Ion Current Rectification Behavior of Bioinspired Nanopores Having a

Mar 10, 2017 - The ion current rectification behavior of bioinspired nanopores is modeled by adopting a bullet-shaped nanopore having a pH-tunable zwi...
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Ion Current Rectification Behavior of Bioinspired Nanopores Having a pH-Tunable Zwitterionic Surface Jyh-Ping Hsu,†,‡ Hou-Hsueh Wu,† Chih-Yuan Lin,† and Shiojenn Tseng*,§ †

Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617 Department of Chemical Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan 10607 § Department of Mathematics, Tamkang University, New Taipei City, Taiwan 25137 ‡

S Supporting Information *

ABSTRACT: The ion current rectification behavior of bioinspired nanopores is modeled by adopting a bullet-shaped nanopore having a pH-tunable zwitterionic surface, focusing on discussing the underlying mechanisms. We show that with its specific geometry, such nanopore is capable of exhibiting several interesting behaviors, including ion concentration polarization and ion current rectification. The influences of the nanopore shape, solution pH, and bulk salt concentration on the associated ion current rectification behavior are examined. We found that if pH exceeds the isoelectric point, the rectification factor has a local maximum as the curvature of the nanopore surface varies, and if it is lower than the isoelectric point, that factor increases (rectification effect decreases) monotonically with increasing surface curvature. In addition to capable of interpreting relevant electrokinetic phenomena, the results gathered also provide necessary information for a sophisticated design of relevant devices.

R

EOF effect should be considered. However, this effect was often neglected for simplicity in relevant studies.35,36,51 Several techniques have been proposed for fabricating nanopores/nanochannels made of organic,16,51 inorganic,15,32 biological,52 and composite materials.53,54 These include, for instance, track-etching, electrochemical etching, electron beam, laser, and self-assembly of biological molecules.55 It was shown that the shape and the structure of polymeric nanopores can be properly controlled through various etching conditions. Ali et al.,51 for example, found that the PET nanopores fabricated by track-etching technique are bullet shaped, instead of conical, as observed by scanning electron microscopy images. Adopting Poisson−Nernst−Planck (PNP) equations with constant ́ et al.35 investigated compresurface charge density, Ramirez hensively the effect of nanopore shape on the rectification factor and the ion selectivity. They found that as a bullet-like nanopore approaches a conical nanopore, both the rectification factor and the conductance decrease, but the selectivity increases. These behaviors were explained by the effects of ionic concentration and electric potential profiles. However, the effect of the charge-regulated nature of a surface and that of electroosmotic flow were neglected. These two effects are expected to influence significantly the ion transport inside a nanopore as its shape varies. Recently, conical nanopores having zwitterionic groups were studied both experimentally16,56 and theoretically.41 For example, Ali et al.16 designed a nanofluidic diode by

ecent advances in nanomaterials preparation and fabrication technology make relevant applications versatile. Synthetic nanopores, nanochannels, and nanopipettes with functionalized surfaces are widely adopted in modeling biological ion channels, manipulating ion, and particle transport,1−7 biosensing8−12 for ions and biomolecules, ionic gates,13,14 nanofluidic diodes/devices,15−17 energy conversion,18−21 and water desalination.22,23 Although relevant results about ion transport modeling are ample in the literature,24−27 available theoretical results taking account of the effects of pH and pore geometry simultaneously are limited. This is highly desirable and necessary, however, for a sophisticated design of relevant devices. For nanoscaled devices, the nature of Debye length28 (or thickness of electric double layer, EDL) plays the key role. In particular, its overlapping yields distinctive and profound electrokinetic phenomena, including ion concentration polarization (ICP),29 ion selectivity,30 and ionic current rectification (ICR).31,32 ICR is a diode-like current−voltage behavior exhibiting a preferred direction in ionic current. These phenomena resemble to those observed in biological ion channels.33,34 In a nanoscaled system, ICR might arise from several factors including, for example, asymmetric geometry,31,35,36 asymmetric charge distribution,17,37−39 imposing salt gradient,40,41 scan rate,42,43 nanoparticle blocking,44 and environmental conditions.45−47 Several studies48,49 examined the influence of electric osmotic flow (EOF)50 in rectifying ionic current. For instance, in an analysis of the ICR behavior of a pH-regulated conical nanopore, Lin et al.49 found that if the solution pH deviates appreciably from the iso-electric point, the © 2017 American Chemical Society

Received: November 4, 2016 Accepted: March 10, 2017 Published: March 10, 2017 3952

DOI: 10.1021/acs.analchem.6b04325 Anal. Chem. 2017, 89, 3952−3958

Article

Analytical Chemistry functionalizing a polymeric nanopore with lysine and histidine, capable of displaying pH sensitive ICR behavior. Since its surface charge density is pH-tunable, it has potential applications in ionic gating, sensing biomolecules, and regulating ion selectivity,30,52,53 to name a few. In this study, the ICR behavior of a bioinspired nanopore having a zwitterionic surface is discussed in detail by systematically examining the influence of the main factors. These include the nanopore shape, the solution pH, and the bulk salt concentration. In particular, since the pH-regulation and the EOF effects are taking into account, the present model is closer to reality than most of the previous studies.

RB − R T exp[− (L N /h)] − (RB − R T)exp[− (− z /L N)(L N /h)] 1 − exp[− (L N /h)]

(1)

where LN/h is a shape-controlling parameter. As LN/h → 0, the nanopore approaches a conical one, and as LN/h increases, it becomes more bullet-like. Suppose that the nanopore surface has both acidic functional groups and basic functional groups with associated dissociation reactions

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+

RESULTS AND DISCUSSION The influences of the solution pH, the bulk salt concentration C0, the applied voltage Vapp, and the shape parameter LN/h on the ICR behaviors of the nanopore considered are examined through numerical simulation. For illustration, we assume RT = 5.5 nm, RB = 98 nm, LN = 1000 nm, LR = 200 nm, and RR = 200 nm. In addition, we set Γt = 1.5 nm−2, pKA = 3, and pKB = 9.16,30,41 Therefore, the isoelectric point (pI) is 6. The rectification factor Rf is defined as Rf = |I(−1 V)/I(+1 V)|. A small value is assumed for LN/h (10−7) when a conical nanopore needs be simulated. The physical parameters used are ε = 6.95 × 10−10 F m−1, T = 298.15 K, F = 96500 C mol−1, R = 8.3145 J K−1 mol−1, and η = 1 × 10−3 Pa·s. D1(K+), D2(Cl−), D3(H+), and D4(OH−) are 1.957, 2.032, 9.312, and 5.26 (×10−9) m2 s−1, respectively. pH 10 (Cation-Selective Nanopore). The I−V curves illustrated in Figure 1 reveal that at this level of pH the nanopore has a rectification factor Rf larger than unity for all the levels of the bulk salt concentration C0 examined, implying its preference for ionic current at Vapp = −1 V. As reported previously,62 the dependence of Rf on bulk salt concentration is not monotonic, and exhibits a local maximum. Furthermore, at each value of C0, Rf shows a local maximum when LN/h varies, as will be discussed later. For all the values of C0 examined, the larger the LN/h the larger the ionic current. This is because the larger the LN/h the greater the space of a nanopore, so that a more amount of ions is able to diffuse through it, yielding a larger current.

(3)

For illustration, we consider an aqueous KCl solution, and the solution pH is adjusted by KOH or HCl, so that there are four kinds of ionic species: K+, Cl−, H+, and OH−, for j = 1, 2, 3, and 4, respectively, and pH = −log([H+]0/1000) with [H+]0 being the bulk molar concentration of H+ (mM). Note that if C0 is the bulk concentration of KCl (mM), then electric neutrality implies C30 = 10−pH+3, C40 = 10−(14−pH)+3, C10 = C0 − 10−pH+3 + 10−(14−pH)+3, and C20 = C0 for pH ≥ 7, and C10 = C0 and C20 = C0 + 10−pH+3 − 10−(14−pH)+3 for pH < 7.49,57 Let ϕ, u, Nj, and p be the electrical potential, the fluid velocity, the flux of the jth ionic species, and the hydrodynamic pressure, respectively. Then at steady state the present problem can be described by49

−ε∇2 ϕ = ρe

(8)

The present problem is solved numerically by COMSOL MultiPhysics (version 4.3a, www.comsol.com) operated in a high performance cluster. Typically, using a total number of 170000 mesh elements is sufficient. Code validation has been conducted in the Supporting Information.

and equilibrium constants KA = ΓA [H ]/ ΓAH and KB = ΓB[H+]/ΓBH+. [H+] and Γm are the molar concentration of H+ (mM) and the surface site density of species m (AH, A−, BH+, or B), respectively. We assume that the total site density of the acidic and basic functional groups are the same, ΓA− + ΓAH = ΓBH+ + ΓB = Γt. If e denotes the elementary charge, the surface charge density of the nanopore, σw (C/m2), can be expressed as σw = 1018e(ΓBH+ − ΓA−)

∫S F(∑ νj Nj)·ndS j=1

(2a) −

(7)

I=

(2)

BH+ ↔ B + H+

η∇2 u − ∇p − ρe ∇ϕ = 0 4

THEORY The system under consideration comprises a nanopore of tip radius RT, base radius RB, and length LN connecting two identical, large, cylindrical reservoirs. Both the nanopore and the two reservoirs are filled with an aqueous salt solution. The cylindrical coordinates, r, θ, z, are adopted with the origin at the tip end center of the nanopore. The nanopore has a bullet-like shape with its surface described by35

AH ↔ A− + H+

(6)

ρe = ∑ j = 1 νjFcj is the space charge density of the mobile ions with νj, cj, and Dj being the valence, the concentration, and the diffusivity of the jth ionic species, respectively. ε, F, R, T, and η are the fluid permittivity, Faraday constant, gas constant, the absolute temperature, and fluid viscosity, respectively. The presence of Stern layer58 and the viscoelectric59 effect are neglected in our work since their contributions on ionic current are relatively minor, in general. To specify the boundary conditions associated with eqs 4−7, we assume the following. (i) Surface Ω1 is nonslip (u = 0), ionimpenetrable (n · Nj = 0, n being the unit outer normal vector), and has a pH-regulated charge density. (ii) Surface Ω2 is grounded, and an electric potential Vapp is applied on Ω3. (iii) Surface Ω4 is free of charge (n · ∇ ϕ = 0), free of normal flux (n · Nj = 0), and slip. (iv) The ionic concentration at a point far away from the nanopore reaches the bulk values (i.e., cj = Cj0). (v) No external pressure is applied to the system. (ii)−(iv) are based on that the computation domain is sufficiently large.60,61 If we let S be the cross sectional area of either reservoir, then the ionic current I can be evaluated by49

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r(z) =

∇·u = 0

(4)

⎛ ⎞ νjFcj ∇·Nj = ∇·⎜cj u − Dj∇cj − Dj ∇ϕ⎟ = 0, j = 1, 2, 3, 4 RT ⎝ ⎠ (5) 3953

DOI: 10.1021/acs.analchem.6b04325 Anal. Chem. 2017, 89, 3952−3958

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Analytical Chemistry

Figure 1. Schematic representation of the bullet-like nanopore considered. r, θ, z, are the cylindrical coordinates adopted with the origin at the tip end center of the nanopore, which is axial symmetry. Ω1 denotes the inner and outer surface of the nanopore, Ω2 (Ω3) the surface of a computation domain perpendicular to the nanopore axis in the tip (base) end reservoir, and Ω4 the surface of the computation domain parallel to the nanopore axis in both reservoirs.

Figure 3. Axial variation in the cross sectional averaged ionic conductivity at pH 10 for various levels of bulk salt concentration C0. (a) C0 = 1 mM, (b) C0 = 10 mM, (c) C0 = 100 mM, (d) C0 = 500 mM. Solid curve Vapp = +1 V; dashed curve: Vapp = −1 V. Shaded region denotes the nanopore interior.

If we let λj be the molar conductivity of the jth ionic species, then the cross sectional averaged axial concentration cj and the corresponding ionic conductivity Λ can be evaluated by63 cj =

∫

Λ=

∑ cjλj

cjdA /

∫

dA , j = 1, 2, 3, 4

electric field is appreciable only near the tip end of the nanopore, and for both Vapp = +1 and −1 V, the larger the LN/ h, the stronger that field, which is consistent with the work of ́ et al.35 As shown in Figure 3a (C0 = 1 mM), the Ramirez differences of the ionic conductivity at Vapp = +1 and that at −1 V is relatively small for each level of LN/h, so that Rf is close to unity, as shown in the inset of Figure 2a. Note that in Figure S1, an opposite electric field occurs near the nanopore tip for Vapp = +1 V as LN/h is sufficiently small, implying that the ICP effect is more appreciable in conical nanopore. As in the case of Figure S1, if C0 is raised to 10 mM (Figure S2), an opposite electric field is also present right outside the nanopore tip end when LN/h is small (0 and 2), making the ionic current more preferable for Vapp = −1 V. However, as LN/ h increases from 2 to 8, that opposite electric field disappears at Vapp = +1 V (Figure S2c), and the degree of increase in the strength of the electric field as LN/h increases is larger than at Vapp = −1 V. For both Vapp = +1 and −1 V, the ionic conductivity shown in Figure 3b is also seen to decrease with increasing LN/h with the degree of decrease at Vapp = −1 V larger than that at Vapp = +1 V. The combined effect of the strength of electric field and the ionic conductivity yields a decrease in Rf as LN/h increases from 2 to 8. At C0 = 100 mM (Figure S3), the axial electric field near the nanopore tip for both Vapp = +1 and −1 V increase with increasing LN/h. However, Figure 3c shows that, as LN/h varies from 0 to 4, the peak value of the ionic conductivity at Vapp = −1 V shifts toward the tip end, where the electric field is the most appreciable, yielding a larger ionic current. In contrast, as LN/h varies from 0 to 4, the ionic conductivity at Vapp = +1 V varies inappreciably. This makes I(−1 V) more preferable, so that Rf increases with increasing LN/h. If LN/h is raised from 4 to 8, the axial electric field at Vapp = +1 V (Figure S3d) increases at a greater degree than that of Vapp = −1 V, a decline of Rf is observed, accordingly. As seen in Figure 3d, where C0 = 500 mM, if LN/h is raised from 0 to 8, the ionic conductivity at Vapp = −1 V increases significantly, but that at Vapp = +1 V decreases. Since ionic

(9)

4

(10)

j=1 −

Under the conditions considered, λ1 (K ), λ2 (Cl ), λ3 (H+), and λ4 (OH−) are 7.352, 7.654, 35, and 19.8 (×10−3) S m2 mol−1, respectively. To further explain the ICR behavior in Figure 2, we plot the cross sectional averaged ionic conductivity in Figure 3c and the cross sectional averaged axial electric field in Figures S1−S4 of the Supporting Information. The latter reveals that the axial +

Figure 2. Simulated I−Vapp curves for various levels of bulk salt concentrations C0 at at pH 10. (a) C0 = 1 mM, (b) C0 = 10 mM, (c) C0 = 100 mM, (d) C0 = 500 mM. Insets with blue bars show the variation of Rf with LN/h. 3954

DOI: 10.1021/acs.analchem.6b04325 Anal. Chem. 2017, 89, 3952−3958

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Analytical Chemistry current is proportional to ionic conductivity, Rf increases appreciably from 1.93 to 3.68, accordingly. An increase of LN/h from 8 to 12 causes a decline in Rf. This can be explained by Figure S4d,e, where the strength of the electric field at Vapp = +1 V increases with increasing LN/h, but that at Vapp = −1 V remains roughly constant. pH 4 (Anion-Selective Nanopore). As can be inferred from the I−V curves shown in Figure 4, the Rf at pH 4 is less

Figure 5. Axial variation in the cross sectional averaged ionic conductivity at pH 4 for various levels of bulk salt concentration C0. (a) C0 = 1 mM, (b) C0 = 10 mM, (c) C0 = 100 mM, (d) C0 = 500 mM. Solid curve Vapp = +1 V; dashed curve: Vapp = −1 V. Shaded region denotes the nanopore interior.

Supporting Information, if C0 is low, the electric field at Vapp = +1 V is much stronger than that at Vapp = −1 V, but if C0 is high, the former is only slightly stronger than the latter, as shown in Figures S7 and S8. The ionic conductivity illustrated in Figure 5 exhibits the similar behavior as that electric field. It is worth noting that if C0 is low, the ionic current at LN/h = 0 approaches a plateau value at a high level of Vapp. This is because if Vapp is sufficiently high and EDL overlapping serious, the net anionic difference (i.e., (c2 + c4) − (c1 + c3)) is large inside the nanopore, yielding a more significant ICP, thereby retarding the increase of the ionic current. As can be seen in Figure S9, the scaled net anionic difference is appreciable at a lower C0 and a higher Vapp. It also reveals that the ICP effect in a conical nanopore (LN/h = 0) is more appreciable than that in a bullet-shaped nanopre (LN/h = 8) since the EDL overlapping is insignificant in the latter. Different from that at pH 10, where Rf has a local maximum as LN/h varies, Figure 4 indicates that at pH 4 Rf increases monotonically with increasing LN/h. As will be explained later, this might arise from the pH-regulated nature of the nanopore surface. Influence of pH on Rf. Figure 6 indicates that if pH is larger (smaller) than 6, Rf is larger (smaller) than unity. This is because the pI of the nanopore is 6, so that it is negatively charged for pH > 6, and positively charged for pH < 6, as seen

Figure 4. Simulated I−Vapp curves for various levels of bulk salt concentrations C0 at pH 4. (a) C0 = 1 mM, (b) C0 = 10 mM, (c) C0 = 100 mM, (d) C0 = 500 mM. Insets with blue bars show the variation of Rf with LN/h.

than unity for all the cases examined, implying the preference for ionic current at Vapp = +1 V. The dependence of Rf on the bulk salt concentration is more appreciable when it is low ( Rf(LN/h = 2) > Rf(LN/h = 8). However, if both pH and C0 are sufficiently high (ca. 9.5 and 200 mM, respectively), they rank as Rf(LN/h = 2) > Rf(LN/h = 8) > Rf(LN/h = 0). To further explain the influence of pH and C0 on Rf, the spatial distributions of H+ and OH− are examined for various combinations of C0 and LN/h in Figures 8 (pH 10) and 9 (pH

overlapping the more important the influence of OH− (H+) on Rf .

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CONCLUSION

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ASSOCIATED CONTENT

The ionic current rectification (ICR) behavior of a bulletshaped nanopore is investigated for the first time by considering the pH-regulated nature of its surface and the effect of electroosmotic flow. Here, the nanopore shape can be adjusted by varying a shape-controlling factor LN/h: the larger this factor the more serious the nanopore deviates from a conical one. As LN/h increases, a more amount of ions is contained in the nanopore, yielding a larger ionic current. The ICR factor, Rf, of a negatively charged nanopore (pH 10) is larger than unity, implying that it has a preference for a negatively applied potential bias at the base end. For the range of bulk salt concentration considered (1−500 mM), if LN/h is sufficiently large, Rf shows a local maximum as LN/h varies. This can be explained by the variations in the strength of the axial electric field and the ionic conductivity inside the nanopore. The Rf of a positively charged nanopore (pH 4) is smaller than unity, implying that it has a preference for positive applied potential bias. In this case, Rf is lower than unity, and increases (decreasing rectification effect) with increasing LN/h. Again, these can be explained by the behaviors of the axial electric field and the ionic conductivity of the nanopore. As the solution pH varies, oppositely charged nanopores show opposite ion current preference. For a negatively charged nanopore (pH 10), Rf shows a local maximum as LN/h varies, but this is not the case for a positively charged nanopore (pH 4), where Rf increases monotonically with increasing LN/h. Both OH− and H+ influence significantly on Rf if the bulk salt concentration is sufficiently low. This arises from that the accumulation of these ions inside a nanopore is enhanced when the EDL overlapping is significant.

Figure 8. Concentration profiles of OH− for various values of LN/h at C0 = 10 mM (a) and 200 mM (b). The levels of LN/h are 0, 2, and 8, respectively, from left to right.

4). Figure 8b shows that the amount of OH− accumulated inside nanopore increases significantly with increasing LN/h, yielding a higher surface charge density and, therefore, a larger Rf. Figure 6a shows that if C0 is low, the influence of LN/h on Rf is significant, and this influence becomes less significant as C0 gets high, as seen in Figure 6b. This results from the accumulation of H+ near the nanopore tip. As illustrated in Figure 9a,b that at C0 = 10 mM the concentration of H+ near the nanopore tip decreases remarkably with increasing LN/h, but is almost independent of LN/h at C0 = 200 mM. As seen in Figures 8a and 9a, the accumulation of OH− (H+) at C0 = 10 mM is more significant than that at C0 = 200 mM, shown in Figures 8b and 9b, implying that the more significant the EDL

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.analchem.6b04325. Cross sectional averaged axial electric field at pH 10, that at pH 4, profiles of the scaled net anionic difference, as well as model validation (PDF). 3956

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AUTHOR INFORMATION

Corresponding Author

*Tel.: 886-2-26215656, ext. 2508. Fax: 886-2-26209916. Email: [email protected]. ORCID

Jyh-Ping Hsu: 0000-0002-4162-1394 Shiojenn Tseng: 0000-0001-6679-6337 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is supported by the Ministry of Science and Technology, Republic of China.

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