Downstream Impedance in Microfluidic Channels
Authors: Holm Thomas , Ingdal Mats , Strobl Jonathan R , Fanavoll Espen Vinge , Sunde Svein , Seland Frode , Harrington David A. 
Published: February 2020

Channel electrodes, in the wall of a microfluidic channel, allow generator-collector schemes with solution species generated at an upstream electrode and collected at a downstream electrode, possibly with homogenous reactions occurring during the mass transport between the electrodes. These are useful devices to study electrocatalytic reaction mechanisms, especially for oxidation of small organic molecules, where there are multiple pathways and multiple soluble intermediates and products. We briefly review our recent experimental and theoretical advances toward this goal, which include the implementation of a PdH reference in a side channel [1], a new semianalytical method for convective diffusion in rectangular channels [2], and a numerical investigation of the validity of the Lévêque approximation and neglect of axial diffusion for impedance at a single channel electrode [3].
The experimental and theoretical study of the double channel generator-sensor impedance are then described for a reversible solution redox couple (Ru(II/III) hexammine complex) [4]. By using galvanostatic generation of the ac signal at the upstream working electrode and the working sense connection to measure the ac potential at the downstream sensor electrode, we were able to implement a generator-sensor electrode scheme with a single potentiostat. We define the "downstream impedance" as the ratio of the ac potential measured downstream to the ac current measured upstream.
The downstream impedance shows beautiful spirals. The phase may be interpreted simply in terms of the propagation time between the two electrodes relative to the period of the a.c. signal. This simple picture predicts that the phase is linear with the frequency. If there were no diffusive spread as the concentration wave moved downstream, the impedance would be a circle in the complex plane. However, diffusion across the channel during downstream propagation leads to a decreasing amplitude with frequency and therefore spirals rather than circles.
The full solution of the convective-diffusion problem was solved numerically using COMSOL and showed spirals in good agreement with the experimental results. A dimensional analysis for the simplifying assumptions of the Lévêque approximation and neglect of axial diffusion shows that the impedance depends only on a single parameter, a dimensionless frequency Ω. An analytical solution of the downstream impedance under these assumptions was possible for the case of zero frequency, which was found to agree with the numerical solution within 10% except at the lowest flow rates. As predicted by this model, when normalized by the zero-frequency impedance, the complex plane plots for all flow rates fall on a common curve. Bode plots of the normalized impedance also fall on common curves. The slope of log(phase) vs log(Ω) plots is 1 for Ω<1 as predicted by the simplistic model.
We thank the Research Council of Norway, the Natural Science and Engineering Research Council of Norway, and our respective institutions for financial support.
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