Theory and Experiments of Transport at Channel Microband

Anal. Chem. 2007, 79, 8502-8510

Theory and Experiments of Transport at Channel Microband Electrodes under Laminar Flows. 1. Steady-State Regimes at a Single Electrode Christian Amatore,* Nicolas Da Mota, Catherine Sella, and Laurent Thouin*

Ecole Normale Supe´ rieure, De´ partement de Chimie, UMR CNRS-ENS-UPMC 8640 “ Pasteur ”, 24 rue Lhomond, F-75231 Paris Cedex 05, France

Integration of microelectrodes in microfluidic devices has attracted significant attention during the past years, in particular for analytical detections performed by direct or indirect electrochemical techniques. In contrast there is a lack of general theoretical treatments of the difficult diffusion-convection problems which are borne by such devices. In this context, we investigated the influence of the confining effect and hydrodynamic conditions on the steady-state amperometric responses monitored at a microband electrode embedded within a microchannel. Several convective-diffusive mass transport regimes were thus identified under laminar flow on the basis of numerical simulations performed as a function of geometrical and hydrodynamic parameters. A rationalization of these results has been proposed by establishing a zone diagram describing all the limiting and intermediate regimes. Concentration profiles generated by the electrode across the microchannel section were also simulated according to the experimental conditions. Their investigation allowed us to evaluate the thickness of the diffusive-convective layer probed by the electrode as well as the distance downstream from which the solution becomes again homogeneous across the whole microchannel section. Experimental checks of the theoretical principles delineated here have validated the present results. Experiments were performed at microband electrodes integrated in microchannels with aqueous solutions of ferrocene methanol under pressure-driven flow. The development of microfluidic analysis systems1-3 is a field of intense research activities. The miniaturization of modules performing a sequence of analytical procedures is an obvious challenge. The great advantage resulting from the size reduction is the ability to solve many problems related to separation and other physicochemical treatments of analytes. By increasing the surface-to-volume ratio, it allows not only a downsizing of samples and reagent consumptions but also faster analyses. In this context, integration of electrodes in microfluidic devices has attracted * Corresponding authors. E-mail: [email protected] and [email protected]. Fax:+331 44 32 38 63. (1) Erickson, D.; Li, D. Anal. Chim. Acta 2004, 507, 11-26. (2) Uchiyama, K.; Nakajima, H.; Hobo, T. Anal. Bioanal. Chem. 2004, 379, 375-382. (3) Dittrich, P. S.; Tachikawa, K.; Manz, A. Anal. Chem. 2006, 78, 3887-3907.

8502 Analytical Chemistry, Vol. 79, No. 22, November 15, 2007

significant attention involving either direct electrochemical detections4-9 or indirect ones by means of electrogenerated titrating species.10 Other electroanalytical strategies have been proposed, e.g., electrochemical tagging of molecules,11,12 electrochemical elimination of interfering species,13-15 or novel and clean processes in electrosynthesis.16-18 The use of micrometric electrodes offers many advantages for detection of analytes in miniaturized devices.19 First, the current responses recorded at microelectrodes remain proportional to the concentration of the electroactive materials initially present, irrespective of the fact that the quantity of analyte may become extremely minute. Second, electrochemical responses depend strongly on the convective-diffusive mass transport regimes which are generally large and well-controlled in microfluidic systems so that faster dynamic processes may be monitored.20,21 Additional advantages are related to the very small size of these electrodes leading to a higher signal-to-noise ratio22 and requiring (4) Woolley, A. T.; Lao, K.; Glazer, A. N.; Mathies, R. A. Anal. Chem. 1998, 70, 684-688. (5) Baldwin, R. P. Electrophoresis 2000, 21, 4017-4028. (6) Vandaveer, W. R.; Pasas, S. A.; Martin, R. S.; Lunte, S. M. Electrophoresis 2002, 23, 3667-3677. (7) Pumera, M.; Merkoci, A.; Alegret, S. TrAC, Trends Anal. Chem. 2006, 25, 219-235. (8) Yi, C.; Qi, Z.; Li, C.-W.; Yang, J.; Zhao, J.; Yang, M. Anal. Bioanal. Chem. 2006, 384, 1259-1268. (9) Goral, V. N.; Zaytseva, N. V.; Baeumner, A. J. Lab Chip 2006, 6, 414-421. (10) Paixao, T. R. L. C.; Matos, R. C.; Bertotti, M. Electrochim. Acta 2003, 48, 691-698. (11) Dayon, L.; Roussel, C.; Girault, H. H. J. Proteome Res. 2006, 5, 793-800. (12) Zettersten, C.; Lomoth, R.; Hammarstrom, L.; Sjoberg, P. J. R.; Nyholm, L. J. Electroanal. Chem. 2006, 590, 90-99. (13) Zhao, M.; Hibbert, D. B.; Gooding, J. J. Anal. Chem. 2003, 75, 593-600. (14) Hayashi, K.; Iwasaki, Y.; Kurita, R.; Sunagawa, K.; Niwa, O.; Tate, A. J. Electroanal. Chem. 2005, 579, 215-222. (15) Hashimoto, M.; Upadhyay, S.; Suzuki, H. Biosens. Bioelectron. 2006, 21, 2224-2231. (16) Ferrigno, R.; Josserand, J.; Brevet, P. F.; Girault, H. H. Electrochim. Acta 1998, 44, 587-595. (17) Horii, D.; Atobe, M.; Fuchigami, T.; Marken, F. Electrochem. Commun. 2005, 7, 35. (18) Paddon, C. A.; Atobe, M.; Fuchigami, T.; He, P.; Watts, P.; Haswell, S. J.; Pritchard, G. J.; Bull, S. D.; Marken, F. J. Appl. Electrochem. 2006, 36, 617634. (19) Amatore, C. In Physical Electrochemistry; Rubinstein, I., Ed.; M. Dekker: New York, 1995; pp 131-208. (20) Macpherson, J. V.; Marcar, S.; Unwin, P. R. Anal. Chem. 1994, 66, 21752179. (21) Rees, N. V.; Dryfe, R. A. W.; Cooper, J. A.; Coles, B. A.; Compton, R. G.; Davies, S. G.; McCarthy, T. D. J. Phys. Chem. 1995, 99, 7096-7101. (22) Morgan, D. M.; Weber, S. G. Anal. Chem. 1984, 56, 2560-2567. 10.1021/ac070971y CCC: $37.00

© 2007 American Chemical Society Published on Web 10/17/2007

low contents of supporting electrolyte to perform locally amperometric detections.23 Whatever the application considered, the design of electrodes is crucial in order to fulfill all the criteria required. This is the reason why mass transport at band electrodes in restricted volumes has been extensively studied in the literature either in the absence24-29 or the presence16,27,28,30-37 of various hydrodynamic flows. These studies were principally devoted to the examination of the behavior(s) manifested by one microband electrode placed within a microfluidic channel in order to enhance the transport to and from the electrode surface. Accordingly, different methods have been developed by several groups, (particularly worth to mention are those of Girault, Fisher and Compton16,27,34,35,38-43) to calculate and predict the current responses of electrodes as a function of geometrical and hydrodynamic parameters, either in a two-dimensional (2D) approximation or for the effective threedimensional (3D) situation. In addition, another series of theoretical investigations37,44,45 have been performed to examine in full detail how an electrode placed on the floor of a microchannel may probe the complete pattern occurring within the channel core. This was performed in order to demonstrate the feasibility of flow profiles reconstruction from electrochemical signals. The present work belongs to this series of investigations, yet, it is specifically developed to exploit steady-state electrochemical components for one microband electrodes to examine how simple and efficient electrochemical detectors could be designed. In this view, although this study may bear some reminiscences with the above ones due to the very nature of the physical problem (viz., band electrodes located in microchannels), it differs fundamentally through its scope and purpose. (23) Amatore, C.; Knobloch, K.; Thouin, L. J. Electroanal. Chem. 2007, 601, 17-28. (24) Rossier, J. S.; Roberts, M. A.; Ferrigno, R.; Girault, H. H. Anal. Chem. 1999, 71, 4294-4299. (25) Arkoub, I. A.; Amatore, C.; Sella, C.; Thouin, L.; Warkocz, J.-S. J. Phys. Chem. B 2001, 105, 8694-8703. (26) Amatore, C.; Sella, C.; Thouin, L. J. Phys. Chem. B 2002, 106, 11565-11571. (27) Henley, I. E.; Yunus, K.; Fisher, A. C. J. Phys. Chem. B 2003, 107, 38783884. (28) Yunus, K.; Fisher, A. C. Electroanalysis 2003, 15, 1782-1786. (29) Amatore, C.; Sella, C.; Thouin, L. J. Electroanal. Chem. 2006, 593, 194202. (30) Muller, O. H. J. Am. Chem. Soc. 1947, 69, 2992-2997. (31) Blaedel, W. J.; Sharma, L. R.; Olson, C. L. Anal. Chem. 1963, 35, 21002103. (32) Ackerberg, R. C.; Patel, R. D.; Gupta, S. K. J. Fluid Mech. 1978, 86, 49-65. (33) Pastore, P.; Magno, F.; Lavagnini, I.; Amatore, C. J. Electroanal. Chem. 1991, 301, 1-13. (34) Compton, R. G.; Fisher, A. C.; Wellington, R. G.; Dobson, P. J.; Leigh, P. A. J. Phys. Chem. B 1993, 97, 10410-10415. (35) Alden, J. A.; Compton, R. G. J. Electroanal. Chem. 1996, 404, 27-35. (36) Cooper, J. A.; Compton, R. G. Electroanalysis 1998, 10, 141-155. (37) Amatore, C.; Klymenko, O. V.; Svir, I. ChemPhysChem 2006, 7, 482-487. (38) Stevens, N. P. C.; Fisher, A. C. J. Phys. Chem. B 1997, 101, 8259-8263. (39) Stevens, N. P. C.; Fisher, A. C. Electroanalysis 1998, 10, 16-20. (40) Sullivan, S. P.; Johns, M. L.; Matthews, S. M.; Fisher, A. C. Electrochem. Commun. 2005, 7, 1323-1328. (41) Matthews, S. M.; Du, G. Q.; Fisher, A. C. J. Solid State Electrochem. 2006, 10, 817-825. (42) Compton, R. G.; Pilkington, M. B. G.; Stearn, G. M. J. Chem. Soc., Faraday Trans. 1 1988, 84, 2155-2171. (43) Alden, J. A.; Compton, R. G. J. Electroanal. Chem. 1996, 402, 1-10. (44) Amatore, C.; Oleinick, A.; Klymenko, O. V.; Svir, I. ChemPhysChem 2005, 6, 1581-1589. (45) Klymenko, O. V.; Oleinick, A. I.; Amatore, C.; Svir, I. Electrochim. Acta, in press.

Figure 1. (A) 3D scheme and top view of the microfluidic device showing the relative positions of the working electrode (WE), pseudoreference electrode (REF) and counter electrode (CE) within the microchannel. (B) Corresponding 2D scheme of the device used for simulations when a parabolic flow propagates across the microchannel. w is the electrode width. h and L are the height and width of the microchannel, respectively, with L . h. ux(y) is the flow velocity in the x-direction.

In the following, a wide rectangular microchannel (Figure 1) is considered and the steady-state amperometric responses of microband electrodes are discussed as a function of the device geometry and flow velocities. As established by a series of studies from Fisher’s group,27,40,41 this situation allows a drastic simplification of the physical problem by permitting its formulation in 2D space, the distortions at the lateral walls being negligible. The flow was pressure-driven and strictly laminar (viz., characterized by very low Reynolds numbers Re ,1). Since the microband electrodes are located on one side of the channel, the quantitative detection of a redox species depends on its confining effect25 which is mainly controlled by the relative dimensions of the microchannel versus the microelectrode. To establish the relevance of these parameters, steady-state currents were simulated for different geometrical and hydrodynamic conditions and their experimental validity was tested based on measurements performed in microfluidic devices in which aqueous solutions of ferrocene methanol were continuously transferred. PRINCIPLE In this section, we consider the mass transport of a reversible redox species A at a microband electrode:

A ( n e- a B

(1)

The geometry of the device is reported in Figure 1 where h is the height of the microchannel and w the microband electrode width. L corresponds simultaneously to the microband length and to the microchannel width. L being much larger than w, the formulation of problem reduces to a two-dimension problem. Under these conditions, diffusion and convection are governed by the following mass transport equation: Analytical Chemistry, Vol. 79, No. 22, November 15, 2007

8503

(

)

∂c ∂2c ∂2c ∂c ) D 2 + 2 - ux(y) ∂t ∂x ∂x ∂y

(2)

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, see eq 2. ux(y) is the flow velocity whose profile is assumed to be parabolic across the microchannel section:

y y ux(y) ) 6uav 1 h h

(

)

(3)

where uav is the average flow velocity. To give more general scope to the following, several dimensionless parameters are introduced.

x y w geometrical parameters: X ) , Y ) and W ) h h h concentration: C )

c c°

(4) (5)

where c° is the bulk concentration of the redox species in solution upstream far away from the electrode.

time: τ )

flow velocity: VX(Y) )

Dt h2

ux(y)h ) 6PeY(1 - Y) D

(6)

(7)

Pe is the Peclet number matching the time scales of convective and diffusional transport.

Pe )

uavh D

current: Ψ )

(8)

i nFLDc°

(9)

where i is the current and F the Faraday. Introducing eqs 4-7 into eq 2 affords the dimensionless mass transport equation:

∂C ∂2C ∂2C ∂C + - VX(Y) ) ∂τ ∂X2 ∂Y2 ∂X At steady-state,

∂C )0 ∂τ

(10) (11)

If the mass transport is the rate determining step, the corresponding boundary conditions are given by

C ) 0 at the electrode surface

(12)

C ) 1 at the microchannel entrance, i.e., upstream far away from the electrode location (13) 8504

Analytical Chemistry, Vol. 79, No. 22, November 15, 2007

Equation 10 may then be solved numerically in association with its boundary conditions in eqs 11-13 by finite differences. According to the dimensionless parameters introduced, the steadystate current is calculated as

ψ)

∫

W

0

∂C dX | ∂Y Y)0

|

(14)

EXPERIMENTAL SECTION Two schematic views of microfluidic devices used are presented in Figure 1A. Their fabrication was performed as already reported.46,47 It is based on two parts which are assembled together. The first one is a polydimethylsiloxane (PDMS) block. One side comprises a linear channel and reservoir elements which are engraved on its surface and connected to the inlet and outlet tubes across the PDMS matrix. The second part is a glass substrate on which a series of different size independent platinum or gold microband electrodes (15 µm < w < 500 µm wide) are patterned. 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). All the characteristic dimensions of microbands (w) and microchannels (h, L) were controlled by optical microscopy before use. The liquid 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 were systematically monitored in situ by direct measurements following a procedure previously described.47,48 Experimentally, the values of Peclet number were varied between 10 and 100. All electrochemical experiments were performed at room temperature using a homemade multipotentiostat adapted from an original design.49 The counter electrode (CE) consisted of one large platinum or gold band electrode (w ∼ 500 µm) located downstream from the working electrode (WE). The pseudo-reference electrode (REF) was a platinum or gold band located upstream from the working electrode. The working electrode was biased at a constant potential of 0.4 V/REF. The amperometric responses were monitored after sufficiently long time durations to ensure that steady-state limiting currents were always achieved. Since we wished to ensure the absence of any migration contribution, aqueous solutions of 1 mM ferrocene methanol were prepared with 0.1 M potassium chloride as the supporting electrolyte. The solutions were degassed with argon before experiments. In such conditions, the diffusion coefficient of ferrocene methanol was measured independently to be D ) (7.6 ( 0.4) × 10-6 cm2 s-1. The diffusion-convection equation was solved numerically in the 2D space described in Figure 1B using the FEMLAB software (46) Pepin, A.; Youinou, P.; Studer, V.; Lebib, A.; Chen, Y. Microelectron. Eng. 2002, 61-62, 927-932. (47) Amatore, C.; Belotti, M.; Chen, Y.; Roy, E.; Sella, C.; Thouin, L. J. Electroanal. Chem. 2004, 573, 333-343. (48) Amatore, C.; Chen, Y.; Sella, C.; Thouin, L. Houille Blanche 2006, 60-64. (49) Maisonhaute, E.; White, P. C.; Compton, R. G. J. Phys. Chem. B 2001, 105, 12087-12091.

Figure 2. Simulated steady-state current Ψ as a function of the electrode width W and Peclet number Pe. ΨLevich and Ψthinlayer correspond to eqs 17 and 18, respectively. 1 e W e 100 and 1 e Pe e 150.

(Comsol) controlled by a MATLAB interface (Mathworks). All computations were performed on a PC workstation. RESULTS AND DISCUSSION Influence of the Geometric and Hydrodynamic Parameters on the Steady-State Current. Figure 2A shows the steadystate current Ψ evaluated numerically as a function of W and Pe for values ranging between 1 e W e 100 and 1 e Pe e 150. These values encompass all experiments considered here. Two limiting behaviors were identified (Figure 2B,C) corresponding to the limiting cases where convection and diffusion contributions in the mass transport become predominant, respectively. When convection dominates, the steady-state current at a channel microband electrode verifies the well-known Levich expression:50

iLevich ) 0.925nFLc°(wD)2/3(4uav/h)1/3

(15)

Conversely, when diffusion is the main transport regime, at steadystate, the thickness of the diffusion layer above the electrode is limited by the channel height and the steady-state current follows the “thin-layer” limiting behavior:51

ithinlayer ) nFc°uavLh

(16)

(50) Levich, V. G. In Physicochemical Hydrodynamics; Prentice Hall: Englewood Cliffs, NJ, 1962; pp 112-116. (51) Weber, S. G.; Purdy, W. C. Anal. Chim. Acta 1978, 100, 531.

By considering the dimensionless parameters introduced previously (eqs 4-9), eqs 15 and 16 may be rewritten as

ΨLevich ) 1.468W2/3Pe1/3

(17)

Ψthinlayer ) Pe

(18)

In contrast to Ψthinlayer, ΨLevich is a function of W since the thickness of the diffusion-convection layer is partially controlled by the electrode dimension and remains smaller than the microchannel height. As evidenced in Figure 2B, the Levich regime takes place (i.e., Ψ/ΨLevich tends to 1) at almost any Pe value whenever W is small enough. This regime cannot be reached at large W values whatever the range of Pe considered. The conditions favoring the thin-layer regime are shown in Figure 2C (i.e., Ψ/Ψthinlayer tends to 1). They correspond to low Pe numbers provided that W is not too small. In order to evaluate specifically the role played by W and Pe for the predominance of each regime, the current variations may be analyzed by plotting Ψ/Ψthinlayer and Ψ/ΨLevich together as a function of the ratio W/Pe. As shown in Figure 2A, one observes that W and Pe have opposite-mirror effects on controlling the physical nature of the steady-state currents Ψ. Figure 3A makes clear that the simulated currents (solid symbols) follow in each case a similar variation. Three main limiting domains may be thus clearly distinguished. Zones I and III correspond, respectively, to the thin-layer and Levich regimes. Zone II is a transition between these two domains so that mixed regimes take place. As an illustration, characteristic concentration profiles are reported for each zone in Figure 4. In zone I, the convectiveAnalytical Chemistry, Vol. 79, No. 22, November 15, 2007

8505

WPe )

wuav w2 uav ) D D w

(19)

With w/uav ) tw as the average time for the flow to pass over the electrode span, the term

βcyl ) Figure 3. (A) Simulated variations of log(Ψ/Ψthinlayer) and log(Ψ/ ΨLevich) versus log(W/Pe) for WPe > 15 (b) and WPe < 15 (O when W/Pe < 0.56 and 4 when W/Pe > 0.56). (B) Zone diagram showing the locations of zones I, II, III, IVa and IVb according to the (W,Pe) coordinates. See Table 1 for the definition criteria of each zone. In part B, - - - corresponds to the case where Ψ/Ψthinlayer ) Ψ/ΨLevich ∼ 0.78 and W/Pe ∼ 0.56 (see text).

diffusion layer reaches the channel height and expands laterally by diffusion against the flow, whereas in zone III, the thickness of the diffusion layer is limited by convection. The boundaries between each zone were evaluated so that the current intensities in zones I and III did not exceed within 5% error the limiting values resulting from application of eqs 17 and 18. Moreover, the relevance of the diffusion or convection contribution in zone II can be established by estimating the W/Pe value for which this specific transition occurs. Equality between eqs 17 and 18 affords W/Pe ≈ 0.56. All the parameters (Ψ, W, and Pe) characterizing the three zones are reported in Table 1. At very low W and/or Pe values, the simulated currents (empty symbols in Figure 3A) diverge from the main variations examined previously though they ultimately reach the asymptotic behaviors defined above when W/Pe . 0.56 or W/Pe , 0.56. As a result, two additional zones must be introduced to characterize these deviations and complete the previous ones depending on the fact that the current is either mostly controlled by diffusion (zone IVa, 4 in Figure 3A) or convection (zone IVb, O in Figure 3A). In zone IVa, the currents are lower than the asymptotic behaviors for any given W/Pe ratio. They correspond to a situation that is very close to the thin-layer regime but still diverges from it appreciably. Indeed, at very low Pe, the convective-diffusion layer may also reach the channel ceiling (part IVa of Figure 4) and expands laterally by diffusion against the flow in a way reminiscent of the full “thin-layer diffusional” regime (compare part I of Figure 4). Conversely, in zone IVb, the situation is reminiscent from that achieved under the Levich regime. However, the currents are higher than predicted by the Levich equation. This regime corresponds to the case where the electrode dimension is small compared to the size of the convectivediffusion layer so that cylindrical diffusion takes place significantly at both edges of the electrode (compare parts IVb and III of Figure 4). This situation is at the origin of the increase in the current response.35,52 The transition between zones II-III and IVa-IVb is estimated by the arbitrary constraint WPe < 15 which was obtained empirically from the observation of current variations. One may explain the physical meaning of this condition based on the formulation of the product WPe. Indeed, from the dimensionless definitions in eqs 4 and 8, one obtains (52) Aoki, K.; Tokuda, K.; Matsuda, H. J. Electroanal. Chem. 1987, 217, 33.

8506

Analytical Chemistry, Vol. 79, No. 22, November 15, 2007

Dtw w2

(20)

represents the cylindricity of diffusion over the band electrode in the would-be absence of any flow,53,54 so that

WPe )

1 βcyl

(21)

In other words, when βcyl is rather large, the band electrode imposes a strong cylindricity component to diffusion at its two edges. When simultaneously W/Pe . 0.56, viz., the flow rate is small, the electrode tends to behave according to a thin-layer behavior, though with a marked cylindricity at both of its edges. This defines the situation IVa by opposition to case I where the cylindricity is much smaller (compare parts IVa and I of Figure 4). Conversely, when W/Pe , 0.56, the flow rate tends to impose a Levich-type condition though the cylindrical diffusion is extremely strong. This differs from a classical Levich diffusion where diffusion is essentially planar (part III of Figure 4), although only the solution in the close vicinity of the electrode is affected (compare parts IVb and III of Figure 4). This defines case IVb. The characteristic parameters describing and defining these two zones are also reported in Table 1. This dimensional analysis explains why the classical Levich/non-Levich view does not apply strictly when the cylindricity of the band electrode is large. This in turns explains why the open symbols in Figure 3A behave roughly according to the ideal behavior (viz., zones I f II f III) but shows nevertheless significant deviations due to the effect of the strong cylindricity of diffusion over the electrode when WPe is small. These different regimes can be represented as in Figure 3B using a zone diagram (W,Pe) describing all the contributions which may govern mass transport at a single band electrode embedded in a microchannel floor. As already observed in Figure 2B,C, zone II is prevalent. The validity of these simulations was tested experimentally by monitoring steady-state currents at microfluidic devices involving electrodes with different W values as a function of the flow velocity Pe. As shown in Figure 5A, a very good agreement was found between theoretical predictions and experimental data, i.e., within the accuracy of their determination. The experimental and theoretical values obtained for WPe > 15 were also reported in Figure 5B by plotting Ψ/Pe as a function of log W/Pe. Note that in this case, the variation of Ψ/Pe is identical to that of Ψ/Ψthinlayer (Figure 3A) since Ψthinlayer ) Pe (eq 18). We will establish below that the ratio Ψ/Pe represents in fact the efficiency of the electrode reaction, viz., the relative quantity of analyte detected (53) Deakin, M. R.; Wightman, R. M.; Amatore, C. J. Electroanal. Chem. 1986, 215, 49-61. (54) Aoki, K.; Honda, K.; Tokuda, K.; Matsuda, H. J. Electroanal. Chem. 1985, 182, 267-279.

Figure 4. Characteristic concentration profiles simulated in zones I (W ) 7, Pe ) 3), II (W ) 7, Pe ) 40), III (W ) 2, Pe ) 90), IVa (W ) 2, Pe ) 3), and IVb (W ) 0.2, Pe ) 30). The curves correspond to a series of isoconcentration lines prevailing above a single microband electrode. Numbers indicate the concentration ratios C/co for some isoconcentration lines (10 curves are shown for each situation, corresponding to C/C0 varying from 0.05 to 0.95). Table 1. Definition of Zones I, II, III, IVa, and IVb zone boundaries

zone I zone II zone III transition I/III

zone definition

W/Pea

WPeb

Wc

0.95 < Ψ/Ψthinlayer 0.95 > Ψ/Ψthinlayer 0.95 > Ψ/ΨLevich 0.95 < Ψ/ΨLevich Ψ/Ψthinlayer ) 0.78 Ψ/ΨLevich ) 0.78

W/Pe > 1.2 1.2 > W/Pe > 0.04

none WPe > 15

W > 4.2 none

W/Pe < 0.04 W/Pe ) 0.56

WPe > 15 none

none none

W/Pe > 0.56 W/Pe < 0.56

WPe < 15 WPe < 15

W < 4.2 none

zone IVa zone IVb

a Values deduced from Ψ criterion. b Values deduced from variations of Ψ/Ψ c Levich and Ψ/Ψthinlayer versus W/Pe. Values deduced from concentration profiles.

by the electrode within the microchannel. One notes in Figure 5B a slight deviation from theory at large values of W/Pe. This may reflect a systematic experimental bias on the experimental Pe determination due to their very small values under these conditions. As expected, most of the experimental results reported in Figure 5B correspond to zone II with an overall predominance of convection mass transport (solid symbols). Evaluation of the Thickness of the Diffusion-Convection Layer under Steady-State. The average thickness of the diffusion-convection layer developed at steady-state over the microband electrode controls quantitatively the conversion efficiency Ψ/Pe of the analyte flowing across the microchannel by the electrode. This thickness was estimated graphically by analyzing the streamline layers deduced from the simulated concentration profiles (Figure 6). Streamlines are defined to characterize locally all the tangents to the vectors of the concentration gradients in such a way that any particle located initially above a given streamline cannot penetrate below it while the solution proceeds above the electrode range. As a result, the streamline linking the

entrance of the microchannel to the downstream electrode edge determines the solution volume probed by the electrode, viz., its equivalent diffusion layer thickness δ across the channel (see Figure 6). In dimensionless parameter, this thickness is given by

H)

δ h

(22)

The numerical evaluation of H has been performed in each zone investigated above. These values are shown as a function of log W/Pe in Figure 7A. As for the variation of Ψ/Pe versus W/Pe (Figure 3A), a continuous behavior is observed for zones I-III, being common with zone IVa (4). Conversely, data from zone IVb (O for W/Pe < 0.56) diverge. In this case, H is influenced by the hemicylindrical-type expansion of the layer. In zone IVa, H is not significantly altered by such hemicylindrical diffusion since it is already limited by the channel height, H, being close to 1. These experimental results are identical to those shown in Figure 3A though their presentation differs. Indeed, the relationship between Analytical Chemistry, Vol. 79, No. 22, November 15, 2007

8507

Figure 5. (A) Experimental (symbols) and simulated (lines) steady-state current Ψ as a function of Pe for various values of W: (b,O) 0.8; ([,]) 2.6; (2,∆) 4.4; (1,3) 8.9; (9,0) 13.5. (B) Corresponding variation of Ψ/Pe versus log W/Pe. In parts A and B, empty symbols correspond to conditions when diffusion is dominant (W/Pe > 0.56) and solid symbols when convection prevails (i.e., W/Pe < 0.56). In all cases, WPe > 15.

Figure 6. Simulated concentration profiles and related streamlines (white curves, see text) across the microchannel when performing in zone II where convection is dominant. In this representation the Y scale was expanded twice. The thicker streamline linking the microchannel entrance to the downstream edge of the electrode is used to evaluate the average thickness of the diffusion-convection layer H. In this case H ) 0.42 for W ) 3 and Pe ) 20. The black curves represent 10 isoconcentration lines corresponding to C/co varying from 0.05 to 0.95.

H and Ψ/Pe is almost linear whatever the values of W and Pe (Figure 7B). It can be demonstrated (see Appendix) that in this case

Ψ ) 3H2 - 2H3 with Ψ/Pe and H∈ [0,1] Pe

(23)

Note that this equation is characteristic of the parabolic flow profile that was assumed to prevail across the microchannel (eq 7). It affords a way to estimate either the collection efficiency Ψ/Pe from the thickness H in zones I-III where a very good agreement is observed (Figure 7B). All the characteristic values of Ψ/Pe and H are summarized in Table 2. When the mass transport is mainly controlled by convection, an approximation based on the Nernst layer concept affords also a simple relation between the current and the layer thickness since

i)

nFLwDc° δ

(24)

With the use of the present dimensionless parameters and identifying δ with H, eq 21 gives then

H) 8508

W Ψ

(25)

Analytical Chemistry, Vol. 79, No. 22, November 15, 2007

Figure 7. (A) Variation of H as a function of log W/Pe in zones I-III (b), zone IVa (4), and zone IVb (O). (B) Corresponding variation of H as a function of Ψ/Pe based on data reported in part A. The curve corresponds to eq 23. (C) Variation of H as a function of W/Ψ in zone III (b), in zone II (O for W/Pe < 0.56 and 0 for W/Pe > 0.56) and zones IV (+). The straight line corresponds to a strict application of the Nernst hypothesis (eq 25). (D) Values of H estimated from experimental currents through application of eq 25 (same symbols as in Figure 5B). The curve corresponds to the common variation displayed in part A for zones I-III.

The gross validity of this relation was checked in Figure 7C by plotting the values of H and Ψ obtained numerically for different values of W and Pe. As expected, eq 25 is valid in zone I for the Levich regime but also for the mixed regime identified in zone II provided convection plays a major role (i.e., when W/Pe < 0.56, see Table 1 and Figure 3B). For the other regimes, deviations from eq 25 arise since diffusion controls partially or totally the expansion of the layer above the electrode. In this case, the Nernst layer approximation is no longer valid. Therefore, eq 25 provides a means to estimate H from the current monitored. Figure 7D shows the comparison between values of H (simulated and determined from experimental currents) as a function of W/Pe. As expected, a very good agreement is observed as long as convection predominates in the mass transport (i.e., for W/Pe < 0.56). For other conditions, experimental results confirm that the equivalent concept of the Nernst layer does not strictly apply.

Table 2. Performances of the Electrochemical Detection in Zones I-III

collection efficiency Ψ/Pe thickness of probed layer H concentration Ch

zone I

zone II

zone III

0.95 < Ψ/Pe 0.86 < H 0.05 > Ch

0.17 < Ψ/Pe < 0.95 0.24 < H < 0.86 0.05 < Ch < 0.83

Ψ/Pe < 0.17 H < 0.24 Ch > 0.83

transition I/III Ψ/Pe ∼ 0.78 H ∼ 0.68 Ch ∼ 0.22

Evaluation of the Homogeneous Concentration Downstream the Electrode. When the electrode operates, concentration gradients generated at the interface dissipate downstream until the solution becomes again homogeneous over the entire section of the microchannel. The propagation of depletion effects downstream of a channel microband electrode has been already described.55 In the present work, two parameters characterize this effect: (i) the homogeneous concentration Ch reached at the exit of the microchannel and (ii) the minimal distance Gh in the microchannel at which the condition of homogeneity is reached. Note that (1 - Ch) is the electrochemical conversion efficiency of the device since (c° - ch)/c° ) 1 - Ch. The concentration Ch can be easily evaluated from the steadystate current since application of the mass conservation law gives

Ch ) 1 -

Ψ Ψ or ) 1 - Ch Pe Pe

(26)

As mentioned previously, this shows that the quantity Ψ/Pe represents the conversion efficiency of the electrode, i.e., the relative quantity of analyte detected across the microchannel. As a consequence, variations of Ch versus W/Pe are similar to those reported for Ψ/Pe in Figure 5B. Equation 26 is entirely valid in zones I-III where no lateral diffusion takes place in the mass transport (i.e., for WPe > 15). Under these conditions, it must be highlighted that contrary to eq 23, eq 26 does not depend on the flow velocity profile which is assumed to prevail across the microchannel and remains valid for other experimental conditions. Indicative values of Ch are reported in Table 2 for each of the three zones. Gh can be estimated by monitoring the variation of concentration within the microchannel downstream (Figure 8A). The criterion required to determine Gh is the degree of homogeneity of the solution or the concentration ratio C/Ch reached across the channel height. Since the two extreme situations are obtained along the floor (Y ) 0) and the ceiling (Y ) 1) of the channel, Ch is the common limit of the two variations depicted in Figure 8A. Figure 8B shows the corresponding variation of Gh as a function of Pe for different geometries W and selected C/Ch values along the channel floor (Y ) 0). No common behavior can be observed, though at low Pe, variations of Gh become almost constant versus Pe. This limit corresponds approximately to zone I where the thinlayer regime takes place. In this case, a simple relationship is obtained between C/Ch and G as shown in Figure 8C which displays the limiting normalized concentration profiles obtained downstream at very low Pe along the floor of the microchannel. This result is indicative of the progressive homogenization of the solution when the electrode is probing almost quantitatively the front of analyte passing above its surface. It must be pointed out (55) Bidwell, M. J.; Alden, J. A.; Compton, R. G. Electroanalysis 1997, 9, 383389.

Figure 8. (A) Determination of Gh from the concentration profiles. G is the distance along the downstream X-direction measured from the downstream edge of the electrode and Ch is the homogeneous concentration reached at the end of the microchannel. The two curves shown correspond to the concentration profiles simulated along each wall of the microchannel (ceiling of the microchannel in dashed line and floor in solid line where the electrode is embedded). In this example, W ) 5 and Pe ) 30 (zone II). Gh is the distance for which a specified concentration ratio C/Ch is reached (e.g., 8.6 in this figure). (B) Variation of log Gh as a function of Pe for different C/Ch ratios and W. (C) Limiting concentration profiles simulated at low Pe (zone I) whatever the geometry W provided that W is not vanishingly small.

that no correlation could be found between Gh and W/Pe, suggesting that no systematic variation of Gh exists with W/Pe when the channel operates either in zone II or III. CONCLUSION This work presented an exhaustive theoretical analysis of the steady-state regimes taking place at a single microband electrode embedded within a linear microchannel and their experimental validation. The influence of the channel and electrode geometries and of the hydrodynamic conditions was investigated on the basis of predictions of steady-state currents based on numerical simulations. Comparison between theoretical predictions and experimental data shows a very good agreement which validate a posteriori the model developed here and in particular the initial assumption of a parabolic flow profile for the pressure-driven flow conditions used here. This theoretical approach allowed a zone diagram to be constructed which delimits several (W,Pe) areas in which limiting identifiable regimes take place for the mass transport. Examination of the simulated concentration profiles and steady-state currents within these zones afford the electrode streamlines, i.e., the thickness of the convective-diffusion layer, viz., the conversion efficiency of analyte probed by the electrode across the microAnalytical Chemistry, Vol. 79, No. 22, November 15, 2007

8509

channel. Perturbations generated downstream by the electrode operation were also investigated by analyzing how the concentration homogenizes while the solution flows toward the end of the microchannel. This study allows detailed predictions for optimizing electrochemical detection devices within a microchannel. Criteria may be readily derived from this zone diagram to optimize electrochemical microfluidic devices and select the best hydrodynamic conditions for any particular analytical or preparative application involving an electrochemical sampling by a microband within a microfluidic pressure-driven microchannel. ACKNOWLEDGMENT This work has been supported in part by the CNRS (UMR 8640), the Ecole Normale Supe´rieure, UPMC, and the French Ministry of Research. The authors thank Prof. Y. Chen and C. Pebay for their kind help in lithography and lift-off techniques.

8510

Analytical Chemistry, Vol. 79, No. 22, November 15, 2007

APPENDIX By considering the thickness H of the diffusion-convection layer and the electrode width W, the flux conservation gives

Ψ)

∫

W

0

∂C | dX ) ∂Y Y)0

∫

H

0

PeX(Y) dY

(A1)

Indeed, the integration of concentration gradient over the electrode width W is equivalent to the integration of motion quantity over the layer thickness H. By taking into account the parabolic flow velocity profile, resolution of eq A1 in combination with eq 7 leads to

Ψ ) 3H2 - 2H3 with Ψ/Pe and H∈ [0,1] Pe

(A2)

Received for review May 14, 2007. Accepted September 7, 2007. AC070971Y