Channel Microband Chronoamperometry: From Transient to Steady

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Channel Microband Chronoamperometry: From Transient to Steady-State Regimes Christian Amatore,* Celia Lemmer, Catherine Sella, and Laurent Thouin* Departement de Chimie, Ecole Normale Superieure, UMR CNRS-ENS-UPMC 8640 “Pasteur”, 24 rue Lhomond, F-75231 Paris Cedex 05, France ABSTRACT: Chronoamperometric transient regimes were investigated at a single channel microband electrode during chronoamperometric measurements in a microchannel continuously filled by a redox solution. Simulations were performed by spanning a wide range of conditions according to the geometry of microdevices, flow velocity, and time scale of experiments. Boundary conditions were identified and zone diagrams were established showing the predominance areas of transient and steady-state regimes. The predictions were compared to chronoamperometric experiments performed with microdevices of various geometries. The good agreement observed between data validated the predictions.

T

he use of microfluidics in the area of analytical chemistry has received a great deal of interest. Numerous applications have been reported in biological and environmental analysis based on the development of microfluidic analysis systems.1 Since the standard photolithographic techniques allow the miniaturization and the implementation of electrodes in microdevices, some of these applications include electrochemistry as the monitoring technique.24 Potentialities, viability, and performances of electrochemical detections in flowing systems have been already demonstrated.510 For example, a facile experimental method is the determination of mean flow rate velocities inside microchannels, based on electrochemical measurements performed between two adjacent electrodes.11 Electrochemistry offers many advantages in comparison to other analytical methods. The electrode response is proportional to the concentration and not to the quantity of samples, which allows measurements in very low volumes. Measurements of electroactive species can be performed without labeling the species while keeping low sensitivities. In addition, channel microelectrodes exhibit fast detections and high performances because of the enhancement of mass transport by convection.7,12 In this context, we demonstrated that several convective-diffusive regimes need to be recognized in laminar flow conditions, either at single electrode,13 double electrode assembly,14,15 or electrode arrays16,17 under steady-state regime. However, transient measurements at these assemblies may be also attractive for several reasons. A transient technique is always useful for controlling the amount of species detected (even adsorbed or deposited) or monitoring the dynamics of a physicochemical process at the interface. It is also commonly used as electrochemical detection in a flowing stream because of its time dynamic characteristics, such as the response time and sampling frequency (number of measurements in a flow analysis system in a given period of time r 2011 American Chemical Society

without any significant interference by the preceding samples). Yet, the treatment and the resolution of the diffusion-convection problem remains challenging since it involves many parameters related to the geometry of the devices, hydrodynamic flow conditions, and time scale of the experiment.1826 Apart from efforts made to improve the computational efficiency,2628 there is still a lack of a general overview describing and delineating the transient regimes at channel microband electrodes as reported for microband electrodes in infinite still solutions.2934 In particular, the conditions under which steady-state regimes are achieved have not been yet established for all the regimes previously identified.13 The purpose of the present work was then to investigate the amperometric responses of a channel microband electrode over a wide range of experimental conditions. The electrode is considered to be embedded in a microchannel continuously filled by a solution of electroactive species. The currents were simulated for different geometries of the devices and flow velocities to establish the relevance of each parameter on the electrode response. The validity of the predictions was tested based on amperometric measurements performed at channel microband electrodes of various dimensions.

’ PRINCIPLE The formulation of the problem is identical to that described as in our previous work.13 The channel is filled continuously with a solution of an electroactive species at the concentration c0. The electron transfer is supposed to be fast, and the overall electrochemical reaction is fully controlled by the mass transport. The Received: February 22, 2011 Accepted: April 16, 2011 Published: April 17, 2011 4170

dx.doi.org/10.1021/ac2004604 | Anal. Chem. 2011, 83, 4170–4177

Analytical Chemistry microchannel is a wide rectangular channel of width L and height h with L . h. The width and length of the microband

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are w and L, respectively. Since L is much larger than w, the formulation of problem reduces to a two-dimensional problem as reported in Figure 1A. Transient mass transport is given by the classical convection-diffusion equation: ! Dc D2 c D2 c Dc ¼D ð1Þ þ 2 ux ðyÞ 2 Dt Dx Dy Dx where D and c are the diffusion coefficient and concentration of the redox species, respectively. Diffusion results from contributions in the two directions (x-axis and y-axis) whereas convection operates only along the x-axis. ux(y) is the flow velocity whose profile is assumed to be parabolic across the microchannel section: y y 1 ð2Þ ux ðyÞ ¼ 6uav h h

Figure 1. (A) 2D scheme of the microfluidic device used for numerical simulations. The flow velocity profile is parabolic across the microchannel. W is the electrode width and Pe is the Peclet number of the average flow velocity in the X-direction. (B) Concentration profiles simulated for W = 2, Pe = 20, and τ = 0.12 (zone II). The white streamline linking the microchannel entrance to the downstream edge of the microband electrode is used to evaluate the average thickness H of the channel area probed by the electrode. In this case H = Hs = 0.36.

where uav is the average flow velocity over the section of the microchannel. To simplify the resolution of the problem, dimensionless parameters were introduced. Geometrical parameters are normalized by the microchannel height h with X = x/h, Y = y/h and W = w/h. Concentrations are normalized with respect to the bulk concentration c0 with C = c/c0. The average flow velocity is introduced through the definition of a Peclet number: Pe ¼

uav h D

ð3Þ

Figure 2. Simulated concentration profiles for several conditions (W,Pe). (A) W = 2, Pe = 0. (B) zone I, W = 5, Pe = 2. (C) zone II, W = 2, Pe = 20. (D) zone III, W = 2, Pe = 100. (E) zone IVa, W = 2, Pe = 1. (F) zone IVb, W = 2, Pe = 5. 4171

dx.doi.org/10.1021/ac2004604 |Anal. Chem. 2011, 83, 4170–4177

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The time scale is given by Dt ð4Þ h2 The simulations were performed by solving numerically eq 1 in association with the following boundary conditions: τ¼

τ < 0, τ g 0, τ g 0,

C¼1

ð5Þ

C ¼ 0 at the electrode surface

ð6Þ

C ¼ 1 at the microchannel entrance

ð7Þ

The dimensionless current Ψ is calculated by Z W iðtÞ DCðτÞ ¼ dX ΨðτÞ ¼ nFDLc° DY Y ¼ 0 0

ð8Þ

’ EXPERIMENTAL SECTION Millimolar aqueous solutions of ferrocene methanol were prepared with 0.1 M potassium chloride as the supporting electrolyte. The solutions were degassed with argon before experiments to keep the ratio of ferricinium methanol to ferrocene methanol very low. Under such conditions, the diffusion coefficient of ferrocene methanol was D = 7.6 106 cm2 s1.13 The fabrication of microdevices was performed as already reported.13,15,17 It is based on two parts which are assembled together. The first one is a polydimethylsiloxane (PDMS) block that comprises a linear channel and reservoir elements which are engraved on its surface. The second part is a glass substrate on which a platinum microband electrodes (15 < w < 300 μm) are patterned by soft lithography and liftoff procedure. Since the linear microchannel was set perpendicularly to the microbands, the effective microband lengths were delimited by the microchannel width (L ∼ 500 μm), and the volume of solution above the microband was restricted by the microchannel height (h ∼ 20 μm). The flow in the microchannel was pressure driven by a syringe pump (Harvard Apparatus, type 11 Pico Plus). The values of average flow velocities inside the microchannels (0.06 < uav < 0.4 cm s1) were systematically monitored in situ by direct measurements following a procedure previously described.11,14 All electrochemical experiments were performed at room temperature (1821 °C) using a homemade multipotentiostat. The counter electrode (CE) consisted of one large platinum band electrode (w ∼ 500 μm) located downstream from the working electrode (WE). The pseudoreference electrode (REF) was a platinum located upstream from the working electrode with gap distances ranging from 500 to 2000 μm depending on the microdevices used. In such configurations, steady-state voltammetry was used to check that the ohmic drop remained negligible for supporting electrolyte concentrations down to 0.05 M. The working electrode was biased initially at 0 V/REF during sufficient time duration to ensure that a steady-state current was achieved. At t g 0, the working electrode was then biased at 0.4 V/REF on the oxidation plateau of the ferrocene methanol. The amperometric responses were monitored during all the processes. During all sets of measurements, steady-state voltammetry was used to check that the oxygen permeability of PDMS did not impact the initial concentration of ferrocene methanol in the solution and the subsequent ratio of ferricinium methanol to ferrocene methanol.35,36

Figure 3. (A) Variation of log Ψs/Pe as a function of log W/Pe . (B) Variation of log τs as a function of log W/Pe; the straight line corresponds to τs = 0.23 (W/Pe)2/3 (eq 10). (C) Variation of log τs as a function of log Pe; the straight line corresponds to τs = 4/Pe2 (eq 18). In (AC), data were simulated in zone I (0), zone II (Δ), zone III (1), zone IVa (b), and zone IVb (O). In (BC), W = 0.5 (a), 1 (b), 2 (c), 5 (d), 20 (e), and 100 (f).

The diffusion-convection equation was solved numerically using the COMSOL Multiphysics 3.5a software controlled by a MATLAB R2010b interface (Mathworks). All computations were performed on a PC equipped by a Intel Core 2 Duo E8400 processor (3 Ghz) with 3 GB RAM. The purpose of this work was not to optimize the computation time. However, given the total number of mesh points in the grid (up to 200 000) and solver used, computations usually took several seconds to simulate a chronoamperogram with adequate precision within millisecond time scales.

’ RESULTS AND DISCUSSION From Transient to Steady-State Regimes. Figure 2 shows some concentration profiles simulated as a function of time for 4172



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Figure 4. Zone diagrams (Pe,τ) describing the different mass transport regimes taking place at channel microband electrodes for several W: planar diffusion (P), hemicylindrical diffusion (HC), planar lateral diffusion (PL), and steady-state diffusion convection regime (S). The curves correspond to the following conditions: |1 Ψ /Ψs| = 0.05 (solid line), |1 Ψ /Ψdiff| = 0.05 (dotted lines), and |1 Ψ /ΨPe=0| = 0.05 (dashed line). Corresponding transitions between zones I, II, III, IVa, and IVb are indicated on the Pe axis (on the right side).

several conditions (W,Pe). Without any flow, mass transport is fully controlled by diffusion (Figure 2A). With time, diffusion may be successively planar, hemicylindrical, and planar lateral according to the geometry W. In such a case, no steady-state regime is reached even at infinite time. In contrast, three steadystate regimes may be achieved under flow conditions. We demonstrated previously that their characteristics depend on the ratio W/Pe .13 Indeed, W and Pe have opposite-mirror effects on controlling the physical nature of these steady-state regimes. When W/Pe > 1.2 (zone I in Figure 2B), a thin layer regime is set. In this case, the thickness of the diffusion-convection layer is the highest and limited by the channel height. Conversely, when W/Pe < 0.04 (zone III in Figure 2D), the thickness of the layer is mainly controlled by convection, and a Levich regime is quickly established. When 0.04 < W/Pe < 1.2 (zone II in Figure 2C), the behavior is intermediate between the two previous situations. Two other domains are also shown, zones IVa and IVb, that may be distinguished at low Pe when diffusion is able to proceed against the flow.13 This process is characterized by the expansion of concentration profiles at the upstream edge of the microband (zone IVa in Figure 2E and zone IVb in Figure 2F). In every one of the zones (from zones I to IV), the current Ψ reaches a steady-state limit Ψs (Figure 3A) whereas the concentration profiles still expand far from the downstream edge of the microband along the microchannel. Accordingly, the time τs required for the current to attain its steady state value Ψs increases with the ratio W/Pe. It can be evaluated numerically

from simulations by setting a relative threshold on Ψ such as Ψ ð9Þ 1 ¼ 0:05 Ψs Figure 3B shows the variation of τs as a function of W/Pe for conditions fitting each domain (zones I to IV). As expected, a limiting behavior is observed for conditions W/Pe < 0.1 when convection prevails over diffusion (zone III and part of zone II). A simple relationship is obtained with 2=3 W ð10Þ τs ¼ 0:23 Pe The proportionality of τs versus (W/Pe)2/3 can be easily established by considering the thickness Hs of the diffusionconvection layer at steady state (Figure 1B) and by introducing the Levêque approximation of the flow velocity profile at small Y values: PeX ðY Þ ¼ 6PeY

ð11Þ

Indeed, τs corresponds either to the time required for a species to diffuse vertically from the distance Hs to the electrode surface τ s ¼ Hs 2

ð12Þ

and to the time needed for a same species to be transported by convection over the electrode width W at the constant vertical 4173



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height Hs τs ¼

W PeX ðHs Þ

ð13Þ

Within the framework of the Levêque approximation, combination of eqs 1113 gives 2=3 2=3 W W τs ¼ ¼ 0:303 ð14Þ 6Pe Pe Simulations reported in Figure 3B show that the proportionality between τs and (W/Pe)2/3 applies not only to the Levich regime (zone III). It is also verified to apply to a broader range of conditions (i.e., W/Pe < 0.1) that expand into zone II when convection still prevails on diffusion. The difference between the two constants 0.23 (eq 10) and 0.303 (eq 14) comes, on the one hand, from the relative threshold (eq 9) used to evaluate τs and, on the other hand, from the approximation introduced in eq 12 to estimate the diffusion length. Similar relations were proposed in the literature to predict the time required for the steady-state regimes to be achieved.19,23 However, they were estimated by considering other definitions or criteria of τs, and/or smaller ranges of conditions (W,Pe). When W/Pe > 0.1, eq 10 is no longer valid, but τs still depends on W/Pe (Figure 3B). τs varies monotonously with W/Pe from zones II to I, except for data in zone IV that clearly deviate from the common variation. Another limiting behavior is then expected at low flow velocities, in particular in zone IVa where lateral diffusion prevails (Figure 2E). In this case, the current due to planar lateral diffusion on both edges of the electrodes tends to37 2 Ψ ¼ pffiffiffiffiffiffi πτ

ð15Þ

The steady-state limit of the current is close to the one given in the thin-layer regime (zone I) by13 Ψs ¼ Pe

Figure 5. Variation of the thickness H of the diffusion-convection layer as a function of time. (A) W/Pe = 0.02 (a); 0.1 (b); 0.2 (c); 0.8 (d); 1(e); 2 (f); 4 (g); 20 (h); 200 (i); and 2000 (j); the dotted line corresponds to the H = 4(πτ)1/2(eq 24). (B) Values of H as a function of τ and W/Pe; 10 isolines are plotted from H = 0.1 to 1; the solid line corresponds to the variation τs in zones I, II, and III; transitions between these zones are indicated on the W/Pe axis.

or hemicylindrical at longer times so that31,32,38

ð16Þ

Ψ¼

Combination of eqs 15 and 16 thus provides an approximate evaluation of τs as 4 ð17Þ πPe2 Indeed, the proportionality between τs and Pe-2 is verified in Figure 3C for data simulated with Pe < 1. The fitting of the data gives τs ¼

4 ð18Þ Pe2 The difference between eqs 17 and 18 results mainly from the relative threshold set on Ψ (eq 9) for τs evaluation and from the crude hypothesis made by considering eq 15 as the transient current variation. Current Responses. Other specific transition times can be evaluated based on the transient regimes previously identified. Indeed, according to the time scale of the experiment and flow velocity, mass transport at the electrode can be controlled mainly by diffusion. As already shown in (Figure 2A), at very short times diffusion can be planar leading to38 τs ¼

W Ψ ¼ pffiffiffiffiffiffi πτ

ð19Þ

2π 64τ ln W2

ð20Þ

In addition, when diffusion transport is limited by the microchannel height, lateral planar diffusion may operate. In this later case, the current is provided by eq 15. The transition time τdiff from one of these specific regimes to another, can be evaluated by setting a similar threshold on Ψ such as Ψ ð21Þ ¼ 0:05 1 Ψdiff where Ψdiff is given either by eqs 15, 19, or 20. The zone diagrams reported in Figure 4 show together the variations of τdiff and τs as a function of Pe. On one side, variations of τdiff delineate three zones corresponding respectively to planar (P), hemicylindrical (HC), and planar lateral diffusion (PL). On the other side, variation of τs delimits in the diagrams the domain (S) where steady-state regimes are achieved under mixed control by diffusion and convection. The other domains (white areas in Figure 4) are transitions taking place between the four previous domains. To implement the zone diagram and to evaluate accurately the conditions under which diffusion dominates in 4174



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Figure 6. Experimental and simulated current responses versus time for several channel microband electrodes. (AD) uav = 0.144 cm s1 (Pe = 38), w = 16 (W = 0.8) (a), 57 (W = 2.85) (b), 97 (W = 4.85) (c), 197 (W = 9.85) (d), and 297 μm (W = 14.85) (e). (E) w = 17 μm (W = 0.85). (F) w = 97 μm (W = 4.85). (EF) uav = 0.066 (Pe = 17.5) (a0 ), 0.144 (Pe = 38) (b0 ) and 0.387 cm s1 (Pe = 102) (c0 ). In (A, C, D, E, F) 0.53 mM ferrocene methanol/ 0.1 M KCl. In (B) 0.1 M KCl. In (D) the dashed line corresponds to i/FLDc0 = w/(πDt)1/2. In (AF) h = 20 μm and L = 500 μm.

the transition areas, another threshold on Ψ was introduced: Ψ ð22Þ ¼ 0:05 1 ΨPe ¼ 0 to define the limits where ΨPe=0 is the current simulated under pure diffusion control, i.e., at Pe = 0. Accordingly, the zone diagrams illustrate the three diffusion regimes (P, HC, and PL) that may operate at low Pe. As expected, planar diffusion prevails at very short times. In particular, when Pe < 1 and W < 20, the fitting of τdiff variation affords τdiff ¼ 0:001W 2 where τdiff is proportional to W2 and independent of Pe.

ð23Þ

In comparison to planar diffusion, hemicylindrical diffusion and planar lateral diffusion take place respectively at intermediate and longer times according to the value of W. It must be underlined that all properties of microchannel electrodes previously characterized under the steady-state regime apply in the area delineated by τs. Therefore, the subdomains related to zones I to IV in the steady-state regime were also reported in each zone diagram of Figure 4. Thickness of Diffusion-Convection Layer. Simulations allowed also the determination of the thickness of the layer H probed by the electrode in the flow presented at the entrance of the microchannel.39 The definition of H used in this work is reported in Figure 1B. Figure 5A shows the corresponding variations of H as a function of time for several W/Pe ratios. Two distinct behaviors are observed according to the thickness 4175



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Hs of the steady-state diffusion-convection layer. When Hs < 0.5, H increases monotonically with time and tends to Hs. When Hs > 0.5, variation of H as function of time displays a peak shape. This later characteristic is related to the parabolic nature of the flow profile across the microchannel section. Indeed, when H becomes higher than 0.5, the upper part of the diffusion-convection layer is submitted to a decreasing flow velocity. As a result, a relaxation phenomenon occurs due locally to the inversion of velocity gradients along the Y-direction. After reaching a maximum value, H decreases before reaching Hs. For higher W/Pe ratios, evolution of H is quickly limited by the microchannel height leading to a decrease of the peak amplitude. When diffusion dominates, a boundary condition is reached at high W/Pe. Increase of H with time is no longer dependent on W and Pe. Simulated data follow the equation pffiffiffiffiffiffi H ¼ 4 πτ

ð24Þ

Figure 5B displays isolines of H as a function of time and W/Pe. The upper part of the diagram corresponds to steady-state values Hs while the lower part describes transient data. The slight curvatures of the isolines in the upper part of the diagram observed for H > 0.5 are related to the peak-shaped variation of H (Figure 5A). In comparison, the transition time τs previously evaluated from current in zones I, II, and III (Figure 3B) was also reported in this diagram. It can be observed that this boundary matches perfectly the transition between transient and steadystate values of H when H < 0.5. For higher values, the situation becomes more complex since in these conditions steady-state regimes may be reached for H and not for Ψ. This is the case when lateral planar diffusion operates in zones IVa and IVb. Experimental Results. Several chronoamperometric measurements were performed at a constant flow velocity (i.e., Pe = 38) for several geometries of microdevices W ranging from 0.85 to 14.85 (Figure 6AD). These conditions encompassed essentially zones III and II. Transitions were thus predicted essentially from planar diffusion to steady-state regimes fully controlled by diffusion and convection. In addition, to eliminate the capacitive contribution of the currents at short times and keep only their faradaic component, currents were measured in the presence (Figure 6A) and in the absence (Figure 6B) of redox species in the flowing solution. We checked that without any redox species, the experimental currents i0 did not depend on the flux velocity (i.e., mass transport), but only on the electrode width w (i.e., surface area) at a fixed concentration of supporting electrolyte ([KCl] = 0.1 M, data not shown). Under these conditions, values of the cell constant were estimated between 1 to 20 ms for w ranging from 17 to 297 μm. Figure 6A, C, E, and F show the comparisons between simulated and experimental data, with and without subtraction of the current i0.40 These results show a very good agreement between data when the capacitive and residual contributions of the current were eliminated, whatever W or Pe. One must underline that this procedure is actually approximate since Faradaic and capacitive contributions are physicochemical processes which are intimately convoluted. Nevertheless, it provided good results within the accuracy of the experimental measurements. The transitions from planar diffusion to steady-state regimes is better illustrated by plotting in Figure 6D the currents as a function of w/(πDt)1/2(see the corresponding dimensionless equation eq 19). At short times, all the currents are superimposed and follow the same trend (eq 19) over different time

windows. The larger W, the larger the time window is. Similar results were obtained for other flow velocities ranging from Pe = 17.5 to 102, for two geometries W = 0.85 (Figure 6E) and 4.85 (Figure 6F).

’ CONCLUSION In the present paper, we characterized the different regimes that may be achieved during chronoamperometric measurements at a single channel microband electrode. These results extend and generalize those reported previously under steadystate regime. Indeed, we established some zone diagrams describing respectively the predominance areas of steady-state and transient regimes. A wide range of conditions (W,Pe,τ) was investigated. Boundary conditions were identified, and some simple equations were proposed to delineate specific domains and related properties of the mass transport. On the one hand, steady-state regimes are both controlled by diffusion and convection. Their relative contributions depend only on the ratio W/Pe. On the other side, transient regimes that may involve a pure diffusion contribution are functions of W, Pe, and τ. According to the flow velocity, three diffusion modes and their transition regimes may be observed at small electrodes before a steady state is reached. Some of these predictions were compared to chronoamperometric experiments performed with microdevices of various geometries. The good agreement observed with the data demonstrated the validity and the accuracy of these predictions. ’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected] (C.A.), laurent.thouin@ens. fr (L.T.).

’ ACKNOWLEDGMENT This work has been supported in part by the CNRS (UMR8640), Ecole Normale Superieure, UPMC and French Ministry of Research. ’ REFERENCES (1) Arora, A.; Simone, G.; Salieb-Beugelaar, G. B.; Kim, J. T.; Manz, A. Anal. Chem. 2010, 82, 4830–4847. (2) Yi, C. Q.; Zhang, Q.; Li, C. W.; Yang, J.; Zhao, J. L.; Yang, M. S. Anal. Bioanal. Chem. 2006, 384, 1259–1268. (3) Xu, X.; Zhang, S.; Chen, H.; Kong, J. Talanta 2009, 80, 8–18. (4) Trojanowicz, M. Anal. Chim. Acta 2009, 653, 36–58. (5) Weber, S. G.; Purdy, W. C. Ind. Eng. Chem. Prod. R&D 1981, 20, 593–598. (6) Johnson, D. C.; Weber, S. G.; Bond, A. M.; Wightman, R. M.; Shoup, R. E.; Krull, I. S. Anal. Chim. Acta 1986, 180, 187–250. (7) Cooper, J. A.; Compton, R. G. Electroanalysis 1998, 10, 141–155. (8) Trojanowicz, M.; Szewczynska, M.; Wcislo, M. Electroanalysis 2003, 15, 347–365. (9) Perez-Olmos, R.; Soto, J. C.; Zarate, N.; Araujo, A. N.; Montenegro, M. C. B. S. M. Anal. Chim. Acta 2005, 554, 1–16. (10) Economou, A. Anal. Chim. Acta 2010, 683, 38–51. (11) Amatore, C.; Chen, Y.; Sella, C.; Thouin, L. Houille BlancheRevue Internationale De L Eau 2006, 60–64. (12) Compton, R. G.; Unwin, P. R. J. Electroanal. Chem. 1986, 205, 1–20. (13) Amatore, C.; Da Mota, N.; Sella, C.; Thouin, L. Anal. Chem. 2007, 79, 8502–8510. 4176



ARTICLE

(14) Amatore, C.; Belotti, M.; Chen, Y.; Roy, E.; Sella, C.; Thouin, L. J. Electroanal. Chem. 2004, 573, 333–343. (15) Amatore, C.; Da Mota, N.; Lemmer, C.; Pebay, C.; Sella, C.; Thouin, L. Anal. Chem. 2008, 80, 9483–9490. (16) Amatore, C.; Da Mota, N.; Sella, C.; Thouin, L. Anal. Chem. 2008, 80, 4976–4985. (17) Amatore, C.; Da Mota, N.; Sella, C.; Thouin, L. Anal. Chem. 2010, 82, 2434–2440. (18) Compton, R. G.; Daly, P. J. J. Electroanal. Chem. 1984, 178, 45–52. (19) Aoki, K.; Tokuda, K.; Matsuda, H. J. Electroanal. Chem. 1986, 209, 247–258. (20) Aoki, K.; Tokuda, K.; Matsuda, H. J. Electroanal. Chem. 1987, 217, 33–47. (21) Morris, R. B.; Franta, D. J.; White, H. S. J. Phys. Chem. 1987, 91, 3559–3564. (22) Pastore, P.; Magno, F.; Lavagnini, I.; Amatore, C. J. Electroanal. Chem. 1991, 301, 1–13. (23) Compton, R. G.; Dryfe, R. A. W.; Alden, J. A.; Rees, N. V.; Dobson, P. J.; Leigh, P. A. J. Phys. Chem. B 1994, 98, 1270–1275. (24) Aixill, W. J.; Fisher, A. C.; Fulian, Q. J. Phys. Chem. 1996, 100, 14067–14073. (25) Alden, J. A.; Compton, R. G. J. Electroanal. Chem. 1996, 404, 27–35. (26) Matthews, S. M.; Du, G. Q.; Fisher, A. C. J. Solid State Electrochem. 2006, 10, 817–825. (27) Compton, R. G.; Dryfe, R. A. W.; Wellington, R. G.; Hirst, J. J. Electroanal. Chem. 1995, 383, 13–19. (28) Amatore, C.; Klymenko, O. V.; Oleinick, A.; Svir, I. ChemPhysChem 2007, 8, 1870–1874. (29) Deakin, M. R.; Wightman, R. M.; Amatore, C. A. J. Electroanal. Chem. 1986, 215, 49–61. (30) Coen, S.; Cope, D. K.; Tallman, D. E. J. Electroanal. Chem. 1986, 215, 29–48. (31) Aoki, K.; Tokuda, K.; Matsuda, H. J. Electroanal. Chem. 1987, 225, 19–32. (32) Szabo, A.; Cope, D. K.; Tallman, D. E.; Kovach, P. M.; Wightman, R. M. J. Electroanal. Chem. 1987, 217, 417–423. (33) Amatore, C.; Fosset, B. Anal. Chem. 1996, 68, 4377–4388. (34) Cutress, I. J.; Compton, R. G. J. Electroanal. Chem. 2010, 645, 159–166. (35) Hannon, T. E.; Chah, S. W.; Zare, R. N. J. Phys. Chem. B 2005, 109, 7435–7442. (36) Mitrovski, S. M.; Nuzzo, R. G. Lab Chip 2005, 5, 634–645. (37) Arkoub, I. A.; Amatore, C.; Sella, C.; Thouin, L.; Warkocz, J.-S. J. Phys. Chem. B 2001, 105, 8694–8703. (38) Amatore, C. . In Physical Electrochemistry: Principles, Methods and Applications; Rubinstein, I., Ed.; Marcel Dekker: New York, 1995; pp 131208. (39) Weber, S. G.; Purdy, W. C. Anal. Chim. Acta 1978, 100, 531–544. (40) Fosdick, L. E.; Anderson, J. L.; Baginski, T. A.; Jaeger, R. C. Anal. Chem. 1986, 58, 2750–2756.

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Channel Microband Chronoamperometry: From Transient to Steady

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