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Difference between Ultramicroelectrodes andMicroelectrodes: Influence of Natural Convection

Christian Amatore,* Cecile Pebay, Laurent Thouin,* Aifang Wang, and J-S. Warkocz

Ecole Normale Superieure, Departement de Chimie, UMR CNRS-ENS-UPMC 8640 Pasteur, 24 rue Lhomond,F-75231 Paris Cedex 05, France

Natural convection in macroscopically immobile solutions

may still alter electrochemical experiments performed

with electrodes of micrometric dimensions. A model

accounting for the influence of natural convection allowed

delineating conditions under which it interferes with mass

transport. Several electrochemical behaviors may be

observed according to the time scale of the experiment,

electrode dimensions, and intensity of natural convection.

The range of parameters in which ultramicrelectrodes

behave under a true diffusional steady state was identified.Mapping of concentration profiles was performed experi-

mentally by scanning electrochemical microscopy in the

vicinity of microelectrodes of various radii. The results

validated remarkably the predictions of the model, evi-

dencing in particular the alteration of the diffusional

steady state by natural convection.

Microelectrodes are versatile tools for the study of electro-

chemical processes of mechanistic and/or analytical interest. Their

advantageous properties stem from their small size. Microelec-

trodes may be used in highly resistive environments and in very

small sample volumes. They enable the detection of very smallamounts of material and allow short time responses.1-9 However,

the definition of a microelectrode is still nowadays ambiguous.

Actually, the notion of a microelectrode differs greatly according

to the particular origin of electrochemists, i.e., electroanalytical

chemists or molecular electrochemists. The term microelectrode

may then encompass electrodes of either millimetric or micro-

metric dimensions. Electrodes of smaller sizes are referred to as

ultramicroelectrodes. Such definitions, based mainly on historical

criteria, may appear useless since they better define the origin of

the users than the object itself. A better classification of these

electrodes would be obtained if it were based on their particular

properties. Since electrochemical reactions are interfacial reac-

tions, mass transport is one of the key processes to consider.10

In a liquid, elementary contributions in the mass transport are

diffusion, migration, and convection. Under most circumstances,

migration is suppressed by adding a large excess of dissociated

inert salt or supporting electrolyte. Convection is often neglected

at electrodes of micrometric dimensions in macroscopically still

solutions. Indeed, convection originates from movement of fluid

packets of micrometric size.11 It necessarily vanishes close to the

electrode interface over distances where concentrations differ

significantly from their bulk values.12,13 In such a case, only

diffusion is assumed to govern the final approach of an electro-

active molecule toward the electrode. However, according to the

size of these electrodes and time scale of the experiments,

convective fluxes due to natural convection may still compete with

diffusional fluxes in motionless solutions. This may occur even

in the absence of any density gradients14 or effects induced by a

magnetic field.15

These situations arise as soon as the thicknessof the diffusion layer becomes comparable to the thickness of the

convection-free domain.7 Under such conditions, the responses

do not follow the classical relationships given for currents in

dynamic and steady-state regimes. Therefore, under given ex-

perimental conditions, it is of importance to decide the largest

size of an electrode for eliminating any influence of natural

convection.16,17 Such a criterion may then allow distinguishing

properties of ultramicroelectrodes from those of other electrodes

of micrometric sizes.

To assess the conditions of convection-free regimes at elec-

trodes of micrometric dimensions, we investigated in some

previous studies the current responses of micrometric disk

* To whom correspondence should be addressed. E-mail: christian.amatore@

ens.fr (C.A.); [email protected].

(1) Fleischmann, M.; Pons, S.; Rolison, D. R. Ultramicroelectrodes; Datatech

Systems, Inc.: Morgantown, NC, 1987.(2) Bond, A. M.; Oldham, K. B.; Zoski, C. G. Anal. Chim. Acta 1989, 216,

177230.(3) Wightman, R. M.; Wipf, D. O. Electroanalytical Chemistry; Marcel Dekker:

New York, 1989; Vol. 15, pp 267-353.

(4) Montenegro, M. I.; Queiros, M. A.; Daschbach, J. L. Microelectrodes: Theory

and Applications; Kluwer Academic Press: Dordrecht, The Netherlands,

1991; Vol. 197.

(5) Aoki, K. Electroanalysis 1993, 5, 627639.(6) Heinze, J. Angew. Chem., Int. Ed. 1993, 32, 12681288.(7) Amatore, C. Electrochemistry at ultramicroelectrodes. In Physical Electro-

chemistry; Rubinstein, I., Ed.; Marcel Dekker: New York, 1995.

(8) Stulik, K.; Amatore, C.; Holub, K.; Marecek, V.; Kutner, W. Pure Appl. Chem.

2000, 72, 14831492.(9) Forster, R. J. Encyclopedia of Electrochemistry; John Wiley & Sons: New

York, 2003; Vol. 3, pp 160-195.

(10) Bard, A. J.; Faulkner, L. R. Electrochemical Methods, 2nd ed.; John Wiley &

Sons: New York, 2001.

(11) Moreau, M.; Turq, P. Chemical Reactivity in Liquids: Fundamental Aspects;

Kluwer Academic/Plenum Press: New York, 1988; pp 561-606.

(12) Levich, V. G. Physicochemical Hydrodynamics; Prentice Hall: Englewoods

Cliffs, NJ, 1962.

(13) Davies, J. E. Turbulence Phenomena; Academic Press: New York, 1972.

(14) Li, Q. G.; White, H. S. Anal. Chem. 1995, 67, 561569.(15) Grant, K. M.; Hemmert, J. W.; White, H. S. J. Electroanal. Chem. 2001,

500, 9599.(16) Hapiot, P.; Lagrost, C. Chem. Rev. 2008, 108, 22382264.(17) Molina, A.; Gonzalez, J.; Martinez-Ortiz, F.; Compton, R. G. J. Phys. Chem.

C2010, 114, 40934099.

Anal. Chem. 2010, 82, 69336939

10.1021/ac101210r 2010 American Chemical Society 6933Analytical Chemistry, Vol. 82, No. 16, August 15, 2010Published on Web 07/26/2010


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electrodes in various conditions, in chronoamperometry18 and

cyclic voltammetry.19 The mapping of concentration profiles was

alsoperformedintheirvicinityusingamethodalreadydescribed. 18-23

At the same time, we proposed a theoretical model to evaluate

the influence of natural convection on mass transport in still

media.18 The good agreement observed between theory and

experiments demonstrated the validity of this model over a wide

range of experimental conditions.18,22-26The purpose of this study

is then to take the benefit of this model to delineate the

experimental conditions that allow convection-free regimes to beobserved in dynamic and steady-state regimes. These conditions

are better presented in a zone diagram showing the influence of

all the parameters: time scale of the experiment, electrode radius,

and thickness of the convection-free domain. Comparison with

experimental data will also serve to illustrate the relative contribu-

tions of convection and diffusion at electrodes of different sizes

performing under the steady-state regime.

MODEL OF NATURAL CONVECTION

The influence of convection on the electrode responses can

be quantified from deviations of their diffusive currents or from

alteration of their diffusion layers. Under pure diffusional condi-tions, the concentration profile of a species at a disk electrode is

given by integration of Ficks second law:10

c(r,z, t)t

) D(2c(r,z, t)

r2+

2c(r,z, t)

z2+

1r

c(r,z, t)r )

(1)

where r describes the radial position normal to the axis of

symmetry atr) 0 and zdescribes the linear displacement normal

to the plane of the electrode atz) 0. D is the diffusion coefficient.

For a chronoamperometric experiment, the pertinent boundary

conditions are

t< 0; r,zg 0; c(r,z, t) ) c (2)

tg 0; re r0; c(r, 0, t) ) 0 (3)

r,zf ; c(r,z, t) ) c (4)

The current is readily obtained from integration of the concentra-

tion gradients at the electrode surface with

i ) (2nFD0

r0 c(r,z, t)z

rr (5)

In still solutions, natural convection operates perpendicularly

to the electrode surface. It is based on microscopic motions of

the solution except in the very near vicinity of the electrodes,

where it vanishes. Experimentally, the resulting velocity field is

extremely difficult to estimate. Moreover, it is almost impossible

to master mathematically since it depends on many parameters

which are not easy to control (vibrations, temperature gradients,movement of the cell atmosphere, etc.). However, beyond these

difficulties, we demonstrated successfully that, for electroactive

species, the influence of natural convection can be assimilated to

that of an apparent diffusion coefficient depending on the

orthogonal distance zfrom the electrode plane.18 Moreover, since

the electrochemical perturbation affects only the viscous sublayer

adjacent to the electrode, we showed thatDapp could be evaluated

by

Dapp ) D(1 + 1.522( zconv)4

) (6)

where conv is the thickness of the convection-free layer. Thisis the only parameter introduced into the model to account for

the effects of natural convection. It is possible to evaluate its

influence on the electrode response by replacing D byDapp in

eq 1 and solving numerically the new mass transport equation in

association with the same boundary conditions (eqs 2-4).

EXPERIMENTAL SECTION

All the solutions were prepared in purified water (Milli-Q,

Millipore). A 10 mM concentration of K4Fe(CN)6 (Acros) was

dissolved in 1 M KCl (Aldrich), which was used as the

supporting electrolyte. Reciprocally 2 mM FcCH2OH (Acros)

was prepared in 0.1 M KNO3 (Fluka). The diffusion coefficientswere DFe ) (6.0 0.5) 10

-6 cm2 s-1 27 for Fe(CN)64-/

Fe(CN)63- and DFc ) (7.6 0.5) 10

-6 cm2 s-1 for FcCH2OH/

FcCH2OH+.

The working electrodes were Pt disk electrodes of 12.5, 25,

62.5, 125, 250, and 500 m radii. They were obtained from the

cross section of Pt wires (Goodfellow) sealed into soft glass. The

reference electrode was a Ag/AgCl electrode, and the counter

electrode was a platinum coil. A scanning electrochemical micro-

scope (910B CH Instruments) was used to establish the concen-

tration profiles. The amperometric probe was a Pt disk electrode

of submicrometric dimension (500 nm radius). Its fabrication

and the related procedure to map the concentrations have already

been reported.23 For Fe(CN)64-/Fe(CN)63- experiments, the

working electrode was biased at +0.6 V/ref on the oxidation

plateau of Fe(CN)64-. The probe was biased at +0.6 V/ref to

collect Fe(CN)64- or -0.1 V/ref to collect Fe(CN)6

3-. For

FcCH2OH/FcCH2OH+ experiments, the working electrode was

biased at +0.25 V/ref. In this case, the probe was biased at

+0.25 V/ref to collect FcCH2OH and -0.1 V/ref to collect

FcCH2OH+.

The mass transport equation was solved numerically in the

conformal space adapted to the geometry of a microdisk elec-

(18) Amatore, C.; Szunerits, S.; Thouin, L.; Warkocz, J. S. J. Electroanal. Chem.

2001, 500, 6270.(19) Amatore, C.; Pebay, C.; Thouin, L.; Wang, A. F. Electrochem. Commun.

2009, 11, 12691272.(20) Amatore, C.; Pebay, C.; Scialdone, O.; Szunerits, S.; Thouin, L. Chem.sEur.

J. 2001, 7, 29332939.(21) Amatore, C.; Szunerits, S.; Thouin, L.; Warkocz, J. S. Electroanalysis2001,

13, 646652.(22) Amatore, C.; Knobloch, K.; Thouin, L. Electrochem. Commun. 2004, 6, 887

891.(23) Baltes, N.; Thouin, L.; Amatore, C.; Heinze, J. Angew. Chem., Int. Ed. 2004,

43, 14311435.(24) Rudd, N. C.; Cannan, S.; Bitziou, E.; Ciani, L.; Whitworth, A. L.; Unwin,

P. R. Anal. Chem. 2005, 77, 62056217.(25) Amatore, C.; Sella, C.; Thouin, L. J. Electroanal. Chem. 2006, 593, 194

202.(26) Amatore, C.; Knobloch, K.; Thouin, L. J. Electroanal. Chem. 2007, 601,

1728.

(27) Amatore, C.; Szunerits, S.; Thouin, L.; Warkocz, J.-S. Electrochem. Commun.

2000, 2, 353358.

6934 Analytical Chemistry, Vol. 82, No. 16, August 15, 2010


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Figure 2A displays isoconcentration lines c/c) 0.9 calculated

under the influence of natural convection for electrodes of various

radii. They were obtained from combination of eqs 1-6 under

the steady-state regime. To provide a more general representation

of the problem, the spatial coordinates r and z were normalized

byconv. When the electrode radii are small enough (i.e., r0/

conv < 0.5), one observes that the diffusion layers retain their

quasi-hemispherical shapes, like those previously simulated inthe absence of natural convection (Figure 1A). For larger

electrode radii, the concentration profiles become flattened, their

expansion along the z axis being restricted by the boundary at

conv. Therefore, when convection operates, it has two major

effects depending on the ratio r0/conv. On one hand, concentra-

tion gradients (c/z)z)0 become more uniform over the central

area of the electrodes than when natural convection is absent.

On the other hand, the development of the diffusion layers still

operates laterally along the raxis. This can be easily observed

in parts B and C of Figure 2, where the variations of/conv and

z/conv, respectively, are reported as a function of r0/conv. In

particular, one observes that when r0/conv)

4, z has reached

its limitconv whereas is only equal to 0.8conv. Accordingly,

the convolution of these two effects on the development of

diffusion layers leads to a deviation of the currents from eq

10. This alteration may be drastic since the relative error | i -

ihemisph|/|ihemisph| increases almost linearly with r0/conv (Figure2D). In particular, |i- ihemisph|/|ihemisph| 0.65 when conv r0.

A first attempt to summarize all these situations is to establish

a zone diagram describing the boundary condition imposed by

conv on , whether the diffusion is planar or hemispherical.

As previously mentioned, two other parameters have to be

considered: the electrode radius, r0, and the diffusion length,

(Dt)1/2. The transition between planar and quasi-hemispherical

diffusion depends on the ratio r0/(Dt)1/2, whereas the influ-

ence of convection is fixed by r0/conv. Therefore, a diagram

with two coordinates, r0/(Dt)1/2 and r0/conv, allows plotting

the limit, which differentiates the domains where convection

or diffusion prevails independently. This limit is then )

conv.

Figure 2. Steady-state concentration profiles and steady-state

currents simulated at disk electrodes of different radii under the

influence of natural convection. (A) Isoconcentration lines c/c ) 0.9

for disk electrodes of various radii. From left to right, r0/conv ) 0.05,

0.5, 1.0, 1.5, 2.0, 4.2, and 6.0. (B, C) Variations of the diffusion layerthicknesses and z as a function of the electrode radius, with (solid

curves) and without (dashed curves) the influence of natural convec-

tion. (D) Error on steady-state currents due to the influence of natural

convection as a function of r0/conv.

Figure 3. Zone diagrams describing the influence of natural convection

on planar and hemispherical diffusion at disk electrodes. (A) Zone

diagram established from two boundary conditions, ) conv (solid curve)

and r0/(Dt)1/2 ) 4/ (vertical straight line). Note that this diagram isindependent of the model of convection. (B) Zone diagram established

on the basis of the present model of natural convection with four

boundary conditions: | - diff|/ or |i - idiff|/|idiff| ) 0.1 (curve 1), | -

conv|/ or |i - iconv|/|iconv| ) 0.1 (curve 2), | - planar|/ or |i - iplanar|/

|iplanar| ) 0.1 (curve 3), and | - hemisph|/ or |i - ihemisph|/|ihemisph| ) 0.1

(curve 4). The black symbols correspond to the experimental conditions

considered in Figure 4: from left to right, r0 ) 12.5, 25, 62.5, 125, 250,

and 500 m, with conv ) 200-250 m.



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The diagram is reported in Figure 3A, where the zones above

and below this limit correspond to the control of convection and

diffusion, respectively. In the lower zone, the equality between

eqs 7 and 10 discriminates by a vertical line located atr0/(Dt)1/2

) 4/ two other domains where planar diffusion (i.e., r0/(Dt)1/2 > 4/) and quasi-hemispherical diffusion (i.e., r0/

(Dt)1/2 < 4/) dominate.

One must note that this diagram is independent of the model

of natural convection since was calculated without considering

eq 6 and by only assuming ) conv. In this context, the model

enables the transitions between the three regimes of the

diagram to be determined. For this purpose, the model is used

to evaluate and to compare its value to a given reference

thickness, ref, predicted by considering only one specific

regime: (1) ref ) diff for pure diffusion control without any

influence of convection, (2) ref ) conv, (3) ref ) planar for

planar diffusion with planar)

(Dt)1/2

, and (4) ref)

hemisph

for hemispherical diffusion with hemisph ) r0/4. The transition

from one of these specific regimes to the others may then be

estimated by setting a relative threshold on such as

| - ref|

) 0.1 (12)

Note that eq 12 is equivalent to

|i - iref

iref| ) 0.1 (13)

where iref is the reference current obtained from eq 8 with

iref ) (nFADc

ref

(14)

Figure 4. Comparison between simulated (curves) and experimental (symbols) steady-state concentration profiles along the vertical axis of

symmetry at disk electrodes of different radii when the electrode potential is poised on the oxidation plateau of Fe(CN) 64-. Concentration profiles

simulated without (dashed curves) or with (solid curves) natural convection (conv ) 200-250 m). Experimental concentration profiles of the

substrate Fe(CN)64- (0) and product Fe(CN)6

3- (O). t ) 60 s. [Fe(CN)64-] ) 10 mM in 1 M KCl.

6937Analytical Chemistry, Vol. 82, No. 16, August 15, 2010


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The zone diagram built from eq 12 or 13 is reported in Figure

3B. Thus, three curves (curves 2-4) delineate a new domain

corresponding to a mixed regime between the limiting ones

previously identified (Figure 3A). In particular, the transitions from

hemispherical diffusion to convection (i.e., vertical displacement

on the diagram) and from hemispherical diffusion to planar

diffusion (i.e., horizontal displacement) are relatively broad since

they occur approximately over 2 orders of magnitude on the r0/

conv and r0/(Dt)1/2 scales, respectively. Only the transition

from planar diffusion to convection is very sharp. Indeed, assoon as planar conv, the diffusion layer reaches its steady-

state limit and mass transport is fully controlled by convection.

In contrast, when the condition z conv is met for hemi-

spherical-type diffusion, the layer may still expand laterally

along the raxis until reaching its steady-state limit (see Figure

2A-C). This latter condition corresponds to curve 1 in Figure

3B when | - diff|/ ) 0.1 or |i - idiff|/|idiff| ) 0.1. It allows

delineating the upper zone of the diagram where convection

starts to interfere in the mass transport.

A chronoamperometric experiment can be represented on the

diagram by a horizontal straight line described from the right to

the left when the time duration increases. According to the sizeof the electrode, r0, and thickness, conv, the nature of the steady-

state regime reached at longer time may be different. On the

one hand, if log(r0/conv) > 0.95, a sharp transition from planar

diffusion to convection occurs. On the other hand, if log(r0/

conv) < -0.7, a broad transition with a mixed regime from

planar diffusion to quasi-hemispherical diffusion operates

without any influence of natural convection. When log(r0/

(Dt)1/2) < -0.75, a steady-state regime is always observed

though its nature (diffusional or convective) only depends on

the ratio r0/conv.

Under given experimental conditions (i.e., the same position

of the electrode in the cell, temperature, viscosity of the electrolyte,environment, etc.), conv is approximately constant so that the

mass transport regime under steady state depends only on the

electrode dimension. This was checked experimentally by

mapping diffusion layers in the vicinity of electrodes of various

radii. Figure 4 shows the concentration profiles along the vertical

axis of symmetry of the electrodes for both the reactant and

product. conv was evaluated independently by chronoamper-

ometry at a large electrode18 and was found to range from 200

to 250 m. It was thus possible to compare the experimental

data with concentration profiles predicted with or without

natural convection. A very good agreement was observed in

Figure 4 whatever the size of the electrodes between experimentaldata and predictions issued from the model when natural convec-

tion was taken into account. Alterations on the concentration

profiles due to convection were apparent as soon as r0 ) 25 m.

The experimental conditions pertaining to each concentration

profile in Figure 4 are reported as symbols in the zone diagram

of Figure 3B. According to the threshold previously defined with

| - hemisph|/ ) 0.1 or |i - ihemisph|/|ihemisph| ) 0.1, the results

show that a hemispherical diffusion regime was reached forr0) 12.5 and 25 m while a mixed regime was achieved for the

other radii (r0 ) 62.5-500 m).

These experimental data validate the predictions of the present

model, yet they involved only the effect of natural convection along

the axis of symmetry of the electrodes. Conversely, we showed

above (see Figure 2) that this effect is also effective along lateral

directions due to the compensation of transport between verticaland lateral fluxes. In the following, we investigated this latter issue

experimentally by performing 2D imaging. Figure 5 reports the

mapping of concentration profiles established in the steady-state

regime along the zaxis and raxis when r0 ) 25 m. As in Figure

4, the concentration profiles were compared to the predictions

established with and without the influence of convection. Apart

from the good agreement obtained between the data and predic-

tions, these results clearly illustrate the fact that convection may

still alter the diffusion layers even when quasi-hemispherical

diffusion is expected to prevail (Figure 3B). In the present case,

the concentration profiles are distorted over distances zequivalent

to 10 times the electrode radius,r

0. Simultaneously, the relative

Figure 5. Comparison between simulated (curves) and experimental

(symbols) steady-state concentration profiles at a disk electrode ofradius r0 ) 25 m when the electrode potential is poised onto the

oxidation plateau of FeCH2OH. (A) Experimental concentration profiles

of the product FcCH2OH+ along the vertical axis of symmetry (circles).

(B) Experimental concentration profiles of FcCH2OH+ along the raxis

at various vertical distances z: z ) 6 (O), 16 (0), 26 (]), 36 (), 46

(+), and 56 m (). The black area indicates the extent of the

electrode coordinates along the raxis. The concentration profiles are

simulated without (dashed curves) and with (solid curves) consider-

ation of the influence of natural convection (conv ) 200 m).

[FeCH2OH] ) 2 mM in 0.1 KNO3.



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error in the current obtained by the model is |i - ihemisph|/

|ihemisph| ) 0.07.

Finally, variation ofz issued from the mapping of concentra-

tion profiles in Figure 4 is reported in Figure 6 as a function of

the electrode size and then compared to the predicted one. The

diffusion layer thicknesses, , estimated from the experimental

steady-state currents (through eq 8) are also plotted. As observed,

all these data show that the model applies satisfactorily under the

steady-state regime to predict the influence of natural convection

on current responses or concentration profiles.

CONCLUSION

The model elaborated in this work predicts within a very

good accuracy the relative contributions of diffusion and natural

convection to the mass transport at disk electrodes. The

electrochemical behaviors of the electrodes not only are related

to their dimensions but also depend on the time scale of the

experiment and thickness of the convection-free layer (i.e.,

conv). These results stress once more the futili ty of trying

to propose an absolute definition of ultramicroelectrodes

based on the objects themselves. Indeed, the same electrode

may behave as a microelectrode or an ultramicroelectrode,

depending on these parameters. Our model allowed us to

clearly delineate the situations where natural convection

alters both the dynamic and steady-state regimes at disk

electrodes. The properties of ultramicroelectrodes are mainly

achieved when r0/conv < 0.2. This condition has practical

consequences if one needs, for example, to exploit the

characteristics of ultramicroelectrodes to detect or measure

concentrations in restricted volumes, without any alteration

of natural convection on the measurements.

ACKNOWLEDGMENT

This work has been supported in part by the CNRS (Grant

UMR8640), Ecole Normale Superieure, UPMC, and French

Ministry of Research.

Received for review May 7, 2010. Accepted July 9, 2010.

AC101210R

Figure 6. Comparison between simulated (curves) and experimental

(symbols) thicknesses of the diffusion layer at disk electrodes of

various radii: z/conf (dashed lines, O) and /conf (solid curve, 0).

6939Analytical Chemistry, Vol. 82, No. 16, August 15, 2010

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