J Electroanal. Chem., 202 (1986) 23-36 Elsevler Sequoia S.A., Lausanne - Printed I The Netherlands

23

SQUARE WAVE VOLTAMMETRY AT SMALL DISK ELECTRODES

THEORY AND EXPERIMENT

DAVID P. WHELAN, JOHN J. ODEA and JANET OSTERYOUNG

Department of Chemrstty State Unwerstty of New York at BuJJulo. Buffalo. NY 14214 (U S.A )

KOICHI AOKI

Department OJ Electrontc Chemtstry, Gruduute Shoal ut Nqutrutu, Tokyo Instrtute OJ Technoloxv. Nogcrtsuta. Mtdorr-ku, Yokohama 227 (Jupon)

(Received 18th June 1985; m revtsed form 15th January 1986)

ABSTRACT

An equation IS derived for the current response of a reversible electron transfer reactjon for square wave voltammetry at an embedded disk electrode. Peak shape and posItIon are mvarlant to the dlmenslonless parameter D7/r2 where D ~b the dlffusmn coefficient. 7 the square wave pertod. and r the radius of the disk, whereas peak current density mcreases wlthout hmlt with Increaamg 07/r? A simple empirlcal equation predicts the peak current for any value of Or/r for square wave amplitude 50/n mV and step height 10/n mV Expertmental results for oxldatlon of ferrocyamde at small platinum electrodes agree well with the theory and demonstrate the practical utihty of the experiment.

INTRODUCTION

Solid electrodes of micrometer size have been used as powerful tools for measure- ments in vivo [l-5]. Many fundamental electrochemical studies on very small electrodes have also been published [6-111 (see also references cited in refs. 10 and 11). Electrochemical techniques employed with these electrodes include chronoam- perometry, cyclic voltammetry, and chronopotentiometry. Recently ODea et al. have demonstrated the feasibility of carrying out square wave voltammetry at micro-electrodes of various geometries [12]. Square wave voltammetry appeared to yield a more readily analyzed voltammetric response than does cyclic voltammetry. In addition, like cyclic voltammetry. it is a fast technique in comparison with other pulse techniques such as normal pulse voltammetry. It also offers good subtraction of background currents. Thus we have undertaken development of quantitative theories for the square wave response at microelectrodes of various geometries.

Square wave voltammetry has been developed and applied to reverstble systems at planar electrodes [13] and the DME [14], and to the determination of electrode kinetics [15.16]. For reversible reactions under conditions of semi-infinite linear

0022-0728,86/$03.50 ~ 1986 Elsevler Sequoia S.A

diffusion, the differential or net current response is peak-shaped, symmetrical, and centered on the reversible half-wave potential. In addition to providing a quantita- tive description of the corresponding response at disk electrodes of arbitrary size, we wished to employ the result to see to what extent these properttes of the net current response were maintained under conditions of non-planar diffusion. In the follow- ing. an exact equation is developed for the square wave current at a disk electrode for a reversible electrode reaction. A simple approximate equation is derived for the relationship between the peak current and the radius of the electrode. The experi- mental investigation employs the oxidation of ferrocyanide at platinum disk elec- trodes with radii in the range of lo-40 pm.

THEORETICAL

Square wave voltammetry has been investigated in detail by a number of authors from this laboratory [15,16]. The details of the square wave waveform can be seen in Fig. 1 in ref. [15] and the nomenclature used by ODea et al. [15] is also used here. The important parameters characterizing the waveform are the step height of the base staircase, A E,, the square wave amplitude, E,,,, and the square wave period, T.

Consider the simple electrode reaction. R * 0 + tr e-, that occurs at the sta- tionary disk micro-electrode with radius r. It is assumed that the reaction takes place so rapidly that the Nernst equation holds at the electrode surface, and that the initial concentration of the oxidized form is zero. If the diffusion coefficients of substances 0 and R have a common value of D, the total current, I, passing through the electrode is given by [6]

4D(t-n)

rz I[ d(l+e icr dir du - 1 (1) where c.* denotes the bulk concentration of species R. t is the electrolysis time and

l(r) = (nF/~7-)( I - ~0') (2)

Here E" is the formal potential of the reversible couple. and E(f) is the electrode potential which varies with t according to the applied square wave waveform. The function f is defined by

f(I) = 1 + 0.71835Y + 0.05626~~~ - 0.00646; 5,

for z > 0.88 and

f(~)=(Y7/4-_)~7+7r/4+0.094-_ Z (3)

for z < 1.44. As the electrode potential is stepped through the square wave profile. functions

[l + emit]- for the duration of the X forward (k~ < t < (k + l/2)7) and reverse ((/i + l/2)7 < I < (li + 1)~) square waves are given by

C, = l/(l+exp[-rrF( E,+E,, +kAE,-EE"')/RT]) (4)

and

C; = l/(1 + exp[ -nF( E, - E,, + kAE, - E)/RT] ) (5)

respectively. where E, is the initial potential. Substituting eqns. (4) and (5) into eqn. (I) and carrying out the integration with attention to the step functions in eqns. (4) and (5) yields [17]

h-1

+(c~-c,,,lf(5-(j+l)p)] for kr < t < (k + l/2)7, and

h-l

1/(4nFc*Dr) = Cf(t) - c [CC,- C/ME- (j+ 1/2)P) , =o

(6)

+(c;-c,+,)f(~-(/+l)P)] -(C,-C~)S(5-(k+1/2)p) (7)

for (k + l/2), < t < (k + l)~, where 5 = 4Dt/r and p = 4DT/r. The drmension- less quantity p relates the size of the diffusion layer to the size of the electrode and can be used as a dimensionless square wave period. The forward current, I,. is sampled just before the change over point at the center of the square wave period i.e. at t = (k + l/2)7. Therefore eqn. (6) can be rewritten as

+K,-c,+,M-I- 1/W} I (8) Similarly the reverse current. I,, is measured at the end of the square wave period. i.e. at t = (k + l)~, and eqn. (7) can be rewritten as

+(c,-c,+,)f((li-J)P)) -(c,-c;)J(P/2) 1 (9) The net or difference current is given by

AI = I, - I, (10)

Equations (X), (9) and (10) are functions of p, nA E,/RT and rzE,,./RT alone. if the Initial potential is set sufficiently away from E.

26

i -200 -100 0 100 200

n[E - E1 / mV

Fig. 1. TheoretIcal square wave voltammograma at micro disk electrodes. showmg forward. reverse and

net currents. for nAE = 10 mV. nE,, = 50 mV; T= 25C and p = 4D~/r =(a) 10. (b) 5. (c) 2, (d) 1, (e) 0.3. (f) 0.001. The dashed. the dotted and the soltd curves represent the forward. the rewrse and the net currents, respectwely.

Square wave voltammograms calculated using eqns. (8)-(10) are shown in Fig. 1 for several values of p, with nAE, = 10 mV and nE,, = 50 mV at 25C. The net current profiles are bell shaped and symmetrical for all values of p. For 07/r < lo- (large radii or high frequencies) the forward, reverse and net currents are the same as those at a planar electrode with semi-infinite linear diffusion [3.15]. However, for larger values of 07/r, the current densities are much larger than thts. e.g. 2%, 5% and 10% for 07/r= 4 X 10e4, 2.5 X 10-j and 0.01 respectively. because of the edge effect. At larger p values the shape of the forward current-potential curve tends to resemble that of the reverse until finally the difference current may be expressed as

AI/4nFc*Dr = CA - CL

for p + ~10. The dimensionless normalised net current, 4. is defined as

(11)

while the maximum value, 4,. is given by eqn. (12) using the maximum net current, 1 I,. The ratio of #,, to the peak width at half height. W, ,?, depends upon nE,, as shown in Fig. 2. This ratio exhibrts a maximum at nE,w = 50 mV, regardless of the value of p and the potential step height. The step height has little effect on the peak current. Therefore the optimum square wave amplitude and step height at a stationary micro-disk electrode are the same as at an electrode with planar diffusion [15]. Our selected values are nAE, = 10 mV and nEIu = 50 mV at 25C. unless otherwise specified.

21

i0 0

Fig. 2. TheoretIcal variations of the ratm of the net peak height to peak wdth at half height. wth the square wave amplitude, E,,. for p = (a) 10. (b) 5, (c) 2, (d) 1, (e) 0.3, (f) 0.001.

The net current peak potential remains at E;,* - nAE, (note E;,z2 = E since D,, = DR) and is largely unaffected by variations in p and nE,,. There is a slight shift to positive potential with a decrease in either p or nE,,. However for nAE, < 20 mV and nE,, > 40 mV this shift can be approximated to be -nAE,, within n mV error, for all values of p.

The variation of the dimensionless net peak currents with p for nA E, = 10 mV, nEIw = 50 mV, is shown in Fig. 3. Since the peak potentials do not depend on p, the peak currents can be evaluated by summing together each term of function f in eqn.

Fig. 3. TheoretIcal dependence of the normahsed net peak current on p = (4D7/r)*. (--- ) at macro-disk. (. .) asymptotic hne. and ( - - - ) for hnear dlffuslon at large electrodes.

28

(3). For nAE, = 10 mV, nE,, = 50 mV the maximum value of C, - C, is 0.7500. Two equations describing the behaviour in the limiting regions can be evaluated. For p = 0. i.e. without edge effects, we use eqn. (3) for 2 < 1.44, retaining only the first term, to obtain 4, = 1.313. which is the value obtained for semi-infinite linear diffusion [15]. The peak current observed at a relatively large electrode with a small edge effect is obtained by retaining the first two terms in the same equation for f and is given by $r = 1.313 + 0.686~. If one uses an extremely small electrode, the predicted peak current is AZ,, = 3nFc*Dr, which has been derived from eqn. (11). The peak current becomes proportional to r, unlike the case for a large electrode, where it is proportional to the area. Therefore the decrease in the peak current resulting from decreasing the size of the micro-electrode will not be as great as with an electrode of conventional size.

An approximate equation for dimensionless peak current which describes the behaviour over the entire range of p is

4, = 0.846~ + 1.06 + 0.25 exp( -0.8p) (13)

which is valid for nA E, = 10 mV, nE,, = 50 mV with a relative error involved in the approximation of less than 0.3%.

The peak width at half height, W,,z, also stays constant, varying only from 126 mV to 124 mV, for nE,, = 50 mV and nAE, = 10 mV at 25C, with increasing p. It is quite insensitive to changes in nAE,.

In order to evaluate the parameter p for real square wave voltammograms it is necessary to know the values of D and r. Assuming r can be measured geometri- cally, multiplying eqns. (12) and (13) by p yields

and

p2~P = 0.846~ + 1.06~ + 0.25p, exp( -0.8~) (15)

respectively. Knowing ~/~#r from eqn. (14) p can be evaluated by inverting eqn.

(15). Similarly, if D is known, r can be eliminated by dividing eqns. (12) and (13) by

p, giving

Ic, P= AZ,

P 4nFc*( mD3)2

and

$,p- = 0.846~-~ + 1.06~~ + 0.25p- exp( -0.8~~) (17)

Having calculated #,/p from eqn. (16) p can then be calculated by inverting eqn. (17). Note that only p is specified by experiment, and therefore either D or r must be known in order to evaluate the results quantitatively. Inversion of eqn. (15) or (17) may be accomplished by any suitable algorithm [17].

29

EXPERIMENTAL

The working micro-electrodes consisted of platinum wires (25 ym diam., 99.95%; 76 pm diam., 99.99%; Johnson Matthey Inc.). The wires were heat sealed into 2 ~1 Drummond Microcap capillaries, which were subsequently mounted in larger glass tubing, using an insulating epoxy, for ease of handling. The tip of the electrode was either abraded using emery paper or cut back using a diamond saw so as to expose only a disk of platinum to the solution. The surface was then polished with a Buehler Minimet polisher [18] starting with 45 pm diamond paste and working down in steps to 0.05 pm alumina. The geometric radius of the electrode was determined through an optical microscope (Leitz Epivert 500 x , 1000 X ), and electron microscopy was used to confirm the disk shape. An attempt to use gold micro wires had to be abandoned because the shape was distorted too easily on sealing them into the capillaries. It should be emphasized that the phenomenon being examined depends on the edge. Slight imperfections, including deviations from planarity, at the edge can affect seriously the quantitative response. Electrodes which are quite acceptable for analytical work in which concentration is the only parameter varied, or for mechanistic studies in the limiting ranges of behavior (e.g., p < 0.01, p > loo), are unacceptable for applications such as the present one which involve quantitative comparison with theory. The electrodes used here have constant diameter to within 0.1% and the uncertainty in the radius is 1 X lo- m.

All experimental control, data collection and calculations were carried out on a DEC PDP 8/e laboratory minicomputer, interfaced to a home-made potentiostat described elsewhere [19]. The reference electrode was an EG&G PARC saturated calomel electrode, and all potentials are quoted with respect to this. A platinum counter electrode was used.

All chemicals were reagent grade and used without further purification. The experimental system was K,Fe(CN), in 2 mol dme3 KNO,. Water was purified by passing distilled water through a Millipore Mini-Q purification system. The solu- tions were purged for 15-20 n-tin with argon which had been purified by passing it over a BASF catalyst, to remove residual oxygen before analysis.

Before each experiment the electrodes were repolished with 0.05 pm alumina, rinsed briefly with 95% ethanol and finally washed with purified water in an ultrasonic bath. The electrode potential was swept from 1.2 V (SCE) to -1.0 V (SCE) several times to precondition the electrode surface before each experimental run.

RESULTS AND DISCUSSION

The shapes of the finished micro electrodes, seen in the electron micrographs in Fig. 4, are disks to very high precision.

Typical square wave voltammograms, for the oxidation of 20 X 10m3 mol dm-3 K,Fe(CN), in 2 mol dmm3 KNO, on a 25 X 10m6 m diameter platinum micro-elec- trode for different values of the square wave frequency, are shown in Fig. 5. Figure

Fig. 4. Electron micrographs of typlcal platinum microdisk electrodes; (a) r =13.06X 10m6 m. (b)

r=37.05X10-6 m.

400- @J

I I 0.00 0.25 0.50

POTENTIAL / Vk3.C.E.)

Fig. 5. Square wave voltammograms for the oxidation of 20X 1O-3 mol dme3 K,Fe(CN), m 2.0 mol dmm3 KNO, at a 13.06 X 10m6 m radius platmum disk electrode; AE = 10 mV, E,, = 50 mV. T = 25C, f = (a) 5 Hz. (b) 60 Hz, (c) 500 Hz; (0) experimental data, (- ) theoretical fit.

31

0.00 ~OTENT,A$ v(S.C.E.) Fig. 6. Square wave voltammograms for the oxidation of 20X lo- mol dmm3 K,Fe(CN), in 2 mol dm-- KNO, at a 37.05 x 10mh m radius platmum disk electrode: AE = 10 mV, E,, = 50 mV. T = 25C. f = (a) 0.5 Hz, (b) 5 Hz, (c) 60 Hz: (0) experimental data. (- ) theoretical fit.

6 shows similar results for a 75 x 10-h m diameter electrode. The symbols represent the experimentally determined forward, reverse and net currents. The continuous curves represent computer-generated theoretical currents using the optimum values of the parameters chosen by numerical analysis of the data.

As predicted by the theory, at lower square wave frequencies and smaller electrodes the forward current profile tends to resemble that of the reverse current profile. At higher frequencies and larger electrodes the voltammograms resemble those at a planar electrode with semi-infinite linear diffusion [3,15]. The net current profiles in all cases are bell-shaped and symmetrical.

Numerical analysis, using a simplex optimisation routine, of eqns. (8)-(10) yielded a,theoretical fit to the experimental data. This is achieved by testing different values of the variables in the equations and choosing the optimum values. The optimum values are those which give the minimum values of the objective function (1 - R), where R is the correlation coefficient of a linear regression of Z on +. The details of this procedure are discussed elsewhere [16]. Results obtained from the numerical analysis are presented in Table 1. In this case the diffusion coefficient is assumed to be known and the numerical analysis yields estimates of the radius of the electrode and the reversible half wave potential (E;,,). It also yields the 95% confidence interval for the variables and the objective function value which gives a measure of the fit between theory and experiment. There is quite good agreement between the geometric radii of the electrodes and those determined by the simplex

32

TABLE 1

Results from simplex optimlsatlon on the data of Figs. 5 and 6

106r/m a f/Hz 106r/m b

G,,/mV 104(1- R)

13.06 5.0 12.4 (-0.8. +0.9) 239.4(-1.5. +1.5)d 6.0 13.06 60.0 13.7 (-0.7, +0.7) 238.0 ( - 1.4. + 1.5) 10.5 13.06 500.0 14.5 (- 1.9, + 2.4) 239.8 ( - 2.0, + 2.0) 26 4 37.05 05 26.5 ( - 2.7, + 2.7) 234.7 ( - 1.6, + 1.6) 6.8 37.05 5.0 36.7 ( - 1.1, + 1.1) 235 5 (-0.8. +0.8) 3.0 37.05 60.0 36.1 ( - 2.9, + 3.6) 232.5 ( - 1.4, + 1.4) 12.5

From optlcal measurements. From voltammetric data; D = 6.9 x IO- lo m2 5- . R is the correlation coefficient of the plot of I vs. 4 for the optimum values of p and Et,, d Size of the confidence elhpsold m the r dimension. Size of the confidence ellipsoid m the El,* dimension.

optimisation technique for the results shown. However. the agreement is poorer at higher and lower square wave frequencies also investigated for this series of experiments. The range of frequencies examined was 0.5 Hz to 1500 Hz. At the higher square wave frequencies, the confidence interval for r is broader, indicating a smaller dependence on r, and the calculation tends to fail once the value of p is such ( < 10d2) that the enhancement of the currents, in comparison to those with semi-infinite linear diffusion, is comparable to the noise level. Also, at frequencies below 1.5 Hz, the agreement between the radii becomes poorer (see Table 1). probably because of problems with convection. (At f= 1 Hz and AE, = 10 mV. it takes 40 s to cover the range of 400 mV.)

It is well established that micro-electrodes are more sensitive to electron transfer rates than larger electrodes because of the large edge effect. However, no simple criterion of reversibility has been established for square wave voltammetry at micro-electrodes. In order to elucidate the reversibility of the system under study, especially at higher frequencies, variations of the forward, reverse and net peak potential with square wave frequency are shown in Fig. 7. Also shown is the variation of the net peak width at half height. The separation between the forward and reverse peaks is quite large for low frequencies and small electrodes. However the peak positions become closer at higher frequencies and tend to approach to within AE, of the net peak position. The net peak position stays constant at all frequencies at 240 k 4 mV. The width at half height. W,,2. of the net peak also stays constant at 126 + 2 mV, which agrees with the value determined from theoretical curves, 125 mV. Similar results were obtained using the larger electrodes.

At the highest frequencies the effects of non-reversibility should be most preva- lent and the micro-electrode effects should be least. Therefore, reversibility, or non-reversibilify, may be determined by comparing the results obtained at high frequencies with those expected for macro electrodes. The lack of deviation of the net peak position and the peak width at half height at higher frequencies in Fig. 7 confirms the reversibility of the system under the conditions employed.

33

Fig. 7. Effect of the square wave frequency on the positions of the forward (0). reverse (*) and net (0) peaks, and the net peak width at half height ( + ). m the square wave voltammograms for the condltlons outlined m Fig. 5. at a 13.06 x lo- m platinum disk electrode.

The analysis described above gives a picture of the detailed correspondence between theory and experiment for individual voltammograms. It does not show directly how the experimental response can be expected to vary with the parameter p. Furthermore, the simplex optimization procedure itself required considerable time to develop and therefore is not now practically accessible. It also requires for this case both forward and reverse currents in digital form, which are not necessarily obtainable from commercial instruments. For these reasons, we present also analyses of data based on eqns. (14)-(17) which require only that one determine peak currents. For this approach, it is necessary to estimate either values of r or D in advance. In this case we assume that the values of r can be determined from geometrical measurements while the values of D are unknown. Values of I, were determined for a variety of square wave frequencies at electrodes of different radii. The quantity P,~$~ is calculated using eqn. (14) for each frequency and a corresponding value of p can be determined from eqn. (15) by using a trial and error method. These experimentally determined values of pelp should be propor- tional to 7 or the inverse of the frequency, f- . The resulting plots were linear (R > 0.9999) for both electrodes over the entire range of frequencies examined and from the slopes (16.42 and 1.97 ss) the value of D is determined to be (6.9 + 0.1) X 10. "' m SC. This value agrees with that quoted by other authors [11,20].

A further check can be made by plotting p_, against T/r?. This should combine the data onto a single line regardless of the values of r, f and c*. In Fig. 8, log( p,,,) is plotted against log( T/r ) for the different-sized electrodes. A straight line with

34

r

lOg(Y rg/amd*)-6

Fig. 8. Variation of experimental pmax values with r/r2 for the data in Fig. 7. (*) r =13.06~10-6 m, (0) r = 37.05 X 10e6 m. The straight line has a slope of unity.

slope of unity is drawn through the points. The fit is excellent, indicating good agreement between the experimentally determined results and those predicted by the theory.

Having calculated a value for D, it is interesting to use this value to determine the variation of the normalised net peak currents, qP. experimentally evaluated from eqn. (12) using the normalised square wave period, p, and to compare this with the theoretical variation predicted by eqn. (13) for these conditions. In Fig. 9, the curve, which is the same as the solid curve of Fig. 4, shows the theoretical variation. There is good agreement between it and the experimentally determined values (0).

Fig. 9. Dependence of the normalised net peak currents, +,, on the dimensionless square wave period. p: r=13.06~10-~ m, D=6.9X10- m2 SC, other conditions as in Fig. 5; (0) geometric radii, (+ ) calculated radii and normalization coefficients, (- ) the theoretical curve.

35

Fig. 10. Effect of the square wave amplitude, E,,, on the ratio of the normalised net peak height to the

peak width at half height, IJ,/W,,~. Data from square wave voltammograms at a 13.06X 1O-6 m platinum disk, and a square wave frequency of 5 Hz. Other conditions as in Fig. 5. The theoretical curve

(- ) is taken from Fig. 2

Alternatively one can plot the experimentally determined +, against those calcu- lated using eqn. (13). The fit is excellent (R = 0.9998) and the slope is 0.988, again showing the agreement between the theoretical predictions and the experimental results.

Also shown in Fig. 9 are the data obtained by using the radii and normalisation coefficients calculated by the simplex optimisation technique (cf. Table 1). Again the agreement is excellent. The fitting procedure which employs all of the voltam- metric data yields values that agree with theory just as well as the peak currents alone. The advantage of the former procedure is that it yields an unambiguous estimate of the uncertainty of the derived parameters. The derived result, e.g. Fig. 9, can be misleading in suggesting how precisely the value of r can be inferred from the voltammetric response. Clearly the relative uncertainties in the r values of Table 1 are as much as ten-times as large as the deviations of theoretical and experimental values displayed in Fig. 9.

Figure 10 shows the variation of the ratio of the normalized net peak height to half peak width with square wave amplitude, using the conditions outlined in the legend of the graph. This may be compared with the theoretical prediction of Fig. 2. The theoretical curve is bell-shaped with a maximum at 50 mV. The data generally agree with the prediction, but the precision is very poor. Thus the recommendation to choose nE,, = 50 mV for optimum response is verified, but square wave ampli- tude does not appear to be a useful experimental parameter.

In conclusion, square wave voltammetry can be readily applied to micro disk electrodes. The experimental behaviour agrees quite well with the theoretical predic- tions. This is a useful electrochemical technique which combines together the

advantages of square wave voltammetry and microelectrodes. In particular, in sharp contrast with staircase or linear scan voltammetry, the voltammograms are largely invariant in shape over the entire range of the dimensionless parameter, p. which characterises the response. Thus they are well-suited to measurements of formal potential. This last finding can be generalised to other electrode geometries [12].

ACKNOWLEDGEMENTS

This work was supported in part by the U.S. National Science Foundation under grant number CHE8305748 and by the U.S. Office of Naval Research. J.O. gratefully acknowledges support by the Guggenheim Foundation.

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