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OPEN Ion transport and current

rectification in a charged conical
nanopore filled with viscoelastic
fluids
Mohit Trivedi & Neelkanth Nirmalkar*
The ionic current rectification (ICR) is a non-linear current-voltage response upon switching the
polarity of the potential across nanopore which is similar to the I–V response in the semiconductor
diode. The ICR phenomenon finds several potential applications in micro/nano-fluidics (e.g., Bio-
sensors and Lab-on-Chip applications). From a biological application viewpoint, most biological fluids
(e.g., blood, saliva, mucus, etc.) exhibit non-Newtonian visco-elastic behavior; their rheological
properties differ from Newtonian fluids. Therefore, the resultant flow-field should show an additional
dependence on the rheological material properties of viscoelastic fluids such as fluid relaxation time ()
and fluid extensibility (ε). Despite numerous potential applications, the comprehensive investigation
of the viscoelastic behavior of the fluid on ionic concentration profile and ICR phenomena has not
been attempted. ICR phenomena occur when the length scale and Debye layer thickness approaches
to the same order. Therefore, this work extensively investigates the effect of visco-elasticity on the
flow and ionic mass transfer along with the ICR phenomena in a single conical nanopore. The Poisson–
Nernst–Planck (P–N–P) model coupled with momentum equations have been solved for a wide range
of conditions such as, Deborah number, 1 ≤ De ≤ 100, Debye length parameter, 1 ≤ κRt ≤ 50, fluid
extensibility parameter, 0.05 ≤ ε ≤ 0.25, applied electric potential, −40 ≤ V ≤ 40, and surface
charge density σ = −10 and −50. Limited results for Newtonian fluid (De = 0, and ε = 0) have also
been shown in order to demonstrate the effectiveness of non-Newtonian fluid behaviour over the
Newtonian fluid behaviour. Four distinct novel characteristics of electro-osmotic flow (EOF) in a
conical nanopore have been investigated here, namely (1) detailed structure of flow field and velocity
distribution in viscoelastic fluids (2) influence of Deborah number and fluid extensibility parameter on
ionic current rectification (ICR) (3) volumetric flow rate calculation as a function of Deborah number
and fluid extensibility parameter (4) effect of viscoelastic parameters on concentration distribution of
ions in the nanopore. At high applied voltage, both the extensibility parameter and Deborah number
facilitate the ICR phenomena. In addition, the ICR phenomena are observed to be more pronounced at
low values of κRt than the high values of κRt . This effect is due to the overlapping of the electric double
layer at low values of κRt.

Over the years, synthetic nanopores have attracted significant research attention due to their relevance and poten-
tial application in bio-sensing devices, DNA ­sequencing1, micro-nano pumping ­devices2, mixing ­applications3,
and polymer t­ ranslocation4–9. A detailed understanding of flow and ionic transport in these nanopores is impera-
tive not only in the investigation and interpretation of physiological processes/mechanisms in living beings but
also in the development of smart sensing devices based on the electrochemical properties of the flow s­ ystem10.
The ionic current through a conical nanopore at a negative applied voltage bias differs from that at a positive
applied voltage bias, implying that ion transport is preferential in one direction. It is widely noticed that when the
Debye length is comparable to the diameter of a conical nanopore, the ionic current rectification (ICR) phenom-
ena occurs due to ionic concentration imbalance across the electrical double l­ ayer11. Thus, the ICR phenomenon
in a nanopore is analogous to an asymmetric diode-like ionic current-voltage behaviour. In addition, the ionic
selectivity and ICR properties of the nanopore can also be utilized to detect and quantify the characteristics of the
flow field, and the ionic concentration ­distribution12. Therefore, the ICR phenomena are fundamentally related

Department of Chemical Engineering, Indian Institute of Technology Ropar, Rupnagar 140001, India. * email: n.
nirmalkar@iitrpr.ac.in

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to the viscous and elastic behavior of the fluid and the electrical properties (relative permittivity, ionic diffusivity,
and the valency of the cation and anions) in the fl ­ uid13,14. In this work, elastic effects on ICR phone have been
investigated in a single conical nanopore. The Poisson–Nernst–Planck (P–N–P) model coupled with momentum
equations have been solved using finite volume method. Five distinct novel features of electro-osmotic flow
(EOF) in a conical nanopore have been investigated here, namely (1) the effects of the memory of visco-elastic
fluid and extensibility has been investigated in terms of detailed structure of flow field and velocity distribution
(2) analysis of visco-elastic effects on volumetric flow rate (3) detailed findings on the influence of the Debye
length, Deborah number and fluid extensibility parameter on ionic current rectification (ICR) and discussions
on the applied potential over the ionic current behavior (4) volumetric flow rate calculation and analysis as a
function of Deborah number and fluid extensibility parameter (5) Effects of visco-elastic parameters on ionic
concentration distribution. In a pioneer study of Wei et al.15, the ICR phenomena have been observed for the
flow of a KCl solution in a silica nanopore.
Several ­studies11,16–21 have since then reported the ICR in synthetic nanopores. It was concluded that the resist-
ance to the ionic current depends on the flow direction, surface charge, and nonuniformity of cross-section of the
­nanopore11,18,22,23. This feature of nanopores can potentially be utilized for separation ­processes24 or lab-on-a-chip
­applications25. The nanopores can be both symmetric or asymmetric in shape. Example of symmetric shapes are
cylindrical nanopores while a conical nanopore is known as a asymmetrical shape. When such nanopore surfaces
are given a finite charge and introduced to an ionic fluid, an immobile and stationary electric double layer (EDL)
is developed over the surface. The thickness of this layer is based on several factors like the charge density of the
pore-wall, the valency and concentration of the ions present in the fluid etc. Moreover, these EDLs formed at
nanopore walls can be carefully tuned to vary the thickness of the EDL. As the value of EDL thickness becomes
equivalent to the nonpores radius the flow through the nanopore is suppressed because of the blockage in the
cylindrical nanopore. This property of the nanopore could be very interesting for an asymmetric shaped nanopore
or a conical nanopore. The gradual increase/decrease in the flow area coupled with the varying EDL thickness
can act as a valve-like nature for a conical nanopore. Moreover, on the contrary to the cylindrical nanopore,
for a given positive voltage bias, a conical nanopore will exhibit higher flow-rate and ionic current than that at
a negative potential bias of the same magnitude. These nano-scale valves can be used to design and control the
value of the flow and ionic current passing through them in a micro-fluidic pumping devices or in Lab-in-Chip
applications. Therefore, since the ionic current flowing through the nanopore is modulated and rectified due to
the asymmetric (diverging and/or converging) shape of the nanopore, it holds an analogy with the function of an
electrical ­transistor26 too. A positive applied potential bias in a conical nanopore offers a high ionic current under
electro-osmotic flow (EOF), whereas a negative applied potential bias reduces the ionic current significantly,
suggesting higher electrical resistivity in the nanopore for negative or backward applied potential ­bias27. While
the ICR phenomenon is quantified in terms of the net rate of ion transfer or the ionic current, it also yields an
effective electro-osmotic flow rectification (EFR)28. EFR property of a nanopore can be utilized to design nano-
micro scale pumping ­devices29. Some s­ tudies13,30 present excellent comparisons between the experiment and the
analytical solutions based on Poisson-Nernst-Planck (PNP) equations.
These results have consolidated that if the diameter of the pore is larger than 10% of the Debye l­ ength31,32 the
applicability of the continuum hypothesis is still valid. Therefore, numerical studies have also been performed
for the flow and ion transfer through a nanopore over recent years. Though, few of these studies have incorpo-
rated the effect of the electro-osmotic flow induced from the interaction of the charge density and the applied
electric field via the PNP model. The findings of such studies are limited to some special c­ ases13,16,33. It has been
observed from these studies that the applied electric field in conjunction with the surface charge on the nanopore
significantly influence the flow field and consequently the ionic current distribution in the n ­ anopore17,34. Fur-
35,36
thermore, since most of the biological fluids, such as mucus, saliva, blood and other ­lipids exhibit viscoelastic
flow behavior. Therefore, their rheological flow properties exhibit additional features, i.e., the memory effect and
principal stress distribution in addition to shear-dependent v­ iscosity37. Hence, the electro-osmotic flow and the
ICR phenomena of such fluids are expected to differ from the conventional generalized Newtonian fluid (GNF)
behavior. Some recent studies on micro/nanofluidics have considered non-Newtonian flow behavior, e.g., an ana-
lytical solution of the EOF of power-law fluids in a rectangular micro-channel has been investigated by Das and
­Chakraborty38. They have observed that the non-Newtonian shear-thinning nature facilitates the concentration
distribution, whereas it also has insignificant influence over the concentration distribution. Zimmerman et al.39
have numerically studied the electro-osmotic flow in a T-junction for Carreu-Yasuda fluids, and the resulting
end wall pressure profile has been found to be dependent on the relaxation time and power-law index. The EOF
in rectangular and cylindrical tubes for various (Power-law fluids, Bingham plastic fluids, and Eyring fluids)
non-Newtonian models have been studied by Berli and O ­ livares40, it has been concluded that the effect of non-
Newtonian rheology is confined to only the diffusive ionic flux through micro-channel. Analytical results for
power-law flow behavior and power-law index, n has been investigated by Zhao and Y ­ ang41, and it has been shown
that increasing the electrokinetic parameter κH or decreasing the power-law index n can lead to an increase in
the volumetric flow rate Q for the electro-osmotic flow of power-law fluids. A few notable studies on viscoelastic
flow in micro and nano-slits/circular cross-section have also been presented by Li et al.42, and Wang et al.43 for
Maxwell fluids, while Mei et al.44, Chen et al.45 and Park and L ­ ee46 have investigated Phan–Thien–Tanner (PTT)
fluids. While no prior studies on the effect of visco-elasticity on ICR phenomena have been reported yet. Hence,
the present study is dedicated to investigate the flow of simplified Phan–Thien–Tanner fluids through a coni-
cal nanopore and the ICR phenomena. The results are presented in the form of streamlines, velocity contours,
velocity profiles, and current vs. potential (I vs V) plots.

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Figure 1.  (a) Conical nanopore with a forward potential bias. The electric double layer is highlighted with
the red shading. The blue shading represents the bulk solution. (i) When electric double layers are thick and
interaction occurs at the nanopore tip (ii) When they are thin and there is no interaction. Dimensions are not to
scale. (b) Schematic diagram of computational domain.

Problem description
The steady-state electro-osmotic flow of visco-elastic (sPTT) binary electrolyte has been considered in a conical
nanopore (Fig. 1a) with a tip Radius Rt and axial length H = 200Rt and divergent ­angle47 α = 1.432◦. The flow
field has been considered to be 2-D axi-symmetric. Since the flow, concentration, and potential field have been
expected to be symmetric along the θ direction. Therefore, a thin 2D slice of the domain is considered here in
r-z co-ordinate frame for the numerical study (Fig. 1b). The nanopore sizes typically range from 2 to 150 ­nm48,
and the influences of gravitational force can be neglected in such small size ranges, therefore the gravitational
force has been considered to have no effect on flow, concentration, and potential fields. The effect of variation
in cone angle has been also discussed in the terms of radial velocity profiles and I vs V curves in this study and
the corresponding results (Figs. 7 and 11). Due to the interaction of the negatively charged pore surface and the
counter charged (positive) ions, a layer of positively charged ions forms on the charged wall surface, followed
by a relatively thick convective layer of counter-charged ions known as the electric double layer (EDL). At the
nanopore surface, the positive ions are concentrated near the charged wall, and the co-ions (negative) remain
mostly close to the core of the flow. It is also considered that the ionic concentration is very low in the solution
(i.e., dilute electrolyte). Therefore, the Boltzmann equilibrium can be considered. EDL thickness is often given
by D = 1/κ , which can be scaled by the length scale, Rt as κRt . An electric field E is applied in the z-direction
by applying a potential bias V across the nanopore. The ICR phenomena occur when the electric double layer
overlaps at the tip (convergent end) of the nanopore. The electric double layer does not overlap when κRt is high
and consequently, the ion migration does not gets affected by the double layer. Therefore, the ions at the core of
the fluid remain solely driven by the applied potential bias. As the value of the κRt decreases (i.e., the thickness
of the double layer increases), the ionic current through the pore gets dampened by the resisting forces, arising
due to the ionic charge density near the tip of the nanopore. This behavior is analogous with the characteristics of
a p-n type electric ­transistor26. Therefore, at a forward potential bias (i.e., the higher potential at the convergent
end of the pore), the flow of cations is directed towards the divergent end of the pore. The flow configuration
of ions in forward potential bias assures that the double layers do not overlap, and thus the resulting electrical
resistance reduces significantly. At the reverse bias, the cations are forced to flow towards the convergent end of
the nanopore, and these cations get accumulated at the end of the nanopore due to the overlapped electric dou-
ble layer. Thus, in order to delineate the effect of the interaction of the electric double layer and applied electric
field at the tip of the nanopore, the coupled potential, concentration, and velocity fields are solved using Pois-
son–Nernst–Planck (P–N–P) model.

Governing equations. The flow induced by applied potential difference is given by the equations of conti-
nuity and momentum and it is written as follows:
∇ ·u=0 (1)

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− ∇p + ∇ · τ + ρe E = 0 (2)
the potential field is given by Poisson’s equation as follows:
ρe
∇2ψ = − (3)
εp

where the space charge density ρe is defined as:

2

ρe = F zi ci (4)
i=1

where: zi and ci are the valance and concentration of ith species respectively. The externally applied electric field
is expressed in terms of the potential gradient as follows:
E = −∇ψ (5)
Similarly, the ionic flux balance across the nanopore is expressed by the Nernst–Planck equation:
∇ · Ni = 0, i = 1, 2 (6)
where
ezi Di ci
Ni = uci − Di ∇ci − ∇i ψ i = 1, 2 (7)
kB T

The constitutive equation of the sPTT ­model49,50 has been given by:
∇
f (τkk )τ + τ = 2ηP D (8)

where ηP is the polymeric viscosity coefficient,  is the relaxation time of the viscoelastic fluid and
D = (∇uT + ∇u)/2 is the rate of deformation tensor, the f (τkk ) is called the stress coefficient and can be defined
as:
f (τkk ) = 1 + (ε/ηP )τkk (9)
where, τkk denotes the trace of the extra stress tensor and ε is known as the PTT parameter. It governs the exten-
sibility and elongation property of the PTT fl ­ uid49,50. The ∇
τ is the upper convective derivative of the polymeric
tensor and can be defined as:
∇ Dτ
τ = − ∇uT · τ − τ .∇u (10)
DY
Eqs. (1–7) have been rendered dimensionless using the corresponding scaling variables as listed in Table 1. They
are written in their respective dimensionless forms as follows:
∇ ∗ .u∗ = 0 (11)

2
(κRt )2 
− ∇ ∗ p∗ + ∇ ∗ · τ ∗ + zi ci ∗ ∇ ∗ ψ ∗ = 0 (12)
2
i=1

2
∗ (κRt )2 
∇2 ψ∗ = zi ci ∗ (13)
2
i=1

∇∗ · N∗ = 0 (14)

N ∗ = u∗ ci∗ − Di∗ ∇ ∗ ci∗ − zi Di∗ ∇ ∗ ψ ∗ i = 1, 2 (15)

Dimensionless constitutive equation of the sPTT fluids (i.e., Eqs. 8–10) can be written as follows:
De ∇∗
τ∗ + (τ + ετkk ∗ τ ∗ ) = 2D∗ (16)
κRt
where u∗, p∗, τ ∗ and ψ ∗ are the dimensionless velocity, pressure, the polymeric tensor, and electric potential
respectively.
Here De is the Deborah number and it is defined as,
De = Uo κ (17)

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where
1/2
2Co ez 2 F

1
κ= = (18)
D εp kB T

which is known as the Debye-Hückel parameter. Here D has units of length. The Rt /D or κRt gives the meas-
ure of the thickness of the electric double layer (EDL) developed near the charged pore walls and known as the
dimensionless Debye length.
The migration of ions, in turn leads to the flow of electrical current in the nanopore, thus the surface average
current across the nanopore is written as follows:
  2

∗ I 1 ∗
I = = 2 zi Ni · ndS (19)
Io Rt S i

The dimensionless surface charge density is defined a­ s51:

 
ρs ez
σ = = −2 sinh ψo (20)
zFRt 2 Co kB T
The boundary conditions are given below.

• At the boundary (1)

  A uniform pressure (i.e., p∗ = 0) with zero stress and fixed potential V ∗ is imposed and ionic concentration
is fixed as ’1’.
p∗ = 0, ∇ ∗ .u∗ = 0, ci ∗ = 1, ψ ∗ = 0.
• At the boundary (2)
  The gradients of potential and concentration are zero and the slip condition for velocity has been imposed.
n · u∗ = 0, ∇ ∗ .ψ ∗ = 0, ∇ ∗ .ci ∗ = 0.
• At the boundary (3)
  Similarly, the gradients of potential and concentration are zero and no-slip velocity condition at the surface
is applied.
uj ∗ = 0, ∇ ∗ .ψ ∗ = 0, ∇ ∗ .ci ∗ = 0.
• At the boundary (4)
  The no-slip velocity condition at the surface with zero concentration gradient is considered and a fixed
surface charge density σ has been imposed, which has been expressed in terms of fixed applied potential ψo
by Eq. (19).
σ
uj ∗ = 0, ∇ ∗ .ψ ∗ = − , ∇ ∗ .ci ∗ = 0.
εp
• At the boundary (5)
  A uniform pressure (i.e., p∗ = 0) with zero stress and electric potential is fixed as zero (ground) and ionic
concentration is fixed as ’Co ∗’.
p∗ = 0, ∇ ∗ .u∗ = 0, ci = 1, ψ ∗ = ±V .
• At the boundary (6)
  For velocity, concentration and potential fields axi-symmetric boundary condition has been imposed.
n · ∇ ∗ .u∗ = 0, n · ∇ ∗ .ψ ∗ = 0, n · ∇ ∗ .ci ∗ = 0.

Results and discussion

Validation. In order to ascertain the accuracy and precision of the numerical results, it is imperative to per-
form a few benchmark comparisons. Therefore, the chosen numerical scheme (see the supporting information)
has been thoroughly scrutinized by comparing the present results with established e­ xperimental19, ­analytical17,
and ­numerical18 studies. A scant number of theoretical fi
­ ndings16,17,30,40,52,53 are available on the electro-osmotic
flow. The potential profile for a non-convective flow of binary electrolyte (KCl) over a charged surface has been
compared with analytical solution (Eq. 21) derived by the Gouy–Chapman t­heory17,54 for two values of bulk
ionic concentration, Co of 1 mM and 100 mM and a fixed surface charge density of σ = −0.001C/m3 (see Fig. 2).
2kB T 1 − K exp(−x/Lo )
φ(x) = ln (21)
ez 1 + K exp(−x/Lo )
 1/2 
where x represents distance along the charged wall; K = S/ 2 + 4 + S2 and S = −ezσ/(kB Tε). The


results from the applied numerical schemes are validated with the analytical solutions with an excellent

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1.2
Lo = 500 nm
1
200 nm
0.8 y
x
1 mM

x/L = 0
0.6
100 mM
0.4

/
17
Symbols: White and Bund
Lines: Present
0.2

-0.2
-0.05 0 0.05 0.1 0.15 0.2
x/L

Figure 2.  Comparison of the variation of cross-sectional potential profile with White and B
­ und17 at
ρs = −1 mC/m2.

15
Current, I (nA)

2
-240 mC/m2
-170 mC/m2 10
-80 mC/m

-0.5 -0.25 0 0.25 0.5

Potential, V (V)
-5
Symbols: Liu et al.18
-10 Lines: Present

-15

Figure 3.  Comparison of the variation in average current with applied voltage for a bulk concentration of KCl
at 0.05 M with Liu et al.18.

agreement, as demonstrated by Fig. 2. In addition, the present numerical scheme has been validated with experi-
mental (Petrossian et al.19) and numerical (Liu et al.18) studies, and the corresponding comparisons are presented
in Figs. 3 and 4. The present results agree well with the respective experimental results (max. difference 8%, in
the case of Petrossian et al.19). Thus, the current numerical settings are used to perform extensive numerical
simulations, and the resultant velocity, potential, and concentration fields are further derived in terms of velocity
contours, streamlines and the velocity profiles, volumetric flow rate, average concentration profiles, and profiles
of applied potential with net ionic current passing through the nanopore in terms of the Debye length ( κRt ),
Deborah number (De), sPTT extensibility parameter (ε), applied potential difference (V) and the surface charge
density (σ ).

Streamlines and the velocity contours. The velocity magnitude contours and streamlines demonstrate
the features of the flow, such as the spatial variations in velocity distribution and flow field developments with
the varying governing parameters. The electro-osmotic force bears a positive dependence on the bulk concen-
tration. Thus, it is expected that the flow field may intensify with the bulk concentration. Similarly, the applied
potential difference also strengthens the volumetric electro-osmotic f­ orce55, and again it can be speculated that
the higher the applied potential difference, the stronger the flow fields. In the EOF, the velocity and stress fields
depend upon both the rheological as well as the electrochemical properties of the fluid, i.e., the density, viscos-
ity, the applied electric potential, the surface charge density on the wall, and the bulk ionic concentration. For
instance, stability criteria for the flow of viscoelastic Oldroyd-B fluid in a microchannel depends on the bulk
electrolyte concentration as well as the applied potential Ji et al.56. The further discussion over the flow field in
EOF of viscoelastic fluids is limited by a few scarce ­studies42,44,45. These studies include some bounded insights
to the viscoelastic flow behavior for flow in microchannel/microtube for few specific sets of governing variables,
e.g., bulk concentration, applied potential etc., and the properties and characteristics of viscoelastic EOF have
not yet been comprehensively studied for a conical nanopore with a given surface charge.

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103
19
Symbols: Petrossian et al.
Lines: Present

Cunductance, I/V (nS)

102

101

100

10-1 0 1 2 3
10 10 10 10
Concentration, Co (mM)

Figure 4.  Comparison of the variation in average conductance (I/V) with bulk average concentration with
Petrossian et al.19.

Scale Symbol Definition

Length scale Rt Rt
Velocity scale Uo εp kB2 T 2
ηP R t e 2 z 2
Potential scale VT kB T
ez
Ionic concentration scale Co Co
Ionic current Io FCo Uo Rt 2

Table 1.  The scaling variables and their definition.

Furthermore, the rheological behavior of viscoelastic sPTT fluids depends upon dimensionless relaxation
time, De and extensibility parameter, ε49,50. Therefore, it is important to inspect and investigate the streamlines
and velocity contours to visualize the developments in the flow field with the varying governing parameters.
Figure S1 (see the supporting information) and Fig. 5 present the streamlines and the dimensionless velocity
contours considering two extreme values of κRt and different values of ε for V = 40, σ = −50 and De = 1 and
De = 100 respectively. It can be deduced from these plots that in EOF with smaller relaxation time, i.e., De = 1
(Fig. S1), and for thin Debye length, i.e., κRt = 50, the effective fluid viscosity near the nanopore wall diminishes
as the fluid extensibility (ε) increases. Thus, the magnitude of the velocity field strengthens with the increase
in the value of ε. In other words, fluid extensibility decreases the effective viscosity, which mimics the so-called
shear-thinning nature of the fluid. Therefore, it is concluded that the overall velocity magnitude increases with
the extensibility of viscoelastic (sPTT) fluids. At the highest Debye length (i.e., κRt = 1), a thick electric double
layer forms adjacent to the wall. This electric double layer overlaps and engulfs the flow area at the convergent
end of the nanopore (i.e. nanopore tip), and thus, the flow is expected to be constricted. Therefore, the velocity
magnitude near the walls is observed to be weaker than that of the core of the flow. The overlap of the electric
double layer gives rise to a low shear region, and therefore the influence of fluid extensibility (ε) on the flow field
is observed to be negligible in this limit. In other words, the ε has little to no influence on the flow field at such
low values of κRt (i.e., κRt = 1) and De (i.e., De = 1) at the nanopore tip. Essentially, the rheological parameters
i.e. the Deborah number De, the sPTT extensibility parameter ε and the Debye length κRt exhibit a positive
influence over the spatial flow distribution.

Volumetric flow rate: effect of De, ε and κRt. From the discussion of the velocity field contours
inside the conical nanopore presented in the previous section, it could be inferred that, the overlapping of the
electric double layer at the convergent end (the tip) of the nanopore affects the momentum flux through the
­nanopore45,52,57. The insights about the characteristics of the flow field in EOF for a micro-channel/micro or
nanopore can also be visualized by the corresponding radial velocity profiles as presented in Figs. 6 and 7. The
velocity profiles in a 2D micro-geometry (micro-channel/nozzle) has been investigated in terms of the Debye
length, κRt , Deborah number, De, cone angle, α and rheological constraints by several numerical studies, such
as Bezerra et al.58, Chen et al.45, Mei et al.44, and Tseng et al.47 as well as analytical studies e.g. Afonso et al.52,57
and Wang et al.43.
As discussed in previous section, the Debye parameter, κRt and Deborah number, De exhibit a positive influ-
ence on the flow field. In other words, an increase in κRt and De increases the average and maximum velocity.
Therefore, the radial velocity profiles at the tip of the nanopore are shown in Fig. 6, for Vo = 40, σ = −50, four

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Figure 5.  Representative streamlines and velocity contours at De = 100, V = 40, σ = −50 and extreme values
of κRt and ε.

different values of Debye length κRt , and various values of ε with two extreme values of the Deborah number
i.e., De = 1 and De = 100, respectively. Moreover, results of Newtonian fluids are also added for the sake of
comparison. The trends in Fig. 6 can be illustrated as follows: the velocity profile of Newtonian fluids has been
found to be a lower than that of a visco-elastic fluid. The difference between the maximum velocity profile of the
Newtonian fluid and for the fluid with the lowest relaxation time and PTT parameter is found to be proportional
to the Debye length and varies roughly from 20% to 700% as Debye length is varied from highest value (i.e.
κRt = 1) to the lowest value ( κRt = 50). Furthermore moving towards the purely visco-elastic behaviour, as for
the lowest Deborah number, i.e., De = 1 and for the highest Debye length ( κRt = 1), the velocity profile cor-
responds to the different values of ε overlap to each other at the nanopore tip. This effect suggests that the fluid
extensibility (ε) has a negligible effect on the velocity field. Upon gradual reduction in Debye length, the value
of the parameter κRt increases. The maximum velocity shows a positive dependence on the value of κRt , for a
given value of ε. In brief, the lower the thickness of the electric double layer, the higher the maximum velocity. In
addition, the influence of the fluid extensibility on κRt (ε) also reflects in the flow field and the maximum velocity,
it increases by ≈ 170% for κRt = 10 and by ≈ 470% for κRt = 50 as the value of ε varies from 0.05 to 0.25. This
clearly suggests a significant influence of the value of ε on the maximum velocity at a smaller Debye length. For
a fixed value of ε, the velocity distribution across the nanopore gradually shifts from a parabolic-like profile to
the plug-like profile upon increasing De (Fig. 5) and κRt as shown in Fig. 6. Clearly, the proportion of the solid-
like behavior increases with the Deborah number, and thus the flow at the center of the nanopore behaves like
a plug while the fluid-like response is seen towards the wall with a steep velocity gradient. Similarly, the thinner
electric double layer also gives rise to a plug-like velocity profile, irrespective of Deborah number. This is simply
because of the low flow resistance for the migration of ions outside the electric double layer. At De = 100, the
effect of ε45 on the resultant velocity profiles is significant even at κRt = 1. Furthermore, the maximum velocity
exhibits a similar dependence on the values of ε and κRt , as illustrated before. Precisely, the maximum velocity
is observed to enhance with the value of ε by ≈ 300% for κRt = 10 and by ≈ 2000% for κRt = 50. All in all, it
has been observed that the momentum flux near the nanopore wall enhances more profoundly with the PTT
parameter, ε than with the variation of the Debye parameter, κRt . In summary, it is concluded that the fluid exten-
sibility (ε) and the Deborah number (De) strengthen the flow field (Fig. 5), while the value of κRt and σ controls
the thickness of the electric double layer in the nanopore. Furthermore, It has been reported that the average
velocity varies linearly with the applied electric potential and exhibits an inverse dependence on the polymeric
­viscosity56. Furthermore, in a conical nanopore, an increase in the cone angle accompanies the increase in the
associated flow area. Also, the body force exerted on the fluid by the applied potential bias is proportional to the
flow area. Thereby, suggesting the corresponding trends of velocity, concentration and ionic current profiles with

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Figure 6.  Radial velocity variation with ε at De = 1 (empty symbols) and De = 100 (filled symbols), V = 40,
σ = −50, and Newtonian fluid for various values of κRt.

3.5 4.5 800 800

Rt = 1 Rt = 50 α = 1.432
o

3.0 4.0 α = 2.864o

ε = 0.05
Radial velocity profile U/Uo

600
Radial velocity profile U/Uo

2.5 3.5 α = 4.296o 600

o
α = 5.728
3.0 400
2.0
2.5 400
1.5
2.0 200
1.0
1.5 200
0.5 0
1.0
0.0 0.5 0
-200
-0.5 ε = 0.25 0.0
-1.0 -0.5 -400 -200
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
r/Rt r/Rt

Figure 7.  Radial velocity variation with α at ǫ = 0.05 (red symbols) and ǫ= 0.25 (blue symbols), V = 40, De =
100 and σ = −50 and for two extreme values of κRt.

respect to the variation in the cone angle. Tseng et al.47 have discussed the effect of the cone angle on the velocity
as well as the concentration fields.They have demonstrated that, for a fixed charge density and the applied volt-
age, as the cone angle increases, the velocity field and ionic current within the nanopore intensifies whereas the
average cation concentration depletes. Therefore, the variation in the cone angle has been analysed for four suc-
cessively increasing values of α, starting from 1.432o and upto 5.728◦. Figure 7 presents the radialvelocity profile
variation with the cone angle α, for a fixed applied potential bias (V = 40) and surface charge density (σ = -50),
the maximum Deborah number ( De = 100), and two extreme values of the Debye length and PTT parameter.
The maximum velocity and therefore the average volumetric flow rate is found to be positively dependent on
the cone angle. As the cone angle, α increases from 1.432◦ to 5.728◦, for lowest value of κRt or the highest Debye
length, the corresponding results show an 88.46% increase in the maximum velocity for ε = 0.05 and an 91.23%

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250 80

200 κRt = 1 70
κRt = 10 60
150
κRt = 30
50
100 κRt = 50
40
50

Q
30
0
20
-50
10
-100 0

-150 -10
0.05 0.1 0.15 0.2 0.25
PTT fluid extensibility parameter, ε

Figure 8.  Dimensionless volumetric flow rate variation with ε for various values of κRt for De = 1 (dashed line)
and De = 100 (solid line) at V = 40, and σ = −50.

increase for ε = 0.25. When the Debye length is minimum and κRt = 50, the maximum velocity increases by
138.45% for ε = 0.05 and 197.65% for the highest fluid extensibility i.e. ε = 0.25. Thus it could be concluded
that the maximum velocity is directly dependent on the cone angle and the highest variation is achieved at the
maximum fluid extensibility or the highest value of PTT parameter and the lowest value of Debye length or
the highest value of κRt . Moreover, the average velocity can also be quantified in terms of volumetric flow rate.
Therefore, the variation of the dimensionless volumetric flow rate with the values of ε and for various values of
κRt at Vo = 40, σ = −50, De = 1 and De = 100, is shown by Fig. 8.
Similar to the discussion given for the radial velocity profiles, the dimensionless volumetric flow rate, Q is
found to positively correlate with the value of the κRt and De. Also, the volumetric flow rate monotonically
increases with the value of ε. Furthermore, the increase in the flow rate is observed to be more pronounced at
De = 100 than that at De = 1, at ε >= 0.17. In summary, the volumetric flow rate shows a positive correlation
with the κRt , De, and ε.These trends are in line with results reported for the straight channel by Ji et al.56. It is
also to be noted that the average velocity varies linearly with the applied electric potential and exhibits an inverse
dependence on the polymeric viscosity.

Concentration distribution: effect of κRt , De and ε. The axial concentration distribution provides
useful insights into the EOF phenomena in a conical shape nanopore. In the absence of elastic effects (i.e.,
De = 0)16,18,22,23,55 increasing the applied potential increases the concentration of ions near the nanopore tip.
The maximum in the concentration profile occurs near the nanopore tip, and it increases with the increasing
electro-negativity of the cation of salt. The pH of the medium also exhibits an aiding effect on the concentration
distribution. Figure 9, represent the variation of the average ionic concentration for the anions and cations along
the length of nanopore, for two values of Debye length parameter (i.e., κRt = 1 and 50), the extreme values of
Deborah number (De), V = 40 and for a fixed surface charge density σ = −50. The results corresponding to
the Newtonian fluids have also been added for the sake of comparison in the enhancement in the concentration
distribution by the viscoelastic fluids over the Newtonian fluids. It has been observed that at the Highest value of
Debye length (i.e. κRt ), difference between the concentration profiles of Newtonian and the visco elastic fluid is
almost negligible, thus all the plots appear to merge into one curve. Whereas, an augmentation of 16% has been
observed for the Newtonian fluid for κRt = 50 in comparison with the visco-elastic fluids. Since an electric dou-
ble layer exists near the negatively charged pore surface, cations in the electrolyte migrate under the electrostatic
force due to the charged nanopore wall. In contrast, the anions migrate away from the wall and merge with the
bulk flow towards the nanopore tip. Therefore, the average anion concentration is found to be maximum near
the tip of the nanopore and gradually decreases towards the base of the nanopore. Since the radius of the base
end of the nanopore is significantly larger than the Debye length (i.e., κRt ), thus the local electric double layer
does not interact at the base. The overlapping phenomenon of the double layer is only relevant at the convergent
end of the nanopore (i.e., the tip of the nanopore). At De = 1 (see Fig. 9a), for κRt = 1, the value of ε has no effect
on the the dimensionless anion or cation concentration profiles. On the other hand, at κRt = 50, which mim-
ics the lowest Debye length, the flow resistance is expected to be much smaller than that at κRt = 1. At De = 1,
the migration of anion and cation towards the surface of the nanopore and the nanopore core is facilitated by
the EOF and found to be moderately influenced as the value of ε increases. Also, the effect of the extensibility
parameter (i.e., ε) is significantly visible at De = 100 (Fig. 9b), since there are much sharper velocity gradients
near the wall compared to the case at De = 1. Precisely, the ionic concentration distribution of both cations and
anions is found to be maximum at the tip of the nanopore and decline towards the nanopore base. Moreover,
the ionic concentration distribution is found to be invariant with the ε and De at κRt = 1. While variations with
ε are observed at κRt = 100. The concentration profile flattens with an increase in the value of ε. On the other
hand, De exhibits an inverse dependence on the concentration distribution at κRt = 50. In a nutshell, it has been
found that at high values of Debye length, i.e., κRt = 1, relatively lower concentration gradients observed at the
tip of the conical nanopore for both the anions and cations, indicating the retarding of the ionic mass transport

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κRt = 1 κRt = 50
(a) 50 8 1.02 1.08
40
1.01 1.06
30 6
20 1
1.04
10 0.99
4

Ccation
Ccation
Canion

Canion
0 1.02
-10 0.98
2 1
-20 0.97
-30
0.96 0.98
0 =0
-40
-50 0.95 0.96
0 50 100 150 200 250 300 0 50 100 150 200 250 300
z/Rt z/Rt
(b) 8 1.02 1.06
40 7
1.01 1.04
6
20 1
5 1.02
4 0.99

Ccation
Ccation
Canion

Canion
0 1
3 0.98
= 0.25
2 = 0.2 0.98
-20 0.97 = 0.15
1 = 0.1
0.96 = 0.05 0.96
-40 0
-1 0.95 0.94
0 50 100 150 200 250 300 0 50 100 150 200 250 300
z/Rt z/Rt

Figure 9.  Axial average concentration variation of anion (empty symbols) and cation (filled symbols) with
scores of values of ε including the Newtonian fluids, for V = 40, σ = −50 and two extreme values of κRt (a)
De = 1 and (b) De = 100.

due to the interacting electric double layer at the tip of the nanopore. On the other hand, decreasing the Debye
length sharpens the ionic concentration gradients indicating the strong ionic current flow. In addition, the rate
of ion transport also shows a positive dependence on Deborah number. Therefore, it can be deduced that the
ionic mass flux is significantly dependent upon the Debye length κRt , fluid extensibility ε and the Deborah
number De.

Viscoelastic effect on ICR and CRR​. The total ionic flux in EOF constitutes three components: the con-
vective, the diffusive, and the electro-osmotic flux, respectively. The total ionic flux is integrated across the area
of the nanopore tip and quantified as the total ionic current through a nanopore. The corresponding definition
is expressed by Eq. 18. To the d­ ate16,18,22,23,55 the ICR phenomena and its characteristics are investigated in terms
of the governing parameters such as the bulk ionic c­ oncentration15,16,18,23,59, Co, the surface charge d ­ ensity22,30,
55 11
σ , salt ­type and the pH of the ­electrolyte . All in all, it is reported that the ICR phenomena accentuate with
the increasing value of pH, σ , Co, and the electronegativity of the salt cation. Here we have demonstrated the
individual and/or combined effects of the Deborah number De, the PTT parameter ε, the Debye length κRt and
the surface charge density σ on the ICR phenomena, the total ionic current through the nanopore has been plot-
ted against the corresponding applied potential bias V as I vs V graphs. Figure 10 shows the current vs potential
relationship for two extreme values of the Deborah number (De = 1 and 100), and two values of surface charge
density (σ = -10 and -50), at a fixed Debye length (κRt = 50) and for scores of values of ε. These figures also
incorporate the ionic current values for Newtonian fluids and it has been observed that Newtonian fluids cor-
respond to the lowest ionic current in the nanopore and the ionic current shows the least rectification (ICR) in
the case of Newtonian fluids. The ionic current is found to vary with the applied potential monotonically. The
ionic current increases with the value of ε. At a low Deborah number (i.e., De = 1) (Fig. 10a), the surface charge
density, σ has a negligible influence on the ionic current. Thus it is concluded that σ does not have a significant
influence on the ICR at De = 1. While, as the value of De increases, it intensifies the flow rate, which, in turn,
augments the rate of total ionic flux. Moreover, the surface charge density, σ demonstrates a positive depend-
ence on the ICR phenomena at a higher Deborah number ( De = 100) (Fig. 10b). This effect is similar to the
reported results on ICR in the absence of elastic effects. Figure 11 demonstrates the corresponding relationship
of the ionic currant and the applied potential for a fixed surface charge density σ = -50, De = 100 and κRt = 50
for two extreme values of fluid extensibility. It can be inferred from the results that the value of ionic current as
well as the rectification (ICR) is maximum for the maximum value of cone angle. Moreover the fluid extensibil-
ity facilitates the ionic current at the maximum value of cone angle. With the increasing value of α, the ICR is

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(a) 60 60
σ = -10 σ = -50

I
40 40

20 20
V V
-60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60

= 0.25
= 0.2
-20 = 0.15 -20
= 0.1
= 0.05
-40 De = 0, = 0 -40

(b) 60 60
I

I
40 40

20 20
V V
-60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60

-20 -20

-40 -40

Figure 10.  I vs V curve at different values of ε and Newtonian fluid at two extreme values of σ at (a) De = 1 and
(b) De = 100 and κRt = 50.

(a) (b)
ε = 0.05 80 ε = 0.25 120
I

I
100
60
80
40 60
40
20
V 20 V
-50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50
o
α = 1.432
α = 2.864
o -20
-20
α = 4.296o -40
α = 5.728o
-40 -60

Figure 11.  I vs V curve at different values of α at two extreme values of PTT parameter (a) ε = 0.05, and (b) ε =
0.25 at κRt = 50, De = 100, and σ = -50.

shown to be more pronounced at ε = 0.25 than at ε = 0.05, suggesting that the PTT parameter has a significant
influence on ionic current and the rectification phenomena at elevated cone angle. Figure 12 shows the variation
of the current rectification ratio (CRR)22 with the extensibility parameter (ε). The CRR is defined as the ratio
of the value of ionic current for the forward and backward potential bias of the same magnitude. The resulting
trends are observed to agree with the previous discussion on the velocity field. The CRR demonstrates a positive
dependence on the extensibility parameter (i.e., ε), the Deborah number, De, the surface charge density, σ , and
the applied potential bias V. In addition, an increase of 30% in the CRR value has been observed as the ε varies
from its lowest to its highest value for a maximum value of V, De and σ . Figure 13 and Figure S2 demonstrate the
I–V curves to delineate the effect of κRt and De respectively. Figure 13 shows the influence of the Debye length
(κRt ) on the ICR behavior of the nanopore, for two extreme values of the value of ε and the value of σ and a given
fixed value of Deborah number De = 100. Evidently, the ionic current rectification is facilitated by the decreas-
ing value of κRt or the increasing value of the Debye length. As noted earlier, at a low value of κRt , overlapping of

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σ = -10 σ = -50
(a) 1.6 2
V = 40
1.5 V = 30 1.8
V = 20
1.4 V = 10
1.6

CRR

CRR
1.3
1.4
1.2

1.1 1.2

1 1
0 0.1 0.2 0.3 0 0.1 0.2 0.3
PTT parameter, ε PTT parameter, ε
(b) 1.6 1.6
V = 40
1.5 V = 30 1.5
V = 20
1.4 V = 10 1.4
CRR

CRR
1.3 1.3

1.2 1.2

1.1 1.1

1 1
0 0.1 0.2 0.3 0 0.1 0.2 0.3
PTT parameter, ε PTT parameter, ε

Figure 12.  Variation of current rectification ratio (CRR​) with ε for two values of σ and κRt = 50 at (a) De = 1
and (b) De = 100.

(a) 80 80
σ = -10 σ = -50
I

60 I 60

40 40

20 20
V V
-50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50

Rt = 1
Rt = 10
-20 Rt = 30 -20
Rt = 50
-40 -40

(b) 80 80
I

60 60

40 40

20 20
V V
-50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50

-20 -20

-40 -40

Figure 13.  I vs V curve at different values of κRt at De = 100 (a) ε = 0.05 and (b) ε = 0.25.

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the electric double layers occurs at the tip of the conical nanopore. Thus, it can be deduced that the ICR is higher
for the lower values of κRt (Fig. 13a). As it can be observed from Figure S2 (see the supporting information), the
effect of the De is only found to be significant for ε = 0.25 and σ = −50 (Fig. S2b) for a given value of κRt = 50.
The ICR is found to be enhanced with the increasing values of ε and σ.

Conclusions
The flow of viscoelastic sPTT fluids through a conical nanopore has been numerically analyzed for following
ranges of conditions: 1 ≤ De ≤ 100, 1 ≤ κR ≤ 50, 0.05 ≤ ε ≤ 0.25, −40 ≤ V ≤ 40, and σ = −10 and −50. The
P-N-P model has been used to couple the velocity, concentration, and potential fields, and the numerical results
have been delineated in terms of streamlines and velocity contours, radial velocity profiles, volumetric flow rate,
surface averaged concentration profiles, and ionic current with the applied potential and current rectification
ratio plots. The following conclusions are derived from the present results.

• At a low value of κRt (i.e.,κRt = 1 ), the Debye length is comparable to the radius of the nanopore tip, and
the electric double layer overlaps at the tip of the nanopore. Under these conditions, the electrostatic force
between ions and the charged surface becomes stronger than the electro-osmotic force. Therefore, fluid
rheology has little to no influence on the flow field, the concentration, and the potential field.
• The flow extensibility parameter exhibits a positive dependence on the velocity, concentration, and potential
field. The flow extensibility parameter offers a shear-thinning-like behavior where the effective viscosity
decreases with the increased flow extensibility parameter. This, in turn, leads to enhanced momentum and
ion transport across the nanopore in comparison to the respective Newtonian fluid behaviour.
• Deborah number (De) increases the contribution of solid-like behavior. Thus, the plug-like velocity profile
across the nanopore increases with the Deborah number accompanied by high-velocity gradients near the
nanopore wall.
• The ionic current through the nanopore is found to be proportional to extensibility parameter (ε), surface
charge density ( σ ), the Deborah number (De) and applied potential bias V, while it exhibits an inverse
dependence on the κRt , Such inverse trend is attributed to the fact that at high values of κRt there is no
overlap of the electric double layer at the tip of the nanopore. Moreover, the cone angle ( α) also exhibits a
positive influence over the radial velocity and values of ionic current.
• The CRR is the measure of the extent of current rectification. It has been found to increase with the increase
in PTT parameter (i.e., ε), surface charge density, the Deborah number, and the applied potential.

Received: 31 October 2021; Accepted: 10 January 2022

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Acknowledgements
This work is funded by DST Inspire scheme (DST/ INSPIRE/04/2016/001163), and for the Technology Innova-
tion Hub at the Indian Institute of Technology Ropar in the framework of National Mission on Interdisciplinary
Cyber-Physical Systems (NM - ICPS) by the Department of Science & Technology, Government of India.

Author contributions
M.T.: Conceptualization, Methodology, Validation, Data curation, Writing—original draft. N.N.: Writing—origi-
nal draft, Review & editing, Supervision. All authors reviewed the manuscript.

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