Comparison of voltammetric steady states at hemispherical and disc microelectrodes

11

J. Electroanal. Chem., 256 (1988) 11-19 Elsevier Sequoia S.A., Lausanne - Printed

in The Netherlands

Comparison of voltammetric and disc microelectrodes

steady states at hemispherical

Keith B. Oldham Trent Unroersrty, Peierborough (Canada)

Cynthia G. Zoski Deakin University, Geelong (Austraha) (Received

20 July 1988)

ABSTRACT Explicit formulae are presented for the shapes of steady-state voltammograms at mlaid disc and at shrouded hemispherical microelectrodes. For electrodes of equal superficml diameters, microdiscs and microhemispheres give identical steady-state voltammograms under reversible conditions. The two voltammograms are not identical if the electron transfer occurs irreversibly. Quasi-reversible processes are intermediate.

INTRODUCTION

Inlaid disc electrodes are easily made [l-5] but are difficult to treat mathematically. Microelectrodes having the geometry of a hemisphere resting on an insulating plane are difficult to fabricate, but their behaviour is easily predicted. Accordingly, experiments conducted at microdisc electrodes have sometimes [6-81 been analyzed in terms of shrouded hemispherical geometry, and more modelling of this sort is likely to be made in the future. The purpose of this article is to test the validity of using a microhemisphere to model a disc microelectrode by comparing the behaviour of the two electrodes under conditions in which exact solutions are known for both geometries, namely for the electrode reaction O(soln)

+ n e- z R(soln)

(1)

in the steady state. Other situations under which similar comparisons have been made are: the initial response to a concentration-polarizing potential step [9]; and the duration of the transient state in a variety of circumstances [lo]. OC22-0728/88/$03.50

0 1988 Elsevier Sequoia

S.A.

12

We shall consider reaction (1) occurring at the interface between a microelectrode and a solution that contains a bulk concentration & of the electroreactant but from which species R is initially absent. The solution is unstirred and contains excess supporting electrolyte, so that the only transport mechanism that needs to be considered is diffusion, the diffusion coefficients of the two species being Do and D,. The heterogeneous rate constants k, and k, will be assumed to be dependent upon potential via the Butler-Volmer equations k,=k”

exp{-cmP(E-EO)/RT}

(2)

exp{(l-ar)nF(E-EO)/RT}

(3)

and k,=k”

in terms of the three constant parameters: E O, a and k O. A valuable attribute of steady-state voltammetry is that the shapes of the steady-state voltammograms do not depend in any way on the experimental technique (be it potentiostatic, galvanostatic, potentiodynamic, or whatever) used to record them. Thus we can pursue our comparative analysis without making reference to the specific experimental method that might have been employed to construct the voltammograms. MICROELECTRODE

SIZES

We shall need to compare the behaviours of microdisc and microhemispherical electrodes when subjected to steady-state voltammetry. Because their responses depend on electrode size, we must establish some criterion upon which to ascribe equivalent sizes to a microdisc and a microhemisphere. One could consider that the two microelectrodes are equivalent when their radii are equal r0 = a

(4

(see Fig. l), or when their exposed areas are equal 28rg2 = 7ra2

(5)

but it turns out that neither of these criteria is as convenient as a third possibility, which is the one that we adopt. We shall consider a microdisc and a microhemisphere to be equivalent in size when their “superficial diameters” are equal. The

Fig. 1. An inlaid microdisc and a shrouded microhemispherical electrode in cross-section. The two are of equivalent size when their superficial diameters d are equal.

13

superficial diameter of an electrode is the distance d from one edge of the electrode to the other, measured along the electrode surface. From Fig. 1 we see that this means that a microdisc and a microhemisphere are of equivalent size when

sr,=d=2a

(6)

This criterion of equivalence is appropriate because microelectrodes of equal superficial diameters support equal diffusion currents [9], given by eqns. (8) and (12) below. STEADY-STATE

VOLTAMMETRY

AT HEMISPHERICAL

MICROELECTRODES

The most comprehensive theory of voltammetry at electrodes of spherical symmetry is that published first by Delmastro and Smith [ll] and later by Bond and Oldham [12]. These workers treated potentiostatic chrono-amperometry for a whole sphere of radius r,, but, by taking the t + cc limit and rewriting for a hemisphere of superficial diameter d, the Delmastro-Smith relationship simplifies to

2rFc&kfr0 i = (k/Do)

+ (k,/D,)

2nFc;ktd f (l/r,)

= (kJDo)

+ (k,/D,)

+ (s/d)

(7)

As the potential becomes progressively negative, the current approaches its diffusion-limited steady-state value i,, which is seen to be

i, = 2mF&Dor0

(8)

= 2nFckD,d

because k, --, 00 as E --, - CO. The combination of eqns. (7) and (8) may be written as i=

(9)

8+ (:,8r)

in terms of two potential-dependent B=l+$exp

R

parameters

[%(E-E~)]=I+$$ R

b

(10)

and

01) which will play an important role in our development. Notice that Bi/i, is a function only of the ~4 product; this is a feature that the microhemisphere shares with the microdisc, though the functions differ. STEADY-STATE

VOLTAMMETRY

AT INLAID

DISC MICROELECTRODES

Saito [13] was the first to derive an expression for the diffusion-limited current at a disc of radius a. This may be rewritten as

i, = 4nF&D,a

= 2nFc&D,d

steady-state

(12)

14 TABLE 1 Data from which the shapes of steady-state voltammograms at microdisc electrodes can be constructed. The data for the first two columns are taken from ref. 14, which should be consulted for details and further values. The third column was constructed using qn. (13) and the r0 values from the fit column

ue

Bi/i,

0 0.0100 0.1000 0.3420 1.0723 3.5400 10.0000 100.0000 cc

ref. 14

eqn. 13

0 0.0099 0.0903 0.2500 0.5000 0.7500 0.8824 0.9846 1

0 0.0099 0.0903 0.2501 0.5000 0.7493 0.8828 0.9849 1

in terms of the disc’s superficial diameter and is seen to be identical with eqn. (8) for a hemisphere of the same superficial diameter. Bond et al. [14] have recently published a table that describes the general behaviour of microdisc electrodes by listing values of the quantity IA/i, as a function of the product ~0 of the parameters defined in eqns. (10) and (11). A small portion of that table is reproduced as the first two columns of Table 1. The construction of the original table involved extrapolations to an infinitely-sized matrix, so that the tabled numbers doubtlessly incorporate small errors. More recently we have discovered that the empirical formula

reproduces the entries in the published table with a maximum error of less than 0.3%. This formula is probably as accurate as, and certainly a lot more convenient than, the tabulation and we shall use it henceforth to replace the table. The final column in Table 1 contains values calculated via eqn. (13) to emphasize the accuracy of the formula. REVERSIBLE

STEADY-STATE

VOLTAMMETRY

As a function of potential, the

KB

product exhibits a minimum value of

rk “d 8[(1-

(14

o)DO]i-=[a&]”

If even this minimum value is much greater than unity, the electrode reaction occurs reversibly. The condition KB 53 1 leads to the result 2nFc&D,d i=%=

l+(D,/D,)exp{nF(E-E”)/RT}

(15)

15

when applied to either eqn. (9) or eqn. (13), showing that this expression describes the shape of a reversible steady-state voltammogram at either a hemispherical or a disc microelectrode. The halfwave potential, where i = i,/2, is seen to be E 1,* = E o + ( RT/~F)

ln( DJDo)

(16)

and eqn. (15) can be inverted to E=E,,,+(RT/nF)ln{(i,-i)/i}

(17)

in its terms. These equations all resemble classical polarographic results [15] except that the diffusion coefficient ratio DJD, does not appear under a square-root, as is the case for reversible transient voltammetries. The Tomb criterion [16] of wave shape is found to be E l/4

-

=

E3,4

RT/~F)

(

ln(9) = (56.45 mV)/n

(18)

at 25.0 o C. IRREVERSIBLE

STEADY-STATE

VOLTAMMETRY

Total voltammetric irreversibility arises from k” having such a small value that the reduction wave is shifted so far negatively from the standard potential that the exponential term exp{ nF(E - E “)/RT} is negligible in comparison with unity at all potentials at which significant current flows. Thus 8 = 1 characterizes irreversibility. Setting 8 = 1 in eqn. (9) leads to jhem

-

2nFckD,d

id

_

I+ (r2/g~)

= 1+ (sD,/k”d)

exp{anF(E-

as the equation of an irreversible steady-state microelectrode. The four equations

voltammogram

E~~~=E”-(RT/anF)h{‘lTDg/k”d} ihemi

(19) at a hemispherical

(20)

‘d

-_

(21)

l+exp(anF(E-E$y)RT) Eheti

E”)/RT}

=

Ehem’ 1,2

+

(RT/anF)

ln{(id - i)/i}

(22)

-

(RT/anF)

ln(9) = (56.45 mV)/an

(23)

and Eh”“‘_Ehemi_ l/4

3,4

follow directly from eqn. (19). We have used a superscript “hem?’ because, as we shall see, discs do not emulate hemispheres when the voltammetry is irreversible.

16 I 200

100

I

-E

- El~'/rnV

I

0

Rg. 2. The shapes of irreversible steady-state voltammograms from rmcrodisc and microhemispherical electrodes of equivalent size. The lower abscissa is scaled logarithmically with K being the quantity defined in eqn. (11). The values an = 0.5 and T = 298.15 K were assumed in scaling the upper abscissa, whose zero has been made to coincide with the halfwave point of the microhemispherical electrode.

To describe irreversible behaviour in eqn. (13). Thereby the expression

at an inlaid

disc microelectrode,

one sets 8 = 1

(24) for the current in terms of K is produced, (11). Alternatively, this pair of equations equation for the potential

being related to potential via definition may be inverted to produce an explicit

K

(25) in terms of current. Equation (24) or (25) describes the shape of an irreversible steady-state voltammogram at an inlaid disc electrode. An example is graphed in Fig. 2, where the wave shape is compared with that of the equivalent hemisphere, as given by eqn. (19). Evidently the disc voltammogram is less steep than that of its hemispherical model and is, for the most part, displaced towards positive potentials. Setting i/(id - i) equal to unity in eqn. (25) leads to the expression E f;s2c= E” - (RT/anF)

ln{2.730Do/k”d}

(26)

17

for the halfwave potential. hemisphere produces Ed&c

_

l/2

Ehemi 1,2

_ -

RZ’/(7.13cunF)

Comparison

of this result with eqn. (20)

for the

= (3.60 mV)/an

(27)

at 25.0 o C. Thus for a typical reaction (a = 0.5, n = l), the model predicts a halfwave point more negative by some 7.2 mV than is actually found for an inlaid disc microelectrode. This means that, if an irreversible steady-state disc voltammogram were analyzed as if it had been obtained at the equivalent microhemisphere, the calculated standard rate constant k” would be high by some 15%. Similarly, one may set i/(id - i) equal to l/3 or 3 in eqn. (25) to obtain expressions for the one-quarter and three-quarters wave potentials. Subtraction then gives Ed= _ Eti” 3,4 - 2.34RT/( anF) = (60.2 mV)/cYn (28) l/4 at 25.0°C for the Tom& criterion of wave steepness at the microdisc. Comparison with eqn. (23) for the microhemispherical electrode shows that E&X _ E&SC l/4 3/4 = 1.066 (29) Ehemi _ Ehemi l/4

3/4

which quantifies the greater steepness of the hemispherical model that is evident in Fig. 2. Though the irreversible wave at a microdisc electrode is asymmetric, in that it is steeper above the halfwave point than at a corresponding point below, the asymmetry is mild and it has been shown [14] that the voltammogram almost satisfies the symmetrical equation Edis = E$“;‘+ (l.O63RT/anF)

ln{(i,

- i)/i}

(30)

The corresponding equation for the hemisphere, eqn. (22), lacks the 1.063 factor. Therefore the transfer coefficient (r, if calculated from the “log plot slope” of a disc voltammogram on the basis of the hemispherical model, would be 6.3% low. Equation (29) shows that a similar calculation based on the Tom& criterion would be 6.6% low. On the other hand, the transfer coefficient would be calculated correctly from the shift of halfwave potential with electrode diameter, because the result A&,2

A In(d)

RT =- anF

(31)

is common to both microelectrodes. QUASI-REVERSIBLE

STEADY-STATE

VOLTAMMETRY

For the most general, or quasi-reversible, case of reaction (1) occurring microelectrode of superficial diameter d, the steady-state current is given by

at a

-1

(32)

18

Fig. 3. The shapes of quasi-reversible steady-state voltammograms for a microdisc (- - -) and a microhemispherical electrode (), each of superficial diameter d. The parameter values a = 0.5, DJD,=n=l, k” - SD/lrd and T= 298.15 K were assumed. ( . . . . ) Reversible wave for comparison.

for the shrouded hemisphere and by (33) for the inlaid disc. For both electrodes the diffusion current id is equal to ZnF&D,d and the potential-dependent parameters 8 and K are given by eqns. (10) and (11). It is not possible to invert eqn. (32) or (33) to express E in terms of i/i, and hence explicit equations for E,,, and E,,4 - E3,4 cannot be written. Nevertheless, these equations may be used to construct voltammograms numerically, as was done in generating Fig. 3. One finds, as expected, that quasi-reversible steady-state voltammograms at inlaid disc electrodes agree less well with their hemispherical counterparts than do reversible voltammograms but better than in the irreversible limit. The voltammetric waves for discs are somewhat less steep than for hemispheres and lie at slightly more positive potentials. There is interest in using quasi-reversible steady-state voltammograms obtained at inlaid disc electrodes to determine the kinetics of fast electrontransfer reactions and this topic is the subject of recent publications [17,18].

ACKNOWLEDGEMENTS

We express sincere thanks to Alan Bond and Jan Myland, as well as to our financial supporters: the Natural Sciences and Engineering Research Council of Canada, and the Australian Research Grants Committee.

19 REFERENCES 1 R.M. Wightman and D.O. Wipf in A.J. Bard (Ed.), Electroanalytical Chemistry, Vol. 16, Marcel Dekker, in press. 2 C.D. Baer, M.J. Stone and D.A. Sweigart, Anal. Chem., 60 (1988) 188. 3 D.R. Rolison in M. Fleischmann, S. Pans, D.R. Rohson and P.P. Schmidt (Eds.), Ultramicroelectrades, Datatech Systems Publishers, Morganton, N.C., 1987, p. 65. 4 A.M. Bond, D. Luscombe, K.B. Oldham and C.G. Zoski, J. Electroanal. Chem., 249 (1988) 1. 5 A.M. Bond, M Fleishmann and J. Robinson, J. Electroanal. Chem., 168 (1984) 299. 6 J.O. HoweB and R.M. Wightman, Anal. Chem., 56 (1984) 524. 7 Z. Gahts, J.O. Schenk and R.N. Adams, J. Electroanal. Chem., 135 (1982) 1. 8 M.A. Dayton A.G. Ewing and R.M. Wightman, Anal. Chem., 52 (1980) 2392. 9 K.B. Oldham, J. Eleetroanal. Chem., 122 (1981) 1. 10 K.B. Oldham, C.G. Zoski and A.M. Bond, in preparation. 11 J.R. Delmastro and D.E. Smith, J. Phys. Chem., 71 (1967) 2138. 12 A.M. Bond and K.B. Oldham, J. Electroanal. Chem., 158 (1983) 193. 13 Y. Saito, Rev. Polarogr. (Jpn.), 15 (1968) 177. 14 A.M. Bond, K.B. Oldham and C.G. Zoski, J. Electroanal. Chem., 245 (1988) 71. 15 A.J. Bard and L.R. Faulkner, Electrochemical Methods: Fundamentals and Applications, John Wiley, New York, 1980, p. 160. 16 J. Tom&s, Co&et. Czech. Chem. Commun., 9 (1937) 150. 17 Z. Gahrs, J. Galas and J. Osteryoung, J. Phys. Chem., 92 (1988) 1103. 18 K.B. Oldham, C.G. Zoski, A.M. Bond and D.A. Sweigart, J. Electroanal. Chem., 248 (1988) 467.