Short communication

A simulation of a square electrode con®gurationfor electroanalysis

John Cassidy* and John O'Gorman

School of Chemistry, Dublin Institute of Technology, Kevin Street, Dublin, 8, Ireland

(Received 7 November 1997; in revised form 17 December 1997)

AbstractÐA simulation has been carried out for a potential step experiment at a rectangular shaped geo-metry where there are four internal electrodes. Two contiguous electrodes form the working electrode andthe facing two electrodes form the auxiliary electrode. Both forms of a reversible electrochemical coupleare present in solution and an ultimate steady state current is obtained which equals 2anFADCb/d, wherethe electrode area is 2A, the fraction of the total couple concentration Cb, which is in the oxidized form isa and the di�usion coe�cient of the couple is taken to be D. d is the width of the cross-section of the rec-tangle. At short times there is an enhanced current observed due to the recycling of the couple at the cor-ners between the working and auxiliary electrodes. # 1998 Elsevier Science Ltd. All rights reserved

Key words: amperometry, steady-state response, electrode con®guration, di�usion.

INTRODUCTION

A twin electrode con®guration consisting of a work-

ing electrode (WE) and auxiliary electrode (AE) in

a thin layer of a solution of a reversible mediating

couple ([Fe(CN)6]4 ÿ /3ÿ) to which analyte has been

added (cholesterol in the presence of cholesterol

oxidase and oxygen) has been shown to yield a

steady state current, ilim, proportional to the con-

centration of cholesterol added [1]. The aim of this

paper is to extend this concept [2] to a second

dimension [3] in the form of a square geometry

where two contiguous inner faces of a channel

shown in Fig. 1 comprise the working electrode and

the facing two sides the auxiliary electrode. The rec-

tangular channel is dipped into a solution and may

rest at the bottom of the beaker and the electrode

area is de®ned by the height of the solution. In this

way there is no di�usion up or down. Ultimately

the system envisaged is a cuvette-type arrangement

where the sample is deliberately added to dry re-

agents in the cell. Physically these shapes can be

screenprinted [4, 5] or microlithographically

produced [6].

THEORY

In this simulation, sides (a) and (b) in Fig. 1, areelectrically linked as the working electrode andsides (c) and (d) are the auxiliary electrode. The

point where x and y are zero is taken to be the cor-ner between the electrodes (a) and (b). The refer-ence electrode is elsewhere in the system. Thelateral cross-section of the system is a square of

side (d). The electrodes could be screen printed on a¯exible plastic which is then folded into a rectangu-lar channel. The extensions at the top allow connec-

tion. This paper discusses the simulation of asimple di�usion controlled process where a fractiona of the total mediator concentration Cb is oxidized

through the addition of the analyte. A fraction a ofthe total concentration Cb is in an oxidized form Oand the remainder is in the reduced form R. The

potential of the working electrode, of total area 2A(sides (a) and (b)) is held at a value where the oxi-dized form of the mediator, is completely reducedwith simultaneous oxidation occurring at the auxili-

Electrochimica Acta, Vol. 43, Nos 21±22, pp. 3385±3387, 1998# 1998 Elsevier Science Ltd. All rights reserved

Printed in Great Britain0013±4686/98 $19.00+0.00PII: S0013-4686(97)10196-7

*Author to whom correspondence should be addressed.

3385

ary electrode ultimately yielding a steadystate cur-rent ilim.

The following are the equations which representthe system

@CO

@ t� DO

�@ 2CO

@x 2� @

2CO

@y2

��1�

@CR

@ t� DR

�@ 2CR

@x 2� @

2CR

@y2

��2�

DR and DO are the di�usion coe�cients of the oxi-

dized and reduced species which here are taken tobe equal to D. The initial conditions are that

CO�x, y, 0� � aCb and CR�x, y, 0� � �1ÿ a�Cb

�3�where Cb is the total concentration. At time zero a

potential step is applied to electrodes (a) and (b) sothat

CO�0, y, t� � 0 and CO�x, 0, t� � 0: �4�The following are the boundary conditions thatapply

dCO

dx

����x�0�ÿ dCR

dx

����x�0

anddCO

dy

����y�0

�ÿ dCR

dy

����y�0

�5�

dCO

dx

����x�0�ÿ dCR

dx

����x�d

anddCO

dy

����y�0

�ÿ dCR

dy

����y�d

�6�

The latter two conditions arise from symmetry con-siderations. The symmetry of the system allows the

slopes at particular points to be equal though the

current is not uniform over the electrode surface. Inorder to solve equations (1) and (2) by orthogonalcollocation, the concentrations are ratio-ed to thetotal concentration, CO* = CO/Cb and CR* = CR/

Cb. The distances are also normalized to d the edgelength (X = x/d and Y= y/d) and the dimension-less time T* = Dt/(d)2.

RESULTS AND DISCUSSION

The overall current is calculated through inte-gration because of the nonuniform current distri-bution.

i � naFADCb

��10

dCO*

dX

����X�0

dX ��10

dCO*

dY

����Y�0

dY

��7�

In order to simplify the presentation of results thecurrent is ratio-ed to the limiting current for a onedimensional system where the working and auxili-

Fig. 1. Schematic of the electrode system.

Fig. 2. Plot of simulated dimensionless current calculated

using equation (10), (full line), against (T*)ÿ1/2. Conditionsused for generation of plot, D= 3.0� 10ÿ6 cm2 sÿ1,d= 10ÿ3 cm, a= 0.1, Cb=10ÿ3 mol dmÿ3. The number of

internal collocation points is 6 and the time interval is

10ÿ4. The dashed line is a plot of the current calculated

using equation (9) for the one dimensional situation.

J. Cassidy and J O'Gorman3386

ary electrodes are facing each other where theirareas are each 2A. This would be where side (a) is

the working electrode and side (d) the auxiliaryelectrode and in this case the electrode areas of (a)and (d) are 2A each and they are spaced a distance

d apart. For this situation the limiting current is [1];

ilim � 2anFADCb=d �8�Also the early time current decay for a one dimen-

sional system, which is provided here for compari-son, is given by [1]

i

ilim� a

2���������pT*p

�1� 2

X1j�1

exp�ÿ� j�2=�4T*���

�9�

Ultimately the current ratio (equation (7) dividedby equation (8)), solved by Orthogonal

Collocation [7, 8] becomes in terms of the dimen-sionless units:

i

ilim� a

2

�XN�2l�1

Ql

XN�1j�2

A1,jCO*�Xj , Yk, T*��

�10�

where Ql and A1,j are quadrature and collocationparameters respectively [7].Dimensionless current is plotted in Fig. 2 as a

function of dimensionless time (T*)ÿ1/2 and it isapparent that the transient current is greater for thetwo dimensional con®guration (Fig. 1 when (a), (b)are the WE and (c) and (d) are the AE) than for

the one dimensional con®guration geometry wherethe WE and AE are facing each other. Theenhancement is modest. The calculated response is

similar when a di�erent number of internal colloca-tion points are used, i.e. when N= 8 or 10 andalso with di�erent time intervals (5�10ÿ5 to

5�10ÿ4). The dip in current in the two dimensionalplot below that of the one dimensional plot is not anumerical problem since oscillation occurs most fre-quently at short time scales. Perhaps it is due to a

momentary lack of reagent when the concentrationpro®les in the two directions merge.It is interesting to note that the steady state cur-

rent for the one dimensional and two dimensionalsituations are the same. This implies that the recy-cling e�ect has no net current contribution once

steady state conditions have developed. The concen-

tration pro®les at this stage are such that the con-centration is pinned to zero at faces (a) and (b) and

the concentrations rise linearly across faces (c) and(d) to reach a peak dimensionless concentration of0.4 when a= 0.1.

The square geometry would therefore yield agreater charge (as has previously been used [4]) perquantity of analyte added owing to the recycling of

mediator at the junction of the WE/AE electrodes.This charge enhancement is merely due to the par-ticular cell geometry. However with advances in

screenprinting and punch recessing it should bephysically possible to produce these devices easilyallowing this enhancement to be observed.If two electrodes were connected facing each

other with (a) as WE and (d) as AE, leaving theother electrodes unconnected, an enhancementshould also be seen from recycling at the isolated

electrodes as has been seen in feedback current formicroelectrodes [9, 10]

ACKNOWLEDGEMENTS

J. O. G. would like to thank D. I. T. for a MScscholarship.

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Simulation of a square electrode con®guration 3387