MASS TRANSPORT
IN ELECTROCHEMICAL
NANOFLUIDIC DETECTORS
MASS TRANSPORT IN
ELECTROCHEMICAL NANOFLUIDIC
DETECTORS
The graduation committee consists of:
Chairman and Secretary
Prof. Dr. ir. J.W.M. Hilgenkamp University of Twente, the Netherlands
Promotor
Prof. Dr. S.J.G. Lemay University of Twente, the Netherlands
Members
Prof. Dr. W.J. Briels University of Twente, the Netherlands Prof. Dr. J.C.T. Eijkel University of Twente, the Netherlands Prof. Dr. ir. C.R. Kleijn Delft University of Technology,
the Netherlands
Prof. Dr. H. Schiessel Leiden University, the Netherlands Prof. Dr. K. Tschulik Ruhr-University Bochum, Germany
This research was financially supported by the National Institutes of Health (NIH, USA) and carried out at the Nanoionics group, MESA+ Institute for Nanotech-nology, Faculty of Science and TechNanotech-nology, University of Twente, The Netherlands.
Title: Mass transport in electrochemical nanofluidic detectors Author: Zinaida Kostiuchenko
ISBN: 978-90-365-4546-4 DOI: 10.3990/1.9789036545464
MASS TRANSPORT IN
ELECTROCHEMICAL NANOFLUIDIC
DETECTORS
PROEFSCHRIFT
ter verkrijging van
de graad van doctor aan de Universiteit Twente, op gezag van de rector magnificus,
prof.dr. T.T.M. Palstra,
volgens besluit van het College voor Promoties, in het openbaar te verdedigen
op woensdag 2 mai 2018 om 14.45 uur
door
Zinaida Kostiuchenko geboren op 23 November 1988
This dissertation is approved by: Prof. Dr. S.J.G. Lemay (promotor)
C
ONTENTS
1 Introduction 1
1.1 Electrochemical detection . . . 2
1.2 Nanofluidics . . . 4
1.3 The content of the thesis . . . 6
References . . . 7
2 Ionic transport in nanoscale systems 9 2.1 Introduction . . . 10
2.2 Ion concentration polarization . . . 10
2.3 Electrochemical nanofluidics devices . . . 13
2.4 Electrochemically induced concentration polarization . . . 15
2.5 Conclusions . . . 19
References . . . 20
3 Generator-collector experiment 25 3.1 Introduction . . . 26
3.2 System description . . . 27
3.3 Mass transport description . . . 29
3.4 Experimental methods . . . 30
3.5 Results – Collector current . . . 31
3.6 Results – Generator current . . . 35
3.7 Discussion . . . 36
3.8 Summary . . . 38
References . . . 39
4 Nonlocal interactions between electrodes in nanofluidic devices 43 4.1 Introduction . . . 44
4.2 Materials and methods . . . 44
4.2.1 Description of the experimental system . . . 44
ii
4.2.2 Chemicals . . . 47
4.2.3 Flow actuation . . . 47
4.3 Experimental results . . . 48
4.3.1 Redox-cycling current in nanofluidic devices under flow control 48 4.3.2 Local changes of the redox current . . . 49
4.3.3 Nonlocal changes of the redox current . . . 50
4.4 Discussion . . . 54
4.4.1 Ohmic drops . . . 54
4.4.2 Adsorption . . . 56
4.4.3 Unequal diffusion coefficients . . . 59
4.5 Summary . . . 64
References . . . 65
5 The streaming effects in the nanogap fluidic system 67 5.1 Introduction . . . 68 5.2 Theoretical background . . . 69 5.3 Experimental methods . . . 71 5.3.1 Experimental system . . . 71 5.3.2 Chemicals . . . 72 5.3.3 Flow actuation . . . 73
5.3.4 Measurement of the streaming potential . . . 73
5.4 Experimental results . . . 74
5.5 Discussion . . . 76
5.6 Summary . . . 82
References . . . 84
6 Stochastic Charge Fluctuations in Bipolar Electrodes 87 6.1 Introduction . . . 88
6.2 Theory . . . 88
6.3 Results and Discussion . . . 98
6.4 Conclusions . . . 101 References . . . 102 Appendix 107 Samenvatting 115 Summary 117 Acknowledgements 119
1
1
I
NTRODUCTION
The research presented in this thesis is devoted to understanding electrochemical processes occurring in nanofluidic systems. Both fields are very broad and we do not aim to cover them fully. In this chapter we briefly describe the main concepts used throughout the thesis, indicating the scope of the problem. We also introduce the terminology employed afterwards, anticipating that it facilitates further reading and allows focusing on the main conclusions rather than on the technical details.
1
2 Chapter 1. Introduction
1.1
Electrochemical detection
Electrochemistry is a domain of science where chemical and electrical processes are coupled. In the core of the field lies a process of an electron transfer that takes place between two molecules or between a molecule and an electrode. The latter is most relevant for the analytical application considered in this thesis. Hence, here we focus on processes at the boundary between solution and a metallic electrode.
When an appropriate potential difference is applied between a liquid phase (the solution) and a conducting solid phase (the electrode), a molecule in the solution can be oxidized (give an electron) or reduced (accept an electron) at the electrode surface. This heterogeneous electron transfer can occur in two ways. In an outer sphere reaction a molecule stays at some distance from the electrode surface and hardly interacts with it (Figure 1.1a). This is the case, for example, for the oxidation of ferrocenedimethanol (Fc(MeOH)2) or the reduction of hexaamineruthenium(III)
(Ru(NH3)3+6 ) at a Pt surface. In an inner sphere reaction a molecule approaches
the electrode surface closely or even chemically adsorbs on it, changing the original electron configuration of the reactants (Figure 1.1b). It is a case, for example, of oxygen or hydrogen oxidation at a Pt surface in the sulfuric acid (H2SO4) solution.
Figure 1.1: (a) Outer sphere and (b) inner sphere electrode reactions. M is a metal ion surrounded by ligands. In the outer sphere reaction the complex is separated from the electrode by the layer of solvent molecules. In the inner sphere reaction one of the ligands (black) adsorbs on the electrode.1
The potential of a metallic electrode, which drives a redox reaction, is established with respect to a so-called reference electrode, which defines the potential of the solution. The exact value of the potential required to induce an oxidation or reduction process depends on the chemical properties of the substance. We call molecules electroactive if they exchange electrons with an electrode in a potential range
1
1.1 Electrochemical detection 3
accessible in the measurement (Figure 1.2c,d). An electrolyte that is inert under the experimental conditions (Figure 1.2b) is called a supporting electrolyte, it serves to increase the solution conductivity and screen any electric field originated from the electrode.
Figure 1.2: Voltammograms obtained by applying (a) a potential sweep to a solution containing (b) only supporting electrolyte, (c) supporting electrolyte and oxidizable species, (d) supporting electrolyte and reducible species.2
The selectivity of particular species to the oxidation and reduction potentials al-lows to use electrochemical processes for analyte detection by converting information about the chemical composition and concentration of the substance into an electrical signal. Here we mention two approaches related to redox current measurements. In amperometry, a constant potential is applied and the redox current is recorded as a function of time. In voltammetry, the value of the potential changes and the current is recorded as a function of the potential (Figure 1.2a). The most commonly used format of this technique is cyclic voltammetry, in which the potential is changed linearly at a constant rate within a chosen range. The magnitude of the current naturally is proportional to the amount of analyte reacting with the electrode per time unit. For microsized electrodes an on-going redox process consumes the reactant slowly enough that it is continuously replenished and the flux of the electroactive molecules to the electrode surface remains constant. That prevents current from declining after the potential is applied to the electrode unlike for the measurements
1
4 Chapter 1. Introduction
involving macroelectrodes. The amperometric trace stays constant and the shape of the cyclic voltammetric curve depends on the electron transfer rate. For fast reversible outer sphere reactions it actually replicates the density of states in the metal, i.e. it corresponds to the Fermi-Dirac distribution.
We can see that the miniaturization of electrodes size, which can be easily per-formed due to modern progress in fabrication technologies, facilitates interpretation of the measurement outcome. At the same time, when the surface area becomes very small, the current magnitude decreases and some tricks may have to be introduced to amplify the signal. Nanogap devices, which we consider in this thesis, are a beautiful realization of the redox signal amplification in a miniaturized system and will be introduced in due time. The second aspect of the story relates to the delivery of analyte to the detector.
1.2
Nanofluidics
The term "nanofluidics" refers to the fluid motion in the systems with a characteristic dimension lower than 100 nm like carbon nanotubes, nanopipets, nanopores and nanochannels.3 Creation and investigation of these structures became possible due to the significant progress in fabrication technologies and characterization techniques in the past three decades. It especially relates to nanochannels, which have become the most common object of nanofluidics research and a prospective component for the development of lab-on-a-chip technology. This is the structure we keep in mind discussing the features of nanofluidic systems.
While the general chase for miniaturization is the most obvious factor that led to the birth of nanofluidics, the expectations of new phenomena associated with the scaling to nanosizes also should not be ignored.4, 5Highly enhanced surface-to-volume ratios increase the role of the interface and cause severe changes to the mass transport. Reversible adsorption to the walls slows down molecules diffusion and results in effective diffusion coefficients in nanoconfinement that are lower comparing to their bulk values.6–9Hydrophobic interactions of solvent molecules with the walls can lead to hydrodynamic slip lengths of tens of nanometers that is comparable with the size of nanochannels and cannot be neglected. The effect noticeably modifies not only convectional velocity, inducing deviations from continuum hydrodynamic description, but also changes the value of the effective zeta potential.10
The fact that most surfaces are charged in liquid leads to the entire spectrum of phenomena called electrokinetic effects. To maintain a solid-liquid interface neutral, ions in the solution redistribute: oppositely charged counterions move towards the
1
1.2 Nanofluidics 5
surface and coions - away from it. The region where the screening takes place is called the electrical double layer (EDL) and characterized by the scale parameter known as the Debye length:
lD= s ε0εrkBT e2N aPiniz2i (1.1)
whereε0is the electric permittivity of vacuum,εrthe relative electric permittivity of
the solvent, kBthe Boltzmann constant, T the absolute temperature, e the charge of
the electron, Nathe Avogadro number, nithe concentration of species i, zivalence
of species i. This value is negligibly small for concentrated electrolytes (lD= 0.3 nm
for 1 M KCl), but reaches tens of nanometers for dilute ones (lD= 90 nm for 10 µM
KCl), which is comparable with the dimensions of a nanochannel.
There are two ways to drive the movement of the liquid through a nanochannel. First is to apply an electric field parallel to its surface. Mobile ions in the EDL start to move in the electric field and drag neighbor fluid by means of viscous interaction, causing the type of transport called electroosmosis. The velocity increases from a zero value at the slip plane across the EDL and into the bulk where it remains constant. This velocity is given by the expression11
veo= −ε0εrζE
η , (1.2)
where E is the electrical field along the channel,η is the dynamic viscosity of the fluid and ζ is the zeta potential. The zeta potential is the potential on the shear
plane, an imaginary surface next to the real solid-liquid interface at which the fluid is stationary. In the absence of the hydrodynamic slip it is equal to the surface electrostatic potential, but when slippage takes place, the shear plane moves beyond the surface and Eq. 1.2 acquires an extra factor 1 + b/lD, where b is the slip length.10
In microchannels, the EDL is much smaller than the width of the channel and the electroosmotic flow takes the form of a plug, but in nanochannels the EDL can occupy a significant part of the cross-section and the velocity profile changes across the channel following the potential distribution.12, 13
Another way to move liquid through a nanochannel is to apply a pressure difference∆p between inlet and outlet regions. For laminar flow the velocity has a parabolic shape with the value in the center given by
vmax= ∆
p RhydroA
1
6 Chapter 1. Introduction
where Rhydro is the hydrolic resistance and A is the cross-sectional area of the
channel. Uncompensated ions in the EDL are transported together with the rest of fluid resulting in a convective streaming current. Providing an external path for this constantly supplied uncompensated charge, it is possible to measure the magnitude of the streaming current and study the electric properties of the channel surface. One can also use an external load for the practical purposes and convert mechanical pressure into electrical energy.14–18
In the absence of an external path, a reverse conduction current flows through the channel producing a voltage drop. For poorly conducting liquids this can lead to a significant difference in solution potentials between the inlet and the outlet of a channel.
1.3
The content of the thesis
This thesis explores the coupling between electrochemical processes and mass transport in nanochannels. The study is performed with fluidic nanogap devices, which detect electroactive species by repetitive redox cycling between two closely spaced electrodes.
Chapter 2 discusses the concepts of ion concentration polarization of chemically
inert species due to surface effects and ion concentration polarization of electrochem-ically active species due to difference in diffusion coefficients. The nanogap device and principle of redox cycling amplification are introduced.
Chapter 3describes the study of mass transport in nanofluidic channels under conditions of pressure driven flow in a generator-collector system.
Chapter 4 reports unexpected, apparently non-local features in the redox
current measured in the nanogap devices under flow conditions and analyses their possible origin.
Chapter 5 investigates the development of streaming potentials in the fluidic
system associated with nanochannels at low concentrations of the supporting electrolyte.
Chapter 6 is off the main direction of the thesis. It explores the fluctuations
of the number of electrons at a bipolar electrode when reduction and oxidation reactions happen at its opposite ends by means of analytical theory and numerical calculations.
1
REFERENCES 7
References
[1] A.J. Bard and L.R. Faulkner. Electrochemical Methods: Fundamentals and Applications, 2nd Edition. John Wiley & Sons, 2000.
[2] Chapter 10 electrochemical detection (amperometry, voltammetry and coulometry). In Paul R. Haddad and Paul R. Haddad, editors, ion chromatography, volume 46 of Journal of Chromatography Library, pages 291 – 321. Elsevier, 1990.
[3] Jan C. T. Eijkel and Albert van den Berg. Nanofluidics: what is it and what can we expect from it? Microfluidics and Nanofluidics, 1(3):249–267, Jul 2005.
[4] W. Sparreboom, A. van den Berg, and J. C. T. Eijkel. Principles and applications of nanofluidic transport. Nature Nanotechnology, 4:713, nov 2009.
[5] Lyderic Bocquet and Patrick Tabeling. Physics and technological aspects of nanofluidics. Lab Chip, 14:3143–3158, 2014.
[6] Nicolas F. Y. Durand, Arnaud Bertsch, Mina Todorova, and Philippe Renaud. Direct measurement of effective diffusion coefficients in nanochannels using steady-state dispersion effects. Applied Physics Letters, 91(20):203106, 2007.
[7] Y. Y. Kievsky, B. Carey, S. Naik, N. Mangan, D. ben Avraham, and I. Sokolov. Dynamics of molecular diffusion of rhodamine 6g in silica nanochannels. The Journal of Chemical Physics, 128(15):151102, 2008.
[8] Nicolas F.Y. Durand, Claudio Dellagiacoma, Raphaël Goetschmann, Arnaud Bertsch, Iwan Märki, Theo Lasser, and Philippe Renaud. Direct observation of transitions between surface-dominated and bulk diffusion regimes in nanochannels. Analytical Chemistry, 81(13):5407–5412, 2009.
[9] Shuo Kang, Ab F. Nieuwenhuis, Klaus Mathwig, Dileep Mampallil, Zinaida A. Kostiuchenko, and Serge G. Lemay. Single-molecule electrochemistry in nanochannels: probing the time of first passage. Faraday Discuss., 193:41–50, 2016.
1
8 REFERENCES
[10] Lyderic Bocquet and Elisabeth Charlaix. Nanofluidics, from bulk to interfaces. Chem. Soc. Rev., 39:1073–1095, 2010.
[11] R.J. Hunter, R.H. Ottewill, and R.L. Rowell. Zeta Potential in Colloid Science: Principles and Applications. Colloid science. Elsevier Science, 2013.
[12] D. Burgreen and F. R. Nakache. Electrokinetic flow in ultrafine capillary slits1. The Journal of Physical Chemistry, 68(5):1084–1091, 1964.
[13] Samuel Levine, John R. Marriott, and Kenneth Robinson. Theory of electrokinetic flow in a narrow parallel-plate channel. J. Chem. Soc., Faraday Trans. 2, 71:1–11, 1975. [14] Jun Yang, Fuzhi Lu, Larry W Kostiuk, and Daniel Y Kwok. Electrokinetic microchannel
battery by means of electrokinetic and microfluidic phenomena. Journal of Microme-chanics and Microengineering, 13(6):963, 2003.
[15] Yongqiang Ren and Derek Stein. Slip-enhanced electrokinetic energy conversion in nanofluidic channels. Nanotechnology, 19(19):195707, 2008.
[16] Chih-Chang Chang and Ruey-Jen Yang. Electrokinetic energy conversion in micrometer-length nanofluidic channels. Microfluidics and Nanofluidics, 9(2):225–241, Aug 2010. [17] Trieu Nguyen, Yanbo Xie, Lennart J. de Vreede, Albert van den Berg, and Jan C. T. Eijkel.
Highly enhanced energy conversion from the streaming current by polymer addition. Lab Chip, 13:3210–3216, 2013.
[18] Hsin-Fu Huang and Pao-Wen Yang. Electrokinetic streaming power generation using squeezing liquid flows in slit channels with wall slip. Colloids and Surfaces A: Physicochemical and Engineering Aspects, 514:192 – 208, 2017.
1
2
I
ONIC TRANSPOR T IN NANOSCALE SYSTEMS
Ionic transport in nanochannels to a great extent depends on the condition of the solid-liquid interface due to the high surface-to-volume ratio. The electrical double layer is also able to occupy a significant part of the nanochannel, resulting in transport selective to ion polarity. The interface between domains with different polarity, for example, exhibits rectifying behavior analogous to a p-n junction in a semiconductor. The introduction of electrodes and electrochemically active species inside nanochannels opens a door to a series of additional effects such as unbalances in concentrations between molecules in different redox states caused by differences in diffusion coefficients.
∗The contents of this chapter have been published previously as part of Zinaida A. Kostiuchenko,
Piotr J. Glazer, Eduardo Mendes and Serge G. Lemay, Chemical physics of electroactive materials – the oft-overlooked faces of electrochemistry, Faraday Discuss, 199, 9-28 (2017)
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10 Chapter 2. Ionic transport in nanoscale systems
2.1
Introduction
The transport of ions in liquids includes both hydrodynamic and electrostatic components. While the former becomes relatively simplified when working in the laminar flow regime, the latter creates a bouquet of intertwined effects which are not so easy to account for or control. Even in the absence of external electric fields, the formation of the electrical double layer (EDL) at the solid-liquid interfaces leads to the presence of uncompensated charge that can influence fluid transport by means of streaming effects. Introduction of electrodes into the system allows additional mechanisms for moving charge entities, such as electoosmosis and electrophoresis, as well as for manipulation of properties of the molecules, such as their charge state. In this chapter we introduce some of the main processes associated with charge transport in nanofluidic devices. The physics of ionic transport bear many similarities to electron and hole transport in semiconductors. Just as the ability to locally dope semiconductors allows creating devices such as diodes and bipolar transistors, ma-nipulation of fixed charges in electrolyte systems leads to rich behavior characterized by spatially inhomogeneous ion distributions and tunable transport properties. We first briefly summarize well-known results on ion concentration polarization, then introduce electrochemical nanofluidics devices and discuss possible analogies involving faradaic processes.
2.2
Ion concentration polarization
The transference number, ti, is the fraction of the total current density carried in
an electrolyte by ionic species i. In a uniform bulk electrolyte, this quantity depends only on the composition of the electrolyte and the mobilities of the different species i. Transference numbers that differ strongly from bulk electrolyte values can occur in permeable materials such as the polyelectrolytes used in ion-exchange membranes. By virtue of its definition, ti does not in general lend itself readily to describing
systems in which significant concentration gradients occur. In micro- and nanosized channels, however, where equilibration across the cross-section of the channel can take place rapidly, it is convenient to think of the transference number as reflecting the fraction of the total current through the channel carried by species i (that is, the current density associated with the flux of species i integrated over the channel cross-section). In this case tirepresents an effective, one-dimensional transference
number that can differ from bulk values and vary with position since charge present on the channel walls leads to an EDL and an excess of counterions.1The magnitude
1
2.2 Ion concentration polarization 11
increase in surface-to-volume ratio; it becomes particularly pronounced when the diffuse components of the EDLs start to overlap (for weakly charged surfaces) or, more generally, when the counterion charge in the compact layer starts to exceed that in the bulk.2
Interestingly, interfaces between domains with different transference numbers, independently of how this is physically achieved, tend to exhibit the same basic ion transport behavior. First, their current-voltage characteristics exhibit rectifying behavior. That is, the magnitude of the current is larger for one polarity of the voltage ("forward bias") compared to its opposite ("reverse bias"). Second, and often more significant than the rectifying behavior, the concentration of both cations and anions near the interface simultaneously increases (in the forward bias case) or decreases (for reverse bias).
Why this occurs is illustrated in Figure 2.1 for the case of a nanochannel contain-ing a junction between positively and negatively charged segments (Figure 2.1a). Starting from an equilibrium situation, the application of a positive potential to the right reservoir causes an electrical current to flow from right to left. While the total electrical current is independent of position along the channel, it is carried preferentially by cations moving to the left in the right half of the channel and by anions moving to the right in the left half of the channel. The concentration of both cations and anions near the junction tends to increase as a result of this current. This buildup of ions continues until diffusive gradients compensate for the differences in ion concentration, as shown in Figure 2.1b, left panel. If an electric field is applied in the opposite direction, the converse occurs and the junction between the segments becomes depleted of salt (Figure 2.1b, right panel). This is directly reflected in the current-voltage characteristics of the junction: while the current tends to increase superlinearly with increasing forward bias due to a decrease in resistivity caused by carrier accumulation, it is highly suppressed in the reverse-biased direction due to depletion (Figure 2.1c). This behavior is closely related to that of solid-state p-n junctions, but significant differences exist at the quantitative level. In particular, giant enrichment of both carrier types as shown in Figure 2.1b is not as readily realized in semiconductors due to electron-hole recombination, and migration (i.e., ohmic drops, and thus also transference numbers) usually play a more prominent role relative to interfacial potentials in the fluidic case.
The device in Figure 2.1 has been coined an ionic or nanofluidic diode,3–5a term which has also been applied6to conical nanopores,7–11asymmetric nanochannels12 and straight nanochannels connected to reservoirs with asymmetries in solution com-position.13, 14But electroactive systems where domains with different transference
1
12 Chapter 2. Ionic transport in nanoscale systems
Figure 2.1: (a) Sketch of a bipolar nanochannel with positively and negatively charged regions. (b) Calculated concentration profiles for K+cations and Cl−anions under forward (left) and reverse (right) bias of 5 V in a nanochannel of 30 nm height with surface charge densities of 2 mC/m2(left half) and −2mC/m2(right half).3Reprinted with permission from H. Daiguji, Y. Oka and K. Shirono, Nano Lett., 2005, 5, 2274-2280.3Copyright 2005 American Chemical Society. (c) Measured current-voltage characteristics of a nanofluidic diode based on a 20 nm height bipolar channel with surface charge densities of +1.3 and −4mC2at bulk KCl concentrations of 0.1 mM (left), 10 mM(middle) and 1 M (right).4Reprinted with permission from L.-J. Cheng and L. J. Guo, ACS Nano, 2009, 3, 575-584.4Copyright 2009 American Chemical Society.
numbers are in contact arise much more frequently. Perhaps the simplest example is the interface between a nanochannel and a larger channel or reservoir,15, 16which provides a broadly applicable route for concentrating analyte molecules based on their charge.17–22Conversely, the depletion effect at the micro/nanochannel interface has been exploited in, for example, depletion zone isotachophoresis.23More broadly, assemblies of ion-selective porous materials such as microparticle assemblies24, 25 and polyelectrolyte gels26–28exhibit similar behavior. These have been exploited for electrical actuation of responsive gels29, 30and in ion-exchange membranes.31
1
2.3 Electrochemical nanofluidics devices 13
transference numbers can exhibit more complex behavior than simple rectification, including multistability.32
2.3
Electrochemical nanofluidics devices
We have seen in Section 2.1 that static spatial inhomogeneities in the cation/anion transport ratio of a medium, as represented by spatially varying transference numbers, can have dramatic and counterintuitive consequences for charge transport, and that these effects are enhanced for high surface-to-volume ratios such as occur in nanochannels. The introduction of embedded electrodes to perform electrochemical reactions in nanoscale channels further complicates this picture in two main ways. First, the surface charge at an electrode and the associated influence on transference numbers are no longer static and can instead be tuned via an external potential. Other properties such as reversible adsorption of ionic species can also depend on potential.33, 34Second, ionic species are no longer conserved since redox reactions
allow the local creation, annihilation and/or transmutation of ionic species to take place. What are the consequences of these new ingredients is a question that has hardly been broached so far.
For concreteness we focus here on one particular class of electrochemical nanofluidic systems, namely, devices that enable efficient redox cycling between two closely spaced electrodes. When one electrode is kept at a reducing potential and another at an oxidizing potential, an electroactive molecule, trapped in this narrow region, is reduced and oxidized multiple times, shuttling electrons between the electrodes. This leads to orders-of-magnitude signal enhancement compared to conventional voltammetry or amperometry. Such geometries featuring nanoscale distances between the electrodes have been realized using a number of strategies that include, for example, interdigitated electrodes35–39and ring-disk electrodes.40, 41 Our group’s approach,42–45 which relies on lithography-based microfabrication
techniques, is illustrated in Figure 2.2. The core device consists of a nanochannel (height ∼ 40−70nm, width ∼ 2−5µm, length ∼ 10−100µm) whose floor and ceiling consist of separately addressable electrodes. Details of the device fabrication process have been described elsewhere.34, 46In such a geometry, the diffusion-limited redox-cycling current between the two electrodes (i.e., the expected current assuming high overpotentials for both reduction and oxidation) is
Irc= ne A z µ2DredDox Dred+ Dox ¶ C (2.1)
1
14 Chapter 2. Ionic transport in nanoscale systems
Here, n is the number of electrons shuttled per cycle, e is the electronic charge, A is the area of overlap of the two facing electrodes, z is the separation between the electrodes, C is the average concentration of redox molecules in the volume between the electrodes, and Dred and Doxare the diffusion coefficients of the reduced and oxidized species, respectively. This expression motivates the interest in nanodevices as the detected signal is inversely proportional to the electrode spacing z.
Figure 2.2: (a) Schematic cross section of a nanogap electrochemical detector. Redox-active molecules shuttle diffusively between the closely spaced (< 100nm) Pt electrodes, where they are repeatedly oxidized and reduced and generate a highly amplified electrochemical current. (b) Optical micrograph (top view) of a lithographically fabricated device consisting of a 50 µm long and 5 µm wide detection region. The two access holes can be connected to external microfluidic channels to allow pressure-driven flow through the device.47, 48Reprinted from Electrochimica Acta 112, K. Mathwig and S. G. Lemay, Mass transport in electrochemical nanogap sensors, 943-949, Copyright (2013), with permission from Elsevier.48
Bringing a target analyte into such a device can rely either on passive diffusion from an external "bulk" reservoir or on an actively controlled advective flow along the channel.47In the latter case, an important property resulting from the high aspect ratio between the height of the channel and the length of the electrodes is that the redox-cycling gradient between the electrodes is established on much faster time scales than advection through the device. Consequently, the redox-cycling gradient is not significantly disturbed by convection, except at extreme (and so far unachievable) flow rates.48More precisely, the Graetz number, Gz, is a dimensionless quantity that characterizes the ratio of the diffusion time across the cross-section of the channel to the advection time along the length of the electrode. For our rectangular cross-section its value is given by G z = z2v/DL, where v is the average flow speed and L is
1
2.4 Electrochemically induced concentration polarization 15
the length of the detection region. For typical parameters z = 50nm, v = 100µm/s, D = 0.5×10−9m2s−1and L = 100µm, this yields Gz = 5 × 10−6¿ 1 and the diffusion profile between the electrodes is fully established.
Early experiments using such nanoscale devices focused on exploiting the high charge amplification resulting from redox cycling to probe the underlying microscopic charge transport dynamics. Relatively small numbers of molecules are present in the detection region between the two electrodes at any given time because of the small volume involved (order of a few femtoliters). Since the redox-cycling current is directly proportional to the number of such molecules, equilibrium statistical fluctuations around the mean number of molecules can be detected as current noise. An important signature of such noise, illustrated in Figure 2.3a, is that the current fluctuations are essentially identical at the reducing and oxidizing electrodes since both currents are caused by the same molecules shuttling between the electrodes. The rate at which such fluctuations take place, as characterized by the noise spectral density44, 46or its time-domain equivalent, the autocorrelation function,47reflects the speed with which molecules enter and leave the detection volume. This approach can be extended down to pM concentrations, at which point the noise consists of telegraph-like signals corresponding to individual molecules entering and exiting the detection region (Figure 2.3b).
A surprising – and still somewhat controversial – finding from these measure-ments was that outer-sphere redox couples such as ferrocene derivatives appear to exhibit a significant degree of reversible adsorption to metal electrodes.33, 46, 49These observations are also consistent with studies of the temporal response of nanogap sensors34and the smaller-than-expected current signature of single molecules during redox cycling.44, 45Although such molecule-surface interactions are too weak to be readily detected in conventional micro- or macroscale systems, they become very significant in < 100nm high channels due to the high surface-to-volume ratio.
The analysis performed to date to interpret microscopic data such as those sketched here has focused on simple models based on diffusion and advection. As we discuss in the next section, however, this neglects indirect forms of interactions that may play a role in more complex multi-electrode fluidic systems.
2.4
Electrochemically induced concentration polarization
To illustrate the parallel between electrochemical reactions and the enrich-ment/depletion effects discussed in Section 2.1, we first consider the trivial case of two conventional Ag/AgCl electrodes immersed in two large reservoirs that are connected
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16 Chapter 2. Ionic transport in nanoscale systems
Figure 2.3: (a) Fluctuations in the redox-cycling current obtained for 100 µM Fc(MeOH)2in
an aqueous electrolyte due to equilibrium statistical fluctuations of the number of molecules present in the detection volume. The DC component has been offset to focus on the fluctuations. The currents for the oxidizing (blue) and reducing (red) electrodes are strongly correlated but have opposite signs, as expected for redox cycling.33Reprinted with permission from P. S. Singh et al., J. Am. Chem. Soc, 2011, 133, 18289-18295. Copyright 2011 American Chemical Society. (b) Corresponding current-time traces for 10 pM FcTMA+. At such low concentrations, less than one redox molecule is present on average in the detection volume. The entry and departure of individual molecules appear as plateaus in the ultra-low (fA-level) current signals. Here two particularly long events can be observed near 40 s and 95 s.45 Reprinted with permission from S. Kang et al., ACS Nano 2013, 7, 10931-10937. Copyright 2013 American Chemical Society.
via a small channel and filled with KCl. Within the electrolyte, the contributions to the total electrical current carried by K+cations and Cl−anions are comparable (tK+ ≈ tCl−≈ 0.5). At the electrolyte/electrode interface, on the other hand, only
Cl−ions are exchanged and the transference numbers characterizing this interface are thus tK+≈ 0, tCl−≈ 1. In analogy with the mechanism of Figure 2.1, driving a
DC current through the channel results in the net accumulation of both cations and anions at one liquid-electrode interface and their depletion at the other. This is usually negligible on the scale of most nanofluidics experiments, in which the currents, being limited by the resistance of the nanochannel, are small and the concentrations in the reservoirs are sufficiently large that they are not significantly perturbed on experimental scales. For example, a 1 nA current, which represents a
1
2.4 Electrochemically induced concentration polarization 17
significant faradaic current at a microscale electrode, takes ∼ 107s to perturb the concentration of a 100 µL, 0.1 M KCl reservoir by only 1%.
What if the reservoirs are miniaturized or if redox reactions take place in a small isolated volume? Consider a simple thought experiment, illustrated in Figure 2.4, in which a long, thin, sealed channel contains a simple electroactive species that can exist in two charge states, reduced (red) and oxidized (ox). As sketched in Figure 2.4a, electrodes of length Leare situated in the walls of this channel so as
to define two domains separated by a gap of length Lg in which no electrode is
present. Switching the potential of the electrodes between two voltages, Vredand Vox, insures that the molecules near each electrode quickly reach electrochemical equilibrium and attain a well-defined state: red or ox. Any change in the total charge within the system as a result of switching the potential of the electrodes is assumed to be compensated by inert supporting salt ions diffusing into the channel, also screening out electric fields beyond the electrodes’ EDLs. In the initial phase of the experiment, both the left and the right domain are biased at potential Vred. All molecules are in the red state, and the concentration of molecules is uniform throughout the system. At the particular moment t = 0, the potential of the left electrode is switched to potential Vox, switching the redox molecules in this domain to the ox state. Counterbalancing concentration gradients of molecules in red and ox states are set up between the two electrode domains. So long as the molecules are indistinguishable apart from their redox state, the resulting concentration profiles are symmetric, as sketched in Figure 2.4b. In practice, however, the diffusion coefficients Dredand Doxfor the two redox states are usually not identical. In this case the situation in Figure 2.4b is not in steady state, as the flux of molecules in the ox state diffusing from left to right (Dox(dCox/dx)) is less than that of particles in the red state diffusing from right to left (Dred(dCred/dx)). A net migration of particles from left to right then takes place until a new equilibrium is achieved in which the concentration in the left reservoir is lower than that in the right reservoir, as sketched in Figure 2.4c. This illustrates how electrochemical reactions provide a pathway for the establishment of concentration imbalances. Importantly, the relative imbalance in the equilibrium concentrations, Cox/Cred= Dred/Dox, is independent of the device dimensions. The characteristic relaxation time toward equilibrium, which is diffusion limited and thus given roughly by [max(Lg, Ld)]2/[Dox+ Dred], on
the other hand, is geometry dependent. For Le= Lg= 10 µm and typical diffusion
coefficients of order 0.5×10−9m2s−1, the relaxation time yields an order of 0.1 s, a
time scale which is short enough to interfere with many measurements.
1
18 Chapter 2. Ionic transport in nanoscale systems
Figure 2.4: Thought experiment illustrating electrochemically induced concentration po-larization. (a) Sketch of a sealed nanochannel in which the redox state of molecules can be switched between a reduced (red) and an oxidized (ox) state using electrodes (red and blue bands). Here we take the left electrode as oxidizing and the right electrode as reducing. (b) Concentration of red and ox molecules versus longitudinal position along the channel when the diffusion coefficients Dredand Doxare equal. The total concentration of redox molecules is uniform throughout the channel. (c) Same as (b) but with Dred> Dox. An asymmetry develops in the concentration profiles with a higher total concentration being present at the oxidizing electrode.
and is therefore difficult to implement in practice. This because in this case contact with a reference electrode to set the solution potential is questionable, while the electrochemical measurements does not induce a redox-cycling process that would allow determining the concentration. Open systems, however, add an extra level of complexity for the interpretation as interplay with external reservoir must be taken into account. Such an analysis was executed previously50for the case where interaction of nanochannel with bulk reservoir was purely diffusive. Introduction of hydrodynamic flow into the system changes further causes steady state to depend now on both diffusion and convection. In the following chapters we will discuss effects arising when this type of transport of electroactive species takes place inside a nanogap device.
1
2.5 Conclusions 19
2.5
Conclusions
The majority of studies of electroactive materials and systems concentrate on scenarios in which ions play the role of indestructible mobile charge carriers. The existence of cations and anions as separate, parallel pathways for charge transport, the inherent coupling between ionic charge transport and solvent motion, and the ability to control either surface charge (for channel-based systems) or bulk charge density (for permeable materials) leads to a rich range of physical effects that spans classic electroosmosis, ion-selective membranes, tunable tribological and optical properties, nonlinear transport at interfaces, actuation, and more.
In this context, electrochemical reactions often represent unwanted parasitic pathways for charge that one attempts to eliminate or circumvent through a judicious choice of materials, solution composition, or applied potentials. Conceptually, however, electrochemical reactions represent an extension of the range of available boundary conditions for charge transport. Do qualitatively new effects exist as a result of these additional forms of coupling? Do these provide a significant advantage for applications? What seems certain at this moment is that, insofar as such effects exist, they can be expected to become increasingly relevant upon downscaling the dimensions of a system and thus increasing its surface-to-volume ratio.
1
20 REFERENCES
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1
3
G
ENERATOR
-
COLLECTOR EXPERIMENT
Mass transport inside fluidic channels under conditions of pressure-driven flow is determined by the superposition of convection and diffusion. Typically, the height of a channel of the same order of magnitude as its length and 2D calculations are required to quantify the process. Nanofluidic devices, however, have such a low height-to-length ratio that they can be considered as 1D systems, which significantly simplifies their mathematical description. In this chapter we study mass transport in nanochannels using electrodes in a generator-collector configuration and measuring the flux of redox molecules amperometrically. We observe the transition between diffusion-dominated and convection-dominated transport with the variation of the flow velocity and the distance between the electrodes.
1
26 Chapter 3. Generator-collector experiment
3.1
Introduction
In nanofluidic channels, ionic transport driven by externally imposed electric fields can lead to complex behavior such as current rectification1–3due to asym-metries between cations and anions caused by surface charge. The introduction of electrodes inside such a nanochannel complicates the picture even further because of the interconversion of electroactive species, which can further lead to concentration polarization and potential-dependent adsorption as discussed in the previous chapter.
We take a systematic approach to tackle this complexity. In this chapter, we present direct measurements of mass transport in nanochannels under the conditions of pressure-driven flow. We start with the simplest possible system, which we coin generator-collector. The generator is an electrode converting electrochemically active species present in solution into a different redox state, while the collector is a second electrode turning them back into their original state. The measured faradaic current represents the rate of mass transport of the redox species between the electrodes.
The first kind of setup exploited this principle of species conversion was introduced in 1959 and consisted of a rotating disk electrode surrounded by a concentric ring electrode, the two electrodes being separated by a dielectric layer.4 Double electrode system, consisting of two closely spaced flat electrodes embedded into the wall of a channel through which the sample flowed, appeared shortly afterward.5More recently, double electrodes in a channel were applied for the study of electrode dissolution processes,6–9mechanisms and kinetics of electrochemical reactions10–16and in situ velocimetry.17–21Understanding mass transport in these type of systems became an issue from the beginning and led to extensive calculations accounting for complex geometries and various transport mechanisms.22–28Further miniaturization resulted in microchannel structures where one of the lateral dimensions was essentially homogeneous, which allowed a relative simplification to a 2D case. However, interplay between diffusion and convection, where change in flow velocity alters the concentration profile in both dimensions, continued to require extensive calculations and simulations.
The systems employed in the present research consists of two electrodes embedded in a nanochannel. The ratio between channel height and electrode length effectively removes one more dimension, which significantly simplifies the theoretical description of mass transport. To our knowledge, no double electrode systems with one-dimensional concentration distributions in the full accessible fluid velocity range were introduced previously. In this chapter we demonstrate that mass transport in nanochannels can be described by the 1D Nernst-Planck equation, where the relative
1
3.2 System description 27
contributions of diffusion and convection depend on flow velocity and separation between the generator and collector electrodes.
3.2
System description
A sketch of the fluidic system is depicted in Figure 3.1a. It consists of nano- and microchannels connected in parallel.
The nanochannel is fabricated using the same procedure as all our nanogap devices (see Appendix 6 for details), but without top electrodes. In short, a Si wafer with thermally grown 500 nm SiO2is taken as substrate. Pt electrodes 20 nm thick
are defined by photolithography using photoresist OIR 907-12 and deposited by e-beam evaporation. These electrodes have a length of 11 µm and are separated by 2, 5 or 50 µm. Cr sacrificial layer 86 µm long, 5 µm wide, 90 nm high is patterned on top with the same techniques to form the nanochannel. The entire wafer with metal structures is passivated with a layer of SiO2using the chemical vapor deposition
method to isolate the leading wires from the analyte. Then two holes are etched through the dielectric with reactive plasma at the ends of the nanochannel to provide access to the sacrificial layer. Just before the experiment, the Cr is etched to create the nanochannel.
A microchannel in parallel provides transport of fluid through the system on reasonable time scales (the nanochannel alone would take almost a month to pass 1 µL of liquid, making it practically impossible to exchange solutions due to volume in the microfluidic interconnects). To create this additional pathway, a microfluidic structure is formed on the bottom of a block of PDMS (polydimethysiloxane) using an SU-8 mold. This structure consists of two microchannels, each 90 µm long, 5 µm wide and 3 µm high, connecting two spacious pillars regions. An image of this structure is shown in Figure 3.1b. Punching holes through the PDMS in the regions of the large reservoirs using a hollow needle allows inserting external PFTE (polytetrafluoroethylene) microtubes. The inlet of the system is connected to a 500 µL ILS microsyringe driven by a syringe pump (Pump 11 Pico Plus Elite) and the outlet to a reservoir with a Ag/AgCl reference electrode (BASi, MF 2079, RE-5B) immersed in it.
According to Poiseuille’s law, the pressure difference∆p caused by a flow rate Q in such a system is defined as ∆p = QRtotal, where Rtotal is the total hydraulic resistance. For a channel with a rectangular cross-section, this value can be estimated as29
1
28 Chapter 3. Generator-collector experiment
Figure 3.1: Fluidic system. (a) Schematic side view of the nanochannel and microchannel in parallel. (b) Microphotograph of a nanochannel for generator-collector experiments with PDMS microstructure bonded on top.
Rhydrolic=
12ηL 1 − 0.63(h/w)
1
h3w, (3.1)
where h, w and L are the height, width and length of the channel, respectively, and
η is the dynamic viscosity of water. The total flow rate is divided between the
micro-and nanochannels in an inverse proportion to their hydraulic resistances:
Qnano= Q Rtotal Rnano ≈ Q Rmicro 2Rnano , (3.2)
where we made use of the fact that Rmicro¿ Rnanoand that there are two channels.
For the channel geometries employed here, the maximum velocity vmaxin the
nanochannel is about 1000 µm/s, which corresponds to Reynolds number 10−4. Hence, even for our highest operational velocities, the flow remains laminar.
We will call ’upstream’ the electrode that is the closest to the inlet and ’downstream’ that which is closest to the outlet.
1
3.3 Mass transport description 29
3.3
Mass transport description
Three main parameters will determine the nature of the analyte transport inside the nanochannel.
The first parameter is the transverse Peclet number (Petr = vh/D), which indicates how efficient diffusion is at mixing molecules across the height of the nanochannel while being transported along it. Petr= vh/D, where D is the diffusion coefficient of the redox active species. For typical parameters in our experiment: h = 100nm, D = 10−9m2/s and v = 1000µm/s, this yields Petr
= 0.1. This means that the diffusion across the nanochannel happens sufficiently fast that we can neglect the parabolic shape of the convective flow and consider that all molecules move with the same average speed along the channel.
The second parameter, the Graetz number (Gz), also compares diffusion per-pendicular to the nanochannel with convection along it, but at the distinct length scales characteristic for each direction. It reflects how many times a redox molecule crosses the nanochannel vertically while moving along the electrode. Here the importance of the high ratio between electrode length and channel height comes to light. For v = 1000µm/s, which is maximum velocity used in the experiment, Gz = Petrh/L = 10−7. It means that in a nanochannel the vertical mass transport
equilibrates much more rapid than the horizontal one, whatever (realistic) flow rate is applied. This is a crucial distinction with the case of a microchannel, where change in flow velocity alters concentration profiles in both longitudinal and perpendicular directions (Figure 3.2). This allows us to utilize a one-dimensional model for calculating redox species concentrations in the nanochannel, in contrast with microchannels where two dimensions must be considered.
The third parameter is a point of a particular interest here as it characterizes the dominant form of cross talk between the electrodes during generator-collector experiments. Transport of analyte along the nanochannel involves both diffusion and convection caused by externally applied pressure. The longitudinal Peclet number, Pel= vs/D (where s is the spacing between the electrodes) describes the ratio of each component’s contribution. When it is lower than one, diffusion dominates over convection, otherwise transport along the channel is controlled by the flow rate established externally. In the experiments described herein we will access both of these regimes.
1
30 Chapter 3. Generator-collector experiment
Figure 3.2: Qualitative representation of concentration profiles under high flow rate conditions for (a) a nanochannel and (b) a microchannel.
3.4
Experimental methods
In a typical experiment, Cr is first removed from the device with a wet etchant (Selectipur, BASF) to release the nanochannel. This procedure takes 40-80 minutes. The chip is then flushed with water, dried with nitrogen, placed inside a plasma cleaner together with the PDMS block and treated with oxygen plasma at 1 mbar for 70 s to activate the surfaces for bonding. The microstructure on the PDMS block is then aligned with the nanofluidic channel under a microscope and pressed against the chip. The assembled system is thereafter put into an oven at 70◦C for
15-20 min to enhance bonding strength. After this, we place the chip with its assembled microfluidic structure in a custom probe station and insert microtubes into the inlet and outlet holes.
All further chemicals were purchased from Sigma-Aldrich and solutions were prepared with Milli-Q water with a resistivity of 18.2 MΩ· cm. Prior to measurements with redox active species, the electrodes are cleaned with H2SO4 until the cyclic
voltammetry pattern becomes reproducible and corresponds to that expected for clean Pt. Finally a syringe containing an analyte is connected to the inlet. For the experiments described in this chapter we used aqueous solution of 1 mM Fc(MeOH)2
and 0.1 M KCl as supporting electrolyte (half-wave potential is 0.265 V under these conditions). The latter screens the electric field at the electrodes and insures exclusively diffusive-convective transport.
1
3.5 Results – Collector current 31
The reduced form of Fc(MeOH)2is a neutral species. Slight partial oxidation is
however possible due to acid left over in the microfluidic system.30The generator electrode is kept at a high oxidizing potential of 0.5 V vs Ag/AgCl, which converts all molecules coming into contact with this electrode into the oxidized state. The collector electrode, at a high reducing potential of 0 V, returns them back to the reduced state. We measure the reduction current to determine the rate at which molecules reach the collector as a function of the flow velocity and spacing between the electrodes.
The experiments are performed as follows. The two electrodes are first appointed the roles of generator and collector. A certain pump rate is then applied to the system. Initially, both generator and collector are at reducing potentials and any residual currents are considered as baseline currents. We then apply a potential step to the generator electrode to an oxidizing potential and measure both the oxidation current at this electrode (generator current) and the reduction current at the collector electrode (collector current). The average values of these currents during the potential step period and after baseline subtraction yield values for the generator and collector currents for the chosen flow velocity. The pump rate is then switched to the next value. After completing a set of measurements, the inlet and outlet are swapped and the measurements are repeated for the same range of flow rates. The roles of electrodes stay the same, however upstream electrode became downstream and vice versa; these results are represented as negative velocities.
When the generator electrode is upstream, both diffusion and convection assist redox species in reaching the collector. When the generator is downstream, on the other hand, only diffusion from the generator can bring oxidized molecules to the collector, and moreover this has to counteract the pressure-driven flow (Figure 3.3).
3.5
Results – Collector current
In Section 3.3 we argued that downscaling the channel height from the microscale to the nanoscale affects mass transport qualitatively. In the nanochannel, the distribution of the species in the perpendicular direction is essentially independent of the flow rate and we can describe the longitudinal flux with the 1D Nernst-Planck equation,
Jcol= −Doxd c
ox
dx + vc
ox. (3.3)
Here c is an average concentration of molecules across the nanochannel and index "ox" indicates that species are oxidized. The value for the diffusion coefficient of oxidized Fc(MeOH)2Doxis taken 5.4×10−10m2/s.31
1
32 Chapter 3. Generator-collector experiment
Figure 3.3: The reduction current at the collector for several flow velocities when the generator located (a) upstream and (b) downstream
Here we focus on the region between generator and collector, as sketched in Figure 3.4. Considering that molecules interact with an electrode almost instantly as they reach it in the longitudinal direction due to the extremely low Graetz number, all species passed though the generator zone are oxidized. Similarly, they are turned back into the reduced form at the boundary of the collector:
cox|x=−s/2= cb cox|x=s/2= 0.
Here cbis bulk concentration of redox species. The solution to Eq. 3.3 is then simply
cox= cb
1 − exp(−vs/Dox)−
cbexp (vx/Dox)
1
3.5 Results – Collector current 33
which corresponds to a flux
Jcol= cbv
1 − exp(−vs/Dox)=
cbv
1 − exp(−Pel). (3.5)
Figure 3.4: Calculated concentration distribution of oxidized species in the space between the generator and collector electrodes for different flow rates. This shows the qualitative difference between the diffusion-limited regime (Pel¿ 1, red) and convection-limited regime (PelÀ 1, blue).
The electrical current is directly proportional to the flux, yielding
icol= nF A Jcol= nF Acb
D s
Pel
1 − exp(−Pel). (3.6) Here n is a number of electrons transfered per oxidation or reduction event, F is the Faraday constant and A is the cross-section of the nanochannel.
In the diffusion-dominated regime Pel<< 1, corresponding to low flow rates, mass transport is dominated by diffusion and the current is inversely proportional to the spacing between the electrodes:
icol≈ nF Acb
D s.
1
34 Chapter 3. Generator-collector experiment
Basically, this represents a redox-cycling diffusion-limited current analogous to the one we introduced for the system of two parallel electrodes in Chapter 2, see Equation 2.1.
In convection-dominated regime Pel>> 1 the collector current at the downstream electrode is defined exclusively by the bulk concentration and flow velocity:
icol≈ nF Acbv,
while for the collector current at the upstream electrode Pel< 0 and icol≈ 0.
Figure 3.5: Collector currents versus flow rate. Experimental data (symbols) and solutions to the Nernst-Planck equation (solid lines). Positive and negative velocities correspond to the collector being located downstream and upstream, respectively.
Experimental results are presented at the Figure 3.5 for devices with electrodes spacings of 2 µm, 5 µm and 50 µm. The crossover value of Pel= 1 is reached for these spacings at velocities 270, 180 and 18 µm/s correspondingly. As expected, the current near zero velocity scales as 1/s and at high velocities the dependence on velocity becomes linear. Furthermore, the collector current has a pronounced value at the upstream electrode even at high flow velocities in the device with 2-µm spacing between the electrodes while for a 50-µm spacing there is no measurable current upstream at all. A quantitative comparison with based on Eq. 3.6 is shown as solid
