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2-1. Introduction
Many recent applications of nanotechnology including, for example, energy convertion,1,2 particle transportion,3-8 biosensing,9-12 membranes,13,14 ionic circuits,15 ionic gates,16 and many others,17-23 result in vigorous studies of the electrokinetic phenomena occurring in nanoscaled devices. Indeed, a detailed understanding of these phenomena and the associated mechanisms are highly desirable for both elaborating the results gathered and designing relevant devices.
A charged nanopore is capable of exhibiting interesting and significant behaviors that are not observed in the corresponding uncharged nanopore. For instance, in a study of the
conductance of a nanocapillary having a constant surface charge Steinbock et al.24found that if the bulk KCl concentration is low, the conductance is dominated by the surface charge.
They found that the conductance varies nonlinearly with the bulk salt concentration. Taking account of the effect electroosmotic flow (EOF) in a cylindrical nanochannel, Daiguji et al.25 concluded that the higher its surface charge density the more significant that effect is. Ai et al.26 found that the preferential current in a conical nanochannel reverses when Cbulk is below a certain level. In addition, the higher the applied voltage the more important the EOF effect.
Both of these two studies assumed a constant charge density on the nanochannel surface.
Zeng et al.,27 studied the ICR behavior of a conical nanochannel surface modified by a layer of polyelectrolyte brushes. Since this layer is pH-tunable, the sign of its charge depends upon
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of the solution pH, so is the ICR behavior of the nanochannel.
If the solute or particle concentration in a Newtonian fluid is sufficiently high, it will exhibit non-Newtonian behavior. Organic solutions, which are ubiquitous in the applications of nanotechnology, are also of non-Newtonian nature. Unfortunately, relevant previous studies usually based on an aqueous Newtonian solution; seldom of them focused on non-Newtonian solutions. Yin et al.28 found that the direction of the ICR occurring in an organic solution can be different from that in an aqueous solution, thereby developing a new method for detecting the water in an organic solution. Qiu et al.29 considered a single mesopore connecting two chambers filled with an aqueous solution of KCl and an aqua/gel solution of KCl, respectively. Through adjusting the gel percentage, and therefore, the liquid viscosity and conductivity, they were able to tune the current-voltage curve and the resulting degree of current rectification.
Adopting LiClO4, Plett et al.30 examined the influence of solvent on the rectification behavior of conical polymer pores and glass nanopipettes filled with an aprotic solvent. They showed that the adsorption of Li+ to the pore wall and the resulting finite dipole moment is capable of influencing the ionic current rectification of the system considered. Plett et al.31 studied experimentally the ion transport inside a nanopore coated with a LiClO4/polymethyl methacrylate gel. Three cases are considered: (i) an aqueous KCl solution in an uncoated nanopore, (ii) a LiClO4/propylene carbonate solution in a gel-coated nanopore, and (iii) a
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LiClO4/propylene carbonate solution in an uncoated. They showed the emergence of
rectification at high salt concentrations for non-aqueous solutions, which is absent in aqueous solutions. In particular, the gelled electrolyte has a better cycle stability and longer service life, and therefore, appropriate for energy storage. In addition, using the LiClO4/propylene carbonate solution can alleviate the decrease in the conductivity and the migration of electrons in the gel type electrolyte.
Assuming constant surface charge density, Matin et al.32 examined the ionic transport behavior in a conical nanopore for the case of a power-law fluid. They found that the scaled ionic current follows the order pseudoplastic fluid<Newtonian fluid<dilatant fluid. However, the ICR factor follows the reverse order.
Up to now, the ionic transport in a nanoscaled device containing an non-Newtonian solutions taking account of the effect of pH has not been analyzed. This effect is of
fundamental significance in designing sophisticated analytic devices.33,34 The present study is aimed to understand in detail the role that this key effect plays and the associated
mechanisms. For illustration, we consider a pH-regulated conical nanopore containing a power-law fluid. A thorough numerical simulation is conducted to examine the influence of bulk salt concentration, solution pH, and the power-law index on the ICR behavior and the ion selectivity of the system under consideration.
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2-2. Theory
As presented schematically in Figure 2-1, we consider a conical nanopore having the axial length LN, tip radius Rt, and base radius Rb, connecting to two large, identical reservoirs.
The system under consideration is filled with an aqueous salt solution, and the nanopore surface is charged. A potential bias V is applied across the nanopore with the tip end reservoir grounded. The present system is axisymmetric, and it is described by the cylindrical
coordinates (r,θ ,z) with the origin at the nanopore center.
Suppose that the liquid phase in the system is a non-Newtonian, incompressible fluid, and φ, u, and p are the electric potential, fluid velocity, and pressure, respectively. Then at steady state the present problem can be described by the set of equations
2
denote the valence, the molar concentration, the flux, and the diffusivity of ionic species j, respectively. F, and R are Faraday constant and gas constant, respectively; e and
ρ
e are thedoi:10.6342/NTU201801261
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elementary charge and the density of mobile ions in the liquid phase, respectively.
ε
andμ the permittivity of the liquid phase and its viscosity, respectively. N, T, and T denote the
number of ionic species, the absolute temperature, and matrix transpose, respectively.
Suppose that the liquid phase is a power law fluid
1 n
m γ
μ =
• − (6)γ• is the shear rate, n the power law index, and m the viscosity of the corresponding
Newtonian fluid when n=1. The present fluid is dilatant for n>1, and pseudo plastic for n<1.
The nanopore surface is charge-regulated with its charge density varies with the solution pH through regulating the dissociation of the acidic and basic functional groups AH and B, respectively,
AH ⇔ A +H− + (7)
BH+ ⇔ B+H+ (8)
Let Nt =NAH+NA− =NBH+ +NB, and KA =NA-[H ]+ S NAH and KB= NB[H ]+ S NBH+ the corresponding equilibrium constants, where [H ]+ S (mol/m3) is the molar concentration of H+ on the surface. Then it can be shown that the surface charge density of the nanopore,
σ
w(C/m2), is
For illustration, we assume that the salt in the liquid phase is KCl, and its solution adjusted by introducing HCl and KOH. Therefore, four kinds of ionic species need be
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considered (i.e., N=4). The square brackets denote the molar concentration of a species and the subscript the bulk value, then [H ]+ 0 =10−pH+3, [OH ]− 0 =10−(14 pH)+3− , [K ]+ 0 =Cbulk, and
To specify the boundary conditions associated with Eqs. (1)-(5), we define a
computational domain, which includes the nanopore a sufficiently large cylindrical region in each reservoir. We assume that the surface of the nanopore is nonslip and impenetrable so that u=0 and n N
⋅
j= 0
. The surface charge densityσ
w can be expressed asσ
w = − ⋅∇ε
nφ
. The computational domain is sufficiently large so that the ionicconcentration at their ends reaches essentially the bulk value, and the electric potentials there the applied values. In addition, no external pressure gradient is applied.
2-3. Results and Discussion
COMSOL MulitiPhysics (version 4.3a, www.comsol.com) is adopted to solving Eqs. (1)-(5) subject to the boundary conditions assumed. In general, using ca. 150,000 nodes is sufficient for retrieving reliable and sufficiently accurate results, as checked by mesh independence. The
results obtained are then used to evaluate the ionic current I through the nanopore
4
( 1 j j)
I F j z d
Ω =
= N ⋅n Ω, (10)
where Ω denotes a surface perpendicular to the nanopore axis.
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An example are adopted to verify the applicability of our solution procedure. The first example is a conical nanopore having a constant surface charge density filled with an aqueous Newtonian fluid. The following two empirical relationships correlating the conductance G and the bulk salt concentration Cbulk were proposed by Steinbock et al.:10
+ - bulk procedure reproduces successfully the result evaluated by Eq. (12). The deviation of our result from the result predicted by Eq. (12) at low levels of Cbulk arises from the effect of double layer overlapping, which is neglected in the latter.
In this section we discuss in detail the influence of the solution pH, the bulk salt
concentration C0, the functional group density Nt, and the power law index n on the ICR behaviors of the nanopore. For illustration, we assume the following: Rt=6 nm, Rb=90 nm, LN=3000 nm, LR=300 nm, pKA=3, pKB=9, and the salt in the liquid phase is KCl. At T=300 K, we have Nt =2.5 10× −6 mol/m2, DH+ =9.31 10× −9 m2/s, DOH− =5.26 10× −9 m2/s,
9 K 1.96 10
D + = × − m2/s, DCl− =2.03 10× −9 m2/s, and ε =6.95 10× −10 F/m.
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2-3.1 Influence of Bulk Salt Concentration
The simulated variation of the ionic current rectification factor, Rf =|I( 1 V) (1 V)− I |, with the bulk salt concentration, Cbulk, for pH>IEP is illustrated in Figure 2-3 where the nanopore is positively charged. This figure reveals that, for each value of n, Rf has a local maximum occurring at a moderate level of Cbulk. As Cbulk increases the thickness of double layer decreases accordingly, so that the contribution of cations to ionic current approaches to that of anions. It is interesting to note that if Cbulk is lower than ca. 20 mol/m3, Rf (n=0.9) is larger than both Rf (n=1.0) and Rf (n=1.1), but becomes smaller than Rf (n=1.0) and Rf
(n=1.1) if Cbulk exceeds that level. This can be explained by the averaged strength of the axial electric field Λ shown in Figure 2-4. Figure 2-4(a) and 2-4(c) reveals that if Cbulk=10 mol/m3, Λ (n=0.9) is larger than both Λ (n=1.0) and Λ (n=1.1) at V=-1 V, in general, but the situation is reversed at V=1 V. However, as seen in Figure 2-4(b) and 2-4(d), if Cbulk is raised to 100 mol/m3 opposite trends in Λ are observed. Since the ionic current I is proportional to Λ , the definition of Rf yields the qualitative trends seen in Figure 2-3.
Figure 2-5 illustrates the simulated variation in the ionic current rectification factor Rf
with the bulk salt concentration Cbulk for pH<IEP, where the nanopore is positively charged.
In this case, Rf shows a local minimum occurring at a moderate level Cbulk for all of the values of n assumed. Note that in contrast to the case of Figure 2-4, the curve of Rf (n=0.9) does not intersect with that of Rf (n=1.0) or Rf (n=1.1) as Cbulk is higher than ca. 1 mol/m3.
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This can be explained by the results presented in Figure 2-6, where the variation of the cross sectional averaged ionic conductivity Λ under various conditions are shown. Figure 2-6(a) reveals that Λ (n=0.9)< Λ (n=1.0)< Λ (n=1.1) at V=1 V, but Λ (n=0.9)> Λ (n=1.0)> Λ
(n=1.1)at V=1 V. As in the discussion of Figure 2-3, since the ionic current I is proportional to E, the behavior of Rf seen in Figure 2-5 is expected.
at V=+1 V is illustrated in Figure 2-7 for various values of n at two levels of pH. As seen in Figure 2-7(a), if pH<IEP, S decreases with increasing Cbulk, that is, the lower the Cbulk the more anion-selective the nanopore is. This figure also indicates that for a fixed level of Cbulk
the smaller the n the more anion-selective the nanopore is, which will be explained later. In Figure 2-7(b) because pH>IEP, the nanopore is cation-selective. In this case, S deceases with increasing Cbulk, that is, the lower the Cbulk the more cation-selective the nanopore is. Figure 2-7(b) also suggests that the smaller the n the more cation-selective the nanopore is.
2-3.2 Influence of pH
As seen in Figure 2-8, the qualitative behavior of the dependence of the ionic current rectification Rf on pH for pH<IEP is quite different from that for pH>IEP. In the former, the smaller the n the larger the Rf is. To explain this, we plot the axial variation of the cross sectional averaged ionic conductivity Λ at pH 4 in Figure 2-9. Figure 2-9(a) suggests that
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at V=1 V, Λ (n=0.9)< Λ (n=1.0)< Λ (n=1.1). However, Figure 2-9(b) shows that Λ (n=0.9)≅
Λ (n=1.0)≅ Λ (n=1.1) at V=-1 V. Therefore, the behavior of Rf for pH<IEP seen in Figure 2-8 is reasonable.
Demonstrated in Figure 2-8, if pH>IEP, where the nanopore is negatively charged,
Rf(n=0.9)>Rf(n=1.0)>Rf(n=1.1), where the value for n=1.0 n=1.1 is nearly the same.
We illustrate the cross sectional averaged ionic conductivity Λ in Figure 2-10, showing that that the difference of ion conductivity between n=0.9 and n=1.0 at V=1 V in Figure 2-10(a) is apparently smaller than the ion conductivity at V=-1 V in Figure 2-10(b). The illustration reveals the relation of Rf for different n as pH>IEP. Moreover, Rf increases monotonically
with increasing pH at n=0.9, but shows a local maximum at n=1.0 and n=1.1. Again, these behaviors can be explained by the cross sectional averaged ionic conductivity Λ plotted in Figure 2-11. Figure 2-11(a) and 2-11(b) shows that at n=0.9, the difference between Λ (pH 8) and Λ (pH 9) for V=-1 V is much larger than that for V=1 V. Since Rf is defined as
f |I( 1 V) (1 V)I
R = − | in our case, the behavior of Rf seen in Figure 2-8 pH>IEP is
reasonable. Figure 2-11(c) and 2-11(d) indicates that the difference between Λ (pH 8) and Λ (pH 9) at V=-1 V is about the same as that at V=1 V. This explains the behavior of
Rf(n=1.0) and Rf(n=1.1) seen in Figure 2-8 for pH>IEP.
Figure 2-12 suggests that for the levels of the bulk salt concentration examined, the value of the selectivity S at V=+1 V as parameter n varies ranks as |S(n=0.9)|>|S(n=1.0)|>|S(n=1.1)|,
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as is also observed in Figure 2-7, where pH 7 and pH 5.5 are chosen to represent the condition for pH>IEP and pH<IEP, respectively.
As seen in Figure 2-13(a), where pH>IEP, the ionic current contributed by cations, I , is c insensitive to the variation in n. In contrast, the ionic current contributed by anions, |I , a|
increases appreciably with increasing n. If pH<IEP, these trends are reversed, as seen in Figure 2-13(b). Figure 2-14(a) reveals that the cross sectional averaged concentration of anions, Canions, at pH 7 (>IEP) increases with increasing n, in general. This explains the trend of the anion current observed in Figure 2-13(a). Similarly, Figure 2-14(b) indicates that at pH 5 (<IEP), the cross sectional averaged concentration of cations, Ccations, also increases with increasing n, in general. The results illustrated in Figure 2-13 well explained the dependence of the ion selectivity S on the parameter n shown in Figures 2-12 and 2-7. Note that as seen in Figure 2-12(b) if pH>IEP, S has a local maximum, and a local minimum if pH<IEP. This is because if pH<IEP, the lower the pH the higher the concentration of H+, and since DH+ is much larger than DCl−, I increases, thereby lowering S. On the other hand, if pH>IEP, the c higher the pH the higher the concentration of OH-, and since DOH− is much larger thanDK+ ,
|I increases. a|
2-4. Conclusions
We modeled the ionic transport in a pH-regulated conical nanopore connecting two large
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reservoirs containing a power-law fluid. In particular, the influences of the bulk salt concentration Cbulk, the solution pH, and the power-law index n on the ionic current rectification, measured by the current rectification factor Rf, and the ion selectivity S are investigated. The influence of Cbulk on Rf can be summarized as following. (i) If pH exceeds the isoelectric point (IEP), Rf has a local maximum as Cbulk varies. If Cbulk is low,
Rf(pseudoplastic fluid)>Rf(Newtonian fluid)>Rf(dilatant fluid), and if Cbulk is high,
Rf(Newtonian fluid)>Rf(dilatant fluid)>Rf(pseudoplastic fluid). (ii) If pH is lower than IEP, Rf
has a local minimum as Cbulk varies. (iii) For all range of Cbulk, Rf(pseudoplastic fluid)>Rf (Newtonian fluid)>Rf(dilatant fluid). These behaviors can be attributed to the axial variation in Cross sectional averaged ionic conductivity Λ . Regarding the influence of pH on Rf we
conclude the following. (i) If pH is lower than IEP, the smaller the n (i.e., more shear
thinning the fluid) the larger theRf. (ii) If pH exceeds IEP, Rfincreases monotonically with
increasing pH for a pseudoplastic fluid, but shows a local maximum for Newtonian and dilatant fluids. These behaviors can be attributed to the axial variation in the cross sectional averaged ionic conductivity. Regarding the influence of pH on S we conclude the following.
(i) If pH exceeds IEP, the ion selectivity S decreases with increasing Cbulk, and the more shear thing the fluid the larger the S (i.e., the more the nanopore is cation-selective). (ii) If pH is lower than IEP, |S| decreases with increasing Cbulk, and the more shear thing the fluid the larger the |S| (i.e., the more the nanopore is anion-selective). These behaviors can be
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attributed to the relative magnitude of the ionic conductivities.
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Figure 2-1. Ionic transport in a pH-regulated conical nanopore of axial length, tip radius,
base radius, and surface charge density, LN, Rt, Rb, and σw, respectively, connecting two
base radius, and surface charge density, LN, Rt, Rb, and σw, respectively, connecting two