Ion transport in a charged conical nanopore lled with viscoelastic uids: ion current recti cation

The ionic current rectification (ICR) is a non-linear current-voltage response upon switching the polarity of the potential across nanopore, 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 (λ ) 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 of the same order. Therefore, this work extensively investigates the effect of viscoelasticity 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 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. 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 . Introduction Over the years, synthetic nanopores have attracted significant research attention due to their relevance and potential application in bio-sensing devices, DNA sequencing, micro-nano pumping devices, mixing applications, and polymer translocation. A detailed understanding of flow and ionic transport in these nanopores is imperative 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 system. 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) phenomena occurs due to ionic concentration imbalance across the electrical double layer. 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 distribution. Therefore, the ICR phenomena are fundamentally related to the viscous behavior of the fluid and the electrical properties (Relative permittivity, ionic diffusivity, and the valency of the cation and anions) of the fluid . 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. 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 discussion 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) Effect of visco-elastic parameters on ionic concentration distribution. In a pioneer study of Wei et al., the ICR phenomena have been observed for the flow of a KCl solution in a silica nanopore.


Introduction
Over the years, synthetic nanopores have attracted significant research attention due to their relevance and potential application in bio-sensing devices, DNA sequencing 1 , micro-nano pumping devices 2 , mixing applications 3 , and polymer translocation [4][5][6][7][8][9] . A detailed understanding of flow and ionic transport in these nanopores is imperative 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 system 10 . 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) phenomena occurs due to ionic concentration imbalance across the electrical double layer 11 . 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 distribution 12 . Therefore, the ICR phenomena are fundamentally related to the viscous behavior of the fluid and the electrical properties (Relative permittivity, ionic diffusivity, and the valency of the cation and anions) of the fluid 13,14 . 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. 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 discussion 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) Effect 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 studies 11,[16][17][18][19][20][21] have since then reported the ICR in synthetic nanopores. It was concluded that the resistance to the ionic current depends on the flow direction, surface charge, and nonuniform cross-section of the nanopore. 11,18,22,23 . This feature of nanopores can potentially be utilized for separation processes 24 or lab-on-a-chip application 25 . Since the ionic current flowing through the nanopore is modulated and rectified due to the asymmetric (diverging and/or converging) shape of the nanopore, thus it holds an analogy with the function of an electrical transistor 26 . 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 bias 27 . 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 devices 29 .
Some studies 13,30 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 length 31, 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 incorporated 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 cases 13,16,33 . 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 nanopore 17,34 .
Furthermore, since most of the biological fluids, such as mucus, saliva, blood and other lipids 35,36 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 viscosity 37 . 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 analytical solution of the EOF of power-law fluids in a rectangular micro-channel has been investigated by Das  that the non-Newtonian shear-thinning nature facilitates the concentration distribution, whereas it 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 Olivares 40 , 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 Yang 41 , 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 Lee 46 have investigated Phan-Thien-Tanner (PTT) fluids. No prior studies on the effect of viscoelasticity on ICR phenomena have been reported. Hence, the present study is dedicated to investigate the flow of simplified Phan-Thien-Tanner fluids through a conical 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.

Problem Description
The steady-state electro-osmotic flow of visco-elastic (sPTT) binary electrolyte has been considered in a conical nanopore ( Figure 1a) with Radius R t and axial length H = 200R t and divergent angle α = 1.432 o . The flow field has been considered to be 2-D axisymmetric. 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 ( Figure  1b). 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 λ D = 1/κ, which can be scaled by the length scale, R t as κR t . 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 κR t is high. Consequently, the ion migration does not get 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 κR t 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 electric transistor 26 . 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 double 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 Poisson-Nernst-Planck (P-N-P) model.

Governing Equations
The flow induced by applied potential difference is given by the equations of continuity and momentum and it is written as follows: the potential field is given by Poisson's equation as follows: where the space charge density ρ e is defined as: where: z i and c i are the valance and concentration of i th species respectively. The externally applied electric field is expressed in terms of the potential gradient as follows: Similarly, the ionic flux balance across the nanopore is expressed by the Nernst-Planck equation: where The constitutive equation of the sPT T model 47,48 has been given by:

Cunductance, I/V (nS)
where η P is the polymeric viscosity coefficient, λ is the relaxation time of the viscoelastic fluid and D = (∇u T + ∇u)/2 is the rate of deformation tensor, the f (τ kk ) is called the stress coefficient and can be defined as: where, τ kk denotes the trace of the extra stress tensor and ε is known as the PTT parameter. It governs the extensibility and elongation property of the PTT fluid 47,48 . The ∇ τ is the upper convective derivative of the polymeric tensor and can be defined as: Eq. 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: Dimensionless constitutive equation of the sPTT fluids (i.e., Eq. 8-10) can be written as follows: where u * , p * , τ * and ψ * are the dimensionless velocity, pressure, the polymeric tensor, and electric potential respectively. Where De is the Deborah number and it is defined as, The migration of ions, in turn leads to the flow of electrical current in the nanopore the surface average current across the nanopore is written as follows: where which is known as the Debye-Hückel parameter.
The dimensionless surface charge density is defined as 49 : 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'.
• At the boundary 2 The gradients of potential and concentration are zero and the slip condition for velocity has been imposed.
• At the boundary 3 Similarly, the gradients of potential and concentration are zero and no-slip velocity condition at the surface is applied.
• 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 ψ o by Eq. 19. • At the boundary 5 A uniform pressure (i.e., p * = 0) with zero stress and electric potential is fixed as zero and ionic concentration is fixed as 'C o * '.
• At the boundary 6 For velocity, concentration and potential fields axi-symmetric boundary condition has been imposed.

Results and discussion Validation
In order to ascertain the accuracy and precision of the numerical results, it is imperative to perform a few benchmark comparisons. Therefore, the chosen numerical scheme (see the supporting information) has been thoroughly scrutinized by comparing the present results with established experimental 19 , analytical 17 , and numerical 18 studies. A scant number of theoretical findings 16,17,30,40,50,51 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 theory 17,52 for two values of bulk ionic concentration, C o of 1mM and 100mM and a fixed surface charge density of σ = −0.001C/m 3 (see Figure 2).
The results from the applied numerical schemes are validated with the analytical solutions with an excellent agreement, as demonstrated by Figure 2.
In addition, the present numerical scheme has been validated with experimental (Petrossian et al. 19 ) and numerical (Liu et al. 18 ) studies, and the corresponding comparisons are presented in Figure 3 and Figure 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 net ionic current passing through the nanopore in terms of the Debye length (κR t ), 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 concentration. 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 force 53 , 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, viscosity, 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. 54 . The further discussion over the flow field in EOF of viscoelastic fluids is limited by a few scarce studies 42,44,45 . 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. Furthermore, the rheological behavior of viscoelastic sPTT fluids depends upon dimensionless relaxation time, De and extensibility parameter, ε 47, 48 . 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 Figure 5 present the streamlines and the dimensionless velocity contours considering two extreme values of κR t 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 ( Figure S1), and for thin Debye length, i.e., κR t = 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., κR t = 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 κR t (i.e., κR t = 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 κR t exhibit a positive influence over the spatial flow distribution.
Volumetric flow rate: effect of De, ε and κR t 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 nanopore 45,50,55 . 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 Figure 6. The velocity profiles in a 2D micro-geometry (micro-channel/nozzle) has been investigated in terms of the Debye length, κR t , Deborah number, De, and rheological constraints by several numerical studies, such as Bezerra et al. 56 , Chen et al. 45 ,and Mei et al. 44 , as well as analytical studies e.g. Afonso et al. 50,55 and Wang et al. 43 . As discussed in previous section, the Debye parameter, κR t and Deborah number, De exhibit a positive influence on the flow field. In other words, an increase in κR t and De increases the average and maximum velocity. Therefore, the radial velocity profiles at the tip of the nanopore are shown in Figure 6 The trends in Figure 6 can be illustrated as follows: for the fluid with the lowest relaxation time, i.e., De = 1 and for the highest Debye length (κR t = 1), the velocity profile corresponds 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 κR t increases. The maximum velocity shows a positive dependence on the value of κR t , 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 κR t (ε) also reflects in the flow field and the maximum velocity, it increases by ≈ 170% for κR t = 10 and by ≈ 470% for κR t = 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 ( Figure 5) and κR t as shown in Figure 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 ε 45 on the resultant velocity profiles is significant even at κR t = 1. Furthermore, the maximum velocity exhibits a similar dependence on the values of ε and κR t , as illustrated before. Precisely, the maximum velocity is observed to enhance with the value of ε by ≈ 300% for κR t = 10 and by ≈ 2000% for κR t = 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, κR t . In summary, it is concluded that the fluid extensibility (ε) and the Deborah number (De) strengthen the flow field ( Figure 5), while the value of κR t 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 viscosity 54 . 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 κR t at V o = 40, σ = −50, De = 1 and De = 100, is shown by Figure 7.
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 κR t 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 κR t , De, and ε.These trends are in line with results reported for the straight channel by Ji et al. 54 . 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 κR t , 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) 16,18,22,23,53 increasing the applied potential increases the concentration of ions near the nanopore tip. The maxima 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. Since an electric double 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., κR t ), 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 Figure  8a), for κR t = 1, the value of ε has no effect on the the dimensionless anion or cation concentration profiles. On the other hand, at κR t = 50, which mimics the lowest Debye length, the flow resistance is expected to be much smaller than that at κR t = 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 (Figure 8b), 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 κR t = 1. While variations with ε are observed at κR t = 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 κR t = 50.
In a nutshell, it has been found that at high values of Debye length, i.e., κR t = 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 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 κR t , fluid extensibility ε and the Deborah number De.

Viscoelastic effect on ICR and CRR
The total ionic flux in EOF constitutes three components: the convective, 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 date 16,18,22,23,53 the ICR phenomena and its characteristics are 12/16 investigated in terms of the governing parameters such as the bulk ionic concentration 15,16,18,23,57 , C o , the surface charge density 22,30 , σ , salt type 53 and the pH of the electrolyte 11 . All in all, it is reported that the ICR phenomena accentuate with the increasing value of pH, σ , C o , 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 κR t and the surface charge density σ on the ICR phenomena, the total ionic current through the nanopore has been plotted against the corresponding applied potential bias V as I vs V graphs. Figure 9 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 (κR t = 50) and for scores of values of ε. 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) (Figure 9a), 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 dependence on the ICR phenomena at a higher Deborah number (De = 100) (Figure 9b). This effect is similar to the reported results on ICR in the absence of elastic effects. Figure 10 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 11 and Figure S2 demonstrate the I −V curves to delineate the effect of κR t and De respectively. Figure 11 shows the influence of the Debye length (κR t ) 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 decreasing value of κR t or the increasing value of the Debye length. As noted earlier, at a low value of κR t , overlapping of 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 κR t (Figure 11a). 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 ( Figure S2b) for a given value of κR t = 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 κR t (i.e.,κR t = 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.
• 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 κR t , Such inverse trend is attributed to the fact that at high values of κR t there is no overlap of the electric double layer at the tip of the nanopore.
• 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.