Electrochemical Determination of Flow Velocity Proﬁle in a Microﬂuidic Channel from Steady-State Currents: Numerical Approach and Optimization of Electrode Layout Christian Amatore,* ,† Oleksiy V. Klymenko, ‡ Alexander I. Oleinick, ‡ and Irina Svir* ,†,‡ De ´ partement de Chimie, Ecole Normale Supe ´ rieure, UMR CNRS-ENS-UPMC 8640 “PASTEUR”, 24 rue Lhomond, 75231 Paris Cedex 05, France, and Mathematical and Computer Modelling Laboratory, Kharkov National University of Radioelectronics, 14 Lenin Avenue, 61166 Kharkov, Ukraine In this article, the numerical approach for ﬂow proﬁle reconstruction in a microﬂuidic channel equipped with band microelectrodes introduced previously by the au- thors, based on transient currents, is extended to the exclusive use of steady-state currents. It is shown that, although the currents obey steady state, the ﬂow velocity proﬁle in the channel may be reconstructed rapidly with a high accuracy, provided a sufﬁcient number of elec- trodes performing under steady state are considered. The present theory demonstrates how the electrode widths and sizes of gaps separating them can be optimized to achieve better performance of the method. This approach has been evaluated theoretically for band microelectrode arrays embedded into one wall of a rectangular channel consisting of three, four, or ﬁve electrodes, all of which are operated in the generator mode. The results prove that the proposed approach is able to accurately recover the shape of the ﬂow proﬁle in a wide range of Peclet numbers and ﬂow types ranging from the classical parabolic Poi- seuille ﬂow to constant electro-osmotic-type ﬂow. The miniaturization of analytical devices has been shown to provide many speciﬁc advantages, 1 mostly because (i) the amount of sample required for the analysis is minimized and (ii) the ﬂow velocity proﬁles of the solution carriers are expected to be better controlled. However, the reduction of the amount of solution makes precise monitoring of the ﬂow an extremely difﬁcult challenge. When this is a critical issue, one must resort to sophisticated methods, most of them generally involving the direct tracking of particles carried by the ﬂow. 2 Such procedures require obviously good optical quality of the material deﬁning the channel in which the ﬂow is monitored and the use of sophisticated algorithms to follow three-dimensional (3D) displacements of particles from a series of frames. Furthermore, one cannot envision a continuous monitoring of the ﬂow while the microﬂuidic chip is performing its analytical function, so that feedback control of the ﬂow is prevented. In previous work, we have examined how arrays of microband electrodes placed transversely on the microchannel ﬂoor could supply important information about the ﬂow velocity. 3 In a series of attempts, some of us established, theoretically and experimen- tally, that the average ﬂow velocity could be measured by a pair of microbands performing in generator-collector mode. However, such procedures did not supply any information about the effective ﬂow velocity distribution, and it was assumed that this obeyed a classical parabolic Poiseuille regime. In a second series of works, we established theoretically that not only the average ﬂow velocity but its distribution could be extracted from transient electrochemi- cal currents. 4 A related work involved voltammetry for monitoring transient hydrodynamic ﬂow proﬁles in microﬂuidic ﬂow cells. 5 However, despite many attempts in our laboratory and by others, it seems that the accurate monitoring of transient electrochemical chronoamperometric currents is extremely difﬁcult and certainly may not be used in a routine way, because the transient components are altered by capacitive and ohmic phenomena that cannot be controlled with the proper precision. Conversely, steady- state currents can be monitored with an adequate precision without signiﬁcant experimental difﬁculties. * Authors to whom correspondence should be addressed. E-mails: christian. amatore@ens.fr (C.A.); irina.svir@ens.fr (I.S.). † De ´ partement de Chimie, Ecole Normale Supe ´ rieure. ‡ Mathematical and Computer Modelling Laboratory, Kharkov National University of Radioelectronics. (1) (a) Rossier, J. S.; Girault, H. H. Lab Chip 2001, 1, 153–157. (b) Kan, C.-W.; Fredlake, C. P.; Doherty, E. A. S.; Barron, A. E. Electrophoresis 2004, 25, 3564–3588. (c) Dayon, L.; Josserand, J.; Girault, H. H. Phys. Chem. Chem. Phys. 2005, 7, 4054–4060. (d) Ordeig, O.; Godino, N.; Del Campo, J.; Muno ˜z, F. X.; Nikolajeff, F.; Nyholm, L. Anal. Chem. 2008, 80, 3622–3632. (2) For recent examples, see:(a) Plecis, A.; Malaquin, L.; Chen, Y. J. Appl. Phys. 2008, 104, 124909. (b) Boxx, I.; Heeger, C.; Gordon, R.; Bohm, B.; Aigner, M.; Dreizler, A.; Meier, W. Proc. Combust. Inst. 2009, 32, 905–912. (c) Li, C.-T.; Chang, K.-C.; Wang, M.-R. Exp. Therm. Fluid Sci. 2009, 33, 527–537. (d) Wu, S.-C. Exp. Therm. Fluid Sci. 2009, 33 (5), 875-882. (3) (a) Amatore, C.; Belotti, M.; Chen, Y.; Roy, E.; Sella, C.; Thouin, L. J. Electroanal. Chem. 2004, 573, 333–343. (b) Amatore, C.; Oleinick, A.; Svir, I. Electrochem. Commun. 2004, 6, 1123–1130. (c) Amatore, C.; Chen, Y.; Sella, C.; Thouin, L. Houille Blanche 2006, 2, 60–64. (d) Amatore, C.; Sella, C.; Thouin, L. J. Electroanal. Chem. 2006, 593, 194–202. (e) Amatore, C.; Da Mota, N.; Sella, C.; Thouin, L. Anal. Chem. 2007, 79, 8502–8510. (f) Amatore, C.; Da Mota, N.; Sella, C.; Thouin, L. Anal. Chem. 2008, 80, 4976– 4985. (4) (a) Amatore, C.; Oleinick, A.; Klymenko, O. V.; Svir, I. ChemPhysChem 2005, 6, 1581–1589. (b) Amatore, C.; Klymenko, O. V.; Svir, I. ChemPhysChem 2006, 7, 482–487. (c) Amatore, C.; Klymenko, O. V.; Oleinick, A.; Svir, I. ChemPhysChem 2007, 8, 1870–1874. (d) Klymenko, O. V.; Oleinick, A. I.; Amatore, C.; Svir, I. Electrochim. Acta 2007, 53, 1100–1106. (5) Thompson, M.; Compton, R. G. Anal. Chem. 2007, 79, 626–631. Anal. Chem. 2009, 81, 7667–7676 10.1021/ac9010827 CCC: $40.75  2009 American Chemical Society 7667 Analytical Chemistry, Vol. 81, No. 18, September 15, 2009 Published on Web 08/21/2009