Ionic Current Rectification in a pH-Tunable Polyelectrolyte Brushes

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Ionic Current Rectification in a pH-tunable Polyelectrolyte Brushes Functionalized Conical Nanopore: Effect of Salt Gradient Jeng-Yang Lin, Chih Yuan Lin, Jyh-Ping Hsu, and Shiojenn Tseng Anal. Chem., Just Accepted Manuscript • DOI: 10.1021/acs.analchem.5b03074 • Publication Date (Web): 07 Dec 2015 Downloaded from http://pubs.acs.org on December 10, 2015

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Analytical Chemistry

Ionic Current Rectification in a pH-tunable Polyelectrolyte Brushes Functionalized Conical Nanopore: Effect of Salt Gradient Jeng-Yang Lin,1 Chih-Yuan Lin,1 Jyh-Ping Hsu1,* Shiojenn Tseng2,* 1

Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617

2

Department of Mathematics, Tamkang University, Tamsui, Taipei, Taiwan 25137, e-mail:

[email protected]

Tel: 886-2-23637448, Fax: 886-2-23623040, e-mail: [email protected]

1

ACS Paragon Plus Environment


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ABSTRACT

The behavior of ionic current rectification (ICR) in a conical nanopore with its surface modified by pH-tunable polyelectrolyte (PE) brushes connecting two large reservoirs subject to an applied electric field and a salt gradient is investigated. Parameters including the solution pH, types of ionic species, strength of applied salt gradient, and applied potential bias are examined for their influences on the ionic current and rectification factor, and the mechanisms involved investigated comprehensively. The ICR behavior depends highly on the charged conditions of the PE layer, the level of pH, the geometry of nanopore, and the thickness of double layer. In particular, the distribution of ionic species and the local electric field near the nanopore openings play the key role, yielding profound and interesting results that are informative to device design as well as experimental data interpretation.

INTRODUCTION The ion channels1,2 in living organisms are pore-forming membrane proteins capable of selecting specific ionic species to across cell membrane. These channels can be simulated by synthetic conical nanopores3-5 having functionalized surfaces, thereby providing a convenient model to study the mechanisms involved in ion transport in living organisms. Xia et al.,6 for

2


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example, use synthetic conical nanopores immobilized with DNA motors to investigate the gating mechanism of biological ion channels. Since these motors become folding or unfolding depending upon the level of pH, the gating behavior

5,7-9

can be observed by

measuring the associated ionic current. Recently, surface modified synthetic conical nanopores have also drawn the attention of both experimentalists and theoreticians due to their ion selectivity10,11 and ion current rectification (ICR)12-14 behaviors. ICR, a nonlinear current-voltage behavior resulting from switching the applied electrical potential bias across the two ends of a nanochannel, is one of the most important features in a nanofluidic system14,15. This specific behavior can be explained by the enrichment and depletion of ions arising from asymmetric transport of cations and anions through a nanopore due to electric double layer (EDL) overlapping.16,17 ICR is found to be influenced significantly by the ion concentration polarization (ICP)13,18 and the charge density on the surface of a nanopore.19,20 Wang et al.21 summarized theoretical approaches for the ICR phenomenon in conical nanopores based on Poisson-Nernst-Plank (PNP) equations.22 Ai et al.23 considered the influence of electroosmotic flow on the ICR phenomenon by solving simultaneously PNP equations and Navier-Stokes equations (PNP-NS). Applying a PNP-NS model, Zeng et al.,24 analyzed theoretically the ICR phenomenon in a conical nanopore functionalized with pH-tunable polyelectrolyte (PE) brushes.25,26 Experimentally, conical 3



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nanopores with their surface modified by proteins, PEs, and antibodies are widely applied to biosensing devices.4,27-29 For instance, Vlassiouk et al.27 adopted a conical nanopore modified with monoclonal antibody (F26G3) to detect the polypeptide ( γ DPGA) on the bacterial pathogen Bacillus. anthracis. They found that the binding of polypeptide with antibody changes the charge density of nanopore surface, yielding different rectification factor (RF), defined as the ratio of ion currents at the same magnitude of applied electrical potential bias with opposite polarities. The degree of ICR can be measured by RF. Gao et al.

29

illustrated

that ultrasensitive detection of Hg2+ can be achieved through measuring the RF in a functionalized glass nanopore. The lipid bilayer of living cell membrane separates two conducting solutions of different concentration. The concentration gradient across the interior and the exterior of the membrane is significant to the transmembrane potential1 and ionic current. It also controls the function of ion channels and rectifies the transport ionic species through cell membrane.12,30 Cao et al.31 investigated both experimentally and theoretically the ICR behavior in a conical nanopore having fixed surface charge for the case where a salt gradient is applied. They showed that in this case the rectification factor can be inverted by varying the direction of the salt gradient. A PNP model was applied in their theoretical analysis for the case where only two types of ions are present in the liquid phase. Nanopore-based 4


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techniques have also emerged as a powerful tool for sequencing/detecting biomolecules whose properties (e.g., size and structure) can be characterized by their translocation events.32-39 For example, by using biological (α-hemolysin) nanopores, Wang et al.32 were able to discriminate both the aggregation state and the conformation of peptide. In the present study, we study theoretically the ion transport in a conical nanopore modified with pH-tunable PE brushes under the condition where a salt gradient17,40,41 is applied. This extends previous studies in several aspects. Firstly, the charge-regulated25,26 nature of the nanopore surface is considered, where the surface charge density varies with surrounding conditions such as pH and ionic concentrations. Secondly, our model takes account of the presence of multiple ionic species. This is much more realistic than previous models, where only binary ionic species coming from the background salt are considered, since the presence of H+ (OH-) can be significant when pH is low (high). A complete PNP-NS model is adopted, which is capable of modeling the influence of electrokinetic flow(EKF).42 The behavior of the ionic current-voltage curve and the associated ICR phenomenon under various conditions are examined by varying the strength as well as the direction of the applied salt gradient and electric potential bias, and the solution pH. The distributions of ionic species and the local electric field, the thickness of EDL, the EKF velocity, and the charge density of the PE layer are discussed for their influences on the ICR 5



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behavior of the nanopore. This provides a complete picture for the phenomenon under consideration and the associated mechanisms. The results gathered also provide valuable information not only for interpreting the ion transport mechanisms in biological channels but also for designing synthetic nanopore devices. THEORY As illustrated schematically in Figure 1, we consider the transport of ions in a conical nanopore of axial length L N , tip radius RT and base radius RB embedded in a membrane driven by an applied potential bias V. The nanopore surface is functionalized by pH-tunable polyelectrolyte (PE) brushes of uniform thickness RS . The nanopore connects two identical, much larger reservoirs of the axial length LR and radius RR . The tip (base) side reservoir is filled with an aqueous salt solution of bulk concentration Ctip (Cbase), establishing a concentration gradient ∇C . E is established by applying an electric potential bias on the electrodes placed in the reservoirs sufficiently far from the nanopore. The ions in the system under consideration are driven by E and well as ∇C from one reservoir through the nanopore to the other reservoir, yielding an ionic current. The PE layer of the nanopore has both acidic functional groups AH and basic functional groups B with associated dissociation/association K A = [A − ][H + ] /[ AH]

reactions and

AH ⇔ A- + H +

K B = [B][H + ]/[BH + ]

and

BH + ⇔ B + H +

.

Let

be the corresponding equilibrium

6


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constant of these reactions, where a symbol with square brackets denotes its molar concentration. If NA and NB are the total concentrations of the acidic and basic functional groups, respectively, then the definitions of KA and KB yield [AH] = [ B] =

K B [BH + ] [H + ]

.

N B = [BH + ] + [B]=(1 +

N A = [AH] + [A - ] = (

Therefore,

[A − ][H + ] and KA

[H + ] + 1)[A - ] KA

and

NA K A N B[H + ] KB + + [A ] = [BH ] = , so , yielding and )[BH ] [H + ] K A + [H + ] K B + [H + ]

that the charge density of the PE layer is ρ PE = [BH+ ] − [A - ] =

N B[H + ] NA K A − . The + K B + [H ] K A + [ H + ]

PE layer is capable of simulating, for example, proteins, biomolecules, and synthetic polymers. For biological PEs, NA and NB range typically from 0.01 to 2.635 mol/L.43,44 If we let F be Faraday constant, then

ρ PE

N B [H + ] N A KA = 1000 F ([BH ] − [A ]) = 1000 F ( − ) + K B + [H ] K A + [H + ] +

−

(1)

This expression suggests that the charged conditions of the PE layer are pH dependent. Suppose that the background salt is KCl, and the solution pH adjusted by HCl and KOH. Then four kinds of ionic species need be considered: K+, Cl-, H+, and OH-.

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Figure 1. Transport of ions in a conical nanopore of axial length L N , tip radius RT and base radius RB embedded in a membrane connecting two identical, much larger reservoirs of axial length LR and radius RR driven by an applied potential bias V. The nanopore surface is functionalized by pH-tunable polyelectrolyte (PE) brushes of uniform thickness RS . The tip (base) side reservoir is filled with an aqueous salt solution of bulk concentration Ctip (Cbase). The liquid phase contains K+, Cl-, H+, and OH-. The system under consideration mimics biological ionic channels, and the experimental measurement of the ionic current-voltage data can be used to gain information on these channels.6,27

Governing equations. Considering the nature of the problem under consideration, we choose to work on the cylindrical coordinates with the origin at the center of the nanopore tip, and since it is axial symmetry, only the radial and the axial domain (r, z) need be considered. We assume that the system is at a quasi-steady state and the liquid flow is in the creeping flow regime. For simplicity, the PE deformation due to fluid flow is neglected. In addition, the liquid viscosity η , its permittivity ε , and the diffusivity of the jth ionic species Dj in the 8


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PE layer are the same as those outside it. If we let p, u, and φ be the hydrodynamic pressure, the fluid velocity, and the electric potential, respectively, the phenomenon under consideration can be described by21-24 −∇p + η∇ 2u − ρ∇ϕ − hγ u = 0

(2)

∇ ⋅u = 0

(3)

∇ 2φ = −

ρ + hρ PE ε

(4)

∇ ⋅ N j = ∇ ⋅ (u c j − D j ∇ c j − z j

Dj RT

Fc j ∇φ ) = 0 , j = 1,...,4

(5)

Here, h is 0 (1) for the region outside (inside) the PE layer, and γ its hydrodynamic 4

frictional coefficient. ρ = F ∑ z j c j is the space charge density of mobile ions with z j and j =1

cj being the valence and the concentration of ionic species j, respectively. N j is the flux ionic species j.

Boundary conditions. To specify the boundary condition associated with eqs 1-5, we assume the following. (i) The reservoir wall is free of charge, n ⋅ ∇φ = 0 , impenetrable to ions, n ⋅ N j = 0 , and slip to fluid, where n is the unit outer normal vector of a surface, so is the membrane surface. (ii) The electric potential, the electric field, the ionic concentration, the fluid velocity, and the shear stress are all continuous on the PE layer-liquid interface. (iii) All the dependent variables are symmetric about the nanopore axis. (iv) A normal flow without external pressure gradient (i.e., p=0) is specified at both reservoir ends. The electric 9



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potential at the nanopore tip (base) is 0 (V), and the corresponding ionic concentration cj there is Ctip,j0 (Cbase,j0). The solutions in the two reservoirs have the same pH, but different salt concentration, Ctip and Cbase. Let Cm,j0 be the concentration of ionic species j at position m, m=nanopore tip or base end, and j=1,2,3,4, representing K+, Cl-, H+, and OH-, respectively. If we let Cm ,10 = Cm (mM), the electroneutrality requires Cm , 20 = Cm + 10− ( pH+3) − 10− (14−pH)+3 ,

Cm ,30 = 10− ( pH+3) , and Cm , 40 = 10− (14−pH)+3 for pH ≤ 7 ; Cm ,10 = Cm − 10−( pH+3) + 10− (14−pH)+3 , Cm , 20 = Cm , Cm ,30 = 10−( pH+3) , and Cm ,40 = 10−(14−pH)+3 , for pH ≥ 7 . Solving eqs 1-5 subject to the conditions assumed and substituting the Nj obtained into the expression below for the current I:

 4  I = ∫ F  ∑ z j N j  ⋅ n dA A  j =1 

(6)

where, A denotes the anode or the cathode surface. Solution procedure. Equations 1-5 are a set of coupled, highly nonlinear equations. Solving them analytically for the general case is almost impossible. Here, they are solved numerically by COMSOL (version 4.3a, www.comsol.com), a finite-element method based commercial software operating in a high performance cluster. This approach was adopted to solving the ion current rectification problem for a charge-regulated cylindrical nanopore (Yeh et al.),45 and for a solid-state nanopore (Yeh et al.).40 Our model is fitted to the experimental data of Ali et al.,25 the results are presented in Supporting Information. Mesh refinement test 10


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has been performed to ensure the results gathered are convergent and mesh-independent. Because the typical ionic size (0.03~0.2 nm) is small compared with the nanopore size, the ionic volume effect is neglected, for simplicity. RESULTS AND DISCUSSION Numerical simulation is conducted to examine the ICR behavior of the system under consideration. For illustration, we assume LN=500 nm, RT=10 nm, RB=65 nm, LR=RR=200 nm, and RS=5 nm. The PE is lysine with one α -carboxylic groups (pKA=2.2), one α -amino groups (pKB=9),25 and NA=NB=300 mM. The softness parameter of the PE layer, λ −1 = (η / γ )1/ 2 , is chosen as 1 nm. At T=298 K, ε = 6.95 × 10−10 Fm-1, η = 1 × 10 −3 Pa ⋅ s , and the diffusivities of K + , Cl − , H + , and OH − are D1 = 1.96 × 10 −9 , D2 = 2.03 × 10 −9 ,

D3 = 9.38 × 10 −9 , and D4 = 5.29 × 10 −9

m 2 s -1 , respectively; R = 8.314 JK-1mol-1. The

voltage bias applied across the nanopore ranges from -1 V to 1 V. Two types of concentration

gradient

are

applied:

(Cbase | Ctip)=(1

mM | 1,10,100

mM)

(Cbase | Ctip)=(1,10,100 mM | 1 mM). For convenience, we define the following parameters:

C ref = C tip C base Ca , j = ∫ c j dΛ Λ

E a , z = ∫ E Z dΛ Λ

(7)

∫ dΛ

(8)

Λ

∫ dΛ

(9)

Λ

11


and


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RF = I (−V )

I (V )

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(10)

C ref describes the directions of the applied concentration gradient. If C ref ≤ 1 , the concentration gradient points from the nanopore base to its tip, and if C ref ≥ 1 that direction is inverted. Ca. j and E a , z are the axial variations of the cross sectional averaged concentration of the jth ionic species and the z component of the electric field, respectively. Λ is the cross sectional area of the nanopore, which is z-dependent. The rectification

characteristics of the nanopore is measured by the rectification factor (RF) defined in eq 10. The thickness of EDL, measured by Debye length, depends mainly on the bulk salt concentration. For the case of an aqueous KCl solution at room temperature, the Debye length is ca. 9.6, 3.1, and 1 nm when the bulk salt concentrations are 1, 10, and 100 mM, respectively. The simulated I-V curves for both C ref ≤ 1 and C ref ≥ 1 at various levels of pH are presented in Figure 2. The results for pH 10 and 3 suggest that the nanopore behaves like a diode,15 where ICR phenomenon is observed. However, it is more like an Ohmic resister1 at pH 6. In addition, I is seen to increase with increasing ∇C , in general, except at C ref = 10 and pH 10. These results will be discussed later in detail.

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Figure 2. I-V curves at pH 10, (a), pH 6, (b), and pH 3, (c). Dash-dotted curves: without concentration gradient; dash-dotted-dotted curves: 10-fold concentration gradient; solid curves: 100-fold concentration gradient. Column (I): C ref ≤ 1 ; (II): C ref ≥ 1 . pH 10. Because pH is 10 in Figure 2(a), K+ and Cl-, are the major ionic species and, therefore, the behaviors of the I-V curves are influenced mainly by these ions, as illustrated in Figure 3. Since the IEP of the PE layer is 5.6, it is negatively charged at pH 10, and therefore, 13



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C a , K + > C a , Cl − inside the nanopore. If concentration gradient is not applied ( C ref = 1 ), I (V = −1V) > I (V = 1V) , so that RF ≅ 1.24. This can be explained by the results shown in Figure 3(a) and 3(b), where the curves of C a , K + and C a , Cl − indicate an ionic enrichment at V=-1 V and an ionic depletion at V=1 V in the nanopore, verifying the presence of ICR.24 In the case of Figure 3(b) at Cref = 1, counterions (K+) are attracted by the negatively charged PE layer to accumulate in the nanopore, and their distribution depends upon the nanopore shape. On the other hand, coions (Cl-) show a depletion at V=1 V, having a local minimum near z=-550 nm. This is because the nanopore base end has a positive potential (1V), driving the Cl- in the nanopore towards its base end (-z direction) and out of it. The negatively charged PE layer also hinders the Cl- near the nanopore tip end from entering the nanopore. Therefore, the concentration of Cl- inside the nanopore is lower than that in Figure 3(a), where V=-1 V. It is worth noting in Figure 3(a) that an anion-rich ( C a , Cl − > C a , K + ) occurs near z=65 nm. This is because the EDL overlapping is significant near the nanopore tip end, so that a more amount of anions are expelled from the nanopore into the reservoir. In a study of the behavior of a charged cylindrical nanopore subject to an applied salt gradient, Yeh et al.40 also observed the similar behavior. The anion-rich phenomenon in their study results from the applied salt gradient. However, this phenomenon can be observed in our case when no salt gradient is applied (Cbase | Ctip=1 mM or Cref = 1 in Figure 3(a)), suggesting that 14


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it results from the geometry of the nanopore. In Figure 2(a)-(I) ( C ref ≤ 1 ), the I-V curves show an unusual rectification inversion as Cbase increases. As illustrated in Figure 3(a)-(I), this is because the distributions of C a , K + and C a , Cl − at C ref ≤ 1 are influenced by the applied concentration gradient as well as the nanopore shape. For the case of Figure 3(b)-(I) ( C ref ≤ 1 ), because the concentrations of K+ and Cl- near the nanopore base end increase with increasing strength of ∇C , raising appreciably the originally depleted Cl- at V=1 V in the nanopore. Therefore, the minimum value of C a , Cl − increases with increasing strength of ∇C , and shifts towards the tip end of the nanopore. The increase in the ionic current at V=1 V due to the applied concentration gradient ( C ref ≤ 1 ) makes the trend of ICR inversed : a RF larger than unity becomes smaller than unity (RF ≅ 0.56).31 In general, the ICR behavior can be explained by the ionic concentration (or conductivity). However, as shown in Figure S1(a), most of the total ionic concentrations inside the nanopore at V=-1 V are higher than those at V=1 V, implying that explanation for an inverse ICR behavior30 needs modification. Here, we propose that, in addition to the total ionic concentration, the local electric field might also play a role. As shown in Figure S2(a)-(I) and S2(b)-(I) of Supporting Information, if C ref ≤ 1 , the magnitude of the local electric field in the nanopore at V=1 V for a sufficiently large ∇C is more than ten times stronger than that at V=-1 V, so that the ionic current at V=1 V is 15



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larger than that at V=-1 V. Note that the total ionic concentration in the nanopore at V=-1 V is only ca. two times higher than that at V=1 V. Therefore, the inverse ICR phenomenon can be observed only if the applied concentration gradient is sufficiently high.46 The flow field, which is neglected in PNP model, near the nanopore tip are presented in Figures S3 and S4, which shows that EKF velocity also contribute to the phenomenon of RF inversion.

Figure 3. Axial variation in the cross sectional averaged ionic concentration at pH 10 for V=-1 V, (a), and V=1 V, (b). Solid curves: K+; dash-dotted-dotted curves: Cl-. Column (I): C ref ≤ 1 ; (II): C ref ≥ 1 . In the case of Figure 2(a)-(II), where pH=10 and C ref ≥ 1 , if V=-1 V, I increases with increasing Ctip. This can be explained by Figure 3(a)-(II) which shows the distributions of K+ 16


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and Cl-. Unexpectedly, if Ctip is raised from 1 mM to 10 mM, the C a , K + in the nanopore decreases because the increase in the degree of EKF at Ctip=10 mM might also play a role, as can be inferred from Figure S2(a)-(II), which shows that the nanopore tip end has a stronger local electric field at Ctip=10 mM. Note that due to the increase in ∇C the C a , Cl − in the nanopore at Ctip=10 mM is higher than that Ctip=1 mM, so that the total ionic concentration 4

( ∑ C a , j ) in the nanopore at Ctip=10 mM is higher than that at Ctip=1 mM, yielding a larger j =1

I . For both the cases of Ctip=1 mM and 10 mM, the distribution of ionic concentration near the nanopore tip end is influenced significantly by its shape, and the difference between

C a , K + and C a , Cl− is more appreciable than that for the case where Ctip=100 mM. As Ctip is raised to 100 mM, both C a , K + and C a , Cl− increase significantly, yielding an appreciable increase in I . Figure 3(b)-(II) reveals that for the case of V=1 V, if Ctip is raised to 10 mM, the C a , K + in the nanopore decreases accordingly. Therefore, although C a , Cl− increases due 4

to ∇C , the total ionic concentration ( ∑ C a , j ) in the nanopore decreases. This explains the j =1

decrease of I in Figure 2(a)-(II) at V=1 V, and the behavior of I as Ctip varies seen in Figure S6(a). In addition, the decrease in the total ionic concentration shown in Figure S6(b) can be explained by Figure 4. As Ctip increases from 1 mM to 10 mM, the local electric field is enhanced and the increase in C a , K + lowers the H+ concentration. Therefore, the degree of dissociation of the PE negatively functional group ( α -carboxylic groups) increases and 17



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ρ PE increases accordingly,26 yielding a faster EKF velocity near nanopore tip. The raise of Ctip from 1 mM to 10 mM also makes the EDL near the nanopore tip thinner and the flow of the ionic species outside it through EKF out of the nanopore lowers the total amount of the ionic species in the nanopore, I decreases accordingly. In the present case, since V=1 V and the PE layer is negatively charged, the EKF is towards the nanopore tip end, as shown in Figure 4. If Ctip is further raised to 100 mM, both the ρPE and the local electric field near the nanopore tip end become smaller, as seen in Figure S5(b)-(II) and Figure S2(b)-(II), respectively, yielding a slower fluid velocity near the nanopore tip. In this case, both C a , K + and C a , Cl− are influenced mainly by ∇C and, therefore, I increases with increasing Ctip. Figure 2(a)-(II) reveals that I (V = −1 V) > I (V = 1 V) , and the value of RF increases with increasing ∇C , which is consistent with the experimental observations.31 ρ PE , the cross sectional averaged EKF velocity, and the total ionic concentration profile are presented in Figures S5 and S6 in supporting information.

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Figure 4. The flow field near nanopore tip for Ctip=1 mM, (a), 10 mM, (b), and 100 mM, (c), at Cbase=1 mM, pH 10, and V=1 V. Figure 5 illustrates the axial variations of the concentrations of H+ and OH- at pH 10; the concentrations of these ions far away from the nanopore are 10−7 and 10 −1 mM, respectively. The nature of the PE layer considered implies that its ρ PE depends upon the concentration of H+. As Ca , H + decreases, the degree of the dissociation of its acidic functional groups increases, so that ρPE increases. Figure 5(a) and (b) indicates that the behaviors of Ca , H + and C a , OH − at C ref = 1 are similar to those of C a , K + and C a , Cl− , respectively. The counterions H+ accumulate in the nanopore and the coions OH- are expelled out. In addition, at V=-1 V, both C a , OH − and C a , Cl− show a local minimum near z=-550 nm.

19



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Figure 5. Axial variation in the cross sectional averaged ionic concentration at pH 10 for V=-1 V, (a), and V=1 V, (b). Solid curves (left scale): H+ ; dash-dotted-dotted curves (right scale): OH-. Column (I): C ref ≤ 1 ; (II): C ref ≥ 1 . In Figure 5(a)-(I) ( C ref ≤ 1 ) because V=-1 V, the H+ in the nanopore is driven towards the

-z direction. In addition, since C ref ≤ 1 , K+ is driven by ∇C (z direction) to enter the nanopore base, so that H+ is trapped and accumulates in the nanopore. In this case, Ca , H + increases with increasing ∇C , thereby lowering the ρPE in the nanopore, as seen in Figures S2. In addition, since V=-1 V the OH- in the nanopore is driven towards the z direction, and because Cl- is driven by ∇C (z direction) to enter the nanopore base, repelling OH- out of the nanopore, so that the C a , OH − in the nanopore decreases with 20


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increasing ∇C . In Figure 5(b)-(I), where V=1 V, the H+ in the nanopore is driven towards the z direction. In addition, K+ is driven by ∇C to enter the nanopore base, pushing the H+ in the nanopore out of it. Therefore, the Ca , H + ( ρPE ) in the nanopore decreases (increases) with increasing ∇C . However, H+ is also attracted by the negatively charged PE layer and, together with the influence of nanopore shape, it accumulates near the nanopore tip end (z=0 nm). In contrast, if V=1 V, the OH- in the nanopore is driven towards the -z direction and Clis driven by ∇C to enter the nanopore base, thereby repelling OH- back into the nanopore. In addition, the OH- in the tip end reservoir is driven by the applied electric field towards the

-z direction, but repelled by the negatively charged PE, so that it accumulates near the exterior of the nanopore tip end. In the case of Figure 5(a)-(II) ( C ref ≥ 1 ) because V=-1 V, the H+ in the nanopore is driven towards the -z direction, and since C ref ≥ 1 , K+ is driven by ∇C (-z direction) into the nanopore tip, driving H+ towards the -z direction and out of the nanopore, so that Ca , H + decreases and ρPE increases with increasing ∇C . On the other hand, since V=-1 V, the OH- in the nanopore is driven towards the z direction, and Cl- is driven by ∇C into the nanopore tip, so that OH- is trapped and accumulates in the nanopore; the larger the ∇C the higher the C a , OH − . The apparent increase of I at V=-1 V and Ctip=100 mM seen in Figure 2(a)-(II) attributed partially to the appreciable amount of OH- accumulated in the 21



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nanopore. Because V=1 V in Figure 5 (b)-(II) ( C ref ≥ 1 ), H+ is driven towards the z direction. At the same time, K+ is driven by ∇C (-z direction) into the nanopore tip, trapping H+ in the nanopore. As Ctip increases from 1 mM to 10 mM, the Ca , H + in the nanopore decreases, which can be explained by the same reasoning as that employed for the behavior of C a , K + seen in Figure 3(b)-(II). If Ctip is further increased to 100 mM, because the EKF is dominated by ∇C , the H+ inside the nanopore tends to accumulate near its tip. On the other hand, since V=1 V, OH- is driven towards the -z direction, and Cl- by ∇C (-z direction) into the nanopore tip, repelling OH- out of it, so that the C a , OH − in the nanopore decreases with increasing ∇C . pH 6. The axial variations of C a , K + and C a , Cl− at pH 6 are presented in Figure 6. In this case, since pH is only slightly higher than IEP (=5.6), the PE layer of the nanopore is weakly negatively charged. Therefore, the ICR behavior at C ref = 1 is similar to that at pH 10. However, because ρPE is small, the ion accumulation (depletion) at V=-1 V (V=1 V) in Figure 6(a) and (b) is unimportant, so that the I in Figure 2(b) follows roughly Ohm’s law and, therefore, RF ≅ 1 .

22


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Figure 6. Axial variation in the cross sectional averaged ionic concentration at pH 6 for V=-1 V, (a), and V=1 V, (b). Solid curves: K+; dash-dotted-dotted curves: Cl-. Column (I): C ref ≤ 1 ; (II): C ref ≥ 1 . In Figure 6(a)-(I) and 6(b)-(I), where C ref ≤ 1 , C a , K + ≅ C a , Cl − in the range -550 nm I (V = 1V) with RF ≅ 1.2 , as seen in Figure 2(a)-(I). If C ref ≥ 1 , the C a , K + at V=-1 V (Figure 6(a)-(II) is also about the same as that at V=1 V (Figure 6(b)-(II)), yielding an Ohm’s law-like 1 behavior of I ( RF ≅ 1 ) seen in Figure 2(b)-(II). These behaviors result 23



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from the complicated nature of ρ PE at pH 6, and will be discussed in detailed later. A comparison between Figure 6(I) and 6(II) reveals that if C ref ≤ 1 , the C a , K + and C a , Cl− near the nanopore tip vary drastically. This is because ions are driven by ∇C to enter the base end of the nanopore, experiencing a decrease in the cross section as they approach its tip end. In contrast, if C ref ≥ 1 , ions are driven by ∇C to enter the tip end of the nanopore, experiencing an increase in the cross section as they approach its base end, so that both

C a , K + and C a , Cl− decrease with increasing z. As illustrated in Figures 7 and S4 the degree of dissociation of the functional groups of the nanopore PE layer at pH 6 varies with V and ∇C , yielding nonuniform charged conditions. Since ρ PE is pH-tunable, Ca , H + = 10 − IEP+3 (mM) is labeled in Figure 7, for comparison. Figures 7 and S4 reveal that if Ca , H + > 10 − IEP+3 , the PE layer is positively charged (equivalent to pHpI). Figure 7(a) and 7(b) reveals that if C ref = 1 , the distributions of

Ca , H + and C a , OH − are similar to those of C a , K + and C a , Cl− , respectively, in Figure 6(a) and 6(b)

24


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Figure 7. Axial variation in the cross sectional averaged ionic concentration at pH 6 for V=-1 V, (a), and V=1 V, (b). Solid curves (left scale): H+; dash-dotted-dotted curves (right scale): OH-. Column (I): C ref ≤ 1 ; (II): C ref ≥ 1 . In Figure 7(a)-(I), the H+ (OH-) in nanopore is driven by V=-1 V towards the –z (z) direction. However, the K+ (Cl-) in the base end reservoir is driven by ∇C (z direction) to enter the nanopore base end, thereby pushing H+ (OH-) back into (out from) the nanopore, thereby accumulating (depleting) in the nanopore. At Cbase=10 mM, H+ accumulates near the nanopore center and Ca , H + > 10 − IEP+3 mM, so that Ca , H + < 10 − IEP+3 mM near its tip and base ends and, therefore, the PE layer near nanopore center is positively charged, but that near its two ends is negatively charged, yielding a temporarily bipolar-like27, 47 charged conditions, as 25



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Page 26 of 44

shown in Figure S7(a)-(I). Note that this occurs because the PE layer is pH-tunable and ∇C

∇C

is present. If Cbase=100 mM, because

is large, a considerable amount of H+

accumulates in the nanopore, so that its PE layer is positively charged, and the distribution of

ρ PE in Figure S7 is similar to that of Ca , H . Therefore, in Figure 6(a)-(I), C a , Cl > C a , K +

−

+

near the nanopore tip end results from the accumulation of H+ in the nanopore due to ∇C , yielding a positively charged PE layer. Figure 7(b)-(I) shows that at C ref ≤ 1 , the H+ (OH-) in the nanopore is driven by V=1 V towards the z (-z) direction, and K+ (Cl-) is driven by ∇C (z direction) to enter the nanopore base end, pushing H+ (OH-) out from (back into) the nanopore, so that C a , H + ( Ca , OH − ) in the nanopore decreases (increases) with increasing

∇C , and as ∇C increases, the position at which the minimum of Ca , H + ( Ca , OH − ) occurs shifts to the tip end. Since Ca , H + < 10 − IEP+3 mM in the nanopore, its PE layer is negatively charged, and ρ PE increases with increasing ∇C . Because the Ca , H + near the nanopore tip decreases appreciably and ρ PE increases accordingly, the C a , K + − C a , Cl −

in Figure

6(b)-(I) is larger than that in Figure 6(a)-(I). In the case of Figure 7(a)-(II), the H+ (OH-) in the nanopore is driven by V=-1 V towards the -z (z) direction, and K+ (Cl-) driven by ∇C (-z direction) to enter the nanopore tip end, pushing H+ (OH-) out of (back into) it, so that the Ca , H + ( C a , OH − ) in the nanopore decreases (increases) with increasing ∇C . Therefore, the nanopore PE layer is negatively charged, 26


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and ρ PE increases with increasing ∇C . In this case, the OH- in the nanopore is driven by V=-1 V towards the z direction, and Cl- by ∇C (-z direction) to enter the nanopore tip end, pushing OH- back into the nanopore, so that the C a , OH − in the nanopore increases with increasing ∇C . In Figure 7(b)-(II), the H+ in the nanopore is driven by V=1 V towards the z (-z) direction, and K+ (Cl-) driven by ∇C (-z direction) to enter the nanopore tip end, pushing H+ (OH-) back into the nanopore, so that the Ca , H + ( C a , OH − ) in the nanopore increases (decreases) with increasing ∇C . If Ctip=10 mM, because Ca , H + > 10 − IEP+3 mM ( Ca , H + < 10 − IEP+3 mM) in the region near the nanopore tip (base) end, the PE layer in that region is positively (negatively) charged. If Ctip=100 mM, the Ca , H + in the nanopore exceeds 10 − IEP+3 mM, the entire PE layer is negatively charged. In this case, the OH- in the nanopore is driven by V=1 V towards the -z direction, and Cl- by ∇C (-z direction) to enter the tip end, pushing OH- back into the nanopore, so that the C a , OH − in the nanopore decreases with increasing ∇C . We conclude from Figure 7 that if a concentration gradient ∇C is applied and its direction the same as that of the applied electric field (Figure 7(b)-(I) and 7(a)-(II)), H+ (OH-) depletes (accumulates) in the nanopore. In this case, the ρ PE of the nanopore PE layer increases with increasing ∇C . On the other hand, if the direction of ∇C is opposite to that of the applied electric field, H+ (OH-) accumulates (depletes) in the nanopore (Figure 27



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7(a)-(I) and 7(b)-(II)). In this case, an increase in ∇C turns a negatively charged PE layer into a positively charged one, or even a bipolar one. At pH 6, the charged conditions of the PE layer of a nanopore can be tuned by an applying simultaneously an electric field and ∇C . The variation of ρ PE in this case is presented in Figure S7 of Supporting Information. pH 3. The results for the case of pH 3 are summarized in Figure 8. In this case, since pH C a , K + inside. The result in Figure 2(c) that at C ref = 1 , I (V = 1V) > I (V = −1V) , yielding RF ≅ 0.267 can be explained by Figure 8(a) and 8(b), where both Cl- and K+ deplete at V=-1 V and accumulate at V=1 V, yielding the ICR phenomenon. In Figure 8(a), the Cl- in the nanopore is driven by V=-1 V towards the z direction and out of it, so that C a , Cl− ( z ≅ −125 nm) decreases appreciably, but some Cl- accumulates near its tip end due to its geometry and positively charged PE layer. In contrast, the K+ in the nanopore is driven by V=-1 V towards the -z direction and out of it, so that C a , K + ( z ≅ −125 nm) decreases. The C a , Cl− in Figure 8(b) is influenced mainly by the nanopore geometry: Cl- accumulates near the nanopore tip; the K+ in the nanopore is driven by V=1 V towards the z direction, so that C a , K + has a local minimum near the tip end ( z ≅ −40 nm) . Figure 8(a) and 8(b) reveals that at C ref = 1 , the concentration of counterions Cl- at a point far away from the nanopore is about two times that of coions K+. This is because at pH 3, both H+ and K+ have the same concentration of 1 28


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mM, and the distribution of Ca , H + in Figure 9 at C ref = 1 is almost the same as that of

C a , K + . In this case, the dependence of I on Ca , H + is similar to that on C a , K + , implying that the contribution of H+ to I is comparable to that of K+.

Figure 8. Axial variation in the cross sectional averaged ionic concentration at pH 3 for V=-1V, (a), and V=1 V, (b). Solid curves: K+; dash-dotted-dotted curves: Cl-. Column (I): ∇C ≤ 1 ; (II): ∇C ≥ 1 .

In Figure 8(a)-(I) ( C ref ≤ 1 and V=-1 V), because a higher salt concentration is imposed on the nanopore based end ( ∇C in the z direction), both C a , K + and C a , Cl− in the nanopore increase appreciably, and the points at which their minimum values occur shift to the tip end as ∇C increases. In this case, the K+ in the tip end reservoir is driven by V=-1 V towards 29



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the -z direction, but is repelled by the positively charged PE layer and, therefore, accumulates outside the tip, yielding a large C a , K + there. At V=1 V, Figure 8(b)-(I) reveals that if Cbase=10 mM, due to a decrease in ρ PE , the C a , Cl− near the nanopore tip end (z=-100 nm) decreases accordingly. In contrast, due to the application of ∇C , the C a , K + in the nanopore increases. The net result is that the sum ( C a , K + + C a , Cl− ) in the nanopore at V=1 V is larger than that at V=-1 V. If Cbase is raised to 100 mM, because some of the coions K+ in the nanopore are driven out of it by ∇C and the applied voltage, V=1 V, C a , K + > C a , Cl− immediately outside the nanopore tip end. In Figure 8(a)-(II) and 8(b)-(II) ( C ref ≥ 1 ) because

C a , K + and C a , Cl− depend mainly on the diffusive flux when the applied salt gradient is sufficiently large, the ion accumulation and depletion near the nanopore tip end are unimportant. In Figure 8(b)-(II), K+ is driven by V=1 V towards the z direction and, since ∇C is in the - z direction, it accumulates in the nanopore, so that C a , K + (V=1

V)> C a , K + (V=-1 V). Figure 9 illustrates the axial variations of Ca , H + and C a , OH − at pH 3. In this case,

Ca , H + =1 mM and C a , OH − =10-8 mM at a point far away from the nanopore. The distributions of Ca , H + and C a , OH − in Figure 9(a) and 9(b) at C ref = 1 are similar to those of C a , K + and

C a , Cl− shown in Figure 8(a) and 8(b), respectively, where ions deplete at V=-1 V, and accumulate at V=1 V. The distribution of ρ PE

illustrated in Figure S7 of Supporting

30


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Information is similar to that of Ca , H + .

Figure 9. Axial variation in the cross sectional averaged ionic concentration at pH 3 for V=-1 V, (a), and V=1 V, (b). Solid curves (left scale): H+ ; dash-dotted-dotted curves (right scale): OH-. Column (I): C ref ≤ 1 ; (II): C ref ≥ 1 . In Figure 9(a)-(I) ( C ref ≤ 1 ), the H+ (OH-) in the nanopore is driven by V=-1 V towards the -z (z) direction, and K+ (Cl-) by ∇C to enter the nanopore base end, pushing H+ (OH-) back into (out of) the nanopore, so that H+ accumulates in the nanopore. In this case, the

Ca , H + ( C a , OH − ) in the nanopore increases (decreases) with increasing ∇C . Meanwhile, the H+ in the tip end reservoir is also driven by V=-1 V towards the nanopore tip (-z direction), but is repelled by its positively charged PE layer, so that it accumulates near the exterior of 31



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Page 32 of 44

the nanopore tip, yielding a local minimum in Ca , H + near the interior of the nanopore tip. The larger the ∇C the closer the point at which the local minimum occurs to the nanopore tip. This phenomenon is similar to the accumulation of OH- near the exterior of nanopore tip seen in Figure 5(b)-(I) (pH 10, C ref ≤ 1 ), and results mainly from the nanopore geometry and the direction of the applied potential bias. Because the axial variation of ρ PE is similar to that of Ca , H + (Figure S8(a)-(I)), so is the concentration of counterions, C a , Cl− , near the nanopore tip end (Figure 8(a)-(I)). This also influences the shape of the I–V curve in Figure 2(c)-(I) at Cbase=100 mM, details are presented in Figure S9 of Supporting Information. In Figure 9(b)-(I), the H+ (OH-) in nanopore is driven by V=1 V towards the z (-z) direction, and K+ (Cl-) by ∇C to enter the nanopore base end, pushing H+ (OH-) out of (back into) it, so that the larger the ∇C the smaller (larger) the Ca , H + ( C a , OH − ). Because the I at pH 3 is influenced mainly by C a , K + , C a , Cl− , and Ca , H + , the behavior of RF in Figure 2(c) due to the application of ∇C can be explained by the results shown in Figures 8 and 9. For C ref ≤ 1 and Cbase=10 mM (Figure 2(c)-(I), 8(I), and 9(I)), because C a , K + + C a , Cl − V)> C a , K + + C a , Cl −

(V=1

(V=-1 V) in the nanopore, I (V = 1V) > I (V = −1V) . If Cbase is raised

to 100 mM, since the local electric field near the nanopore tip increases significantly (Figure S11(a)-(i)), I (V = −1 V) > I (V = 1 V) . Therefore, the associated RF is inversed, as in the case of pH 10 at Cref C a , K + + C a , Cl − (V=-1 V) and the Ca , H + in the nanopore increases

drastically

at

V=1

V

due

to

the

accumulation

of

H+

(Figure

S10),

I (V = 1 V ) > I (V = −1 V ) , and the rectification effect is better (RF ≅ 0.2). Furthermore, as H+ accumulates (depletes) in the nanopore, the competition of H+ and K+ makes C a , Cl − − C a , K +

larger (smaller), as seen in Figure 8(a)-(I) and 8(b)-(II) (Figure 8(a)-(II) and 33



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Page 34 of 44

8(b)-(I)). Note that the above-mentioned accumulation of H+ plays a role in ionic current, as will be discussed later. Rectification Factor. In Figure 10, for C ref ≤ 1 , Ctip is fixed at 1 mM, and for C ref ≥ 1 , Cbase is fixed at 1 mM. Note that the larger the deviation of Cref from unity the larger the

∇C , and the closer the Cref to unity the smaller the ∇C . Figure 10(a) indicates that if C ref ≤ 1 , RF shows an inverse behavior31 at Cref ≅ 0.6 and 0.03 for pH 10 and 3, respectively. As stated in the discussion in Sections 3.1 and 3.3, this arises from the application of ∇C . As Cref decreases from these critical values, the larger the ∇C the more significant the degree of dissociation of the PE functional groups, ρPE

increases accordingly, and

therefore, RF deviates from unity. At C ref ≅ 20 and pH 10 (3), the level of Cref makes EDL overlapping insignificant, so that RF is close to unity, yielding a local minimum (maximum). If Cref is increased further, although EDL overlapping is insignificant, the increase in ρPE with increasing ∇C makes RF deviates unity again. On the other hand, RF ≅ 1 at pH 6, because ρ PE is small in this case. Referring to the discussion in Sections 3.2, the behavior of RF can be explained by the enrichment and depletion of both K+ and Cl- , and the charged conditions of the PE layer.

To further elaborate the effect of salt gradient on current rectification, we show the

34


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variation of the magnitude of ionic current, I , with Cref in Figure 10(b) and (c) for two levels of pH. Note that the coions at pH 10 (3) are Cl- and OH- (K+ and H+). As expected, I is dominated by counterions, and the contribution of coions becomes significant only if

∇C is sufficiently high. Note that, however, under appropriate conditions the ionic current can be contributed mainly by coions. In the case of Figure 10(b) at +1 V and C ref ≥ 1 , because the direction of the applied electric field is the same as (opposite to) that of the applied salt gradient for anions (cations), the I contributed by anions is more significant than that by cations, suggesting that the Cl- driven by the applied potential bias as well as the salt gradient dominates. The same reasoning can be applied to the case of pH 3. In Figure 10(c), where +1 V (-1V) and C ref ≤ 1 ( C ref ≥ 1 ), the I contributed by cations is more significant than that by anions because the directions of the applied electric field and the salt gradient are the same for cations (anions), implying that K+ dominates. Note that even if the direction of the applied electric field is opposite to that of the applied salt gradient, at +1 V and C ref ≥ 1 in Figure 10(c), the I contributed by cations is still more significant than that by anions. This is due to a significant accumulation of H+ in nanopore, as shown in Figure 9(b)-(II), so that the ionic current contributed by cations is more significant than that by anions. It should be noted in Figure 10(b) that, at -1 V and C ref ≤ 1 , the EDL overlapping is significant and the ρPE near the nanopore tip is sufficiently high and, therefore, the I 35



contributed by anions is less than that by cations.

(a)10

RF

inversion

1 pH 10 pH 6 pH 3

0.1 0.01

0.1

1

Cref

(b)

10

10

|I| (nA)

100

----

+1 V

1V

1

0.1

pH 10 0.01 0.01

0.1

(c)10

100

|I| (nA)

+1 V

+1 V

0.1 0.01

Cref

0.1

----

|I| (nA)

10

1V

1

1

1

Cref

----

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1V

0.1

pH 3 0.01 0.01

0.1

1

Cref

10

100

Figure 10. Variation of the rectification factor as a function of Cref, (a), and the

36


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corresponding I at the opposite potential bias at pH 10, (b), and pH 3, (c). Curves in (b) and (c): total ionic current; symbols in (b) and (c): ionic current contributed by cations (anions), △ and ○ (▲ and ●), respectively. The I-V curves at several other pH levels are presented in Supporting Information.

CONCLUSION We studied theoretically the transport of ions and the ICR phenomenon in a conical nanopore with its surface covered by a pH-regulated PE layer for the case of salt gradient is applied. In particular, the behavior of the system under consideration at various levels of pH and applied salt concentrations are examined, and explained by the ionic distribution, the volume charge density of the PE layer ρ PE , and the associated EKF. The results obtained can be summarized as following. Firstly, at pH 10 and C ref ≤ 1 , RF shows an inverse behavior due to the application of ∇C . In this case, I increases with increasing ∇C . If C ref ≥ 1 , due to the influences of EKF and EDL thickness, the I at V=1 V shows a local minimum as Cref varies (Figure S7(a)). Secondly, the sign of ρ PE at pH 6 depends upon the accumulation or depletion of H+ in the nanopore: if the direction of the applied electric field is the same as (opposite to) that of the applied salt gradient, H+ depletes (accumulates), so that the nanopore PE layer is negatively (positively) charged. In this case, the ionic transport in nanopore depends mainly upon the sign of the PE layer charge. Thirdly, because the concentration of H+ at pH 3 is appreciable and its diffusivity larger than that of other ions, so is its 37



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contribution to I and influence of other properties. For example, if C ref ≤ 1 and a negative V is applied, the Ca , H + near the nanopore tip has a local minimum, so are ρ PE , C a , Cl− , and

C a , K + , thereby influencing the shape of the associated I-V curve. Furthermore, if C ref ≥ 1 and V=1 V, an appreciable amount of H+ accumulates in the nanopore due to the applied electric field and ∇C , yielding a larger I and a more significant RF. Fourthly, for C ref ≤ 1 , RF shows an inverse behavior at pH 3 and 10; for C ref ≥ 1 , the RF at pH 3 (10) has a local maximum (minimum) as Cref varies, due to the influences of ρ PE and EDL overlapping. At pH 6, since ρ PE is small, RF ≅ 1 .

ACKNOWLEDGMENT This work is sponsored by the Ministry of Science and Technology, Republic of China.

SUPPORTING INFORMATION Details of the axial variations of the ionic concentration, local electric field, volume charge density, the flow field near the nanopore tip, the spatial variation of total ionic concentration and H+, the I-V curve at several pH levels, and the code verification.

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Ionic Current Rectification in a pH-Tunable Polyelectrolyte Brushes

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