Abstract
Magnetic fields induced by currents created in pressure driven flows inside a solidstate charged nanopore were modeled by numerically solving a system of steady state continuum partial differential equations, i.e., Poisson, NernstPlanck, Ampere and NavierStokes equations (PNPANS). This analysis was based on nondimensional transport governing equations that were scaled using Debye length as the characteristic length scale, and applied to a finite length cylindrical nanochannel. The comparison of numerical and analytical studies shows an excellent agreement and verified the magnetic fields density both inside and outside the nanopore. The radially nonuniform currents resulted in highly nonuniform magnetic fields within the nanopore that decay as 1/r outside the nanopore. It is worth noting that for either streaming currents or streaming potential cases, the maximum magnetic field occurred inside the pore in the vicinity of nanopore wall, as opposed to a cylindrical conductor that carries a steady electric current where the maximum magnetic fields occur at the perimeter of conductor. Based on these results, it is suggested and envisaged that noninvasive external magnetic fields readouts generated by streaming/ionic currents may be viewed as secondary electronic signatures of biomolecules to complement and enhance current DNA nanopore sequencing techniques.
Introduction
Voltage driven electrokinetic flows inside solidstate charged nanopores, with promising applications such as DNA sequencing, medical diagnostic, and genetics research, have been an area of intense research for the past three decades^{1,2,3}. The idea of nanopore sensing is to translocate single charged biomolecules, e.g., DNA, RNA, proteins, and peptides, through the nanopore under an applied electric potential to generate unique electronic signals by modulating the ionic currents. The duration and amplitude of such transient ionic currents reveal chemiophysical properties of the biomolecules^{4,5}. One major drawback of nanopore sequencing is high translocation velocity of biomolecules. Several approaches have been explored to reduce the translocation velocity inside nanopores, for instance, pressure gradients were introduced recently as a new counterbalance driving force. Slowing down translocation velocity by pressure enhances the capabilities of nanopore sensing by enabling the system to distinguish short length DNA molecules without sacrificing the capture rate and signaltonoise ratios^{6,7,8,9,10}. Despite these recent advancements in nanopore sequencing, new sensing techniques are necessary to complement transient ionic current readouts for direct labelfree sequencing of biomolecules, allowing the ionic current and a secondary signal to be detected simultaneously.
In this work, we contribute to advancing the understanding of the internal and external magnetic fields in charged nanopores created in the cases of pressure driven streaming current and streaming potential. We also suggest and envision that sensing external magnetic fields generated by the ionic currents may be viewed as secondary electronic signatures of biomolecules to noninvasively complement and enhance current DNA nanopore sequencing techniques.
Electrokinetic flows inside nanopores are a coupled problem between the hydrodynamics of electrolyte solutions as described by NavierStokes, and the transport of ions by convection, diffusion, and migration as described by PoissonNernstPlanck^{11,12,13,14}. There is a considerable amount of literature regarding pressuredriven electrokinetic flows in nano and microsize pores^{15,16,17,18}, however the external, as well as the internal nonuniform magnetic fields, induced by such electrokinetic flows in charged nanopores (governed by Ampere’s law) remains unexplored. Streaming currents, similar to electric currents, generate magnetic fields even though these currents flow within a volume of fluid and are not confined to a linear path as in a cylindrical conductor. The system of continuum partial differential equations that captures the detailed physics of magnetic fields generated by moving ion inside a nanopore is consist of Poisson, NernstPlanck, Ampere and NavierStokes equations; together they form the PNPANS system of equations. The main objectives and novelty of the current work are three folds: a) construct a model for the steady state electrokinetic phenomena inside a charged nanopore in streaming current and streaming potential modes using the PNPANS system of equations, b) conduct a series of numerical simulations and analytically verify the external magnetic fields induced by the ions flowing inside a charged nanopore, and c) conduct numerical simulation and analytical verifications of nonuniform internal magnetic fields induced by ions flowing inside a charged nanopore. The paper is organized as follows. In Section 2, a steady state modeling approach is described in terms of a classical geometry, governing equations in nondimensional forms, pertinent boundary conditions, and the simulation parameters. In Section 3, we discuss the numerical solution methodology and describe the approach that will be used to validate the magnitude of the magnetic field available from steady state analytical results from the idealized geometry of having a channel of infinite length. In Section 4, the model results are presented for streaming current and streaming potential phenomena, and comparisons are made with streaming and conduction current profiles for different pore diameters and pressure differences. Results are then presented for internal and external magnetic fields in a charged nanopore at steady state for a system with a pore diameter such that the electric double layer spans the pore. These solutions are then analyzed in terms of their magnetic fields. Finally, in Section 5, the key conclusions from the present study are summarized.
Problem Statement
Geometry
The flow considered was of an aqueous electrolyte solution passing through a straight circular nanochannel of radius “a” and length “L” constructed of a dielectric material that connects two reservoirs of radius “b” and length “L/2”, as shown in Fig. 1. The whole system shown in Fig. 1 will be referred to as the nanopore, while sometimes it will be necessary to restrict discussion specifically to just the channel and that will be referred to as the nanochannel. The surface on the inside of the pore was assumed to have an electrostatic surface potential due to the interaction between the electrolyte and the dielectric. Besides the relative simplicity of the flow, this geometry was used to avoid the formulation of artificial boundary conditions at the nanochannel’s inlet and outlet. This model geometry allows for development of the appropriate composition and flux conditions at the nanochannel entrance and exit that develop out of the “bulk” conditions, which were defined sufficiently far away from nanochannel entrance or exit. Daiguji^{17} first incorporated bulk reservoirs for this purpose in the case of steady state flow, and was subsequently adopted by Mansouri et al. for unsteady problems^{19,20}.
Outside of the liquid flow volume is the confining dielectric material to create walls of thickness “c” and “d” for the nanochannel and front/back faces, respectively. In this exploratory work, c → 0 and d → 0 in order to keep the focus on the formation of the magnetic fields and allow for comparison to analytical models. Lastly, the domain defined by CDEFC in Fig. 1 was assumed to be air, and was considered to capture external magnetic fields that encircle the ion flows inside the nanopore.
Governing Equations
Electrochemical transport phenomena inside a nanopore were modeled by simultaneously solving the coupled equations of fluid motion (NavierStokes) and the ion transport (PoissonNernstPlank)^{21}. The hydrodynamic problem was modeled in the framework of the continuity and NavierStokes equations. The general form of the steady state momentum equations with an electrical body force, typically used in electrokinetic problems, is
where ρ is the fluid density, u = (u, v) is the velocity vector with u and v being the radial and axial components, respectively, p is pressure, ρ_{f} is the free electrical charge density (C/m^{3}), ψ is the electrical potential. In this study there were no external magnetic fields, as well any magnetic fields induced by the small streaming currents were considered to have negligible impact on fluid flows inside nanopore. As a result, the Lorentz force term (i.e., , where is the ionic current density, and is the magnetic flux density) contribution was not considered^{22}.
For calculation of charge distribution, electrical potential, and ion transport, the PoissonNernstPlanck (PNP) system of equations was used. The free charge density was related to the electrical potential by the Poisson equation
where ε is the dielectric permittivity of the fluid, and ρ_{f} was calculated from where e is the elementary charge, and υ_{i} and n_{i} are the valence and number concentration of the i^{th} electrolytic species. Ion transport in the electrolyte solution subjected to induced electrical fields was described by the NernstPlanck equation
where D_{i} is the diffusivity of i^{th} electrolytic species, T is temperature and k_{B} Boltzmann’s constant. Furthermore, it was assumed that the electrolyte solution is very dilute such that all fluid properties were considered to be uniform and constant throughout the liquid domain. It should be emphasized here that in our simulations, no timedependent terms were retained in the governing transport equations.
The potential sources of magnetic fields are flows of either electrons or ionic species. In the absence of a timechanging electric flux density, a flowing electric current or in this study an ionic current density give rise to magnetic fields both inside and around the nanopore. According to Ampere’s circuital law, curl of a magnetic field strength () is equal to the current density, such that
Magnetic field strength is related to magnetic flux density by, where μ_{0} is the vacuum permeability (4π × 10^{−7} N/A^{2}). For air the relative permeability μ_{r} ~ 1. Conservation of current, or the equation of current continuity, is given by ∇·J = 0. The total flux of ions in the solution is represented as a sum of convective, diffusive, and migratory fluxes, as shown in Equation 3.
For pressure driven flows inside charged nanopores there are two limiting steady state scenarios worth considering. The first scenario is the streaming potential mode (i.e., when there is no external electrical connection between the two reservoirs) where pressure driven flow acts tangentially on the mobile portion of electric double layer and generates a streaming current. However, since the external circuit is open, the imbalance of net charges in the vicinity of the entrance and exit of nanochannel induces a streaming potential that generates a conduction current of a proportional magnitude opposite to streaming current. Assuming diffusion current is negligible, the total flux of ions in the solution is given by , where summed only over the positively charged, and n_{n} is the same but summed over the negative species. The two terms that make up the total flux of ions have opposite directions in the streaming potential mode resulting a zero net current.
The second scenario is the streaming current mode where an external shortcircuited electrical connection is made between the reservoirs so that the reservoirs are maintained at the same potential. This is approximated experimentally by placing a standard platinum electrode in each reservoir and connecting them by a wire. Any imbalance of net charge instantaneously induces chemical reactions at electrodes so, no flow induced electrical field will exist in nanopore if polarization of electrodes is neglected. In this mode, the streaming current, which is the product of net charge density and velocity field becomes J = (n_{p} − n_{n})u and is nonzero.
NonDimensional Equations
To facilitate the solution, and to accommodate consideration of the electric double layer effects in the simulations, all the governing equations were nondimensionalized the same as Mansouri et al.^{3}, and the characteristic length for this study is chosen to be the Debye length, κ^{−1}. The definition of the Debye length for a symmetric binary electrolyte is
All length parameters are scaled with respect to the Debye length. The scaled parameters used in the present model are shown in Table 1. It should be noted that we also assume the diffusivities of the different ions in the electrolyte solution to be equal (henceforth, D_{i} = D). While it is quite straightforward to incorporate different diffusivities of ions in the numerical model, we use a single diffusivity in our subsequent analysis.
By substituting the nondimensionalized parameters into the momentum equation, one can obtain the nondimensional form of the momentum equation
The nondimensional form of the Poisson equation is
The nondimensional NernstPlank equations for each ionic species for the positive and negative ions, respectively, are:
The nondimensional form of Ampere’s Law is given as
Boundary Conditions
The boundary conditions applied to the geometry shown in Fig. 1 are based on flow going from left to right. Irrespective of whether the flow considered is to represent streaming current or streaming potential modes the inflow boundary (AB) was specified at a constant pressure in order to enable a fix total flow rate through the nanopore. The inflow was also an electrically neutral electrolytic solute with a specified ion concentration (n_{∞}), and at zero electrical potential. The nanopore walls (i.e., CD, front face; DE, nanochannel; and EF, back face) were set impermeable and noslip for the solvent and solutes. The boundaries shown at r = b (i.e., BC and FG) in the reservoirs are impermeable to solvent and solutes, but have slip conditions to mimic semiinfinity reservoirs. The outflow boundary (GH) was specified to be at ambient pressure. The nanopore centerline (AH) was assigned symmetric conditions of zero gradients in all quantities.
For the streaming current mode, instead of modeling the complex electrodes electrochemistry and their polarization, the outflow boundary (GH) was just specified to be at zero electrical potential and specified ion concentration (neutral electrolyte). For this case the electrical characteristics of nanochannel was given the classical model of a specified surface potential. For the streaming potential mode, zero gradients in electrical potential and concentration of the electrolytes were applied at GH. For this case the electrical characteristics of nanocannel was given a specified surface charge density (of a magnitude equivalent to surface potential assigned in the streaming current case if the surface were planar and exposed to the same electrolyte) to allow the local potential to adapt to the potential difference between the two reservoirs.
All of the calculations were performed using a scaled nanochannel length of κL = 50, while the scaled radius and length of the reservoirs were both 25. The following physical parameters were employed in the calculations to normalize different quantities according to Table 1: e = 1.6021×10^{−19} C, ρ = 10^{3} kgm^{−3}, n_{∞} = 6.022×10^{21} m^{−3}, , υ = 1 T = 298 K, k_{B} = 1.38×10^{−23} JK^{−1}, and surface potential for streaming current case = −0.025 V (or surface charge density for streaming potential case = −0.00019 μC/cm^{2}), D = 10^{−9} m^{2}s^{−1}. With the choice of these properties, the Debye length becomes 96 nm. Parameters that were varied in the results section were the channel radius and the upstream reservoir pressure to allow the free charge density in the EDL and flow rate to be altered, which affects the magnitude and spatial distribution of the transport of charge by convection.
Numerical Methodology and Validation
Numerical Solution Methodology
In the present work, fully coupled stationary solver integrated in COMSOL (V5.2) finite element software, was used to solve the nonlinear partial differential equations. The solution methodology involved a coupled solution of the PoissonNernstPlanck (convection, migration, and diffusion), Ampere, and NavierStokes (PNPANS) partial differential equations in a 2D axisymmetric computational domain. The finite element calculations were performed using quadratic triangular elements. The accuracy of the numerical results depended strongly on the finite element mesh. Near the nanochannel wall, where the electric field was pronounced, a refined mesh was necessary to ensure accurate modeling of these gradients. Also, near the nanopore centerline, velocities were relatively large and highly dependent on the accuracy of the electrical body force term. Therefore, one needed to have a sufficiently refined mesh to capture the subtle changes in the electric and magnetic fields. Consequently, a finer mesh was used near the nanochannel walls as well as along the center of the nanopore. Independence of the results to mesh refinement was studied and all results reported here are independent of measurable influence of mesh size. It was observed that the solution became mesh independent with approximately 26490 elements.
Validation
To validate the finite element formulation, we compared the predictions from the steadystate numerical model against existing analytical results for streaming potential flow in an infinitely long nanochannel. The analytical results were obtained for the transport of a symmetric electrolyte in a straight circular cylindrical nanopore of infinite length with low surface potentials on the nanopore wall. Comparisons were made between scaled velocity at the midplane of the nanopore (i.e., ), streaming potential across the length of nanopore and analytical solutions based on infinitely long pores. The choice of was to be sufficiently far removed from the nanopore entrance and exit regions, and most likely to emulate an infinitely long nanochannel. These velocity comparisons were indistinguishable and in agreement with Mansouri et al.^{19}. Furthermore both internal and external magnetic fields are in excellent agreements with analytical solutions according to Ampere’s law applied to an infinitely long nanochannel in streaming current case. Details of both these comparisons will be shown in Section 4 – Results and Discussions, while what is described here is the methods used to estimate the magnetic field for the infinitely long nanochannel.
The origin of streaming current is the electrical state of the fluidsubstrate interface that creates a spatial distribution in the free charge density, ρ_{f}, which is then transported by the fluid velocity, v, parallel to the walls. The streaming current is the product of velocity field and net charge density and is given by
By substituting the velocity field and net charge density from NavierStokes and Poisson equations in the absence of any electrical field (i.e., for the streaming current mode) for a circular nanopore with radius “a” we find
I_{0} is the first kind modified Bessel function, ε is the permittivity, ζ is the surface potential, κ is inverse of Debye length, μ is the viscosity and dp/dz is the pressure gradient. To determine the magnetic fields outside of an infinitely long wire according to Ampere’s law we consider a closed circular loop of radius “r” around the wire, and due to symmetry, magnetic field must have a constant magnitude around the loop
Similarly outside the nanopore, the magnetic field varies as 1/r where I_{SC} is the streaming current (which is the integral of J(r) over the whole cross section), r is radial distance from nanochannel wall, μ is the permeability and B is magnetic flux density. The magnetic field inside the nanopore however varies nonlinearly with magnetic flux density. This is caused by nonuniform streaming current profiles. Here we show the analytical approach to generate internal magnetic fields. In order to find the magnetic field inside the nanopore, we need to integrate the nonuniform streaming current density for r < R, as shown here
The magnetic field inside the nanopore can be calculated by
These analytical expressions for the magnetic field arrive from the assumption of infinite pore length, so the results are only a function of radius, and will be compared in the next section to the 2D numerical results found at the midlength of the nanopore.
Results and Discussions
The results presented here are divided into two sections. The first section focuses on the electrokinetics aspects of the flow and set the stage for interpreting the magnet fields that are created due to the currents throughout the nanopore system. The second section focuses on the magnetic fields that exist inside the nanopore system that could affect the transport trajectory of any biomolecules with induced dipoles, and outside the nanopore system that could be monitored as a secondary electromagnetic signal as part of a sequencing technique. As mentioned previously, pressure driven electrokinetic flows in nanopores have two limiting modes of operation, i.e. the streaming current mode (no net electrical potential between the reservoirs) and the streaming potential mode (no net current between the reservoirs), and results are presented for both these modes.
Currents and potentials within electrokinetic nanopores
In the streaming current mode, current in the nanochannel exists due to the convective flux of free charge, and that streaming current is calculated by integrating over the whole cross section of the nanochannel. Figure 2 shows the scaled convection current flux for κa = 1, 5 and 25 at the midlength of the nanochannel for a single pressure gradient of 10^{9} Pa/m. Figure 3 shows the corresponding electrical field along the axis of the nanopores for the particular case of κa = 1. The basic observations associated with these two figures are:

1
There is essentially no electric field inside the nanochannel in streaming current mode, but there are spikes in the field of opposite sign that appear at inlet and outlet of the nanochannel. Without the electric field in the nanochannel means that the only mechanism of ion transport is convection.

2
The magnitude and radial distribution of current flux within the nanochannel is dependent on κa, as κa increases (by varying nanochannel radius) from 1 to 5 and to 25 the nondimensional current increases from 0.22 pA to 15 pA and to 0.3 nA, respectively. (It should noted that the actual current per unit flow rate drops). For κa = 1 the maximum current flux is along the central axis of the nanopore, but as κa increases the maximum current flux shifts to nearer the wall.
Also shown in Fig. 3 is the electric field for the streaming potential mode. While this mode has almost identical spikes at the inlet and outlet of the nanochannel, there is a finite and uniform field along the length of the nanochannel. The magnitude of electrical field (0.42 × 10^{5} V/m) is strongly dependent on the pressure gradient and κa, and is shown here only for the same pressure gradient (10^{9} Pa/m) and κa = 1. This electric field induces a conduction current of the ions, which in steady state balances the convection current. Figure 4 shows the scaled net current flux in the streaming potential mode for κa = 1, 5 and 25 at the midlength of the nanochannel. The key observation associated with Fig. 4 is that while the net current (i.e., the current flux integrated over any crosssection in the nanochannel) is zero, the flux is highly nonuniform. For κa = 1, the inner three quarters of the radius has a positive current flux and the outer quarter is negative. While for the κa = 25 case, the inner 85% of the radius has a negative current flux there is a significant positive flux near the nanochannel wall^{23}.
All of these different spatially different current fluxes, as well as the magnitude of the net current, have important implications for the magnetic fields that exist either within and external to the nanopore.
Magnetic fields within and around electrokinetic nanopores
We now turn our attention to magnetic fields induced by the various current fluxes presented in the previous section. Figure 5 shows a 2D plot of the magnetic field strength for the case of streaming current and κa = 1 for a pressure gradient of 10^{8} Pa/m. The important qualitative observations are that, other than very near the inlet and outlet of the nanochannel, the gradients in the magnetic field strength are only a function of the radial coordinate, that maximum value of the magnetic field exist near the wall of the nanochannel, and extends outside the nanochannel where it can potentially be measured. Overlaid onto the magnetic field strength are vectors with magnitudes scaled to the local current flux, which shows that while the total current inside any radius continues to increase until the full radius is reach, the magnetic field strength reaches a maximum within the nanochannel because of the 1/r dependency from the various convection current streaming tubes.
More quantitative measures of the magnetic field are shown in Figs 6 and 7 for the streaming current and streaming potential modes, respectively. These figures are the radial profile of the magnetic flux density on a log scale for κa = 1 at the midlength of the nanochannel, and for two different pressure gradients (5 × 10^{8} and 10^{9} Pa/m, which produce convection currents of 0.11 and 0.22 pA, respectively). The log scale is necessary to capture the range of spatial gradients. Before discussing these results it is worth comparing the magnitude of these fields with the analytical results of the idealized case of an infinitely long channel described in section 3.2 for streaming current case. In Fig. 6 there is excellent agreement between the numerical and analytical results at this location in the nanochannel. In reference to Fig. 5, it is clear that this level of agreement is due to making the comparison at the midlength of a relatively long the nanochannel.
With respect to the magnetic flux density for κa = 1, it is noted that its peak value in all case is very close to the nanochannel wall that decays to zero at the centerline (symmetry requirement) and in the far field (as 1/r). Since there is no net current in the streaming current mode, the magnitude of the magnetic field decays at a relatively modest rate compared to the streaming potential case with zero net current across the nanochannel. It’s interesting to note that due to nonuniform distribution of net current in the center of nanochannel in streaming potential case (Fig. 4), internal and external magnetic fields are not zero, however external magnetic fields are notably lower than the streaming current case in Fig. 6.
With the existence of these magnetic fields two possible interactions with molecules that can have an induced dipole are envisioned. First, Zhai et al. has shown the external magnetic fields as small as pico Tesla, which is of the order that was modeled here, are detectable and measurable in voltage/pressure driven electrokinetic flow in nanopores^{24,25,26}. As a result, we envision that noninvasive external magnetic fields readouts generated by streaming/ionic currents may be viewed as secondary electronic signatures of biomolecules to complement and enhance current DNA nanopore sequencing techniques. Second, depending on the trajectory that a biomolecule enters the nanochannel it could be displaced to a preferred radial location and become subjected to that local axial flow velocity. As a result, the magnetic fields generated by these electrokinetic flows could be used as a separate management tool for the transport of biomolecules. Our postulate is also supported by experimental evidences in the field of biology and medicine. It is known that several tissues and living organ systems in human body that are electrically excitable induce external magnetic fields. Therefore, biomagnetic field measurements employing magnetic sensors with high sensitivity provide a noninvasive detection of living system activity. Most recent advances in magnetic sensors technology include a superconducting quantum interference device (SQUID) with the sensitivity of a femto tesla (fT) to detect magnetic activity in the brain and heart of humans^{27}.
Conclusions
In this study, a set of PNPANS equations, consisting of Poisson, NernstPlanck, Ampere and NavierStokes equations were solved numerically to simulate induced internal and external magnetic fields created by streaming current/potentials inside a charged nanopore at κa = 1. The magnetic fields outside the nanopore decays rapidly and varies with inverse of distance according the Ampere’s law. However the magnetic fields inside the nanopore are highly nonuniform. The maximum of magnetic field occurs inside the nanopore and in the vicinity of nanopore wall contrary to magnetic fields of electric currents inside conductive wires. We suggest that noninvasive external magnetic fields readouts generated by streaming/ionic currents are of great importance and may be viewed as secondary electronic signatures of biomolecules to complement and enhance current nanopore sequencing techniques.
Additional Information
How to cite this article: Mansouri, A. et al. Streaming current magnetic fields in a charged nanopore. Sci. Rep. 6, 36771; doi: 10.1038/srep36771 (2016).
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A.M. developed the concept and numerical model. L.W.K. and P.T. analyzed the results. A.M. and L.W.K. wrote the main manuscript text. All authors reviewed the manuscript.
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Mansouri, A., Taheri, P. & Kostiuk, L. Streaming current magnetic fields in a charged nanopore. Sci Rep 6, 36771 (2016). https://doi.org/10.1038/srep36771
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DOI: https://doi.org/10.1038/srep36771
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