Anal. Chem. 2010, 82, 2434–2440
Theory and Experiments of Transport at Channel Microband Electrodes Under Laminar Flow. 3. Electrochemical Detection at Electrode Arrays under Steady State 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 Microband arrays improve the analytical performance and information content of electrochemical detection in flow channel relative to single-electrode configurations. However, exploiting their full advantages requires a detailed understanding of the properties of arrays, which depend on their geometry and on the hydrodynamic regimes established inside the microfluidic channel. This paper investigates the influence of two main operating situations (sequential and coupling regimes) on steady-state amperometric responses of microband arrays performing under laminar flow conditions. Simulations and experimental measurements showed that the resulting properties of the arrays are a function of the number of electrodes and average ratio between gaps and electrode widths, whether the layout of the arrays is regular or not. Since the contribution of each electrode can be finely tailored, this allows the arrays to be designed and adapted to a wide variety of experimental demands. Since the 1980s, multiple microband arrays have been investigated in order to improve sensitivity and selectivity of electrochemical detection for a wide range of electroactive compounds,1 in particular for flow injection analysis (FIA) or high-performance liquid chromatography (HPLC).2-12 As noise generally increases * To whom correspondence should be addressed: e-mail christian.amatore@ ens.fr (C.A.) or
[email protected] (L.T.). (1) Bartelt, J. E.; Deakin, M. R.; Amatore, C.; Wightman, R. M. Anal. Chem. 1988, 60, 2167–2169. (2) Roston, D. A.; Kissinger, P. T. Anal. Chem. 1982, 54, 429–434. (3) Caudill, W. L.; Howell, J. O.; Wightman, R. M. Anal. Chem. 1982, 54, 2532– 2535. (4) Matson, W. R.; Langlais, P.; Volicer, L.; Gamache, P. H.; Bird, E.; Mark, K. A. Clin. Chem. 1984, 30, 1477–1488. (5) Aoki, A.; Matsue, T.; Uchida, I. Anal. Chem. 1990, 62, 2206–2210. (6) Hoogvliet, J. C.; Reijn, J. M.; Vanbennekom, W. P. Anal. Chem. 1991, 63, 2418–2423. (7) Takahashi, M.; Morita, M.; Niwa, O.; Tabei, H. J. Electroanal. Chem. 1992, 335, 253–263. (8) Niwa, O.; Tabei, H.; Solomon, B. P.; Xie, F. M.; Kissinger, P. T. J. Chromatogr. B 1995, 670, 21–28. (9) Chao, M. H.; Huang, H. J. Anal. Chem. 1997, 69, 463–470. (10) Bjorefors, F.; Strandman, C.; Nyholm, L. Electroanalysis 2000, 12, 255– 261. (11) Lawrence, N. S.; Beckett, E. L.; Davis, J.; Compton, R. G. Anal. Biochem. 2002, 303, 1–16. (12) Shi, H. L.; Vigneau-Callahan, K. E.; Matson, W. R.; Kristal, B. S. Anal. Biochem. 2002, 302, 239–245.
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with electrode area more rapidly than signal does,13-15 microelectrode arrays exhibit higher signal-to-noise ratio.16,17 Their electroanalytical performance depends on their design or geometry.18 A first category is represented by interdigitated arrays (IDA), which are composed of two comb-type electrode arrays opposed to one other and individually addressed. One of the most attractive aspects of IDA electrodes is their ability to enhance the redox currents for reversible species, owing to the redox cycling occurring between two adjacent electrodes separated by a narrow gap distance. This process has been extensively used in flow and microfluidic systems.5,7,8,10,16,18-20 The other type is independently addressable arrays (IAA) composed of a series of independent electrodes held at identical or different potentials, the individual currents being recorded independently. As they are attractive in separation systems, many coulometric-amperometric detectors4,12 and multiple amperometric or multichannel detectors6,18,21-23 were also proposed by use of this operating mode. The main advantage of microelectrode arrays in flow channel is the improvement of information content relative to single and dual-electrode configurations. However, their performances depend both on their geometry and on hydrodynamic regimes inside the microchannel.17,24-27 Mass transfer to and from the electrodes is mainly controlled by the convection at high flow rate, while radial diffusion at both edges of the electrodes dominates at lower flow rates.16,17,24,27,28 Simultaneously, reloading of the diffusion layers between each electrode depends partially on the insulating (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28)
Lankelma, J.; Poppe, H. J. Chromatogr. A 1976, 125, 375–388. Weber, S. G.; Purdy, W. C. Anal. Chim. Acta 1978, 100, 531–544. Morgan, D. M.; Weber, S. G. Anal. Chem. 1984, 56, 2560–2567. Anderson, J. L.; Ou, T. Y.; Moldoveanu, S. J. Electroanal. Chem. 1985, 196, 213–226. Fosdick, L. E.; Anderson, J. L. Anal. Chem. 1986, 58, 2481–2485. Matsue, T. Trends Anal. Chem. 1993, 12, 100–108. Morita, M.; Niwa, O.; Horiuchi, T. Electrochim. Acta 1997, 42, 3177–3183. Daniel, D.; Gutz, I. G. R. Talanta 2005, 68, 429–436. Aoki, A.; Matsue, T.; Uchida, I. Anal. Chem. 1992, 64, 44–49. Holcomb, R. E.; Kraly, J. R.; Henry, C. S. Analyst 2009, 134, 486–492. Ordeig, O.; Godino, N.; del Campo, J.; Mun ˜oz, F. X.; Nikolajeff, F.; Nyholm, L. Anal. Chem. 2008, 80, 3622–3632. Cope, D. K.; Tallman, D. E. J. Electroanal. Chem. 1985, 188, 21–31. Aoki, K.; Tokuda, K.; Matsuda, H. J. Electroanal. Chem. 1987, 217, 33– 47. Alden, J. A.; Compton, R. G. J. Electroanal. Chem. 1996, 404, 27–35. Amatore, C.; Da Mota, N.; Sella, C.; Thouin, L. Anal. Chem. 2007, 79, 8502–8510. Moldoveanu, S.; Anderson, J. L. J. Electroanal. Chem. 1985, 185, 239– 252. 10.1021/ac902788v 2010 American Chemical Society Published on Web 02/25/2010
Figure 1. (A) Two-dimensional schematic view of a longitudinal cross section of the microchannel. ux(y) is the parabolic flow velocity directed along the x-axis, wi is the width of successive microband electrodes, gi is the insulating gap distance separating each pair of electrodes, and h is the channel height. (B) Simulated concentration profiles over a regular array performing in the sequential regime. (C) Simulated concentration profiles over a regular array performing in the coupling regime. In panels B and C, n ) 6, Wi ) 1.5, Gi ) 10, and Pe ) 40. The black curves represent 10 isoconcentration lines ranging from C ) 0.05 to 0.95 in steps of 0.1. The white curves represent the streamlines linking the microchannel entrance to the downstream edge of each electrode, Hi being the contribution of each electrode i to the total thickness of solution ∑Hi probed by the array when referred to the channel entrance. The Y scale is expanded 10 times compared to the X scale.
gaps24 and is predominant for widely spaced electrodes.16,17,28,29 Numerical simulations of the convective diffusion transport at channel microelectrode arrays have been then performed to evaluate their subsequent properties.16,17,24,28,30,31 Concentration profiles and currents have been predicted as a function of their geometries, channel dimensions, and flow conditions. Nevertheless, all these studies were mostly dedicated to solve particular problems that called in turn for specific solutions. To enlarge channel electrode applications, we studied all possible regimes that could take place at a single electrode27 and paired electrodes operating under steady state.32 In addition, we defined a new concept of amperometric detectors.33 The aim of the present paper is to implement this fundamental approach by evaluating the influence of key parameters (i.e., number of electrodes, electrode dimension, insulating gap, and flow velocity), controlling both the performance of each electrode and the global response of the array (i.e., current response and thickness of solution layer probed by the electrodes or the array across the channel). In the following, a model was developed in which all electrodes were addressed individually and held at the same constant potential. This model was used to solve numerically the mass transport equation and to analyze the performances of the arrays according to a wide range of geometries and hydrodynamic conditions. The treatment (29) Ou, T. Y.; Anderson, J. L. Anal. Chem. 1991, 63, 1651–1658. (30) Klymenko, O. V.; Oleinick, A. I.; Amatore, C.; Svir, I. Electrochim. Acta 2007, 53, 1100–1106. (31) Amatore, C.; Klymenko, O. V.; Oleinick, A. I.; Svir, I. Anal. Chem. 2009, 81, 7667–7676. (32) Amatore, C.; Da Mota, N.; Lemmer, C.; Pebay, C.; Sella, C.; Thouin, L. Anal. Chem. 2008, 80, 9483–9490. (33) Amatore, C.; Da Mota, N.; Sella, C.; Thouin, L. Anal. Chem. 2008, 80, 4976–4985.
of data obtained from experimental systems was also performed, thus showing extremely good agreement. PRINCIPLE We considered an array composed of a series of n independent parallel microband electrodes placed transversally across the floor of a rectangular microchannel. The electrodes are separated by (n - 1) insulating gaps of individual width gi. When the channel width L is much larger than its height h (which is the most common experimental situation), the device can be described in two dimensions (Figure 1A).34 In the all-generator mode, all the electrodes of the array are biased at a same potential located on the oxidation or reduction plateau of the wave of the redox species to be detected. When the corresponding couple is chemically reversible, mass transport at the electrodes is the only rate-limiting step. Under laminar flow conditions, it is governed by the following equation:
(
)
∂c ∂2c ∂2c ∂c ) D 2 + 2 - ux(y) ∂t ∂x ∂x ∂y
(1)
where D and c are the diffusion coefficient and concentration of the redox species, respectively (see Figure 1A for the definition of x- and y-axis). Under such conditions, diffusion operates along the two directions whereas convection proceeds exclusively along the x-axis. ux(y) is the flow velocity whose profile is assumed to be parabolic across the microchannel section as occurs in Poiseuille flow: (34) Henley, I. E.; Yunus, K.; Fisher, A. C. J. Phys. Chem. B 2003, 107, 3878– 3884.
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2435
ux(y) ) 6uav
y y 1h h
(
)
(2)
uav is the average flow velocity with uav ) 2umax/3. umax is the maximum flow velocity that prevails at y ) h/2. To solve this problem in a most general way, dimensionless parameters need to be introduced. Geometrical parameters are normalized by the microchannel height h with X ) x/h, Y ) y/h, Wi ) wi /h, and Gi ) gi /h. Concentrations are normalized with respect to the homogeneous concentration c0 of the redox species at the entrance of the microchannel: C ) c/c0. The dimensionless time scale is defined as τ ) Dt/h2, which compares the diffusion layer thickness to the channel height. The dimensionless average flow velocity is represented by the Peclet number:
Pe )
uavh D
(3)
Accordingly, eqs 1 and 2 become respectively
(
)
∂C ∂2C ∂2C ∂C ) + - Vx(Y) 2 ∂τ ∂X ∂X ∂Y2
(4)
ux(y)h ) 6PeY(1 - Y) D
(5)
and
Vx(Y) )
Equation 4 is solved numerically at steady state (∂C/∂τ ) 0) by finite differences in association with the following boundary conditions: C ) 1 at the microchannel entrance, C ) 0 at the electrode surfaces, and ∂C/∂Y ) 0 at the insulating surfaces. The dimensionless current Ψi at each electrode Ei is given by
Ψi )
ii FLDc0
)
∫
Wi
0
∂C ∂Y
|
Y)0
dX
(6)
where ii is the corresponding current and F is the Faraday constant. EXPERIMENTAL SECTION The electrochemical reaction consisted of the one-electron oxidation of ferrocene methanol (FcOH, 97%; Acros Organics). Its concentration was about 1 mM in aqueous solution. To ensure the absence of any migration contribution onto the mass transport, 0.1 M potassium chloride was added as the supporting electrolyte.35,36 This solution was previously degassed with argon prior to any experiments. Under these conditions, the diffusion coefficient of FcOH was determined previously to be D ) (7.6 ± 0.4) 10-6 cm2 s-1.27 The fabrication of the devices was performed as already reported.32 They consisted of two separate parts assembled together. The top one, made in polydimethylsiloxane (PDMS), comprised a series of independent channels with their reservoir (35) Bento, M. F.; Thouin, L.; Amatore, C.; Montenegro, I. J. Electroanal. Chem. 1998, 443, 137–148. (36) Drew, S. M.; Wightman, R. M.; Amatore, C. J. Electroanal. Chem. 1991, 317, 117–124.
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elements. This was then fixed onto the channel floor, consisting of a glass substrate on which a series of parallel platinum or gold microband electrodes (Ti/Pt with 5 nm/20 nm or Cr/Au with 5 nm/30 nm) were patterned transversally to the channel x-axis. Each effective microband length was delimited by the microchannel width (L ∼ 510 µm). The volume of solution flowing above the microband was restricted by the microchannel height (h ∼ 20 µm). All the characteristic dimensions (wi, gi, h, and L) were confirmed optically after the device assembling and before use. During each experiment, the microchannel was filled with solution and the liquid flow was pressure-driven by use of either a syringe pump (Harvard Apparatus, type 11 Pico Plus) or a controlled gas flux (argon) imposed over the input reservoir of solution. The values of average flow velocities inside the microchannel were systematically monitored in situ by direct measurements following a procedure previously described.37,38 The Peclet number ranged experimentally from 5 to 320 (eq 3). All electrochemical experiments were performed at room temperature by use of a homemade multipotentiostat (eight recording channels). The counterelectrode (CE) consisted of one large platinum or gold band electrode (w ∼ 500 µm) located downstream, far from the working electrodes, whereas the platinum or gold band used as pseudoreference electrode (REF) was located upstream. The working electrodes Ei were poised at the same potential +0.4 V/REF. This potential was approximately 0.3 V higher than the half-wave potential of FcOH oxidation and high enough to avoid any ohmic drop contribution on the current responses. Their individual currents were monitored except for arrays with more than eight working electrodes. Amperometric measurements were carried out after sufficiently long time had elapsed since the potential application to ensure that a steady-state regime was always achieved. Numerical simulations based on eqs 3-6 were performed with Comsol Multiphysics 3.5 software controlled by the Matlab 7.2 interface. RESULTS AND DISCUSSION Sequential Regime. We first considered the case of arrays in which the gap between electrodes was sufficiently large to prevent any overlap between concentration gradients generated at each electrode (Figure 1B). This situation corresponds exactly to one of the three regimes previously identified in the case of a pair of electrodes E1 and E2.32 Indeed, provided that G1/Pe > 0.25 and (W1 + W2)/G < 1, a sequential regime is established where the composition of flow, after passing over one microband, again becomes homogeneous across the microchannel section before reaching the next one placed downstream. In this situation, each electrode operates individually as in the case of single electrodes (i.e., when only one electrode is performing in the microchannel) since they probe solutions that are homogeneous over their upstream edge. However, the difference is that each electrode analyzes a solution of lower concentration than those placed upstream. For a two-electrode configuration, it was previously established (eq 26 in ref 27) that the concentration reached in the flow after passing over the first electrode was (37) Amatore, C.; Belotti, M.; Chen, Y.; Roy, E.; Sella, C.; Thouin, L. J. Electroanal. Chem. 2004, 573, 333–343. (38) Amatore, C.; Chen, Y.; Sella, C.; Thouin, L. Houille Blanche 2006, 60–64.
Ch,2 ) C0 -
Ψ1 Ψ1 )1Pe Pe
(7)
as follows from straight application of the mass conservation law. Since under this regime all electrodes are independent, iterative application of eq 7 shows that the concentration Ch,i probed by the ith electrode is related to that probed by the (i - 1)th one by Ch,i ) Ch,i-1 -
Ψi-1 Pe
with Ch,1 ) C0
(8)
where Ψi-1 is the current flowing at the (i - 1)th electrode. For n electrodes, eq 8 gives the concentration Cexit of the solution achieved after passing over the whole array: n
Ψi
∑ Pe
Cexit ) 1 -
(9)
i)1
On the other side, upon defining Ψi0 as the current monitored at the ith electrode when those upstream are not active (i.e., when probing a concentration C0), one obtains n
Cexit )
∏ i)1
(
1-
Ψi0 Pe
)
(10)
Combination of eqs 9 and 10 affords n
∑Ψ
i
i)1
Pe
n
)1-
∏ i)1
(
1-
Ψi0 Pe
)
(11)
Note that when all electrodes Ei have identical dimensions, Wi ) W, then Ψi0 ) Ψ1 and eq 11 becomes n
∑Ψ i)1
Pe
i
(
)1- 1-
Ψ1 Pe
)
n
(12)
The term ∑Ψi /Pe in eqs 11 and 12 is equal to (1 - Cexit) (eq 9). Thus, owing to the formulations used in this work, it represents the detection efficiency of the array. This extends the equivalent formula that can be established in the case of a pair of electrodes. When the redox species is quantitatively detected, Cexit is equal to 0 and ∑Ψi /Pe is maximal and equal to 1. Equations 11 and 12 allow ∑Ψi /Pe to be estimated in a sequential regime on the basis of variations of Ψ1/Pe versus W1/Pe simulated in the case of a single electrode.27 In the following, most of the calculations were performed based on eq 12, that is, when all electrode widths are equal. However, it was checked independently that eq 11 gives the same results for arrays presenting nonregular electrode widths. When ∑Ψi /Pe variations as a function of ∑Wi /Pe are plotted for different n values (Figure 2A), it is observed that the plots distribute smoothly between the case n ) 1 and the step function achieved for n f ∞. Since ∑Ψi /Pe increases with both n and the total active surface area (∑Wi ), identical detection efficiencies ∑Ψi /
Figure 2. Global array efficiency ∑Ψi /Pe in the sequential regime. (A) Variation of ∑Ψi /Pe as a function of ∑Wi /Pe for n ) 1, 2, 4, 6, 20, 50, 100, 500, and ∞. (B) Variation of log (∑Ψi /Pe) as a function of log (∑Wi /Pe) for n ) 20. The dashed lines correspond to the Levich regime (eq 14) and the dotted line corresponds to the thin-layer regime (eq 13).
Pe may be achieved upon either increasing n or decreasing ∑Wi. Yet larger n values allow a more precise time definition,33 though this cannot be evaluated in this work through considering a constant concentration flow. Similarly, larger n values provide better signal-to-noise ratios due to the enhancement of current densities at smaller microelectrodes. This is presently one of the main advantages of using microelectrode arrays in comparison to a single electrode of larger dimension. If we now consider the hydrodynamic regimes, two limiting behaviors may be distinguished according to the parameter ∑Wi / Pe(Figure 2B). Indeed, a thin-layer regime may be reached when the redox species entering the microchannel is quantitatively detected.39 This situation observed for single-electrode configuration (eq 18 in ref 27) is fulfilled when the detection efficiency of the array is maximal and equal to 1: n
∑Ψ i)1
Pe
i
)1
(13)
Similarly, the Levich regime observed for a single electrode (eq 17 in ref 27) is achieved at the array when
i)1
Pe
( )
2/3
n
n
∑Ψ
i
) 1.468n1/3
∑W
i
i)1
Pe
(14)
The transition between these two regimes was estimated according to the following criteria. The thin-layer regime was considered (39) Scialdone, O.; Guarisco, C.; Galia, A.; Filardo, G.; Silvestri, G.; Amatore, C.; Sella, C.; Thouin, L. J. Electroanal. Chem. 2010, 638, 293–296.
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Figure 3. Zone diagram (n, ∑Wi /Pe) showing the different hydrodynamic regimes observed for a sequential regime: zone I, thin-layer regime with ∑Wi /Pe > 0.95; zone II, intermediate regime; and zone III, Levich regime with ∑Ψi /[1.468 n1/3 Pe1/3(∑Wi)2/3] < 0.95. The dashed curve corresponds to eq 15 and represents equal contributions of each limiting regime.
to be reached for ∑Ψi /Pe g 0.95 (see eq 13). Reciprocally, the Levich regime was considered to be established when ∑Ψi /[1.468 n1/3 Pe1/3(∑Wi )2/3] e 0.95 (see eq 14). When these data were reported on a diagram (n, ∑Wi /Pe), three zones were clearly distinguished (Figure 3). Zones I and III correspond respectively to the thin-layer and Levich regimes, whereas zone II featured an intermediate behavior between the two others due to a mixed regime. When n increases, all transitions take place at lower values of ∑Wi /Pe. This result demonstrates again that if hydrodynamic regimes equivalent to single-electrode configurations may be achieved for arrays, this happens upon involving lower electroactive surface areas. In zone II, the proportion between fulldiffusive (as in zone I) and diffusive-convective (as in zone III) components could be evaluated upon considering the equality between eqs 13 and 14. This condition amounts to
n
i)1
Wi
i)1
Pe
Pe -1/2
) 0.56n
W ) 0.56n-3/2 Pe
(16)
Thus, diffusion dominates when ∑Wi /Pe > 0.56n-1/2, whereas convection does so when ∑Wi /Pe < 0.56n-1/2. This specific transition was also reported in Figure 3. It shows that when 10 < n < 100, the limit occurs mainly for ∑Wi /Pe ≈ 0.1. This corresponds to an average efficiency of ∑Ψi /Pe ≈ 0.65 (Figure 2A). Note that one still obtains from both sides of the transition a signal proportional to c0 (see eq 6). In contrast to zones I and III, for which expressions of ∑Ψi /Pe as a function of ∑Wi /Pe are available (eqs 13 and 14), no explicit relation between current and concentration is obtained in zone II. Only eq 11 allows ∑Ψi / Pe to be evaluated from individual contributions Ψi0/Pe. Values of Ψi0/Pe are those simulated and reported in Figure 2A with n ) 1. As in our previous works,27,32 simulations allowed also the determination of the thickness of solution ∑Hi probed by the array at the entrance of the microchannel (Figure 1B). Figure Analytical Chemistry, Vol. 82, No. 6, March 15, 2010
( ) ( )
i
2
n
)3
(15)
For an array with electrodes having the same dimension Wi, this equation simplifies to
2438
4A shows that the variation of ∑Hi as a function of ∑Wi /Pe or ∑Ψi /Pe is similar to the ones already reported for singleelectrode configurations. As for eq 23 in ref 27, the relationship between ∑Ψi /Pe and ∑Hi for arrays is
∑Ψ
n
∑
Figure 4. (A) Variation of ∑Ψi /Pe vs ∑Hi in sequential or coupling regime. The solid curve corresponds to eq 17. (B) Variation of ∑Hi /H vs log (∑Wi /Pe) in sequential regime for different values of n. Η is the thickness of probed solution layer by a single electrode of dimension W ) ∑Wi, that is, of identical electroactive surface area.
∑H
i
3
n
-2
i)1
∑H
i
(17)
i)1
if one considers H as the thickness of solution probed by a single electrode of dimension W equal to the total active surface area of an array (i.e., W ) ∑Wi). Simulations showed that the ratio ∑Hi /H varies with ∑Wi /Pe and is maximal when ∑Wi /Pe is sufficiently low (Figure 4B). This means that the gain in ∑Hi is the highest when convection dominates (i.e., in zone III). The same trend was observed when the efficiency of the array ∑Ψi / Pe was compared to that of a single electrode of the same active surface area (data not shown). Simulations showed also that, for a fixed value of ∑Wi /Pe, the ratio ∑Hi /H varied as a function of n1/6. Coupling Regime. This regime refers to the situation where the gaps between the electrodes are not sufficiently large to allow the solution to be rehomogenized between two successive electrodes (Figure 1C).27,32 In this case, eq 17 still applies. However, due to the overlap between the concentration gradient patterns between successive electrodes, the properties of the arrays depend also on the gaps and dimensions of the electrodes through a parameter Λ, defined as
Λ)
2 n-1
n-1
∑W i)1
i
Gi + Wi+1
(18)
Figure 5. Influence of Λ on the conversion efficiency ∑Ψi /Pe for arrays operating in coupling or sequential regimes for n ) 20. (A) Variation of log (∑Ψi /Pe) as a function of log (∑Wi /Pe) for Λ ) 0, 0.2, 0.5, 1, 2, 4, 15, and ∞. (B) Variation of ∑Ψi /Pe as a function of log Λ for ∑Wi /Pe ) 0.05 and 0.5. The dotted line corresponds to the variation of Λh given in eq 20.
This parameter integrates the ratios between gaps and electrode widths to provide the corresponding average value over the microchannel. Note that when the layout of the array is regular, eq 18 becomes Λ)
G W
(19)
Λ is equal to 0 in case of single-electrode configurations. The influence of Λ is reported in Figure 5A for a large number of electrodes (n ) 20). In comparison to Figure 2B, similar variations of ∑Ψi /Pe versus ∑Wi /Pe were observed whatever the arrays were regular or not. However, ∑Ψi /Pe increased with Λ up to a given limiting behavior imposed by n (Figure 5A). This limiting behavior corresponds to the situation where the sequential regime is reached, that is, when ∑Ψi /Pe does not depend any longer on the gap distances separating the electrodes. These results confirmed that the efficiencies were the highest in sequential regime. For paired electrodes (n ) 2), the transition between the two regimes was shown previously to be achieved when G/Pe ) 0.25 and (W1 + W2)/G < 1.32 For electrode arrays, this condition remained fulfilled provided that Λh )
nPe 4
∑W
> 10
(20)
i
When 0 < Λ < Λh or Λh < 10, coupling regimes prevail, thus leading to lower efficiencies for given values of ∑Wi /Pe (Figure 5B). Experimental Illustration. The experimental validity of the above model was examined by comparing the simulated currents with the experimental responses measured for two regular arrays. They had the same number of electrodes (n ) 5) but two different
Figure 6. Comparison between experimental (symbols) and simulated (solid lines) currents for two regular arrays equipped with n ) 5 band electrodes. (A) Variation of the individual currents Ψi as a function of Pe with i ) 1 (b), 2 (1), 3 (9), 4 ([), and 5 (2) for Λ ) 15 (w ) 40 µm, g ) 600 µm). (B) Same for Λ ) 0.7 (w ) 30 µm, g ) 21 µm) with i ) 1 (O), 2 (∇), 3 (0), 4 (]), and 5 (∆). (C) Variation of log (∑Ψi /Pe) vs log (∑Wi /Pe) for the data in panel A (solid symbols) or panel B (open symbols). The dashed line corresponds to Λ ) 0 and the dotted line to Λ f ∞. Experimental data: h ) 20 µm and L ) 510 µm; FcOH, 1 mM in 0.1 KCl.
G/W values, Λ ) 15 or Λ ) 0.7. As illustrated in Figure 6, very good agreement was observed between data, whatever the array considered. Under these conditions, Λh could not exceed 15 according to the range of ∑Wi /Pe values investigated experimentally. Therefore, a sequential regime was always observed when Λ ) 15 (Figure 6A), whereas a coupling regime prevailed when Λ ) 0.7 (Figure 6B). Figure 6C compares the efficiencies ∑Ψi /Pe of each array as a function of ∑Wi /Pe. It shows that within the accuracy of measurements, the differences between the two regimes could be well monitored. Such results established the extremely good agreement between predictions and experimental measurements over the whole span between the two limiting regimes (Figure 6C). Comparison between Regular and Nonregular Arrays. All regular arrays characterized by the same set of parameters (n, Λ) displayed the same variations of ∑Ψi /Pe as a function of ∑Wi / Pe (Figure 5A), for sequential or coupling regime. The above property was true for irregular arrays even if a given set of parameters (n, Λ) corresponded to an infinity of layouts (i.e., to various dimensions of electrodes and gaps distributed along the microchannel). This characteristic is of fundamental interest since it shows that the properties of an electrode array may be tuned locally according to the contribution of each current Ψi or thickness Hi to the global response.31 According to the applications envisaged,17,28,31 the optimization of the layouts may depends strongly on whether the electrodes have the same current Ψi, have the same current density Ψi /Wi, or probe the same fraction Hi of the channel height. Indeed, the current at each electrode may differ for arrays displaying the same efficiency ∑Ψi /Pe or the same ∑Hi. For example, when n ) 5 and Λ ) 15, the currents Ψi or efficiencies ∑Ψi /Pe may decrease or increase from upstream to downstream electrodes according Analytical Chemistry, Vol. 82, No. 6, March 15, 2010
2439
solution entering into the microchannel or for reconstructing the flow rate across the channel height.31
Figure 7. Contributions of each electrode to the global response for regular and nonregular arrays with n ) 5, R ) 15, and ∑Ψi /Pe ) 0.43: (A) current Ψi /∑Ψi; (B) thickness Hi /∑Hi. (- - -) Regular array with W ) 2, G ) 30, and Pe ) 94. (s) Array with decreasing electrode widths Wi ) 5, 4, 3, 2, and 1; Gi ) 65, 50, 40, and 23; and Pe ) 142. (---) Array with increasing electrode widths Wi ) 1, 2, 3, 4, and 5; Gi ) 23, 40, 50, and 65; and Pe ) 142. ( · · · ) Array with increasing electrode widths Wi ) 0.3, 1.2, 3, 5, and 7; Gi ) 12, 33, 60, and 84; and Pe ) 156. Solid circles in panel A correspond to the experimental data shown in Figure 6A for ∑Wi /Pe ) 0.106.
to the design of the array (Figure 7A). In the case of regular arrays, the current Ψi always decreases (Figure 7A). A comparison between simulated and experimental currents showed very good agreement. The same trends were observed for simulated Hi (Figure 7B), though this could not be tested experimentally. Interestingly, arrays with increasing electrode widths along the microchannel may allow identical thickness Hi to be probed. This can be relevant to test, for example, the homogeneity of the
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Analytical Chemistry, Vol. 82, No. 6, March 15, 2010
CONCLUSION The present model and ensuing simulations under steady-state regime showed that different hydrodynamic regimes can be achieved when all the electrodes operate at a same constant potential. This result extends and generalizes that reported previously for single electrodes. Indeed, a zone diagram can be established in sequential regime showing respectively the predominance areas of the Levich and thin-layer regimes on the global response. As for single electrodes, arrays may probe a fraction or the whole height of the microchannel. Despite this generalization, microelectrode arrays always exhibit advanced electrochemical performance. The detection efficiency of an array remains always higher than that of a single electrode under similar flow conditions, thus leading to better signal-to-noise ratios. However, several designs or geometries of arrays with the same global performance may be considered, depending on the individual electrode contributions. If one needs to optimize the detection, sequential regimes will be preferred in order to reach higher efficiencies. The number of electrode and their relative sizes will determine in turn the individual thicknesses of solution layer probed upstream. This modulation can be achieved according to numerous and specific criteria. ACKNOWLEDGMENT This work has been supported in part by the CNRS (UMR8640), Ecole Normale Superieure, UPMC French Ministry of Research, and ANR µPHYSCHEMBIO. The group of Professor R. Compton is gratefully acknowledged for providing help in the design of the multipotentiostat. Received for review December 8, 2009. Accepted February 12, 2010. AC902788V
