Interplay between Nanochannel and Microchannel ResistancesYoav Green,* Ran Eshel, Sinwook Park, and Gilad Yossifon*

Faculty of Mechanical Engineering, Micro- and Nanofluidics Laboratory, Technion−Israel Institute of Technology, Technion City32000, Israel

ABSTRACT: Current nanochannel system paradigm commonly neglects therole of the interfacing microchannels and assumes that the ohmic electricalresponse of a microchannel−nanochannel system is solely determined by thegeometric properties of the nanochannel. In this work, we demonstrate that theoverall response is determined by the interplay between the nanochannelresistance and various microchannel attributed resistances. Our experimentsconfirm a recent theoretical prediction that in contrast to what was previouslyassumed at very low concentrations the role of the interfacing microchannels onthe overall resistance becomes increasingly important. We argue that the currentnanochannel-dominated conductance paradigm can be replaced with a morecorrect and intuitive microchannel−nanochannel-resistance-model-based para-digm.

KEYWORDS: Nanofluidics, electrokinetics, fluid-based circuits, concentration polarization

The pioneering work of Stein et al.1 showed that the ohmicresponse of nanofluidic systems, such as the one in Figure

1, exhibited a very peculiar behavior. At very high saltconcentrations, the conductance increased linearly with theconcentration while at very low salt concentrations it appearedthat the conductance saturated to a constant value (solid blueline in Figure 2). The change in the characteristic response wasattributed to the change in the nanochannel’s ion permse-lectivity. Simply put, at high concentrations when the electricdouble layer (EDL), which scales inversely to the bulk ionicconcentration (c0

−1/2), of the system is substantially smaller thanthe height of the channel then the top and bottom EDLs do notoverlap. This results in a nanochannel behaving like any typicalmacroscale channel. In contrast, at low concentrations theEDLs overlap intensity increases and the system approachesideal selectivity behavior wherein only counterions residewithin the nanochannel while co-ions are excluded. Thesymmetry breaking phenomena due to permselectivity hasbecome quite ubiquitous2−4 since its first observation. In lowconcentration regimes, in nanochannel systems wherepermselectivity is often prevalent understanding the ohmicresponse of microchannel−nanochannel systems is of muchimportance. Especially because such systems can be used fornumerous applications such as DNA biosensors,5−7 nano-fluidics based diodes,3,8−12 and energy harvesting.9,13−16 Thisalso remains true for other nanoporous permselective materialsthat are also used for ion transport such as graphene oxide,17

mesoporous silica films,18 and exfoliated layers of a claymineral.19

Conductance and Resistance Models. It was rationalizedthat in three-layered systems, micro-nano−microchannelsystems (Figure 1), the extremely small size of the nano-channel’s cross section would result in the nanochannel’sresistance being the system’s dominant resistance. This led to

the simplifying approach of taking the system’s conductance(the inverse of the resistance, σ = R−1) to be solely dependenton the nanochannel’s geometry. In the remainder of this workwe shall argue otherwise, however it will be beneficial to discussthe popular models and their shortcomings in order tohighlight the subtleties of a newer model that has alleviatedsuch an assumption. Under the assumption of nanochanneldominated resistance, two models were proposed for theconductance of these systems with a binary (z± = ±1) andsymmetric electrolyte (D± = D)20−23

σ = = + ⎜ ⎟⎛⎝

⎞⎠

IV

DFT

hwd

cN

22I

2

02

2

(1)

σ σ σ= + = +DFT

hwd

N c( 2 )II ideal vanishing

2

0 (2)

with being the universal gas constant, T is the absolutetemperature, F is the Faraday constant, and D is the diffusioncoefficient. Also, c0 is the unstirred bulk concentration while Nis the average excess counterion concentration within thenanochannels due to the surface charge, σs. In our previouswork,24 we provided relations for N and σs for the case of ananochannel (eq 32) and nanoporous medium (eq 33). For ananochannel, whose dimensions are such that w ≫ h, therelation is N = −2σs/Fh. At the extreme limits of idealpermselectivity (complete EDLs overlap), N ≫ c0, andvanishing permselectivity (no EDLs overlap), N ≪ c0, bothmodels give identical values for the conductance yet atintermediate concentrations, N ∼ c0, these models differ.

Received: February 1, 2016Revised: February 29, 2016Published: March 9, 2016

Letter

pubs.acs.org/NanoLett

© 2016 American Chemical Society 2744 DOI: 10.1021/acs.nanolett.6b00429Nano Lett. 2016, 16, 2744−2748

Model I (eq 1) assumes that Donnan equilibrium holds withinthe nanochannel at all concentrations.20,21 In contrast, model II(eq 2) was heuristically derived under the ad-hoc assumption

that the total conductance is the superposition of two extremecases of the conductance, ideal and vanishing permselectivity.23

The subtleties of this assumption will be discussed shortly.In a previous theoretical endeavor, we derived the relation for

the ohmic response for a three-layered microchannel−nano-channel system (Figure 1) at the two extreme cases of ideal andvanishing permselectivity.24 We showed that the overallresistance R = I/V is given by

=+ +

RR R R2 2

2vanishingnano micro ff

(3)

= ̅ + +R R R R2 2ideal nano micro ff (4)

ρ ρ

ρ

= ̅ = =

=̅

Rd

hwR R

cN

RL

HW

RfL

, , ,nano res nano nano0

micro res

ff res (5)

with ρ = T DF c/res2

0, being the resistivity and Rff is the fieldfocusing resistor discussed in our previous works,9,24−26

[ ̅ = ̅f f w W h H L( /2, /2, , , ) is a nondimensional functiongiven in eq 26 of ref 24] which represents the resistance thatcan be attributed to field lines focusing from the largermicrochannel into the smaller nanochannel. In the case ofvanishing permselectivity (eq 3), both the counterion and co-ions are transported, thus the resistance drops by a factor of 2.The form of eqs 3 and 4 suggests that the overall resistance is

the sum of a resistors connected in series, R = ΣRi, can bemodeled as a simple equivalent circuit (Figure 1c). Such acircuit suggests two important outcomes that are intricatelyconnected: (1) the total resistance is not necessarilydetermined solely by the nanochannel but by all thecomponents; (2) for the case of a truly nanochannel dominatedresistance, σ = R−1 are equivalent, otherwise it is preferable todiscuss the results in term of the resistance. Hence, from thispoint on we will only refer to the resistances. This will beshown in the Experimental Setup, where we will keep Rmicroconstant while varying the remaining resistors. It is moreintuitive to analyze how R = ΣRi changes as the Rnano changesthen understanding how σ = (ΣRi)

−1 changes. When̅ ≫R R R R, ,nano nano micro ff , eqs 3 and 4 reduces to the

respective extreme limits of vanishing and ideal permselectivityof eqs 1 and 2. This indicates the validity of eqs 1 and 2 at theselimits. So even though at the extreme cases of ideal andvanishing permselectivity, when Rnano, R̅nano ≫ Rmicro, Rff, thesum of eqs 3 and 4 gives eq 2, this is entirely circumstantial.Moreover, it should be stated that each of the ideal andvanishing permselective models was derived under completelydifferent assumptions.24 Hence, using the superpositiontheorem of ideal and vanishing permselectivity resistors isclearly incorrect. While this last point appears to be trivial,despites model II inadequacies, model II has become thedefault model of choice.11,15,17,27−31

Inspection of eq 3 provides a simple criteria for determiningwhen the nanochannel resistance is the dominating resistor:d/hw ≫ L/HW, f/̅L. Simply put, the length over cross-sectionarea ratio of the nanochannel is larger than that of themicrochannel. Indeed this is the case of many micro-channel−nanochannel systems including the geometry consid-ered in this work. However, eq 4 is no longer just a ratio ofgeometries as in eq 3 but rather includes a dependence on theaverage excess counterion concentration N (or surface charge,

Figure 1. (a) A 3D schematic representation of our experimentalmicrochannel−nanochannel system. The geometric details are given inTable 1. The nanochannel’s height, h, is not to scale and has beenexaggerated for presentation purposes. (b) Cross section of our systemconnected to a power source where the blue semicircles are schematicrepresentations of the field-focusing resistors. (c) An equivalentelectrical circuit of the system comprised of resistors in series (eqs 3and 4).

Figure 2. A log−log plot of the 1D conductance versus the saltconcentration. The 1D model assumes that w = W, h = H and theconductance has been normalized by hw. The theoretical models arecompared with simulations (marked by symbols) of a one layered(nanochannel only) and three layered model (with microchannels).The resistance models turn into conductance models through σ = R−1.

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σs). At ideal permselectivity, R̅nano in eq 4 is independent of theconcentration (similar to eqs 1 and 2), however the terms Rmicroand Rff are not. They are increasing functions with decreasingconcentrations. This indicates that at sufficiently lowconcentrations, they can surpass the saturated value of R̅nano,that is, the nanochannel is no longer the dominating resistance.As a result, we will be able to observe an interesting interplaybetween the nanochannel and the microchannel for dominanceof the system’s resistance.To illustrate our above arguments, in Figure 2 we compare

the one-dimensional (1D) conductance models, eqs 1−4, with1D numerical simulations (see Methods) with w = W, h = H,and Rff = 0, for two cases: (1) a standard three-layered system(Figure 1) ; (2) a one-layered system comprised solely of apermselective medium (this is equivalent to taking L = 0). Forthe case of the one layered system, we observe that eq 1 followsthe simulations at all concentration. In contrast, eq 2 has goodcorrespondence only at very low and very high concentrations.At intermediate concentrations, eq 2 overpredicts the value ofthe conductance. This is because the superposition theoremcannot be used to for two different cases (ideal and vanishingpermselectivity) that have inherently contradicting underlyingassumptions.24 We can then compare the results with the three-layered system. We can immediately see that the one-layeredmodels (eqs 1 and 2) are inadequate to describe theconductance of a realistic three-layered system at allconcentrations. In contrast, eqs 3 and 4 correspond nicely tothe simulated data. The inadequacies of models I and II is thatthey do not account for the microchannel resistors (Rmicro) aswell as the shape of the nanochannel (through Rff). A completediscussion regarding the underlying assumptions of the idealand vanishing permselection model, as well as the effects of Rff,is provided in ref 24. Also, for a thorough discussion on theeffects of geometric heterogeneity see the recent work of abu-Rjal et al.32 and our recent works24−26 and references therein.Two of our recent works,9,24 which focused on three-dimensional (3D) systems, examined the extreme cases ofeither ideal or vanishing permselectivity and compare well withthe study of abu-Rjal et al.,32 which accounts for nonidealpermselectivity but requires numerical evaluation and was onlytwo-dimensional (2D).Experimental Setup. Our experimental setup is comprised

of a simple three-layered setup described in Figure 1 with thegeometric details given in Table 1. In all of our experiments, wekept the rectangular microchannels constant in size but variedthe width of the connecting nanochannels (Table 1). Byvarying the nanochannel width, we varied the relative influenceof the nanochannel’s resistance to the total response. Flexible

silicone reservoirs (Grace Bio-Laboratories) were placed on topof the drilled holes inlets. Ag/AgCl electrodes were inserted inthe drilled holes and connected to an electrical voltage source(Keithley 2636). The fabrication, cleaning, and measurementprocesses are discussed in Methods.From Table 1 it is quite evident that d/hw ≫ L/HW, f/̅L,

which would indicate that nanochannel resistor is indeed thedominating resistor in the vanishing permselectivity limit.However, as we have stated previously, this is not strictly thecase when nanochannel is behaving as an ideal permselectivechannel. There, the nanochannel resistance is not necessarilythe dominating resistor in the system at all concentrations. Infact, in the ideal permselection limit the microchannel and fieldfocusing resistors can be of the same order of magnitude orlarger.

Current−Voltage Analysis. To illustrate the totalresponse dependence on the various resistors in Figure 3a,b

we plot the current−voltage (I−V) curves of all of the channelsat two concentrations that can be attributed to the ideal andvanishing permselectivity limits. We then divide the totalmeasured current by the nanochannel cross-sectional area andcalculate the current density, j = I/hw. The current density−voltage, j−V, curves are plotted in Figure 3c,d. In the vanishingpermselectivity case, where Rnano ≫ Rmicro, Rff such that Rvanishing≅ Rnano, we indeed get the expected result that all the currentdensity plots collapse onto each other Figure 3d. We can nowconsider the low concentration case. First, we note that unlikethe high concentration case, the low concentration I−Vmeasurements (Figure 3a) can also exhibit additional behaviorsuch as a limiting current.33,34,32 The change from an ohmicresponse to a limiting current response is most apparent in thewidest channel, w1. The narrow channels do not have a limitingcurrent window due to electroconvective effects that resultfrom the formation of strong tangential electrical field.21,35−37

The disappearance of the limiting current window due to

Table 1. Geometric Information and Measurementsa

width [mm]⎡⎣⎢

⎤⎦⎥

dhw m

105 ̅ ⎡⎣⎢

⎤⎦⎥

fL m

105⟨ ⟩⎡⎣ ⎤⎦N mol

m3⎡⎣ ⎤⎦N mol

mstd 3

w1 = 2.37 8.29 0.033 2.1 0.28w2 = 0.94 20.9 0.112 1.66 0.24w3 = 0.092 197 0.5921 2.75 0.37w4 = 0.046 422 1.03 2.42 0.13

aThe geometric details for experimental setup are L = 2 mm, H = 48μm, W = 3.3 mm and D = 2 × 10−9 m2/s. The nanochannel heights ofchannels 1,2,4 was h = 178 nm whereas the height for channel 3 is h =190 nm. The mean and standard deviation of the computed excesscounterion concentrations are given in units of mol/m3. The ratioL/HW = 0.124 [105/m].

Figure 3. Experimentally measured I−V measurements are shown in(a,b) and j−V measurements are shown in (c,d). On the basis ofFigure 4 and Table 1, we can estimate that N ≈ 2 mol/m3. Theconcentrations for the ideal permselectivity case is c0,ideal = 0.2 mol/m3

≪ N and vanishing permselectivity concentration is c0,vanishing = 190mol/m3 ≫ N. (d) In the vanishing permselectivity limit, thenanochannel is the dominating resistor leading to the collapse of allthe j−V measurements on a single curve.

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increasing geometric heterogeneity and convective effects hasbeen addressed and reported in our previous work.37 However,the j−V curves (Figure 3c) also display additional differentbehavior than their high concentration counterparts; the curvesno longer collapse on to each other. We note that the twonarrow channels, w3 and w4, have similar current densitiesbecause the nanochannel resistor at the measured concen-tration is indeed the dominating resistor. In contrast, in thewider channels the microchannel and field-focusing resistors areno longer negligible. As result, the narrowest channels have ahigh current density and the wider channels have a lowercurrent density. The reversed behavior of the current density tothat of the current was theoretically predicted in our previousworks.24,25 This result further strengthens our previous workwhere we measured the I−V response an array of identicalnanochannels with varying interchannel spacing.37 In that workwe conducted experiments only at a single concentration andfocused on the high voltage response, that is, the limiting andoverlimiting regions, where electroconvection effects arepredominant. For further discussions on the resistance ofpermselective systems in the overlimiting region see ref 37 fornanochannels and ref 38 for membranes.Resistance Measurements. To measure the resistance of

each of the channels, we typically measured 7−10 low-voltageI−V measurements in the Ohmic region at each concentration.For each measurement we calculated the slope of the I−V andextracted the resistance R. We then calculated the meanresistance and the standard deviations at each concentration.For the very low concentrations, N ≫ c0, we used eq 4 and theaverage measured ⟨R⟩ to calculate N. We then calculated theaverage, ⟨N⟩, and standard deviation, Nstd, for each channel(Table 1).Figure 4 depicts a log−log plot of the resistance versus the

concentration. Each point on this plot is the calculated average,⟨R⟩. The error bars of the standard deviation have been addedas well but are not visible in log−log scale. The lowconcentration/ideal permselectivity branch is plotted usingthe extracted ⟨N⟩ of each channel. The correspondencebetween the theoretical models (eqs 3 and 4) and experiments

is striking for all concentrations. Moreover, the transition fromnanochannel dominated resistance to microchannel and field-focusing resistor is strongly dependent on the geometry of thesystem. This is best observed in the inset of Figure 4 where wehave normalized each resistance by its value at c0 = 0.5 mol/m3.This normalization is done to allow for better observation andcomparison of the interplay between the various resistors. Itbecomes quite obvious that once more the narrowest channels,w3 and w4, which have a large length to cross-section-area ratio(Table 1) show the least change. Yet it should be noted thatthis change is still non-trivial and indeed follows the expectedcurves. In contrast, the wider channels have a more apparentchange in the resistance and transition to microchannel andfield-focusing dominated regime. It is likely that thisphenomena has been overlooked in previous works that usedsystems with a large nanochannel length to cross-section-arearatio such that nanochannel was the dominating resistor over alarger concentration range.1,8,21,22

Conclusions. Finally, we would like to reiterate the keypoints of this work. A problem in the current nanochannelconductance paradigm has been identified, namely, thenanochannel is not always the dominating resistance in thesystem. We have highlighted the inadequacies of previousoversimplified models that fail to explain our measurements.The models given by eqs 3 and 4 propose that the totalresistance be viewed as the sum of resistors, associated with thedifferent system components, connected in series (Figure 1c).At low concentrations, it has been shown that even when thenanochannel resistance is extremely large, the microchanneland field focusing resistors will become the dominatingresistors. Our analysis then suggests a two-fold shift inparadigm: (1) accounting for the microchannel and fieldfocusing is essential; (2) the conductance paradigm be changedto a simpler and more intuitive resistance-based paradigm.Our model (eqs 3 and 4) holds not only for nanochannels

but rather for any nanoporous material that is permselective atlow concentrations. The applicative outcome of this works alsoneeds to be emphasized that regardless of the application(biosensing, nanofluidics circuits, and so forth), preliminarywork should be taken to assess and calibrate the contribution ofeach of the resistors in the system and to fully understand theirinterplay.

Methods. Numerical Simulations. To verify our results, wesolved the fully coupled Poission-Nernst−Planck (PNP)equations using the finite elements program Comsol. ThePNP equations were solved using the Transport of DilutedSpecies and Electrostatic modules in Comsol. More details on a1D implementation can be found in our recent work.9 Here weprovide the parameters used for the simulations and thattypically represent our real life system: D = 2 × 10−9 m/s2, T =298 K, L = 10−1 mm, d = 10−3, N = 0.76 mol/m3, εr = 80 andthe valency z = ±1. We applied an electric potential of V = Vth/10 with the thermal potential being Vth = RT/F, we thenevaluate the resultant current I and calculated the conductanceσ = I/V.

Chip Fabrication. The fabrication technique is the samedescribed in our previous work.37 The microchambers werewet-etched into a 1 mm thick Pyrex glass slide where inlet/outlet access holes were mechanically drilled into each of them.The nanoslot were dry-etched (RIE) into a deposited sacrificialpolysilicon layer on top of a second 1 mm Pyrex glass slide. Theanodic bonding of both slides formed the nanoslots thatconnect the two microchannels.

Figure 4. A log−log plot of the resistance versus the concentration.Theoretical models (eqs 3 and 4) are marked by lines whileexperiments are marked by symbols. The inset shows the lowconcentration behavior of the resistances that have been normalized bytheir respective experimental value at c0 = 0.5 mol/m3.

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Measurements. Before the first experiment, the channelswere cleaned of ionic contaminants (fabrication residues). Thiswas done by applying a voltage difference of approximately 10−20 V and periodically flushing the reservoirs with fresh DI wateruntil the current equilibrated to a minimum. This typically tookan hour. From this point on, the channels were kept wet withDI water. Prior to the experiments, we introduced KClsolutions of varying concentrations in the following manner.The electrolyte was pumped into the channels. The fluid wasset to rest for approximately an hour. This ensured that ionscould diffuse into the nanochannel and reach their Donnanequilibrium. After, we took numerous current−voltage (I−V)sweeps in the Ohmic region, between 0 to 1 V at a sweep-rateof 0.11 V every 20 s. We also measured I−Vs between 0 to 20 Vat a sweep-rate of 0.25 V every 30 s for two concentrations inthe ideal and vanishing permselection regions. At theconclusion of each set of experiments, we either introducedthe next concentration and followed the same procedure orflushed the reservoirs with DI water and let rest until the nextexperiment.

■ AUTHOR INFORMATIONCorresponding Authors*(Y.G.) E-mail: yoavgr@tx.technion.ac.il.*(G.Y.) E-mail: yossifon@tx.technion.ac.il.Author ContributionsY.G. and G.Y. conceived the idea and designed the experiment.S.P. fabricated the channels. Y.G. and R.E. carried out theexperiments and conducted the analysis. Y.G. conducted thesimulations. The manuscript was written by all authors. G.Y.supervised the study.NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSThis work was supported by ISF Grant 1078/10. We thank theTechnion RBNI (Russell Berrie Nanotechnology Institute) andthe Technion GWRI (Grand Water Research Institute) fortheir financial support.

■ REFERENCES(1) Stein, D.; Kruithof, M.; Dekker, C. Phys. Rev. Lett. 2004, 93,035901.(2) Schoch, R. B.; Han, J.; Renaud, P. Rev. Mod. Phys. 2008, 80, 839−883.(3) Siwy, Z. S.; Howorka, S. Chem. Soc. Rev. 2010, 39, 1115−1132.(4) Sparreboom, W.; van den Berg, A.; Eijkel, J. C. T. Nat.Nanotechnol. 2009, 4, 713−720.(5) Branton, D.; Deamer, D. W.; Marziali, A.; Bayley, H.; Benner, S.A.; Butler, T.; Di Ventra, M.; Garaj, S.; Hibbs, A.; Huang, X.; et al. Nat.Biotechnol. 2008, 26, 1146−1153.(6) Meller, A.; Nivon, L.; Branton, D. Phys. Rev. Lett. 2001, 86,3435−3438.(7) Nadler, B.; Schuss, Z.; Hollerbach, U.; Eisenberg, R. S. Phys. Rev.E 2004, 70, 051912.(8) Karnik, R.; Fan, R.; Yue, M.; Li, D.; Yang, P.; Majumdar, A. NanoLett. 2005, 5, 943−948.(9) Green, Y.; Edri, Y.; Yossifon, G. Phys. Rev. E 2015, 92, 033018.(10) Karnik, R.; Duan, C.; Castelino, K.; Daiguji, H.; Majumdar, A.Nano Lett. 2007, 7, 547−551.(11) Guan, W.; Fan, R.; Reed, M. A. Nat. Commun. 2011, 2, 506.(12) Yan, R.; Liang, W.; Fan, R.; Yang, P. Nano Lett. 2009, 9, 3820−3825.(13) Brogioli, D. Phys. Rev. Lett. 2009, 103, 058501.

(14) La Mantia, F.; Pasta, M.; Deshazer, H. D.; Logan, B. E.; Cui, Y.Nano Lett. 2011, 11, 1810−1813.(15) Cheng, L.-J.; Guo, L. J. Nano Lett. 2007, 7, 3165−3171.(16) van der Heyden, F. H. J.; Bonthuis, D. J.; Stein, D.; Meyer, C.;Dekker, C. Nano Lett. 2007, 7, 1022−1025.(17) Raidongia, K.; Huang, J. J. Am. Chem. Soc. 2012, 134, 16528−16531.(18) Fan, R.; Huh, S.; Yan, R.; Arnold, J.; Yang, P. Nat. Mater. 2008,7, 303−307.(19) Shao, J.-J.; Raidongia, K.; Koltonow, A. R.; Huang, J. Nat.Commun. 2015, 6, 7602.(20) Manzanares, J. A.; Murphy, W. D.; Mafe, S.; Reiss, H. J. Phys.Chem. 1993, 97, 8524−8530.(21) Yossifon, G.; Mushenheim, P.; Chang, Y.-C.; Chang, H.-C. Phys.Rev. E 2010, 81, 046301.(22) Schoch, R. B.; van Lintel, H.; Renaud, P. Phys. Fluids 2005, 17,100604.(23) Schoch, R. B.; Renaud, P. Appl. Phys. Lett. 2005, 86, 253111.(24) Green, Y.; Shloush, S.; Yossifon, G. Phys. Rev. E 2014, 89,043015.(25) Green, Y.; Yossifon, G. Phys. Rev. E 2014, 89, 013024.(26) Green, Y.; Yossifon, G. Phys. Rev. E 2015, 91, 063001.(27) Chein, R.; Chung, B. J. Appl. Electrochem. 2013, 43, 1197−1206.(28) Li, S. X.; Guan, W.; Weiner, B.; Reed, M. A. Nano Lett. 2015, 15,5046−5051.(29) Wang, J.; Zhang, L.; Xue, J.; Hu, G. Biomicrofluidics 2014, 8,024118.(30) Choi, E.; Kwon, K.; Kim, D.; Park, J. Lab Chip 2015, 15, 512−523.(31) Choi, E.; Kwon, K.; Kim, D.; Park, J. Lab Chip 2015, 15, 168−178.(32) abu-Rjal, R.; Chinaryan, V.; Bazant, M. Z.; Rubinstein, I.;Zaltzman, B. Phys. Rev. E 2014, 89, 012302.(33) Levich, V. G. Physicochemical Hydrodynamics; Prentice-Hall:New-York, 1962.(34) Rubinstein, I.; Shtilman, L. J. Chem. Soc., Faraday Trans. 2 1979,75, 231.(35) Zaltzman, B.; Rubinstein, I. J. Fluid Mech. 2007, 579, 173−226.(36) Dukhin, S. S. Adv. Colloid Interface Sci. 1991, 35, 173−196.(37) Green, Y.; Park, S.; Yossifon, G. Phys. Rev. E 2015, 91, 011002.(38) Nam, S.; Cho, I.; Heo, J.; Lim, G.; Bazant, M. Z.; Moon, D. J.;Sung, G. Y.; Kim, S. J. Phys. Rev. Lett. 2015, 114, 114501.

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