pH-Regulated Ionic Conductance in a Nanochannel with Overlapped

Mar 24, 2015 - To achieve this, an analytical model for the surface charge properties and ionic current/conductance in a pH-regulated nanochannel is ...
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pH-Regulated Ionic Conductance in a Nanochannel with Overlapped Electric Double Layers Yu Ma,†,⊥ Li-Hsien Yeh,*,‡,⊥ Chih-Yuan Lin,§ Lanju Mei,∥ and Shizhi Qian∥ †

School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, P. R. China Department of Chemical and Materials Engineering, National Yunlin University of Science and Technology, Yunlin 64002, Taiwan § Institute of Atomic and Molecular Sciences, Academia Sinica, Taipei 10617, Taiwan ∥ Institute of Micro/Nanotechnology, Old Dominion University, Norfolk, Virginia 23529, United States ‡

S Supporting Information *

ABSTRACT: Accurately and rapidly analyzing the ionic current/ conductance in a nanochannel, especially under the condition of overlapped electric double layers (EDLs), is of fundamental significance for the design and development of novel nanofluidic devices. To achieve this, an analytical model for the surface charge properties and ionic current/conductance in a pH-regulated nanochannel is developed for the first time. The developed model takes into account the effects of the EDL overlap, electroosmotic flow, Stern layer, multiple ionic species, and the site dissociation/association reactions on the channel walls. In addition to good agreement with the existing experimental data of nanochannel conductance available from the literature, our analytical model is also validated by the full model with the Poisson−Nernst−Planck and Navier−Stokes equations. The EDL overlap effect is significant at small nanochannel height, low salt concentration, and medium low pH. Neglecting the EDL overlap effect could result in a wrong estimation in the zeta potential and conductance of the nanochannel in a single measurement.

T

numerically for the ionic current and selectivity of single nanochannels. They concluded that EOF contributes noticeably for highly charged nanochannels. Taking into account the site dissociation/association reactions, the Stern layer and EOF effects, Ma et al.30 and Mei et al.31 derived analytical models for the ionic current/conductance in a nanochannel without and with buffers, respectively. Until recently, the model of Schoch and Renaud28 was extended by Taghipoor et al.32 with further considering the surface chemical reactions. These analytical models, however, neglected the effect of EDLs overlapping. They are nevertheless simple and informative for some nanofluidic experiments but might be inappropriate and incorrect for the nanofluidic channels with height comparable to the EDL thickness, which are typical in modern nanofluidic applications.2,16,21,33−35 The current study investigates the effects of the EDL overlap on the surface charge property and ion transport in a pHregulated nanochannel filled with multiple ionic species. Compared with most of the previously reported numerical results16−27 and limited analytical models ignoring the EDL overlap effect,28−32,36 analytical expressions are derived for the

he transport of ionic species within a nanoscale channel (or pore) driven by an applied electric field generates an ionic current (or conductance),1 which is of fundamental significance in widespread nanofluidic applications ranging from biosensing to regulation of charged species.2−8 It has been demonstrated that the solution properties (e.g., pH and salt concentration) are key factors for affecting the ionic conductance and performance of nanofluidic devices.9−16 Knowledge and method of precisely analyzing ionic current/ conductance in nanofluidic devices in various solution properties are thus crucial to the development, design, and interpretation of nanofluidic experiments. Many theoretical efforts, most of which focused on the numerical simulations,16−27 have been made to reveal how the solution properties affect the electrokinetic ion transport through nanochannels. Compared to a multitude of numerical studies, efforts on analytical models, which are more useful for experimentalists and need lower computational cost, are nevertheless limited. For example, assuming a constant surface charge density and neglecting the effects of the Stern layer and electroosmotic flow (EOF), Schoch and Renaud,28 first proposed an analytical expression to describe the conductance of a nanoslit. They showed that a conductance plateau, as a result of the surface-charge-governed phenomenon,9 was observed at low salt concentration. Vlassiouk et al.29 further considered the EOF effect and solved both analytically and © 2015 American Chemical Society

Received: February 9, 2015 Accepted: March 24, 2015 Published: March 24, 2015 4508

DOI: 10.1021/acs.analchem.5b00536 Anal. Chem. 2015, 87, 4508−4514

Article

Analytical Chemistry first time to predict the zeta potential, surface charge density, and ionic current/conductance with consideration of the effects of EDL overlap, EOF, the Stern layer, multiple ionic species, and the site dissociation/association reactions on the channel walls. The model is validated by the existing experimental data of ionic conductance in a 35 nm silica nanochannel. Because most of the practical effects are considered, the developed model is suitable for any values of pH, salt concentration, and channel height in nanofluidic experiments.

In the above, δs is the thickness of the Stern layer; ε and μ are the permittivity and dynamic viscosity of the liquid phase, respectively; u is the fully developed EOF velocity in the ydirection; and ρe is the mobile space charge density and can be expressed as 4 ⎡ ziF(ψ − ψ ) ⎤ c ⎥ ρe = 1000 ∑ FziCic exp⎢ RT ⎦ ⎣ i=1

■

(4)

where ψc is the electric potential at the center of the nanochannel; zi is the valence of the ith ionic species; F, R, and T are the Faraday constant, universal gas constant, and absolute fluid temperature, respectively. Note that the factor of 1000 in eq 4 is due to the use of the molar ionic concentrations. Cic is the concentration of the ith ionic species at the center of the nanochannel and can be described by

MATHEMATICAL MODEL The problem under consideration is depicted in Figure 1. A long nanochannel of length L, width W, and height H connects

⎡ ziF(ψ − ψ ) ⎤ c 0 ⎥ Cic = Ci0exp⎢ − RT ⎣ ⎦

(5)

where ψ0 is the reference potential in the reservoirs and typically vanishes (i.e., ψ0 = 0) since it is far away from the charged nanochannel. Substituting eqs 4 and 5 into eq 2, we obtain ⎛ zFψ ⎞⎤ ⎛ ⎞ d2ψ RTκ 2 ⎡ ⎛⎜ zFψ ⎞⎟ H ⎜0 ≤ x ≤ ⎟ = − exp⎜ − − δs⎟ ⎢⎣exp⎝ 2 ⎠ ⎠ ⎝ RT ⎠⎥⎦ ⎝ RT 2zF 2 dx

(6)

where z = |zi|; κ = 1/λD = (2000z2F2C0/εRT)1/2 is the reciprocal of Debye length with C0 = C10 + C30 = C20+C40. Because the presence of H+ and OH− is taken into account in the calculation of κ, the present multi-ion model (MIM) is applicable to the entire pH range in experiments. The boundary conditions for eqs 1, 3, and 6 are the following: at the center of the nanochannel (x = 0)

Figure 1. Schematic depiction of the electrokinetic ion and fluid transport in a pH-regulated nanochannel containing multiple ionic species, K+, Cl−, H+, and OH−. ψd and ψs are the zeta and surface potentials of the nanochannel, respectively.

two large, identical reservoirs filled with an electrolyte solution containing four kinds of ionic species, namely, K+, Cl−, H+, and OH− with Ci0, i = 1, 2, 3, and 4, being their bulk molar concentrations. Electroneutrality gives C10 = Cb, C20 = Cb + 10(−pH) − 10−(14−pH), C30 = 10(−pH), and C40 = 10−(14−pH) for pH ≤ 7; C10 = Cb − 10(−pH) + 10−(14−pH), C20 = Cb, C30 = 10(−pH), and C40 = 10−(14−pH) for pH > 7.37 Here, Cb is the background salt concentration of KCl. Assuming that both width and length of the nanochannel are much larger than its height (W ≫ H and L ≫ H), the considered problem can be simplified as a nanoslit. The Cartesian coordinate system, (x, y), with the origin located at the center of the nanochannel, is adopted. The electrode in the left (right) reservoir is applied at a positive potential bias V (grounded, GND), resulting in an axial electric field E = V/L, which drives the EOF and ionic current, directed in the y-direction. Assuming that ions and fluid inside the Stern layer are immobile, the electric potentials within the Stern layer and liquid, ϕ and ψ, respectively, and the flow field can be described by d 2ϕ =0 dx 2 ε

⎛H H⎞ within the Stern layer ⎜ − δs ≤ x ≤ ⎟ ⎝2 2⎠

d 2ψ = − ρe dx 2

μ

⎛ ⎞ H within the liquid phase ⎜0 ≤ x ≤ − δs⎟ ⎝ ⎠ 2

d 2u + ρe E = 0 dx 2

ψ = ψc and

dψ =0 dx

(7a)

du =0 dx

(7b)

at the Stern layer/diffusive layer interface (x = H/2 − δs) ϕ = ψ = ψd

(8a)

dϕ dψ =ε dx dx

(8b)

ε

(8c)

u=0 and at the dielectric nanochannel surface (x = H/2)

ϕ = ψs ε

dϕ = σs dx

(9a)

(9b)

In the above, ψd, ψs and σs are the zeta potential, surface potential, and surface charge density of the nanochannel, respectively. The wall of the dielectric nanochannel materials (e.g., SiO2, Si3N4, and Al2O3) in contact with aqueous solution is generally of charge-regulated nature38−40 due to the occurrence of chemical reactions of functional groups on that surface. If the dielectric nanochannel wall has dissociable groups MOH,

(1)

(2)

⎛ ⎞ H within the liquid phase ⎜0 ≤ x ≤ − δs⎟ ⎝ ⎠ 2 (3) 4509

DOI: 10.1021/acs.analchem.5b00536 Anal. Chem. 2015, 87, 4508−4514

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Analytical Chemistry ⎛ 2000WEF 2 ⎞ 4 I=⎜ ⎟∑ RT ⎠ i=1 ⎝

capable of proceeding the following two dissociation/ association reactions: MOH ↔ MO−+H+ and MOH+H+ ↔ MOH2+, σs in eq 9b can be described by41 2 ⎫ ⎛ F Γ × 1018 ⎞⎧ KA − KB[H+]s ⎬ ⎟⎨ σs = −⎜ t 2⎪ ⎪ Na ⎝ ⎠⎩ KA + [H+]s + KB[H+]s ⎭ ⎪

+ 2000W

⎪

2δsψs + Hψd − Hψs 2δs

−

ψd − ψs δs

x

⎛ 2000WVF Ie = ⎜ ⎟⎨ ⎝ RTL ⎠⎪ ⎪+ (D2C20 ⎩ ⎛ 8000εWVRTC0 ⎞ ⎡ Iv = ⎜ ⎟⎢ μL ⎝ ⎠⎣

∫0

(14) ⎛ F Γ × 1018 ⎞ ⎟ Cs(ψd − ψs) = ⎜ t Na ⎠ ⎝

( )

(15)

In the above, sign(ψd)=1 for ψd > 0 and sign(ψd)= −1 for ψd < 0. Typically, Cs for nanochannels made of silica ranges from 0.15 to 2.9 F/m2.42,44 For the flow field, the EOF velocity, u, can be determined by solving eq 3 subject to eqs 7b and 8c u=

εV (ψ − ψd) μL

⎛ Fψ ⎞⎤ ⎛ Fψ ⎞ H cosh⎜ ⎟dx − cosh⎜ c ⎟⎥ ⎝ RT ⎠ ⎝ RT ⎠⎦ 2

RESULTS AND DISCUSSION Verification by Experimental Data. The derived analytical model is first verified by the experimental data of Karnik et al.,2 who measured the ionic conductance (G) of an array of 30 silica nanochannels with height H = 35 nm in the deionized (DI) water containing background salt KCl with concentration Cb ranging from 10−6 ∼ 1 M. Because the background solution is DI water, which is slightly acidic due to the dissolution of atmosphere CO2, we assume that the background pH = 5.5. Because the EDL thickness under considered Cb is apparently larger than the channel height, the EDL overlap effect in the nanochannel is significant and should not be negligible. The ionic diffusivities D1 (K+), D2 (Cl−), D3 (H+), and D4 (OH−) are 1.96 × 10−9, 2.03 × 10−9, 9.31 × 10−9, and 5.30 × 10−9 m2/s, respectively.45 The results predicted from our analytical MIM (solid line) with pKA = 6.45, pKB = 2.45, Cs = 0.17 F/m2, and Γt = 8 sites/nm2, along with the experimental data of Karnik et al.2 (symbols), are shown in Figure 2, revealing that our model agrees well with their experimental data. In addition to the fitted parameters, pKA, pKB, Cs, and Γt, the resulting point of zero charge (PZC = 2) of the dielectric nanochannel made of silica is also consistent with that reported in the literatures (e.g., PZC = 1.6−3.5).46 It is interesting to note in Figure 2 that G shows a local minimum as Cb varies in the low regime of Cb. The presence of the local minimum in G stems from the more significant EOF effect (Iv) at sufficiently low Cb due to a combined effect of larger zeta potential and mobile space charge density ρe. Similar phenomenon, which only occurs for pH < 7 and will be shown and explained later, has also been observed by Kim et al.,47 and Ouyang and Wang48 for the conductance of a triangular nanochannel and a silica nanochannel, respectively, when the EDLs are highly overlapped at low salt concentration region. Since the conductance behavior can be well described by our analytical MIM (Figure 2), the predicted values of pKA = 6.45, pKB = 2.45, Cs = 0.17 F/m2, and Γt = 8 sites/nm2 are used in

⎡ ⎛ Fψ ⎞ ⎛ Fψ ⎞⎤ Cs(ψd − ψs) + sign(ψd) 4000εRTC0⎢cosh⎜ d ⎟ − cosh⎜ c ⎟⎥ = 0 ⎝ RT ⎠⎦ ⎝ RT ⎠ ⎣

( )

H /2 − δs

■

By substituting eqs 10−12 into eqs 8b and 9b, and letting the surface capacitance of the Stern layer, Cs = ε/δs, we obtain the following implicit expressions connecting ψs and ψd:

( )

∫

In the above, Di is the diffusivity of the ith ionic species; Ie and Iv are, respectively, the ionic currents contributed from the electromigrative (imposed electric field) and the convective (EOF) fluxes. For given conditions, ψd, ψs, and ψc can be determined by simultaneously solving eqs 13−15 using, for example, the Matlab function fsolve. Substituting the obtained ψc and ψd into eqs 17−19 and using eqs 12 and 13, we obtain the ionic current, I, and the conductance, G = I/V, of the nanochannel by integrating the resultant eqs 17−19.

(13)

2 ⎧ ⎫ Fψ ⎤ ⎡ KA − KB⎢⎣C30exp − RTs ⎥⎦ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ 2 Fψs Fψs ⎤ ⎪ ⎡ ⎪ ⎪ KA + C30exp − RT + KB⎣⎢C30exp − RT ⎦⎥ ⎪ ⎩ ⎭

H /2 − δs

(19)

(12)

2RT ln{cd[f (x → H /2)|g ]} zF

⎡ ⎛ Fψ ⎞⎤ ⎫ ⎟ dx ⎪ ⎢exp⎝⎜− ⎥ ⎣ RT ⎠⎦ ⎪ ⎬ H /2 − δs ⎡ ⎛ Fψ ⎞⎤ ⎪ ⎜ ⎟ + D4 C40) ⎢exp⎝ ⎠⎥dx ⎪ ⎣ RT ⎦ ⎭ 0

∫0

(18)

where cd( f|g) is a Jacobian elliptic function of argument f = (κx)/[2exp(zFψc/2RT)] and parameter g = exp(2zFψc/RT). Note that the Jacobian elliptic function is available in Matlab. Suppose that the thickness of the Stern layer δs is very thin (i.e., 1−3 times ionic sizes1) and can be neglected (δs → 0). By letting x = H/2 − δs → H/2, one gets the relationship between ψd and ψc from eq 12: ψd = ψc +

(17) ⎧ ⎪(D1C10 + D3C30)

9,43

2RT ln(cd(f |g )) zF

ψ = ψc +

(ρe u)dx ,

2 ⎞⎪

(11)

Solving eq 6 subject to eqs 7a and 8a yields

H /2 − δs

⎡ ⎛ ziFψ ⎞⎤ ⎟⎥dx ⎢DiCi0exp⎜− ⎝ RT ⎠⎦ ⎣

= Ie + Iv

(10)

where Na is the Avogadaros’ constant; Γt = ΓMO− + ΓMOH+2 + ΓMOH (in the unit of sites/nm2) is the total number site density of functional groups on the dielectric nanochannel surface with Γj being the surface site densities of functional groups j (j = MO−, MOH2+, and MOH); KA = (ΓMO−[H+]s)/ΓMOH and KB = ΓMOH+2 /(ΓMOH[H+]s) are, respectively, the equilibrium constants of the dissociation and association reactions of MOH; [H+]s = C30exp(−Fψs/RT) is the molar concentration of H+ at the nanochannel surface. Typical values for Γt, pKA, and pKB are in the range of 3.8−8 sites/nm2, 6−8, and 0−3, respectively,42 where pKA = −logKA and pKB = −log KB. Solving eq 1 subject to eqs 8a and 9a, we obtain ϕ=

∫0

H /2 − δs

∫0

(16)

The ionic current through the nanochannel, I, can be evaluated by (see the detailed derivation in the Supporting Information) 4510

DOI: 10.1021/acs.analchem.5b00536 Anal. Chem. 2015, 87, 4508−4514

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ionic conductance of the nanochannel under various conditions. Influence of the Nanochannel Height. Figure 4 illustrates the influence of the nanochannel height (H) on the zeta potential, ψd, and conductance, G, of the nanochannel for various background salt concentrations Cb at two levels of pH. For comparison, both results with (symbols) and without (lines) considering the EDL overlap inside the nanochannel are all presented. As seen in Figure 4a,b, if the EDL overlap is neglected (lines), the behavior of ψd is independent of H regardless of the levels of Cb and pH; however, it becomes highly dependent if the EDL overlap is taken into account (symbols). On the other hand, Figure 4c,d depicts that the nanochannel conductance, if the EDL overlap is considered, depends less significantly on H, particularly at the regime of low salt concentration. The magnitudes of ψd and G with considering the EDL overlap are larger than those without considering EDL overlap, as shown in Figure 4. Note that at fixed pH, the percentage deviations of ψd and G between the results with (w/) and without (w/o) the EDL overlap, calculated by PDψd = |[ψd(w/) − ψd(w/o)]/ψd(w/o)| × 100% and PDG = |[G(w/)−G(w/o)]/G(w/o)| × 100%, respectively, are remarkable for small H and Cb. This is expected because the smaller the nanochannel height and/or the lower the salt concentration (the thicker the EDL thickness), the more significant is the EDL overlap effect. It should be pointed out in Figures 4c,d that if the nanochannel height is small, the relative deviation of G between the results with and without the EDL overlap at pH = 4 (Figure 4c) are larger than those at pH = 8 (Figure 4d), except when the background salt concentration is sufficiently high (Cb = 10−2 M). For example, PDG are ca. 96, 26, and 0.44% (34, 23, and 10%) for Cb = 10−4, 10−3, and 10−2 M, respectively, at pH = 4 (pH = 8) and H = 10 nm. This surprised behavior implies that if the background salt concentration is relatively low (high), the EDL overlap effect on the nanochannel conductance is significant at low (high) pH. As shown in Figure 4a,b, the surface charges of the nanochannel at pH = 8 are significantly higher than those at pH = 4, yielding more counterions electrostatically attracted into the nanochannel and, therefore, a thinner apparent EDL thickness and less significant EDL overlap effect. However, if the salt concentration is high, the nanochannel conductance is dominated primarily by its geometry and second by the surface charges of the nanochannel,1,11 the latter of which affects the convective current (Iv). Because the nanochannel geometries are identical for different pH considered in Figure 4c,d, the behaviors of G between the results with and without EDL overlap are majorly dominated by the zeta potential of the nanochannel. Because the magnitudes of ψd for these two cases at pH = 4 (Figure 4a) are all very small, they contribute negligibly on the nanochannel conductance. On the other hand, the negative zeta potential of the nanochannel at pH = 8 (Figure 4b) are much higher than that at pH = 4, which affects more significantly on its conductance and makes the EDL overlap effect remarkable. The maximum percentage differences of PDψd and PDG are ca. 85 and 166%, respectively, at pH = 4, Cb = 10−4 M, and H = 5 nm. This implies that neglecting the EDL overlap results in a significantly wrong estimation in the zeta potential and conductance of the nanochannel, especially for thin nanochannel and low slat concentration.

Figure 2. Dependence of the conductance (G) in a silica nanochannel of H = 35 nm, W = 1 μm, and L = 120 μm on the background KCl concentration Cb at pH = 5.5. Symbols: experimental data of Karnik et al.;2 Solid line: result of the present analytical MIM at pKA = 6.45, pKB = 2.45, Cs = 0.17 F/m2, and Γt = 8 sites/nm2. An amplification of 30 times is applied to the solid line due to the nanofluidic device of Karnik et al. consisting of 30 single nanochannels.2

the following examinations. Unless otherwise specified, the geometry of the nanochannel is fixed at H = 10 nm, W = 1 μm, and L = 10 μm. Comparison with Numerical Results of Full Model. To further validate the present analytical MIM, we compare its results with those obtained from the full mathematical model including the multi-ion Poisson−Nernst−Planck (PNP) coupled with Navier−Stokes (NS) equations, the so-called PNP−NS model.11,29 The detailed PNP−NS model is described in the Supporting Information. The full model takes into account the concentration polarization,49,50 which is not considered in our analytical model. However, the Stern layer effect is not considered in the full model. Figure 3 shows

Figure 3. Comparison of the nanochannel conductance G as a function of its length L at pH = 5.5 and Cb = 0.1 M without considering the Stern layer effect (ψd = ψs at Cs → ∞). Symbols and line denote, respectively, the results of the present analytical MIM and the numerical PNP−NS.

the variation of the nanochannel conductance as a function of its length. As seen in Figure 3, the result of the analytical MIM (solid line) is in good agreement with the result from the full model (circles), especially when the nanochannel length is relatively large. The larger deviation between the results of the two models at smaller nanochannel length is expected and can be attributed to the more significant ion concentration polarization effect.49,50 Figure 3 thus shows the present model is valid for a relatively long nanochannel. Consequently, the present analytical MIM is used to comprehensively discuss the EDL overlap effect on the surface charge properties and 4511

DOI: 10.1021/acs.analchem.5b00536 Anal. Chem. 2015, 87, 4508−4514

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Figure 4. Zeta potential ψd, (a) and (b), and conductance G, (c) and (d), of the nanochannel as a function of its height H for various Cb at pH = 4, (a) and (c), and 8, (b) and (d). Symbols and lines denote the results with and without considering the EDL overlap, respectively.

Influence of Background Solution Properties. Figure 5 summarizes the influence of the background salt concentration,

the difference between G with and without considering the EDL overlap is remarkable when Cb is low and pH is medium low, and unremarkable when Cb is high and pH is sufficiently low and high. It is expected that when there is a higher salt concentration, then there is a less significant EDL overlap effect and, therefore, a smaller percentage deviation (PDG) of results from both models. If pH is sufficiently high, the surface charges of the nanochannel become large as well (Figure 6a,c), thus attracting more counterions inside the nanochannel and accordingly a thinner apparent EDL thickness. Therefore, PDG becomes small for sufficiently high pH, as depicted in Figure 5. On the other hand, if pH is sufficiently low, although the nanochannel’s surface charges are small (Figure 6a,c), a significant deviation of pH from neutral also results in an increase in the net ionic strength. This leads to a decrease in the EDL thickness as mentioned previously and, therefore, a smaller PDG. It is interesting to note in Figure 5b that if the EDL overlap effect is considered (symbols), a nonlinear dependence of G on Cb is observed at low Cb, and depends on the levels of pH. For example, if pH is sufficiently low (e.g., 4), G decreases with decreasing Cb, attains a minimum at a critical Cb, and then increases with a further decrease in Cb. For a medium pH (e.g., 8), G decreases monotonically with decreasing Cb. If pH is sufficiently high (e.g., 10), as Cb decreases, G decreases first, and then reaches a minimum plateau at a sufficiently low Cb. These nonlinear salt-concentration-dependent behaviors of G at the low salt concentration regime have been experimentally observed in nanochannels and nanopores,2,9,19,44,51 and they can be attributed to the surface-charge-governed ion transport phenomena.1 For pH > 7 (e.g., 8 and 10), the behaviors of G on Cb at low Cb are similar to those of |σs|, as shown in Figure 6d. This suggests that the aforementioned nonlinear salt-concentrationdependence of the nanochannel conductance is primarily dominated by its surface charge density. On the other hand, for pH < 7 (e.g., 4), the dependence of G on Cb without the EDL overlap effect (solid line) at low Cb is also consistent with that

Figure 5. Nanochannel conductance, G, as a function of pH for various background salt concentrations Cb, (a), and as a function of Cb for various pHs, (b). Symbols and lines denote the results with and without considering the EDL overlap, respectively.

Cb, and pH on the nanochannel conductance, G. The corresponding influence on the surface charge properties (e.g., zeta potential, ψd, and surface charge density, σs) and the surface pHs(= −log[H+]s) is depicted in Figure 6. For comparison, both results with (symbols) and without (lines) considering the EDL overlap are presented. Figure 5 shows that 4512

DOI: 10.1021/acs.analchem.5b00536 Anal. Chem. 2015, 87, 4508−4514

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Figure 6. Zeta potential ψd, (a) and (b), surface charge density σs, (c) and (d), and surface pHs, (e) and (f), as a function of pH for various background salt concentrations Cb, (a), (c), and (e), and as a function of Cb for various pHs, (b), (d), and (f). Symbols and lines denote the results with and without considering the EDL overlap, respectively. Dotted line in (e) denotes the result of pHs = pH.

of |σs| in Figure 6d; however, the behavior of |σs| with the EDL overlap effect (squares) is different from that of G. In this case of low pH, |σs| (Figure 6d) and the corresponding |ψd| (Figure 6b) are very small. As Cb decreases, the former decreases, whereas the later increases. Note that if the EDL overlap effect is considered, the increase of |ψd| with decreasing Cb (squares) becomes noticeable, especially in the regime of low Cb, as shown in Figure 6b. This results in an increase in the convective current (Iv) due to EOF and, therefore, the nanochannel conductance. This explains why Figure 5b reveals that the nanochannel conductance with the EDL overlap effect shows an apparent local minimum as the background salt concentration decreases. In short, ignoring both the EDL overlap and pH effects might cause not only an incorrect estimation of the nanochannel conductance but also a wrong description of conductance behaviors. Figure 6 also reveals that the EDL overlap effect on ψd, σs, and pHs is significant when the background salt concentration is low and pH is medium low. The behaviors of pHs seen in Figure 6e,f are consistent with those of |σs| in Figure 6c,d, respectively, but inconsistent with those of |ψd| in Figure 6a,b. These behaviors arise from the fact that when there is a lower pHs, there is a higher surface proton concentration and, therefore, a smaller surface charge density (due to less amount of MO− dissociated from the channel wall) but a larger zeta potential (due to [H+]s = C30exp(−Fψs/RT)) of the nanochannel. Figure 6a,b shows that when there is a higher pH and/ or the lower background salt concentration, then there is a

higher nanochannel zeta potential and, therefore, a more significant deviation of pHs from the bulk pH, as shown in Figure 6e,f. Note that the deviation, if the EDL overlap effect is taken into account, becomes remarkable, which emphasizes again that considering the EDL overlap effect is essential in the estimation of surface charge properties and ion transport phenomenon in the nanochannel.

■

CONCLUSIONS The electric double layer (EDL) overlap effects on the surface charge properties and the ionic conductance in a pH-regulated nanochannel are investigated by considering the electroosmotic flow, Stern layer, multiple ionic species, and the site dissociation/association reactions on the channel walls. Taking account of these effects, analytical expressions without considering the ion concentration polarization effect are derived for the first time for predicting the ionic current/ conductance, zeta potential, and surface charge density of the nanochannel. The developed analytical model is validated by not only the existing experimental data of the conductance in a 35 nm silica nanochannel but also by the numerical results from the full model consisting of Poisson−Nernst−Planck and Navier−Stokes equations. We show that the EDL overlap has a profound effect on the zeta potential, surface proton concentration, and conductance of the nanochannel, particularly for a thin nanochannel, low salt concentration, and medium low pH. If the background salt concentration is low, the EDL overlap effect on the nanochannel conductance at low 4513

DOI: 10.1021/acs.analchem.5b00536 Anal. Chem. 2015, 87, 4508−4514

Article

Analytical Chemistry

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pH is more significant than that at high pH; however, the reversed trend is observed if the salt concentration is sufficiently high. With the EDL overlap effect, the nonlinear salt-concentration-dependence of the nanochannel conductance at the low salt concentration regime depends substantially on the levels of pH. The deviation of the nanochannel conductance between the results with and without the EDL overlap can be up to over one time.

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ASSOCIATED CONTENT

S Supporting Information *

The detailed derivation of eqs 17−19 and description of the numerical modeling based on the full multi-ion PNP−NS model. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Fax: +886-5-5312071. Author Contributions ⊥

Y.M. and L.-H.Y. contributed equally to this work.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is supported, in part, by the Ministry of Science and Technology of the Republic of China under Grants 102-2221E-224-052-MY3 and 103-2221-E-224-039-MY3 (to L.H.Y.), the State Key Program of National Natural Science of China under Grant No. 51436009 (to Y.M.), the China Scholarship Council (to L.M.), and NSF CMMI-1265785 (to S.Q.).

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DOI: 10.1021/acs.analchem.5b00536 Anal. Chem. 2015, 87, 4508−4514