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

Faculty of Mechanical Engineering, Micro- and Nanofluidics Laboratory, TechnionIsrael 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 microchannelnanochannel 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 microchannelnanochannel-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 Figure1, 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 nanochannels 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 ubiquitous24 since its first observation. In lowconcentration regimes, in nanochannel systems wherepermselectivity is often prevalent understanding the ohmicresponse of microchannelnanochannel systems is of muchimportance. Especially because such systems can be used fornumerous applications such as DNA biosensors,57 nano-fluidics based diodes,3,812 and energy harvesting.9,1316 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-nanomicrochannelsystems (Figure 1), the extremely small size of the nano-channels cross section would result in the nanochannelsresistance being the systems dominant resistance. This led to

the simplifying approach of taking the systems conductance(the inverse of the resistance, = R1) to be solely dependenton the nanochannels 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)

2023

= = +

IV

DFT

hwd

cN

22I

2

02

2

(1)

= + = +DFT

hwd

N c( 2 )II ideal vanishing2

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, s. In our previouswork,24 we provided relations for N and s for the case of ananochannel (eq 32) and nanoporous medium (eq 33). For ananochannel, whose dimensions are such that w h, therelation is N = 2s/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, 27442748

pubs.acs.org/NanoLetthttp://dx.doi.org/10.1021/acs.nanolett.6b00429

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 microchannelnano-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)

= = =

=

Rd

hwR R

cN

RL

HW

RfL

, , ,nano res nano nano0

micro res

ff res (5)

with = T DF c/res2

0, being the resistivity and Rff is the fieldfocusing resistor discussed in our previous works,9,2426

[ = 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 = Ri, 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, = R1 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 = Ri changes as the Rnano changesthen understanding how = (Ri)1 changes. When

R R R R, ,nano nano micro ff , eqs 3 and 4 reduces to therespective 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, Rnano 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.11,15,17,2731

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-channelnanochannel 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 experimentalmicrochannelnanochannel system. The geometric details are given inTable 1. The nanochannels 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 loglog 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 = R1.

Nano Letters Letter

DOI: 10.1021/acs.nanolett.6b00429Nano Lett. 2016, 16, 27442748

2745

http://dx.doi.org/10.1021/acs.nanolett.6b00429

s). At ideal permselectivity, Rnano 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 Rnano,that is, the nanochannel is no longer the dominating resistance.As a result, we will be able to observe an interesting interplaybetween the nano

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