reverse, and difference currents vs.
the potential on the corresponding staircase tread.Square wave voltammetry is always performed using a computer-controlled potentiostatic system with functional elements organized essentially as in Figure 7.3.11. The computer provides for operator interaction, synthesizes the waveform, sequences the samplingand logging of data, computes difference currents, and handles reporting of results, eithergraphically or otherwise. In many systems, the computer also controls the electrode, especially if an SMDE is involved.nMany years ago, Barker (51, 52) invented a method that he called "square wave polarography," inwhich a quite different experimental strategy is used. A small-amplitude, high-frequency square wave issuperimposed on the slowly changing ramp or staircase used in polarography, and a current sampling scheme isemployed to detect the averaged response to many cycles of the square wave for each drop at the DME.
Thismethod is based on the idea of achieving a "steady-state" in the form of a repeated current cycle as a response toadditional potential cycles, and it has an interpretation similar to ac polarography (Chapter 10). It has beengeneralized to electrodes other than the DME and is also encountered as "square wave voltammetry." To avoidconfusion, it is sometimes called steady-state square wave voltammetry or Barker square wave voltammetry(BSWV), while the method of interest here is called transient square wave voltammetry or Osteryoung squarewave voltammetry (OSWV). OSWV is far more powerful and widely used. The term "transient" refers to thefact that a steady state is not reached in OSWV because the square wave is applied about a changing centralvalue at a stationary electrode.7.3 Pulse Voltammetry295In general, tp defines the experimental time scale; AES fixes the spacing of datapoints along the potential axis, and these parameters together determine the time required for a full scan.
In normal practice, A£ s is significantly less than A£ p , which defines the span of interrogation in each cycle and therefore determines the resolution ofvoltammetric features along the potential axis. Only tp is varied over a wide range, typically 1-500 ms ( / = 1-500 Hz). Osteryoung and O'Dea (50) suggest that A£ s = 10/wmV and AEp = 50/n mV suffice generally. With A£ s = 10 mV and tp = 1-500 ms, thescan proceeds at 5 V/s to 10 mV/s; thus the recording of a full voltammogram is quickcompared to the performance of most pulse methods and is comparable in duration totypical cyclic voltammetric runs (Chapter 6).(b) Theoretical Prediction of ResponseSince the diffusion layer is not renewed at the beginning of each measurement cycle, it isnot possible to treat each cycle in isolation, and theoretical treatments of SWV are intrinsically much more complex than for other forms of pulse voltammetry.
The initial condition for each cycle is the complex diffusion layer that has evolved from all prior pulses,and it is a function, not only of the details of the waveform, but also of the kinetics andmechanisms of the chemistry linked to the electrode process. The considerations are similar to those that we encountered in Section 5.7 as we treated double-step responses, andthe mathematical device of superposition can be applied to the extended step waveformsof SWV in the simpler cases.Let us now consider the prototypical case in which the electrode reaction О + ne *± Rexhibits reversible kinetics and the solution contains O, but not R, in the bulk.
The solutionhas been homogenized and the initial potential E-x is chosen well positive of is 0 ', so that theconcentration profiles are uniform as the SWV scan begins. The experiment is fast enoughto confine behavior to semi-infinite linear diffusion at most electrodes, and we assume itsapplicability here. These circumstances imply that we can invoke Fick's second law forboth О and R, the usual initial and semi-infinite conditions, and the flux balance at the electrode surface, exactly as in (5.4.2)-(5.4.5). The final boundary condition needed to solvethe problem comes from the potential waveform, which is linked to the concentration profile through the nernstian balance at the electrode.
It is convenient to consider the waveform as consisting of a series of half cycles with index m beginning from the first forwardpulse, which has m — 1. Then,- 1 I A£ s + ( - l)m A£ p(for m > 1)(7.3.27)where Int[(m + l)/2] denotes truncation of the ratio to the highest integer. The nernstianbalance at the surface can be expressed [as in (5.4.6)] for each half cycle in the followingway:вт = cu= ехр[л/(£ т - E°)](7.3.28)The solution can be obtained analytically (49, 50, 53) and the sampled current for the mthhalf cycle turns out to be2'«n=nFADU C% m Gi-i - Gi^ i (m-i+i)i/2nintm( 7 3-2 9 )whereа = т4^тг о>о)eo = o(7.3.30)296Chapter 7. Polarography and Pulse Voltammetryand £ = (D0/DR)l/2.
The sum in (7.3.29), which runs over all half cycles preceding and including the one of interest, manifests the prior history of electrolysis. Odd values of m correspond to forward current samples and even values denote reverse current samples.In much of the theory of SWV, currents are represented dimensionlessly by normalizing with the factor preceding the sum in (7.3.29). This factor is obviously the Cottrell current for time fp, which is the plateau current sampled in an NPV experiment with pulsewidth tp [see (7.3.3)].
Designating this current as zd, we can define the dimensionless current sample, фт, for the mth cycle askк (m - i + 1)1/2(7.3.31)The dimensionless difference current, Дг//т, is given by subtraction of pairs of samples,with the odd m taken first:(7.3.32)where m covers only odd values.Figure 7.3.14 shows dimensionless current transients and current samples for the experiment that we have been discussing. The currents are small in the early cycles, becausethe staircase potential is too far positive of Eo> for the forward pulse to reach the regionwhere electrolysis can occur.
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