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In this work, we established theoretically that amperometric detector arrays consisting of a series of parallel band microelectrodes placed on the wal...
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Anal. Chem. 2008, 80, 4976–4985

General Concept of High-Performance Amperometric Detector for Microfluidic (Bio)Analytical Chips Christian Amatore,* Nicolas Da Mota, Catherine Sella, and Laurent Thouin UMR 8640 “PASTEUR”, De´partement de Chimie, Ecole Normale Supe´rieure, CNRS and Universite´ Pierre et Marie Curie 24, rue Lhomond, 75231 Paris Cedex 05, France In this work, we established theoretically that amperometric detector arrays consisting of a series of parallel band microelectrodes placed on the wall of a microchannel may offer excellent analytical detection performances when implemented onto microfluidic (bio)analytical devices after the separative stages. In combination with the concentration imprinting strategies reported in a previous work, these exceptional performances may be extended to nonelectroactive or poorly diffusing analytes. Using an array of electrodes instead of a large single band allows the whole core of the channel to be probed though keeping an excellent time resolution. Thus, analytes with close retention times may be characterized individually with a resolution which eventually outpaces that of spectroscopic detections. Such important advantages may be obtained only through a complete understanding of the complex coupling between diffusional and convective transport of molecules in microfluidic solutions near an electrochemical detector. As a consequence, the conditions underlying the theoretical data presented in this work have been selected after optimizing procedures rooted on previous theoretical analyses. They will be fully disclosed in a series of further works that will also establish the experimental performances of such amperometric detectors and validate the present concept. The increasing accessibility of microfabrication techniques and the growing panel of active microfluidics components have boosted the development of microlaboratories for total analysis (µ-TAS) particularly for (bio)analytical applications.1–6 Many reviews exist to describe the numerous advantages of microfluidic analytical devices over classical ones and the reader is referred to them.1–7 For our purpose here, it is sufficient to recall that the decrease in length scales associated with the precision of microfabrications allows us to better control separation procedures and hence shortest analysis durations. Another decisive advantage, noteworthy for analyses performed on biological samples, lies in * To whom correspondence should be addressed. Tel: 33-1-4432-3388; Fax: 33-1-4432-3863. E-mail: [email protected] (1) Pumera, M.; Merkoci, A.; Alegret, S. Trends Anal. Chem. 2006, 25, 219– 235. (2) Yi, C.; Qi, Z.; Li, C.-W.; Yang, J.; Zhao, J.; Yang, M. Anal. Bioanal. Chem. 2006, 384, 1259–1268. (3) Dittrich, P. S.; Tachikawa, K.; Manz, A. Anal. Chem. 2006, 78, 3887–3907. (4) Chen, L.; Manz, A.; Day, P. J. R. Lab Chip 2007, 7, 1413–1423.

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the considerable reduction of sample sizes. However, the very source of these great advantages is also the root of a considerable problem: reducing analytical sample size is tantamount to an increasing difficulty during detection of the sample components after their separation due to decreasing signal amplitudes. Most detection methods used presently in microfluidic separative devices rely on analytical properties that scale as the number of target molecules in the detected sample. As a consequence, the analytical signal intensities, S, decrease upon reducing analyzed volumes. This would not create severe difficulty if the noise, N, was decreasing along the same scaling factors or faster. This explains why the search for better detectors, viz., better S/N ratio, has been mostly focused along directions aimed to reduce the intrinsic source of noise in the frequency domain where a given analytical measurement is performed. For example, this is the essential reason that sustains the devising of advanced spectrophotometric procedures (e.g., two-photon spectroscopies, near-field optics, etc.). Increasing the analytical signal S rather than trying only to decrease the measurement noise N provides another strategy for better analytical detectors. One entry consists in remarking that even if analyte quantities are reduced in microchips, their local concentrations, c, are maintained at high levels in most separative devices. Therefore, methods affording signals proportional to concentrations rather than to quantities spontaneously lead to increased S values. To the best of our knowledge, such methods must be based ultimately on kinetics, taking advantage of the fact that the rate of a reaction is a probabilistic property, which depends only on local concentrations and not on global quantities. Nature exploits this strategy in most of its signaling procedures, thus relying on a few molecules or ions but present at high local concentrations near the detecting entity. The neural synapse is an archetype of such approach. Most enzyme-mediated electroanalytical methods use this strategy, hence their high sensitivities. Increasing the analytical signal by measuring local concentrations rather than global quantities may be coupled with noise reduction strategies, as in electrochemiluminescence detection in microchips,5–8 to produce exceptional analytical performances. The same is true for amperometry at ultramicroelectrodes.9 (5) Kuswandi, B.; Nuriman Huskens, J.; Verboom, W. Anal. Chim. Acta 2007, 601, 141–155. (6) Du, Y.; Wang, E. K. J. Sep. Sci. 2007, 30, 875–890. (7) Viskari, P. J.; Landers, J. P. Electrophoresis 2006, 27, 1797–1810. 10.1021/ac800227t CCC: $40.75  2008 American Chemical Society Published on Web 05/10/2008

Though generally not discussed in such terms, dynamic electrochemistry is also directly relevant to this dual strategy. Amperometric signals are proportional to rates (kinetic of electrochemical reactions or kinetics of transport) that scale up as concentrations.10 Simultaneously, electrochemical noise depends on the electrode capacitance, viz., on its surface area, which is reduced upon miniaturizing electrode sizes.9 This is the main fundamental reason that sustains the success of many electroanalytical sensors based on microelectrodes. Within the context of microfluidic analytical devices, amperometry presents an additional advantage rooted in the extreme simplicity of dedicated electrochemical instrumentation when designed for a specific purpose. Electronics stages for control and detection can be integrated directly on the device, e.g., as is performed in present commercial glucose sensors.11 This is an exceptional advantage for analytical devices granting both portability, viz., in situ monitoring, and servicing by nontechnical experts. Note that besides commodity, in situ analyses also decrease risks of sample contamination by eliminating the need for series of manipulations, transfers, and storage procedures. Notwithstanding these extremely important known advantages, amperometry is seldom used in microfluidic analytical devices developed for commercial applications, especially for those devoted to biological samples. They are several reasons, though undue, that explain this puzzling observation. Among nonelectrochemists, it is widely considered that electrochemical measurements are limited to fast-diffusing electroactive molecules. Furthermore, they are alleged be able to only probe thin solution films passing in the proximity of the electrode surfaces so that electrodes are assumed to be “blind” to the analytical contents of interest, which are those flowing in solution cores. Such undue but often accepted views give the impression that electrochemical detection is generally prohibitory particularly for many biological microfluidic analyses. Indeed, these generally involve large biological molecules, e.g., proteins, which give sluggish electrochemical behavior. Even for those molecules best suited to amperometric detection, the alleged constraint of analyses being restricted to small solution fractions adjacent to the electrode surface represents an intrinsic bias. Indeed, the composition of solution films flowing in a microfluidic channel near its walls may not be representative of the main solution, which flows within the microchannel core due to uncontrolled and undesired chemical or physicochemical interactions with the wall materials,12 or due to geometrical imperfections such as expected in mass fabrications. In this work, we wish to establish that all the above disadvantages may be easily bypassed through a careful understanding of analytically related issues of electrochemistry within microfluidic channels. Note that many excellent works have been published to account theoretically and exploit experimentally microelectrochemistry within microfluidic channels.1,2,13–27 These works have (8) Qiu, H. B.; Yan, J. L.; Sun, X. H.; Liu, J. F.; Cao, W. D.; Yang, X. R.; Wang, E. K. Anal. Chem. 2003, 75, 5435–5440. (9) Amatore, C. In Physical Electrochemistry: Principles, Methods and Applications; Rubinstein, I., Ed.; M. Dekker: New York, 1995; pp 131-208. (10) Bard, A. J.; Faulkner, L. R. Electrochemical Methods; J. Wiley & Sons, Inc.; New York, 2001. (11) Newman, J. D.; Turner, A. P. F. Biosens. Bioelectron. 2005, 20, 2435–2453. (12) Amatore, C. Chem. Eur. J. In press. (13) Bidwell, M. J.; Alden, J. A.; Compton, R. G. Electroanalysis 1997, 9, 383– 389.

examined quantitatively many issues regarding the ubiquitous competition between heterogeneous electron transfer, diffusion of molecules, and hydrodynamical transport in microchannels. Yet most of these works have been focused onto pure electrochemical purposes in which microfluidics brings added value and not toward analytically oriented issues that this work is challenging. Conversely, hereafter, we wish to document theoretically a strategy that we hope will establish amperometry as the most competitive solution to the problem of analytical detection and characterization of analytes at the output of microfluidic separative devices. THEORY All the theoretical data shown in this work have been obtained through resolution of the mass-transfer equation described below using the commercial program Femlab (Comsol). This was performed in the 2D cross section of a microchannel (Figure 1a) having a width L (i.e., its dimension perpendicular to the cross section shown in Figure 1a). The electrode(s) consisted of band electrode(s) embedded in the channel floor across the whole channel width. As in most used microfluidic systems, the height h of the channel was considered much smaller than its width L so that the transport and electrochemical phenomena could be investigated in the 2D cross section and be representative of the whole system.21 We considered an electroactive species, A, whose concentration and diffusion coefficient are noted c and D, respectively, being transported into a microchannel submitted to a parabolic flow rate regime (Figure 1a). The flow velocity rate, ux(y), was supposed to depend parabolically on the vertical coordinate, y, perpendicular to the main axis, x, of the channel along which the flows proceeded; see eq 1.

ux(y) ) 4umax

y y y y 1 - ) 6uav 1 h h h h

(

)

(

)

(1)

where umax and uav were respectively the maximum (viz. at y ) h/2) and average flow rates (uav ) 2umax /3), so that the Peclet number pertaining to A was Pe ) uavh/D.28 The ensuing mass (14) Cooper, J. A.; Compton, R. G. Electroanalysis 1998, 10, 141–155. (15) Amatore, C.; Belotti, M.; Chen, Y.; Roy, E.; Sella, C.; Thouin, L. J. Electroanal. Chem. 2004, 573, 333–343. (16) Amatore, C.; Oleinick, A.; Svir, I. Electrochem. Commun. 2004, 6, 1123– 1130. (17) Daniel, D.; Gutz, I. G. R. Talanta 2005, 68, 429. (18) Amatore, C.; Oleinick, A.; Klymenko, O. V.; Svir, I. ChemPhysChem 2005, 6, 1581–1589. (19) Amatore, C.; Chen, Y.; Sella, C.; Thouin, L. La Houille Blanche 2006, 60– 64. (20) Amatore, C.; Klymenko, O. V.; Svir, I. ChemPhysChem 2006, 7, 482–487. (21) Matthews, S. M.; Du, G. Q.; Fisher, A. C. J. Solid State Electrochem. 2006, 10, 817–825. (22) Kwakye, S.; Goral, V. N.; Baeumner, A. J. Biosens. Bioelectron. 2006, 21, 2217–2223. (23) Thompson, M.; Compton, R. G. Anal. Chem. 2007, 79, 626–631. (24) Amatore, C.; Klymenko, O. V.; Oleinick, A.; Svir, I. ChemPhysChem 2007, 8, 1870–1874. (25) Amatore, C.; Da Mota, N.; Sella, C.; Thouin, L. Anal. Chem. 2007, 79, 8502–8510. (26) Klymenko, O. V.; Oleinick, A. I.; Amatore, C.; Svir, I. Electrochim. Acta 2007, 53, 1100–1106. (27) Bai, X. X.; Josserand, J.; Jensen, H.; Rossier, J. S.; Girault, H. H. Anal. Chem. 2002, 74, 6205–6215. (28) Lindeburg, M. R. Engineer In Training Reference Manual, 8th ed.; Professional Publication, Inc.: Belmont, CA, 1992.

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x>0 xe0 Figure 1. Effect of diffusion on a step concentration profile injected at t ) 0, X ) x/h ) 0 (i.e., ct)0 ) 0 and ct)0 ) cmax) and carried through a microfluidic channel by parabolic flow. (a) Schematic description of the channel with its dimensions and of a parabolic flow rate distribution. (b) Series of charts illustrate how the concentration front distorts progressively while the dimensionless times (Dt/h2) increase when the flow proceeds along a microchannel with fully inert walls (Pe´clet number Pe ) uavh/D ) 40). Ten isoconcentration lines corresponding to c/cmax varying from 0.05 to 0.95 are superimposed on each chart (black superimposed curves) to emphasize the progressive distortion. Note that the vertical dimensionless scale (Y ) y/h) is expanded three times with respect to the horizontal one (X ) x/h) to allow a better viewing of the nonparabolic features created near the channel walls by lateral and transverse diffusion. (c) Limiting shapes of the isoconcentration lines achieved at Dt/h2 .1 for different values of Pe. Xisoconc ) (xy - xy ) 0)/h, where xy is the abscissa of the isoconcentration line at the value of y in ordinate. (d) Absolute position of the isoconcentration line c/cmax ) 0.5 when t varies. This is shown at different heights in the channel: (b) y/h ) 0.5; (O) y/h ) 0.25 or 0.75; (2) y/h ) 0 or 1. (e) Variations of the concentration wave width with time at different heights in the channel: (b) y/h ) 0.5; (O) y/h ) 0.25 or 0.75; (2) y/h ) 0 or 1.

transport equation describing the time and space-dependent concentration of A was as follows:

(

)

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

(2)

where t is the time elapsed since the injection of A at x ) 0. Various theoretical concentration signals were introduced in the device to model either initial step concentration profiles (viz., ct x)>0 0 ) 0 and ct x)e0 0 ) cmax) or concentration spikes as those produced by analytical devices after their separative stages. Concentration spikes were modeled through the derivation along x of step concentration profiles (as defined above) to avoid the injection of a Dirac function at x ) 0, which results are extremely delicate in a digital form. 4978

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Species A was considered as being inactive on all channel walls ins ins (viz., (∂c⁄∂y)y)0 ) (∂c⁄∂y)y)h ) 0) except at those places located on the channel floor (i.e., at y ) 0), which were equipped with a band electrode whose potential was set on the plateau of the A electrochemical wave. Over the electrode surfaces (see text), the concentration was then set to zero at any time and the current ij flowing through any particular electrode j was ij ) -nFDL

∫

xj+wj

xj

( ∂c∂y )

y)0

dx

(3)

where n is the number of electrons transferred, F the Faraday, xj the abscissa of the upstream edge of the jth electrode, and wj its width (1 e j e jmax, where jmax is the total number of electrode(s)). Note that, in the following, the abscissa x1 of the first (or single) electrode upstream edge is defined arbitrarily as the distance from

the point where the step concentration profiles were numerically injected. For simplification of the following formulations, x1 is formally considered as the width (g1) of the gap associated to the first electrode of the array or to the single electrode accordingly. Similarly, each insulating gap, gj, is arbitrarily affected to the electrode j, which immediately follows it. Finally, w1 is used in general formulations to represent the width, w, of the electrode for a single-electrode device (Figure 1a). These somewhat abusive notations allow using the same general formulation for an array or a single electrode detector. For offering generality to the results presented, in the following, all currents are reported in dimensionless values: Ij ) ij/ (nFDLcmax). cmax is the maximum concentration of A reached in the microchannel. This allows predicting experimental values for any value of n, L, D, and cmax through a simple scaling of the results presented. For the same reason, all calculations were performed in dimensionless formulations: C ) c/cmax, X ) x/h, Y ) y/h, Wj ) wj/h, and Gj ) gj/h. RESULTS AND DISCUSSION Specific Features of Concentration Profiles in a Microfluidic Channel. Microfluidic solutions may be driven by pressure, which results in parabolic flow rates (eq 1), or electroosmotically, then producing a constant flow rate (viz., ux(y) ) constant) across the channel except over nanometric distances from the channel walls. However, electroosmotic pumping of microfluidic solution requires strong electrical fields, which (or whose noise components) may affect the potential difference between an electrode and its adjacent solution. This forbids maintaining electroosmotic pumping of electrolytic solutions near electrochemical detection stages for all applications envisioned here. Hence, even if electroosmotic conditions applied upstream in the separating stage, thus also providing the pressure pushing the microfluidic flow above the electrochemical detector, the flow profile near an electroanalytical stage is necessarily deteriorated by friction over the channel walls and relaxes to a parabolic flow as occurs with pressure-driven solutions. So eq 1 is valid for all applications envisioned here. However, when a concentration front is propagated along the microchannel by a parabolic flow regime, the gradient of molecules versus the y coordinate is not parabolic. This is not commonly recognized, though it has been suggested on the basis of experimental fluorescence measurements of concentration fronts in microchannels.29 Indeed, the would-be parabolic distribution enforced by the flow velocity creates a concentration gradient along y that tends to shift the analyte from the channel core toward the walls. This process deteriorates partially any initial parabolic front by retarding the analyte progression in the channel central zones (see the shapes of isoconcentration lines in Figure 1b,c). In addition, a well-recognized diffusion component acts along the solution flow direction and smoothes the initial abrupt concentration front while it proceeds in the channel. This combination of three transport modes produces a specific pattern that alters more and more the expected parabolic shape that would be enforced if diffusion did not occur (D f 0). We have established elsewhere15,16,18–20,24–26 that the competition between transverse and normal diffusion, which both evolve as Dt, and the convective (29) Sinton, D.; Li, D. Q. Colloids Surf. A 2003, 222, 273–283.

transport proportional to uavt results rapidly in a quasi-sustained pattern when the travel time, t, increases. This is apparent in the charts shown in Figure 1b, which describe the progressive deformation of a step concentration profile injected at the channel entrance (x ) 0) at t ) 0. The parabolic shape of isoconcentration lines is readily altered by transverse diffusion during the very initial times since t grows faster than t, so that it reaches almost immediately an asymptotic shape that is then retained while the concentration front travels along the channel with the flow (Figure 1c,d). Conversely, the distance between two isoconcentration lines increases as Dt while time elapses but is nearly independent of y (Figure 1e). Note that the shape alteration described above and documented in the series of charts in Figure 1b is valid under any situation that involves a propagation of a step concentration profile in microchannels except for electroosmotically driven ones since these do not produce any transverse concentration gradient. This general feature has several consequences onto the performance of electrochemical detectors. Finally, note also that panels c-e of Figure 1 demonstrate that the wave propagation scales up as Pe. In other words, what matters are not the independent values of uav, h, and D, but the value of Pe. In the following, we take advantage of this to report as much as possible our results as a function of Pe, though a complete optimization of the behavior of each system considered will be provided only for Pe ) 40. Optimization of the single band detector for a wide range of Pe values has already been reported by us and others,1,2,13–27 while the complete optimization of the band array electrode introduced here will be reported in a following work. Electrochemical Detection by a Single Microelectrode Inserted into a Microfluidic Channel. In the following, we wish first to recall the effect produced by a microelectrode acting on the analyte onto an analyte profile distribution such as those in Figure 1b. This has been previously examined by us and by others to characterize the different kinetic regimes obeyed by electrochemical currents monitored under such conditions.14,15,18,20,21,23–26 However, our purpose here is to insist more specifically on the effects produced by the electrochemical detection on the structure of the solution composition since this is central for understanding the behavior of band array detectors. Figure 2b evidence that any electrode of sufficient width may probe the channel composition up to its core. The comparison between Figure 2a and b shows that this occurs because the electrode activity imposes a severe distortion of the analyte isoconcentration lines inside the channel. Though the electrode diffusion field extends only near the channel floor, it couples with the diffusion field created by the flow profile (see above) to act as a boundary condition, which ultimately modifies the concentration pattern inside the whole channel. The analyte is then progressively dragged from the channel core toward the electrode surface where it is analyzed. Figure 2b demonstrates that for probing within the very central section of the channel the electrode width must be extremely large compared to the channel height, and this is even more so when the Peclet number, Pe ) uavh/D, increases.25 This is required because the constraint created by the microelectrode may propagate into the solution only over distances of ∼D∆tw where Analytical Chemistry, Vol. 80, No. 13, July 1, 2008

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Figure 2. Comparison of the concentration profiles across the microchannel at Pe ) uavh/D ) 40 and at Dt/h2 ) 0.5 of an electroactive species in the absence (a) or in the presence (b) of a single band electrode having a dimensionless width W ) w/h ) 7. The electrode in (b) performs amperometrically on the plateau of the detected species. The white curve describes the trajectory of the point located initially at H ) y/h ) 0.43 at the entrance of the channel (X ) x/h ) 0), which hits ultimately the downstream edge of the band electrode (x ) 17h, y ) 0). All points located at X ) x/h ) 0 below y/h ) 0.43 hit the electrode while those located at y/h > 0.43 are carried away by the flow without reaching the electrode. Ten isoconcentration lines corresponding to c/cmax varying from 0.05 to 0.95 are marked by black superimposed curves. Note that the vertical dimensionless scale (Y ) y/h) is expanded three times with respect to the horizontal one (X ) x/h) to allow a better viewing of the features created near the channel floor and the electrode. (c) Resulting time dependence of the current flow at the electrode. (d) Comparison between the normalized current (i/imax) and normalized concentration (c/cmax corresponds to the analyte concentration in the solution passing at the point x ) 10h, y ) 0 in absence of electrochemical detection) variations with time emphasizing the sluggish character of the former.

∆tw ) w/uav is the time duration for the flow to pass over the electrode. Though this is of no serious problem and is even interesting for microchannel-based electrochemical applications, this feature may become prohibitory for detection in an analytical device. Indeed, this means that two species experiencing a difference ∆Tsep between their retention times may be resolved only when ∆Tsep exceeds a few ∆tw ) w/uav. In other words, two species may be detected separately only when the distance separation, ∆xsep ) uav∆Tsep, between their concentration traces in the flow (see Figure 1e) is larger than a few w. Since in turn, w must be several times h, this means that ∆xsep has to greatly exceed the channel height, a condition that may be prohibitory in many circumstances. A second consequence of large electrode widths is to impose a kinetic distortion onto the amperometric measurement. This results from the time lag of the electrochemical detection, which increases along the electrode width and adds to the time width of the analyte concentration profile. Indeed, the electrode can be thought of as a series of smaller electrodes placed in series, each 4980

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one reporting the concentration change passing immediately above it. In other words, an electrode element located at distance ω from the upstream edge of the electrode (0 e ω e w) may detect the analyte only a time duration ∆tω ≈ ω/uav after this concentration presented itself at the vertical of the upstream edge of the electrode. This is physically observable in Figure 2b through the slanted-linear aspect of the isoconcentration lines in the zone located above the electrode surface. The overall distortion thus created by the amperometric measurement is quantitatively apparent in Figure 2d upon comparing the concentration rise measured at the vertical of the upstream edge of electrode of Figure 2b to the electrochemical current monitored by it. One notes immediately the sluggish appearance of the electrochemical current due to the convolution between the real concentration front (i.e., the variations of c/cmax) and the convective and diffusional transport modes required for its detection at the electrode surface. Electrochemical Detection by a Series of Band Microelectrodes in a Microfluidic Channel. The effects that have been recalled and summarized above are intrinsic to the very process of electrochemical detection so they cannot be suppressed except by reducing the microelectrode width. On the other hand, in a flow system, a narrow electrode would be blind to the composition of the solution in the channel core so it could not perform as a reliable analytical detector. In the following, we wish to establish that replacing a large band microelectrode by a series of independent small ones performing at the same potential affords a solution to this conundrum provided each individual electrode current is monitored independently. The principle of this approach relies on the observation described above about the concentration profile alteration induced by a large microelectrode activity (Figure 2b). This occurs also at juxtaposed electrodes of smaller widths. The jth electrode performing in such array probes a solution volume whose size is only ∼(wjL) × Dwj/uav, but the action of the (j - 1) previous electrodes operating in the array fills this volume with a solution that would flow much above Dwj/uav in their absence. This is apparent in Figure 3a, which presents the concentration pattern created by a series of four electrodes of identical widths (wj ) h, 1 e j e 4) separated by three insulating gaps also of identical widths (gj ) h, 2 e j e 4). As evidenced in Figure 3a, each electrode probes a solution layer of identical thickness, Dwj/ uav ) Dh/uav immediately above its surface. However, it is seen that the trajectory joining the downstream edge of the jth electrode to the vertical plane cross-secting the channel at the entrance of the detector (x ) 0) penetrates more and more deeply into the solution core when j increases. Through operation of the electrode(s) placed before it, the jth electrode may then report on the concentration of a solution slice, which, upon entering the detector area, was located between the heights yj-1 and yj. Note that whenever yj is not too small compared to h, one has approximately k)j

yj⁄h ) Hj ≈ ((w1 + ∑ (gk + wk))⁄hPe)1⁄2 as shown in Figure 3a k)2 and e (see Appendix). Conversely, when yj is very small compared to h, this approximation does not apply here since we are interested only in system geometries (i.e., electrodes widths and gaps) allowing the solution composition to be probed deep within the channel core, i.e., yjmax ≈h/2, so that the above approximation applies (compare Figure 3e).

duration ∆tj ) wj/uav, but this occurs after a time delay, tj, required for the solution in the slice [yj-1,yj] to be transported to the k)j-1

upstream edge of the jth electrode, viz., tj ≈ [(gj + ∑ (gk + k)1 wk))/uav] + [yj2/D], since the moment of injection (see Appendix). Note that, in the above expression, under the conditions envisioned here the two terms in the expression of tj are equal, so k)j-1

that tj∝(gj + ∑ (gk+wk))/uav (see Appendix). This time lag is k)1 apparent in Figure 3b, which reports the individual current probed by each electrode of the array, and is quantified in Figure 3c upon plotting θjhalf ) Dtjhalf/h2, the dimensionless time at which the jth electrode experiences a current being half of its plateau, as a function of the distance between the channel entrance (x ) 0) k)j-1

Figure 3. (a) Same as in Figure 2b for an amperometric detector consisting of four band electrodes of identical widths (Wj ) wj/h ) 1) separated by three identical insulating gaps (Gj ) gj/h ) 1) so that the distance between the upstream edge of the first electrode and the downstream one of the last one is equal to the width (W ) w/h ) 7) of the single electrode shown in Figure 2b. The white curves evidence the trajectories of the points located at X ) x/h ) 0, which hit ultimately the downstream edge of each electrode. The points located above H4 ) y4/h ) 0.40 at X ) x/h ) 0 cannot reach the amperometric detector. (b) Current variations monitored by each individual electrode (1 e j e 4) represented together with their direct sum (iΣ) and compared to the current trace (isingle) monitored by a single large band electrode having the same width as the array. (c) Comparison between θjhalf ) Dtjhalf/h2 values (solid symbol) and the k)j-1

time delay θj ) Dtj/h2 ≈ [(gj + ∑ (gk + wk))/hPe] (straight line) as k)j-1

k)1

a function of (gj + ∑ (gk + wk))/h, i.e., the distance between the k)1 channel entrance (x ) 0) and the upstream edge of the jth electrode (see text; note that g1 ) 10h is the abscissa of the first electrode upstream edge). (d) Comparison between the concentration variation at y ) 0 before the first electrode upstream edge, i.e., x ) 10h (solid symbols), and the currents monitored by a single electrode (isingle) or by the array of four electrodes shown in (a): direct current sum (iΣ) or reconstructed (ireconst ) after compensation of the individual electrodes tot time lags according to eq 4. In (d) are also presented the theoretical variations of the solution absorbance (A) that would be monitored by an ideal detector in the vertical plane cross-cutting the microchannel at x ) 10h in the absence of electrochemical detection (see text). All data shown in (d) are normalized to their maximum value achieved at infinite time to allow their comparison on the same scale. (e) Plot of the upper height of solution, Hj ) yj/h, probed by the jth electrode k)j

as a function of ((w1 + ∑ (gk + wk))/(hPe))1/2, for (O) each electrode k)2 of the array shown in Figure 3a, or for (b*) a large single electrode of the same overall width (data for others single electrodes are also included and marked by (b) symbols).25

Therefore, an adequate array consisting of small parallel band electrodes provides a precise representation of the composition of the solution flowing within the microchannel through “slicing” it at different heights. Each slice [yj-1,yj] (note that y0 ) 0 for j ) 1, viz., for the first electrode) is analyzed within a limited time

and its upstream edge, viz., [gj+ ∑ (gk+wk) ]/h ≈ tj × (uav/h) k)1 (see Appendix). It is seen that when j increases (viz., when the travel parallel to the y axis decreases relatively to that parallel to x) the above approximation is excellent. Note that since, in the example of Figure 3a, all electrodes have identical widths and are preceded by identical gaps, the thickness ∆yj ) yj - yj-1 of each probed slice decreases when j increases. With such configuration (i.e., wj ) constant (1 e j e jmax)and gj ) constant (2 e j e jmax)) the jth electrode experiences a smaller current as evidenced by the decreasing ijmax values in Figure 3b. However, this could be adjusted at will by modifying the successive widths of electrodes and gaps to afford, for example, a constancy of ∆yj ) yj - yj-1 (viz., of imax values) versus j j or any other variation that could be of experimental interest. Interestingly, the direct summation of all individual currents (iΣ) in Figure 3b approaches both the intensity and the time dependence of the current (isingle), which would be determined by a single large electrode extending over the whole array, viz., k)jmax

such as w ) w1 + ∑ (wk+gk), establishing that the effect of k)2 gaps tends to be negligible. As established previously (though for still solutions), this is an intrinsic property of arrays of band microelectrodes whenever the diffusion layers of each electrode overlap sufficiently.30–37 In this respect, using the direct sum of individual currents for detection purposes presents no advantage at all over a single electrode configuration (except for increasing moderately the S/N ratio,9 because the electrode capacitance would decrease). However, when each individual current is monitored separately, their summation may be performed after compensating for the time delay ∆tjhalf ) tjhalf - t1half experienced by each successive electrode, e.g., with respect to the first one: k)jmax reconst (t) ) itot

∑ i (t - ∆t j

half

j

)

(4)

k)1

reconst The reconstructed current, viz., itot (t) in eq 4, obtained through this procedure is shown by the solid curve in Figure 3d. It tracks precisely the concentration variations in the solution as

(30) Amatore, C.; Saveant, J. M.; Tessier, D. J. Electroanal. Chem. 1983, 147, 39–51. (31) Arkoub, I. A.; Amatore, C.; Sella, C.; Thouin, L.; Warkocz, J.-S. J. Phys. Chem. B 2001, 105, 8694–8703. (32) Amatore, C.; Sella, C.; Thouin, L. J. Phys. Chem. B. 2002, 106, 11565– 11571. (33) Amatore, C.; Sella, C.; Thouin, L. J. Electroanal. Chem. 2003, 547, 151– 161. (34) Amatore, C.; Sella, C.; Thouin, L. J. Electroanal. Chem. 2006, 593, 194– 202.

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the analyte front passes through the vertical plane cross-cutting the channel above the location of the first electrode upstream edge (see symbols in Figure 3d). The quality of the ensuing measurement is even more apparent when plotting the concentration and current data after normalizing them to their plateau values. To emphasize the quality of this procedure, the normalized direct sum of the individual currents (i.e., without compensation of time lags) or the normalized current monitored by a single electrode k)jmax

(with w ) w1 + ∑ (wk + gk)) are superimposed onto the same k)2 normalized plot. To conclude this section, we will discuss another advantage of such detector. Indeed, since the solution composition is probed virtually in a vertical plane transverse to the microchannel main axis, this is technically equivalent to monitoring it spectroscopically using an extremely thin beam transverse to the channel axis and probing across the solution. Both normalized measurements are compared in Figure 3d, when considering an ideal spectroscopic beam (i.e., having no thickness along x). It is seen that the match is perfect, evidencing the great analytical performance predicted for the amperometric one since any real spectroscopic detection would involve beams with micrometric thickness at least. Such more realistic beam size would result in a more sluggish signal due to the concentration distribution shape in the microchannel (see Figure 1). Besides the fact that concentrations rather than quantities are monitored by the above electrochemical detector, thus providing optimal S/N ratios, a multielectrode amperometric detector offers an exceptional analytical resolution, viz., a crucial property for the characterization of two species with close retention times. Monitoring Two Species with Close Retention Times. The analytical quality of a separating column is ultimately conditioned by the ability of the detector placed at its output to separate two species with close retention times Tsep. Indeed, the larger the minimum ∆Tsep value, the larger must be the separative stage, hence the lower S/N ratio and the lower accuracy due to the larger diffusional spikes enlargement. In the above sections, we used step functions to describe the injected analyte concentration at x ) 0. Indeed, step functions allow the transfer function of any system to be extracted and are then essential for optimizing a device performance. In a real practice, analyte concentrations at the output of a separative device, viz., at x ) 0 in our system, display spiked variations closely modeled by exponentially modified Gaussians, as in eq 5, where σ is the half-width of the original Gaussian and λ represents the distortion degree.38 c(t) ) cmax

σλ √2π

∫

∞

0

e-(t-v)

2 ⁄ 2σ2

e-λv dν

(5)

Since the degree of exponential distortion is characteristic of the species and of the properties of separating devices, we will not consider them hereafter. For our general purpose here, the ensuing concentration traces may therefore be modeled by (35) Szunerits, S.; Thouin, L. In Handbook of Electrochemistry; Zoski, C., Ed.; Elsevier: Amsterdam, 2006; pp 391-428. (36) Streeter, I.; Compton, R. G. J. Phys. Chem. C 2007, 111, 15053–15058. (37) Streeter, I.; Fietkau, N.; Del Campo, J.; Mas, R.; Munoz, F. X.; Compton, R. G. J. Phys. Chem. C 2007, 111, 12058–12066. (38) Grushka, E. Anal. Chem. 1972, 44, 1733–1738.

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Figure 4. Representation at t ) 0.7(h2/D) of the typical concentration traces generated in this work for evaluating the analytical performance of amperometric detection. (a, c) Step concentration profiles initially x>0 xe0 injected at X ) x/h ) 0 at t ) t0 (i.e., ct-t ) 0 and ct-t ) cmax) and 0)0 0g0 carried through a microfluidic channel by parabolic flow corresponding to a Pe´clet number Pe ) uavh/D ) 40 for two species with identical diffusion coefficients but injected at different times (a) t0 ) 0 or (c) t0 ) ∆Tsep ) 0.2(h2/D). (b, d) Normalized concentration spikes generated through derivation along x (see text: (c/cmax)spike ) [d(c/cmax)step/ dx]/[d(c/cmax)step/dx]max) of the concentration traces shown in (a,,c) respectively. (e) Superimposition of the two concentration spikes shown in (b, d) to model the progression in the channel of two species separated with a retention time difference t0 ) ∆Tsep ) 0.2(h2/D) at x ) 0.

Gaussian spikes, viz., by the derivatives of step functions versus the channel axis coordinate. The result of this procedure is shown by comparing Figure 4a (as results from propagation of a step function injected at x ) 0; compare Figure 1) to Figure 4b, which represents the corresponding concentration spike. The presence of two species at the entrance of the detector may then be modeled by the superimposition of two of such spikes injected into the device (viz., at x ) 0) at t ) 0 and at t ) ∆Tsep to figure the difference in retention times ∆Tsep of the two species (Figure 4b, d) so that the overall signal is depicted in Figure 4e. Figure 5 represents the amperometric traces detected when two spikes featuring species with close retention times pass in front of two different electrochemical detectors. One is composed of a single band electrode (e.g., as in Figure 2b) and the other composed of four small electrodes whose cumulative dimension, including their gaps, is identical to that of the single electrode (e.g., as in Figure 3a compared to Figure 2b). Note that, for simplicity, concentration maximums at x ) 0, numbers of electron exchanges of each electrochemical reaction, and diffusion coefficients of each species were assumed identical. In Figure 5a-c, the individual amperometric traces monitored by each electrode of the array are compared to that monitored by the single electrode detector. Though this is not shown in the figures, a direct summation (i.e., without compensation of time lags) of the individual currents of the array electrode leads to a detection signal similar to that recorded by the single electrode

Figure 5. Comparison of the performances of the four-band electrodes amperometric detector shown in Figure 3a to those of the single band one shown in Figure 2b during the analytical resolution of two concentration spikes flowing through each detector (compare Figure 4e) for Pe ) uavh/D ) 40. Left column: comparison between the individual electrode currents (1 e j e 4) of the four-electrode array (without any correction of time lags) and that of a single electrode detector (isingle) of identical overall dimension. Right column: comparison between the currents monitored by the single electrode reconst detector (isingle) and the current variations (itot ) obtained after application of eq (4) to the individual traces (1-4) monitored by the four-electrode detector and shown in the left column. The data in (d-f) are normalized to their maximum to allow their comparison. The data in (a-f) were obtained for two species with identical diffusion coefficients and exchanging an identical number of electron(s) with the electrodes during their detection. The two species experienced different retention times Tsep at the output of the separative stage: ∆Tsep(D/h2) ) 0.12 (a, d), 0.15 (b, e), and 0.20 (c, f).

detector. However, when the current traces of the array detector are summed according to eq 4, their overall trace becomes much thinner (Figure 5d-f) and closely tracks the concentration-time variations of the analyte. This establishes the excellent analytical performance of the amperometric array detector and its superiority over the single-band one. Indeed, as shown in Figure 5d, two species with extremely close retention times may be identified individually by the array while a single-band detector would merge the two traces. Note in this respect that increasing the number of electrodes in the array would provide even better analytical resolution, though in the example considered here four electrodes were sufficient. Again, we want to evaluate the quality of the detection performed by the multielectrode amperometric detector versus that of a virtual (but ideal) spectroscopic one performing as described above for step profiles. This is shown in Figure 6a-c

for the same conditions as those used in Figure 5 and assuming identical absorbances for the two species. Albeit in this comparison the spectroscopic detection is given an exceptional advantage by assuming that the vertical plane in which absorbances are probed has no width at all, one sees that the performance of the fourelectrode detector is equivalent. Yet, as noted above (Figure 3d) for a single step-concentration case, a real spectroscopic beam would have a significant thickness compared to the microchannel dimensions so that the spectroscopic absorbance would be significantly more sluggish and broader than predicted for the ideal situation considered here. This precision in detection is obtained through a full optimization of the band array detector along a procedure that will be reported in a following work. However, the general principle of this process can be summarized through the following principles, which are rooted in elements disclosed in this work. As evidenced for the single electrode detector (Figure 2), for any given Pe value the electrode widths must be sufficient to allow the electrodes to probe within a significant fraction of the channel height, though being small as compared to h. The size of the solution slice that is then probed is fixed by the maximum number, jmax, of electrodes (viz., of solution “slices”). Both factors, viz., Pe and jmax, then determine the electrode widths. Conversely, the size of interelectrode gaps must be fixed adequately for allowing the content of each slice to be transferred to the slice below (i.e., closer to the electrodes) while the solution relaxes (see Figure 2b) during its travel over the gap. This again is function of Pe and jmax. This is evidence that, given Pe and the maximum number of electrodes in the array, electrodes widths and gap may be optimized to yield the best detection performance. CONCLUSION In this work, we have established that amperometric detectors consisting of a series of band electrodes embedded in the floor of a microchannel may be advantageously used at the output of microfluidic (bio)analytical separative devices to offer exceptional analytical performances that may even outpace those of most advanced spectroscopic ones. This is particularly important since electrochemical measurements at microelectrodes have also intrinsically large signal-to-noise ratios. Furthermore, after their optimization for a specific series of tasks, electronics responsible for electrochemical commands and current recording are sufficiently light to be implemented either onto the analytical microchips themselves during the microfabrication stages or integrated in small stations designed for receiving the microchip. This is of extreme interest since it allows portability, thus enabling in situ analyses, and particularly servicing by nonexpert users.11 Obviously such important advantages may be obtained only through a correct understanding and experimental mastering of the complex coupling between diffusional and convective transport of molecules in microfluidic solutions near an electrochemical detector. As a consequence, array detectors must be optimized for specified operations. This does not appear to be a constraint for light portable devices devoted to a specific type of microfluidic analysis. The various data that have been presented here to illustrate this concept have been selected after optimizing procedures rooted on complete theoretical analyses that we and others have reported previously or which will be fully disclosed in a series Analytical Chemistry, Vol. 80, No. 13, July 1, 2008

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Figure 6. Comparison of the analytical performances of the four-band electrodes amperometric detector (thick curves: reconstructed traces reconst itot already shown in Figure 5d-f) with those of an ideal spectroscopic one (A, thin curves) probing the solution absorbance of the same solution within a vertical plane cross-cutting the microchannel at x ) 10h in the absence of electrochemical detection (see text). The comparison is performed for Pe ) 40 and two species with identical diffusion coefficients, identical absorbances, and exchanging an identical number of electron(s) with the electrodes during their detection, but experiencing different retention times Tsep at the output of the separative stage: ∆Tsep(D/ h2) ) 0.12 (a), 0.15 (b), and 0.20 (c).

Figure 7

of further works, which will also report experimental validations of the present theoretical concept. Finally, we wish to emphasize that though the present detector is apparently useful only for monitoring fluxes of electroactive analytes with sufficient diffusive rates, its application may also be envisioned to detect poorly diffusing species or nonelectroactive ones. Indeed, we have shown elsewhere39 that the flux of any species flowing through a microfluidic channel may be “imprinted” on that of an electroactive fast-diffusing mediator prone to react chemically with the species. Therefore, the problem of detection of poorly diffusing species or nonelectroactive ones passing in front of the detector amounts to detect the depletion in the flux of the fast diffusing mediator “imprinted” by the target analyte, a problem strictly identical to that addresses here. In this respect, the present concept has certainly a wider scope than that strictly considered here. APPENDIX In the text and figures, we referred to several time and distance values (yj, tj, θj) through approximate analytical expressions. The purpose of this appendix is to legitimate these approximate analytical expressions based on a simple model. Our simplified model is based on the projection of trajectories such as those represented by the white curves in Figure 2b or 3a along the channel axis (x) and the vertical axis (y) as sketched in Figure 7. Indeed, though the geometrical projection enlarges the length of the trajectory, it allows us to evaluate the time duration of the overall path since each duration along x or y obeys different transport modes. For simplicity of the analytical formulations, we consider that transport occurs exclusively through convection along the channel horizontal axis x and diffusion along the vertical direction y. Based on the results in Figure 1d,e, one observes that this is a correct approximation as soon as Dt/h2 > 0.2. Similarly, (39) Amatore, C.; Oleinick, A.; Svir, I.; DaMota, N.; Thouin, L. Nonlinear Anal.: Model.Contr. 2006, 11, 345–365.

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we do not consider that the fluid velocity varies along y and consider it always equal to its average value uav. Note that one could take the real fluid velocity into account though this would bring a modest benefice to the validity of the approximation and be associated with much more complicated analytical expressions. Indeed, a full analysis would use the same approach as developed here though applying it to a series of incremental steps where the fluid velocity is constant. The projection (Figure 7) of a trajectory m starting from point (x ) 0, ym) at time t ) 0, and hitting the point (xm, y ) 0) at time tm, shows that tm is the sum of a convective transport duration, (tm)conv, and a diffusional one, (tm)dif. One obtains readily (tm)conv ≈ xm/uav, so that tm ) xm/uav + (tm)dif. (tm)dif represents the duration of the diffusional flight across the channel vertical axis over the distance ym; i.e., it may be approximated by ym2/D ) ym2Pe/(uavh). It ensues that 2 2 Pe ⁄ h) ⁄ uav ) (xm ⁄ uav) + ym ⁄D tm ≈ (xm + ym

(A1)

Remarking that the position of the upstream edge of electrode k)j-1

j is located at xj ) (gj + ∑ (gk + wk)), eq A1 gives immediately k)1 the approximate analytical expression of the time lags tj:

tj )

[(

k)j-1

gj +

) ]

∑ (g + w ) k

k

k)1

⁄ uav + [y2j ⁄ D]

(A2)

This time lag tj is used to define θhalf j , which is plotted in Figure 3c when yj is negligible, i.e., when the length of convective transport is large enough (see Figure 3c). To obtain the analytical formulation of θj, one neglects the second term in eq A2, so that

(

θj ) Dtj ⁄ h2 ≈ gj +

k)j-1

)

∑ (g + w ) D ⁄ (u k

k)1

k

(

2 avh

))

k)j-1

gj +

)

∑ (g + w ) k

k)1

k

⁄ (hPe) (A3)

To evaluate Hj (Figure 3e), one remarks that the lateral distance between the upstream edge of the first electrode to the k)j

downstream one of electrode j is then ∆ xj ) (w1 + ∑ (gk+wk)), k)2

so that ∆(tj)conv ≈ ∆ xj/uav is the overall time allocated for the vertical diffusion to occur while molecules travel along the trajectory in Figure 7. Note that in writing ∆(tj)conv ≈ ∆ xj/uav, we implicitly consider that most of the species travel at a velocity comparable to uav, a fact which assumes that the solution is probed within the very core of the flowing solution. Indeed, near the electrode surfaces the flow velocity is close to zero.24,25 The distance across which molecules may diffuse along y k)j

during this time duration is yj ≈ (D∆(tj)conv) ) [D(w1 + ∑ (gk k)2 + wk))/uav]1/2, which establishes that the two bracketed terms in eq A2 are equal. Finally, noting again that D/uav ) h/Pe, it follows that 1/2

Hj ) yj ⁄ h )

((

k)j

w1 +

) )

∑ (g + w ) k

k)2

k

⁄ hPe

1⁄2

(A4)

ACKNOWLEDGMENT This work was supported in part by CNRS, Ecole Normale Supe´rieure, Universite´ Pierre et Marie Curie, and the French Ministry of Research (UMR 8640, FR 2702, ANR µPHYSCHEMBIO, and LIA XiamENS). NOTE ADDED AFTER ASAP PUBLICATION This paper was posted on May 10, 2008. A small error in a text equation and one in eq 5 were corrected. The paper was reposted on May 20, 2008.

Received for review January 31, 2008. Accepted March 31, 2008. AC800227T

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