Effects of Electroosmotic Flow on Ionic Current Rectification in Conical

Feb 12, 2010 - for the ionic concentrations, the Poisson equation for the electric potential, and Navier-Stokes equations for the flow field. It is fo...
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J. Phys. Chem. C 2010, 114, 3883–3890

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Effects of Electroosmotic Flow on Ionic Current Rectification in Conical Nanopores Ye Ai,† Mingkan Zhang,† Sang W. Joo,‡ Marcos A. Cheney,§ and Shizhi Qian*,†,‡ Department of Aerospace Engineering, Old Dominion UniVersity, Norfolk, Virginia 23529, School of Mechanical Engineering, Yeungnam UniVersity, Gyongsan 712-749, South Korea, and Department of Natural Sciences, UniVersity of Maryland Eastern Shore, Princess Anne, Maryland 21853 ReceiVed: December 13, 2009; ReVised Manuscript ReceiVed: January 20, 2010

The effects of electroosmotic flow (EOF) on the ionic current rectification (ICR) phenomenon in conical nanopores are studied comprehensively with use of a continuum model, composed of Nernst-Planck equations for the ionic concentrations, the Poisson equation for the electric potential, and Navier-Stokes equations for the flow field. It is found that the preferential current direction of a negatively charged nanopore is toward the base (tip) under a relatively high (low) κRt, the ratio of the tip radius size to the Debye length. The direction also changes with the charge polarity of the nanopore. The EOF effect on the ionic current rectification ratio in a conical nanopore becomes noticeable at an intermediate κRt and surface charge density of the nanopore, meanwhile increasing significantly with the applied voltage. 1. Introduction Synthetic nanopores are attractive mimetic materials for biological ionic channels.1-7 Study on the ion transport in synthetic nanopores provides a potential way to understand the real physiological processes in living organisms,8-10 and in particular facilitates the development of practical biosensors using synthetic nanopores.11-17 Several fascinating features of nanopores, such as ion selectivity18-22 and ionic current rectification,19,23-36 have been observed. Furthermore, synthetic nanopores have been widely used to detect and count biological entities, including proteins and DNAs, based on the highly sensitive ionic current depression by particles translocating through nanopores.2,4,6,12,13,37-40 Ionic current rectification (ICR) in conical nanopores refers to an asymmetric diode-like current-voltage (I-V) behavior, and has attracted many researchers. The magnitude of ionic current through the nanopore at a negative voltage is different from that at a positive voltage, indicating a preferential current direction. Several mechanisms, such as electric potential barrier inside the pore,24 electrochemical prosperities of the nanopore tip,41,42 and enrichment and depletion of ions,28,32,34 have been proposed to explain the ICR phenomenon, as summarized by Siwy.25 Despite the differences among these mechanisms, it is generally accepted that the electrostatic asymmetry through the nanopore is responsible for the ICR phenomenon when the Debye length becomes comparable to the characteristic length (e.g., the pore diameter). Recently, a theoretical model based on Poisson-Nernst-Planck (PNP) equations has been developed, and widely used to predict the ICR behavior in nanopores.26,28,43-45 Remarkable agreements between theoretical predictions and experimental observations suggest that the continuum model correctly describes the ion transport in nanopores with diameters larger than 0.1 Debye lengths.46-49 However, the electroosmotic flow (EOF), arising from the electrostatic interaction between the spatially dependent electric field and the net ionic charge density within the * To whom correspondence should be addressed. E-mail: [email protected]. † Old Dominion University. ‡ Yeungnam University. § University of Maryland Eastern Shore.

nanopore, has not been taken into account in the PNP model. The induced EOF affects the ion distribution in the nanopore by convection, which may eventually alter the ion transport due to the diffusion and electromigration. It is not appropriate to estimate the EOF effect by using a direct quantitative comparison between the convective flux and the total flux. Indeed, several numerical studies with the PNP model without EOF have achieved acceptable agreement with the existing experiments under certain conditions.26,28,42,44 However, Daiguji et al.50 point out that the EOF effect increases significantly with the surface charge of nanopores. White and Bund29 also state that the EOF may have a noticeable indirect effect on the ion transport in nanopores. However, a comprehensive understanding of the EOF effect on the ion transport in nanopores is still very limited. In this study, the PNP model and that coupled with the Navier-Stokes equations (PNP-NS) are both implemented to study comprehensively the ion transport in conical nanopores with emphasis on the EOF effect on the ICR behavior. Three main factors, namely the applied voltage, the Debye length, and the surface charge of the nanopore, are investigated. Section 2 introduces the full mathematical model for the fluid motion and the ionic mass transport. Detailed numerical implementation and code validation are described in section 3. The EOF effects on the ICR in the conical nanopore are presented and discussed in section 4, followed by concluding remarks in section 5. 2. Mathematical Model We consider a conical nanopore with axial length L and tip (smaller) and base (larger) radii Rt and Rb, respectively, as shown in Figure 1. It is filled with a binary electrolyte solution, such as KCl aqueous solution, with density F, dynamic viscosity µ, and permittivity ε, supplied from two fluid reservoirs connected. The wall of the nanopore (segment DE in Figure 1) bears a uniform surface charge density, σ, and the wall of the reservoirs (segments CD and EF) is not charged. The radius b and the axial length LR of the reservoirs are large enough to maintain the bulk ionic concentration C0 at a constant value far away from the conical nanopore. As a result, counterions accumulate in the vicinity of the charged nanopore wall, forming the electric double layer (EDL). An external electric potential difference is

10.1021/jp911773m  2010 American Chemical Society Published on Web 02/12/2010

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Ai et al. solution, and R is the universal gas constant. The flow field u is determined by eqs 1 and 2. The first, second, and third terms in the right-hand side (RHS) of expression 3 represent respectively the convective, diffusive, and electromigrative flux density. Under steady state, the concentration of each species is governed by the Nernst-Planck (NP) equation

Figure 1. A charged conical nanopore connected to two identical reservoirs filled completely with a binary electrolyte solution. An electric potential difference imposed between the two reservoirs (segments AB and GH) induces an ionic current through the conical nanopore.

applied between the left (segment AB) and right (segment GH) reservoirs to generate the ionic current and EOF through the nanopore. A cylindrical coordinate system (r, z) with the origin fixed at the center of the nanopore is used due to the axisymmetric geometry. The axial z and radial r coordinates are respectively parallel and perpendicular to the axis of the conical nanopore. 2.1. Mathematical Model for Fluid Motion. Since the Reynolds numbers of the EOF in nanopores are extremely small, we neglect the inertial terms in the Navier-Stokes equations and model the fluid motion with the continuity equation

∇•u ) 0

(1)

-∇p + µ∇2u - F(z1c1 + z2c2)∇φ ) 0

(2)

and the Stokes equation

In the above, u ) uer + Vez is the fluid velocity. Hereafter, bold letters denote vectors; er and ez are respectively unit vectors in the r- and z-directions; u and V are respectively the velocity components in the r- and z-directions; p is the pressure; φ is the electric potential; c1 and c2 are respectively the molar concentrations of cations (K+) and anions (Cl-) in the electrolyte solution; z1 and z2 are respectively the valences of cations and anions; and F is the Faraday constant. The last term on the lefthand side (LHS) of eq 2 represents the electrostatic force originating from the interactions between the electric field and the net charge density in the electrolyte solution. To solve eqs 1 and 2, appropriate boundary conditions are required. A nonslip boundary condition (i.e., u ) V ) 0) is specified on segments CD, DE, and EF in Figure 1. A normal flow with p ) 0 is applied to the planes of AB and GH, which are far away from the nanopore. An axisymmetric boundary condition is imposed on AH. Slip boundary conditions are used on segments BC and FG since they are far away from the entrances of the nanopore. 2.2. Mathematical Model for Ionic Mass Transport. A generalmulti-ionicmasstransportmodelincludestheNernst-Planck (NP) equations for the concentration of each ionic species, and the Poisson equation for the electric potential in the electrolyte solution. The flux density of each aqueous species due to the convection, diffusion, and electromigration is given by

Di Ni ) uci - Di∇ci - zi Fci∇φ, RT

i ) 1 and 2

(3)

In the above equation, Di is the diffusion coefficient of the ith ionic species, T is the absolute temperature of the electrolyte

∇•Ni ) 0,

i ) 1 and 2

(4)

The set of the NP eqs 4 consist of three unknown variables, the concentrations of cations and anions and the electric potential. The Poisson equation provides the third equation

-ε∇2φ ) F(z1c1 + z2c2)

(5)

Segments CD, DE, and EF are the walls of the reservoirs and the nanopore, and the solid surface is impervious to ions. Therefore, the normal ionic flux is zero:

n•Ni ) 0,

i ) 1 and 2

(6)

where n is the local unit normal vector of the above segments. Similarly, zero normal ionic flux is used along AH due to the axisymmetric condition and along BC and FG since they are in the bulk electrolyte reservoirs. We assume that the concentration of each species recovers the value for the bulk electrolyte concentration on AB and GH, and write

( (

ci r, ( LR +

L 2

)) ) C , 0

i ) 1 and 2

(7)

The externally imposed electric field is described by the potential difference, V, imposed between AB and GH, which is expressed as

( (

φ r, - LR +

L 2

)) - V ) φ(r, (L

R

+

L 2

)) ) 0

(8)

When V > 0, the imposed electric field is directed from the tip toward the base of the nanopore, and vice versa. The walls of the reservoirs, CD and EF, are uncharged, so that

n•∇φ ) 0

(9)

Since the surfaces of BC and FG are far away from the nanopore and are in the bulk electrolyte reservoirs, no charge boundary condition for the potential is used. The axisymmetric boundary condition is used along the axis of the nanopore, AH. The surface charge density of the nanopore (DE), σ, is described by

n•(-ε∇φ) ) σ

(10)

The resulting ionic current through the nanopore is the integration of the current density along the surface of the anode or cathode:

I)

∫S F(z1N1 + z2N2)•n dS

(11)

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2.3. Dimensionless Form of the Mathematical Models. We use the bulk concentration C0 as the ionic concentration scale, RT/F as the potential scale, the tip radius Rt as the length scale, U0 ) εR2T2/(µRtF2) as the velocity scale, and µU0/Rt as the pressure scale. The dimensionless governing equations are given by:

∇*•u* ) 0

(12)

1 -∇*p* + ∇*2u* - (κRt)2(z1c1* + z2c2*)∇*φ* ) 0 2

(13) ∇* · Ni* ) 0,

i ) 1 and 2

(14)

1 -∇*2φ* ) (κRt)2(z1c1* + z2c2*) 2

(15)

In the above equations, variables with a superscript asterisk are dimensionless. κ-1 ) λ ) [εRT/2F2C0]1/2 is the Debye length. The dimensionless flux density normalized by U0C0 is

Ni* ) u*ci* - Λi∇*ci* - ziΛici*∇*φ*,

i ) 1 and 2 (16)

where Λi ) Di/D0 with D0 ) εR2T2/(µF2). The dimensionless surface charge of nanopore normalized with εRT/(FRt) is

n•(-∇*φ*) )

FRt σ ) σ¯ εRT

(17)

The dimensionless ionic current through the nanopore normalized with FU0C0Rt2 is

I* )

∫ (z1N1* + z2N2*)•n dS*

(18)

The PNP model refers to the system coupling eqs 14 and 15 without the convective flux in the total flux Ni* (i.e., u* ) 0). In contrast, the PNP-NS model, accounting for the EOF effect, includes eqs 12-15. Obviously, the PNP-NS model is strongly coupled. In addition to the direct effect of the convective flux on the total flux density of each aqueous species, the EOF may have considerable effect on the spatial distributions of the ionic concentrations and electric potential, which in turn affects the total flux density Ni* and thus alters the ionic current through the nanopore. 3. Numerical Implementation and Code Validation The nonlinear coupled systems above, including the PNP and the PNP-NS models, are numerically solved by a commercial finite-element package COMSOL (version 3.5a, www.comsol.com) operating in a high-performance cluster. The computational domain, as shown in Figure 1, is discretized into quadratic triangular elements. Nonuniform elements are employed with larger numbers of elements assigned locally as necessary. A minimum of 30 elements are positioned within the EDL, adjacent to the charged wall of the conical nanopore. Finer mesh is used in the regions close to the tip of the nanopore in which the EDL may overlap.51,52 The ionic current through the

Figure 2. Comparisons between the analytical solutions (lines) and numerical results (symbols) of the electric potential near a planar charged surface (σ ) -1 mC/m2) in 1 mM (solid line and circles), 10 mM (dashed line and squares), and 100 mM (dash-dotted line and triangles) KCl solution. The electric potential, φ(x), is normalized by its value at x ) 0. The inset shows a schematic view of the computational domain with the charged planar surface at the left side.

nanopore is obtained by using the weak constrain in COMSOL specially developed for an accurate calculation of flux. Rigorous mesh-refinement tests have been performed to ensure that the solutions obtained are convergent and grid independent. Typically, the number of elements is 2 × 105. A maximum tolerance of 0.01% is imposed on the relative difference |Ia - Ic|/|Ia|, where Ia and Ic are respectively the current entering (anode) and leaving (cathode) the nanopore. The following dimensional parameter values represent cases considered in the present study: T ) 300 K, D(K+) ) 1.95 × 10-9 m2/s, D(Cl-) ) 2.03 × 10-9 m2/s, ε ) 7.08 × 10-10 F/m, µ ) 1 × 10-3 Pa · s, and F ) 1 × 103 kg/m3. Several benchmark tests were carried out to ensure the validity and accuracy of the numerical model. For example, the spatial distribution of the electric potential in a KCl electrolyte solution near a charged planar surface is simulated by using the PNP model without convection. The analytical solution of the electric potential along the direction normal to the charged surface is given by29,53

φ(x) )

2RT 1 - K exp(-x/λ) ln F 1 + K exp(-x/λ)

(19)

where x is the distance from the charged planar surface, K ) Q/[2 + (4 + Q2)1/2], and Q ) -λFσ/(RTε). Figure 2 shows an excellent agreement between the analytical solutions (lines) and the numerical results (symbols) obtained by the PNP model. In addition, the EOF in a cylindrical nanotube filled with 10 mM KCl electrolyte is simulated by using the PNP-NS model. The surface charge and radius of the tube are respectively σ ) -1 mC/m2 and r0 ) 50 nm. The analytical solution of the fully developed axial EOF velocity is given by29,53

Vz(r) ) -

λσE [I (r /λ) - I0(r/λ)] µI1(r0 /λ) 0 0

(20)

where E is the imposed axial electric field, and Ii is the modified Bessel functions of the first kind of order i. Our numerical results (circles) are in good agreement with the analytical solution (solid line), as shown in Figure 3. We also simulated the diffusioosmostic flow in a slit nanochannel connecting to fluid reservoirs using the PNP-NS model, in which the fluid motion is induced

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Figure 3. Comparison between the analytical solution (solid line) and the numerical result (circles) of the axial velocity of an electroosmotic flow (EOF) in a cylindrical nanotube. The bulk electrolyte is 10 mM KCl solution, and the surface charge density of the nanotube is σ ) -1 mC/m2. The externally imposed axial electric field is -50 KV/m. The inset shows a schematic view of the nanotube with dimensions.

Figure 4. I-V curve of a conical nanopore when κRt ) 1 (a), 6 (b), and 10 (c) in a KCl electrolyte solution. The solid line (circles), dashed line (squares), and dash-dotted line (triangles) represent respectively the numerical results with (without) EOF for surface charge densities σ j ) -2.73 (σ ) -10 mC/m2), -13.66 (-50 mC/m2), and -27.32 (-100 mC/m2).

by the imposed concentration gradient. Our numerical results54 agree with the results obtained by Pivonka and Smith.55 These good agreements make us confident of our following computational results. 4. Results and Discussion In this section, we focus on the EOF effects on the ICR in a conical nanopore by comparing the results with (PNP model) and without (PNP-NS model) EOF. For consistency among the cases presented in the following sections, conditions are specified as follows: Rt ) 5 nm, Rb ) 30 nm, and L ) 1 µm. The radius and the length of each reservoir are set to b ) LR ) 0.2 µm, which are confirmed to be sufficiently large for all cases shown. A fillet with a radius of 0.2Rt is applied to smooth the connection between the nanopore and each reservoir. The resulting dimensionless ionic current through the nanopore is investigated as functions of the imposed dimensionless voltage, j , the ratio of the tip size to the Debye length, κRt, and the V dimensionless surface charge density of the nanopore, σ j. 4.1. Effect of the Ratio of the Tip Radius to the Debye Length, KRt. Figure 4 shows the dimensionless I-V curves of a conical nanopore for κRt ) 1 (a), 6 (b), and 10 (c) when the surface charge densities of the nanopore are respectively σ j) -2.73 (σ ) -10 mC/m2, solid lines and circles), -13.66 (-50 mC/m2, dashed lines and squares), and -27.32 (-100 mC/m2, dash-dotted lines and triangles). Lines and symbols are respec-

Ai et al.

Figure 5. Current rectification ratio, IR, as a function of the applied voltage when κRt ) 1 (a), 6 (b), and 10 (c) in a KCl electrolyte solution. The solid line (circles), dashed line (squares), and dash-dotted line (triangles) represent respectively the numerical results with (without) EOF for surface charge densities σ j ) -2.73 (σ ) -10 mC/m2), -13.66 (-50 mC/m2), and -27.32 (-100 mC/m2).

tively the numerical results with and without EOF. The electrolyte solution in the conical nanopore does not behave as an ohmic resistor because diode-like I-V curves are observed. The character of the I-V curves strongly depends on the bulk concentration of the KCl electrolyte solution and the surface charge density of the nanopore. The positive ionic current is defined as a current directed from the tip toward the base when a positive voltage (V ) φAB - φGH > 0) is applied. For a relatively high κRt (κRt ) 6 and 10), the magnitude of a positive ionic current under a positive voltage is higher than that under a negative voltage imposed, which is consistent with existing experimental and numerical studies.23,25 However, the preferential current direction reverses as κRt decreases (κRt ) 1), which will be future discussed later in the subsequent sections. Since the EOF is very small under low voltage, low ionic concentration, and low surface charge density of the nanopore, the EOF effect on the ICR in a conical nanopore is negligible under these conditions, as shown in Figure 4. However, as these parameters increase, the EOF effect may become increasingly important. Figure 5 depicts the current rectification ratio, IR ) |I(V)/ I(- V)|, as a function of the magnitude of the applied voltage under the same conditions as Figure 4. For a relatively high κRt (κRt ) 6 and 10), IR increases with the applied voltage. In contrast, IR decreases with the applied voltage for a relatively low κRt (κRt ) 1). It is consistent with Figure 4 that the EOF j ) -2.73 (solid effect is negligible for all three κRt when σ lines and circles). However, the EOF effect on the current rectification ratio becomes significant as the applied voltage increases. The largest discrepancy between IR in the presence j ) and absence of EOF is 27.8%, presented when κRt ) 6, σ j ) 40, suggesting that the effect of EOF on the -13.66, and V ICR is significant under this condition. Figure 6 depicts the detailed cross-sectional averaged ionic concentrations and electric potential along the axis of the conical nanopore for the aforementioned three different values of κRt in the presence of EOF when σ j ) -13.66. The cross-sectional averaged ionic concentration is defined as the integration of ionic concentration over the cross section divided by the crosssectional area. When κRt ) 1 (Figure 6a), the tip radius is the same as the Debye length resulting in an overlapped EDL in j ) 0 (Figure 6a-II), the electric potential the tip region. For V drops near the negatively charged tip and gradually increases to zero far away from the tip. The generated positive axial electric field at the junction of the left reservoir and the tip drags

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Figure 6. Cross-sectional averaged ionic concentrations (red dashed lines for K+ and blue solid lines for Cl-) and electric potential (black solid lines) along the axis of the conical nanopore in the presence of EOF for σ j ) -13.66 (σ ) -50 mC/m2) when κRt ) 1(a), 6 (b), and 10 (c). The j ) -40, 0, and 40. first, second, and third column represent respectively the applied voltage of V

the cations migrating from the left reservoir into the tip region of the nanopore. Meanwhile, the induced negative axial electric field inside the pore also attracts cations from the right reservoir toward the tip region. As a result, the negatively charged nanopore, especially the tip region, is predominantly filled with the cations (K+), while the anions (Cl-) are depleted from the nanopore. The ionic current through the nanopore is thus mainly carried by the cations. In addition, the ionic concentrations within the nanopore are almost identical for the three different applied voltages, as shown in Figure 6, parts a-I, -II, and -III. For a positive voltage (Figure 6a-III), the potential drops abruptly near the tip, and a positive axial electric field is generated within the nanopore. In contrast, the potential slightly drops near the base, and a negative axial electric field is generated within the pore for a negative voltage imposed (Figure 6a-I). The magnitude of the generated axial electric field within j ) -40 is about the nanopore (-L/2 < z < L/2 in Figure 1) for V j ) 40, leading to a higher electromigrative four times that of V current under a negative voltage. On the basis of the spatial distribution of the concentration of the cations and the electric potential, a positive diffusive current predominately carried by the cations is induced inside the nanopore, regardless of the polarity of the imposed voltage. A positive (negative) electromigrative current is generated for a positive (negative) voltage imposed. However, the magnitude of a negative electromigrative current is much higher than that of a positive one, resulting in a higher negative net ionic current under a negative voltage, as shown in Figure 4a.

For κRt ) 6 and 10, the Debye length is smaller than the tip radius and the EDL is not overlapped. Thus, the nanopore is filled with both cations and anions, as shown in Figure 6b,c. Since the nanopore is negatively charged, the concentration of the cations is higher than that of the anions. Different from the case of κRt )1, the magnitudes of the axial electric fields inside the nanopore are almost the same, and the spatial distributions of the ionic concentrations are significantly different under negative and positive voltages, shown in Figure 6b,c. Comparing j ) 0, when a negative voltage is applied, to the results of V both cations and anions are depleted in the nanopore as shown in Figure 6, parts b-I and c-I. As a result, the electrical conductivity of the electrolyte solution within the nanopore decreases due to the decrease in the ionic concentrations. On the other hand, both cations and anions are significantly enriched within the nanopore when a positive voltage is applied, as shown in Figure 6, parts b-III and c-III. As a result, the ionic current is larger than that under a negative voltage of the same magnitude due to the increase in the electrical conductivity of the electrolyte solution within the nanopores, which is consistent with some previous studies.28,29,41,42,44 Basically, the surface charge of the nanopore generates an asymmetric distribution of ions along the conical nanopore, rendering the conical nanopore similar to a nanofluidic diode.27,28,56,57 Daiguji et al.56 gave a detailed explanation of the ion depletion and enrichment (Figure 1) under different voltage polarities. Therefore, the ICR for a relatively high κRt is attributed to the ion depletion and enrichment within the nanopore.

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Figure 7. Current rectification ratio as a function of κRt with (solid line plus circles) and without (dashed line plus squares) EOF when σ j j ) (40. κRt ) 1.7 represents the ) -13.66 (σ ) -50 mC/m2) and V point at which the ICR vanishes.

Figure 7 shows the current rectification ratio as a function of j ) (40 in the presence (solid line j ) -13.66 and V κRt when σ plus circles) and absence (dashed line plus squares) of EOF. κRt ) 1.7 is predicted as the threshold value, at which the ICR in the conical nanopore vanishes. IR > 1 results for κRt > 1.7, and vice versa. Most existing experiments are conducted in the region κRt > 1.7. The dimensional ionic current in the region κRt < 1.7 is typically on the order of 10-2 nA, which may be quite difficult to detect due to external noises. In the region κRt > 1.7, the maximum IR occurs at an intermediate κRt (κRt ≈ 6 in the present study), which has also been experimentally observed by Schiedt et al.58 and numerically predicted by White and Bund.29 Once κRt is smaller than the threshold value, the preferential current direction reverses due to the dominance of the cations within the conical nanopores. This prediction shows a similarity to the local charge inversion with polyvalent cations, which is strongly related to the thickness of EDL.59 Figure 7 also demonstrates that the EOF effect is negligible in the region of κRt < 1.7; however, it attains the maximum at an intermediate κRt (κRt ≈ 6 in the present study) in the region of κRt > 1.7. The relative difference of IR in the presence and absence EOF is 27.8% when κRt ) 6. Note that the concentrations of the cations and anions are almost the same outside the EDL, referring to a bulk concentration. In the bulk solution, EOF may facilitate the cations transport, but simultaneously retard the anions transport. Accordingly, the convective contribution, in terms of uci, is negligible outside the EDL. When κRt increases further, the EDL becomes thinner and more regions in the nanopore turn to a bulk concentration. As a result, the ionic current is mostly carried by the ions in the bulk solution, which decreases the EOF effect on the ICR in a nanopore. To better understand the EOF effect on the ICR in a conical nanopore, the spatial distributions of the dimensionless concentrations of the cations (K+, Figure 8) and anions (Cl-, Figure 9) in the presence (a) and absence (b) of EOF are examined near the tip of the nanopore for κRt ) 1 (I), 6 (II), and 10 (III) j ) -40. Figure 8a-I reveals that the when σ j ) -13.66 and V nanopore is predominantly filled with the cations. The cations depletion near the tip is observed in Figure 8, parts a-II and -III. In addition, Figure 8 demonstrates that the cations distributions in the presence and absence of EOF are almost the same for κRt ) 1; however, it indicates apparent deviations for κRt ) 6 and 10. Due to the negative voltage applied through the nanopore, the EOF directs from the base toward the tip. Therefore, the cation concentration near the tip in the presence of EOF is lower than that in the absence of EOF, as shown in Figure 8, parts a and b-II and -III. The anions are significantly inhibited in the nanopore for κRt ) 1, as shown in Figure 9a-I.

Figure 8. Distribution of the dimensionless concentration of cation (K+) near the tip of the conical nanopore when κRt ) 1 (I), 6 (II), and j ) -40. 10 (III) with (a) and without (b) EOF. σ j ) -13.66 and V

Figure 9. Distribution of the dimensionless concentration of anion (Cl-) near the tip of the conical nanopore when κRt ) 1 (I), 6 (II), and j ) -40. 10 (III) with (a) and without (b) EOF. σ j ) -13.66 and V

The anions depletion is also observed near the tip in Figure 9, parts a-II and -III. Similar to the EOF effect on the distributions of the cations, EOF also drags the anions out of the nanopore into the bottom reservoir, resulting in a lower anions concentration near the tip in the presence of EOF compared to that in the absence of EOF, as illustrated in Figure 9, parts a and b-II and -III. Figure 10 further shows the cross-sectional averaged ionic concentrations (Figure 10a) and the electric potential (Figure 10b) along the axis of the nanopore with and without of EOF j) j ) -13.66 and V for κRt )1 (I), 6 (II), and 10 (III) when σ -40. The difference between the ionic concentrations with and without EOF is very small for κRt ) 1; however, it becomes

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Figure 12. Ionic current as a function of the surface charge density j ) -40 (a) and 40 (b). The solid lines (circles), of nanopores when V dashed lines (squares), dash-dotted lines (crosses), and dotted lines (triangles) represent respectively the numerical results with (without) EOF when κRt ) 2, 4, 6, and 8.

Figure 10. (a) Cross-sectional averaged ionic concentrations with (black solid line for K+ and black dash-dotted line for Cl-) and without (red dashed line for K+ and red dotted line for Cl-) EOF; (b) electric potential along the axis of the conical nanopore with (black solid lines) and without (red dashed lines) EOF when κRt ) 1 (I), 6 (II), and 10 j ) -40. (III). σ j ) -13.66 and V

Figure 13. Current rectification ratio as a function of the surface charge j ) 40. The solid line (circles) and dashed line (squares) density when V represent respectively the numerical results with (without) EOF when κRt ) 4 and 6.

Figure 11. I-V curve of a negatively charged (σ j ) -13.66, solid line and circles) and positively charged (σ j ) 13.66, dashed line and squares) conical nanopore when κRt ) 1 (a) and 6 (b). The lines and symbols represent respectively the results with and without EOF.

significant for κRt ) 6 and 10. Due to the EOF effect on the spatial distribution of the ionic concentrations, the corresponding electric potentials are also altered accordingly, as shown in Figure 10b. As aforementioned, the convection, diffusion, and electromigration of ions are strongly coupled. The indirect effect of convection may be much more significant than its direct effect on the ionic current through the conical nanopore, also in agreement with White and Bund.29 Therefore, an exclusive comparison between the convective flux and the total flux is not rational to estimate the EOF effect on ICR in nanopores. 4.2. Effect of the Nanopore’s Surface Charge Density, σ j. Figure 11 depicts the I-V curve of a negatively (σ j ) -13.66, solid line and circles) and positively (σ j ) 13.66, dashed line and squares) charged conical nanopore in the presence (lines) and absence (symbols) of EOF for κRt ) 1 (a) and 6 (b). The I-V curve symmetrically reverses as the charge of the nanopore switches from negative to positive, which qualitatively agrees with previous experimental and numerical studies.23,30,59 Therefore, it is able to tune ICR in nanopores by altering the charge polarity of the nanopores, which has been accomplished by using polyvalent cations,59 surfactant in the solution,60 and adjustment of the pH value of the electrolyte solutions.23,30,61 Figure 12 shows the ionic current as a function of the surface charge density of nanopores (I-σ curve) with (lines) and without (symbols) EOF under a negative (a) and positive (b) voltage

for κRt )2 (solid line and circles), 4 (dashed line and squares), 6 (dash-dotted line and crosses), and 8 (dotted line and triangles). It has been revealed that the surface charge density of most materials adopted in the fabrication of nanopores, such as glass and polymer, ranges from -1 to -100 mC/m2.29 In addition, the magnitude and polarity of surface charge density vary with the pH value of the electrolyte solution.23,30,61 Therefore, the surface charge density varies from -100 to 100 mC/m2 in Figure 12. The ionic current increases with the magnitude of surface charge density. The asymmetric I-σ curves are attributed to the inversion of ICR, as shown in Figure 11. In addition, the I-σ curve reverses with the polarity of the applied voltage. When the surface charge is σ j ) -13.66, it has been mentioned that the preferential current direction directs from the tip toward the base when κRt > 1.7 (Figure 7). However, the preferential current direction reverses from the base toward the tip for κRt ) 2 when the surface charge is increased to σ j ) -27.32. Therefore, the threshold value of κRt, determining the preferential current direction, is affected by the surface charge density of the nanopore. It is also found that the EOF is very small when the surface charge of nanopores is low. However, it significantly increases with the magnitude of the surface charge density. The relative difference between the ionic current in the j) presence and absence of the EOF is 15% when κRt ) 8, σ j ) 40. Therefore, the EOF effect should also be -27.32, and V taken into account when the surface charge of the nanopore is relatively high. The effect of the nanopore’s surface charge density on the current rectification ratio, IR, is depicted in Figure 13. When the nanopore is uncharged, the ICR in the nanopore vanishes, as shown by IR ) 1. This confirms that the origin of ICR is the electrostatic interaction between the surface charge and ions in the nanopores. The current rectification ratio is maximized at

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an intermediate surface charge for both negatively and positively charged nanopores. If the surface charge of nanopores increases further, the rectification decreases as IR approaches unity. The EOF effect on the rectification ratio is also maximized at an intermediate surface charge density for both negatively and positively charged nanopores. 5. Concluding Remarks A verified continuum model, composed of the coupled Poisson-Nernst-Planck equations and Navier-Stokes equations, is implemented to study the ionic current rectification (ICR) in a conical nanopore. The predicted diode-like I-V curves are in qualitative agreement with the existing experiment. Typically, the preferential current direction of a negatively charged nanopore directs from the tip toward the base due to the ion depletion under a negative voltage and ion enrichment under a positive voltage. However, the preferential current direction reverses when κRt decreases to a certain value, in which the nanopore is dominated by the cations. Furthermore, the preferential current direction of a nanopore highly depends on the charge polarity of the nanopore, which makes it possible to tune ICR by altering the charge polarity of the nanopore. ICR is maximized at an intermediate κRt and surface charge of the nanopore. The EOF effect on the current rectification ratio is negligible under low applied voltage and surface charge of the nanopore. It becomes significant, however, when the applied voltages are relatively high. In addition, the EOF effect on the current rectification ratio attains the maximum at an intermediate κRt and the surface charge density of the nanopore. Acknowledgment. This work is supported by the World Class University Grant No. R32-2008-000-20082-0 of the Ministry of Education, Science and Technology of Korea. References and Notes (1) Baker, L. A.; Bird, S. P. Nat. Nanotechnol. 2008, 3, 73–74. (2) Li, J. L.; Gershow, M.; Stein, D.; Brandin, E.; Golovchenko, J. A. Nat. Mater. 2003, 2, 611–615. (3) Meller, A.; Nivon, L.; Branton, D. Phys. ReV. Lett. 2001, 86, 3435– 3438. (4) Saleh, O. A.; Sohn, L. L. Nano Lett. 2003, 3, 37–38. (5) Siwy, Z.; Fulinski, A. Phys. ReV. Lett. 2002, 89, 198103. (6) Storm, A. J.; Storm, C.; Chen, J. H.; Zandbergen, H.; Joanny, J. F.; Dekker, C. Nano Lett. 2005, 5, 1193–1197. (7) Zhang, B.; Wood, M.; Lee, H. Anal. Chem. 2009, 81, 5541–5548. (8) Korchev, Y. E.; Bashford, C. L.; Alder, G. M.; Apel, P. Y.; Edmonds, D. T.; Lev, A. A.; Nandi, K.; Zima, A. V.; Pasternak, C. A. FASEB J. 1997, 11, 600–608. (9) Lev, A. A.; Korchev, Y. E.; Rostovtseva, T. K.; Bashford, C. L.; Edmonds, D. T.; Pasternak, C. A. Proc. R. Soc. London, Ser. B 1993, 252, 187–192. (10) Rostovtseva, T. K.; Bashford, C. L.; Lev, A. A.; Pasternak, C. A. J. Membr. Biol. 1994, 141, 83–90. (11) Sexton, L. T.; Horne, L. P.; Martin, C. R. Mol. BioSyst. 2007, 3, 667–685. (12) Martin, C. R.; Siwy, Z. S. Science 2007, 317, 331–332. (13) Howorka, S.; Siwy, Z. Chem. Soc. ReV. 2009, 38, 2360–2384. (14) Baker, L. A.; Choi, Y. S.; Martin, C. R. Curr. Nanosci. 2006, 2, 243–255. (15) Kim, Y. R.; Min, J.; Lee, I. H.; Kim, S.; Kim, A. G.; Kim, K.; Namkoong, K.; Ko, C. Biosens. Bioelectron. 2007, 22, 2926–2931. (16) Rhee, M.; Burns, M. A. Trends Biotechnol. 2006, 24, 580–586. (17) Healy, K.; Schiedt, B.; Morrison, A. P. Nanomedicine 2007, 2, 875– 897. (18) Cervera, J.; Alcaraz, A.; Schiedt, B.; Neumann, R.; Ramirez, P. J. Phys. Chem. C 2007, 111, 12265–12273.

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