Voltammetry at geometrically uneven electrodes

J. ElectroaMl. Chem., 246 (1988) l-14 Elsevier Sequoia !%A., Lausanne - Printed in The Netherlands

VOLTAMMETRY

AT GEOMETRICALLY

UNEVEN ELECTRODES

PART I. CHRONOAMPEROMETRY AT MODEL ELECTRODES RECTANGULAR HOLLOW OR PROTRUSIVE SURFACES

MITSUYA

TANAKA,

KOICHI AOKI, KOICHI TOKUDA * and HIROAKI

Department of Electronic Chemistry, Graduate Nagatsuta, Midori-ku, Yokohama 227 (Japan)

School

at Nhgatsuta,

Tokyo

WITH

MATSUDA Institute

l

*

of Technology,

(Received 13th October 1987; in revised form 8th December 1987)

ABSTRACT An electrode model with a two-dimensionally rectangular, hollow or protrusive surface is presented for uneven surface electrodes. A finite element method is applied to solve the non-linear diffusion problem at this electrode model for chronoamperometry. It is shown that the plot of the current (I) multiplied by the square root of time (fi) against fi reflects well the geometry of the electrode surface. A simple approximate equation is presented for the Zfi vs. fi relationship. Seven model electrodes of different dimensions (depth and width of hollows) are fabricated and the validity of the theoretical results is examined. Good agreement between the theoretical and the experimental curves is obtained and it is shown that the geometric parameters for model electrodes can be determined from the analysis of Z\lT vs. fi curves.

INTRODUCTION

The surface of a solid electrode is microscopically uneven no matter how elaborately it may be polished mechanically or electrochemically. In addition, it is often inhomogeneous because of the presence of inactive parts on it due to the adsorption of electroinactive species or the formation of less conductive oxides. This irregularity on electrode surfaces may cause the voltammograms for even simple redox reactions to be apparently complicated. Some attempts to challenge this subject have been made by Vetter [l], Llopis et al. [2], Nagy et al. [3], Landsberg and co-workers [4-91 and Levart et al. [lo]. However, these studies are qualitative, as

* To whom correspondence should be addressed.. l * Present address: Department of Chemistry, Faculty of Science, Chiba University, Yayoi-cho, Chiba 260, Japan.

0022-0728/88/$03.50

6 1988 Plsevier Sequoia S.A.

2

pointed out by Gueshi et al. [ll]. Gueshi et al. [lf,U], Matsuda [13] and Tokuda et al. [14] developed theoretical approaches to the diffusion problem at electrodes with active and inactive sites distributed periodically, and presented techniques for analysing the voltammetric curves. They also showed that the surface inhomogeneity could be estimated from variations of the diffusion-controlled currents with time. Amatore et al. ]15,16] elucidated sluggish charge-transfer reactions at solid metal electrodes by partial blocking of electroactive species on the electrode surface. Reller et al. [17,18] carried out digital sim~ation for an array of microelectrodes, which is analogous to a partially blocked electrode. Etman et al. [19,20] and Levart [21] treated partially blocked rotating-disk electrodes. Recently, Moldoveanu and Anderson [22] worked on the convective diffusion-controlled current at channel flow electrodes at which active and inactive sites are randomly distributed. On the other hand, the subject of unevenness of the electrode surface has progressed in the fields of el~~oplat~g and el~tropo~s~g. Despic and Popov [23] and Prentice and Tobias [24] computed numerically variations of the electrode geometry in the course of electroplating on the assumption of the steady-state concentration profiles caused by finite diffusion. Their computation resulted in solving the Laplace equation and hence is valid at very long times. Oldham [25] derived an expression for the time-dependent diffusion-controlled current at a wedge-shaped electrode by means of separation of variables and the Fourier expansion. This expression is valid only at times so short that the thickness of the diffusion layer is smaller than the characteristic length of the unevenness. When an electrode has an uneven surface of the order of 10 pm, the area of the diffusion layer may vary from the real area of the electrode surface to the projected area in the time domain employed for usual voltammetric measurements. Therefore this diffusion behaviour cannot be sufficiently covered by Oldham’s equation 1251 or by the steady-state approbation. Gueshi 1261 calculated n~e~cally chrono~perometric and chronopotentiometric curves and the farad& impedance at corrugated sheet electrodes and examined them experimentally. At present, to our knowledge there is no quantitative work except for this, which concentrates on diffusion in a wide time domain ranging from very short to very long times. In this paper, an electrode model with a rectangular, hollow or protrusive surface is presented for such uneven surface electrodes, and the initial and boundary value problem for mass transport caused by time-dependent non-linear diffusion at this electrode model is solved numerically by applying a finite element method. Further, the theoretical results are compared with experimental ones obtained at gold model electrodes fabricated after the geometry of the model. THEORY

Presentation

of the eiectrode model

Let us consider an electrode model with the surface profile shown in Fig. 1A. The characteristic dimensions are specified by the height h, the half-pitch p of the

[A]

y ,__.,‘.

n

s,

,

52

/,

;5

AI31

/

LB1

0

w’

-x

P

Fig. 1. (A) Schematic diagram of the two-dimensional model electrode with a rectangular, hollow protrusive uneven surface. (B) The unit diffusion space used for the finite element method analysis.

or

repeating unit and the half-width w of the hollow. The shape of this electrode model has been selected on the basis of the following considerations: (a) An electrode with cracks, e.g. in Fig. 4A of ref. 27, has a geometry similar to this model electrode. (b) This model is applicable to electrodes having pits although the model is two-dimensional. (c) When an electrode is composed of deposits in a plate or column crystal form, the morphology may be roughly approximated by the geometry of the model. (d) When the width of the hollow is smaller than the depth, diffusion in the hollow, being like finite diffusion, is distinguished from diffusion outside of the hollow, which is similar to semi-infinite diffusion. Therefore it is expected that diffusion at the model electrode is possibly expressed by a simple combination of infinite and finite diffusion. This prediction will be discussed at the end of this section. The repeating unit is shown in Fig. lB, together with the Cartesian axes. It is in this unit region that the boundary value problem is solved. It is assumed that the mass transport of an electroactive species obeys the time-dependent two-dimensional diffusion equation

aqat = D(a2qax2

+ a2qay2)

(1)

where c and D are the concentration and the diffusion coefficient of the electroactive species, respectively, and t is the time elapsed since the beginning of electrolysis. Let us assume that the electroactive species is distributed uniformly prior to electrolysis and that the concentration on the electrode surface drops instanta-

4

neously to zero with the imposition of a large potential step. Then the initial and boundary conditions are given by t=o,

in&l

c=c*

(2)

t>O, t>O,

onS, on&

c=O &~/&x=0

(3) (4)

where Q is the solution phase in which diffusion takes place, S, is the electrode surface and S, is the boundary between the adjacent units. Since it is difficult to solve this boundary value problem analytically, we employed the finite element method which has been described previously [28]. We applied Crank-Nicolson’s difference formula to the time-differentiation term in eqn. (1) and Gale&in’s method to the Laplacian terms. These applications enabled us to transform the integral representation into the matrix one. The discretized elements were rectangular of the same size. At short electrolysis times, it is sufficient to discretize the diffusion space only in the vicinity of the electrode because of the poorly developed diffusion layer. On the contrary, the discretization should be made all over the diffusion space at long electrolysis times. Finer discretization was needed for short electrolysis times than for long times because of the more drastic concentration variation in the smaller region at short times. Thus, the numbers of elements were taken to be 30 to 240 in the x- and y-directions, respectively, for short times, and 10 and 130 for long times. In each case, data for about 1300 and 6000 elements were loaded into the computer memory. The details of the calculation have been described elsewhere [28]. The total currents were evaluated from numerical integration of the values of the current density which vary from point to point on the electrode surface. They are expressed as a function of the dimensionless time, 7 7 = Dt/p2 for various values of two geometrical parameters,

(5) w/p and h/p.

Results of the computation A set of concentration profiles and current lines thus computed for an electrode with w/p = 0.2 and h/p = 0.8 is shown, as an example, for r = 0.008 in Fig. 2. As predicted intuitively, equi-concentration contours for c/c* < 0.2 have a form similar to the electrode surface, whereas those for a large value of c/c* ( = 0.99) are almost parallel to the x-axis, irrespective of the geometry of the electrode surface. Some current lines are also drawn in Fig. 2. One can see from the densities of the equi-concentration contours that the current density is the highest in the vicinity of the salient angle of the electrode surface while it is the lowest near the re-entrant angle.

0.64

0.32

0. IT 0.04

0.8

0.014

0.8 kg

0.004

0

w

P

Fig. 2. Equi-concentration contours (solid lines) and current lines (dashed lines) computed by the finite element method for T = 0.008 at w/p = 0.2 and h/p = 0.8. The numbers on the y-axis denote the values of c/c*. Fig. 3. Variations of the equi-concentration contours of c/c * = 0.5 with T for w/p = 0.2 and h/p The numbers on the y-axis denote the values of T.

= 0.8.

In Fig. 3, e&-concentration lines at c/c * = 0.5 are shown for various values of r for the same electrode geometry as in Fig. 2. These curves in Fig. 3 manifest the variations of the diffusion layer thickness with time. In Fig. 4, the dimensionless currents computed for several combinations of w/p and h/p values are plotted against 6. IWtt involved in the ordinate denotes the diffusion-controlled current given by the Cottrell equation for the projected area: IWtt = nFNpqc * ( D/at )I’* = nFNqDc * ( m ) - “*

(6)

where N is the total number of repeating units and q is the length of the electrode normal to the xy-plane as shown in Fig. 1B. Immediately after the begirming of electrolysis, the area of the diffusion layer is almost the same as the true surface area, as depicted by the curve for the smallest value of T in Fig. 3. Then the current, I 7-07 can be expressed by the Cottrell equation for the electrode area of N( p + h)q, i.e. (7)

6

L ’ ‘1

23.4

:.

JFC =JE/pl

$;

[ o&n,pY

Fig. 4. Variations of I/Z,,,, with fi computed by the finite element method for h/p = 1.0 (A), 0.6 (B) and 0.2 (C) and w/p = 0.2 (l), 0.4 (2), 0.6 (3) and 0.8 (4). The insets show the electrode geometries. The dashed lines have a slope of - 1.17.

The theoretical values of (I/I,,,),, ,, in Figs. 4A, 4B and 4C are 2.0, 1.6 and 1.2, respectively. As the time elapses, the area of the diffusion layer decreases and hence the values of I/& also decrease. When the time is so long that the thickness of the diffusion layer exceeds the depth of the hollow, the area of the diffusion layer tends to Npq (the projected area) and hence the values of I/I,,, become unity. At relatively short times, all the curves fall on a straight line with a slope of - 1.17, as depicted by the dotted lines in Fig. 4. This slope value was found to be independent of p, w and h. It is easily seen that the current-time behaviour at the very beginning of electrolysis at the present model electrode is quite similar to that at a corrugated sheet electrode, which has been treated by Oldham [25]. The value -1.17 is in agreement with the value -1.19 [25] which was calculated from fi{ k( 7~- 8/2) + k(8/2)} with 8 = lr/2 in eqn. (65) of ref. 25, where the function k(B) introduced by Oldham is a complicated function of angle 8 and the values for k(n/4) and k(37r/4) are -1.273 and 0.602, respectively. The value -1.19 represents the sum of the effects of diffusion at the salient right and the re-entrant right-angles. Hence the deviation of the curves from the straight lines in Fig. 4 may be ascribed to interaction between the diffusion layers developing in the x- and y-directions. As the depth of the hollow becomes larger than the width, the curve deviates downward from the line at short times. Conversely, the curve deviates upward from the line as the depth becomes smaller than the width. Thus there may be a critical

geometry for which the curve does not become lower than the straight line nor deviates upward from the line. This is the case when w=O.82

h

(8)

Thus, according to the Ifi- fi curve behaviour we can classify the model electrodes into two groups. We will call those electrodes which exhibit a sudden drop of the 1fi values type A and the remainder type B. Electrodes of type A represent electrodes having narrow hollows in a plane, whereas those of type B represent electrodes having narrow protrusions on a plane. The characteristics of the Ifi vs. fi curves common to the two types are: (1) a constant value of Ifi for sufficiently long times which is equal to nFNpqc * (D/a)l/*; (2) a value of 1fi at the extreme of 6 --, 0 which is given by nFN(p + h)qc * (D/r)‘/*; and (3) a slope value at relatively short times, i.e. - 1.17nFNqc * (D/lr)l/*. For type A electrodes, we can add the following characteristics in addition to (l)-(3) above from inspection of Fig. 4. (4) A characteristic time t, at which the Ifi-fi curve starts to deviate from the line having a slope of - 1.17 and falls rapidly. Let us denote the value of 7 corresponding to t, by ri as shown in Fig. 4B. This rapid decrease of current corresponds to depletion of the electroactive species in the hollows as if it might occur in a thin layer cell due to finite diffusion. (5) A dimensionless time T* which is defined from the intercept of the rapidly falling line with the 6 axis (Fig. 4). This value r2 or t, =p27*/D denotes the time after which the diffusion of electroactive species to the electrode can be regarded as a linear one towards the projected electrode area. In Fig. 5, values of r;/* and $I2 are plotted against w/p for several values of h/p. These plots show good proportionality to w/p, irrespective of the depth of the hollow. The slope of the plot for & is about 0.5. This finding indicates that the

Oo4

:::::::::I 0.2

0.4

0.6

0.6

1.0

W/P

Fig. 5. Plots of 71 ‘I2 (A) and ~2’~ (B) against w/p

%

0.2

0.4

0.6

0.8

1.0

W/P

for h/p = 2.0 (O), 1.0 (o), 0.8 (A),0.6 (m) and 0.4 (0).

8

sudden decrease in Ifi occurs at the time t, =p2r1/D = w2/4D. This would correspond to the time when the thickness of the diffusion layer develops to w if the electrolysis were conducted at a completely planar electrode. The plot of fi vs. w/p has a slope of almost unity, indicating that the diffusion can be regarded as a linear one towards the projected area if f exceeds t,, which is approximately equal to four times the value of t,. It is found from the above characteristics that the geometrical dimensions p and h can be determined from the Ifi-fi plot. The value of l/Lott at short times can be expressed as I/&Ott = (h/p

)“‘/p

+ 1) - 1.17( Dt

Thus, we have + 1) - {l.l7&$,c*D/(pJ;;)}fi

IJi =nF’,,c*(D/~)~‘~(h/p

where A, is the projected area (= Nqp). If A, is known, then p can be obtained from the slope and h can be determined from the intercept. In addition, for type A, w can also be obtained from t, or t,, i.e. or

w=2fi=2p&

Derivation

w=&=pfi

of an approximate

(9

equation

It is desirable to have a simple approximate equation which can express the I-t relationship with reasonable accuracy for various values of the parameters w/p and h/p. Figure 4 suggests that the current at these model electrodes of type A may be qualitatively given as the sum of the Cottrellian current and the current due to the consumption of species originally present in hollows which may act as adsorbed surface excess. The latter current may be evaluated as follows. Consider a hollow space (0 G x G w and 0
C=

CI =

0

CII= C*

(0
and the boundary conditions

acI/ax = 0 at x = 0 CI= CII and ac,/ax = &+/ax

at

x=d

Then the current due to the consumption of species in this limited space is given by I=nFZVqdc*(h/w){B2(O;Dt/w2)

-8,(d/2w;Dt/w2)}

9

where 8, and @, are theta functions defined by [29] m B,(a;x) = (“x)p2 c (-l)mexp{-(a-1/2+m)2/x}

(10)

m=-co

t9,(u;x) =

(“x)-1’2

E

(-1)”

exp{ -(a+m)2/x}

01)

nl=--co

We replaced these theta functions, @,(O; x) and &(d/2w;x), following simple functions f(x) and g(x),

f(x) =

(a~)-~‘~(1

- exp( -x-l))

exp(2 - 2x)/(2rx112)

(x Q 0.87) (x > 0.87)

respectively,

by the

(12)

and 0.660( w2/hp) g(x)=

l+

[b+2{0.660(~~/hp)}~‘~]x~

(13)

x lix3

We have found that the equation = 1+ J;;(7h/W)[f(P2?/W2)

I/L,,

-8(P2++J2)]

04)

is a good approximate equation of the I/Imtt- 6 relation for electrodes of type A when the value of constant b is 0.45. A comparison of the I/I,,,-\/; curves calculated from the finite element method and those obtained from eqn. (14) is

0

T

l/Z

0.8

Fig. 6. Comparison of the Z/Z,,,- \/; curves obtained by the finite element method ( -) with those evaluated from approximate equation (14) (- - -) for electrodes with the following geometries: (a) w/p = 0.2, h/p = 0.4; (b) w/p = 0.6, h/p = 0.4; (c) w/p = 0.2, h/p =l.O; (d) w/p = 0.8, h/p = 1.0.

10

shown in Fig. 6 for four electrode geometries. The dashed curves calculated from eqn. (14) for electrodes of type A (curves a and c) reproduce the essential features of the I/&,-J; relationship. It can be seen that eqn. (14) is a fairly good approximate equation even for the electrode with w/p = 0.8 and h/p = 1.0; in this case, the maximum relative error involved in this equation is less than 4%. Curve b corresponds to an electrode with w/p = 0.6 and h/p = 0.4, which evidently belongs to type B, and thus eqn. (14) cannot be applied. EXPERIMENTAL

Fabrication of the model electrodes A casting epoxy (Araldite CY230 and Hardener HY-956, Nagase-CIBA) was turned on a lathe into a cylinder 3 cm in diameter and then threads were cut. The choice of gearing combination allowed a thread of 0.25 mm pitch to be cut. The tip of a single-point cutting tool was ground with a grinder and was whetted on an oilstone so that the tip had a very thin square nose. Three cutting tools were prepared; the widths of the thin square nose tips were 0.04,O.lO and 0.17 mm. Next the threaded epoxy cylinder was cut into four workpieces. Each workpiece was machined on a milling machine into a piece of the form shown in Fig. 7. Part of the threaded face which served as the working electrode surface had typically an area of about3mmx3mm. The workpieces underwent ultrasonic washing successively in detergent, alkaline degreasing solution and 5% hydrochloric acid, and were then washed thoroughly with distilled water and dried in an air oven at about 110°C for 1 day. Gold was evaporated in vacuum (4 x 10e4 Pa) onto the threaded face and on the adjacent part to which a lead wire would be connected. The evaporation process was performed three times for each piece with a slight change to the direction of the face towards the evaporating source so that deposition of gold was ensured not only on the crest and root of the threads but also on the side walls of the threads. About 1 pm of gold was deposited at the top of the thread so as to ensure sufficient electrical conductivity all over the electrode surface. However, electrodes with poor conductivity were discarded. A lead wire was connected with conductive silver paint. The lead

Fig. 7. Schematic diagram of the form of the workpieces.

11

wire and all parts of the piece except for the threaded face were coated with 5-min epoxy to insulate them from the solution. Furthermore, the periphery of the threaded face was piled up with the epoxy in order to eliminate any edge effects. The geometric dimensions p, h and w of the electrodes were measured by an optical microscope before coating of the periphery. The projected area of the electrodes was determined by the optical microscope after finishing the fabrication.

An HESC-318B high sensitive potentiostat (Huso Ltd.) was employed. Current-time curves were recorded on a Nicolet digital oscilloscope, model 3091, to which an IF-800/30 personal computer (Oki Electric Ltd.) was connected through the interface. Data storage and processing were carried out on a personal computer. The el~tr~he~~~ cell was a conventions one and a spiral platinum wire and a saturated calomel electrode (SCE) were used as the auxiliary and reference electrodes, respectively. All potentials are referred to the SCE. Chemicals

All chemicals were of analytical grade and were used as received. A solution of 10 mmol/dm3 potassium hexacy~ofe~at~I1) in 1.0 mol/dm3 sodium sulphate was freshly prepared with doubly distilled water and purged thoroughly with nitrogen gas before use. Procedure

In order to check the performance of the electrodes, cyclic voltammograms were recorded in a solution of 1 mol/dm3 Na,SO,. Only electrodes exhibiting a small residual current in the potential domain from -0.2 to 0.8 V (vs. SCE) were used for the chronoamperometric measurements. The potential was stepped from -0.1 to 0.7 V, where hexacyanoferrate(II) was oxidized under diffusion-~ontro~ed conditions. Since the current at short times is much larger than that at long times, chrono~peromet~c meas~ements were made separately in the two time domains 0.001-3 s and 0.02260 s in order to retain the sensitivity of the current. RESULTS AND DISCUSSION

Seven electrodes were employed in the present me~urem~ts and their characteristic lengths for the electrode geometry are listed in Table 1. Since the digitized faradaic current signals at long times (over 10 s) were often comparable to the electrical noise level, smoothing of these data was carried out by the linear least-squares method using five adjacent points.

12 TABLE 1 Electrode geometry and characteristic values of the chronoamperometric Electrode

W/P”

h/Pa

(z/L,,),-0

Slope

Twe

1 2 3 4 5 6

0.18 0.43 0.18 0.68 0.70 0.78

0.18 0.51 0.96 0.64 0.94 0.96

1.18 1.42 1.86 1.69 1.97 2.01

-1.13 -1.14 -1.22 -1.11 -1.11 -1.15

B B A B B B

7

0.68

1.31

2.38

-1.24

A

curves fib

J;;”

0.13(0.09)

0.32(0.18)

0.40(0.34)

0.77(0.68)

a p = 0.125 mm. b Theoretical data are given in parentheses.

In Fig. 8, values of l/IWtt are plotted against \/; = (Dt)“*/p for four electrod of different geometries. In the calculation of I/IWtt, the value 7.0 x 10m6 cm*/s w used for the diffusion coefficient of hexacyanoferrate(I1); this value was determin from chronoamperometry at a planar platinum electrode having a shield to ensu linear diffusion. The value p = 0.125 mm was also used. Currents at short tin were very large because they contained a contribution from the double-lay charging and probably a contribution of faradaic current enhanced by surfa asperities much finer than the characteristic length p of the model electrodes. T plots in the time domain 5-50 ms fell on straight lines, whose slopes have values - 1.18 + 0.07. These are in good agreement with the theoretical value - 1.17. In tl plot, the abscissa has a common value of p for all seven electrodes. From the slo of the plot of I\/i against \/1;,the value of p can be obtained as mentioned in t

Fig. 8. Plots of Z/Z,,, against (Dt/p2)‘/2 measured at electrode Nos. 2, 3, 5 and 7 in Table 1 in mmol/dm3 Fe(CN)z- solution containing 1 mol/dm3 Na,SO, at 25 k lo C. The numbers in the fig1 correspond to the electrode number.

13

theoretical section. The values of (I/I,,,) t -t o can be determined from extrapolation of the straight lines to fi -+ 0 and are listed in Table 1 for the seven electrodes. They agree well with the values of (1 + h/p) calculated from the ‘characteristic lengths of the electrodes, irrespective of the types of electrodes mentioned above. Then h is evaluated since the value of p is already known. As expected from the theoretical results, the plots for electrodes 3 and 7 exhibit a sudden drop of the I/& values. These electrodes belong to type A since their w/h values are less than 2/3. The values of 6 and fi were obtained graphically for these two electrodes and are given in Table 1. They are in good agreement with the theoretical values. The experimental I/&6 behaviour for type B electrodes is also as predicted from the theoretical calculation. Although electrode 5 should be classified as type A if one considers the value of w/h (= 0.74) this classification is not so strict, as can be seen from Fig. 4. The profiles of the real electrode surface are not always well defined as expected for the model electrode used in the calculation, and this may be, at least partly, the cause of the ambiguous I/&-h behaviour of electrode 5. In conclusion, the experimental chronoamperometric behaviour of these seven electrodes was exactly as expected from the theoretical calculation and it can be said that as far as type A electrodes are concerned, the characteristic dimensions of the electrodes can be determined from chronoamperometry. Finally, it should be mentioned that the presence of the wall or shield which was prepared around the periphery of each electrode surface with 5-min epoxy is very important. This wall works well to prevent the undesirable effect of diffusion from the surroundings and a possible effect of natural convection. It was found that the Ifi values reached a minimum and then increased linearly with fi at long times (e.g. more than 5 s), unless this shielding was employed. This can be explained by non-linear diffusion at the peripheral edges of the electrodes. It is known that the edge effect at an inlaid disk electrode attains more than 18% of the total current at t = 10 s and exhibits a linear relation between 1fi and fi when the radius of the disk is 5 mm [30]. CONCLUSION

It has been shown that electrodes with uneven surfaces of a two-dimensional square wave profile exhibit characteristic chronoamperometric curves which can be distinguished from curves for simple linear diffusion. The chronoamperometric features come to light when Ifi is plotted against fi. The plot at short times falls on a straight line. Extrapolation of the straight line to fi + 0 allows us to evaluate the real surface area of the electrode. At long times, 1fi approaches a constant value, nFNqpc*m, which corresponds to the Cottrell equation at the electrode with the projected area. For electrodes with narrow and deep hollows, two characteristic times reveal themselves in the plot of 1fi against fi which characterize finite diffusion in the hollow. The Ifi-\/t behaviour at these electrodes is quite similar to that at electrodes with a surface excess of reacting species. Although the

14

pitch p of the electrodes we employed in the present work was 1.25 x lo* pm, this method may be applicable to the evaluation of smaller values of p. If we assume that D = 1 X 10m5 cm*/s and the minimum value of t, which can be determined experimentally is 1 ms, then the minimum value of p lies in the range 2.5-10 pm depending on the value of width w. REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

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