Streaming Potential Revisited: The Influence of Convection on theSurface ConductivityRakesh Saini, Abhinandan Garg, and Dominik P. J. Barz*

Department of Chemical Engineering, Queen’s University, Dupuis Hall 213, Kingston Ontario K7L 3N6, Canada

Queen’s−RMC Fuel Cell Research Centre, Queen’s Innovation Park, 945 Princess Street, Kingston, Ontario K7L 3N6, Canada

*S Supporting Information

ABSTRACT: Electrokinetic phenomena play an importantrole in the electrical characterization of surfaces. In terms ofplanar or porous substrates, streaming potential and/orstreaming current measurements can be used to determinethe zeta potential of the substrates in contact with aqueouselectrolytes. In this work, we perform electrical impedancespectroscopy measurements to infer the electrical resistance ina microchannel with the same conditions as for a streamingpotential experiment. Novel correlations are derived to relatethe streaming current and streaming potential to the Reynolds number of the channel flow. Our results not only quantify theinfluence of surface conductivity, and here especially the contribution of the stagnant layer, but also reveal that channel resistanceand therefore zeta potential are influenced by the flow in the case of low ionic strengths. We conclude that convection can have asignificant impact on the electrical double layer configuration which is reflected by changes in the surfaces conductivity.

I. INTRODUCTION

The formation of an electrical double layer (EDL) occurs whena surface, featuring electrical charges which are characterized bythe so-called zeta potential (ZP), is in contact with a liquidwhich contains mobile charges. The surface charges attractcounterions in the liquid which constitutes an interfacial layerwith an electrical net charge; contrary to the liquid bulk whichremains electrically neutral. Electrokinetic phenomena arise dueto interaction of an EDL and an external force field such aspressure or electrical potential gradient. (A comprehensivereview is available in the references.1,2)There has been a renaissance of electrokinetics research

which is mainly based on two motivations. On the one hand,novel phenomena such as AC-electroosmosis3 have beendiscovered after a century of research in the field. On theother hand, various microfluidic concepts utilize electrokineticphenomena.4 Additionally, the conversion of mechanical intoelectrical energy in nanofluidic channels has recently attractedincreased interest.5,6 More traditionally, electrokinetic phenom-ena play an important role in the electrical characterization ofsurfaces; the motivation ranges from geophysical research overmaterial to colloid and interface science. The electrokineticphenomenon streaming potential (SP) results from a liquidflow along a stationary surface featuring an EDL. Theconvection of the EDL induces a streaming current (SC)which, vice versa, generates an electrical potential differencewhich can be conveniently measured, that is, the SP.The classical correlation which was developed to relate SP

and SC to the ZP is the Helmholtz−Smoluchowski (HS)theory.7 Nowadays, SP measurements are performed withvarious cell designs including microchannels formed by two

parallel flat plates,8 microfluidic on-chip devices,9 asymmetriccells,10 cells coupled with fluorescence microscopy,11 and cellswhere the sample rotates.12 Despite some progress/variationsin cell designs, the vast majority of SP experiments utilize theclassic HS correlation (cf. refs 11 and 13−17).However, even the early work on SP noted that the classical

HS theory has several shortcomings. Essentially, it does notaccount for the so-called surface conductivity, even thoughSmoluchowski himself pointed out that an excess of ions near acharged surface makes a definite contribution to the overallconductivity.18 Hence, considerable errors can occur if theconductance (resistance) of the setup is calculated based on thebulk conductivity. The term bulk conductivity (resistivity)implies that the specific conductivity is not influenced by theboundaries of the measuring volume and, hence, it is measuredex-situ in large volumes such as beakers. In this article, we usethe terms ex-situ and in situ for resistance measurementsoutside and inside of the SP cell, respectively. An earlyapproach to include the effect of surface conductivity is the so-called Fairbrother−Mastin procedure.19 Here, in situ resistancemeasurements for liquids of low and high ionic strengths areemployed with a modified HS equation to include the effect ofsurface conductivity. Generally, these in situ resistancemeasurements are performed with direct current (DC) orvery rarely with alternating current (AC) methods of very lowfrequencies. Additionally, the review of the literature revealsthat the resistance is generally measured with resting liquids

Received: March 5, 2014Revised: August 20, 2014Published: August 22, 2014

Article

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and not under experimental flow conditions (cf. e.g., refs 13 and20−22). We discuss below that this can considerably alter theresults. One exception is the work of Werner et al., whoperformed consecutive SP and SC measurements.23 The ratioof SP and SC gives the in situ cell resistance for experimentalconditions but averaged over the investigated flow range.There are several literature sources that claim that the HS

correlation tends to give imprecise ZPs for other reasons thansurface conductivity as discussed in ref 2. This is in contrast tothe work of Overbeek who, based on the Onsager relation,claimed that one electrokinetic experiment is sufficient to inferall other phenomena.24 That is, the result of a SP experiment isequivalent to the result of an electroosmotic or SC experiment.This is practically not observed since each electrokineticmethods has its very own experimental difficulties and theirinfluence on the results can be significant as shown for PMMAin our previous work.25 Joule heating in electroosmoticexperiments can considerably impact the electroosmoticmobility and thus falsifies the ZP inferred from it. SCmeasurements are challenging since the currents are on theorder of nA and asymmetry potentials induce unwantedconduction currents. SP measurements at low ionic strengthsand narrow gaps require electrical equipment with extremelyhigh internal impedance to obtain reliable results.1

There are also several works which, based on dimensionalreasoning, assert that the SP gradient (and therefore the ZP)depends on the flow conditions.21,26,27 The same situation isclaimed by Lu et al. based on a different motivation. In theirwork, a nonlinear decay of the SP in the downstream directionof their measurement cell is found and explained with aconvective influence on the EDL.28 Such a behavior wouldquestion the applicability of the HS correlation to obtain evenapproximate results, since respective flow parameters are notincluded and all ZPs which are commonly reported from SPexperiments would be only apparent ZPs.Motivated by these open questions, we perform SP and SC

experiments in PMMA microchannels having different channelheights. For a better comparability of the different micro-channel experiments, we derive correlations for the electro-kinetic phenomena depending on the Reynolds number of thechannel flow. In contrast to others, we measure the electricalcharacteristics of the aqueous electrolytes in situ and underexperimental conditions by employing electrical impedancespectroscopy (EIS). The application of this method allows foran evaluation of the influence of surface conductivity and thatof the flow on the ZP. Additionally, we investigate thereversibility of the commonly used silver/silver chloride (Ag/AgCl) electrodes, especially for low ionic strength liquids.The article is structured as follows: First, we review the

theory of SC and SP and derive correlations which relate thephenomena to the Reynolds number. We also discuss someaspects of surface conductivity. Second, we describe theexperimental methodology used in this work. Third, we presentthe experimental results of the electrical and electrokineticexperiments. Finally, the article is completed with theConclusions section.

II. THEORYIn this section, we briefly introduce the general theory of SCand SP. Figure 1 shows a sketch of the electrokinetic cell(EKC) used in this work which illustrates the underlyingphysicochemical phenomena. The EKC consists of tworeservoirs (indexed 1,3) with incorporated electrodes which

are connected by a microchannel (indexed 2). The micro-channel is formed by two parallel plates, made from thesubstrate to be characterized. The microchannel consists ofwidth w2, length l2 and height h2; such that l2 > w2 ≫ h2. TheEDL at the interface liquid/microchannel wall is characterizedby the ZP ζ. A pressure difference Δp drives the liquid throughthe channel. This generates a convective charge transport in theEDL which is called SC ISC. If we operate the EKC in a opencircuit mode, a potential difference between the reservoirsarises. This potential difference is called SP ΔφSP = φ1 − φ3which, in turn, induces the conduction current IC between theelectrodes. Whether we measure SC or SP depends on theelectrical instrument connected to the EKC. This is indicatedby the different symbols which connect the electrodes, such as,ammeter and voltmeter, and an “impedance meter” used for theelectrical characterization.For further description, we analyze the equivalent electrical

circuits (EEC) for the different operation modes as given inFigure 2. The main motivation here is that the EEC of the SP/

Figure 1. Schematic of the electrokinetic cell used in this work tomeasure the electrokinetic phenomena of streaming potential andstreaming current. The dimensions are not to scale.

Figure 2. Equivalent electrical circuits for (a) SC measurements; (b)SP measurements; (c) EIS (left) with a typical Nyquist plot (right)measured in a microchannel of height h2 = 60 μm and ionic strength I= 0.5 mM.

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SC does not correspond to the one for the electricalcharacterization. Generally, the SC and SP circuits are similarand consist of an internal and external circuit. The internalcircuit, indicated by the dotted-dashed line, comprises a currentsource symbolizing the SC. Additionally, there is an internalresistor Ri which is required to generate the SP. The outercircuit consists of the electrical elements symbolizing electrodesand microchannel. We computed that there is practically noinfluence of the liquid reservoirs on the electrical problem. Awell-accepted model for electrodes is a RC element, where theelectron transfer resistance is modeled with an ohmic resistorwhile the electrode’s EDL is considered a capacitor, botharranged in parallel.29 We make the reasonable assumption thatR1 = R3 = Rel and C1 = C3 = Cel. The ionic path in themicrochannel is modeled with ohmic resistors. Here, wedistinguish between the resistor of the bulk liquid R2B andsurface R2S. If we measure the SC, an ammeter is connected tothe electrodes which has practically no internal resistance asdepicted in Figure 2a. Hence, the entire SC flows through theelectrodes and the conduction current is practically zero sincethe microchannel resistance is much higher than that of theammeter. For SP measurements, illustrated in part b, weconnect a voltmeter to the EKC. The voltmeter should have aninternal resistance much higher than that of the channel. Then,the SC in conjunction with the internal resistor Ri induces theSP between the terminals (reservoirs) 1 and 3. Practically, nocurrent flows through the electrodes and the voltmeter. The SPinduces the conduction current which goes through themicrochannel resistors R2B and R2S.The situation changes when we perform an in situ electrical

characterization of the EKC using electrical impedancespectroscopy (EIS) as given in Figure 2c (left). Essentially,we apply an alternating excitation across the EKC and measurethe alternating answer for different excitation frequencies. Atypical result is given as well in Figure 2c (right). We discussthe method in more detail in section III C and give here only abrief summary. For fairly conductive electrolytes, EIS allows fora separate determination of the total microchannel resistor R2 =1/(1/R2B + 1/R2S) and the electrode resistor Rel. However, forelectrolytes with a very low ion content, there is the so-calledbulk capacitance C2 arranged in parallel to R2 (dashed line) andonly the total external resistance R2 + 2Rel can be measured.To summarize the EEC analysis: The current paths in the

EKC depend on the connected instrument. For SC measure-ments, the SC goes through the electrodes while for SP theconduction current goes through the microchannel but notthrough the electrodes. EIS only allows for the measurement ofthe total microchannel resistance but not for the singlecontributions of bulk and surface. For low ionic strengths, theelectrode transfer resistance is included in the measuredresistance.We now derive the electrokinetic and hydrodynamic

correlations in the EKC where we generally follow the standardassumptions as for example, given in ref 30. However, wedeviate in some points: (i) We do not use the pressuredifference since we perform experiments with different channelheights. Hence, we relate the electrokinetic phenomena to theReynolds number, which expresses hydrodynamics independ-ent of the channel geometries. (ii) The hydrodynamiccorrelations are given independent of the channel’s crosssection to ensure a general applicability for laminar flows inarbitrary geometries. (iii) Differentials are used instead ofdifferences to obtain a general applicability. We also do not

account for diffusive transport which has significant contribu-tions in colloidal and membrane systems.31,32 The reader isreferred to the Supporting Information of this article for adetailed analysis showing the insignificance of diffusive(polarization) currents in microchannels with h2/l2 ≪ 1.A differential SC in a microchannel of arbitrary cross section

Ac, using a linearized velocity profile in the EDL, can be writtenas

εζμ

= −⎛⎝⎜

⎞⎠⎟I A

px

d dddSC c

(1)

Here, ε is the liquid permittivity; μ is the dynamic viscosity; andd(dp/dx) is a differential change of the pressure gradient inflow direction x. The differential pressure gradient can beexpressed in terms of a differential volumetric flow rateaccording to d(dp/dx) = Rμ dV, where the viscous flowresistance Rμ depends on the cross section of the microchannel.Using the velocity amplitude u0 of an equivalent plug flow, thehydraulic diameter dh = 4Ac/Pc, where Pc is the wettedperimeter of the cross section, and the liquid density ρ, we canwrite the differential Reynolds number in a channel of arbitrarycross section as 4ρ/(μRμPc)d(dp/dx). After some rearrange-ments, we obtain

εζρ

=− μI

A R

dRed dSC

c2

h (2)

for the differential SC in a channel of arbitrary cross sectiondepending on the differential Reynolds number. Equation 2 canbe employed to obtain the ZP from measurements of ISC vs Re.For a steady state and no current flowing through the

electrodes (cf. Figure 2b), the differential SC corresponds tothe negative differential conduction current which can also beexpressed using Ohm’s law. That is,

φ φ= − = ≈

Δ⎜ ⎟⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟I I

R Rd d d

dd

dSC CSP

2 (3)

where the gradient of the electrical potential with respect to theohmic resistance dφ/dR can also be approximated with theratio of SP and total microchannel resistance. After inserting eq2 in eq 3, integrating over the channel length l2, and furtherrearrangements, we obtain the general expression

ζ ρ ε φ=

Δ

μA R R Red / d

dh

c2

2

SP

(4)

which relates the ZP to the gradient of the SP with respect tothe Reynolds number. The presence of the viscous resistanceRμ indicates that this electrokinetic phenomenon originatesfrom the viscous drag of the electrical charge in the EDL. Toobtain a correlation for a specific channel, the geometricparameters must be customized. In this work, we use amicrochannel of rectangular cross section of area Ac = w2h2.The correlation between volumetric flow rate and pressuregradient is an infinite analytical series solution where the flowresistance can be approximated as Rμ ≈ 12μ/h2

3w2.33 Inserting

the geometric parameters in eq 4, we obtain

ζ ρεμ

φ=

+Δh

h w R Re6( )1 d

d22

2 2 2

SP

(5)

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Note that the correlation depends on the channel resistance R2that we measure in situ under the same conditions as for the SPexperiments.The classical HS theory considers only the bulk liquid

resistance in the microchannel R2B = l2/(w2h2 σB). Hence, theHS equivalent of eq 5 is

ζρσεμ

φ=

+Δh w

l h w Re6 ( )

d

d23

2

2 2 2

B SP

(6)

The modification of the classical HS correlation incorporatesthe surface conductivity (surface resistance R2S) arising fromthe EDL at the interfaces. The ions in the EDL have a differentconcentration than those of the bulk; the mobility may differ aswell. This phenomenon is quantified in terms of the surfaceconductivity σS which is the surface equivalent to the bulkconductivity σB. Whatever the ion distribution in the EDL, σScan always be defined through the 2D analog of Ohm’s law;that is, jS = −σS(dφ/dx) where jS is the surface current densityin A/m.1 The influence of the surface conduction is usuallyexpressed by the dimensionless Dukhin number Du = σS/(σBh2). We realize that the surface conductivity is proportionalto the inverse channel height; that is, it becomes prominent forvery narrow channels as well as for low ionic strengths. Finally,considering the relations above, we use the ohmic resistance R2≡ (1/(1/R2B + 1/R2S)) and obtain the equivalent modified HSeq which accounts for the surface conductance of the upper andlower channel wall according to

ζρσεμ

φ=

++

Δh wl h w

DuRe6 ( )

(1 2 )d

d23

2

2 2 2

B SP

(7)

Generally, the surface conductivity can have contributionsowing to the diffuse layer (DL) charge, adjacent to the plane ofshear, and to the stagnant layer (SL) charge; it is σS = σDL + σSL(see e.g., ref 34). The DL contribution was first derived forsymmetric electrolytes by Bikerman in 1933 (as cited in ref 35)as

σ εν

ζν

ζε

μ ζ

=−

−+

+−

+ −⎛⎝⎜

⎞⎠⎟

R TIA A

R T

zF A

8( ) 1 ( ) 1

4 1( ) 1

DL g

g2

(8)

where Rg is the universal Gas constant; ν+ and ν− are theabsolute values of the ion mobility of cation and anion,respectively; and I and T are the ionic strength and the absolutetemperature of the symmetric electrolyte. The function A(ζ) =coth(zFζ/(4RgT)) derives from the analytical solution of thecharge distribution at an infinite plate and contains the valencyz and the Faraday constant F. This expression includes thecontribution of migration and that of electroosmosis whichresults in an additional mobility of the charges. The expressionis valid for negative surface charges. In case of positive ones, thecationic and anionic mobilities have to be exchanged. There aredifferent assumptions on the contribution of the stagnant layerto the surface conductivity. Historically, the term stagnant hasbeen defined such that neither a hydrodynamic flow nor an ionmigration takes place in the SL. However, in order to explainexperimental discrepancies between surface charge and electro-kinetic charge density, i.e., the charge density at the plane ofshear, it is proposed that the SL conductivity σSL may include acontribution due to the specifically adsorbed charge and

another one due to the part of the DL that may reside behindthe plane of shear.1 The charge on the solid surface is generallyassumed to be immobile and, hence, does not contribute to σS.

1

Here, some authors assume that the solvent moleculesthat is,water in case of an aqueous electrolyteare at rest in the SL,but ion migration as well as ion diffusion is possible.36,37 Otherauthors38,39 assume that the liquid in the SL consists of apermeable gel-like layer. These assumptions are supported byMolecular Dynamic Simulations (MDS) of Lyklema et al.which revealed that the macroscopic structure of the solventmolecules in the SL is indeed gel-like while the mobilities of thedissolved ions are hardly impaired.40

III. EXPERIMENTAL METHODS AND MATERIALSThis section describes details of the experimental setup, materials, andmethods that we use in this work.

A. Experimental Setup. The experimental setup consists of theelectrokinetic cell (EKC) and several fluidic and electrical components.The EKC is connected to a polyethylene vessel containing the aqueouselectrolyte. This vessel can be charged with a controlled pressure by anitrogen gas cylinder (Ultra High Purity 99.999%, MEGS, Canada) topump the aqueous solution through the EKC. The pressure dropacross the EKC is measured by two pressure gauges. Two Ag/AgClelectrodes are inserted in the EKC for the measurement of electricalsignals. Electrodes are made from silver wires (99.99% trace metals,Sigma-Aldrich, Canada) by anodic deposition of silver chloride in a 3.0M KCl solution (ACS reagent grade, Sigma-Aldrich, Canada) at acurrent density of 1 mA/cm2 for 45 min. The electrodes are connectedto a potentiostat (PGSTAT302N, Metrohm Autolab B.V., TheNetherlands) which allows for the measurement of electrical signals;the input impedance of the instrument is >1 TΩ.

The custom-build EKC is constructed from two Teflon blockscontaining fluidic and electrical connections, liquid reservoirs, andallows for the incorporation of two test wafers of dimensions 70 mm ×30 mm × 1.5 mm to form the microchannel. We use test wafers madefrom PMMA sheets since a comprehensive data set which can be usedfor verification is available from our previous work;25 however, anymaterial can be tested as long as it features the specified dimensions.Gaskets are used inside the PTFE blocks to form a well-defined gapbetween the PMMA wafers. Microchannel geometries used in thiswork are of length l2 = 7 cm and width w2 = 2.5 cm, while heights fromh2 ≃ 5 to 300 μm are achieved depending on the gaskets. Two types ofgaskets are used: (i) rigid (spacer) gaskets made of PTFE sheets orKapton (polyimide) which give microchannel heights of h2 ≳ 50 μmand h2 ≲ 50 μm, respectively; (ii) soft gaskets of latex rubber which areused for sealing purposes.

B. Streaming Potential/Current Protocol. SP and SC measure-ments are carried out using the following procedure. First, PMMAwafers are cleaned using a cleaning procedure as described in ref 10.PMMA wafers and gaskets are then installed in the EKC, and theassembled device is connected to the setup. Four different micro-channel heights (h2 = 6, 60, 175, and 300 μm) are used in this work.Verification of the microchannel height is done each time the cell isassembled by using electrical and hydrodynamic resistance measure-ments. The observed accuracies are around 30% and 10% for h2 < 100μm and h2 > 100 μm, respectively. Electrolyte solutions are preparedusing NaCl (ACS reagent, Sigma-Aldrich, Canada) dissolved in a DIwater matrix (σ ≲ 1 μS/cm). The lowest ionic strength that can beprepared is I ≈ 0.002 mM where we consider the contribution of thecarbonic acid equilibrium to the water matrix. There is no bufferreagents added to adjust the pH value since there is no influence of thepH value on the ZP of PMMA in an acidic and neutral regime.25

Hence, the pH value of the electrolytes is around 5.5 ± 1. Ex-situconductivity and pH testings are performed using a modular pH andconductivity meter (Mettler-Toledo, SevenMulti, Switzerland).Subsequently, the electrolyte solution is filled in the vessel which isthen pressurized by the nitrogen gas. Before each experiment, therespective electrolyte is pumped through the EKC for 30 min at a

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pressure difference Δp = 25 kPa. The actual experiment is undertakenby adjusting pressure differences in the range of 10 kPa to 75 kPa; themeasurement of the electrokinetic phenomenon is done for 60seconds at a constant pressure. We use the corresponding time-averaged value for data evaluation. For each ionic strength, 5 pressurevalues are applied to obtain a correlation between SP or SC andReynolds number of the flow. Note that the available range ofReynolds numbers depends on the microchannel height. For thelargest and smallest height the investigated range is Re ≈ 0 − 4000 andRe ≈ 0 − 0.04, respectively. To minimize the asymmetry potential, theelectrodes are short-circuited between two consecutive measurements.Furthermore, the electrodes are renewed between measurements ofdifferent ionic strengths or when the asymmetry potential becomesquite significant.C. Electrical Impedance Spectroscopy. We measure the

channel resistance in situ under operation conditions by employingEIS. The impedance Z = R + jX is a measure of the ability of anelectrical circuit to resist the passage of an alternating current. Itconsists of time-independent, linear elements expressed by an ohmicresistor (real part) R and the reactance (imaginary part) X whichdescribes the time-dependent behavior of nonlinear elements such ascapacitors or inductors. It can be determined by applying analternating voltage U to the EKC and measuring the resultingalternating current I. The impedance is defined likewise to Ohm’s Lawas Z = U/I. If the EEC contains nonlinear elements, the current andimpedance feature a phase shift. For EIS, the impedance is scanned fora range of excitation frequencies which often allows for specification ofthe single EEC elements. The use of EIS, as opposed to DCmeasurements, has a number of advantages. AC signals at appropriatefrequencies eliminate the parasitic influence of liquid viscous relaxationand other capacitive elements in the setup. A typical example is theelectrode capacitance (see Figure 2), but also those of plugs and cableswhich can considerably influence the results if DC or AC voltages oflow frequencies are used. Additionally, DC or low frequencymeasurements may induce electrochemical reactions at the electrodeswhich considerably alter pH and ionic strength of the liquid;information on EIS can be found in the literature.41 For furtherevaluation, the charge relaxation time tr = ε/σ; that is, the ratio of theliquid permittivity and conductivity can be used. The relaxation time isthe time scale at which free charges relax from the liquid bulk to theouter boundaries;42 it is a measure of how long it takes for aperturbated electrical system to become polarized by conductionprocesses. We estimate that we require frequencies higher than f ≫ 1/tr ≃ 100 kHz to exclude polarization effects in the case of very lowionic strengths.Essentially, we measure impedances in the EKC for resting liquids

and various flows in a frequency range of f ≈ 0−106 Hz. Theimpedance data are interpreted on the basis of the EEC in conjunctionwith Nyquist plots as given in Figure 2c. A Nyquist plot containsimpedance data for the measured frequency range; the imaginary andreal parts are plotted on the y-axis and x-axis, respectively. The Nyquistplots measured in this work are full or partial semicircles depending onthe scanned frequency range; this is in accordance to the theory asgiven by the EEC. A full semicircle crosses the x-axis at f = 0 and at f→ ∞.In the case of high ionic strengths, scans are performed over the

entire frequency range. In this case, there is no significant influence ofthe bulk capacitance C2, and the electrode capacitors Cel are short-circuited at high excitation frequencies. Hence, the x-intercept at f →∞ corresponds to the total ohmic resistance of the microchannel R2.The interpretation is more demanding for very low ionic strengthliquids. For this case, the aqueous electrolyte is more or less pure waterand behaves more like a dielectric than a conductive liquid. Analternating electrical field reorientates the polar water moleculesresulting in a displacement current. Hence, a fraction of the high-frequency alternating current goes through C2 and bypasses R2. The x-intercept value at f → ∞ does not correspond to R2 anymore. Theinterpretation of the EEC gives that the x-intercept value at f = 0corresponds to the total ohmic resistance R2 + 2Rel of the externalcircuit, though this value cannot be directly measured since the already

discussed parasitic capacitive effects occur at low frequencies givingvery scattered data. In this case, we measure only in a medium-to-highfrequency range and fit the EIS data to a semicircle correlation. The fitis used to infer R2 + 2Rel from the x-intercept at f = 0. Consequently,further experiments are required to obtain Rel which then allows forthe identification of R2.

Hence, we use a modified setup where the EKC is replaced bynarrow glass capillaries of different lengths l = 3, 6.8, and 9.4 cm. Thecapillaries are connected to the setup by T-junctions with Ag/AgClelectrodes incorporated. Defined electrolyte solutions are pumpedthrough the capillaries, and the respective impedances are measured.The plot of impedance, at constant Reynolds number and ionicstrength, versus the capillary length reveals a linear correlation. Weextrapolate the capillary length to l = 0 to find the impedance of thetwo T-junctions. The ohmic resistance of the electrolyte solution inthe T-junction can be neglected to good approximation; hence, theresistance at l = 0 corresponds to that of both electrodes for the givenionic strength.

IV. EXPERIMENTAL RESULTS AND DISCUSSIONThe experimental results are given in two subsections, namelyEIS and electrokinetic experiments. All results are given in theform of mean values and their standard deviations, based on atleast three replicates or on standard errors.

A. Electrical Impedance Spectroscopy Experiments.The first set of experiments is performed to determine theohmic resistor Rel depending on flow rate and the ionic strengthof the aqueous NaCl electrolyte. The motivation for theseexperiments is manifold. On the one hand, we are not aware ofany literature which confirms the reversibility of Ag/AgClelectrodes at very low ionic strengths. Additionally, we needthese values for the accurate determination of the microchannelresistance as explained in the previous section. On the otherhand, it is important to prove whether the influence of the flowfield, that we observe below, is related to the flow around theelectrodes or to the flow through the microchannel. The resultsare given in Table 1 as electron transfer resistance RT = RelAel

where Ael is the wetted electrode area. The data show that thelower the ionic strength is, the higher is the electrode transferresistance. Note that the values of the electrode resistor at lowionic strength can be up to around 5% of the microchannelresistance. Furthermore, we do not find a dependency of theelectrode resistor on the electrolyte flow. A regression yieldsthe logarithmic correlation log(RT/(Ω m2)) = −1.023log(I/(mol/L)) − 4.001. Table 1 also gives the correspondingexchange current densitiesa measure for the electrodereversibilitywhich are calculated the standard way.19 Forexample, a mercury-sulfate electrode features an exchange

Table 1. Electron Transfer Resistance and Exchange CurrentDensity of an Ag/AgCl Electrode in Aqueous NaClElectrolytes Measured by EISa

ionic strengthI/mM

electron transfer resistanceRT/(kΩ mm2)

exchange current densityj0/(A/cm

2))

0.002 1.30 × 105 ± 11% 1.98 × 10−8

0.01 1.26 × 104 ± 16% 2.01 × 10−7

0.1 1.80 × 103 ± 17% 1.40 × 10−6

1 59.79 ± 23% 4.23 × 10−4

10 22.75 ± 13% 1.11 × 10−4

50 1.53 ± 10% 1.66 × 10−3

aThe standard deviation of an exchange current density at a givenionic strength corresponds to that of the respective electron transferresistance.

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current density of j00 = 10−12 A/cm2 at standard state and is

therefore highly polarizable.19 In terms of a SC experiment,such an electrode cannot be employed since it does not take upthe electrical charges which accumulate in the electrode EDLeven at high overpotentials. In contrast, a Ag/AgCl electrode atstandard state features j0

0 = 13.4 A/cm2 and is thereforeconsidered as fully reversible.19 This means that the SC isentirely consumed by the electrodes at overpotentials which arepractically zero and there is no unwanted potential differencewhich induces a conduction current. However, an electrokineticexperiment is generally conducted under conditions notcomparable to the standard state, and the data in Table 1indicate that a Ag/AgCl electrode at low ionic strengths is faraway from being fully reversible; a fact that is mainly neglectedin the respective literature. The question arises whether thisinduces a significant conduction current during SC measure-ments. To answer this question, we estimate the potentialdifference between the electrodes using the Butler−Volmer eqin conjunction with the equilibrium current densities in Table 1and a charge transfer coefficient (symmetry factor) of β = 0.3.We estimate that the influence on the SC is ≲1%. Hence, theinfluence of the nonideal electrodes can be neglected withrespect to the higher standard deviations of our measurements,though, we assume that the errors become significant for wornout electrodes and low chloride concentrations underlining theimportance to renew the electrodes on a frequent basis.In terms of SP measurements, we subtract the electrode

resistances from the total resistance of the external circuit thatwe measure using in situ EIS. Figure 3 shows the results of

various microchannels for stagnant liquids (Re = 0). In detail,we plot the product of microchannel’s resistance and heightversus the ionic strength of the NaCl solution, both on alogarithmic scale. We also show the corresponding valuesinferred from ex-situ measurements; that is, we use the bulkconductivities σB and compute the corresponding resistancesbased on the microchannel geometries. Then, the motivationfor this representation becomes clear. The correlation betweenthe product of ex-situ resistance and channel height and ionicstrength is linear as indicated by the solid line. The situation isdifferent for the resistances which are measured in situ. Whilewe find a similar linear relationship for intermediate to high

ionic strengths, we find generally lower values for allinvestigated microchannels at low ionic strengths of I ≲ 0.1mM. Generally, the lower the ionic strength and the smaller thechannel height, the higher is the deviation from the ex-situresults. Here, the question arises whether the conductivitydifference can be explained by the dissociation of dissolvedcarbon dioxide. Calculations based on the carbonic acidequilibrium reveal that we would need an atmosphere with aCO2 concentration of more than 10000 ppm to explain theobserved discrepancy. For comparison, the air in theatmosphere contains around 400 ppm and the nitrogen gasthat we use to pressurize the vessel has a concentration of lessthan 1 ppm so that this possibility can be ruled out. Note alsothat we cannot find significant conductivity differences betweenthe aqueous electrolyte which enters and leaves the EKC.Figure 3 also contains an inset which demonstrates that the I ≲0.1 mM data collapses onto a single curve, to a goodapproximation, if the resistances are multiplied with anadequate scale f(h2). Hence, a regression can be performedusing two correlations which depend on the range of the ionicstrength; we obtain

σ

Ω≈

≳

+≲

−

μ

⎧

⎨⎪⎪⎪

⎩⎪⎪⎪

( )R h I

lw h

lI

I

( , )

M

for 1 mM

0.0041 0.433for 0.1 mM

I

h

2,0 2

2

2 2 B

mol/L

0.272

m2

(9)

Further EIS experiments are performed for various Reynoldsnumber flows. For some conditions (discussed below), weobserve that the resistance changes linearly with the Reynoldsnumbers, to a good approximation. Detailed results are given inFigure 4a in the form of relative resistance change per Reynoldsnumber ΔR2/Re versus ionic strength. Here, relative means thatthe absolute resistance change per Reynolds number isnormalized with the respective resistance for the stagnantliquid R2,0. The connections between the data points are for thesake of a better illustration. A twofold behavior of ΔR2/Re interms of the ionic strength can be found. Generally, ΔR2/Re ata given channel height remains more or less constant for I ≲ 0.1mM. For I ≳ 0.1 mM, we observe a decrease of ΔR2/Re with anincreasing ionic strength. We also notice that the smaller is thechannel height, the higher is ΔR2/Re; the difference can beseveral orders of magnitude. In this context, we should recallthat there are considerably different Reynolds number rangesfor the different channel heights. Essentially, the change of themicrochannel resistance is rather insignificant for large channelsand high ionic strengths. A specific example, whichdemonstrates the absolute resistance for different Reynoldsnumbers, is discussed below. To find an empirical correlationfor the relevant data, we divide ΔR2/Re with the magnitude ofthe logarithmic ionic strength and plot it versus the channelheight, both on a logarithmic scale. Figure 4b demonstrates thatthe data collapses onto two different curves, depending on therange of the ionic strength. The slopes of the curves are almostidentical; the difference is in the y-intercept. Finally, we use thecollapsed data to derive an empirical correlation for themicrochannel resistance as a function of ionic strength, channelheight, and Reynolds number consisting of eq 9 and the flowcorrection term

Figure 3. Microchannel resistance times the channel height vs theionic strength for stagnant liquids (Re = 0). For intermediate to highionic strengths, there is no significant difference between in situ andex-situ measured microchannel resistances. For low ionic strengths, insitu measurements reveal significantly lower values heights. The insetshows the collapse of the low ionic strength data if scaledappropriately.

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=

>

μ

≈

μ

≲

−

−

⎧

⎨

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

f h I Re

I

h IRe

I

h IRe

I

( , , )

0, for 1 mM

10m

logmol/L

,

for 1 mM

10m

logmol/L

,

for 0.1 mM

2

2.4153.28

3.393.117

(10)

according to

Ω≈ +

R h I ReR h I f h I Re

( , , )M

( , )(1 ( , , ))22,0 2 (11)

Note that eq 11 is a reasonable quantitative correlation for mostof the investigated range while it gives only qualitative resultsfor the transition between low and high ionic strengths.We employ eq 11 to obtain a deeper insight into the EIS

results. First, we investigate the limiting case of a microchannelof vanishing height; that is, h2 → 0. For I ≳ 1 mM the channelresistance approaches infinity. For I ≲ 1 mM a finite value isobserved which depends on the ionic strength and theReynolds number. Recall that for decreasing microchannelheight, the influence of the surface conductivity increases,whereas the influence of the bulk liquid decreases. Con-sequently, the surface conductivity can be derived from theinverse resistance at h2 → 0. Figure 5a gives the so inferred

surface conductivity σS of the PMMA surface (dotted line)versus the ionic strength and for a stagnant liquid, both plottedon a logarithmic scale. For the sake of comparison, theBikerman surface conductivity σDL is given as well. In detail, wecompute σDL based on eq 8 using ZPs from four SCexperiments as described below. Generally, we find that bothsurface conductivities linearly increase with an increasing ionicstrength. The slopes are fairly similar as well. The maindifference is on the magnitude. While the measured surfaceconductivity is on the order of nano-Siemens, the Bikermansurface conductivity which accounts only for the diffuse layer isabout one order of magnitude less. This would mean that thesurface conductivity is mainly determined by the stagnant layerwhile the double-layer contribution is almost negligible.Assuming that the ionic mobility is not significantly impairedin the SL, this would mean that there is considerably morecharge in the SL than in the DL, which is also indicated by themolecular dynamics simulations of Lyklema et al.40 Ourexperimental observation is also consistent with the work ofWerner et al.23 and Leroy and Revil43 who measured thesurface conductivity of fluor-polymers and clay minerals,respectively. Both studies found high surface conductivities inthe nano-Siemens regime and attributed the main contributionto the SL (inner Helmholtz plane). Nevertheless, the questionarises whether there are further phenomena contributing to thesurface conductivity which are yet to be identified.

Figure 4. Relative change of the microchannel resistance per Reynoldsnumber (a) for different channel heights vs the ionic strength; (b)scaled with the absolute logarithmic ionic strength vs the channelheight. The results demonstrate that the electrical resistance of themicrochannel depends on the flow conditions, especially for smallchannel heights and electrolytes of low ionic strengths.

Figure 5. (a) Surface conductivity vs the ionic strength of the aqueouselectrolyte; (b) Dukhin numbers from in situ and ex-situ measure-ments vs the microchannel height for stagnant liquids (Re = 0). Themeasured surface conductivities are much higher than those predictedfrom theory and their influence can only be neglected for relativelylarge channels of height h2 ≳ 100 μm along with ionic strengths of I ≳1 mM.

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We use the surface conductivities to calculate Dukhinnumbers as defined in section II. Figure 5b shows the Dukhinnumbers based on in situ and ex-situ (i.e., bulk conductivity inconjunction with eq 8) measurements versus the channelheight and for various ionic strengths. Generally, the Dukhinnumber increases with decreasing channel heights anddecreasing ionic strengths. In terms of the ex-situ data, asignificant surface conductivity influence is only predicted forchannel heights of h2 ≲ 100 μm. In contrast, the in situ datareveal that even for the largest channel height, Dukhin numbersof Du ∼ 0.1 for the low ionic strengths are found. For thesmallest channels and lowest ionic strengths, even Dukhinnumbers of Du ≳ 10 occur.We now demonstrate the influence of the channel flow as

predicted by eq 11. Figure 6a plots the resistance R2 of a

microchannel with height h2 = 60 μm for various ionicstrengths and depending on the Reynolds number, both on alogarithmic scale. The solid circles indicate the respective ex-situ resistances which are independent of the flow. We find thatfor low Reynolds numbers of Re ≈ 0 to 10, all channelresistances remain rather constant. However, for higherReynolds numbers, a considerable increase is observed. Theonset of the increase depends on the ionic strength. We observethat the lower the ionic strength is, the higher is the Reynoldsnumber which is required to observe in situ measuredresistances equal to the ex-situ measured ones. This statementdoes not hold for the I = 1 mM data in which the ex situ valueis lower than the in situ value at Re = 0. This is due to thelimited accuracy from the regression of an ensemble of

collapsed data. It should also be noted that the achievableReynolds number range for the given channel height, is Re ≈ 0to 50; that is, higher values are extrapolated.The influence of the Reynolds number on the Dukhin

number is reported in Figure 6b. At the given channel height, anegligible influence of the surface conductivity (Du ≃ 0.01) isonly observed for I = 1 mM. For lower ionic strengths, theinfluence of surface conductivity is mainly significant Du ≃ 0.1.Generally, an increase in Reynolds number decreases theinfluence of the surface conductivity.Finally, we perform a limited set of experiments using

borosilicate glass wafers in the EKC forming a microchannel ofheight h2 = 49 μm. The electrical (as well as the electrokinetic)results show an identical qualitative behavior as for the PMMAwafers (not shown). Equations 9 to 10 can still be employed todescribe the measured microchannel resistance quantitatively.However, some minor parameter fitting is required whichcomes without surprise since the surface conductivity ofPMMA and borosilicate glass should be different.

B. Electrokinetic Experiments. In this section, we discussthe ZPs resulting from various SC and SP measurements usingresistances from in situ and ex situ measurements.

1. Streaming Current. SCs are measured for varyingmicrochannel heights and ionic strengths. We learned in theprevious section that flows through channels of small heightshave an impact on the channel resistance. This behavior mustbe related to a interfacial phenomena as we discuss in moredetail below in the Conclusion section. If the interfacialstructure changes with the flow, this must consequently bereflected in a dependency of the ZP on the flow as well. Hence,we plot the SC versus the Reynolds number (not shown) andperform two types of data regressions, a linear and a quadraticone. The linear regression results in a constant slope andtherefore, in conjunction with eq 2, in a ZP which isindependent of the Reynolds number. The quadratic regressionresults in a linear dependency of ZP and Reynolds number.Note that both regression types generally result in comparableand high coefficients of determination.Figure 7a gives the ZP versus the ionic strength of the

aqueous electrolytes, exemplarily for a channel with h2 = 60 μm.The standard deviation is only given for the linear regressiondata. Generally, we find that the magnitude of the ZP decreaseslinearly with increasing ionic strength. Linear correlationsbetween ZPs and the logarithmic ionic strength are observed bymultiple researchers, even for high ZP magnitudes and forsystems with relatively complex surface chemistries.2,44 PMMAin an acidic and neutral milieu features no relevant surfacechemistry.25 Hence, we assume that the nature of the linearcorrelation between ionic strength and ZP is mainly based onthe shielding of the surface charges. The lowest set of ZPmagnitudes is observed when the SC data are processed with alinear regression. The utilization of the quadratic correlationraises the question of which Reynolds number should be usedfor the calculation of the ZP. We choose Re = 0 and Re = 20,where the latter is roughly the median Reynolds number of theexperiment. We observe the highest magnitudes of the ZPs forRe = 0. The magnitude decreases with increasing Reynoldsnumber. However, if we consider the standard deviations of theexperiments, the differences are minor.Figure 7b shows the ZP versus ionic strength inferred from

SC measurements in microchannels of different heights. Weemploy the quadratic correlation to process the data and givethe ZPs for the respective median Reynolds numbers of the

Figure 6. (a) Microchannel resistance and (b) Dukhin number vs theReynolds number in a microchannel of height h2 = 60 μm for aqueouselectrolytes of various ionic strengths. The total channel resistance and(the influence of) the surface conductivity strongly depend on the flowthrough the channel. The smaller is the channel height and the lower isthe ionic strength, the higher is the influence.

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experiments. The dashed line gives a regression of the ZPsaveraged over all channel heights. Generally, the differencesbetween the ZPs measured in different channels are more orless within the standard deviation of the experiments. We alsoplot the ZP based on the empirical correlation ζ/mV = (28.21− 2.7pH + 0.29pH2) log(I/(mol/L)) received from LaserDoppler Electrophoresis (LDEP) experiments of PMMAparticles as described in ref 25. While the slope of thecorrelation is in good agreement with the present data, themagnitudes of the ZP measured with LDEP are around 20 to30% higher for low ionic strengths. Differences are smaller forhigher ionic strengths.2. Streaming Potential. SP measurements are performed for

varying microchannel heights and ionic strengths. In contrast tothe SC experiments, we generally observe that a quadraticcorrelation reproduces the data with better quality than a linearone. SPs are calculated on the basis of eq 5 and eq 6 toincorporate the findings from section A.Figure 8a shows the ZPs inferred from measurements in the

microchannel with h2 = 60 μm. First, we apply the classical HSeq 6 along with the bulk conductivity and a linear regression ofthe experimental data. The results are labeled as ex-situ andshown with the standard deviations of the experiment.Altogether, we do not observe a linear behavior between ZPand logarithmic ionic strength. We distinguish between tworanges which are in accordance with those observed in the insitu resistance measurements. For I ≳ 1 mM the magnitude ofthe ZP decreases with increasing ionic strength. In contrast, themagnitude of the ZP increases with increasing ionic strength incase of I ≲ 1 mM. This deviation for I < 1 mM has been

observed by different researchers who employ HS correla-tion.45−47 Nevertheless, the large majority of work in literaturedoes not show data for I < 1 mM. Next, we discuss resultsbased on eq 5 using the channel resistances from the in situmeasurements. In this case, the ZP is two-fold influenced by theReynolds number: (i) the channel resistance can depend on theflow (see eq 11) and (ii) the use of the quadratic regressionresults in dΔφSP/dRe = f(Re). We compute the ZPs fordifferent Reynolds numbers. Here, it is striking that we recoverthe linear correlation of ZP and logarithmic ionic strengths forRe = 0 as indicated by the respective regression (solid line). Forincreasing Reynolds numbers, there is no change for I ≳ 1 mM,and the values are nearly identical to the ex-situ ones. For I < 1mM we see that the magnitude of the ZP decreases withincreasing channel flow. That is, the higher the Reynolds

Figure 7. Zeta potential vs ionic strength (a) measured in amicrochannel with h2 = 60 μm based on linear and quadratic dataregression; and (b) for different microchannel heights and a medianReynolds number. Linear correlations between ZP and logarithmicionic strength are observed. Moderate differences exist depending onwhether a linear or a quadratic correlation between SC gradient andReynolds number is used.

Figure 8. Zeta potential vs ionic strength (a) measured in amicrochannel with h2 = 60 μm and different Reynolds numbers; (b)for different microchannel heights and Re = 0; (c) for differentmicrochannel heights and the median Reynolds numbers of theexperiments. In case of low ionic strength electrolyte, the ZPs dependon the microchannel flow. The linear correlation between ZP andlogarithmic ionic strength is recovered for in situ resistances and Re =0.

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number is, the closer are the ZPs of in situ experiments to thoseof the ex-situ experiments.Figure 8b summarizes the results obtained from measure-

ments in microchannels of different heights. All ZPs arecomputed using the results from the in situ resistancemeasurements and for Re = 0. We give the standard deviationof the experiments with channel height h2 = 6 μm, the values ofthe other experiments are similar or less. Only small differencesare observed for ZPs based on measurements with channels ofdifferent heights. Furthermore, we observe a linear correlationbetween the ZP and the logarithm of the ionic strength in allcases. We also plot a regression of the average values over allchannel heights (dot−dash line) and compare it with the LDEPdata (solid line). Surprisingly, we find a very good agreement ofaveraged SP and LDEP data with some differences for low ionicstrengths.Finally, we would like to present in Figure 8c the ZPs based

on experiments with different channel heights and inferredusing the median Reynolds number of the respectiveexperiment. For I ≲ 0.1 mM we find that the ZPs computedwith the median Reynolds numbers are generally lower thanthose at Re = 0 (see Figure 8b). Furthermore, the LDEP resultsand the averaged SC results are given as a solid and dashed line,respectively. We find an interesting aspect: For I ≳ 1 mM theSP results for median Reynolds numbers are in good agreementwith the LDEP results. However, for I ≲ 0.1 mM the SP resultsfor median Reynolds numbers rather correspond to the SCresults.

V. CONCLUSIONSWe perform streaming current and potential measurements inPMMA microchannels of different heights and for aqueouselectrolytes of various ionic strengths. Electrical impedancespectroscopy is applied to measure the resistance of themicrochannel in situ under experimental conditions. Theimpedance measurements show that in the case of liquidshaving low ionic strengths, there is a significant differencebetween the channel resistances obtained from in situ and ex-situ measurements. The differences are significant even forrelatively large channel heights which has not yet been reportedin the literature. In this case, the classical Helmholtz−Smoluchowski correlation fails. We assign the difference tothe influence of surface conductivity. A comparison with theBikerman diffuse layer contribution indicates that the majorityof the surface conductivity must either arise from the stagnantpart of the electrical double layer or from a phenomena whichis not yet specified. We also study the effect of the flow field(Reynolds number) on the channel resistance. For low ionicstrength liquids, the effect significantly increases with thereciprocal channel height. The proportionality with the channelheight once more indicates that this behavior is related to aninterfacial and not to a bulk phenomenon. A limited series ofexperiments using borosilicate glass wafers in the electrokineticcell shows the same qualitative results as for the PMMA wafers.Additionally, minor changes in the empirical equations derivedfrom the PMMA in situ experiments allow for the quantitativedescription of the borosilicate glass experiments.On the basis of our results, we hypothesize that convection

influences the configuration of an EDL which is observed as achange of surface conductivity. Our hypothesis is in contextwith the work of others. Erickson and Li computed thatconvection can influence the ion distribution in the diffuse layerfor Re ≳ 1 and I ≲ 0.01 mM.48 Additionally, Zukoski and

Saville proposed that the ions in the stagnant layer of a particlesurface can respond to lateral diffusion and potential gradients.Hence, migration/diffusion between the stagnant and diffuselayer alters the surface conductivity and the zeta potential,respectively.36 The combination of both effects is conceivableto explain our experimental observations. That is, the flowthrough the microchannel disturbs the (equilibrium) distribu-tion of the ions in the diffuse layer and, in turn, inducespotential and concentration gradients which trigger a lateraltransport of ions between stagnant and diffuse layer. Weobserve that the influence of surface conductivity decreases forincreasing Reynolds number. We hypothesize that the flowinduces gradients for the lateral ion transport which causes a“depletion” of the stagnant layer. It is possible that the ionsextracted from the stagnant layer remain in the diffusive layerand still contribute to the shielding of the zeta potential. Thatis, even though these transport processes have considerableinfluence on the conductivity layer, the impact on the zetapotential is less pronounced. This assumption would alsoexplain why the influence of the flow on the zeta potentialsmeasured with streaming currents is difficult to observe sincethe differences are comparable to the experimental standarddeviation. It would also explain why the influence of flow isconsiderably higher for the zeta potentials measured with thestreaming potential method. In that case, the change of thesurface conductivity additionally influences the microchannelresistance; its correct value is required for the computation ofthe zeta potential.We find that for high ionic strength, that is, for no or only

little influence of surface conductivity, the results of ourdifferent measurement methods are in good agreement. Forlow ionic strength, the zeta potential derived from streamingcurrent and streaming potential experiments, corrected forsurface conductivity and flow, show a reasonable agreement aswell.Finally, our results suggest that the influence of convection

on the surface conductivity, is a new, and at present,unexplained phenomenon which is not covered in the classicalelectrokinetic theories such as the Helmholtz−Smoluchowskicorrelation. Furthermore, the observed phenomenon indicatesanother reason why the Onsager relation between electro-kinetic phenomena is usually not observed in experiments.Detailed insights in the nature of the phenomenon, forexample, the hypothesized lateral ion transport, can only beanswered by complex simulations and is the subject of currentresearch.

■ ASSOCIATED CONTENT

*S Supporting InformationMathematical analysis showing the insignificance of diffusive(polarization) currents in microchannels with height to lengthratios ≪ 1. This material is available free of charge via theInternet at http://pubs.acs.org.

■ AUTHOR INFORMATION

Corresponding Author*E-mail: dominik.barz@chee.queensu.ca.

NotesThe authors declare no competing financial interest.

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■ ACKNOWLEDGMENTS

The Natural Sciences and Engineering Research Council ofCanada (NSERC) and DuPont, Canada, are gratefullyacknowledge for providing financial support.

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