Electrokinetics in Nanochannels: Part I. Electric double layer overlap and channel-to-well equilibrium

Transcription

1 letrokinetis in Nanohannels: Part I. letri double layer overlap and hannel-to-well equilibrium Fabio Baldessari and Juan G. Santiago Department of Mehanial ngineering Stanford University Stanford, CA Abstrat We present a new model for alulating the eletri potential field in a long, thin nanohannel with overlapped eletri double layers. We adopt equilibrium between the ioni solutions in the wells and inside the nanohannel to self-onsistently predit eletrolyte onentration. Differently than published models that require detailed iterative numerial solutions of oupled differential equations, we obtain preditions solving a simple one-dimensional integral. Our derivation learly shows that the eletri potential field depends on three new parameters: the ratio of ion density in the hannel to ion density in the wells; the ratio of free-harge density to bulk ion density within the hannel; and, a modified Debye-Hükel thikness ( λd p ), whih is the relevant sale for shielding of surfae net harge. We analyze three wall-surfae boundary onditions: speified zeta-potential; speified surfae net harge density; and harge regulation. Preditions of experimentally observable quantities suh as depth-averaged eletroosmoti flow and net ioni urrent are signifiantly different than results for weakly interating eletri double-layers, and are also signifiantly different than those from previous overlapped eletri double layer models. In this paper, we present preditions where we vary hannel depth at onstant well onentration. Under onditions of DL overlap, we find that eletroosmosis 1

2 (bulk flow) ontributes only a small fration of the net ioni urrent, and that most of the observable urrent is due to ondution in onditions of inreased ounterion density in the nanohannel. In the seond of this two-paper series, we extend the urrent model to inlude the dependene of ion mobility of ioni strength, and ompare preditions to measurements of ioni urrent as a funtion of hannel depth and ion density. 1 Introdution Reent advanes in nanofabriation have allowed for detailed experimental investigations of eletrokineti fluid flow in long, thin hannels with harateristi dimensions in the range of tens of nanometers (nanohannels). 1-5 At this length sale, eletri double layers (DLs) at the wall/eletrolyte interfae strongly influene ion and bulk transport proesses. DLs are haraterized by the presene of high onentrations of exess ounter harges required to shield surfae net harge. For hannel depths on the order of or smaller than the DL length sale, wall DLs an interat strongly and there is net free harge throughout the hannel ross setion. This harge distribution is ultimately determined by at least a quasi-equilibrium balane between wall surfae harge (and assoiated surfae hemistry) and the ion densities in end-hannel wells. Charge distributions are the key physial desriptors of nanohannel transport as they govern axial ion transport rates, total ioni urrent (both advetive and eletromigration urrent), and bulk flow. Various and diverse models have been proposed to treat the ioni equilibria and eletrokineti transport harateristis of nanohannels with finite and overlapped DLs. In Table I we summarize published ontinuum models desribing liquid flow and ioni onentration 2

3 equilibrium dynamis in the presene of DL overlap. The ten models reflet different assumptions regarding the governing physial mehanisms, whih are imposed (nearly always) through the hoie in boundary onditions for the ioni onentrations and eletri potential field, and the hoie of using eletroneutrality (loal balane between positive and negative harge) and net neutrality (ross-setion-area-averaged balane of harge inluding wall harge) onstraints. Before disussing the key differenes, we point out that just in the last deade a variety of models have been proposed for making preditions of eletrokineti flows in nanohannel, and that to date there is lak of onsensus among researhers of how to univoally desribe nanohannel eletrokinetis in onditions of strong DL overlap. All of the models are based on desriptions of an eletrolyte in thermodynami equilibrium near a harged surfae, and rely on the Poisson equation for the eletrostati potential, (some form of) the Boltzmann distribution for diffuse eletrolyte ions, and the Stokes equations for unidiretional liquid flow in the presene of an eletrial body fore. Differenes emerge from assumptions regarding either the ion density or potential at the nanohannel surfae and the hannel enterline. For slit hannels, Burgreen and Nakahe 6 were the first to model the presene of thik DL and its effet on advetion and ondution of eletrolyte ions. Their model inludes treatment of the (equilibrium) transverse eletri potential distribution, its impliations on eletroosmoti flow, and on ioni urrent and streaming potential measurements. Burgreen and Nakahe assume the following: ion onentration in the bulk an be speified independently of the eletri potential (assumed small at the enter of the hannel); and potential at the shear plane (zeta potential) is a known, fixed value. Their preditions are aurate provided the degree of overlap is small 3

4 (weakly interating DLs). Other investigators proposed similar models with minor variations. 7, 8 In the last few years, some studies have signifiantly extended the theoretial framework of Burgreen and Nakahe and ompared preditions to eletrokineti transport measurements. Stein et al. 1 modified the boundary onditions of the Burgreen and Nakahe model to make preditions for speified (and fixed) surfae net harge density at the hannel walls, instead of speified zeta potential. As we shall desribe in this paper, this hoie of boundary ondition an hange the eletri potential distribution in the hannel (and therefore eletroosmoti flow and ioni urrent) signifiantly. Stein et al. used the value of surfae net harge density as fitting parameter when omparing to measured ioni urrent in 70 to 1015 nm deep silia hannels with aqueous potassium hloride (KCl) solutions and 10 mm TRIS salt. They fit observed trends in eletri ondutane at high and low salt onentrations and onstant hannel height, using surfae net harge density values between -45 and -68 mc/m 2. Qualitatively similar results were obtained by Karnik et al. 9 who found that values of the surfae net harge density between -2 and -100 mc/m 2 adequately fit their experimental results in long, thin 35 nm deep silion dioxide hannels. Van der Heyden et al. 10 measured streaming urrents in nm deep silia hannels as a funtion of applied pressure with aqueous solutions of KCl and TRIS. They observed that streaming urrent inreases as KCl onentration is redued from 1 M to 1 mm, but below 1 mm ondutane saturates to (approximately) a onstant value. To model these experiments, they assume that the ioni onentration in the bulk of the nanohannel an be speified independently of other parameters, but they proposed a different (third) hoie for the wall boundary ondition on the eletri potential distribution: a hemial equilibrium deprotonation reation of the silanol groups on the silia surfae (speifially a boundary 4

5 ondition known as harge regulation (CR)) All these investigations foused on the idea that, at thik and overlapping DL onditions, advetive urrent is a signifiant fration of total ioni urrent in the hannel. Reently, Qu and Li, 14 and Ren et al., 15 proposed a model for overlapped DLs in infinitely long and wide, thin hannels with low values of zeta potential (the Debye-Hükle approximation). As a wall boundary ondition, they use the site-dissoiation model of Healy and White 16 whih predits wall harge based on the ph-dependent ondensation of hydroxyl or hydronium ions. In this approah, a net free-harge develops beause loss of hydroxyl (hydronium) ions to the surfae is only in part ountered by dissoiation of water moleules. Thus, Qu and Li (and Ren et al.) use a oupling of Poisson equation, the site-dissoiation model the water dissoiation equilibrium reation, and onservation of mass for ioni speies. The differenes between their preditions and lassial preditions assuming Boltzmann equilibrium are large when DLs are highly overlapped. They argue that applying Boltzmann equilibrium diretly is inorret beause it assumes that diffuse ion omposition is independent of surfae ions. The Qu and Li model is valid only for infinitely long thin hannels. In reality transport of ions to and from relatively large hannel wells eventually establishes Boltzmann distribution type equilibrium between the hannel wall and wells. Thus, the behavior found by Qu and Li is desriptive of the intermediate state toward final equilibrium, as verified by detailed time dependent alulations of Kwak and Hasselbrink, 17 and Kang and Suh 18, who solve the transient problem starting at the instant when a speified surfae net harge density is instantaneously imposed on the nanohannel wall. Their alulation then proeeds to an equilibrium well desribed by Boltzmann distributions. 5

6 Shoh et al. 4,5 also observed trends of measured ondutane for varying eletrolyte onentration similar to Stein et al. and van der Heyden et al. Their model is based on linear superposition of the expeted bulk ondutivity (proportional to the sum of the produt of onentration and mobility for all ions), and an estimate of the ontribution to ondutane due to the exess ounterions in solution inside the nanohannel. To estimate the density of ounterions, they impose net neutrality (whih we here define as a net zero sum of the area average harge density inluding wall harge) for a speified surfae harge density. Shoh et al. use this simplisti desription, and use the surfae net harge density as a fitting parameter to explain trends in measured ondutane. They find that a surfae harge density of - 53 mc/m 2 fits experimental data. Their approah aptures an important trend that ditates values of the eletrial ondutivity in onfined hannels, namely: ounterions aumulate within the nanohannel in proportion to the loal surfae net harge value (see Figure 9 and assoiated disussion). In a series of three publiations, Conlisk et al developed models where ioni onentrations at the wall (instead of at the midpoint) are either speified, or determined from hannel-to-well equilibrium onsiderations. When the system is open (infinitely long, slit hannel) Conlisk et al. 19 assume that the ion onentrations at the wall are known and that the zeta potential is speified. When the hannel is onneted to large wells at eah end (losed system), Conlisk et al. 20,21 model ioni onentrations inside the nanohannel oupled to onentrations in wells via the Nernst equations. The hannel-to-well fluxes are approximated using a one-dimensional flux balane between the depth-averaged onentrations in the nanohannel and the (large) well. They apply net neutrality at eah ross setion within the nanohannel, and independently apply eletroneutrality to the walls and ontent of wells. We omment further on this issue in setion 6

7 2A where we derive an equation for the self-onsistent oupling of ioni onentrations in the wells and the nanohannel. For now, we note that the Conlisk et al. approah requires intensive, iterative, numerial solutions of the onstrained boundary value problem for the oupled Poisson (differential) equation for the eletrostati potential, and the Nernst (differential) equations for ioni onentration fields. As mentioned above, in their model the onstraints are net neutrality at eah ross-setion and (separately) eletroneutrality in the wells. Finally, Tessier and Slater 22 present a model to desribe the distribution of ions onfined between harged surfaes for losed long, thin hannels. They speify surfae net harge density, and adopt net neutrality at eah ross-setion of the hannel. They show that a losed system is equivalent to the traditional treatment of an open system provided that an effetive length sale is introdued in plae of the Debye length: the Debye length divided by the geometri mean of the normalized densities of ounter and o-ions at the enter of the hannel. In this paper, we propose a new theoretial framework to aurately desribe liquid flow and ion transport in nanohannels in onditions of DL overlap. We address two speifi questions relevant to DL overlap: (1) how an we determine self-onsistently the transverse eletri potential distribution and the ioni onentrations in a nanohannel in equilibrium with endhannel wells? (2) how an we inlude the effet of loal ioni strength on ion mobility? We address question 1. in this paper and question 2. in the seond of this two-paper series (Part II 23 ). To answer the first question we assume that Boltzmann equilibrium is established between the hannel and the wells. We use this idea to develop a model that allows univoal determination of the two oupled variables in the system: eletri potential distribution and ioni onentrations (ion densities). We first give a desription of the theoretial model, and then evaluate the onsequenes of the following assumed boundary onditions for eletri potential: 7

8 speified zeta potential, speified surfae harge density, or harge regulation. The results desribe how one might predit nanohannel behavior given mirohannel measurements of eletroosmoti mobility. Part II of this two-paper series ompares preditions to experimental data. 2 Theoretial formulation A Distribution of ions ( ni () r ) and free-harge ( ρ ( r )) Consider a nanohannel with dieletri, impermeable walls and a native surfae harge as shown shematially in Figure 1. The hannel is bounded by two relatively large eletrolyte wells at either end. At equilibrium gradients in the eletrohemial potential of eah speies i ( μ i ) are zero (the eletrial and diffusion fores must balane): [ kt n zeψ ] μ () r = ln () r + () r = 0, (1) i B i i where ψ () r is the eletri potential in the diffuse harge regions, e is the eletron harge, z i ion valene, kt B is the thermal energy, and r is the position vetor. We denote the ion number density and the eletri potential in the middle of a symmetri hannel as n ( d ) = n and ψ ( d ) = ψ, where the position vetor d indiates the midplane. The ioni onentration profile ( ni ( r )) valid for the three dimensional spae whih inludes the inside of the nanohannel and its onneting wells is then the Boltzmann distribution given by 24 ezi ni() ni exp ( () ) kt ψ ψ r = r. (2) B i i 8

9 The enterline ion onentration ( n ) in the nanohannel is unknown at this stage. Solving for i n i requires treatment of the equilibrium between ions in the hannel and the ions in the endhannel eletrolyte wells. Before ontinuing, the initial ondition of the problem and time to reah an, at least, quasi-steady state solution bears a brief mention. xperimentally, the following proess is typially followed: An eletrolyte solution of speified onentration is added to one of the wells and the hannel is filled (e.g., by apillary ation). After some time, equilibrium is established between the ioni solution in the wells and in the nanohannel. We assume the time required to ahieve equilibrium in the absene of an external axial field an be long, on the order of the diffusion time sale 2 5 along the axial length, L, of the hannel ( τ ~ L D ~10 s for an L = 1 m long hannel and D where D is harateristi ion diffusivity). The time sale for equilibrium an be orders of magnitude shorter than τ D when an external field is applied: presumably the onvetive time sale due to eletroosmosis, τ ~ Lu ~1 10s. For example, a glass nanohannel with OF OF overlapped DLs might be initially rih in protons (low ph) from the dissoiation of wall silanol groups (this has been raised before by, for example, Tas et al. 25 on the proess of filling nanohannels). Subsequent eletroosmoti flow (OF) will then bring fresh eletrolyte into the system and eventually reah some equilibrium. We therefore further assume here that the hannel has been filled, and we have either waited long enough or used OF to reah an approximate equilibrium. This also requires that wells are large and/or strongly buffered suh that their properties and inlet flow onditions remain onstant throughout the time of experiment. 26 In the absene of applied eletri fields, the equilibrium onentration in the nanohannel may differ from that of the wells sine eletri potentials in the hannel may be different than that of 9

10 the well (i.e., when we have signifiant double layer overlap). Satisfying μ i = 0 along the axial hannel diretion allows us to write an equilibrium ondition between the ion onentration in the wells ( n well i ) and the ion onentration at the enterline of the nanohannel well ezi ni ni exp kt ψ = (3) B where we assume ψ well = 0 as our referene eletri potential with respet to whih all other (wall harge related) eletri potentials are measured. Substitution of q. (3) into (2) yields an expression for ion distribution in long thin hannels well ezi ezi n () r i = ni exp ψ exp ( ψ() ψ) kt B kt r. (4) B We hoose to write the ion distribution funtion as shown in (4) instead of the more ompat form n ( ) = n well exp ( ezψ ( ) k T) r r as a reminder that a self-onsistent solution of ψ () r i i i B requires knowledge of ψ whih is non-zero and determined by the boundary onditions at the walls, as will be seen in detail in the next setion. Given q. (4), the distribution of free harge in the eletrolyte is given by ez i ρ() r = zen i i() r = zen i i exp ( ψ() r ψ) i i kt B well ez i ez i = zen i i exp ψ exp ( ψ( r) ψ). i kt B kt B To the best of our knowledge, this is the first time that a ondition of equilibrium between the solutions in the wells and inside a long, thin nanohannel has been used to model selfonsistently the ioni onentration inside the nanohannel as expliit funtions of the enterline eletri potential. For example, models based on the work by Burgreen and Nakahe 6 assume that the exponential fator in q. (3) is unity: i.e. ion densities are speified independently of ψ. (5) 10

11 Conlisk et al. 20,21 also proposed to adopt equilibrium between the wells and the hannel. They postulate that the ross-setional area-averaged eletrohemial potential is uniform along the hannel length, and equal to the eletrohemial potential in the well, and they derive an areaaveraged ion density form of q. (4). However, they then impose a net neutrality onstraint at the hannel ross setion (inluding hannel wall net harge), and a separate, independent eletroneutrality onstraint for the eletrolyte in the wells. These are part of a single net neutrality onstraint for the entire system, and, stritly speaking, should not be imposed separately (see Appendix A). Further, Conlisk et al. eliminate the expliit dependene of ion density on ψ, by expressing eletroneutrality and net neutrality in terms of onentration ratios, obtained dividing q. (4) by the ion density of the most populous speies present in solution. This formulation requires to iteratively solve non-linear, oupled differential equations with integral onstraints. Surprisingly, equations (4) and (5) are therefore new and express the fat that ioni onentrations inside the nanohannel annot in general be speified independently of ψ or independently of the onditions at the sample well. Further, these equations embody an expliit dependene on one variable, ψ. This is one of two main modifiations present in our model relative to past work. Before moving forward with the model desription, we offer a few omparisons to previous formulations of Boltzmann distribution for nanohannel modeling. The urrent analysis shows that any area-averaged ross setion of the hannel (inluding wall harge) need not be net neutral, as has been assumed previously. 2,6,20-22,27 In fat, near hannel inlets and exits, the areaaveraged harge density (inluding the wall) annot be zero as the loal potentials of the hannel need to be different than the well, and eletri fields must typially emerge from the nanohannel. These non-zero axial omponents of the eletri field ross the inlet and exit of the 11

12 hannel, and terminate in the near-hannel regions of the well (.f. insert shemati of Figure 1). This reates a differene between ψ and ψ well that balanes the diffusive flux assoiated with the potentially large hannel-to-well ion density ratios. The urrent model also shows that loal and area-averaged (exluding wall) ion densities in nanohannels are not ditated by well ion densities alone, as has been assumed. 2,6,7,27 Instead, all ion densities are ultimately ditated by the hemial potential balane between hannel walls and the ontents of the well. The well and hannel walls are oupled intimately. B The potential distribution in a wide, shallow hannel ( ψ ( y) ) In this setion, we fous the disussion to the potential field in a wide, shallow nano-sale 1-3, hannel of the type typially reated using planar mirofabriation methods. We define y as the transverse oordinate as in Figure 1. We assume for now that end-effets due to axial gradients in the potential are onfined to small regions near the entranes to the hannel, and an be negleted when studying the potential distribution aross the hannel depth. (We will justify this assumption in detail in setion 2F.) The transverse potential distribution (between the top and bottom walls) is obtained solving the Poisson equation subjet to the ondition of symmetry at the enter (here we assume that the hannel walls are idential), and a seond boundary ondition at one of the walls. From the one-dimensional (1D) Poisson equation for the eletri potential within the double-layer 24 2 d ψ 1 ez i = zen exp 2 i i ( ψ ψ) dy ε i kbt (6) (where ε is the dieletri onstant of the eletrolyte, assumed to be uniform throughout). We an integrate and apply symmetry at the hannel s enter ( dψ dy = 0 ) to obtain = y d 12

13 1/2 dψ kbt ez i =+ 2 ni exp ( ψ ψ) 1 dy ε. (7) i kbt Up to this point in the derivation we have assumed that the potential distribution an be desribed using a planar, 1D geometry and symmetry with respet to the enter plane, otherwise q. (7) is general and desribes the potential distribution far from the ends of the hannel. A seond boundary ondition is required to speify a unique solution for ψ ( y). In the following setions, we derive the governing equations for three hoies of this seond boundary ondition: speified wall-potential, speified surfae harge density, or harge regulation. These boundary onditions reflet ommonly aepted approximations of the behavior of eletrolytes in ontat 1/2 with harged surfaes. 8,11,12,24,34 Our goal is to investigate the onsequenes of eah boundary ondition and their effet on the struture of liquid and urrent flow in the presene of an applied axial field. We are interested in onditions where the DLs may be signifiantly overlapped. From this point onward in the derivation, we make the assumption that the bakground eletrolyte is symmetri and binary ( z+ = z = z) to simplify the mathematial treatment, but the urrent model an be extended to inlude arbitrary eletrolyte solutions. I. Boundary ondition I (BC I): speified wall-potential ( ψ(0) For a symmetri, binary eletrolyte q. (7) an be written as well ( 2n+ ) 1/2 1/2 dψ kt B 1/2 2 ez ez = ( p ) 2sinh ( ψ ψ) Ω sinh ( ψ ψ) dy ε 2kT B kt B where = ζ ) 24 (8) 13

14 r n n well well well + ez well ez n+ + n p exp ψ + r exp ψ = well kt B kt B n+ ez well ez exp ψ r exp ψ kt B kt B n n Ω = ez well ez n + n exp ψ r exp ψ kt + B kt B + + The seond term within urly brakets in q. (8) ensures that equilibrium between the solution in the well and within the nanohannel is satisfied self-onsistently. The parameter r well is the ratio of volume-averaged negative to positive ion densities in the wells, it is typially very lose to unity for nanohannels and relatively large wells, but we shall keep it here for onsisteny. We present a more detailed disussion of r well in Appendix A. The oeffiient Ω measures free harge density at the enterline: Ω= 0 for non-interating DLs, and Ω 1 for strong overlap. p is the ratio of ioni strength of the hannel enterline to that of the well. We will disuss in well more detail p and Ω in setion 4. Here we note that for r = 1, p = 2 and Ω= 0, we reover the formulation of published models for thik, weakly-overlapped DLs where the bulk ioni onentration in the middle of the hannel is not signifiantly hanged by the eletri potential field. 6, 24 We shall here refer to this model as the existing thik DL model. 1,6-10,27. (9) kbt, ψ zeψ B Choosing ψ ze( ψ ψ ) reast qs. (8) and (9) in dimensionless form well, ( ) 1/2 k T 14 2 λ 2 D ε kt B ze(2 n+ ), and ξ y λd, we dψ 1/2 2 ψ = ( p ) 2sinh Ω sinh( ψ ) (10) dξ 2 well ( ψ ) r exp( ψ ) well ( ψ ) r exp( ψ ) well ( ψ ) + r exp( ψ ) p exp + exp Ω exp 1/2 (11).

15 Here λ D is a form of the Debye-Hükel thikness. The speified wall (shear plane) potential is the zeta potential, ψ(0) ( ) = ζ, and the solution of q. (10) must satisfy ψ 0 = ψ (0) = ze ζ ψ kbt. We here define zeta potential, ζ, as the potential at the shear plane measured relative to the well potential. The formal solution to q. (10) is given by integrating from the wall to a position within the hannel y λ D ψ s = 2sinh 1/2 Ω p 2 ψ sinh ( s) 1/2 ds. (12) We determine ψ iteratively by numerially evaluating q. (12) at enterline where y λ D = d λ and ψ = 0 (see Appendix B for details about the numerial approximation of the D integral in q. (13)): d λ D ψ ψ = 2sinh 1/2 Ω p 2 0 1/2 sinh( ψ ) dψ. (13) Note that λd p is now the effetive Debye-Hükel thikness (relevant DL thikness) in the presene of inreased ion density in the hannel. We estimate p an be order 50 or larger (e.g., assuming a wall zeta potential of ~150 mv). When the potential at the enterline is small (i.e., weak DL overlap) the exat solution an be approximated by the superposition of the solution for two isolated walls: 24 ψ ( y) ψ ( y) + ψ (2 d y) no-interation no-interation kt ez y ez 4kBT λd B 1 = 4 tanh tanh ζ exp kt ez (2 d y) + ez 4kBT λd B 1 4 tanh tanh ζ exp. For this speial ase and knowing ζ, the potential at the enter of the hannel an be alulated (14) expliitly 15

16 kt B 1 ez d ψ = 8 tanh tanh ζ exp ez 4kBT λd. (15) II. Boundary ondition II (BC II): speified wall-harge density 24 We again onsider q. (8) (or the dimensionless form q. (10)) for a binary, symmetri eletrolyte, but this time hoose to speify the value of harge density at the wall σ dψ = ε. (16) dy y= 0 From (10) we find that at the wall ( y = 0 ) ez λd 2 ψ 0 2sinh εkt B p 2 σ = Ω sinh ( ψ ) 0. (17) The parameter on the left-hand side of q. (17) measures (( kt B ez)( D p) ) σ ε λ, the apaitane of the DL aross a thikness of order λd p, again the orretion to the Debye- Hükel thikness in the presene of inreased ion density in the hannel. Note that both ψ 0 and ψ are unknown at this stage of the formulation. To obtain a speifi solution we solve the oupled system qs. (12)-(13) subjet the onstraint of q. (17). For a small potential at the enterline, the exat solution an be approximated by the superposition of the solution for two isolated walls, given in q. (14), and q. (17) an be approximated by the Grahame equation: 11 2 kt B 1 1 σ ζ osh 1 4 well ze + n+ εkbt. (18) In this weak overlap regime, quations (14) and (18) an be solved to find ψ 0 and ψ onsistent with σ. 16

17 III. Boundary ondition III (BC III): harge regulation We again onsider q.(8) for a binary, symmetri eletrolyte, and we assume that there is an equilibrium reation for the assoiation and dissoiation of silanol groups at the hannel surfae whih depends on ph and ion onentration. 11 The assumptions impliit here are: 11 1) the deprotonation reation at the fused silia surfae is _ + H3O +SiO SiOH ; 2) ounterions due to the harging of the surfae (i.e., + H3O ) provide a negligible ontribution to the overall ioni strength of the solution; 3) the surfae potential is redued linearly aording to a basi Stern layer apaitane model. 35 The reation kinetis yield the following relations between the potential and harge density near the wall and the potential in the bulk: kt σ kt σ = ln ph pk ln10 ( ) ( ) ζ σ B B e a eγ+ σ e C σ ez σ ψ = ψ + zln z( ph pk ) ln10 k T C 0 a eγ+ σ B (19) where Γ is the fration of hargeable sites that are dissoiated, C is the Stern layer s phenomenologial apaity, 11 + ph log 10 { γ + [H HO 3O ]} = and pka = log10k a where K a is the equilibrium onstant of the surfae deprotonation reation. Here γ + is the ativity of the 3 HO 3 hydronium ion in solution. 36 As before, harge and surfae potential are related aording to ez λd 2 ψ 0 2sinh εkt B p 2 σ = Ω sinh In this formulation, σ, ψ 0 and 17 ( ψ ) 0. (20) ψ are unknown, and must be determined self-onsistently. The formal solution of the differential equation (q. (8)) is still valid, and the solution requires satisfying the set of simultaneous equations qs. (12)-(13) subjet to the onstraints in (19)-(20).

18 When the potential at the enterline is small, q. (14) is a useful approximation, whih must be solved along with the onstraints in (19)-(20) to yield a valid solution. C Ioni mobility dependene on ioni strength and ph At this stage we have defined the model for eletri potential distribution aross the hannel depth, and we have desribed three boundary onditions ommonly used to find solutions for ψ ( y). In order to make preditions for measurable quantities, for example ioni urrent density, we need to provide a framework for understanding the effet of an external field on liquid flow and ion transport. One important physial mehanism that is often ignored in the miro- and nanofluidis ommunity is the dependene of ioni mobility on loal ph and ioni strength. In this setion we offer a brief aount for suh dependenes. This issue is addressed in more detail in Part II of this two-paper series 23 where we present preditions of bulk solution ondutivity and ioni urrent in nanohannels (with and without DL overlap) and ompare these with experimental data. The ioni mobility of any speies varies with ph and ioni strength, ( I Z ), of the eletrolyte due to two known effets First, ionizable speies exist in solution as an ensemble of ioni forms involved in fast (dynami) assoiation and dissoiation reations that determine the equilibrium (ensemble) form. Variations of loal ph influene the effetive ioni mobility by shifting the equilibrium ondition for these reations. For example, for a monovalent weak aid the equilibrium dissoiation reation and equilibrium onstant are given by _ + HA H3O +A (21) 18

19 K a _ + - γ + [H A H 3O ][A ] 3O γ = (22) [HA] where γ j is the ativity oeffiient of speies j (note that we assume γ HA = 1), and + [H3O ], - [A ], [HA] are the equilibrium onentrations of the hydronium ion, the onjugate base, and the undissoiated weak aid, respetively. Continuing with this weak aid example and adopting the + usual definitions of ph log 10 { γ + [H HO 3O ]} equilibrium onstant as = and pka = log10k a, we an express the 3 - [A ] a 10( γ - A ) 10 pk = ph log log. (23) [HA] A weak base will have a reation of the form _ + B +OH BOH, where pkb = log10kb. letrolytes with more omplex ioni equilibria are ommon. 36 For example, sodium borate solutions typially involve two equilibrium reations with six speies in solution: sodium ion, tetraborate ion, borate ion, buri aid, hydroxyl ion, and hydronium ion. This equilibrium an also be affeted by interation of the solution with arbon dioxide in the atmosphere (whih reats with water to form arboni aid). We analyze sodium borate solutions as an example real eletrolyte in Part II of this two-paper series. 23 For now, we simply note that, as explained below, ion mobility of any weak eletrolyte is intimately oupled to the physis of the double layers (whih partly determine ion density) and all reations in the buffer. q. (23) is one suh oupling whih we use here as an illustrative example. Seond, inrease in ioni strength of a solution inreases the effetive eletrostati shielding of ions in solution and dereases their ativity oeffiients. 38,39 Ioni mobility dereases with inreasing ioni strength. This effet an be desribed by a modified Debye-Hükel theory 36 result where finite ion size effets are inluded 19

20 ( γ - ) z where ( εt) 3/2 and ( εt) 1/ I 1 Z log10 =. (24) A 1 + a 2 IZ depend on the absolute temperature and the dieletri onstant of the solvent ( M 1/2 and -1 1/ 2 for water at 25 C ), Å M I = 12 z is the ioni strength, and a is an adjustable parameter related to the ion size 2 Z j j j (expressed in units of Å ). 39 Continuing our example of a simple buffer omposed of single weak aid, we now desribe the dependene of mobility on loal ion densities and ph. The effetive ioni mobility for a weak aid an be written as the produt of the fration of monovalent aid present in solution and its mobility at infinite dilution: and, similarly, for a weak base K 10 ν ν ν pka a = +, = pka ph, [H3O ] + Ka (25) + ph ν [H ] ν 10 = = ν + + +, pka ph +, [H3O ] + K a (26) where ν ±, are the mobilities at infinite-dilution, pk a is given by qs. (23) and (24) (e.g., for the weak aid ase). In experiments, bakground eletrolyte onentrations of interest vary over several orders of magnitude (e.g., from tens of μm to hundreds of mm ), resulting in large hanges in ph and mobility. As mentioned above, we have developed a detailed model for a realisti buffer of interest in Part II of this series. 23 The model effetively gives a funtional relationship between loal ph and ion density for a borate buffer in ontat with the atmosphere. The results of this model are summarized in Figure 2 whih shows sodium ion mobility and solution ph for a sodium borate solution in equilibrium with typial atmospheri CO 2 levels. The details of this model and 20

21 auray of these preditions is also disussed in the seond of this two-paper series (Part II). We here use the preditions to make the point that even buffered solutions (e.g., borate buffer) annot maintain a onstant ph and mobility over ~4 orders of magnitude hanges in onentration. The mobility of m 2 S/mol), and the mobility of BOH ( ) - 4 ( Na is predited to fall to 0.4 of its value at infinite-dilution is predited to rise to 64% of its value at infinite dilution ( m 2 S/mol). In our results below, we inorporate these realisti buffer mobility and ph values in prediting ioni urrent in nanohannels (.f. Figures 9 and 10). Surprisingly, the effets of ion density and ph on mobility have not been widely inorporated into models of eletrophoresis and urrent transport in either miro- or nanohannels. Nanohannels eletrokineti transport in partiular is by definition strongly influened by high ion density DLs, and yet we know of no inorporation of ion density-dependent mobilities. Nanohannel studies whih have assumed mobilities independent of ion density inlude those of Burgreen and Nakahe (1964), Hildreth (1970), Pennathur and Santiago (2005), Garia et al. (2005), Griffiths and Nilson (2006), and others. 2,6-8,19-22,27,34,40 D letroosmoti flow At this point, we will disuss ion and bulk motion due to the appliation of an external field. We make the usual approximation that the eletri potentials and ioni speies distributions of the DL remain unhanged as an external field is applied. This allows us to treat the externally applied eletri field (i.e., from well to well in Figure 1) as a linearly superposable eletri potential. 6-8,27,41-45 We note that this is an assumption we make here for simplifiation of the analysis, but that this is an issue that should be treated more arefully in future work. As disussed by Stone et al. 46,47 and Saville 48 and Santiago and Pennathur, 40 for example, this 21

22 frozen DL assumption assumes we are interested in a regime haraterized by Pe = ucd ν CkBT 1, where Pe is the eletrophoreti Pelet number of ions, u C is a harateristi speed of fluid motion, and ν C is a harateristi ioni mobility. That is, we assume that the transverse distribution of ions of the DL are not affeted by bulk motion. (We, of ourse, treat advetive urrent by aounting for the ontribution of bulk flow to axial ioni motion.) We also assume relatively large end-hannel liquid wells with negligible hanges in ion density over time. This, for example, assumes negligible hanges in wells due to onentration polarization (also alled the exlusion-enrihment effet). 31,49 The latter auses approximately net neutral regions of ion aumulation and depletion (e.g., at nanohannel outlets and inlets, respetively, for negative surfae harge) due to the disrepany between the total flux of positive versus negative harge arriers through hannels with finite DL. 50,51 Clearly, a fully oupled model where hanges in the applied field an perturb harge distribution in and out of the nanohannel, and whih takes into aount end-effets, is omplex and we are hoping to address it in future work. letroosmoti flow is driven by the presene of net harge of the DL. The unidiretional flow framework developed by Burgreen and Nakahe 6 is appliable to eletrokineti flow in nanohannels with a high aspet ratio (width to depth), and when the flow is laminar ( Re 1). Under these onditions visous flow is governed by 2 du dp ρ 0 2 x μ dy = (27) dx where μ is the visosity of the fluid, u is the fluid veloity in the axial diretion, p is pressure, and x is the applied axial eletri field. xpanding in terms of eletroosmoti and pressure- 22

23 driven flow omponents as u = uof + up and exploiting the linearity of the momentum balane we write 2 2 duof ε dψ ; duof dψ = 2 x = = y = d 2 dy μ dy dy dy 2 dup 2 u OF = 0, ψ = y = 0 1 dp dup = ; = y = d dy μ dx dy u p = y = 0. (28) (29) We further note that qs. (27) through (29) apply to regions of long-thin nanohannels away from interfaes. In suh hannels, DL potential gradients that drive flow are solely in the y- diretion as disussed in setion 2F. Solving qs. (29) is straight forward when the pressure gradient is uniform along the hannel. Integrating q. (29) one from y = d to y, integrating the resultant differential equation from y = 0 (the wall) to y, and applying the no-slip boundary ondition yields 1 dp up( y) = y( y 2d). (30) 2 μ dx The impliations of pressure-driven frationation methods were investigated reently by Griffiths and Nilson (2006) 27 and will not be disussed here. We fous instead on the eletroosmoti flow. When the applied eletri field is uniform along the hannel, integrating q. (28) from the enter-line toward the wall, applying the symmetry onditions at the hannel enter-line, and integrating one more from the wall toward the enter of the hannel yields ε ψ( y) uof ( y) = xζ 1 μ ζ ; (31) it follows that the depth averaged eletroosmoti veloity is given by d ε 1 ψ uof = xζ 1 dy μ d. (32) ζ 0 23

24 The veloity profile in (31) depends on the hoie of boundary ondition at the wall (through ζ ), the hannel depth (through d, ψ ), and the onditions in the well (as ψ depends on ψ ). well Net ioni urrent Current is arried by the motion of ions relative to the bulk neutral fluid (ondution) and by the ions adveted by bulk fluid flow. (Away from end effets, net ioni urrent due to diffusion is negligible.) The net urrent density in a binary eletrolyte is given by ( ν ) ( ν ) i( y) = i ( y) i ( y) = n ezu n ez n ezu n ez x ez ez = ρ( yuy ) ( ) K 0 x osh ( ψ ψ) γ sinh ( ψ ψ) kt B kt B where x (33) well ez well ez K0 n+ ez ν+ exp ψ + r ν exp ψ kt B kt B ez well ez ν+ exp ψ r exp kt ν ψ B kt B ν n ν n γ = ez well ez ν n + ν n ν+ exp ψ r ν exp ψ kt + B kt B (34) Here K 0 is the effetive bulk ondutivity of the eletrolyte, and ν ± are the mobilities of the bakground eletrolyte ions. To be exat ν ± are funtions of the loal ioni strength and ph of the solution. The depth-averaged net urrent density is 1 < i >= d d 0 i( y) dy d d 1 1 ez = ( yuydy ) ( ) x K0 osh ( ) dy d ρ ψ ψ d k 0 0 BT sinh. d 1 ez K0 ( ) dy d γ ψ ψ k 0 BT (35) 24

25 In q. (35) K ( y ) and ( y) 0 γ are eah funtions of the loal ioni strength sine, as disussed earlier, ioni mobilities (ν ± ) vary with ioni strength. This makes evaluation of q. (35) omplex as it requires to express ν ± as loal funtions of ψ ( y). All preditions shown in Parts I and II of this two-paper series were obtained using this full q. (35), inluding ν ± whih vary with loal ph and ion density (e.g., vary within the DL). An obvious approximation we ould onsider for q. (35) is to assume that the mobility ν ± do not vary in the transverse diretion (not a funtion of y ), but are exlusively a funtion the area-averaged ioni strength; q. (35) would then simplify to d 1 i( y) ρ ( y) u( y) dy d 0 d d 1 ez 1 ez K0 x osh ( ) dy sinh ( ) dy d ψ ψ γ ψ ψ k 0 BT d k 0 BT (36) where K 0 and γ are area-averaged quantities. We note that this approximation is fairly aurate for low ioni strength solutions. For example, for = 1 mm (the onditions for BG Figure 9 below), aounting for non-uniform ion mobilities redues predited urrent density by about 16% relative to the area-averaged mobility assumption shown in q. (36). In Part II we will show that for high values of the well onentration the approximation given in q. (36) is not satisfatory. The first term on the right-hand side of q. (35) reflets the advetive omponent of the eletri urrent density. The pressure-driven flow and eletroosmoti flow omponents of the veloity field eah ontribute to this. The ontribution due to pressure gradient is d d ε dp ψ ρ ( y) u ( y) dy = ζ d dy ζ (37) p μ dx

26 where the value of ζ depends on the boundary ondition hosen. The ontribution due to eletroosmoti flow is expressed onveniently in terms of integrals on the eletri potential. From q. (8) 1 d d 0 ρ ( yu ) ( ydy ) OF ψ 0 1 1/2 ψ s well ψ s well εxζ λd e e r e e = n+ e ds 1/2 μ d p 2 s 2sinh Ω sinh ( s) 0 2 (38). In setion 4 we will show preditions of urrent density based these equations. Advetive urrent is learly a ritial issue in overlapped DL eletroosmoti flow. F nd effets and onsequenes of equilibrium We have established that hemial equilibrium between a nanohannel and end-hannel wells implies nonzero, axial ion and potential gradients. For simpliity, we assumed long, thin hannel regions away from inlets/outlets in estimating net bulk and ion flow integrals. A fair question is: How far inside the hannel do suh gradients persist? In fat, end effets are important inside the hannel only for axial distanes on the order of the Debye length. An exat expression for this distane x d is given below: ψ 2 1/2 ψ 2 1+ ψ 2 1/2 ψ ln e 1/2 ( 1 e ψ ) xd e e = + +. (39) λd 1 e + ψ well ψ ( e r e ) This expression is derived by first integrating the axial omponent of the Poisson equation along the enterline from x d (a loation far into the hannel where ψ x = 0 ) to a position x in the diretion of the well ( 0 x x, see Figure 1). We then integrate a seond time along the d enterline from x = 0 where ψ ( x = 0) = ψ well 0 to x d. In Figure 3 we plot the (dimensionless) 26 x d

27 axial distane over whih the eletri potential reahes its value at the enterline as a funtion of d λ D, at a fixed ion onentrations in the wells, for the three boundary onditions desribed. The urve shows that at most the axial distane x d is slightly larger than the Debye length, λ D. A maximum ours near d λd 1, where xd λd 1.3. For d λ D > 1, x d dereases beause the potential at the enterline, ψ, deays to zero as hannel depth inreases. The strong overlap ( d λ D < 1) regime is haraterized by weak shielding of surfae harges, and by large (negative) values of the transverse eletri potential at the enter-line. Large ψ therefore strongly affet the axial eletri field and an also impose a short axial length sale over whih transverse equilibrium is attained. Note assumptions regarding the wall boundary onditions do not signifiantly affet x d. At equilibrium, liquid veloity is zero everywhere, u = 0. Therefore, there exists an axial osmoti pressure gradient that balanes the axial variation of eletri potential from ψ well to ψ along the enterline. We have already disussed that suh variation ours over a length sale on the order of the Debye length. For a binary, symmetri eletrolyte, the equilibrium pressure distribution is given by so that p ψ well ez i ψ = ρ = e zn i i exp ψ( x) x x i kbt x (40) ψ well ez i ez i pxy (, = d) = e zn i i exp ψ exp ( ψ ψ) dψ i kt B kt B 0 well well ez well ez = ktn B r exp ψ + r exp ψ kt B kt B (41) 27

28 The pressure distribution (41) does not ause liquid flow. As we have seen, flow is ahievable in the presene of an externally applied axial eletri field (eletroosmoti flow) or in the presene of a net pressure differene between hannel wells. For a hannel of uniform ross setion and wells with equal ion density, p(x,y) is symmetri along x and so exerts no net axial fore on the liquid in the nanohannel. Lastly, we give an expression for area-averaged net harge near the end of the nanohannel, where we add (fixed) wall harge to harge in the bulk of the hannel: 2 σ (2 d + w) + ρ (2 dw) well ez ez (42) = 2 σ (2 d + w) + (2 dw)( ezn+ ) exp ψ( x) exp ψ( x) kt B kt B The expression shows that the sum of harge along the hannel ross setion (inluding the wall), 2 σ (2 d + w) + ρ (2 dw), is not neessarily zero as been assumed by previous models. 17,19-22,52 Area-averaged harge (inluding wall) is zero only for regions far from the inlet. 3 Parameter estimates in thin DL regime: zeta potential, surfae harge density, and fration of hargeable sites In this setion, we summarize parameter values that we will use in setion 4 to make preditions of observable quantities. Our aim is to generate a unique, self-onsistent set of values for zeta potential, surfae harge density, and fration of hargeable sites all of whih give rise to the same observed flow and urrent at one ondition: the thin DL ase. The thin-dl regime is then the ontrol from whih we extrapolate overlapped DL behavior using the various assumptions regarding the surfae onditions. We begin by speifying values of zeta potential determined experimentally via urrent monitoring 3 of eletroosmoti flow in a 20 μm fused silia mirohannel filled with a solution of 28

29 sodium borate buffer in onentration that was varied between 1 and 100 mm. A onvenient power law fit of the experimentally determined zeta potential as a funtion of the BG onentration is shown in Figure 4 with open irles, and was given by 3 b ζ = a BG (43) where a = , b = 0.245, BG is the onentration (in molar units) of the BG, and ζ is alulated in volts. 3 We assume that q. (43) holds when making preditions at speified ζ potential (BC I). Stritly speaking q. (43) is valid only in the range of experimental onentrations studied (1-100 mm). We will show preditions for onentrations outside the experimentally validated range, but these are mere extrapolations from q. (43). Next, we determine the value of surfae harge density that is onsistent with the value of ζ at a speifi onentration of sodium borate. To do so we alulate σ = εdψ dy using q. (8) y= 0 and the value of ζ given by q. (43). Finally, we determine the fration of hargeable sites, Γ, whih is onsistent with ζ, the value of σ at the same onditions, and whih satisfies q. (19), where the remaining parameters take the following values: pk a = 6.57 and 2 C = 3.0 F m. 10,11 In the alulation for Γ we use the value of ph measured in the bulk (ph = 8.25). In the thin DL regime, bulk ph is an aurate estimate of the loal ph in proximity of the surfae. 11 In Figure 4 we plot omputed values of σ and Γ that are onsistent with the given relation for ζ. The values of zeta potential, surfae harge density and fration of hargeable sites in Figures 4 are used in our alulations in setion 4. 29

30 4 Theoretial results for onstant BG well onentration, and varying hannel depth In this setion, we present model preditions for fixed well ion onentration but variable hannel depth. In a follow up paper (Part II), we will present preditions for fixed depth and variable ion density. We stress that these ases are distint as ion density also affets zeta potential, ion mobility, ph, et. We advoate that exploring both is useful as these are perhaps the two most important experimental parameters. We vary hannel depth from the non-overlapped DL regime to strong DL overlap. We fix 1 mm sodium borate as our well onentration (the bakground eletrolyte, BG). (See setion 4 in the seond of Part II for a detailed desription of the buffer). This weak eletrolyte is ommonly used for buffering, has a relatively high ph = 8.25, and a Debye length λd 9.6 nm. The three boundary onditions presented in setion 2B reflet three assumptions regarding surfae onditions. xploring these assumptions vis a viz variations in hannel depth is useful for several reasons. For example, onsider a well-haraterized surfae for whih zeta potential xploring these assumptions vis a viz variations in hannel depth is useful for several reasons. For example, the DL of a substrate may be haraterized in a thin-dl mirohannel. If we use similar fabriation methods to reate nanohannels with the same substrate material, do we assume the nanohannel will have the same zeta potential or surfae harge density? Should we assume the same wall hemistry model? We show here that these hoies lead to qualitatively different preditions when existing thik-dl models are used; but that the urrent model provides very similar preditions. In this setion our aims are: 1) use the model of setion 2 to make preditions for nanohannels based on measured parameters in thin DL regimes; 2) ompare preditions obtained using the 30

31 three boundary onditions we disussed; and, 3) ompare our model preditions to results obtained by the existing theory for thik DLs (Burgreen and Nakahe 6 ). We again note the latter theory and its derivatives are in ommon use. 1,4-10,27 To these aims we adopt values of the parameters (zeta potential, surfae harge density, fration of hargeable sites) that give us onsistent measurable quantities for all models in the thin DL regime, as disussed in setion 3. In our model, equilibrium between the well solutions and the liquid in the nanohannel is satisfied provided that the ion density at the enterline, and the free-harge at the enterline ( p and Ω respetively in q. (9)) are orretly evaluated. For strong degree of overlap, the DL is predominantly made up of ounter-ions, while o-ions are depleted from the hannel. The funtions p and Ω are summarized in Figure 5 versus nondimensional hannel depth for a speified ζ and r well = 1. As DLs overlap, n + beomes larger than n, and Ω saturates to its limiting value, Ω 1. Similarly, p (desribing enterline ion density relative to the well) inreases with stronger overlap. Both parameters are strong funtions of ψ and thus of the degree of overlap. In the thin-dl regime, d λ D >> 1, ψ 0 so Ω 0 and p 2 agrees well with existing models. For stronger overlap, d λ D < 7, both p and Ω depart, whih strongly from the thin-dl limit and urrent models. We note that hoosing an alternative boundary ondition has negligible affet on p and Ω (plots not shown). Next, we present preditions based on the urrent model and ompare these to existing thik DL models. In Figure 6 we plot predited values of zeta potential (eletri potential at the shear plane) as a funtion of d/ λ D. There are six theory urves. We fous first on preditions obtained using our model (open symbols). Results for speified ζ (BC I) obviously fall onto the horizontal line where ζ = 156 mv (open diamonds). In BC II (fixed harge density and 31