The transient current at the disk electrode under diffusion control: a new determination by the...

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Electrochimica Acta 49 (2004) 5041–5048

The transient current at the disk electrode under diffusion control:

a new determination by the Cope–Tallman method

Peter J. Mahona,*, Keith B. Oldhamb

aResearch School of Chemistry, Australian National University, Canberra, ACT 0200, AustraliabDepartment of Chemistry, Trent University, Peterborough, ON K9J 7B8, Canada

Received 2 March 2004; received in revised form 28 May 2004; accepted 6 June 2004

Available online 20 July 2004

Abstract

For purposes of convolutive modelling, a need exists for polynomial functions relating to inlaid disc electrodes operating under diffusion-

controlled conditions. To satisfy this need, the method pioneered by Cope and Tallman has been reimplemented. Data from that exercise, and

from other sources, have been used to construct useful short- and long-time polynomial functions that together span the entire time range. To

the extent possible, these functions have been compared with literature data and tested for internal consistency.

# 2004 Elsevier Ltd. All rights reserved.

Keywords: Disk electrodes; Cope–Tallman method; Chronoamperometry; Chronopotentiometry

1. Introduction

Microelectrodes having the shape of an inlaid circular

disk are easily fabricated and, in consequence, are the most

popular experimentally. Ironically, however, predicting the

current at such electrodes remains a particularly vexing

problem, even in the simplest circumstances, despite many

attempts having been made using a variety of approaches.

Generally the methods, which have had varied success, fall

into three broad categories: elaborate series solutions to the

boundary value problem [1–8]; digital simulation using

various algorithms [2,9–20]; and the integral equation

method [21–29] of Cope and Tallman.

The development of the short-time expression for the

diffusion-controlled current I(t) at an inlaid disk electrode

of radius R has an interesting history. The prime term,

ðnFcbD=2ÞðpR2Þffiffiffiffiffiffiffiffiffiffiffiffiffiffi4=pDt

pis the cottrellian term [30],

proportional to the exposed area. The second term

ðnFcbD=2Þð2pRÞ is a constant proportional to the disk’s

perimeter [31]. The original short-time expression of Aoki

and Osteryoung [1] was revised [4] to three terms, with the

* Corresponding author. Tel.: +61 2 6125 2620; fax: +61 2 6125 3216.

E-mail address: [email protected] (P.J. Mahon).

0013-4686/$ – see front matter # 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.electacta.2004.06.005

third term empirically adjusted so that there was some

overlap with the long-term equation. Phillips and Jansons

[6] then derived the three-term equation

Iðshort tÞnpFcbDR

¼ 1ffiffiffiffiffiffipup þ 1þ 1

2

ffiffiffiffiu

p

rwhere u ¼ Dt

R2(1)

in which the third term ðnFcbD=2ÞffiffiffiffiffiffiffiffiffipDtp

is independent of

the disk’s size, but can be attributed to its topology. In these

formulae, F is Faraday’s constant and D is the diffusivity of

the electroactive substrate, each molecule of which loses n

electrons1 and which has a bulk concentration of cb. Eq. (1)

is useful only if u < 1.

For application to long polarization times, Aoki and

Osteryoung [1] generated an asymptotic expansion that

contained an algebraic error that was later corrected by

Shoup and Szabo [2]. Aoki and Osteryoung then revised

their earlier calculations and generated a four-term expres-

sion with numerical coefficients for the higher terms. A

more general approach was developed by Phillips [7] who

produced expressions for a range of electrode geometries,

1 n is negative for a reduction.

P.J. Mahon, K.B. Oldham / Electrochimica Acta 49 (2004) 5041–50485042

including the equation

Iðlong tÞnpFcbDR

¼ 4

pþ 8ffiffiffiffiffiffiffiffi

p5up þ 1

9� 1

p2

� �16ffiffiffiffiffiffiffiffiffiffip5u3p (2)

for an inlaid disk under diffusion control. This equation

requires u > 1 to be useful. The normalizations in Eqs. (1)

and (2) are specific for use in convolutive modelling but

elsewhere they are normalized differently.2

Unfortunately, there exists a time gap when neither of

Eqs. (1) and (2) provide satisfactory data. Attempts at

bridging the gap between the short- and long-time regimes

have been only moderately successful [4,32].

An ongoing research endeavour in these laboratories is to

extend the semianalytical technique known as ‘‘convolutive

modelling via extended semiintegrals" [33–39] to electrode

geometries for which the diffusion field cannot be described

by a single spatial coordinate. Such modelling permits, for

example, cyclic voltammograms to be predicted under a

range of geometric, chemical and electrochemical condi-

tions. In particular, we seek to apply this technique to the

popular inlaid disk electrode. For this purpose, accurate

knowledge of the diffusion-controlled chronoamperometric

response is necessary and it was this need that prompted the

present research. Specifically we seek accurate access to two

functions. The first of these is defined by

hðtÞ ¼ IðtÞnpFcb

ffiffiffiffiDp

R2(3)

while the second is related to it by the convolution [34]

gðtÞ�hðtÞ ¼Z t

0

gðtÞhðt � tÞ dt ¼ 1 (4)

In convolution voltammetry, the function h(t) has been

used3 to link the change in surface concentration to current

and the complimentary function g(t) has found utility in

extracting the change in surface concentration from the

current. In this mode although not explicitly so, g(t) has

been applied analytically to obtain ‘‘neo-polarograms’’ [40–

42], utilized in ‘‘convolution potential sweep voltammetry’’

[43–47] and has been useful in ‘‘global analysis’’ [48–51].

Ideally one would like single analytical formulae for

these functions, akin to those that exist [33–39] for compu-

tationally simpler electrodes. Failing that, adequate infor-

mation could be provided by polynomial functions offfiffiffiup

expressing the short- and long-time behaviour of two salient

functions h(t) and g(t), provided that there is satisfactory

overlap between the two temporal domains.

We initially explored the use of diffusion-controlled

chronoamperograms derived from digital simulations such

as the two-dimensional quasi-explicit finite-difference

2 The Saito steady-state current equation has also been used for normal-

izing the current.3 We have continued to use h(t) from our previous work on convolutive

modelling but we do note that Aoki and Osteryoung have also used a h(t)

function in their related work. A word of warning, they are not the same.

approach, optimized for the disk geometry [16]. It was

found, however, that the simulations were insufficiently

accurate, particularly at short times, to match Eq. (1) or

for successful non-linear curve fitting. We therefore, turned

to the method of Cope and Tallman, in the hope of better

results. The data reported by these authors [26] were too

sparse for our purpose, so we decided to repeat the calcula-

tions. We followed their procedure closely, but not slavishly.

1.1. The Cope–Tallman method

In a series of ground-breaking publications [21–29],

Cope and coworkers described a versatile, but mathemati-

cally abstruse, method for solving the important problem of

calculating the transient current I(t) at a number of electro-

des of computationally awkward shapes, under a variety of

electrochemical scenarios. Here our concern is with one of

the simpler electrode shapes – that of an inlaid disk – under

the simplest electrochemical conditions – pure diffusion

control. Those authors described their method as ‘‘the

integral equation method’’ but, because that title could

equally well describe several other approaches, including

even the classic research of Nicholson and Shain, we shall

use the unambiguous name ‘‘Cope–Tallman method’’.

The method has three stages: (a) transformation of the

problem to an integral equation in Laplace space (b) solution

of the integral equation, and (c) numerical inversion of that

solution. Of the several publications by Cope and Tallman

that address the method, the one that comes closest to

providing a thorough description of the method is a 1990

article [23] coauthored by Scott, but even this is sketchy on

detail, has some transcription errors, and requires careful

reading to appreciate the finer points of what is a mathema-

tically complicated procedure. Moreover, this seminal article

describes, not the disk electrode, but the more complicated

ring electrode. Below we present a precis of the method as it

applies solely to the disk electrode under diffusion control.

Our notation and approach are not always identical4 to those

of Cope and Tallman. See the original publications for further

details and greater mathematical rigour.

The system of equations that describe the diffusion-

controlled current at an inlaid- disk electrode of radius R,

following the imposition of a large potential step at t = 0, is:

@2

@r2cðr; z; tÞþ1

r

@

@rcðr; z; tÞþ @2

@z2cðr; z; tÞ¼ 1

D

@

@tcðr; z; tÞ (5)

cðr; z; 0Þ ¼ cb (6)

cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ z2

p!1; t

� �! cb (7)

cðr <R; 0; tÞ ¼ 0 (8)

@c ðr >R; 0; tÞ ¼ 0 (9)
@z
4 In particular, we do not here undimensionalize the variables, so that the

physical significance of the equations is less diminished.

P.J. Mahon, K.B. Oldham / Electrochimica Acta 49 (2004) 5041–5048 5043

iðr; tÞ ¼ nFD@c ðr R; 0; tÞ (10)
@z
IðtÞ ¼ 2p

Z R

riðr; tÞ dr (11)
0
In these equations, the concentration c of the electro-

active substrate is treated as a function of time and of the

radial r and axial z coordinates (both nonnegative) of a

cylindrical coordinate system that has its origin at the centre

of the disk. The local current density on the disk at a distance

r from the origin is denoted i(r,t).

The Cope–Tallman procedure starts by performing a

Laplace transformation with respect to t, which converts

Eq. (5) to

@2

@r2Dcðr; z; sÞ þ 1

r

@

@rDcðr; z; sÞ þ @2

@z2Dcðr; z; sÞ

¼ s

DDcðr; z; sÞ (12)

where Dc ¼ cb � c, an overbar being used to represent the

effect of Laplace transformation, for which s is the dummy

variable. Initial condition (6) is incorporated into this equa-

tion. The second step is a double exponential Fourier

transformation with respect to the cartesian coordinates,

which permits incorporation of the boundary conditions

and specialization to z = 0, eventually leading to an integral

equation, namely5

2p

Z R

0

Gðrjr0Þiðr0; sÞr0 dr0 ¼ nFcbD

s(13)

in terms of a Green function Gðrjr0Þ. This function6 involves

the distance l between two points, expressed as a function of

the radial coordinates r and r0 of those points and the angle ’

subtended at the origin by the straight line connecting them:

Gðrjr0Þ ¼ 1

p

Z p

0

expf�lffiffiffiffiffiffiffiffiffis=D

pg

ld’ where

l ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ ðr0Þ2 � 2rr0 cosf’g

q (14)

In common with all Green functions, that in Eq. (14)

displays a singularity when r = r0. This can be detached from

the integral by writing the Green function as

pGðrjr0Þ ¼Z p

0

expf�lffiffiffiffiffiffiffiffiffis=D

pg � cosf’=2g

ld’

þZ p

0

cosf’=2gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðr � r0Þ2 þ 4rr0 sin2f’=2g

q d’ (15)

The first integral now lacks singularities, while the

second may be integrated exactly to ð1=ffiffiffiffiffiffirr0p

Þarsinh

f2ffiffiffiffiffiffirr0p

=ðr � r0Þg or an equivalent logarithm. This latter,

of course, still incorporates the singularity, but its integral in

5 This is equivalent to equation (C4a) of reference [23].6 Its dimensionless equivalent is called a ‘‘Neumann function’’ by Cope

and Tallman because it is a Green function satisfying boundary conditions

of a type identified with the name of Neumann.

(15) does not. In this way, Cope, Scott and Tallman cleverly

evade the singularity and convert Eq. (13+15) into7

Z R

0

Lðr; r0; sÞ�1

plnjr�r0j

R

�� iðr0; sÞ dr0 ¼nFcbD

s(16)

where the quantity in square brackets equals ðr0=2pRÞR 2p0 l�1 expf�l ffiffi

sp g d’ when r0 and r are unequal and

pLðr; r; sÞ ¼ lnf8r=RgþR p=2

0 cos’ exp f�2rffiffiffiffiffiffiffiffis=D

psin’

o�

h1i

d’ for r ¼ r0.For given values of s and r, Eq. (16) is an integral equation

linking i and r0. The Cope–Tallman procedure assumes that

these two variables are related by the expansion

Rsiðu; sÞnFcbD

¼XM�1

k¼0

hkTkðuÞffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� u2p where u ¼ 2r0

R� 1 (17)

Here Tk() is the kth Chebyshev polynomial of the first

kind.8 The hk coefficients are determined numerically, in

effect by solving M simultaneous equations. Transformed

values of the local current density, iðr; sÞ, are not determined

in the Cope–Tallman method, only the transform IðsÞ of the

total current being accessed. Numerous values of s must be

used, ranging over many orders of magnitude, with judicious

selection of the matrix order, M. This exercise leads ulti-

mately to a file of values of the dimensionless current

transform IðsÞ=nFcbR3 corresponding to a second file of

the dimensionless dummy Laplace variable R2s/D.

The final step in the Cope–Tallman procedure is the

numerical inversion of the undimensionalized IðsÞ versus

s data, giving I(t) as a function of the dimensionless time u.

A set of CjðsÞ coefficients is first derived through fitting

the transformed current data to sufficient terms of the series

1ffiffisp

XJ�1

j¼0

CjðsÞTj 1�ffiffiffiffiffiffiffi4s

s

r( )(18)

where T{} is again a Chebyshev polynomial, s an adjustable

scaling factor and J a large integer. Inversion is achieved

through an extension of the Piessens method [21] involving

the series

IðuÞnFcbDR

¼ffiffiffipp XJ�1

j¼0

CjðsÞC j

ffiffiffiffiffiffisupn o

(19)

The polynomial function C{} is defined by

C j


¼ j

4

Xj

k¼1

Gðjþ kÞGðk=2ÞGðj� k þ 1ÞGðk þ 1ÞGð2kÞ �8

ffiffiffiffiffisup� �k

with C0


¼ 1 (20)

where G() is the gamma function [52]. Selection of the best

values of M, s and J is an art described thoroughly in

reference [23].

7 This is equivalent to equations (C8a) and (A1) of reference [23].8 T0ðuÞ ¼ 1; T1ðuÞ ¼ u; TkðuÞ ¼ 2uTk�1ðuÞ � Tk�2ðuÞ for k= 2,3,. . .


Table 1

The effect of J and s on e, the a coefficients and I(u)

Expected s = 0.001 s = 0.00001

J 40 60 80 100 50 100 150 200 250 300

e 10.85 �0.45 0.13 �0.03 1762 11.82 1.03 �0.16 0.13 0.14

a1 0.5642 0.5642 0.5642 0.5642 0.5642 0.5632 0.5642 0.5642 0.5642 0.5642 0.5642

a2 1.0000 1.0011 0.9999 1.0000 1.0000 1.0268 0.9994 1.0001 1.0000 1.0000 1.0001

a3 0.2821 0.2615 0.2865 0.2797 0.2838 0.2190 0.2846 0.2794 0.2840 0.2796 0.2785

a4 �0.0073 �0.1795 �0.0827 �0.1727 �0.0949 �0.1025 �0.1169 �0.1493 �0.0803 �0.0588

a5 �0.2307 0.3332 �0.3117 0.6637 0.0231 �0.0521 0.0832 0.1621 �0.3358 �0.5366

u I(u)/npFcbDR

0.0025 12.298 12.298 12.298 12.298 12.301 12.297 12.298 12.298 12.298 12.298

0.01 6.669 6.669 6.669 6.669 6.680 6.669 6.669 6.669 6.669 6.669

0.0625 3.320 3.320 3.320 3.320 3.329 3.320 3.320 3.320 3.320 3.320

0.16 2.505 2.505 2.505 2.505 2.509 2.506 2.505 2.505 2.505 2.505

0.25 2.242 2.242 2.242 2.242 2.242 2.242 2.242 2.242 2.242 2.242

0.5625 1.902 1.902 1.902 1.902 1.897 1.902 1.902 1.902 1.902 1.902

1.00 1.739 1.739 1.739 1.739 1.734 1.739 1.739 1.739 1.739 1.739

1.69 1.629 1.629 1.629 1.629 1.626 1.629 1.629 1.629 1.629 1.629

6.25 1.457 1.457 1.457 1.457 1.460 1.456 1.457 1.457 1.457 1.457

50.00 1.338 1.337 1.337 1.342 1.338 1.339 1.339 1.338 1.337 1.338

1.2. Our calculations

We adapted the original FORTRAN program for the ring

electrode, kindly made available by the Cope–Tallman team,

to the Visual Basic9 code. The ring-shape parametergwas set

to 0.5000000001, rather than the 1/2 appropriate to the disk,

because the exact value leads to numerical difficulty. This

substitution was shown to be innocuous. A matrix order M of

256 was used after extensive comparisons with M = 128.

It was observed that numerical instability, due to an

insufficient number of valid digits in the computation,10

resulted if the scaling factor s was too large. A valuable

criterion is that the sum of all CjðsÞ coefficients should equal

p and an iterative procedure minimizing the absolute value of

e ¼ 106 1� 1

p

XJ�1

j¼0

CjðsÞ" #

(21)

established s = 0.00001 and J = 250 as optimal values. For

this combination of parametersP

CjðsÞ differed from p by

0.13 parts per million. Table 1 includes a row showing how eis affected by changes in J and s.

The CjðsÞ values were imported into a Maple 8 work-

sheet for calculating the time series. Eq. (19+20) was used to

determine all the coefficients in the polynomial

IðtÞnpFcbDR

¼ a1u�1=2 þ a2u

0 þ a3u1=2 þ a4u

þ a5u3=2 þ � � � þ aJu

ðJ�2Þ=2 (22)

Table 1 reports a subset of the first five coefficients for

different values of J and s including our optimal set. Eq. (1)

shows that the first three coefficients should have the values

1=ffiffiffiffipp

; 1; and 1=ffiffiffiffiffiffiffi4pp

and the table shows that these

9# Microsoft Corporation.

10 Also known as catastrophic cancellation.

expectations are well met. The fourth and fifth coefficients,

however, are inconsistently dependent on the choices made

for J and s. Nevertheless, as the lower panel in Table 1

demonstrates, the current predictions are remarkably inde-

pendent of these choices, provided that a sufficiently large J

is used. Notice that at u = 50 the onset of the aforementioned

‘‘catastrophic cancellation’’ is becoming noticeable.

The message of Table 1 is that the Cope–Tallman method

is incapable, at least in our hands, of generating reliable

values of the aJ coefficients in Eq. (22), beyond those

already established analytically. In spite of this, the method

gives current values that are convincingly independent of the

choice made for the adjustable parameters. We therefore,

decided to make no further use of these aJ coefficients and

only utilize the numerical current values.

The full polynomial from Eq. (19) with s = 0.00001 and J

= 250 was used to generate an evenly spaced data set of

numbers up to u = 25. Various data sets with different cut-off

limits were created from these data with at least 104 values in

each file. However, only one such data set will be discussed

at this stage and it had an upper limit of u = 1.5. During the

fitting procedure the first three coefficients were fixed at the

known values from Eq. (1) and subtracted from the data set

to be fitted. The non-linear least squares curve fitting

application was also encoded using Visual Basic based on

the routines described for the Levenberg–Marquart method

[53,54]. The precision of the fitting was affected by two

factors, namely the domain of u and the number of terms

used in the fitting, with the average deviation being calcu-

lated in each case. The upper panel in Fig. 1 is a plot of the

difference

D¼ðdata from Cope�Tallman methodÞ� 1ffiffiffiffiffiffipup �1�1

2

ffiffiffiffiu

p

r

(23)


Fig. 1. The upper panel shows the fitting of a five-term polynomial to the

Cope–Tallman data where the first three terms, as given in Eq (1), have been

subtracted from the original data. The circles represent the data from the

polynomial fitting equation plotted with an output point density of 1 in 200

with the solid line being the Cope–Tallman data. The difference between the

polynomial and the Cope–Tallman data is shown in the lower panel.

Table 2

A list of some important polynomial coefficients

j h(t) g(t)

a/j bj aj bj

1 1p1=2

4p

1p1=2

14p1=2

2 1 8p5=2

�1 �4:7016� 10�2

3 12p1=2 8:9542� 10�3 3

2p1=2 4:7019� 10�3

4 �0:12003 �2:5664� 10�4 �0:38472 1:6079� 10�2

5 1:3273� 10�2 �2:2312� 10�4 7:7365� 10�2 �2:2686� 10�2

6 2:7628� 10�5 1:2948� 10�2

versus u. It shows a line with such a small curvature that we

deemed it necessary to use only two additional terms,

namely a04u and a05u3=2, to describe the data adequately in

the range 0 u 1.5. With the ‘‘best’’ values, a04 ¼�0:12003 and a05 ¼ 0:013273, the relative deviation is

shown in the lower panel of Fig. 1; it has an average value11

of 2.9 ppm and a maximum at u = 1.5.

1.3. The h(t) and g(t) functions

Recall that the motivation for this study was to derive

reliable expressions for the functions defined in Eqs. (3) and

(4), for subsequent application in convolutive modelling.

Both these functions carry a dimensionality matched by the

(second)�1/2 unit. Note also that, though both functions

approach the cottrellian limit of 1=ffiffiffiffiffiptp

as 0 t, their

behaviours in the t!1 limit are dissimilar. Whereas h(long

t) approaches the steady-state limit of 4ffiffiffiffiDp

=pR, g(long t)

decays rapidly towards zero as the function R2= 4Dffiffiffiffiffiffiffipt3p� �

.

The h(t) function is proportional to the current and so a

short-time expression may be taken over simply from the

five-term polynomial developed in the foregoing discussion:

hðshort tÞ ¼ Iðshort tÞnpFcb

ffiffiffiffiDp

R¼

ffiffiffiffiDp

R

X5

j¼1

a0juðj�2Þ=2

where u ¼ Dt

R2

(24)

11 In fact, these data will be used only in the 0 u 1:281; so the

average relative deviation is smaller still.

Table 2 lists the coefficients for this fitted polynomial, as

well as coefficients for other polynomials that will be

discussed later.

The long-time expression of Aoki and Osteryoung [4]

was expanded to six terms by using their tabulated coeffi-

cients, to produce the following expression

hðlong tÞ ¼ Iðlong tÞnpFcb

ffiffiffiffiDp

R¼

ffiffiffiffiDp

Rb1 þ

X6

j¼2

bjuð3�2jÞ=2

" #(25)

The six b coefficients, of which the first three are analytic,

are listed in Table 2. Note that there are no terms proportional

to u�1, u�2,. . .. Although based on an asymptotic series, it

was observed that the additional terms beyond the four given

by Aoki and Osteryoung [4] decreased the average deviation

when compared to our data from the Cope–Tallman method.

The short-time and long-time expansions cross at u =

1.281 and this is taken as the changeover point between the

two representations. A measure of how smoothly the transi-

tion occurs between formulas (24) and (25) is provided by

the comparison

RffiffiffiffiDp dh

duðu ¼ 1:281Þ

¼ �0:1674 by equation ð24Þ�0:1644 by equation ð25Þ

(26)

Turning now to the complementary function g(t), one can

see that the convolutive relationship in definition (4) implies

that

gðsÞ ¼ 1

shðsÞ(27)

and this provides a route by which a series expansion for h(t)

can be converted into a series expansion in g(t). The

procedure for obtaining g(short t) involved using an

extended h(t) data set from the Cope–Tallman procedure,

with values up to u = 12.5. The extended data set demanded

additional terms in the polynomial and it was found that as

many as eight terms were needed to match the data with an

average relative deviation of 2.2 ppm, which is to be com-

pared to that for a five-term polynomial over 0 u 1.5.

Laplace transformation yielded an eight-term expansion.

This converted to a fourteen-term polynomial in s during


the reciprocation step specified in (27) and hence yielded a

fourteen-term polynomial for g(t) on term-by-term inver-

sion. The g(t) function has a more difficult asymptotic nature

than h(t), manifested by the sluggish decrease in the mag-

nitudes of these fourteen coefficients. The procedure we

followed parallels that described in the context of Eq. (22).

The first three, analytically established, terms were sub-

tracted from the numerical data and the residue was found to

be well fitted, within the 0 u 1.5 domain, by retaining

two only further terms:

gðshort tÞ ¼ffiffiffiffiDp

R

X5

j¼1

ajuðj�2Þ=2 (28)

with carefully assigned values. Adopted values of all five a

coefficients will be found in Table 2.

The six coefficients used for the long-time expansion

gðlong tÞ ¼ffiffiffiffiDp

R

X6

j¼1

bju�ð2jþ1Þ=2 (29)

are also listed in Table 2. These are derived from the

tabulated coefficients for the long-time expansion given

by Aoki and Osteryoung [4].

Fig. 2 shows a smooth transition between the short-, (28),

and long-, (29), time polynomials for g(t). The two curves

intersect at u = 1.414 and we take this as the point of

transition between the two representations. The slopes at

either side of the transition point are

RffiffiffiffiDp dg

duðu ¼ 1:414Þ ¼ �0:0587 by equation ð28Þ

�0:0614 by equation ð29Þ

(30)

Fig. 2. A plot of g(short t) (solid line) and g(long t) (dashed line)

demonstrating the overlap.

showing a significantly greater discontinuity (4.5%) than the

1.8% in (26).

Though we have no way of checking how well our

bipartite polynomials correspond in reality to the g(t) and

h(t) functions, we find that they are better suited for the

modelling applications described in the companion article

[55] than other data we have trialed, possibly because they

are more accurate. Moreover, we can demonstrate that the

bipartite polynomials are mutually consistent by numeri-

cally testing how well they satisfy Eq. (4). Because both h(t)

and g(t) approach infinity at short times, it is more con-

venient to test the equivalent convolution [56]

dhðtÞdt�Z t

0

gðt0Þ dt0 ¼ 1 (31)

Of course dh/dt tends to infinity, too, a fortiori in fact; but

the numerical convolution routine that we employ [57] can

overcome this difficulty provided that only one member of

the convolved pair is unbounded. Fig. 3 displays a plot of

convolution (31) over the critical range 0 u 3, which

embraces both transitions and therefore employs all four

of the polynomials (24), (25), (28) and (29). Except close

to u = 0, where discretization errors are unavoidable, the

result of convolution is virtually indistinguishable from

unity.

1.4. Literature comparisons

We are aware of no experimental data of sufficient quality

to warrant comparison with our results. However, there are a

number of theoretical studies with which we may compare

Fig. 3. A plot of g(t) * h(t) as a function of u.


Table 3

Comparison of IðuÞ=npFcbDR values calculated by various methods

u CM FIFD FQEFD AO CT Eq. (24) Eq. (25)

0.0025 12.335 12.296 12.322 12.296 12.298 12.298

0.0100 6.676 6.665 6.679 6.665 6.669 6.669

0.0225 4.803 4.796 4.808 4.798 4.801 4.801

0.0625 3.319 3.317 3.323 3.317 3.320 3.320

0.1600 2.502 2.507 2.506 2.505 2.505

0.2500 2.238 2.242 2.251 2.242 2.241 (2.237)

0.3025 2.147 2.145 2.148 2.153 2.147 2.147 (2.145)

0.5625 1.901 1.901 1.902 1.903 1.902 1.902 (1.902)

1.0000 1.738 1.741 1.738 1.741 1.739 1.740 (1.739)

1.6900 1.628 1.628 1.627 1.628 1.629 (1.627) 1.629

6.2500 1.458 1.455 1.457 1.457 (1.388) 1.457

25.0000 1.366 1.366 1.365 1.365

Table 4

Values of Rg(t)/ffiffiffiffiDp

with g(t) calculated from three sources

u Eq. (32) Eq. (28) Eq. (29)

0.0025 10.435 10.325

0.0100 4.786 4.723

0.0225 2.916 2.880

0.0625 1.454 1.445

0.1600 0.6885 0.6924

0.2500 0.4603 0.4650

0.3025 0.3835 0.3877

0.5625 0.2019 0.2032 (0.3935)

1.0000 0.1043 0.1031 (0.1051)

1.6900 0.05415 (0.05396) 0.05296

6.2500 0.008829 (0.1457) 0.008557

25.0000 0.001091 0.001113

our results. The investigators in question had goals other

than constructing polynomial approximants to the g(t) and

h(t) functions, so direct comparisons cannot be made. Never-

theless, some useful correlations are possible.

Several authors have employed a variety of methods to

predict the shapes of diffusion-controlled chronoampero-

grams at inlaid disk electrodes. Their results are most

economically reported as u-dependent12 values of the nor-

malized current IðtÞ=npFcbDR. Of course, either of our h(t)

polynomials can also provide these data. Table 3 is a

comparison of the values13 given by our short- and long-

time polynomials with the data of others. The 1984 data of

Aoki and Osteryoung [4] is reported in the column headed

‘‘AO’’. ‘‘CM’’ refers to the conformal mapping study of

Amatore and Fosset [13]. Two other simulation methods

are given; the first is Gavaghan’s work [17–19], using the

fully-implicit finite difference (‘‘FIFD’’) and the other is

the fast quasi-explicit finite difference (‘‘FQEFD’’) simula-

tion [16]. ‘‘CT’’ is data generated by the Cope–Tallman

method in the course of the present study.14 Though there

are minor discrepancies, Table 3 shows that all methods,

12 Sometimes the parameter t, equal to 4u, is used instead.13 Values in parentheses imply a u outside the polynomial’s design range.14 Moreover our CT data agree almost exactly with a recent digital

simulation [J.C. Myland and K.B. Oldham, J. Electroanal. Chem., in press].

including our new polynomials, give essentially the same

values.

It is less easy to find literature data with which to compare

our g(t) values. However, an examination of the research of

Aoki et al [5] on chronopotentiometry at an inlaid disk

reveals15 a function f(u) related to our g(t) by

pR

4ffiffiffiffiDp d

dtfðuÞ ¼ gðtÞ (32)

Table 4 was constructed to compare the values of g(t)

calculated in this way with those from Eqs. (28) and (29).

There is general agreement, with discrepancies of both signs

and of a magnitude averaging about 1.3%.

Acknowledgements

We thank Professor Davis Cope for help and encourage-

ment. An RSC Fellowship from the Research School of

Chemistry at the Australian National University is gratefully

acknowledged, as is financial support from the Natural

Sciences and Engineering Research Council of Canada.

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