Indian Journal of Chemistry Vol .39A, Jan-March 2000, pp. 356-363

Chronoamperometric current at ultramicrospheroidal electrodes for steady state EC' reactions - A two-point Pade' approximant

L Rajendran & M V Sangaranarayanan* Department of Chemistry, Indian Institute of Technology, Madras 600 036, India

Received 2 November 1 999, accepted 22 December 1 999

The steady state chronoamperometric current for EC' reac.tions pertaining to ultramicroelectrodes is estimated using scattering analogue techniques for different ranges of rate constants. The general results valid for disc, sphere, oblate and prolate spheroidal geometries are reported. The polynomial expressions pertaining to two extreme limits of reaction rates are combined using a two-point Pade' approximant. Numerical estimates for current are reported and compared with existing approximate results, wherever available. A simple procedure to estimate steady state chronoamperometric current for electron transfer reactions at spheroidal electrodes usincr , '"

coordinate transformation is also provided.

1. Introduction The coupling of mass transfer with surface reactions

and migration occurs in diverse contexts such as hetero­geneous catalysis 1 .2, electrochemical transport phenom­ena3, protein binding4, etc. The theoretical analysis of such processes is rendered difficult on account of the nature of the boundary conditions. Among a variety of mathematical techniquesS employed to solve such mixed boundary value problems, mention may be made of (i) analysis using dual integral equations involving Bessel functions(,·7, (ii) replacing the stipulated exact boundary conditions by approximate versionsx, (iii) scattel:ing ana­logue techniques9. , 0 and (iv) multidimensional integral equations and digital simulation procedures"" x.

Ultramicroelectrodes of various geometries and di­mensions have become increasingly popular in the past few decades on account of their lower interfacial capaci­tance, reduction in ohmic resistance, measurement of fast electrochemical reactions, etc. However, the analysis of current response, pertaining to ultramicroelectrodes is not straight forward on account of the mixed nature of the boundary value problem, i.e., one boundary condi­tion is specified on the conducting part of the boundary which comprises the electrode surface while the other restriction is formulated on its insulating portion. In view of this, standard mathematical techniques such as sepa­ration of variables, integral transforms, substitutions, etc ., become no longer applicable. In electrochemical trans­port problems involving ultramicroelectrodes, different

geometries such as disc, ring, band, hemisphere, etc., have been employed to obtain current response in po­tential step experiments. However, satisfactory numeri­cal results exist only for disc geometries, even for sim­ple electron transfer reactions. The non-steady state dif­fusion current for ultramicrodisc electrodes has been estimated using Wiener Hopf factorisation 19.20 and Pade approximations2 1 .22 • Among various numerical tech­niques proposed to study ultramicroelectrodes, we may mention the Hopscotch algorithm'7.2J, successive over­relaxation24, conformal mapping versions", etc. How­ever, the calculation of steady state current for EC' re­actions at spheroidal ultramicroelectrodes has been found formidable and till date, no rigorous analysis is avail­able. We note that spheroidal geometry encompasses disc, whisker, oblate and prolate spheres as special cases.

The purpose of this paper is to (i) derive accurate poly­nomial expressions pertaining to chronoamperometric current for EC ' react ions at spheroidal ultramicroelectrodes, (i i) propose a two-point Pade' approximant for evaluating the same over the entire do­main of reaction rates, (iii) indicate the analogy of steady state EC' reactions with non-steady state diffusion ver­sions and (iv) obtain steady state current for simple elec­tron transfer reactions at different ultramicroelectrode geometries using coordinate transformations. Our tour de force consists in employing the scattering analogue techniques, well known in electromagnetic theories'! and soil infiltration studies ' lJ .

RAJENDRAN e/ al. : CHRONOAMPEROMETRIC CURRENT AT ULTRAMICROSPHEROIDAL ELECTRODES 357

2 Formulation of the problem The EC' reaction is represented as

kf O + ne- � R

kh

k and R+Z � 0+ Products

... ( 1 )

. . . (2)

Let SE and Sp refer to conducting and insulating regions respectively. The governing transport equation is as fol­lows (cf Appendix A):

h ( ) I [kf Co (r) - kb CR (r)] w ere g r = - -'----"--------::-"------"'----k C(b) f 0

and

. . . .(3)

. . . (4)

. . . (5)

V2 refers to the Laplacian operator, 'a' denotes the char­acteristic length associated with the geometry under con­sideration Ca' may be identified as the radius for discs and spheres) while DR is the diffusion coefficient of spe­cies R. The mixed boundary conditions are given by

g = I + KI (ag J . z = O' r E S az ' , E

d g - = O· z = o· r E S d z ' , p

g = O as l r l ---7 DO

. . . (6)

. . . (7)

. . . (8)

K is a dimensionless parameter representing the inter­play between charge transfer and mass transfer rate con­stants given by

. . . (9)

Since the quantity of interest in electrochemistry is in general, the flux vis-a-vis current, we write the latter as

i � -nF D, Co'"} a ( I + k. D/ k,D,) , L[;:) ds . . . ( 1 0)

In the diffusion-limited regime, K ---7 DO and hence

i � - n F C,'"} D" a . 1. ( �! ) ds . . . ( I I )

while equation (6) becomes

g = I on z = 0; r E SE . . . ( 1 2)

The above problem is isomorphic with soil-infiltra­tion from spheroidal cavities of different geometries22. Consequently, the expression for current at two extreme values of x (= al12) is written in a straight forward man­ner.

2. 1 Chronoamperometric current at two limits of reac­tion rates

For small values of x, the steady state current is given by

i 5 -:- = L a (0)) x" 10 n=O " X ---7 0 . . . ( 1 3)

where io is giwlI hy 21tnFD"Co(h) while the aspect ratio 0) = bfa. a and b denote respectively the length of the two semi-axis and that of the third axis.

The coefficients ao to as are as follows:

. . . ( 1 4)

a = (_'Z'_)[(al _ l) {92 - l o&.i + 1 6(2w2 +1 ) } 4 1 620 (W_'Z')2 2w2 -3w'Z'+1

- 9 (25 0)1 + 1 20 0) 't - 29 0) + 1 00 't) 't - 1 620 (0) - 't)t1 ] as = (;: J (0)2 - 1 )2 [ 142125 + 2m2 / (0) - 't)2] - (0) 't)

(95 0)2 - 1 07) / 5 + 1 8 (3 0)2 - 4 0) 't1 + 3 't4 - 2)]

We reiterate that the above coefficients an to as are dependent in a general manner on the parameters 0) and

358 INDIAN J CHEM. SEC. A. JAN - MARCH 2000

Table I - Different ultramicroelectrode geometries and their corresponding parameters ro and t

Geometry ro t Disc a 2/rt Oblate spheroid o < ro< 1 ( l -ro2t'/cos·' ro Sphere I I Prolate spheroid ro> I (ro2_ 1 ) Y'/cos· ' ro Whisker ro » 1 ro/ln 2ro

't thereby making Eq. ( 1 3) appl icable for different geometries of spheroidal electrodes. (cf. Table I ) . Analo­gously, when the dimensionless rate constant x is large, current is given by

4 " b I - n L.. n X if x � 00 and oo = 0 n = O

= c x + c X 1 /3 if x � 00 and oo > 0 o I

while the coefficients { b } are : n

. . . ( 1 5)

. . . ( 1 6)

. . . ( 1 7)

Further, Co = 1 12 and c i = 0.996 1 oo2I3 . The essential limitation of the above version for large values of reac­tion rates is that two separate sets of coefficients are needed for oo = 0 (disc geometry) and oo > 0 (oblate and prolate spheroids, spheres, whisker electrodes etc . ) . Nev­ertheless, we emphasize here that Eqs ( 1 3) and ( 1 6) rep­resent the most general expressions available, till date, for ultramicroelectrodes of various shapes and these en­able accurate estimation of steady state current for EC' reactions. This is not all. By proper transcription of vari­ables, such steady state results can be converted into non­steady state diffusion controlled reactions and hence Eqs ( 1 3) and ( 1 6) become applicable mutatis mutandis to evaluate transient chronoamperometric current since the Laplace-transformed equations for reversible charge transfer reactions become isomorphic with the steady state equation for EC' reactions.

2.2. Comparison with existing approaches

The equivalence between steady state diffusion lim­ited current for Ee' reactions and Laplace transformed transient current for disc electrodes has been exploited by Philips2� so as to report the first three terms of Eqs ( 1 3) and ( 1 5) . Fleishchmann et al.27 have obtained the

first two terms of Eq. ( 1 3) by employing an equivalent sphere approximation for disc electrodes. This would imply that the est imation of steady state chronoamperometric current has now been made more accurate due to Eqs ( 1 3) and ( 1 5) . Further, the current expressions for all other geometries (such as sphere, ob­late and prolate spheroids, whisker etc . ) too have been derived here for the first time spanning all values of dimensionless reaction rates.

2.3. Two-point Pade ' approximation

Pade' approximants29 are now routinely employed in diverse contexts so as to overcome problems with slowly convergent or divergent power series expressions. In view of the easy implementation of the algorithm, this tech­nique is widely analysed in phase transitions and criti­cal phenomena3o, virial equation of state3 1 , quartic anharmonic oscillators32 , transient electrochemical tech­niques2", etc. In the present context, partial information at two extreme values of the dimensionless reaction rates is available. Hence, it is imperative to construct a two­point Pade' approximant using the coefficients { a,, } and { cJ of Eqs ( 1 3) and ( 1 6) . Analogous Pade' approxi­mants can be constructed using Eqs ( 1 3) and ( 1 5) for disc geometries (oo = 0)

The [6/3] Pade' approximant obtained using equations ( 1 3) and ( 1 6) is

Po + PI Y + P2 y2 + P3 / + P4 / + p, / + Pc. l 10 I + ql Y + q2 y2 + q3 i . . . ( 1 8)

where y = x 1/3 . The coefficients { p } and { q } are as fol-n n lows: Po = ao ; P I = aoq l ; P2 = q l c l ; Po = Co + q2 c I ; P4 = q l C o + q3 c I ; P, = q2 c o ; Po = q3 c (j" The coefficients q l ' q2 and q, are obtained from the matrices.

[: : ] == [ � � ::' _

,

O

c",] - ' == [c () �o a , ] . . . ( 1 9)

q 1 a , - C o 0

The two-point Pade' approximant given by Eq. 1 8 is ap­plicable to different geometries of ultramicroelectrodes if appropriate values of oo and 't are substituted. As a case study, let us consider the hemispherical electrodes wherein oo = I . On account of the singularity behaviour of coefficients { a } when oo = 't = I , we resort to the

"

limits oo � I and 't � I so as to define

RAJENDRAN et al. : CHRONOAMPEROMETRIC CURRENT AT ULTRAMICROSPHEROIDAL ELECTRODES 359

()) = I + E . . . (20)

and 't = 1 + E 13 - E 2 1 45 + 0 (E 3) . . . (2 1 ) Hence the coefficients become

ao = al = 1 ; a2 = - 1 /3 ; a3 = 1 13 ; a4 = - 1 6/45 ; a5 = 1 7/45 Co = 1 /2 ; c l = 0.996 1

. . . (22) Consequently, the steady state current is

1 + 1 .02y + 1 .0 1 6i + 1 .5 1 2y' + 1 .02l + 0.508i + 0.256/ =

io 1 + 1 .02y + 1 .0 1 6i + 0.5 1 2y)

Table 2 - Steady state diffusion l imited current (il47tnFD"C,,(hla) for EC' reactions at spheroidal ultramicroelectrodes estimated using

Eq. (23). The values reported by Fleischmann et al.27 obtained using equivalent sphere approximation along with the magnitude

of the deviation from Eq. (23) are also shown.

x

o

[6/3]pade' approximation

Eq. 23

Fleischmann et al.(27)

I . 000 (0. 00) 1 .00 (0.73)

1 .200 ( 1 .86) 1 .500 (5.85)

0. 1 0.2 0.5 I

. . . (23) ' . 2

1 .000 1 .092 1 . 1 72 1 .4 1 7 1 .785 2.464 3 . 1 05 3.725 4.329 5.507 7.230 1 2.753

2.000 ( 1 2.04) 3.000 (21 .75) 4.000 (28.82) 5.000 (34.22) 6.000 (38.23) 8.00 (45.26)

where io = 41tnFDoCo(h) a for hemispherical electrodes, y being equal to x 1 13 . Table 2 indicates the ratio ilio esti­mated at different values of dimensionless reaction rates using Eq. (23) along with the values reported by Fleischmann et aF7.When the pseudofirst order rate constants are large,marked deviations from the estimate of Fleischmann et al. are noticed.It is interesting to note that the long time time expression ( 1 6) is reported in terms of x 1/3 whereas the short time Eq. ( 1 3) is given in terms of x . Consequently, the lowest order Pade' approximant is as given by Eq. ( 1 8) which requires only two terms of the associated power series expansions.

Further, Oldham36 has reported the ratio i/io for hemi­spherical electrodes as

I - = 1 + x . . . (24) 1 0

when the dimensionless reaction rate x is small. For other geometries such as oblate, prolate and whisker electrodes, the coefficients Pi ,q i along with the coefficients of the given series are reported in Table 3 .Table 4 provides steady state chronoamperometric current for EC' reac­tions pertaining to the above geometries - hitherto una­vailable .To place the above analysis in the proper per­spective, we reiterate that the calculation of steady state current for spherical electrodes even when the chemical reaction (2) is absent (i .e. , simple electron transfer reac­tions and not EC' mechanism) is rendered difficult. The exist ing formal isms due to Oldham37 employing gudermannian functions and B irke 35 using orthogonal curvilinear coordinates are quite tedious. A simple analy­sis to obtain the steady state current for spherical elec-

3 4 5 7 1 0 20

1 1 .000 (52. 14) 2 1 .000 (64.66)

trodes, using coordinate transformation is presented in Appendix B. This procedure has been outlined by Phillip33 in the study of soil infiltration from spheroidal cavities.

2.4. Non-steady state current for diffusion limited elec­tron transfer reactions

The mathematical formalism of steady state current for EC' reactions is equivalent to the Laplace-trans­formed transient current for diffusion limited electon transfer reactions at ultramicro electrodes. Consequently, Eqs ( 1 3 , 1 5) and ( 1 6) become applicable to the estima­tion of current at ultramicroelectrodes by relating a with Laplace-transformed surface concentrations . Thus, the methodology advocated herein leads to accurate expres­s ions for trans ient current at spheroidal ultramicroelectrodes, pertaining to simple electron trans­fer reactions.

3. Perspectives

As indicated earlier, mixed boundary conditions arise naturally in the case of transport at ultramicroelectrodes. Among various geometries of these, disc electrodes have been thoroughly investigated albeit for simple electron transfer reaction schemes. For EC' reactions, steady state chronoamperometric current expressions have only re­cently been derived, even for disc electrodes. However, judicious combinations of numerical and analytical ap-

360 INDIAN J CHEM, SEC. A, JAN - MARCH 2000

Table 3 - The coefficients of Pade' approximant in Eq.( 1 8) along with coefficients in Eqs ( 1 3 ) and ( 1 6) .

Co- O)

efficients 0. 1 0.2 0.5 2 5 1 0 20

Pn 0.677 0.7 1 6 0.827 1 .3 1 5 2. 1 37 3 . 324 5.4 1 6

PI -0. 1 42 6.23 1 x 1 0-1 0.65 1 1 .842 8.8 1 5 -30.389 - 1 3.783

PI -0.045 2.967 x 1 0-1 0.494 2.2 1 4 1 2.0 1 4 -42.269 - 1 8 .678

P, +0.486 0.5 1 4 0.875 3 . 1 62 1 6.875 -58.29 1 -24.8 1 2

P4 -9 .. 6 1 5 x 1 0-1 4.458 X 1 0 1 0.538 2.422 1 8.837 - 1 0 1 .0 1 7 -74.645

P5 -3.332 x 1 0-1 2.073 X 1 0-1 0.299 0.842 2.8 1 1 -6.358 - 1 .724

P6 2.073.x 1 0-1 1 .526 x 1 0-' 0. 1 1 5 0.545 2.880 - 1 0.430 -4.999

ql -0.2 1 0 8.708 x 1 0-2 0.787 1 .400 4. 1 25 -9. 1 42 -2.545

ql -6.665 x 1 0-2 4. 1 46 X 1 0-2 0.597 1 .684 5 .622 - 1 2.7 1 6 -3 .449

q, 4. 1 46 x 1 02 3.052 X 1 0-' 0.23 1 1 .089 5 .759 -20.860 -9.997

ao 0.677 0.7 1 6 0.827 1 .3 1 5 2. 1 37 3.324 5.4 1 6

al 0.458 0.5 1 2 0.684 1 .730 4.567 1 1 .05�) 29.33 1

Co 0.500 0.500 0.500 0.500 0.500 0.500 0.500

ci 0.2 1 5 0.341 0.628 1 .58 1 2.9 1 3 4.624 7.339

Table 4 - Dimensionless chronoamperometric steady state current for EC' reactions pertaining to different

ultramicrospheroidal electrodes estimated using Eq. ( 1 8) and Pade' coefficients of Table 3

0)

x Oblate Sphere Prolate

0. 1 0.2 0.5 2 5 1 0 20

0 0.6765 0.7 1 55 0.8270 1 .0000 1 .3 1 52 2 . 1 370 3.3241 5.4 1 58

0. 1 0.7222 0.7666 0.8935 1 .0922 1 .4608 2.4390 3 .798 1 5.7374

0.2 0.7678 0.8 1 78 0.9584 1 . 1 780 1 .5864 2.6648 4. 1 368 6.2835

0.5 0.9048 0.9 1 73 1 . 1 476 1 .4 1 72 1 .9 1 45 3 . 1 995 4.9 1 03 7.4607

I 1 . 1 328 1 .2269 1 .4527 1 .7846 2.3858 3 .8996 5.8830 8.8870

2 1 .589 1 1 .7378 2.0424 2.4643 3 .2080 5.0290 7 .3884 1 1 .0 1 28

3 2.0468 2.2485 2.6 1 65 3 . 1 056 3 .9532 5.997 1 8.6342 1 2.7 1 39

4 2.5066 2.7588 3 . 1 808 3 .7248 4.6562 6.8805 9.7444 1 4. 1 940

5 2.9687 3 .2688 3 .7382 4.329 1 5.33 1 7 7.7099 1 0.7682 1 5.534 1

7 3 .90 1 3 4.2882 4.8386 5.5074 6.6286 9.2640 1 2.6489 1 7.9442

1 0 5.32 1 2 5.8 1 60 6.4647 7.2267 8.4892 1 1 .4337 1 5 .2 1 28 2 1 . 1449

20 1 0.2024 1 0.8997 1 1 .7765 1 2 .7537 1 4.3439 1 8.0 1 36 22.72 1 6 30. 1 465

50 25.3333 26. 1 027 27.3505 28.6943 30.8524 35.7929 42. 1 337 52. 1 687

RAJENDRAN et at. : CHRONOAMPEROMETRIC CURRENT AT ULTRAMICROSPHEROIDAL ELECTRODES 361

proaches have been investigated to make partial break­throughs in what is essentially a tedious exercise. The most comprehensive approach to the study of diffusion at ultramicroelectrodes is due to Phillips and Janson 3X

and Alden and Compton 3Y . An alternate approach is to exploit results known elsewhere in mixed boundary value problems and translate them into electrochemical litera­ture. A particularly illuminating example is the soil in­filtration studies wherein accurate polynomial expres­sions have been reported for transient and steady state EC ' reactions using scattering analogue techniques lO . This mapping enables the analysis of steady state EC' reactions as well as non-steady diffusion limited elec­tron transfer schemes at ultramicro e lectrodes . Conse­quently, scattering analogue technique seems to be a powerful methodology to obtain accurate polynomial ex­press ions for curren t perta in ing to d i ffus iona l electrochemical transport studies.

Acknowledgement This work was supported by CSIR, New Delhi.

Appendix A

For the EC' reaction sequence represented by

0 + ne' R

with R+Z k � O+Products

the diffusion equation is given as

D \72 C = -kC () (l R

D \72 C = kC (I R R

along with the boundary conditions

D ac " = _ D ac R = k C k C O S R I' , , - h R Z= ;rE E " az d Z

oC OCR __ 0 = - -- = O· z = O· rE S OZ OZ ' , I'

. . . (A I )

. . . (A2)

. . . (A3 )

. . . (A4)

. . . (AS)

. . . (A6)

C � C ( h) and C � 0 as Irl � 00 () \I R . . . (A7)

The heterogeneous electron transfer rate constants k, and kh are potential - dependent via Butler-Volmer equation3. The dimensionless concentration g(r) is defined as

g (r) - I [k f eo ( r ) - k b e R ( r ) ]

- - k e " f 11

with

a = a2k/D R

. . . (A8)

. . . (A9)

We further define the ratio of charge transfer and mass transfer rate constants via K as

. . . (A l O)

where a denotes the characteri stic length of the geolll­etry (a denotes radius for discs and spheres and equato­rial radius for prolate and oblate spheroids)

D C (r) + D C (r) = D C (h) \I 0 R R 0 (l

. . . (A I I )

The above considerations lead to Eqs (6) - (8 ) of the text. The parameter a indicates the rate of regeneration of the bulk species relative to the time constant of di ffu­sion over distance 'a' .

Appendix B

In this Appendix, we report a simple method of de­riving steady state diffusion limited current at spheroi­dal electrodes, using co-ordinate transformation. (This procedure is identical with that demonstrated by Phi l ip}3 for soil infiltration). In this case, a = 0 since the chemi­cal reaction is absent and g= I , K being 00 . Consequently, the transport equation and the boundary conditions are :

g = 1 z = 0; rE SE

g = 0 r� 00

. . . (B 1 )

. . . (B2)

. . . (B3) The spheroidal electrodes have two semi-axes of

length a and another third semi-axis of length b. The

362 INDIAN J CHEM, SEC. A, JAN - MARCH 2000

aspect ratio OJ = b/a takes values depending upon the particular spheriod geometry under consideration. The spheriod surface is represented in cylindrical coordinates as

. . . (B4)

The prolate spheroidal co-ordinaties Tj, <I> are related as

r = A sinh Tj sin <I> Z = A cosh Tj cos <I>

and

1.2 = b2_a2

By using separation of variables, (B I ) becomes

d ' d � + (coth ry ) � = 0 d 17 - d ry

whose solution is

g = C I + C2 In coth (Tj/2)

. . . (BS)

. . . (B6)

. . . (B7)

. . . (B8)

. . . (B9)

where C and C are integration constants whose values I 2 follow from the Eqs (B2) and (B3) as

C = 0 and C, = l /cosh" (b/a) I _ . . . (B 1 0)

Further, Tj �oc as p�oc being (r2+z2 ) 1 /2 =p.Hence p2 Z 1.2 e2T]/4 and dTj/dp=po' .

The steady state current for prolate spheroidal ge­ometry is

. . . (B I I )

Equation (B9) leads to

( dO ) - Ap - I

d� = cosh - I ( b / a ) . . . (B 1 2)

Hence

I 4n (b2 _ a2 ) 1 / 2 nFDC�b) = 41tAC2 = cosh -' (b / a) . . . (B I 3)

Eq. (B 1 3) i s identical with Eq. (2 1 ) of Oldham et al.37 and is consistent with Eq. ( 1 9) of Birke35

Oblate spheroids

The co-ordinates Tj , <I> are now given as r = A cosh Tj sin <I> Z = A sinh Tj cos <I>

As in the earlier prolate case, we obtain

d 2 d � + (tanh 11) � = O d112 d11

. . . (B I 4)

. . . (B 1 5)

. . . (B 1 6)

whose solution, taking into account the boundary condi-tions is g = corl (sinh Tj)/ coso l (b/a) . . . (B 1 7)

The steady state current becomes

. . . . (B 1 8)

where i is 41tnFDC ( h )a for oblate spheroidal geometries. II II

(B 1 8) is identical with Eq. (2 1 ) of Oldham et a/. 37 .

Disc and whisker electrodes For disc electrodes eo = 0 and Eq. (B 1 8) yields i=

8nFDC a - a well -known result. Whisker electrodes refer () to prolate hemispheroidal geometries in which the as-pect ratio exceeds 6.3 and eo � 00 . Noting that cosh o leo = In [(OJ+eo2_ 1 )] V' , we obtain from Eq. (B 1 3)

i =41tnFDC (h) b/ln (2eo) () . . . (B 1 9)

which is identical with equation (27) of Oldham et al. 37 .

List of symbols: a = Characteristic length of the geometrical shape; equals

radius for discs and spheres; for oblate and prolate geometries a denotes the horizontal equatorial radius.

b = Length of the vertical third semi-axis C",CR = Surface concentrations of 0 and R C.,

lhl = Concentration of 0 D

"DR = Diffusion coefficient of 0 and R

g = Dimensionless concentration k,. kh = Forward and baGkward electron transfer rate constants

RAJENDRAN et al. : CHRONOAMPEROMETRIC CURRENT AT ULTRAMICROSPHEROIDAL ELECTRODES 363

k K

z 0: ill or 11,CP

= Pseudo first order rate constant for the bulk reaction = Dimensionless ratio of mass transfer and charge transfer rate constants = CQefficients of Pade' approximant = Radial distance in cylindrical co-ordinates = Conducting electrode surface and insulating planar re­gion = Direction normal to the plane = Dimensionless bulk reaction coefficient. = Aspect ratio (b/a) = Steady state limit = Spheroidal co-ordinates

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