doi:10.6342/NTU201801250 43
2-1. Introduction
Recent progress in nanotechnology facilitates fabrication of devices used in various applications, including, for example, nanopipettes, nanoelectrodes, and nanopores/nanochannels.
In addition to capable of simulating biological ion channels, surface functionalized
nanopores/nanochannels are also readily applicable in sensing ions and biomolecules[1-5],
modeling nanofluidic diodes[6-11], ionic gates[12-16], and controlling the transport ions and
particles[17-23].Recent applications of nanopores/nanochannels also include renewable energy
storage[24-26], desalination of seawater[27,28], and controlled release of drugs[29]. Moreover,
inspired by biological ion channels and ion pumps[30-32] existing in cell membrane to
intelligently control ions into and out of it, bioinspired artificial functional nanochannels are
essential for various vital processes including maintaining cellular ion homeostasis[33] and
energy conversion[34-36].
For an aqueous, monovalent salt solution of concentration 10-3 M the associated thickness of
electric double layer (EDL), or Debye length[37], is ca. 10 nm at room temperature. This implies
that the overlapping of EDL in a nanopore or nanochannel having a linear size of that order is
significant, yielding several interesting electrokinetic phenomena. These include, for example,
ionic current rectification (ICR)[38,39], ion selectivity[40,41], and ion concentration polarization
(ICP)[42,43]. These phenomena are important in nanopore-based biosensing[44], and the
doi:10.6342/NTU201801250 44
associated mechanisms have many potential applications in regulating ion transport[45], and
separation and detection of bioentities[46], to name a few.
Ionic current rectification (ICR) is an asymmetric diode-like current-voltage behavior, where
the ionic current arising from an applied potential bias exhibits a preferential direction. For a
nano-scaled device, ICR can come, for example, from its asymmetric geometry[38,47],
non-uniform surface charge[48,49], and an imposed salt concentration gradients[50]. Several
mechanisms have been proposed for interpreting this phenomenon, including, for instance, the
electrochemical prosperities of the tip of a nanopore[51], electric potential barrier inside a
nanopore[52],and the enrichment/depletion of ions inside a nanopore[53]. The results of recent
studies reveal that the ICR behavior of a nanopore can be influenced by factors such as the applied
voltage[54,55], bulk salt concentration [54-56], its shape [55,56], electroosmotic flow[57], pH
[56-58], and temperature[59].
Nanopores/nanochannels can be organic, inorganic, biological, or composite. Several
fabrication techniques were proposed including, for instance, track-etching, electrochemical
etching, electron beam, laser, and self-assembly of biological molecules. It was shown that both
the shape and the structure of a polymeric nanochannel can be designed through controlling
etching conditions. Adopting polyethylene terephthalate (PET) membrane, Apel et al.[60]
prepared single nanopore by irradiating it with single swift heavy ions using a
surfactant-doi:10.6342/NTU201801250 45
controlled track-etching technique. A similar approach was adopted by Ali et al.[48] to prepare
cigar-shaped nanopores with its wall functionalized with amphoteric amino acid (lysine) chains.
It was shown that the nanopores obtained can be operated as resistors and diodes with a broad
range of rectifying properties. Their results are informative for interpreting ionic transport in
biological systems such as ion channels, and relevant application of tunable nanopore technology
such as information processing and drug controlled release. However, both the orifice radii of a
nanopore and the shape of its surface are fixed in their analysis, a more detailed discussion on
the influence of these key parameters is highly desirable.
Various types of nanochannel shape and ways that the nanochannel surface are functionalized
have been proposed. The former includes, for example, conical nanopores, concave nanopores
having a bullet-like tip, and convex nanopores having a trumpet-like tip. Assuming constant
surface charge density, Ramiŕez et al.[55] investigated comprehensively the influence of the
shape of a nanopore on its rectification and ionic selectivity behaviors. They found that as a bullet
(trumpet) like nanopore approaches a conical nanopore, both the rectification factor and the
conductance decrease (increase), but the selectivity increases (decreases). Since a
Poisson-Nernst-Planck (PNP) model is adopted in their analysis, the electroosmotic (EOF) effect[57],
which can be significant if the solution pH deviates appreciably from the isoelectric point and the
applied electric potential bias is high, is neglected. In addition, instead of maintaining constant
doi:10.6342/NTU201801250 46
charge density, the surface of a nanopore is usually charge- or pH-regulated. This nature comes
either from the properties of the original material or from that of the materials used to
functionalize the nanopore surface.
Inspired by the ion channels and ion pumps existing in cell membrane, Zhang et al.[61]
showed that through applying a pH gradient a cigar-shaped nanochannel can serve as a
double-gate: it can open and close alternately under a symmetric pH gradient, and simultaneously under
an asymmetric pH gradient. Their study was followed by Liu et al.[15] to develop a double-gated
nanochannel which is dual-responsive to K+ and pH. While these results are very interesting and
valuable, the behavior of the electrokinetic ion transport in the cigar-shaped nanochannel
deserves a more detailed discussion. In fact, the available theoretical results for that behavior in
a nanopore/nanochannel having a pH-regulated surface bearing zwitterionic groups, taking
account of the effects of pH and pore/channel geometry simultaneously are still very limited.
However, obtaining these results is highly desirable and necessary for a sophisticated design of
relevant devices.
In this study we simulate the rectification and ionic selectivity behaviors of a cigar-shaped
nanochannel having a pH-regulated surface bearing zwitterionic groups, taking account of the
EOF effect. A thorough numerical simulation is conducted to illustrate the influences of the
surface curvature, the base opening radii, and the bulk salt concentration and pH. In addition to
doi:10.6342/NTU201801250 47
searching for the optimum conditions, the underlying mechanisms are also elaborated.
2-2. Theory
As shown schematically in Figure 2-1, we consider a cigar-shaped nanochannel having the
tip, base, and center radii Rt, Rb, and Rc, respectively, and axial length LN connecting two large,
identical, cylindrical reservoirs of radius Rr and length Lr. The nanochannel can be formed by, for
example, etching a membrane, and LN corresponds to its thickness. Both the nanochannel and the
two reservoirs are filled with an incompressible, aqueous Newtonian salt solution. The two
reservoirs are large enough so that the concentration of each ionic species at places far away from
the membrane maintains its bulk ionic concentration. A potential bias V is applied across the
nanochannel with the electrode in the upper reservoir ground (0 V). The external electric potential
difference across the nanochannel drives the electroosmotic flow and the ion transport through
the nanochannel, and then it yields an ionic current. The system under consideration is axial
symmetric, and we choose to work on the cylindrical coordinates (r, θ, z) with the origin placed
at the center of the nanochannel tip end.
The surface of the nanochannel can be described by Zhang et al.[61]
doi:10.6342/NTU201801250
curvature of the nanochannel profile, or the curvature radius, and LN/2h is the shape-controlling parameter which depends on degree of etching. As LN/2h→0, the nanochannel approaches a
bi-conical one, and LN/2h increases, it becomes more cigar-like.The opening radii (Rt and Rb) can
be adjusted by modifying.
The surface of the nanochannel is modified by materials having pH-tunable, zwitterionic
functional groups so that it bears charges when immersed in an aqueous solution, implying that
their charges are exquisitely dependent on the local concentration of protons. To mimic the
pH-regulated nature of biological ion channels, let AH and B be the corresponding acidic and basic
functional groups, respectively, with associated reactions AH⇌A-+H+ and BH+⇌B+H+, and equilibrium constants KA=ГA-[H+]/ГAH and KB=ГB[H+]/ГBH+ , respectively. [H+] and Гm are
the molar concentration of H+ (mM) and the surface site density of species m (AH, A-, BH+, and
B), respectively. For simplicity, we assume that the acidic and the basic functional groups have the same density Гt so that ГA-+ГAH=ГBH++ГB=Гt. Therefore, if e is the elementary charge, the
doi:10.6342/NTU201801250 49
surface charge density of the nanochannel, σs (C/m2), can be expressed as σs=1018e(ГBH+−ГA-).
Suppose that the liquid phase is an aqueous KCl solution with its pH adjusted by KOH and
HCl, implying that four kinds of ionic species are present in the liquid phase: K+, Cl−, H+, and
OH−, denoted by species j, j=1, 2, 3, and 4, respectively. Let C10, C20, C30, and C40 be the bulk
concentrations of these ions (mM), respectively, and C0 the bulk concentration of KCl (mM).
Since pH=−log([H+]0/1000) with [H+]0 being the bulk molar concentration of H+ (mM), C30=10
-pH+3, C40=10-(14-pH)+3, C10=C0−10-pH+3+10-(14-pH)+3, and C20=C0 for pH≧7; C30=10-pH+3, C40=10
-(14-pH)+3, C10=C0, and C20=C0+10-pH+3−10-(14-pH)+3 for pH<7.
To fully describe the present problem, the following set of nonlinear, couple differential equations need be solved simultaneously:
[ ( j j )]=0
universal gas constant, the absolute temperature, the fluid velocity, and the electric potential, respectively. ρe and ε are the space charge density of mobile ions and the permittivity of the
liquid phase, respectively.
Since the fluid flow in the present problem is in the creeping flow regime, the equations
doi:10.6342/NTU201801250 50
governing the flow field at steady state can be described by
∇⋅ =u 0 (2.4)
2
p e
μ∇ −∇ − ∇ =u ρ φ 0 (2.5)
μ and p are the fluid viscosity and the hydrodynamic pressure, respectively.
Equations (2.2)-(2.5) need be solved subject to the following conditions: (i) The rigid nanochannel surface Ω1 is non-slip (i.e., u=0), impenetrable to ions (i.e.,n N⋅ j =0, n is the unit
outer normal vector), and has a pH-regulated charge density σs defined previously. (ii) The
boundary Ω2 of the computation domain is grounded, and an electric potential bias V is applied
to its boundary Ω3. All the ionic concentrations reach their bulk values on Ω2 and Ω3, (cj=Cj0). A
normal flow with no external pressure gradient (i.e., p=0) is specified on these two boundaries.
(iii) There is no accumulated charge on the boundary Ω4 of the computation domain (n⋅∇ =φ 0),
the normal flux of ions vanishes on that surface (i.e.,n N⋅ j =0), and it is slip. (iv) Electric
potential, electric field, ionic concentration, and flow field are all continuous on the interface
between the charged wall and liquid. (v) All the dependent variables are symmetric to the
nanochannel axis.
The solution of Eqs. (2.2)-(2.5) can be used to evaluate the ionic current I through the
nanochannel by integrating the normal component of the total ionic flux over a surface S
perpendicular to the nanochannel axis
doi:10.6342/NTU201801250 51
4 1
( j j)
S j
I F z dS
= = N ⋅n (2.6)
2-3. Results and Discussion
A thorough numerical simulation is conducted to illustrate the influence of the parameters
key to the system under consideration on its ion transport behavior. These include the surface
curvature (LN/2h), the base radii Rb, the bulk salt concentration C0, and the solution pH. We assume LN=3000 nm, Lr=300 nm, Rr=300 nm, Rt=5 nm, and Rc=100 nm[61], Гt=1.5 nm-2, and
pKA=3 and pKB=9[41, 53, 56], that is, the isoelectric point (IEP) is 6. In addition, 6.95 1010F/m
ε= × − , F=96500 C/mol, R=8.3145 J/(mol·K),μ=10 Pa s−3 ⋅ , and T=298.15 K. D1(K+),
D2(Cl-), D3(H+), and D4(OH-) are 1.957, 2.032, 9.312, and 5.26(×10-9) m2/s, respectively. We consider the case where Rt≠Rb and LN/2h1=LN/2h2=LN/2h, with Rt=5 nm and Rb is either 15 nm
or 25 nm. Note that the orifice diameter of the cigar-like nanochannel varies along its axis, so are
the degree of EDL overlapping and the ionic distribution, yielding its ICR behavior. Since the
rectification factor Rf is usually defined at a voltage higher than 0.5 V, we choose
Rf=|I(−1V)/I(+1V)| to quantify the rectification characteristics.
Equations (2.2)-(2.5) are solved numerically subject to prespecified boundary conditions by
COMSOL (version 4.3a, http://www.comsol.com ) operated in a high-performance cluster.
Quadratic, triangular elements are used and, if necessary, non-uniform elements with larger
doi:10.6342/NTU201801250 52
number of elements are assigned locally. In our case, using a total number of 200000 mesh
elements is sufficient for retrieving results that are sufficiently accurate and reliable.
2-3.1 Cation-Selective Nanochannel (pH 10)
Let us consider first the case where pH=10. In this case, since the nanochannel is negatively charged, it is generally cation-selective. Figure 2-2 shows the simulated current-voltage (I-V)
curves under various conditions. As can be seen in this figure, there is a preference for ionic
current at a negative applied potential bias at this level of pH so that the rectification factor Rf of
the nanochannel is larger than unity for all the levels of the bulk salt concentration C0 examined.
Note that the higher the bulk salt concentration C0 and the larger the base radii Rb the larger the
ionic current is. Figure 2-2 also shows that the larger the (LN/2h) the larger the ionic current. This
is because the larger the (LN/2h) the greater the inner space of the nanochannel so that a more
amount of ions is able to diffuse through it, yielding a larger current. Due to the enrichment of
ions inside the nanochannel, this phenomenon is more pronounced for V<0.
To explain the influences of the surface curvature (LN/2h), the bulk salt concentration C0, and
the base radii Rb on the ICR behavior of a nanochannel, we plot the variations of Rf with LN/2h
under various conditions at pH 10 in Figure 2-3. This figure reveals that for both values of Rb (15
and 25 nm), Rf increases monotonically with increasing (LN/2h) at all the levels of C0 examined,
except at C0=100 mM, where Rf shows a local maximum at LN/2h=5. Note that both Rf(Rb=25
doi:10.6342/NTU201801250 53
nm) and Rf(Rb=15 nm) are larger than unity, and for the same values of Rt and (LN/2h), Rf(Rb=25
nm)>Rf(Rb=15 nm). This is because the larger the difference between Rt and Rb the greater the
difference in the degree of electric double layer overlapping near the both nanochannel tip
regions, and the more significant the ICR effect.
Figure 2-3 also reveals that Rf has a local maximum as the bulk salt concentration C0 varies.
This behavior was also reported previously for conical nanopores[54], and was explained by the
rates of change in Rf and I(V=±1 V) with respect to C0.
The ICR behavior observed in Figure 2-3 can be explained by the ionic conductivity and the
strength of the electric field in the nanochannel. Letting λj be the molar conductivity of the jth ionic species, we define respectively the axial cross sectional averaged concentration c and j
the corresponding ionic conductivity G as [17]
, 1, 2, 3, 4
j j
c c dA j dA
= =
(2.7)
and
4 1 j j j
G c λ
= = (2.8)
Under the conditions considered, λ1 (K+), λ2 (Cl-), λ3 (H+), and λ4 (OH-) are 7.352×10-3, 7.654×10
-3, 35×10-3, and 19.8×10-3 S m2 mol-1, respectively. Note that the ionic current is proportional to
G.
doi:10.6342/NTU201801250 54
Figure 2-4 illustrates the variations in G for various combinations of C0, (LN/2h), and V; the
corresponding axial variations in the cross sectional averaged strength of the electric field Ez,avg
are summarized in Figures S1−S4 of the Supporting Information. The latter reveals that Ez,avg is
appreciable only near the tip end of the nanochannel, where the effect of surface charge is more
important. In addition, for both V=±1 V, the larger the (LN/2h), the stronger the Ez,avg. Since the
qualitative behaviors of both G and Ez,avg at Rb=15 nm are similar to those at Rb=25 nm, those at
Rb=25 nm are adopted for interpreting the observed ICR behavior seen in Figure 2-3.
Figure 2-4(a) reveals that if C0=1 mM, the difference between G(V=+1 V) and G(V=−1 V) is
relatively small for each level of (LN/2h) so that Rf is close to unity. Note that as shown in Figure
S1 of the Supporting Information, if V=+1 V, a local electric field with its direction opposite to
that of the applied electric field is induced near the nanochannel tip. This phenomenon is more
appreciable as (LN/2h) gets smaller, implying that the smaller the (LN/2h) the more important the
effect of ion concentration polarization (ICP). Figure 2-4(b) indicates that the behavior of G at
C0=10 mM is similar to that at C0=1 mM seen in Figure 2-4(a). That is, for both V=±1 V, G (and
therefore, Rf) is insensitive to the variation in (LN/2h). However, for each level of (LN/2h),
G(V=−1 V) is appreciably larger than G(V=+1 V), so that Rf(C0=10 mM) is larger than Rf(C0=1
mM). On the other hand, as illustrated in Figure S2 of the Supporting Information, the degree of
increase in Ez,avg as (LN/2h) increases at V=−1 V is larger than that at V=+1 V so that Rf increases
doi:10.6342/NTU201801250 55
with increasing (LN/2h).
As seen in Figure S3 of the Supporting Information that for both V=±1 V, the averaged axial
electric field Ez,avg near the nanochannel tip at C0=100 mM increases with increasing (LN/2h).
However, Figure 2-4(c) reveals that as (LN/2h) increases from 2 to 5, the position at which the
local maximum of G(V=−1 V) occurs shifts towards the nanochannel tip end, where Ez,avg has the
largest value. This leads to a larger ionic current. In contrast, G(V=+1 V) varies inappreciably as
(LN/2h) increases from 2 to 5, so is the corresponding ionic current. Therefore, Rf increases with
increasing (LN/2h). If (LN/2h) is further raised from 5 to 11, the degree of increase in Ez,avg(V=+1
V) is larger than that in Ez,avg(V=−1 V), yielding a decrease in Rf. These explain the presence of
a local maximum in Rf as (LN/2h) increases from 2 to 11.
Figure 2-4(d) reveals that at C0=500 mM, G(V=−1 V) increases appreciably when (LN/2h) is
raised from 2 to 11, but G(V=+1 V) follows the reverse trend. Since the ionic current is
proportional to G, Rf also increases appreciably from 1.79 to 4.07.
To further examine the transport properties of the present cigar-shaped nanochannel, we
evaluate its selectivity S defined below:
| | | |
| | | |
I I
S I I
+ −
+ −
= −
+ , (2.9)
where I+=I1+I3 and I-=I2+I4 are the ionic current contributed by cations and that by anions,
respectively. If S>0, a nanochannel is cation-selective; if S<0, it is anion-selective, and
non-doi:10.6342/NTU201801250 56
selective if S=0. For S>0 (S<0), the more appreciable the deviation of S from 0 the more cation-
(anion-) selective it is, implying that the ionic current contributed by cations (anions) is larger
than that that by anions (cations).
Figure 2-5 shows the simulated variation in the selectivity S as a function of the surface
curvature (LN/2h) at pH 10. Since the pH is higher than the IEP (6) of the nanochannel, it is
negatively charged in this case, and therefore, cation-selective (S>0) and the ionic current is
mainly contributed by cations. The variation of S with (LN/2h) at V=+1 V is more sensitive
than that at V=−1 V, and for the same nanochannel shape and bulk salt concentration, S(V=+1
V)>S(V=−1 V), in general. This is because the nanochannel is negatively charged at pH 10,
ion depletion occurs when V=+1 V. In this case EDL is thick and its overlapping is significant
so that the ionic current is mainly contributed by counterions (cations), yielding a large S.
Note that for the same applied voltage and surface curvature S increases with decreasing C0,
and the nanochannel becomes more cation-selective. This is because the lower the C0 the
thicker the EDL and the more significant its overlapping near the nanochannel tip region,
making it more difficult for coions to pass through so that counterions are enriched and coions
depleted, and S deviates more from zero. In addition, since EDL overlapping becomes less
significant as (LN/2h) increases, for fixed values of V and C0, S decreases as (LN/2h) increases
from 2 to 11. A comparison between Figure 2-5(a) and Figure 2-5(b) reveals that S(Rb=15 nm)
doi:10.6342/NTU201801250 57
is slightly larger than S(Rb=25 nm). This is because the internal space of the nanochannel at
Rb=15 nm is smaller than that at Rb=25 nm. Therefore, the effect of EDL overlapping in the
former is more significant than that in the latter so that the amount of counterions (cations) in
the former is greater than that in the latter. Note that both S and Rf are dominated mainly by
the conditions near the nanochannel tip region. This is because the factors key to the present
system, including the analyte size effect[54, 55, 62, 63], the influence of the radial electrical
double layer[62, 64], and the major part of the applied axial voltage drop[54, 55, 65, 66], are
most significant in that region and act together to determine the final device performance.
Since the tip size is fixed, S is insensitive to the variation in (LN/2h) in Figure 2-5, especially
at V=+1 V.
2-3.2 Anion-Selective Nanochannel (pH 4)
At pH 4 the nanochannel is positively charged, and therefore, anion-selective. In this case Figure 2-6 reveals that there is a preference for ionic current at a positive applied potential
bias at this level of pH so that 0<Rf<1 for all the levels of the bulk salt concentration C0
examined. This figure also indicates that the higher the bulk salt concentration C0 and the
larger the base radii Rb the greater the ionic current. In addition, the larger the surface
curvature (LN/2h) the larger the ionic current, which can be explained by the same reasoning
doi:10.6342/NTU201801250 58
as that employed in the discussion of Figure 2-2.
Figure 2-7 summarizes the dependence of the rectification factor Rf on the surface
curvature (LN/2h) under various conditions at pH 4. For the cases of both values of Rb assumed
(15 and 25 nm), Rf is seen to increase monotonically with increasing (LN/2h) at all the levels
of C0 considered. Note that both Rf(Rb=25 nm) and Rf(Rb=15 nm) are lower than unity, and
for the same values of Rt and (LN/2h), Rf(Rb=25 nm)<Rf(Rb=15 nm). That is, the performance
of rectifying ionic current of a nanochannel having a larger Rb (25 nm) is better than that
having a smaller Rb (15 nm). This is because the larger the difference between Rt and Rb the
greater the difference in the degree of electric double layer overlapping near the nanochannel
tip regions, and therefore, the more significant the ICR effect. Figure 2-7 and Figure S5 of the
Supporting Information also suggest that Rf has a local minimum as the bulk salt concentration
C0 varies.
Figure 2-7 indicates that for the level of the bulk salt concentration considered, the larger
the (LN/2h) the closer the Rf to unity, that is, the less significant the ICR effect. This can be
explained by Figure 2-8 and Figures S6-S9 of the Supporting Information. As seen in the latter
Ez,avg is appreciable near the nanochannel tip region, where the effect of surface charge is most
important. In addition, for both V=±1 V, the larger the (LN/2h) the stronger the Ez,avg. Since
the qualitative behaviors of both G and Ez,avg at Rb=15 nm in Figure 2-7 are similar to those at