cyclic voltammetry at a regular microdisc electrode array

Journal of Electroanalytical Chemistry 502 (2001) 138–145www.elsevier.nl/locate/jelechem

Cyclic voltammetry at a regular microdisc electrode array

Hye Jin Lee, Carine Beriet, Rosaria Ferrigno, Hubert H. Girault *Laboratoire d’Electrochimie, Departement de Chimie, Ecole Polytechnique Federale de Lausanne, CH-1015 Lausanne, Switzerland

Abstract

A simulation of cyclic voltammetric responses has been performed to study the diffusion process of a regular array of microdiscinterfaces by the use of finite element software. The effect of centre-to-centre distance between the active electrodes on the currentresponse and also their voltammetric shape and characteristics have been investigated. © 2001 Elsevier Science B.V. All rightsreserved.

Keywords: Finite element method; Hexagonal array; Square array; Microdisc electrode; Cyclic voltammetry

1. Introduction

Nowadays, the design of microelectrode arrays withwell-defined geometries is reported mostly for electro-analytical purposes [1]. The geometries feature an as-sembly of parallel microbands operating at the samepotential or alternatively polarised as cathodes or an-odes [2,3]. This latter configuration is generally knownas the interdigitated band electrode configuration. An-other common geometry is the microdisc array wherethe electrodes are arranged in a hexagonal or a squarelattice [4–7] or even in a random manner [8–10].

A theoretical analysis of ensembles of microelec-trodes has been originally initiated in order to explainthe effect of the partial blocking at a large electrode[11–14]. As already shown, the apparent electron trans-fer rate constant seems slower at a dirty (not polished)electrode than at a cleaned, freshly polished one. Thisphenomenon results from the partial blocking of theelectrode by an insulating material leading to the for-mation of a random array of microelectrodes. Gueshi etal. [11,12] demonstrated theoretically and experimen-tally that a distortion of the cyclic voltammogramappears in this case. A mathematical treatment of thisphenomenon has been reported by Amatore et al. [15]for a hexagonal assembly of microdisc and microband

electrodes. They have shown that depending on thefractional coverage of the array, various voltam-mogram shapes are obtained. For a low coverage byinactive sites, i.e. small inter-electrode distance, a Nern-stian peak-shaped voltammogram is obtained whereas asigmoidal shape results from a large inactive site cover-age. Moreover, between these two limiting cases, theincrease of the peak-to-peak separation with the inter-electrode distance is also noticed even if a fast electrontransfer reaction is assumed.

For arrays of microelectrodes as for the single mi-croelectrode case, the theoretical consideration ismostly cumbersome due to the non-linear diffusionprofiles at the electrodes. Analytical expressions areavailable for the chronoamperometric response of mi-crodisc electrodes alone [16], in a hexagonal [17,18] or asquare lattice [18]. Depending on the inter-electrodedistance, the response of an array can be simply thesum of the individual responses when the inter-elec-trode distance is large compared to the individual diffu-sion layers. Otherwise, when the diffusion fields beginto overlap, a more complex behaviour is observedarising from the mixed diffusion behaviour. Finally, ifthe experimental time scale is sufficiently large, all theindividual diffusion layers merge to form one singlelinear diffusion layer. A proposed condition generallyreported for observing independent behaviour of theactive sites and therefore avoiding a shielding effect isthat [19]:

R0/r\6, i.e. d/r\12 (1)

* Corresponding author. Tel.: +41-21693-3151; fax: + 41-21693-3667.

E-mail address: [email protected] (H.H. Girault).

0022-0728/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved.PII: S 0022 -0728 (01 )00343 -6

H.J. Lee et al. / Journal of Electroanalytical Chemistry 502 (2001) 138–145 139

where 2R0=d is the inter-electrode centre-to-centredistance and r is the radius of the active site.

Condition (1) arises from the fact that 90% of theconcentration lines are located within a distance of 6.3rat a microdisc electrode [19]. This condition is oftenused for the design of a microelectrode array but doesnot seem to be always appropriate experimentally. Foran array with nanometer-scale microdisc electrodes, apure spherical cyclic voltammogram is not observedwhen r=50 nm and 2R0=1.3 mm (i.e. d/r=26) [20].

Cyclic voltammetry treatment at an assembly of mi-croelectrodes is rather more complicated thanchronoamperometry, as one more parameter, the po-tential, is introduced. But cyclic voltammetry has alsoproved over the years to be one of the most ubiquitouselectrochemical methods. In the case of semi-infinitelinear diffusion, the technique is often used to deter-mine the reversibility of the electrode reactions. Whenapplied to a microelectrode or a microelectrode array,cyclic voltammetry is mainly used in the steady statelimit. From an electroanalytical viewpoint, the majorissue is then to choose a scan rate to ensure that asteady state current can be measured. Otherwise, apeak-shaped CV will be recorded, the characterising ofwhich requires a simulation.

In a previous paper [21], we demonstrated a 3D-sim-ulation approach based on a finite element resolution tostudy the diffusion characteristics of a regular array ofdisc electrodes by the use of chronoamperometry. Thiswork mainly assessed the validity of our computationalapproach by comparison with other numerical or ana-lytical methods.

In this paper, the same simulation approach is fur-ther used to investigate the effect of centre-to-centredistances in hexagonal and square arrays of microdiscelectrodes during cyclic voltammetry. Both the simu-lated and experimental results are compared with thoseof the calculated data using analytical equations devel-oped by Amatore et al. [15]. A dimensionless analysis isalso introduced.

2. Numerical details: 3D geometries and numericalparameters

The 3D-simulation of the diffusion processes of anarray of infinite and finite numbers of inlaid microdiscsis performed using a commercial finite element softwarepackage (Flux-Expert®, Simulog Grenoble, Montbon-not St. Martin) and operated on a UNIX workstation(Silicon Graphics Indigo 2 Impact 10000, 640 MBRAM). The diffusion profiles are calculated by solvingFick’s second law in Cartesian coordinates assuming afast electron transfer at the active sites, and written as:

#c#t

=D�#2c#x2+

#2c#y2+

#2c#z2

�(2)

where c is the concentration of the diffusing species andD is its diffusion coefficient.

The geometries simulated in this work are the classi-cal hexagonal and square lattices already presented inour previous paper [21]. The 3D mesh is made of a 2Dmesh on the x–y plane containing the microdiscs ofradius (r) separated from each other by a centre-to-cen-tre distance d=2R0. The 2D mesh is then elevated onthe z-axis in order to create the 3D geometry required.

The boundary and the initial conditions used in thesimulations are:

c(x,y,z,0)=c0=1 mol m−3 (3)�#c#z�

z=0

=0, x2+y2\r2 (4)

limz��

c=c0=1 mol m−3 (5)

At the electroactive sites, due to the potential scan, atime evolution condition is defined as:

tBt c/c0=U e−gt/1+U e−gt (6)

tB tB2t c/c0=U egt−2gt/1+U egt−2gt (7)

where the various parameters, t, U and g read:

t=DE/n (8)

U=ezF/RT(Ei−E°%) (9)

g=zFn/RT (10)

with Ei being the initial potential, E°% the formal poten-tial, DE the potential window, n the potential sweeprate, F the Faraday constant, z the number of ex-changed electrons, R the universal gas constant and Tthe temperature.

The relationships in Eqs. (6) and (7) are valid only ifthe diffusing species and its product compound havethe same diffusion coefficient.

The various geometrical and physical parameter val-ues taken in this work are:

Inter-electrode distance: d=2R0=30, 60, 120, 400,600, 1000, 1500 mm.Microelectrode radius: r=10 or 13 mm.Diffusion coefficient: D=10−9 m2 s−1.Bulk concentration: c0=1 mol m−3.Sweep rate: n=10, 40, 100 mV s−1.All simulations are carried out with dimensional

mass transport parameters and using a transient linearalgorithm. Triangular finite elements are used in theplane of the microdisc with a finer mesh around its edgethan in the bulk region. The mesh away from themicrodisc is kept as fine as possible considering thelimitation imposed by the available computer memory.Extra points are placed in both regions to impose a gridconstruction and the number of nodes as well as the

H.J. Lee et al. / Journal of Electroanalytical Chemistry 502 (2001) 138–145140

node repartition can be modulated on segments linkingtwo such points. A geometrical distribution of thenodes is used near the interface edge, whereas an arith-metic distribution of the nodes is used on lines far fromthe interface. Similarly on the z-axis, the geometry iselevated with closer packed layers near the microinter-face in the same manner as described in a previouspaper [21]. The accuracy of the present simulationmethod is assessed using an infinite number of mi-crodiscs spaced by a large distance (d=1500 mm) in ahexagonal array. A 2% error is obtained on the steadystate current value. In most cases, the potential windowchosen is 1 V and the potential increment is 10 mV forall the scan rates.

3. Results and discussion

3.1. Infinite number of microdisc interfaces in ahexagonal array

3.1.1. Voltammetric responsesFrom the previous results obtained in the case of a

hexagonal array of inlaid microdisc interfaces [21], ithas been confirmed that the spacing distance betweenthe adjacent microinterfaces strongly influences the dif-fusion process of the redox reactions. Thus, the effectof the inter-electrode distances in an infinite number ofmicrodiscs (r=10 mm) arranged in a hexagonal latticeis here investigated by cyclic voltammetry. The voltam-metric responses presented in Fig. 1 are obtained for

centre-to-centre distances ranging from 30 to 1000 mm.The scan rate used is 10 mV s−1 unless otherwisespecified.

The smaller the centre-to-centre distances betweenthe microdiscs, the smaller the measured current is withrespect to the value obtained from Eq. (11). This equa-tion is valid for the case of an array of inlaid solidmicrodiscs when no overlapping of the individual diffu-sion fields is noticed:

Iss=4mzFDc0r (11)

where m is the number of microinterfaces. It can beobserved that the steady state values obtained for largevalues of the ratio d/r exceed the theoretical values by2%.

This result corroborates the shielding effect of theindividual diffusion fields. It can be pointed out thattwo characteristic limiting polarisation curves are ob-tained as follows: (i) in the case of d/r\40, a sigmoidalshape that can be related to microdisc electrode be-haviour (spherical diffusion); and (ii) in the case ofd/rB6, a peak-shaped voltammogram that can corre-spond to reversible linear diffusion. We can alreadyconclude here that the proposed Condition (1), d/r\12, is not satisfactory.

Significant decreases of the current at closely spacedactive sites (e.g. d/r=3) compared with those withlarge centre-to-centre distances can be explained by thefact that the former interfaces cannot be considered asindependent of each other. The border-to-border dis-tance is almost equal to the diameter of the microdiscs,so that the individual diffusion fields may stronglyoverlap. The merged diffusion fields then form an al-most planar diffusion layer and the voltammetric be-haviour of the microinterfaces becomes similar to aconventional macroelectrode.

This finding was investigated using various parame-ters such as the plot of the peak current as a function ofthe square root of the scan rate and also the evolutionof the peak-to-peak separation with the parameter d/r.The slope of the simulated peak current response vs thesquare root of the scan rate (0.043) is in good agree-ment with the Randles–Sevcik equation (0.049) (seeFig. 2) for the case d/r=3. Here the surface area usedin the calculation is the sum of the active microdiscelectrode and the insulating area. Further supportiveevidence can be the peak-to-peak separation on theforward and reverse scans, which should be equal to59/z mV at 298 K for the case of a reversible reactionThe simulated value is close to 60 mV for the case ofd/r=3 and starts deviating from d/r\6.

When the ratio of d/r is larger than 40, the shape ofthe voltammograms tends towards sigmoidal and asteady state current is reached. The ratio of the simu-lated steady state current and the theoretical value (Eq.(11)) is close to unity. This indicates typical behaviour

Fig. 1. Simulated cyclic voltammograms obtained at an infinitenumber of microdiscs in a hexagonal array having various centre-to-centre distances. d/r varied from 3 to 100. Sweep rate was 10 mV s−1,r=10 mm and D=10−9 m2 s−1.


Fig. 2. Simulated cyclic voltammograms obtained at different scanrates for an infinite number of microdiscs in a hexagonal array whered/r=3. Sweep rates varied from 10 to 100 mV s−1, r=10 mm andD=10−9 m2 s−1.

3.1.2. Dimensionless analysisThe aim of this section is to provide a dimensionless

diagram in order to predict the behaviour of an arrayof microelectrodes. The same analysis can be applied toan array of supported liquid � liquid interfaces but sub-ject to specific conditions.

The critical parameters here are the characteristiclengths d, r and the scan rate n. If the scan rate isreduced to a length d=RTD/nFn, the dimensionlessparameters d/r and d/r can be used.

For chronoamperometry, Shoup and Szabo [17] andScharifker [18] defined a parameter ts�p expressing thetransition time between spherical and planar diffusion.These equations read, respectively:

ts�p=(R0−r)2

6D(12)

ts�p=[1−pnr2]2

pD [pnr ]2(13)

where n= (2/3d2) for a hexagonal array. It is worthpointing out that Shoup and Szabo’s Eq. (14) corre-sponds to the beginning of the overlap of the diffusionfields, whereas in Sharifker’s Eq. (15) the current hasalready decreased by more than 50%. The former tran-sition corresponds to the departure from the sphericaldiffusion behaviour whereas the latter corresponds tothe onset of a planar diffusion regime.

From Eqs. (12) and (13) and the definition of thescan rate reduced to a length (d=2Dt),we can asso-ciate d values to these two transition times, and therebydefine two dimensionless parameters associated with themerging of the diffusion fields:

ds�p

r=� d

2r−1

n/3 (14)

ds�p

r='2

p

�3d2

2pr2 −1n

(15)

The plot of log(d/r) vs log(d/r) is presented in Fig. 3.Eqs. (14) and (15) define three main areas where thecentral area delimited by these equations can be re-garded as the domain where a mixed diffusion processis encountered, i.e. where a peak-to-peak separationlarger than 59/z mV is obtained. In the left side aspherical process is established whereas in the right sidelinear diffusion prevails. For the case of a macroelec-trode (i.e. d/r=2), only linear diffusion is exhibited andtherefore Eqs. (14) and (15) tend toward an asymptotictrend. This phenomenon can be easily understood bynoticing that in this case a peak-shaped voltammogramis obtained whatever the scan rate used.

The simulated results reported in Fig. 3 corroboratethe analysis derived above. Spherical, mixed and linearvoltammograrns are located in the respective areasdefined by Eqs. (14) and (15). It is also worth noticingthat Fig. 3 represents only half of the real zone dia-

Fig. 3. Zone diagram of the predominance of the linear, spherical andmixed diffusion processes as a function of the parameters d/r and d/r.

of the inlaid microdisc array having no overlap ofdiffusion fields as already mentioned.

For the case of 65d/r540 a mixed diffusion profilebetween linear and spherical is obtained. The voltam-mograms still present a peak shape but the currentvalue slowly increases with d/r. Moreover, the peak-to-peak separation becomes larger in the same way aswhen the apparent electron transfer process is loweredas already mentioned by Amatore et al. [15]. Ratios ofd/r=6 and 12 yield a peak-to-peak separation of 80and 160 mV, respectively, even if Nernstian behaviouris assumed during the computational calculations. In-creasing the ratio d/r even more results in the slowdisappearance of the return peak as the species pro-duced diffuse away faster by spherical diffusion.


gram. A symmetric representation can be envisagedwhere a transition between spherical and linear takesplace again for microelectrodes when the scan rate isincreased. Fig. 3 clearly shows that the condition tofulfil for independent diffusion fields and pure sphericaldiffusion is given by Eq. (14), i.e. (d/r)B [(d/2r)−1]/3. Generally, the reported condition (1) is d/r\12.Condition (1) is really not satisfactory as it should besweep rate dependent. Fig. 3 clearly shows that thisstatement is correct only for a range of scan rates. Forinstance, for r=10 mm or 5 nm the scan rate valuemust be larger than 30 mV s−1 and 120 kV s−1, respec-tively, to remain in the spherical behaviour area ifd/r=12.3.1.3. Zone diagram

The shapes and characteristics of the voltammogramsfor the various distances, d/r ranging from 3 to 60 arefurther investigated by comparison with the analysisdeveloped by Amatore et al. [15] for charge transfer atpartially blocked surfaces. Two dimensionless parame-ters, Kl1/2 and L(1−u) are considered:

Kl1/2= [(1−u)/u ]

[{(2D/R02)u−1(1−u)−1/ln[1+

[a(1−u)−1/2]}tc]1/2 (16)

L(1−u)=k(tc/D)1/2(1−u) (17)

where R0 is the half distance between the active site, tc

is equal to RT/Fn and a is an empirical adjustableparameter that is taken as equal to 0.27 [15]. u repre-sents the fractional coverage of the insulating sites,which is defined as u=Sins/(Sins+Sactive), Sins and Sactive

are the insulating and electroactive surface areas, re-spectively, and k is the standard rate constant for theelectron transfer at the active site. As a fast electrontransfer reaction is assumed in our case, k must betaken large enough, e.g. ]0.1 cm s−1.

The graphical meaning of these dimensionlessparameters is first described in Fig. 4, representing azone diagram as reported by Amatore et al. [15]. Therepresentation of logL(1−u) vs log Kl1/2 defines fourmain domains where different characteristic behaviouris encountered. The vertical frontier delimits the linearand spherical diffusion fields. In the left hand side ofthe representation, a sigmoidal shape is obtained forthe voltammogram whereas in the right hand side, apeak shape is achieved. Then, a division is also distin-guished between the upper and lower sides. In theupper side, a fast electron transfer is stated whereas inthe lower part, the voltammogram exhibits apparentquasi-irreversible electron transfer behaviour. Thegraphical evidence includes a steady state voltam-mogram, which is not centred on E°% and a peak-to-peak separation larger than the thermodynamic value.This representation was derived for a hexagonal arrayof microdisc electrodes with the assumption of discinactive sites rather than hexagonal ones.

The values calculated for the examples shown in Fig.1 are summarised in Table 1 and are plotted in Fig. 4.In the case of d/r=40, where the fractional coveragewas close to unity (0.998), the location in the log/logplot is in the upper left side, which corresponds to asteady state reversible voltammogram (see Fig. 4). Bysimulation, a polarisation curve having the shape of apolarogram where the plateau height is also indepen-dent of the sweep rate (see Fig. 1) is obtained. Theslope of the plot of ln(Iss−I)/I vs potential yields 24.5mV, which is in good agreement with the theoreticalvalue of a reversible reaction (25 mV at 298 K). Also,the half-wave potential is simply equal to the formalpotential of the reaction.

On the other hand, the case of d/r=3 where thefractional coverage is not close to unity (0.595), isrepresented in the upper right side of the log/log plot(see Fig. 4). The simulated voltammogram presents apeak shape with a peak-to-peak separation correspon-ding to the Nernstian behaviour.

Fig. 4. Zone diagram reported from Ref. [15] representing the maincharacteristics of cyclic voltammograms as a function of two dimen-sionless parameters, Kl1/2 and L(1−u), using Eqs. (15) and (16)taken from Ref. [15].

Table 1Estimated values of two dimensionless parameters, Kl1/2 and L(1−u), using Eqs. (15) and (16) taken from Ref. [15] depending onvarious experimental values of r/d and the corresponding value of u

log Kl1/2Kl1/2u log L(1−u)L(1−u)d/r

0.595 11.21 1.05 1839 2.2630.899 1.146 0.057 45.9 1.66

1.060.975 11.412 −0.700.202.7−1.42 0.430.0390.99424

0.998 0.012 −1.94 0.91 −0.041400.999 0.00506 −2.30 0.45 −0.3560


Fig. 5. Simulated and calculated (Eq. (17)) cyclic voltammogramsobtained at three different geometries including a (2×2) microdisc,an infinite number of microdiscs assembled in a square lattice and asingle microdisc. Sweep rate was 10 mV s−1, r=13 mm, d=45 mm(d/r=3.46) and D=10−9 m2 s−1.

3.2.1. Square lattice microdisc interfacesSimulated current responses as a function of the

applied potential at three different geometries areshown in Fig. 5 for geometries featuring an inter-elec-trode distance of 45 mm and a radius of 13 mm. Theresponse of an individual interface depends on its envi-ronment as it is placed alone in a semi-infinite plane,located within an array containing a finite or an infinitenumber of microinterfaces.

The case of a single microelectrode was modelled bya 0.1 mm recessed microdisc resulting in a small de-crease of the steady state current compared with thepure inlaid case (deviation less than 2% compared toEq. (11)). Moreover, the semi-logarithmic representa-tion of (Iss−I)/I vs (E−E°%) yielded a straight linewith a slope of 24.5 mV per decade (1.1% error).

In the case of an infinite number of microdiscs sepa-rated by d/r=3.46 (45/13 mm), a peak-shaped voltam-metric response was observed at 10 mV s−1 as hasalready been shown in Section 3.1 for a hexagonalarray. When the finite number of electrodes (4 mi-crodiscs, 2×2) is taken into account, a quasi-sig-moidal-shape response was obtained but with a currentvalue lower than that for a single microdisc. Thisdemonstrates the significant influence of the microelec-trodes located at the edges of the array. This observa-tion also allows the predict on that the condition(d/r)B [(d/2r)−1]/3 valid for no diffusion field overlapin an array of a finite number of interfaces must bere-estimated in the case of an array of a finite numberof microdisc electrodes.

A theoretical equation considering the edge effect onthe square lattice array was used to fit the simulateddata. In this case, the current due to a finite number ofmicrodisc interfaces N with the number of microinter-faces on the edge NE can be expressed as [21]:

Ifinite square(E)= [1+NE/2]Isingle(E)

+ [N−NE/2−1]I�(E) (18)

where Isingle(E) and I�(E) represent the current at asingle isolated microelectrode and at a microelectrodein an infinite array determined by simulation during acyclic voltammetry experiment, respectively.

The calculated cyclic voltammogram of four mi-crodisc interfaces in a square array (Eq. (18)) is alsopresented in Fig. 5. As was already reported forchronoamperometry, it can be observed that the calcu-lated voltammetric response is in good agreement withthe simulated current response (within error of 2% ofthe steady state current).

3.2.2. Microdisc interfaces in a hexagonal arrayOne direct application of the semi-analytical ap-

proach derived above is when the simulation limit isreached as in the case of a finite number of interfaces

For both d/r=3 and 40 cases, the present simulationdata obtained for a hexagonal array corroborates theanalysis [15].

Between the two limiting cases, as the inter-electrodedistance is increased, the representation goes from thereversible linear diffusion voltammogram to the sig-moidal shape by increasing both the forward currentvalue and the peak-to-peak separation. The fractionalcoverage strongly influences the apparent standard rateconstant of the reaction.

3.2. Finite number of microdisc interfaces

An infinite number of interfaces is obviously anassumption, and experimental data are generally ob-tained for arrays with a finite number of microdiscinterfaces. Due to the edge effect resulting from theinterfaces located at the edges of the array, the ampero-metric response of an array of a finite number ofinterfaces can deviate drastically from the response of atheoretical lattice with an infinite number of electroac-tive sites. As demonstrated previously for chronoam-perometry [21], the behaviour of an array with a finitenumber of interfaces can be derived by calculation fromboth simulated responses of a single microinterface andan array of an infinite number of interfaces. Thisassumption has been investigated here for the cases ofsquare and hexagonal lattices during cyclic voltamme-try experiments. The same kinds of simulations asshown in Section 3.1 have been carried out for an arrayof a finite number of microdisc interfaces.


assembled in a hexagonal array. In this case, thepresent 3D-simulation approach requires a large com-puter memory. Therefore, the simulation of the finitecase is not achieved here. To circumvent this limitation,Eq. (19) can be used to calculate the current response ofa hexagonal microdisc array.

Ifinite hexagonal(E)=aIsingle(E)+bI�(E) (19)

where a and b are weighting parameters representingthe number of microdisc interfaces responsible for theedge and the array term in the total current,respectively.

The calculated current response for a finite numberof electrodes obtained from the simulated data is shownin Fig. 6 for a hexagonal array (6×11) having acentre-to-centre distance of 120 mm and an interfaceradius of 10 mm. The number of microelectrodes res-ponsible for the edge and array were a=50 and b=16,respectively, in the present array.

For the case of an infinite number of microdiscs in ahexagonal array separated by d=120 mm, the peakshape of the voltammetric response was observed to besimilar to the case of a square array having the sameinter-electrode distance. In the case of the finite numberof interfaces, the calculated voltammogram shows aquasi-sigmoidal response with a current value lowerthan that for a single microdisc. This finding alsosupports the significant influence of the interfaces lo-cated at the edges of the array on the current responseof the finite number of interfaces.

The comparison of the calculated voltammogramwith an experimental one is also performed in Fig. 6.These experimental data were achieved for the case ofthe transfer of tetramethylammonium ions across aliquid � liquid interface featuring the geometry describedabove [5–7]. Liquid interfaces supported on microholescan be easily manufactured by laser ablation of poly-mer films. The amperometric response of an array ofliquid � liquid microinterfaces supported on a microholearray can be compared to an array of solid microdiscelectrodes when symmetric forward and reverse diffu-sion fields are assumed. It can be pointed out that thecalculated voltammetric response is in good agreementwith that of the experimental current response (withinan error of 2%). This demonstrates the validity of thepresent simulation and the semi-analytical method.

The comparison of a hexagonal array with an infinitenumber of interfaces and a (6×11) hexagonal latticehas shown that a smaller d/r condition is determinedfor a sigmoidal shape. In the case presented here, thelimit is pushed to d/r\24 (compared with 40 forSection 3.1). Nevertheless, even for d/r=3 a sigmoidalshape is observed due to the large deviation of thecurrent values between a single microdisc and an arraywith an infinite number of active sites.

4. Conclusions

The study of the diffusion behaviour at a regulararray of microdisc interfaces showed the importance ofthe spacing between the adjacent active sites in optimis-ing the interfacial geometry.

This report clearly shows that the usual conditiond/r\12 is not really satisfactory. A complete charac-terisation can be achieved with the log/log plot of thedimensionless parameters, d/r and d/r. The resultingzone diagram delimits the area where spherical, mixedor linear diffusion fields prevail. This diagram empha-sises that a quasi-irreversible voltammogram can beexplained merely by a mixed diffusion field rather thanby a slow charge transfer rate constant.

For a finite number of interfaces in an array, theoptimum inter-electrode distance can be reduced signifi-cantly due to the edge effect. The validity of Eq. (18), aconcept developed in a previous paper [18], is alsodemonstrated by both the simulated and experimentaldata.

Acknowledgements

The authors wish to acknowledge the financial sup-port given by the Commission pour la Technologie etl’Innovation under grant 32941. Laboratoire d’Elec-trochimie is part of the European Training & Mobility

Fig. 6. Comparison between simulated/calculated (Eq. (18)) andexperimental cyclic voltammograms obtained at two differentgeometries including a single microdisc and a microdisc interfacelocated within an infinite number of microdiscs. Sweep rate was 10mV s−1, r=10 mm, d=60 mm (d/r=6) and D=10−9 m2 s−1. Theexperimental array is composed of 66 microdiscs (6×11).


Network on ‘‘Organisation, Dynamics and Reactivityat Electrified Liquid � Liquid Interfaces’’ (ODRELLI).

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cyclic voltammetry at a regular microdisc electrode array

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