# Pushing the Limits of Electrical Detection of Ultralow Flows in Nanofluidic Channels

^{*}

## Abstract

**:**

## 1. Introduction

**Figure 1.**Schematic of the experimental setup. Fluctuations in the number density of electroactive molecules are used as tracers of liquid flow as solution is transported through a nanochannel. The fluctuations are detected electrically by redox cycling and their time of flight between the detectors—or, equivalently, the flow velocity—is then determined by cross correlation analysis of current-time traces (curves in the insets are a schematic illustration).

## 2. Device Fabrication

**Figure 2.**(

**a**) Schematic cross section of the device along the longitudinal (top) and lateral axis (bottom). (

**b**) Top view micrograph of a 202 μm long nanofluidic device bonded to a polydimethylsiloxane (PDMS) microchannel layer (only two out of four microchannels running in parallel to the device are shown). (

**c**) Micrograph of the overall chip structure. (

**d**) Photography of a chip bonded to PDMS connected to tubing; electrical contact pads are visible on the right.

## 3. Experimental

_{2}from Acros, diffusion coefficient D = 6.7 × 10

^{−10}m

^{2}/s) was chosen as redox-active species and prepared as a 1 mM solution in Milli-Q water with 0.1 M KCl (Sigma-Aldrich) added as background electrolyte together with 5 mM H

_{2}SO

_{4}(Sigma-Aldrich) to prevent electrode degradation. Directly before a measurement, the Cr sacrificial layer was removed and the nanochannel was formed by purging the microchannels with a chromium etchant solution (Selectipur, BASF). The redox active solution was then driven through the channel with varying syringe pump flow rates of up to 60 μL/h.

## 4. Results and Discussion

_{2}molecules from the center of the first transducer to the center of the second one located 102 μm downstream. In Figure 3, cross-correlation functions are shown as a function of the syringe pump flow rate and compared to analytical functions for one-dimensional drift and diffusion in the same device geometry [4].

**Figure 3.**(

**a**) Cross-correlation functions for different syringe pump flow rates determined from current-time traces recorded at both 100 μm long top electrodes of a 202 μm long nanofluidic device. (

**b**) Analytically derived cross-correlation functions for different molecular drift velocities v for the same device geometry and an effective diffusion coefficient of D = 3.4 × 10

^{−10}m

^{2}/s.

**Figure 4.**Nanofluidic flow rates as a function of syringe flow. The adjusted experimental data points are corrected for the shift of the peak times as well as for dynamic adsorption. The dashed line’s slope corresponds to the ratio of the micro- and nanochannels’ resistances of 1/400,000.

^{7}m

^{−1}, the Fc(MeOH)

_{2}molecules undergo pronounced dynamic reversible adsorption at the electrodes and nanochannel walls, which slows down their average transport with respect to the fluid. We estimate the relative number of adsorbed molecules by stochastic chronoamperometry [8]. Its magnitude exhibits considerable scatter but amounts approximately to = 0.5, i.e., the molecules are slowed down to 50% of the mean liquid flow velocity.

^{−10}m

^{2}/s (green curve) and an adjusted effective D

_{eff}= 0.5 D, which takes into account the reduction of diffusive transport by the dynamic adsorption. In Figure 5b, this effect is exemplified by analytical cross-correlation functions with a constant time of flight of 4 s and flow rates ranging from 250 μm/s to 10 μm/s (and corresponding electrode lengths of 40 μm to 1000 μm with no gap in between.)

**Figure 5.**(

**a**) Analytically determined deviation of the cross-correlation function’s peak time from the time of flight as a function of flow velocity. The blue curve is corrected for an effectively slower diffusion of the molecules due to dynamic adsorption. Experimental flow rates range from 15 μm/s to 50 μm/s. (

**b**) Cross-correlation functions for a constant time of flight of 4 s but with different molecular velocities v (and corresponding different electrode lengths L or flight paths, respectively).

#### 4.1. Detection Sensitivity

- The nanochannel has a height of 55 nm instead of 130 nm. For the same average flow velocity, the flow rate is simply reduced by the smaller channel cross-section.
- The length of each electrode is increased from 50 μm to 100 μm. As shown below (Figure 6), this increases the signal strength because the correlation in a longer plug of fluid takes more time to decay. However, a longer electrode also has the undesirable effect of decreasing the SNR according to Equation (2), ( 𝑣/𝐿)
^{1}^{/2}. For our measurement parameters (T > 100 s, v > 10 μm/s, L = 100 μm, SNR > 10), this is however not limiting. - Since our method relies on the detection of purely stochastic noise, sensitivity is increased by sampling traces with longer durations T. Our previous instrumentation (Keithley 6430 sub-femtoamp source meters) limited the current-time traces to a length of 25 s (at a fast = 100 s
^{−1}acquisition rate). The instrumentation used here allowed extending the measurement period to up to 600 s at the same sampling interval. Therefore the traces are considerably longer than the decay time and the SNR is improved.

#### 4.2. Geometric Design Rules

^{−10}m

^{2}/s and no gap between the electrodes . With increasing sensor length the magnitude of the peak in the correlation function increases, which is favorable. The position of the peak however simultaneously shifts to longer times. Therefore, increasing the detector length beyond several hundreds of μm will not deliver a more sensitive flow detection, since long times of flight are obscured by instrumental drift. Furthermore, the pressure necessary to drive a given flow increases linearly with the channel length and can then exceed 1 bar for a typical channel cross-section, which is often impractically high.

**Figure 6.**Analytical cross-correlations function as a function of electrode length L for symmetric transducers for a constant velocity v = 100 μm/s, D = 5 × 10

^{−10}m

^{2}/s and no gap in between the consecutive sensors. The cross-correlation peaks are more pronounced with increasing L and shift to longer .

**Figure 7.**Analytical correlation functions for a varying separation distance L

_{2}- L

_{1}in between consecutive electrodes for v = 100 μm/s, D = 5 × 10

^{−10}m

^{2}/s, 100 μm. Green curve: Autocorrelation, which corresponds to L

_{2}= 0; blue curves: intermediate auto-cross-correlation for a gap length L

_{2}- L

_{1}= −75 μm, −50 μm, −25 μm; red, orange, purple curves: cross-correlation with L

_{2}- L

_{1}= 0 μm…1000 μm.

## 5. Conclusion and Outlook

## Acknowledgments

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**MDPI and ACS Style**

Mathwig, K.; Lemay, S.G.
Pushing the Limits of Electrical Detection of Ultralow Flows in Nanofluidic Channels. *Micromachines* **2013**, *4*, 138-148.
https://doi.org/10.3390/mi4020138

**AMA Style**

Mathwig K, Lemay SG.
Pushing the Limits of Electrical Detection of Ultralow Flows in Nanofluidic Channels. *Micromachines*. 2013; 4(2):138-148.
https://doi.org/10.3390/mi4020138

**Chicago/Turabian Style**

Mathwig, Klaus, and Serge G. Lemay.
2013. "Pushing the Limits of Electrical Detection of Ultralow Flows in Nanofluidic Channels" *Micromachines* 4, no. 2: 138-148.
https://doi.org/10.3390/mi4020138