The Electric Field Strength in Orifice-Like Nanopores of Ultrathin Membranes

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The electric field strength in orifice-like nanopores of ultrathin membranes

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Nanotechnology 26 (2015) 045704 (9pp) doi:10.1088/0957-4484/26/4/045704 The electric ﬁeld strength in oriﬁce-like nanopores of ultrathin membranes

Jirachai Getpreecharsawas1, James L McGrath2 and David A Borkholder1,2

1 Microsystems Engineering, Rochester Institute of Technology, Rochester, New York 14623, USA 2 Department of Biomedical Engineering, University of Rochester, Rochester, New York 14627, USA

E-mail: [email protected]

Received 25 August 2014, revised 20 November 2014 Accepted for publication 24 November 2014 Published 5 January 2015

Abstract Here we show that the electric field inside an ultrathin membrane is weaker than conventional theory would predict, and that the reduced field is predictive of measured electroosmotic flow rates. Our theoretical analysis shows that the electric field inside a charged nanopore is affected by end effects and dependent on the Dukhin number Du when the pore length-to-diameter aspect ratio λ is less than 80 for Du ≪ 1 or 300 for Du ≫ 1. The electric field follows an unconventional scaling law; it no longer scales uniformly with the thickness of membrane, but with the local value of λ for each nanopore.

Keywords: electroosmosis, oriﬁce-like nanopore, short-pore effect, ultrathin membrane (Some ﬁgures may appear in colour only in the online journal)

1. Introduction The conventional long pore (CLP) theory assumes that the pore has very high pore length-to-diameter aspect ratio in The magnitude of the electric field within porous, charged order to simplify its derivation [7]. This assumption under- media determines the flow rate of an electroosmotic pump estimates the end effects inherited to the short nanopore. A (EOP). Requiring no moving parts and causing no dispersion, more generalized model is proposed by compensating for the EOPs are often preferable to pressure-driven transport for hydrodynamic end effects (HEE), which incur the extra miniaturization efforts such as point-of-care diagnostics. The pressure load at either end of the pore [8–10]. Even though growing demand for on-chip EOPs, however, faces an ardu- the HEE model is valid for the nanopore with a lower aspect ous challenge due to its high voltage requirements (typically ratio than the CLP model is, it is still not valid for the orifice- ∼1 kV) [1]. Lowering the operating voltage necessitates a like nanopore with its aspect ratio less than 1. The electric more efficient design of the entire EOP including the transport field inside a short nanopore can be reduced by an access mechanism itself. Previous work has focused on redesigning resistance [11, 12], which is negligible compared to the pore the electroosmosis-inducing media [2], using different elec- resistance for a long pore, but not for a short pore [13]. A trolytes [3], and reducing the spacing between electrodes recent theoretical study also showed that the surface current [4, 5]. We have recently shown that the use of a molecularly density at a discontinuous charged surface, like at the entrance thin (∼15 nm) silicon membrane in an EOP produces the and exit of a charged nanopore, was greatly reduced required field strengths at dramatically lower voltages decreasing the tangential electric field near the discontinuity (∼10 V) [6]. EOPs produced with these highly permeable [14]. It requires the transverse electric field to drive more ions membranes are expected to be useful for on-chip pumping from/to the bulk to maintain the conservation of charge within applications that do not have significant backpressure the nanopore. The tangential electric field regains the strength requirements. of the applied field eventually deep inside the long pore The electric field in lossy media like electrolyte solutions [14, 15]. However, for short nanopores this tangential field depends on both the current density and the gradient of the remains weakened. In this work we analytically combine the electric potential. Thus simply decreasing the membrane effects of the access resistance—discontinuity-induced tan- thickness to achieve higher field strength will not be gential field weakening—and HEE with numerical simulation straightforward if the local current density is also affected. and experimental verification of the comprehensive model.

2.1. Experiment protocol The electric field in a single orifice-like nanopore has not been well examined experimentally and its behavior in a thin porous membrane containing a multiple nanopores is unex- plored. Our work focuses on investigating the electric field behavior near short pores for the purpose of understanding the ability of nanoporous ultrathin membranes to support electroosmosis (EO) [6]. In the experimental investigation, the electric field strength is deduced from the EO flow rates induced by a 15 nm thick porous nanocrystalline silicon (pnc- Si) membrane [16] with a mean pore length-to-diameter aspect ratio of 0.7 under various potentiostatic conditions (see appendix (A) for the detailed experiment protocol). The measured flow rates are then compared with those predicted by the CLP transport theory and more advanced models that Figure 1. The schematic shows the axisymmetric domain and include end effects for short pores. In addition to the access boundary conditions used in the numerical study of the electric field E fi resistance which affects the electric field in pores, our models inside a single ori ce-like nanopore with the pore length lp and – radius rp. The voltage of Vapp is applied across the boundaries AB include the HEE [8 10] to account for an extra pressure drop and GH. The membrane and its single nanopore are defined by the at the entrance and exit of the short pores. We find that the boundaries CD, DE, and EF with the negative surface charge density inclusion of both electrical and hydrodynamic short-pore of ρs. The boundaries BC and FG contain no surface charge. The effects (SPE) is essential to predicting EO rates through intrinsic potential ψ caused by the charged surface is defined with ultrathin membranes. respect to that on the boundaries AB and GH, assumed to have a constant bulk concentration cb. All the boundaries except AB, GH, fl and AH are assumed to have no ionic ux j± and no current density J n 2.2. Numerical simulation . The symbol denotes a unit vector normal to a given boundary and pointing into the domain. The electric field inside the orifice-like nanopores of the ultrathin membrane is suspected to be less than the conven- creeping flow is negligible and is omitted from the simula- fi tional calculation. The nite element simulations clarify this tions to reduce the computational time and complexity. suspicion and elucidate the physics underlying the attenuated Figure 1 illustrates the domain, which is axisymmetric fi eld strength. This numerical study uses electrokinetic theory along the boundary AH, used in the simulations and all the fi to examine the concentration of ions and how the electric eld respective boundary conditions. The electrolyte is aqueous near the membrane and within the nanopore is affected by the NaCl with the bulk concentration of cb, which is held constant conductivity of the solution inside the nanopore. The con- at boundaries AB and GH to which the intrinsic potential ψ is fi centration pro les of ions are prescribed by the charged sur- referred. Also along the boundaries AB and GH, the poten- face of pore wall and membrane, and can be determined by tials of –V /2 and V /2 are applied. The membrane is – – app app solving Poisson Nernst Planck (PNP) equations. Over the defined by the boundaries CD and EF with its nanopore wall charged surface, cations and anions distribute differently along the boundary DE; all contain the negative surface under the influence of diffusion from the gradient of their charge density of ρs with no flux and no current flowing concentration c±, migration under the gradient of the intrinsic through. The boundaries BC and FG representing the ψ fi v potential , and convection due to the velocity pro le of extending reservoir are assumed to have no flux, no current, – solvent according to the Nernst Planck (NP) equation, and no electric field in the direction perpendicular to these j± =−D±± c − z ±±() D RT Fc ±ψ +v c ±, describing their boundaries. fl fi ionic ux j± where D± and z± denote their diffusion coef - cients and valence numbers, and R, T, and F stand for the ’ universal gas constant, absolute temperature, and Faraday s 3. Results and discussion constant, respectively. Influenced by the electrostatics, the ε ρ permittivity of solvent and the volume charge density, = F 3.1. SPE on EO flow rates (z+c+ + z−c−), of ions in the electrolyte solution dictates the intrinsic potential ψ as described by the Poisson equation, It was found that the measured flow rates of EO induced by  ⋅=−()εψ ρ. Unlike the intrinsic potential, the applied the 15 nm thick pnc-Si membrane using 1 mM NaCl aqueous potential ϕ produces the electric field, E =−ϕ, corre- solution with neutral pH were uncharacteristically lower than sponding to the current density, J = σE, with the conductivity, the conventionally predicted by CLP and HEE transport fi σ = Λ+c+ + Λ−c−, of the profiles obtained from solving PNP models as shown in gure 2. fi equations, where Λ± represent the molar conductivities. It is The conventional calculation of the electric eld across worth noting that the convective flux of ions due to the the membrane is found by dividing the Ohmic potential drop

2 Nanotechnology 26 (2015) 045704 J Getpreecharsawas et al with the bulk. The conductivity of the mid-pore region is therefore diverse from that of pore-end regions. At high Du (the third row of figure 3), the EDL becomes overlapped and screens the electrolyte inside the charged orifice-like and long nanopores from the bulk; the conductivity in the regions near the nanopore entrance and exit is influenced by that in the mid-pore region instead. Therefore, the conductivity across the membrane varies slightly from that in the mid-pore region. Regardless of the Dukhin number, the axial component of the electric field is significantly weaker along the centreline of an orifice-like nanopore than near the pore walls. In con- trast, the axial component of the electric field is nearly uniform across the pore cross-section for long pores (figure 3). The first column of figure 3 illustrates how converging and fi Figure 2. Accounting for short-pore effects results in good diverging eld lines at the pore entrance and exit affect the agreement between theoretical predictions of EO flow rates and field lines in the mid-pore region to give diminished field experiments. Each measured flow rate is plotted with respect to the strength at the pore centreline. Therefore, the average electric electric field across the membrane, which is estimated from the field over the cross-sectional area of a short pore will be Ohmic potential drop divided by the membrane thickness. The significantly lower than the conventional estimate assuming a symbols CLP, HEE, and SPE represent the predicted results from the fi conventional long pore theory, additionally with hydrodynamic end uniform eld. effects compensated, and with short-pore effects compensated (this work), respectively. 3.3. SPE analytical model for field strength within a nanopore In addition to our computational model, we have developed [17] across the membrane by the membrane thickness [18]. and solved an analytical model describing the impact of pore The potential drop across the membrane in turn was found by geometry on field strength. We begin by neglecting SPE, as in multiplying the current measured between electrodes with the CLP [7] and charge regulation [20–24] theories. With this total membrane resistance which includes the access resis- assumption, the transport of solvent and ions in a single tance of nanopores. The total resistance of each nanopore is charged nanopore can be derived by the one-dimensional predicted from the bulk solution conductivity and surface Navier–Stokes (NS) and NP equations, respectively. Fluid conductivity of charged walls of membrane nanopores and ion transport are described in terms of the volumetric flow according to the well-established theory [13]. Taking the rate Qi of the solvent, ionic flux Ni, and ionic current Ii as a pore-size distribution of the membrane into account yields the function of the gradients of hydrostatic pressure Pi and total resistance of the membrane [6]. osmotic pressure πi in addition to electric potential ϕi being applied externally. Only the axial components of NS equations are needed to prescribe the velocity profile vi of 3.2. Numerical simulations of concentration profiles and solvent, which is driven by Coulomb forces on the ions in the electric field within a single nanopore diffuse layer. The transport equation for the flow rate is then ri fi found with Qi = 2dπ ∫ vri r. It is important to note that the The simulations reveal that the electric eld inside the pore is 0 impacted by both Dukhin number, Du =(σs/σb)/rp– the ratio hydrostatic pressure term in the NS equations is modeled as a of the surface conductivity σs to the bulk conductivity σb sum of the partial pressure of solvent and the osmotic pressure fl divided by the pore radius [19]—and pore aspect ratio, λ = lp/ of ions [7]. Similarly, the NP equations for the ux density of 2rp,(figure 3). This phenomenon occurs because the electric cation j+ and anion j− in the binary symmetric electrolyte field is sensitive to the conductivity of medium; the ability of solution, with valence number z+ = −z− = z, give rise to ri ions to diffuse from the bulk to the interior of short pores transport equations for the ionic flux Ni =+2dπ∫ jjrr 0 ()+− impacts the conductivity of the pore interior. This effect is ri and the ionic current Ii =−2dπzF∫ j j r r. All three more pronounced at higher bulk ion concentrations as seen in 0 ()+− CLP transport equations are expressed in matrix form as the first and second rows of figure 3, than it is at low bulk concentrations (the third row of figure 3). ⎡ ⎤ ⎡ ⎤⎡ ⎤ Qi KKK11 12 13 ΔPi Dukhin number indicates the ratio of the electric double ⎢ ⎥ 1 ⎢ ⎥⎢ ⎥ N = KKK Δπi ,(1) layer (EDL) thickness to the pore radius [13]. When Du is ⎢ i ⎥ ⎢ 21 22 23 ⎥⎢ ⎥ ⎣⎢ ⎦⎥ lp ⎣⎢ ⎦⎥⎢ ⎥ much smaller than unity, usually at high bulk concentrations Ii KKK31 32 33 ⎣Δϕi ⎦ (the first row of figure 3), the majority of the nanopore is filled with the bulk solution and the electrolyte is nearly uniform where the K coefficients are defined in appendix (B). Notably, fi across the orifice-like and both long nanopores. At moderate the coef cient K33 characterizes the conductivity of a long, Du (the second row of figure 3), the EDL extends more charged nanopore emphasizing the contribution from the prominently into the mid-pore region than in the regions near convection K33,conv,(B.10), in addition to those from the fi 2 the nanopore entrance and exit, which are in direct contact concentration pro les, πrci (Λ+Λ++¯¯ −− c), where Λ± denotes

3 Nanotechnology 26 (2015) 045704 J Getpreecharsawas et al

Figure 3. Electric field in a charged nanopore is impacted by short-pore effects and the uniformity of electrolyte solution inside the pore. COMSOL simulations for an axisymmetric nanopore, showing a close-up, half model with its wall and centreline, indicate that converging and diverging vectors of electric field at pore entrance and exit affect the field lines in the central region of an orifice-like nanopore (λ <1)morethan the long counterparts (λ⩾ 1). Representing their axial component, a shorter arrow indicates a weaker field at the centreline than near the wall of the orifice-like nanopore while uniform for the long ones. In the simulations, the orifice-like and both long nanopores have the same pore radius of 10.55 nm—the mean pore radius of the 15 nm thick pnc-Si membrane used in the experiment—with the pore lengths of 15 nm, 150 nm, and 1500 nm, respectively. The color gradient signifies the concentration profile of counterions relative to bulk concentration, demonstrating the impact of the profiles near the entrance and exit of nanopore on the nanopore conductivity. Such impact can be characterized by Dukhin number, Du: the ratio of the electric double layer (EDL) thickness to the pore radius. (First row) for the high bulk concentration of 100 mM and low Dukhin number of Du = 0.23, the nanopore conductivity is governed by the conductivity of bulk solution; (second row) for the moderate concentration of 1 mM and Du = 4.58, with EDL starting to envelope the membrane, the mid-pore region becomes more conductive than the bulk, but the regions near the entrance and exit still have the influence of bulk solution on their conductivity; (third row) for the low concentration of 0.1 mM and high Dukhin number of Du = 18, with EDL strongly enveloping the entire membrane, the regions near the entrance and exit are screened from the bulk and their conductivity is governed by the conductivity of the mid-pore region.

rr the molar conductivity and c¯ = ∫∫iicrdd r r r repre- extra potential drop associated with Ohmic resistances, these ±±00 sents the average concentration over the pore cross-sec- fluidic transitions result in additional extra pressure loss tional area. across the membrane. Assuming low Reynolds number flow, δ To advance this model to short pores, the influence of the the pressure loss Pp,i can be approximated as the pressure electric field at either end of the nanopore on the field in the drop across an orifice of zero thickness [8, 9], which is given fl η mid-pore region is taken into account, as observed in figure 3. in terms of the uid viscosity , pore radius ri, and volumetric fl 3 Considering the impact of entrance effects on the Ohmic ow rate Qi as 2δηPrQp,i = ()3 i i [25]. The total pressure potential first, we consider the equivalent RC parallel circuit differential ΔPm across the membrane is given by the suc- of the porous membrane. The pore resistance Rp,i in series cessive pressure drops at the entrance δPp,i, across the pore with access resistance Ra,i at either end constitutes the total ΔPi, and at the exit δPp,i so that resistance of each nanopore. The potential drop across the 3η Δϕ Δ=Δ−PP Q .(3) capacitor-like membrane m is equal to the sum of Ohmic i m 3 i ri potential drop across the pore Δϕi = Rp,i Ii and those contributed by both access resistances, 2 Ra,i Ii, where Ii is the Incorporating the SPE described in (2) and (3) into the current flowing in the ith nanopore. For a charged nanopore CLP transport equation (1) gives a set of generalized of radius ri, the access resistance is related to the bulk con- equations (SPE) describing fluid and ion transport in pores σ σ ductivity b and surface conductivity s,i of the pore wall via with low aspect ratio λ, while using (3) alone gives HEE σ σ Dukhin number, Dui =( s/ b)/ri,[19]asRa, transport equations. Expressed in the matrix form similar to σ −1 i =[2ri b(2 + Dui)] [13]. Accordingly, the potential differ- (1), the SPE transport equations are ence across each nanopore can be estimated as ⎡ ⎤ ⎡ ⎤⎡ ⎤ Qi MMM11 12 13 ΔPm 1 ⎢ ⎥ 1 ⎢ ⎥⎢ ⎥ Δ=Δ−ϕϕi m Ii .(2) N = MMM Δπ (4) rDuσ 2 + ⎢ i ⎥ ⎢ 21 22 23 ⎥⎢ m ⎥ iib () ⎢ ⎥ lp ⎢ ⎥⎢ ⎥ ⎣ Ii ⎦ ⎣ MMM31 32 33 ⎦⎣Δϕm ⎦ HEE originate from the converging and diverging streamlines of fluid passing through the sudden contraction with the omission of the end effects on the osmotic pressure, and expansion at the entrance and exit of the pore. Like the Δπi ≈ Δπm, for the weakly charged surface of silicon (see

4 Nanotechnology 26 (2015) 045704 J Getpreecharsawas et al appendix (C) for the definition of M coefficients). Likewise, the coefficient M33 characterizes the conductivity of a short, charged nanopore. As an illustration of the utility of (4), consider the generalized transport equation for the ionic current Ii in the absence of the applied hydrostatic and osmotic pressure differentials. In this case only the electric field across the membrane governs the flow of ionic current in each nanopore and (4) reduces to

Δϕm IMi = 33 .(5) lp

Given the pore tortuosity of one without loss of generality, the influence of this predicted current on the electric field Ei = Δϕi/lp inside each nanopore indicates a possible reduction from the electric field Em = Δϕm/lp across the membrane according to (2) as

E M i =−1 33 .(6) Em lrpbiiσ ()2 + Du

Figure 4 examines the dependence of the normalized electric field Ei/Em on the pore length-to-diameter aspect ratio λ and Dukhin number Du as predicted by the SPE model. The results illustrate how the effective electric field is reduced as λ decreases or the EDL thickness increases (figure 4(a)). In either case, end effects have a greater influence on the central regions of the pore. The actual physical dimensions of a pore are not a significant determinant of the field strength. As fi shown in gure 4(b), the pore aspect ratio and Dukhin Figure 4. According to the SPE model for a single nanopore, the number nearly collapse the behavior of several curves with electric field Ei inside the nanopore normalized with its conventional λ = 1 despite physical dimensions that range from nm to μm. estimate Em across the membrane is a function of both the pore λ λ A slight difference in the electric field for different physical length-to-diameter aspect ratio and the Dukhin number Du. (a) If is changed, the normalized electric field is higher for the nanopore dimensions arises from the increasing conductivity of elec- with highest value of λ; (b) in case of a fixed value of λ = 1 and trolyte caused by the electroosmosis-induced convection K33, multiple pore sizes, all characteristic curves are closely similar; a conv; the larger pore tends to have higher convection because slight difference arises from the enhanced conductivity contributed of a greater hydrodynamic permeability. Note that a deviation by electroosmosis-induced convection, which more prominent for the larger pore diameter. Note that all characteristic curves saturate as high as 70% from the conventional estimate Em is possible when the EDL is very thin or strongly overlapped. for the 1-to-1 aspect ratio at high Dukhin number. The reduced electric field (6) leads to simpler expressions under a number of limiting conditions. Approximating the fi total conductivity of a charged nanopore as a sum of the normalized electric eld in (8) is then bounded by contributions from the bulk and its charged surface [19] leads ⎡ ⎤−1 to Eii12+ Du π ⩽+⎢1 ⎥ .(9) Em ⎣ 22+ Duiiλ ⎦ Λ+Λ≈++cc¯¯ −− σb ()12 + Dui . (7) From the inequality in (9), the threshold values of the Using this approximation in (6) results in a compact form of pore aspect ratio λi can be given as a function of the nor- the normalized electric field as malized electric field and Dukhin number as

⎡ ⎤−1 ⎛ ⎞⎛ ⎞ 12++Du Kπσ r2 π EEi m 12+ Dui Ei ⎢ i ()33,conv i b π ⎥ λi ⩾ ⎜ ⎟⎜ ⎟. (10) =+⎢1 ⎥ (8) 21⎝ − EEi m ⎠⎝ 2 + Dui ⎠ Em ⎣ 22+ Duiiλ ⎦

For the normalized ﬁeld of at least 0.99, λi must be at least (see appendix (D) for the derivation). Since the K33,conv term 49.5π (1 + 2Dui)/(2 + Dui), which is about 78 for Dui ≪ 1or in (B.10) always possesses a nonnegative value, the 311 for Dui ≫ 1.

5 Nanotechnology 26 (2015) 045704 J Getpreecharsawas et al equations, ⎡ ⎤ ⎡ ⎤⎡ ⎤ QΣ MMMΣΣΣ11 12 13 ΔPm ⎢ ⎥ 1 ⎢ ⎥⎢ ⎥ ⎢ NΣ ⎥ = ⎢ MMMΣΣΣ21 22 23 ⎥⎢Δπm ⎥ (11) ⎢ ⎥ lp ⎢ ⎥⎢ ⎥ ⎣ IΣ ⎦ ⎣ MMMΣΣΣ31 32 33 ⎦⎣Δϕm ⎦

for the total flow rate QΣ, total solute flux NΣ, and total current r max * IΣ where Mljk = n∫ frMrlr()jk , d for j, k =1, Σ ()p00 R (p ) 2, 3 with the total number of pores n0 derived in terms of the porosity φ and area Am of the membrane as φA n = m . (12) 0 rmax * 2 ∫ frrrR ()π d 0 In the absence of the externally applied pressure and concentration differences across the membrane, the electric field Em can be deduced from the flow rate QΣ,EO of the purely EO transport via the coefficient MΣ13 from (11) Figure 5. Localized electric fields exhibit a distribution corre- as sponding to the pore size statistic of nanoporous membranes. For illustration purposes, assume a log-normal pore-size distribution for rmax E the 15 nm, 150 nm, and 1500 nm thick membranes. A majority of * m QnΣ,EO= 0 ∫ frKrR ()13 () dr , (13) nanopores of thinner membrane have very low normalized electric 0 Yrl(), p fields; however, increased as the mean pore aspect ratio increases. The pore-size distribution is very important for models of ultrathin η 3 η 3 − membranes, but less of a concern for those of thick membranes—as where Y =1+(3 /lpr ) K11 +[K33 +(3 /lpr )(K11K33 K13 σ the probability density function (pdf) shifts to the right. K31)]/lpr b(2 + Du) for the ensemble SPE model. In case of the derivations based on the CLP and HEE transport models, the denominator in the integral of (13) has Y = 1 and 3 Y =1+(3η/lpr ) K11, respectively. The denominator Y indicates the reduction in EO flow rate due to end effects. Note 3.4. Electric field strength and EO flow rate for ultrathin that the theoretical pore-size distribution governing all these membranes with distribution of pore sizes predictions was based on the log-normal probability density For an ultrathin membrane with a distribution of pore sizes, function [26], which was truncated to represent the finite, the local electric field depends distinctively on the aspect ratio maximum pore size as described previously. The EO flow of each nanopore. Figure 5 examines the impact of a pore rates predicted by (13) for the CLP, HEE, and SPE models are distribution on electric field over the membrane cross-section. compared with the experimentally measured EO flow rates Assuming a log-normal probability density function (pdf) for and were shown in figure 2. 15 nm, 150 nm, and 1500 nm thick membranes, the figure shows electric field reduction as a function of pore length-to- diameter aspect ratios. As the membrane becomes thicker, the 4. Conclusions effective electric field becomes more uniform, and the pore- size distribution is less of a concern. The SPE model is more accurate than CLP and HEE models The distribution of effective electric field in ultrathin in predicting EO induced flow rates in ultrathin membranes. membranes has a direct impact on the overall electroosmosis The model improvement arises from the proper estimation of induced. This impact can be predicted by the ensemble SPE the electric fields inside the membrane’s orifice-like nano- transport model that takes the pore-size distribution into pores. Use of the standard electric field calculation Δϕ account. To derive this result we modify the standard prob- Em = m/lp commonly associated with EO data and CLP theory underestimates the total membrane resistance. The ability density function fR(r) with a probability density function fr* ()that describes the realistic distribution of pore underestimate originates from the fact that CLP theory R neglects the pore access resistance, which in the case of sizes with a finite range and maximum pore radius rmax. This modification involves truncation and rescaling such that ultrathin membranes becomes comparable to the resistance r fr* ()= fr ()∫ max frr ()d satisfies the normalization of electrolyte inside the nanopore itself. Both types of R R 0 R r resistance coexist, contribute to the total resistance across condition ∫ max frr* ()d = 1, while maintaining the same Δϕ 0 R membrane, and result in the Ohmic potential m. The r mean r¯ = ∫ max frrr* ()d and standard deviation agreement between the SPE-based prediction and experi- 0 R r mental measurement substantiates the novel scaling laws r =−∫ max frr* ()( r¯)d2 rof the original distribution. fi std 0 R that the electric eld is scaled not only with Dukhin number Incorporating this pore-size distribution into the pore-specific as described by the electrokinetic theory, but also with the transport equation (4) results in the ensemble transport pore length-to-diameter aspect ratio. This finding will

6 Nanotechnology 26 (2015) 045704 J Getpreecharsawas et al contribute significantly to the proper design of lab-on-a-chip Appendix B. K coefficients for the conventional systems that aim to capitalize on the low-voltage electro- transport equations osmosis from the ultrathin membranes. K coefficients for the CLP transport equation (1) characterize three important lump parameters that describe the electrokinetic transports of liquid and ions in a single, infinitely long Acknowledgments pore of radius ri [7]. The coefficients K11, K12, and K13 characterize the flow rate Qi of liquid solvent transport driven This work was supported by the National Science by the gradients of hydrostatic pressure, osmotic pressure, and Foundation Partnership for Innovation Program (Grant No. applied potential, respectively. Derived by integrating the 11P-1237699) and National Science Foundation (PFI-BIC flow velocity profile over the pore cross-sectional area, these 1237699). three coefficients are given as 4 πri Kr11()i = ,(B.1) 8η Appendix A. Experiment protocol 2π Kr12 ()i = η We measured electroosmosis through pnc-Si membranes r r α ⎡ ⎤ 4 ii zF β πri (SiMPore, West Henrietta, NY, USA) that had a thickness of ×−∫∫∫cosh⎣⎢ ψβ()⎦⎥ rrdddβα ,(B.2) 2 00r RT α η 15 nm and total area of 0.8 mm with the porosity of 8.1%, 8 mean pore radius of 10.55 nm, maximum pore radius of 2πzF Kr13 ()i = 18.1 nm, and standard deviation of 4.3 nm. The aqueous η electrolyte solution was 1 mM NaCl and degased prior to use. riir α β ×−∫∫∫[]ccrr+−()ββ () ddd.βα (B.3) The test device was custom-made to hold a 7 mm circular 00r α silicon membrane chip and allow the insertion of both elec- The second set of K coefficients includes K , K , and trodes into its chambers [6]. Both electrodes were made of 21 22 K . These three coefficients characterize the ionic flux N pure silver wire coiled and coated with AGCL-675C Ag/ 23 i contributed by every cation and anion that involve in the ionic AgCl ink (Conductive Compounds, Hudson, NH, USA). The transport inside the pore. Obtained by integrating the sum of ink was dried in the oven at 80 °C for an hour to remove all their flux densities over the pore cross-sectional area, these fl the solvents. A foot-long Te on 1548 tubing (Upchurch three coefficients are Scientific, Oak Harbor, WA, USA) with an inner diameter of π ri 500 μm was attached to the each port of the test device. All 2 2 Kr21()i =+−∫ [] cr+−() crr ()()i rrr d, (B.4) the parts were assembled under the electrolyte solution after 2η 0 degasing. The spacing between each electrode and the 2π riir α fi Kr22 ()i =+∫∫∫[] cr+−() cr () membrane was about 2 mm. Each tube was half lled with η 00r electrolyte. The assembled test device was padded dry and left ⎡ zF ⎤ β stationary for 12 hours to allow the membrane to soak in the × cosh⎢ ψβ()⎥ rrdddβα ⎣ RT ⎦ α electrolyte solution. Before the measurement, the membrane π ri ⎧ ⎡ zF ⎤ pores were rinsed by pressure-driven flow with the flow rate ⎨ +−∫ ⎩D+ exp⎣⎢ ψ (r )⎦⎥ about 1–2 μL min–1 at 0.3–0.5 psi of the applied pressure for RT 0 RT ⎡ ⎤⎫ at least two hours. Comparing between the results from the zF ⎬ + D− exp⎢ ψ (rrr )⎥ d repeated measurements with and without pore rinsing, it was ⎣ RT ⎦⎭ found that after pore rinsing, the measured flow rates were π ri fl −+∫ []cr+−() cr () consistent and repeatable for a given membrane. The ow 2η 0 velocity of liquid was measured by tracing the motion of both ×−rrrr2 2 d, (B.5) menisci, which were recorded under a microscope equipped ()i with a CCD camera (Motic, British Columbia, Canada). The 2πzF riir α Kr23 ()i =+∫∫∫[]cr+−() cr () velocity was then multiplied by the cross-sectional area of the η 00r fl tube to obtain the volumetric ow rate. The DC voltage, β ranging from 0.25 V to 1.5 V, was applied to the electrodes ×−[]ccrr+−()ββ () dddβα α via an Agilent 33220A arbitrary waveform generator (Agi- ri lent, Santa Clara, CA, USA). This waveform generator served 2πzF +−∫ []Dc++() r Dc −− () r rr d. (B.6) as an adjustable constant voltage source. The electric current RT 0 fl ow across the membrane was monitored and recorded by an The last three coefficients K31, K32, and K33 characterize Agilent 34410A digital multimeter (Agilent, Santa Clara, the ionic current Ii flowing through the pore and signify the CA, USA). contributions from the hydrostatic pressure gradient, osmotic

7 Nanotechnology 26 (2015) 045704 J Getpreecharsawas et al pressure gradient, and electric field, respectively. Derived by coefficients taking the form of integrating the net current density over the pore cross-sec- M tional area, they are given as ⎛ ⎡ ⎤⎞−1 ηr−3 ri ⎜ ⎢30i 0 ⎥⎟ πzF 2 2 11 Kr31()i =−−∫ []cr+−() cr () ri rrr d, (B.7) ⎜ ⎢ 00 0 ⎥⎟ 2η 0 () =+IK K,(C.2) llpp⎜ ⎢ ⎡ ⎤−1⎥⎟ ⎝ ⎣ 00⎣rDuiiσb () 2+ ⎦ ⎦⎠ 2πzF riir α Kr32 ()i =−∫∫∫[]cr+−() cr () η 00r where I, M, and K represent an identity matrix, the matrix of ⎡ zF ⎤ β M coefficients in (4), and the matrix of K coefficients in (1) × cosh⎢ ψβ()⎥ rrdddβα ⎣ RT ⎦ α and listed in appendix B, respectively. r ⎧ ⎡ ⎤ πzF i ⎨ zF +−∫ D+ exp⎣⎢ ψ (r )⎦⎥ RT 0 ⎩ RT Appendix D. Derivation of the approximated ⎡ ⎤⎫ zF ⎬ normalized electric field − D− exp⎢ ψ (rrr )⎥ d ⎣ RT ⎦⎭ r πzF i Substituting coefficient M33 in (6), the normalized electric −−∫ []cr+−() cr () fi fi fi 2η 0 eld is given in terms of K coef cients de ned in 2 2 appendix (B)as ×−()rrrri d, (B.8) ⎡ 24η ⎤−1 Kr()=+ K () r K (), r (B.9) ⎢ K33 − KK13 31⎥ 33iii 33,conv 33,em 83lr−14+ π r Ei ⎢ ()p ii⎥ =+⎢1 ⎥ .(D.1) 22 riir α 2πzF Em ⎢ lrpbiiσ ()2 + Du ⎥ Kr33,conv ()i =−∫∫∫[]cr+−() cr () η 00r ⎣⎢ ⎦⎥ β ×−[]ccrr+−()ββ () dβα d d , (B.10) α Since the conductivity contributed by the concentration profiles inside a nanopore is usually much larger than that con- 2πzF22 ri tributed by the hydrodynamic convection, the normalized Kr33,em ()i =+∫ []Dc++( r ) Dc −− ( r ) rr d . (B.11) RT 0 field can be approximated as

Note that K33 represents the conductivity of electrolyte ⎡ ⎤−1 Ei K33 solution inside a charged pore where K33,conv and K33,em ≈+⎢1 ⎥ .(D.2) ⎢ ⎥ denote the contributions from the convection and electro- Em ⎣ lrpbiiσ ()2 + Du ⎦ migration, respectively. From (B.9) and (B.11), the coefficient K33 can be given as 2 Krc33 =Λ+Λ+π i ()++¯¯ −− cK33,conv, where the average M fi concentrations of cation and anion is defined as Appendix C. coef cients for the generalized rr c¯ = ∫∫iicrdd r r r with their molar conductivities, transport equations ±±00 2 Λ±±= FD RT. Using the approximated conductivity of a M coefficients for the generalized transport equations based charged nanopore (7) the normalized electric field has a on the model of SPE (4) describes the electrokinetic trans- compact form as ports of liquid and ions in a single pore of radius ri and finite ⎡ 2 ⎤−1 length l . Deriving from the CLP transport equation (1), E 12++Dui K33,conv πσ r b π p i ≈+⎢1 i ⎥ ,(D.3) taking SPE given by (2) and (3) into account transforms the Em ⎣⎢ 22+ Duiiλ ⎦⎥ pore-specific to membrane-oriented parameters as λ ⎡ ⎤ where i = lp/2ri represents the pore length-to-diameter aspect ⎡ ΔPP⎤ ⎡ Δ ⎤ 30ηr−3 0 ⎢ i ⎥ ⎢ m ⎥ ⎢ i ⎥ ratio. ⎢Δπi ⎥ = ⎢Δπm ⎥ − ⎢ 00 0 ⎥ ⎢ ⎥ ⎣⎢Δϕ ⎦⎥ ⎣⎢Δϕ ⎦⎥ ⎡ ⎤−1 i m ⎣ 00⎣rDuiiσb () 2+ ⎦ ⎦ References ⎡Q ⎤ ⎢ i ⎥ × ⎢ Ni ⎥ .(C.1)[1] Lee C-Y, Lee G-B, Lin J-L, Huang F-C and Liao C-S 2005 ⎢ ⎥ 15 – ⎣ Ii ⎦ J. Micromech. Microeng. 1215 23 [2] Wang C, Wang L, Zhu X, Wang Y and Xue J 2012 Lab Chip 12 – Note the omission of the end effects on the osmotic 1710 6 Δπ ≈ Δπ [3] Litster S, Suss M E and Santiago J G 2010 Sensors Actuators pressure, i m, for the weakly charged surface of silicon. A 163 311–4 Substituting this transformation back into (1) and rearranging [4] Guo Q, Liu Y, Wu X and Yang J 2009 Microsyst. Technol. 15 terms yield the generalized transport equation (4) with the M 1009–15

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[5] Ai Y, Yalcin S E, Gu D, Baysal O, Baumgart H, Qian S and [16] Striemer C C, Gaborski T R, McGrath J L and Fauchet P M Beskok A 2010 J. Colloid Interface Sci. 350 465–70 2007 Nature 445 749–53 [6] Snyder J L, Getpreecharsawas J, Fang D Z, Gaborski T R, [17] Fievet P, Szymczyk A, Aoubiza B and Pagetti J 2000 Striemer C C, Fauchet P M, Borkholder D A and J. Membr. Sci. 168 87–100 McGrath J L 2013 Proc. Natl. Acad. Sci. USA 110 [18] Yao S, Hertzog D E, Zeng S, Mikkelsen J C Jr and 18425–30 Santiago J G 2003 J. Colloid Interface Sci. 268 143–53 [7] Gross R J and Osterle J F 1968 J. Chem. Phys. 49 228–34 [19] Delgado A V, González-Caballero F, Hunter R J, [8] Dagan Z, Weinbaum S and Pfeffer R 1982 J. Fluid Mech. 115 Koopal L K and Lyklema J 2007 J. Colloid Interface Sci. 505–23 309 194–224 [9] Zhang Y, Gu X-J, Barber R W and Emerson D R 2004 [20] Yates D E, Levine S and Healy T W 1974 J. Chem. Soc. J. Colloid Interface Sci. 275 670–8 Faraday Trans. 1 1807–18 [10] Chakraborty S and Padhy S 2008 J. Phys. D: Appl. Phys. 41 [21] Ninham B W and Parsegian V A 1971 J. Theor. Biol. 31 065502 405–28 [11] Hille B 1968 J. Gen. Physiol. 51 199–219 [22] Behrens S H and Grier D G 2001 J. Chem. Phys. 115 6716–21 [12] Hall J E 1975 J. Gen. Physiol. 66 531–2 [23] Grahame D C 1947 Chem. Rev. 41 441–501 [13] Lee C, Joly L, Siria A, Biance A-L, Fulcrand R and Bocquet L [24] Westall J and Hohl H 1980 Adv. Colloid Interface Sci. 12 2012 Nano Lett. 12 4037–44 265–94 [14] Khair A S and Squires T M 2008 J. Fluid Mech. 615 323–34 [25] Sampson R A 1891 Phil. Trans. R. Soc. A 182 449–518 [15] Baldessari F and Santiago J G 2008 J. Colloid Interface Sci. [26] Zydney A L, Aimar P, Meireles M, Pimbley J M and Belfort G 325 526–38 1994 J. Membr. Sci. 91 293–8