Ionic Current Rectification in a Conical Nanopore: Influences of Electroosmotic Flow and Type of Salt
1-1. Introduction
Due to its potential in simulating biological ion channels and devices in various
applications, synthetic nanopores have drawn the attention of researches of various
fields.1-10 For example, the sequence of DNAs can be determined by measuring the ionic
current signals as they translocate through a nanopore.11-18 A detailed understanding of
the electrokinetics and the associated mechanisms for the transport phenomena occur in
nanopores is of practical significance.
Ionic current rectification (ICR),19-22 where the ionic current arising from an applied
potential bias exhibits a preferential direction or diode-like behavior, is one of the
interesting and important electrokinetics behaviors of nanopores. Several mechanisms
have been proposed to explain this phenomenon, including, for example, ion
enrichment/depletion,23-24 influence of nanopore tip,25-26 and electrokinetic trapping of
mobile ions.27-28 Ion selectivity29-32 is another specific behavior of conical nanopores.
This behavior is significant when the nanopore radius is comparable to Debye length, so
that electric double layer (EDL) overlapping is important. In this case, because coions
are repelled electrically by the nanopore the measured current is mostly contributed by
counterions. Ion selectivity is influenced by factors such as the level of applied potential
bias, tip radius, and nanopore length.25-26, 33-34
Often, a model comprising Poisson and Nernst-Planck (PNP) equations is adopted to
describe the experimentally observed data for many reserches.25 Since the effect of
electroosmotic flow (EOF)35-36 is neglected in this model, it can be inapplicable under
certain circumstances. For instance, in a study of the ion transport in nanofluidic
channels, Daiguji et al.37 concluded that the higher the surface charge density of a
channel the more important the EOF effect is, and the ion transport inside is influenced
appreciably by the channel surface. Assuming constant surface charge density, Ai et al.38
solved a set of Poisson and Nernst-Planck and Navier-Stokes (PNP+NS) equations. They
concluded that the EOF effect is significant if the applied voltage is high, and both the
surface charge and the thickness of double layer take a medium large value. The analysis
of Ai et al.38 was extended to the case of channels having a charge-regulated surface by
Lin et al.39 They illustrated that the EOF effect should not be neglected if the bulk salt
concentration takes a medium high level and the applied potential bias is high.
Many attempts have been made on examining the influence of the nanopore
properties, such as its surface charge and geometry, and solution properties including pH
and salt concentration, on the associated ICR behavior. In contrast, that influence of the
types of ionic species in the liquid phase receives much less attention. The influence of
multivalent cations on the ICR behavior of a biological pore was studied both
experimentally40-43 and theoretically.44-45 It was found that these cations are capable of
inducing a charge inversion on the pore walls.40 Valisko et al.46 reported that due to the
binding of Ca2+, an originally highly negatively charged silica surface might become
positively charged, known as overcharging, and anion-selective.
For simplicity, relevant previous studies, both experimental and theoretical, adopted
an aqueous KCl solution as the model system because the diffusivity of K+ is essentially
the same as that of Cl-, so that the electrophoresis effect47 can be neglected. However, it
was reported that the translocation of DNA through a nanopore is influenced by the
types of salt solution,48 and was explained by the interaction of cations with DNA
molecules. In an attempt to understand how cations affect the ICR behavior of
α-hemolysin channel, Bhattacharya et al.49 considered various alkali chloride salts. They
found that the magnitude of the rectification factors for the cations considered follows
the order: Li+<Na+<K+<Rb+<Cs+, and the difference was attributed to the difference in
the affinity of cations to the charged residues of the channel. The interaction of cations
with the charged channel surface yields a depletion in mobile cations inside the channel.
A detailed understanding of the ions-pore surface interaction is helpful for enhancing the
detection capability of nanopore sensors. Piguet et al.48 examined the influence of EOF
on the electrokinetic flow of aqueous LiCl and KCl solutions in the α-hemolysin
channel. Due to a more significant EOF, LiCl solution shows a larger anion selectivity
than KCl. Adopting a PET nanopore, Gamble et al.50 examined the influence of the types
of ionic species on its rectification behavior. A molecular dynamics (MD) simulation
was used to explain the difference between the experimentally measured rectification
factor and that predicted by a PNP model. The result of MD simulation reveals that
different cations have different binding ability to the nanopore surface, yielding different
degree of surface charge reduction. Although this is interesting, the ionic current
predicted by the MD simulation deviates appreciably from the experiment data,
presumably due to the limitation in the simulation scale of the approach adopted.
Adopting a continuum model, the ionic transport in a conical nanopore is analyzed in
this study, taking account of the effects of surface charge density and EOF. Three types
of typical monovalent aqueous salt solutions are considered: KCl, NaCl, and LiCl.
1-2. Theoretical model
We consider a conical nanopore of length Ln, tip radius Rt, base radius Rb connecting
two identical large reservoirs of length Lr and radius Rr, as shown in Figure 1-1. The
system is filled with an incompressible aqueous salt solution. An electric potential bias V
is applied across the two reservoirs with the furthest surface of the tip side reservoir
grounded. The wall of the nanopore has a fixed charge density σw.
The ions in the system are driven by from one reservoir to the other. At steady
state, their transport can be described by the Nernst-Planck equations below:
0
Nj (1.1)
Nj, Cj, Dj, and zj represent the ionic flux, the concentration, the diffusion coefficient, and
the valence of ionic species j. u, F, R, and T are the fluid velocity, Faraday constant, gas
constant, and the absolute temperature, respectively. The electric potential ϕ in Eq. (1.2)
is described by the following Poisson equation:
2
ε and ρe are the fluid permittivity and the space charge density, respectively.
Since the fluid flow in the nanopore driven by is in the creeping flow region,
the corresponding flow field can be described by the equation of continuity and the
Navier-Stokes equation
2
p e
u 0 (1.4)
u 0 (1.5)
μ is fluid viscosity, and p the hydrodynamic pressure.
If we let S be the surface of the reservoir end, the ionic current I can be evaluated by
Suppose that no external pressure gradient is applied across the two reservoirs, and
they are large enough so that the salt concentration on the furthest surface of the tip-end
reservoir reaches the bulk value (Cj=C0). In addition, the electric potential on that
surface vanishes, ϕ(z=0.5Ln+Lr)=0, and that on the furthest surface of the base-end
reservoir ϕ(z=-0.5Ln-Lr)=V. The side boundaries of reservoir are rigid (n·Nj=0), and free
of charge (-n·∇ϕ=0). The wall of the conical nanopore is impermeable to ions (n·Nj=0),
no slip (u=0), and has a fix surface charge σw (=-n·(ε∇ϕ)).
Note that if the effect of electroosmotic flow is neglected, Eqs. (1.4) and (1.5) need
not be solved, defined as PNP model, for convenience. Otherwise, we have to solve Eqs.
(1)-(5), defined as (PNP+NS) model in subsequent discussion.
1-3. Results and discussion
This model is solved numerically by a finite element based commercial software,
COMSOL (version 4.3a, http://www.comsol.com). The behaviors of the ionic transport
in the nanopore under various conditions are examined. The present (PNP+NS) model is
first calibrated by fitting it to the experiment data of Liu et al.51 for a conical nanopore
and an aqueous KCl solution. The results obtained are shown in Figure S1 of Supporting
Information. For illustration, we assume pH=7, Ln=1000 nm, Rt=5 nm, Rb=28 nm,
Lr=Rr=800 nm, σw=-1 e/nm2 (e.g., a PET nanopore50, 52), and C0=50 mM. The half cone angle θ=1.32˚ is similar with the angle of the nanopore used in the experiment of
Gamble et al.50 Three types of salts are considered: KCl, NaCl, and LiCl. Other
parameter values adopted are F=96500 C/mol, R=8.314 J/mol K, T=298 K, μ=0.001 Pa
s, ε=6.95×10−10 F/m, DK+=1.96×10−9 m2/s, DNa+=1.33×10−9 m2/s, DLi+=1.03×10−9 m2/s,
and DCl-=2.03×10−9 m2/s.
In subsequent discussion, , V, and the types of cation are examined for their w
influences on current rectification. Figure 1-2 shows the simulated current-voltage (I-V)
curves for various types of salt. Both the results for the case where EOF effect is
considered and the corresponding results for the case where it is neglected are presented.
This figure shows the preferential direction of the ionic current from the nanopore tip to
its base, regardless the EOF effect is considered or not. This is because if V>0 (electric
field directs from nanopore base to its tip), the cation-selective nature of the nanopore
yields ion depletion inside.53 In contrast, if V<0, both cations and anions are enriched in
the nanopore, thereby raising the ionic current. This phenomena is known as ionic
current rectification (ICR). For both positive and negative voltage bias, the magnitude of
ionic current follows the order (KCl)I I(NaCl) I(LiCl) , which can be explained
by the difference in the mobility of the ionic species considered. Figure 1-2 reveals that
the presence of EOF has the effect of raising the ionic current, especially when a
negative voltage bias is applied. This is because EOF facilitates the ion transport.38
To measure the current rectification effect of a nanopore, we define the rectification
factor Rf= ( = ) / (I V v I V .v) 50 Figure 1-3 summarizes the simulated variation in Rf
with the applied voltage bias V for various types of salt. Both the results for the case
where EOF is neglected (i.e., PNP model), and those for the case where EOF is
considered (i.e., (PNP+NS) model) are presented. As seen in Figure 1-3, EOF is able to
influence both quantitatively and qualitatively the behavior of Rf. For example, Figure
1-3a reveals that if EOF is neglected, the values of Rf for various salt follow the order
Rf(LiCl)>Rf(NaCl)>Rf(KCl). This behavior is also reported by Gamble et al.50 in their
theoretical work, and was explained by the difference in the bulk conductivity. However,
as seen in Figure 1-3b, if EOF is taken into account, then for V lower than ca. 0.9 V,
Rf(LiCl)>Rf(NaCl)>Rf(KCl), but this order is reversed when V exceeds ca. 0.9 V. This
phenomenon is consistent with the experimental observation of Gamble et al.,50 where it
was explained by a molecular dynamics (MD) simulation based on a model PET pore.
They concluded that the binding abilities of the types of cation examined to the PET
membrane surface follow the order: Li+>Na+>K+. Since the higher the binding ability of
cations the lower the effective surface charge density of the nanopore, the resultant
values of Rf are different. However, the simulated values of Rf based on MD deviate
appreciably from the experimental data. This might arise from the limitation of MD: the
thickness of PET membrane was assumed the value of 10 nm, which is too thin to
observe the effect of ion depletion/enrichment.26
To explain the influence of EOF on Rf seen in Figure 1-3b, we examine the ionic
distributions in the nanopore. Figure 1-4 reveals that if the applied voltage bias is low
(V=-0.2V, Figure 1-4a and Figure 1-4b), the ionic distributions for the case where EOF is
neglected (i.e., PNP model) are similar to those for the case where it is taken into
account (i.e., (PNP+NS) model). That is, if the applied voltage bias is low, the EOF
effect can be neglected, as observed in several previous works.38 However, as indicated
in Figure 1-4c and Figure 1-4d, if the applied voltage bias is sufficiently high, this effect
becomes significant. In this case, the behavior of the ionic distributions based on a PNP
model remains about the same as that in Figure 1-4a, but becomes quite different if a
(PNP+NS) model is applied. This is because the degree of EOF increases with
increasing with applied voltage bias,38 as illustrated in Figure S2 of Supporting
Information. This figure also reveals that, due to EOF effect, the concentration profile is
influenced appreciably by the applied voltage bias. In addition, as seen in Figure S3 of
Supporting Information, if a positive voltage bias is applied, the ionic distributions for
the types of ionic species considered are about the same. This suggests that the ICR
behavior of the nanopore for types of salt examined are dominated mainly by the ionic
current at negative voltage bias. However, this fails to explain the rectification behavior
when the applied voltage bias is sufficiently small since the ionic distributions are almost
the same for the types of salt examined for both PNP and (PNP+NS) models, as
illustrated in Figure 1-4a, 1-4b, S3a, and S3b.
In the absence of an applied voltage bias (V=0 V), the attraction of counterions
(cations) by the charged wall of the nanopore establishes naturally an ion distribution in
its interior. When an applied voltage bias is applied this distribution is influenced,
yielding ion enrichment/depletion in the nanopore. To illustrate this, we plot the
difference in the ionic distribution, Ct (C1C10)(C2C20), due to the application
of the voltage bias in Figure 1-5, where Ci0 denotes the C in the nanopore at 0 V. i
This figure reveals that the degree of ion enrichment/depletion depends upon the types
of cation. As seen in Figure 1-5a, if a small negative voltage bias (V=-0.2 V) is applied,
an ion enrichment arising from the applied voltage bias occurs, and its degree follows
the order LiCl>NaCl>KCl. However, this order is reversed (KCl>NaCl>LiCl) if a large
negative voltage bias (V=-2 V) is applied, as shown in Figure 1-5b. This reasonably
explains the observed voltage-dependent ICR behaviors of the salts examined. Note that
as shown in Figure 1-5c and 1-4d that if a positive voltage bias is applied, the ionic
distributions for the types of salt examined are essentially the same. For comparison, the
concentration difference for the corresponding PNP model is also plotted in Ct
Figure S4 of Supporting Information, which reveals that the influence of the type of salt
is negligible. We conclude that the difference of the ionic distribution in the nanopore
among the salts examined mainly arises from the effect of EOF.
Figure 1-6 reveals that for the types of salt considered, the current rectification factor
Rf exhibits a local maximum as the surface charge density varies. The presence of the
local maximum in Rf for the case of KCl was also observed and thoroughly discussed in
other theoretical studies.38,54 Note that the rectification effect must vanish (i.e., Rf =1) at
zero surface charge.55 Previous studies for the influence of the types of salt on the ICR
behavior of a nanopore focused on the affinity between ions and its surface. Gamble et
al.50, for example, suggested that the difference between the experimentally measured
rectification factor and that predicted by a PNP model arises from a reduction in surface
charge due to surface binding of ions. Adopting a (PNP+NS) model, we show that the
experimentally observed behavior of Rf can also be explained by the presence of EOF.
Figure 1-6a also reveals that at |V|=0.2 V, Rf follows the order LiCl>NaCl>KCl for the
range of the surface charge density examined. However, if |V| is raised to 2 V, this order
is reversed when the surface charge density is sufficiently large, as seen in Figure 1-6b.
Again, this suggests that EOF plays a critical role in ion-species rectification.
1-4. Conclusions
The influence of electroosmotic flow (EOF) on the ionic current rectification (ICR)
behavior of a conical nanopore is investigated by considering three types of aqueous salt
solution, namely, LiCl, NaCl, and KCl. We show that if the EOF effect is neglected, the
magnitudes of the rectification factor Rf of these solutions rank as LiCl>NaCl>KCl at
each level of the applied potential bias V. However, if that effect is considered, the
relative magnitudes of the Rf of the salts examined depend upon the level of V. If |V| is
lower than ca. 0.9 V, Rf follows the order of LiCl>NaCl>KCl, but if |V| is higher than ca.
0.9 V, the order becomes KCl>NaCl>LiCl, which is consistent with experimental
observation. At a higher V, the EOF effect is more significant, thereby influencing more
appreciably the ionic concentration inside the nanopore. If V<0, the degrees of
concentration enrichment for different types of ions differ appreciably. The concentration
enrichment of LiCl increases most if V is low, while that of KCl increases most if V is
high, yielding the inversion of the order of Rf mentioned above. We conclude that, in
addition to ionic binding, EOF also plays a crucial role in ion-species current
rectification, especially when V is high.
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