Difference between Ultramicroelectrodes and Microelectrodes

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Difference between Ultramicroelectrodes and Microelectrodes: Influence of Natural Convection Christian Amatore,* Ce´cile Pebay, Laurent Thouin,* Aifang Wang, and J-S. Warkocz Ecole Normale Supe´rieure, De´partement 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 experimentally by scanning electrochemical microscopy in the vicinity of microelectrodes of various radii. The results validated remarkably the predictions of the model, evidencing in particular the alteration of the diffusional steady state by natural convection. Microelectrodes are versatile tools for the study of electrochemical processes of mechanistic and/or analytical interest. Their advantageous properties stem from their small size. Microelectrodes may be used in highly resistive environments and in very small sample volumes. They enable the detection of very small amounts 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 micrometric dimensions. Electrodes of smaller sizes are referred to as ultramicroelectrodes. Such definitions, based mainly on historical * 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, 177–230. (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, 627–639. (6) Heinze, J. Angew. Chem., Int. Ed. 1993, 32, 1268–1288. (7) Amatore, C. Electrochemistry at ultramicroelectrodes. In Physical Electrochemistry; Rubinstein, I., Ed.; Marcel Dekker: New York, 1995. (8) Stulik, K.; Amatore, C.; Holub, K.; Marecek, V.; Kutner, W. Pure Appl. Chem. 2000, 72, 1483–1492. (9) Forster, R. J. Encyclopedia of Electrochemistry; John Wiley & Sons: New York, 2003; Vol. 3, pp 160-195. 10.1021/ac101210r  2010 American Chemical Society Published on Web 07/26/2010

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 reactions, 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 electroactive 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 thickness of 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 experimental 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 electrodes of micrometric dimensions, we investigated in some previous studies the current responses of micrometric disk (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, 561–569. (15) Grant, K. M.; Hemmert, J. W.; White, H. S. J. Electroanal. Chem. 2001, 500, 95–99. (16) Hapiot, P.; Lagrost, C. Chem. Rev. 2008, 108, 2238–2264. (17) Molina, A.; Gonzalez, J.; Martinez-Ortiz, F.; Compton, R. G. J. Phys. Chem. C 2010, 114, 4093–4099.

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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.18,22-26 The purpose of this study is then to take the benefit of this model to delineate the experimental conditions that allow convection-free regimes to be observed 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 contributions 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 conditions, the concentration profile of a species at a disk electrode is given by integration of Fick’s second law:10

(

∂c(r, z, t) ∂2c(r, z, t) ∂2c(r, z, t) 1 ∂c(r, z, t) )D + + 2 2 ∂t r ∂r ∂r ∂z

)

i ) (2πnFD

∫

r0

0

∂c(r, z, t) r ∂r ∂z

(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 z from the electrode plane.18 Moreover, since the electrochemical perturbation affects only the viscous sublayer adjacent to the electrode, we showed that Dapp could be evaluated by

(

Dapp ) D 1 + 1.522

( )) z δconv

4

(6)

where δconv is the thickness of the convection-free layer. This is 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 by Dapp in eq 1 and solving numerically the new mass transport equation in association with the same boundary conditions (eqs 2-4).

(1) where r describes the radial position normal to the axis of symmetry at r ) 0 and z describes the linear displacement normal to the plane of the electrode at z ) 0. D is the diffusion coefficient. For a chronoamperometric experiment, the pertinent boundary conditions are t < 0; t g 0;

r, z g 0;

c(r, z, t) ) c°

(2)

r e r0 ;

c(r, 0, t) ) 0

(3)

r, z f ∞;

c(r, z, t) ) c°

(4)

The current is readily obtained from integration of the concentration gradients at the electrode surface with (18) Amatore, C.; Szunerits, S.; Thouin, L.; Warkocz, J. S. J. Electroanal. Chem. 2001, 500, 62–70. (19) Amatore, C.; Pebay, C.; Thouin, L.; Wang, A. F. Electrochem. Commun. 2009, 11, 1269–1272. (20) Amatore, C.; Pebay, C.; Scialdone, O.; Szunerits, S.; Thouin, L. Chem.sEur. J. 2001, 7, 2933–2939. (21) Amatore, C.; Szunerits, S.; Thouin, L.; Warkocz, J. S. Electroanalysis 2001, 13, 646–652. (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, 1431–1435. (24) Rudd, N. C.; Cannan, S.; Bitziou, E.; Ciani, L.; Whitworth, A. L.; Unwin, P. R. Anal. Chem. 2005, 77, 6205–6217. (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, 17–28.

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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 coefficients were 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 µm 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 microscope (910B CH Instruments) was used to establish the concentration 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)63-. 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(27) Amatore, C.; Szunerits, S.; Thouin, L.; Warkocz, J.-S. Electrochem. Commun. 2000, 2, 353–358.

trode28 by a finite element using Comsol Multiphysics software. The thickness of the convection-free layer, δconv, was determined before each experiment by chronoamperometry as described previously.18 RESULTS AND DISCUSSION An important property of disk ultramicroelectrodes is that their diffusion layers develop with time until reaching a steady-state limit imposed by hemispherical-type diffusion. However, natural convection may interfere with the mass transport as soon as the thickness of the expanding diffusion layer becomes comparable to δconv. In such a case, a steady-state regime is still achieved but is then controlled by the respective contributions of diffusion and natural convection. Depending on the electrode radius, r0, and the thickness of the convection-free layer, δconv, two situations may be encountered, whether the diffusion at the electrode surface is planar or not. Indeed, at short time scales, the thickness of the diffusion layer is considerably smaller than the electrode radius. The electrodes then behave as electrodes of infinite dimensions, and planar diffusion operates. In that particular situation, the Cottrell equation applies with

iplanar ) (

nFADc° √πDt

(7)

for electrodes of surface area A. Using the Nernst formulation, eq 7 is similar to that given in hydrodynamic electrochemical methods:10

i)(

nFADc° δ

(8)

where δ ) (πDt)1/2. Therefore, natural convection interferes significantly with the mass transport as soon as δconv ≈ (πDt)1/2. Whenever this condition is not met, the diffusion layer may develop, possibly reaching a hemispherical behavior until being eventually limited by δconv. However, since diffusion and natural convection occur together in steady-state regimes, their contributions in mass transport remain difficult to comprehend. To solve this problem, one needs first to investigate the concentration profiles established under the pure diffusional steady-state regime, i.e., without any influence of natural convection. In such a case, solution of eq 1 shows that most of the concentration gradients operate over a distance comparable to the electrode radius (Figure 1A). Concentration along the z axis (Figure 1B) varies according to

()

2 c z ) arctan c° π r0

(9)

The diffusion layer presents a hemispherical shape, and the steady-state current is given by: ihemisph ) (4nFr0Dc° (28) Amatore, C.; Fosset, B. J. Electroanal. Chem. 1992, 328, 21.

(10)

Figure 1. Steady-state concentration profile simulated at a disk electrode without considering the influence of natural convection: (A) 2D concentration profile with isoconcentration lines ranging from c/c° ) 0.1 to c/c° ) 0.9. (B) Concentration profile along the vertical axis of symmetry.

Using the Nernst formulation again, comparison between eqs 8 and 10 leads to an equivalent diffusion layer thickness, δ ) πr0/ 4. This thickness differs slightly from δz, which may be obtained by extrapolating the concentration gradient at z ) 0, r ) 0 along the z axis. Indeed, when z f 0, the concentration profile at r ) 0 tends to (Figure 1B) 2 z c ) c° π r0

(11)

which gives δz ) πr0/2. The difference in the δ and δz values results from the nonradial distribution of diffusion fields at disk electrodes. Concentration gradients are higher at the electrode edges than at the center (Figure 1A). Since δ is evaluated from the integration of concentration gradients over the whole electrode surface (eq 5), it is necessarily smaller than δz. In the following, the variation of these two parameters will provide an accurate estimation of the influence of natural convection, either from the current (i.e., through δ) or from the alteration of concentration profiles in the z direction (i.e., through δz), where natural convection prevails. Analytical Chemistry, Vol. 82, No. 16, August 15, 2010

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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/δconv ) 0.05, 0.5, 1.0, 1.5, 2.0, 4.2, and 6.0. (B, C) Variations of the diffusion layer thicknesses δ and δz as a function of the electrode radius, with (solid curves) and without (dashed curves) the influence of natural convection. (D) Error on steady-state currents due to the influence of natural convection as a function of r0/δconv.

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 by δconv. When the electrode radii are small enough (i.e., r0/ δconv < 0.5), one observes that the diffusion layers retain their quasi-hemispherical shapes, like those previously simulated in the 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 δconv. Therefore, when convection operates, it has two major effects depending on the ratio r0/δconv. On one hand, concentration 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 r axis. This can be easily observed in parts B and C of Figure 2, where the variations of δ/δconv and δz/δconv, respectively, are reported as a function of r0/δconv. In particular, one observes that when r0/δconv ) 4, δz has reached 6936

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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, δ ) δconv (solid curve) and r0/(πDt)1/2 ) 4/π (vertical straight line). Note that this diagram is independent of the model of convection. (B) Zone diagram established on the basis of the present model of natural convection with four boundary conditions: |δ - δdiff|/δ or |i - idiff|/|idiff| ) 0.1 (curve 1), |δ δconv|/δ or |i - iconv|/|iconv| ) 0.1 (curve 2), |δ - δplanar|/δ or |i - iplanar|/ |iplanar| ) 0.1 (curve 3), and |δ - δhemisph|/δ 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 µm, with δconv ) 200-250 µm.

its limit δconv whereas δ is only equal to 0.8δconv. 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/δconv (Figure 2D). In particular, |i - ihemisph|/|ihemisph| ≈ 0.65 when δconv ≈ r0. A first attempt to summarize all these situations is to establish a zone diagram describing the boundary condition imposed by δconv on δ, 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, (πDt)1/2. The transition between planar and quasi-hemispherical diffusion depends on the ratio r0/(πDt)1/2, whereas the influence of convection is fixed by r0/δconv. Therefore, a diagram with two coordinates, r0/(πDt)1/2 and r0/δconv, allows plotting the limit, which differentiates the domains where convection or diffusion prevails independently. This limit is then δ ) δconv.

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 (δconv ) 200-250 µm). Experimental concentration profiles of the substrate Fe(CN)64- (0) and product Fe(CN)63- (O). t ) 60 s. [Fe(CN)64-] ) 10 mM in 1 M KCl.

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 at r0/(πDt)1/2 ) 4/π two other domains where planar diffusion (i.e., r0/ (πDt)1/2 > 4/π) and quasi-hemispherical diffusion (i.e., r0/ (πDt)1/2 < 4/π) dominate. One must note that this diagram is independent of the model of natural convection since δ was calculated without considering eq 6 and by only assuming δ ) δconv. 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 δ and to compare its value to a given reference thickness, δref, predicted by considering only one specific regime: (1) δref ) δdiff for pure diffusion control without any influence of convection, (2) δref ) δconv, (3) δref ) δplanar for planar diffusion with δplanar ) (πDt)1/2, and (4) δref ) δhemisph

for hemispherical diffusion with δhemisph ) πr0/4. The transition from one of these specific regimes to the others may then be estimated by setting a relative threshold on δ such as |δ - δref | ) 0.1 δ

(12)

Note that eq 12 is equivalent to

|

|

i - iref ) 0.1 iref

(13)

where iref is the reference current obtained from eq 8 with iref ) (

nFADc° δref

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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/ δconv and r0/(πDt)1/2 scales, respectively. Only the transition from planar diffusion to convection is very sharp. Indeed, as soon as δplanar ≈ δconv, the diffusion layer reaches its steadystate limit and mass transport is fully controlled by convection. In contrast, when the condition δz ≈ δconv is met for hemispherical-type diffusion, the layer may still expand laterally along the r axis until reaching its steady-state limit (see Figure 2A-C). This latter condition corresponds to curve 1 in Figure 3B when |δ - δdiff|/δ ) 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 size of the electrode, r0, and thickness, δconv, the nature of the steadystate regime reached at longer time may be different. On the one hand, if log(r0/δconv) > 0.95, a sharp transition from planar diffusion to convection occurs. On the other hand, if log(r0/ δconv) < -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/ (πDt)1/2) < -0.75, a steady-state regime is always observed though its nature (diffusional or convective) only depends on the ratio r0/δconv. Under given experimental conditions (i.e., the same position of the electrode in the cell, temperature, viscosity of the electrolyte, environment, etc.), δconv 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. δconv was evaluated independently by chronoamperometry at a large electrode18 and was found to range from 200 to 250 µm. 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 experimental data and predictions issued from the model when natural convection was taken into account. Alterations on the concentration profiles due to convection were apparent as soon as r0 ) 25 µm. 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 |δ - δhemisph|/δ ) 0.1 or |i - ihemisph|/|ihemisph| ) 0.1, the results show that a hemispherical diffusion regime was reached for r0 ) 12.5 and 25 µm while a mixed regime was achieved for the other radii (r0 ) 62.5-500 µm). These experimental data validate the predictions of the present model, yet they involved only the effect of natural convection along 6938

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Figure 5. Comparison between simulated (curves) and experimental (symbols) steady-state concentration profiles at a disk electrode of radius r0 ) 25 µm 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 r axis at various vertical distances z: z ) 6 (O), 16 (0), 26 (]), 36 (×), 46 (+), and 56 µm (∆). The black area indicates the extent of the electrode coordinates along the r axis. The concentration profiles are simulated without (dashed curves) and with (solid curves) consideration of the influence of natural convection (δconv ) 200 µm). [FeCH2OH] ) 2 mM in 0.1 KNO3.

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 vertical and 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 z axis and r axis when r0 ) 25 µm. 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 predictions, 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 z equivalent to 10 times the electrode radius, r0. Simultaneously, the relative

CONCLUSION

Figure 6. Comparison between simulated (curves) and experimental (symbols) thicknesses of the diffusion layer at disk electrodes of various radii: δz/δconf (dashed lines, O) and δ/δconf (solid curve, 0).

error in the current obtained by the model is |i - ihemisph|/ |ihemisph| ) 0.07. Finally, variation of δz issued from the mapping of concentration 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, δ, 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.

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., δconv). These results stress once more the futility 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/δconv < 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

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