Small-Scale and Low-Cost Electrodes for "Standard" Reduction

Proceedings of the ASME 2017 Fluids Engineering Division Summer Meeting FEDSM2017 July 30-August 3, 2017, Waikoloa, Hawaii, USA

FEDSM2017-69194 EFFECT OF CHARGE DENSITY ON ELECTROKINETIC IONS AND FLUID FLOW THROUGH POLYELECTROLYTE COATED NANOPORE

Subrata Bera ∗

S. Bhattacharyya

Department of Mathematics National Institute of Technology Silchar Silchar, Assam,788010 India. Email: [email protected]

Department of Mathematics Indian Institute of Technology Khargapur Khargapur, West Bengal, 721302 India Email: [email protected]

ABSTRACT We have studied the electroosmotic flow (EOF) and its effect through a polyelectrolyte coated conical nanopore. The nanopore wall bears a uniform negative surface charge while charged density of the polyelectrolyte layer (PEL) bears positive charge. The degree of softness in the PEL is mainly affects the hydrodynamic field inside the nanopore while ionic current is not affected significantly by flow field. The characteristic of electrokinetic flow is based on the nonlinear Nernst-Planck equations for the ion transport coupled with the Brinkman extended Navier-Stokes equations for fluid flow and the Poisson equation for induced electric potential. The coupled set of governing nonlinear equations for fluid flow and ionic species concentration are solved through a finite volume method on a staggered grid system. A numerical method based on the pressure correction iterative algorithm is adopted to compute the flow field. This study investigated the importance of the bulk concentration of the electrolyte, the geometries of the nanopore and both the thickness and the charged density of PEL on the electrokinetic ion and fluid transport. The ratio of the cross-sectional average flow of the present model with plane cylindrical channel, decreases with the increase of the scaled charge density of PEL for both low and high ionic concentration cases when softness parameter and thickness of PEL are fixed. The average flow rate decreases with the increase of the PEL sealed charge density in both low and high ionic concentration cases for fixed PEL thick-

∗ Address

ness. The increase of nanopore radius increases the cross sectional averaged flow for fixed scaled charged density and PEL thickness. The average flow rate decreases with the increase of the PEL thickness for fixed charged density of PEL. The critical value of scaled charge density of PEL is defined for which there is no flow through the nanopore. The average current density increases with the increase of applied electric field for different charged density of PEL. But there is no different of average current density for different charge density of PEL in high ionic concentration cases.

NOMENCLATURE List of symbols a cylinder half radius. Di diffusivity of ’i’ type ion. E0 applied electric field strength. e elementary charge. F Farad’s constant. Iz cross sectional average current density. J current density. kB Boltzmann constant. ni (= g, f ) number concentration of ’i’ type ions. n0 bulk ionic concentration. Q f ix scaled fixed charge density. q(= v, u) velocity along cross radial and radial direction. UHS Helmholtz-Smoluchowski velocity.

all correspondence to this author.

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Um dimensional average flow. R gas constant. rs uniform thickness of polyelectrolyte layer. T absolute temperature. zi valance. Greek Letters β non-dimensional softness parameter. γ hydrodynamics frictional coefficient of the PEL. ε permittivity. εr dielectric constant of the solution. ε0 permittivity of vacuum. κ Debye length. λ0−1 softness degree. Λ non-dimensional strength of applied electric field. λ EDL thickness. µ viscosity. ρ density. ρe net charge density. ρ f ix fixed charge density of polyelectrolyte chain. σs surface charged density of polyelectrolyte chain. φ induced potential. φ0 scaled potential. ωi mobility of ’i’ type ion. Subscripts e net charge f ix fixed charge density HS Helmholtz-Smoluchowski i ’i’th ion s surface

cant effects in certain technologically important cases for nonhomogeneous EDL structure. The EOF through an annulus is studied by Kang et al. [2] under the situation when the two cylindrical walls carry high zeta potentials. Berg and Findlay [3] presented a complete analytical solution for electrokinetic flow of an acidic solution in an infinite circular channel. A simple algorithm is proposed by Ohshima [4] for obtaining an approximate analytic solution to the Poisson-Boltzmann equation for the EDL potential distribution in a charged cylindrical narrow pore which filled with an electrolyte solution. Bhattacharyya and Bera [5] have studied the electrokinetic transport and mixing through a wall roughness in the form of a rectangular block mounted on the one wall of infinity long micro channel. Chen et al. [6] numerically investigated the effects of a step change in zeta potential on EOF in a cylindrical microchannel. In many ways of modulating electroosmosis, polymer coatings are often employed to control effectively the EOF or minimize wall-analyte interactions which is important for biomolecule separations using electrophoresis technology. Experimentally, grafting polymer chains to a solid surface can be irreversible or reversible. For example, the polymer chains can be chemically bonded to substrate or physically adsorbed onto the surface. EOF in a nanofluidic channel coated by grafting polymers onto two parallel channel walls has been studied by Cao et al. [7] using Molecular Dynamics simulations. Zuo et al. [8] demonstrated the suppression of EOF by polyampholyte brushes onto two parallel channel walls using Molecular Dynamics simulations. EOF in porous media through the cylindrical and annular geometrical models are investigated by Wu et al. [9] using Poisson-Boltzmann equation. Cao at al. [10] have performed dissipative particle dynamics(DPD) simulations of EOF through a polymer-grafted nanopore. A mathematical model is developed by Scales and Tait [11] on electroosmotic and pressure-driven flow in porous media of different geometries for microfluidic Applications. Different interesting features of EOF through microchannel can be observed for the channel grafted with polyelectrolyte layer (PEL). The modeling of EOF through PEL have been used in a Lab-on-a chip devices for a number of different advantages. PEL in a micro fluidic devices can be used to create a region of high fluidic resistance. It can also the capability of filtering analytes based upon the selection of an appropriate pore size and it offer a large surface area for chemical reaction. A relatively small electric filed can be used to generate significant pressure, as high pressure are required to force the fluid through very small pores. The electrokinetic transport of ions and fluid in a nanoscale pore or channel plays an important role to control the ion transport and to detect and analyze individual biomolecules in modern biophysics and biochemistry. Kim at al. [12] studied electrokinetic separation of biomolecules through multiple nanopores on membrane. EOF through a nanopore have been studied by Mao

INTRODUCTION The burgeoning field of constructing nanofluidic devices to control ion transport as well as to detect and analyze biomolecules through them, electrokinetic transport of ions and fluid in nanoscaled pores or channels plays an important role in modern biophysics and biochemistry. Electroosmotic flow (EOF) is the bulk fluid motion driven by the electrokinetic force acting on the net charged ions in the diffuse layer, the outer part of an electrical double layer (EDL). The thickness of the EDL characterised by the Debye length which is in the order of nanometers. In nanofluidic devices, where the characteristic length is comparable to the thickness of electric double layer (EDL), several fascinating features were observed experimentally. When EOF is modeled by the thin EDL approximation using slip velocity condition is known as the HelmholtzSmoluchowski velocity [1]. In most analyses of EOF, the ionic distribution in the EDL is assumed to follow the equilibrium Boltzmann distribution, resulting in the Poisson-Boltzmann equation for the electric potential induced by theses ions. However, the convective transport of these ions may have signifi-

2



et al. [13] for uniform and low surface charged using PoissonBoltzmann equation. Berg and Ladipoer [14] proposed the exact solution of an EOF problem in an infinite cylindrical polymer electrolyte membranes with a uniform surface charge density using Poisson-Boltzmann-Stokes model. A three-stage model is proposed by Ohshima [15] for the electrostatic repulsion between two parallel plates covered with polyelectrolyte brushes in an electrolyte solution. Yeh et al. [16] theoretically investigated on the electrokinetic ion and fluid transport in polyelectrolyte brush functionalized nanopores. Tessier and Slater [17] investigated on coarse-grained molecular dynamics simulations of the EOF of an electrolyte confined in a cylindrical nanoscopic pore and present the equilibrium distribution of fluid particles and ions in the electrolyte. The effects of EOF on the ionic current rectification phenomenon in conical nanopores connected with reserver are studied comprehensively by Ai et al. [18] using of a continuum model. Yeh et al. [19] theoretically studied the ion transport and the resulting conductance behavior in a polyelectrolyte modified nanopore using a continuum-based model. Li at al. [20] investigated the time periodic EOF of an electrolyte solution through a slit polyelectrolyte-grafted nanochannel under applied alternating current electric field.

z Wall

r=1

rs

Polyelectrolyte Layer

fix

r Cathode

Anode

r=0

r

Polyelectrolyte Layer Wall

FIGURE 1. SCHEMATIC REPRESENTATION OF THE POLYELECTROLYTE LAYER IN A CANONICAL NANOPORE.

Mathematical Model We consider a canonical nanopore with axial length z and radius r be filled with an incompressible Newtonian electrolyte of uniform permittivity εe and viscosity µ (Fig.1). The wall of the nanopore bears a uniform surface potential ζ . A PEL is embedded in the wall surface of nanopore. For simplicity, we assume that the PEL is ion-penetrable, homogeneously structured, highly (nonregulated) charged and of uniform thickness as which yields a fixed charge density ρ f ix ∼ = (eZ σs /as ) with e, Z and σs beings the elementary charge, the valence of the dissociable groups per polyelectrolyte chain, and surface charged density of polyelectrolyte chain grafted to the membrane respectively. The electric field E (= Er , Ez , Eθ ) is determined by the by the uniform external electric field E0 along the z-direction of nanopore. The charge density ρe is related to the electric field as

The modelling of the electrokinetic flow in a micro and nanochannel is usually based on the Poisson-Boltzmann equation where the ionic species concentration follows the Boltzmann distribution. This approach leads the assumption of equilibrium EDL, electronically is reached at a point faraway from charged surface and non-overlapping EDLs. Thus, it fails to describe the fascinating features mentioned earlier due to significant overlapping EDLs, nonequilibrium EDLs arising from uneven distribution of ions and counterions inside a nanopore. Most of the existing studies are based on the equilibrium Boltzmann distribution without considering the convection and electro-migration effects of ions. To capture the effect of convection, diffusion, and electromigration of ions, the Nernst-Plank equation is considered to describe the ionic species distribution in the exterior and interior of the PEL. All of the existing studies are based on the Stokes and Brinkman equations for the hydrodynamic flow field without considering the inertia effects. The present model deals with the full set of Brinkman extended Navier-Stoke equations with modified bodyforce term by considering inertia effects for fluid transport in the exterior and interior of the PEL. The poisson equations gives the distribution of induced electric filed. The characteristics for the electrokinetic flow are obtained by solving numerically the Poisson equation, the Nernst-Planck equation, and the Navier-Stokes equations in a coupled manner through a control volume approach in a staggered grid system using higher order upwind schemes.

∇ · (εe E) = −εe ∇2 φ = ρe + hρ f ix

(1)

Here, φ is the induced electric potential and εe = ε0 εr , where ε0 is the permittivity of vacuum and εr , the dielectric constant of the solution. ρe = ∑i zi eni is the space charge density of mobile ions; zi and ni are respectively, the valance and number concentration of the i type ion. In present case, we consider a symmetric electrolyte of valance zi = ±1. Here, h is a unit region function, assume 1 within the region inside PEL and 0 outside it. We consider φ0 (= kB T /e) and bulk number concentration n0 , as the potential and concentration scale respectively. Because of the symmetric nature of the present problem, we adopt the cylindrical coordinate system (r, z) with origin fixed at the center of the nanopore. The non-dimensional form of the Poisson equation of electric potential is

∂ 2φ 1 ∂ ∂φ (κ a)2 + r = − (g − f ) − hQ f ix ∂ z2 r ∂r ∂r 2

(2)

We have taken radius of the cylinder, a as a length scale. The parameter κ is the reciprocal of the characteristic EDL thickness (λ ), where the EDL thickness (λ ) is defined as λ =

3



p εe kB T /Σi (zi e)2 ni0 and κ a = a/λ . The scaled fixed charge density Q f ix = ρ f ix a2 /εe φ0 . We denote the non-dimensional concentration of cation by g and anion by f . The transport equation of the ionic species i is governed by the Nernst-Planck equation as

∂ ni + ∇ · Ni = 0 ∂t

Here, ρ and µ are density and viscosity of the fluid respectively and γ is the hydrodynamics frictional coefficient of the PEL. Here, Pressure is scaled by µ UHS /a. The non-dimensional equations for fluid flow along z and r direction are given by respectively as,

(3)

Re

where the net ionic flux of each ionic species due to convection, dissuasion and electro-migration is given by Ni = −Di ∇ni + ni ωi zi FE + ni q. Here, Di and ωi are respectively, the diffusivity and mobility of i type species and F is the Faraday’s constant. q = (v, u) is the velocity field of the fluid with v and u are the velocity components in the r and z directions respectively. Here, velocity field q is scaled by the Helmholtz-Smoluchowski velocity UHS (= εe E0 φ0 /µ ) and time t by a/UHS . The Reynolds number based on UHS is defined as Re = UHS a/ν , Schmidt number Sc = ν /Di , Peclet number Pe = ReSc and ν = µ /ρ . Here, R is the gas constant and µ is the viscosity of the electrolyte. The non-dimensional equations for ionic transport are given by

+

∂g ∂φ ∂g ∂φ (κ a)2 − + + g (g − f ) − hgQ f ix = 0 ∂r ∂r ∂z ∂z 2

Pe

∂f 1 ∂ ∂f − + r 2 ∂t ∂z r ∂r ∂r

∂2 f

+Pe

(4)

∂ f ∂φ ∂ f ∂φ (κ a)2 + + − f (g − f ) + h f Q f ix = 0 ∂r ∂r ∂z ∂z 2

(5)

The Brinkman extended Navier-Stokes equation with modified bodyforce term for electroosmotic flow in the exterior and interior of the nanopore of a constant property of Newtonian fluid is ∂q ρ + (q · ∇)q = −∇p + µ ∇2q + ρe E − hγ q ∂t

(6)

∇·q = 0

(7)

∂ 2u 1 ∂ ∂u + r − hβ 2 u ∂ z2 r ∂ r ∂r

(8)

∂ 2v 1 ∂ ∂v v + + r − 2 − hβ 2 v ∂ z2 r ∂ r ∂r r

(9)

∂u ∂v v + + =0 ∂z ∂r r

(10)

∂ (u f ) 1 ∂ (rv f ) ∂g + +Λ ∂z r ∂r ∂z

∂v ∂v ∂v ∂ p (κ a)2 ∂ φ =− − (g − f ) Re + Re u + v ∂t ∂z ∂r ∂r 2Λ ∂ r

2 ∂g ∂ g 1 ∂ ∂g ∂ (ug) 1 ∂ (rvg) ∂g Pe − + r +Pe + +Λ ∂t ∂ z2 r ∂ r ∂r ∂z r ∂r ∂z

∂u ∂u ∂u ∂ p (κ a)2 ∂φ +Re u + v =− − −Λ + (g − f ) ∂t ∂z ∂r ∂z 2Λ ∂z

Here, β is the non-dimensional softness parameter which can be −1 −1 related the softness degree of PEL p (λ0 ) as β = a/λ0 . The −1 softness degree of PEL, λ0 (= µ /γ ) mainly affects the hydrodynamic field inside the nanopore while the conductance is not affected significantly by the flow field. Fully developed boundary condition is imposed at the inlet and outlet boundaries of the computational domain. We consider the no-slip boundary condition on the channel wall. The rigid surface of the membrane is assumed to be ion-impenetrable i.e., n · Ni = 0 and constant surface potential ζ along the nanopore wall where n is the unit normal vector on the outward the surface . Axisymmetric boundary condition is imposed along the axis of the nanopore.

Numerical methods We solved the coupled set of governing non-linear equations for fluid flow and ionic species concentration through a finite volume method on a staggered grid system. In the staggered grid system, the scalar quantities are evaluated at each cell center and the velocity components are evaluated at the midpoint of

4



Start

Solve the discretized potential and ionic concentration equations to obtain the intermediate values

µ

Guess the pressure

Solve the discretized momentum equations to obtain the intermediate

Solve the pressure correction equation using the intermediate velocity fields obtained in the previous

Correct the velocities and pressure using the

Solution converge?

FIGURE 3. COMPARISON OF THE PRESENT SOLUTION WITH ANALYTIC SOLUTION Ai et al. [18] AND THE EFFECTS OF GRID SIZE FOR THE AXIAL VELOCITY u OF AN EOF IN A CYLINDRICAL CHANNEL WITH h = 0. THE BULK ELECTROLYTE IS 10 mM IN KCl SOLUTION AND SURFACE CHARGE DENSITY (σ ) = −1 mC/m2 (i.e., CORRESPONDING ζ = −0.11) AND THE EXTERNAL ELECTRIC FIELD E0 =106 V/m.

Stop

FIGURE 2. FLOWCHART OF THE NUMERICAL METHOD.

the cell sides to which they are normal. The discretized form of the governing equations is obtained by integrating the governing equations over each control volumes. Different control volumes are used to integrate different equations. The equations for fluid flow and ion transport involves first order derivatives of electric potential. In order to capture the sharp change in variable values accurately, we use the higher-order upwind scheme, QUICK (Quadratic Upwind Interpolation Convective Kinematics, Leonard [21]) to discretize the convective and electromigration terms in both concentration and Navier-Stokes equations. The QUICK scheme uses a quadratic interpolation/ exterpolation between the three nodal values of variables to estimate its value at the interface of the control volume. The upwind scheme imparts stability to the numerical solution in the region where a steep gradient in variables occur. An implicit first-order scheme is used for discretising the time derivative terms. The resulting discretized equations are solved iteratively through the pressure correction based iterative algorithm SIMPLE (Fletcher [22]). The iteration starts by assuming the induced electric potential φ at every cell center. In this algorithm, the pressure link between the continuity and momentum equations is accomplished by transforming the continuity equation into the Poisson equation for pressure. The Poisson equation implements a pressure correction for a divergent velocity field. The pressure Poisson equation is given by ′

′

′

pi+1, j − 2pi, j + pi−1, j (∆z)2

1 − ∆t

′

+

(∆r)2

−

v∗i, j − v∗i, j−1 ∆r

1. Implicit calculations of the momentum and mass transfer equations are performed. Due to coupling of equations, we solve the system of linear algebraic equations through a block elimination method (Varga [23]). 2. The Poisson equation for pressure correction is solved using the successive under relaxation method. 3. The velocity field at each cell is updated using the pressure correction. Iteration at each time step is continued until the divergence-free velocity field is obtained. The divergence in each cell is towed below a preassigned small quantity (≤ 10−5). The iteration starts by computing the Poisson Eq. 2 for induced electric potential (φ ). A flowchart of the numerical methods is shown in Fig. 2. We consider a non-uniform grid distribution along rdirection but uniform grid is considered along z-direction and δ t was taken as 0.0001 in Fig. 3. To check the effects of grid spacing, computations have been performed for three different meshes with grid 1: 160 ×250, grid 2: 400×250 and grid 3: 400×500 for EOF in cylindrical channel and compared with the results due to the Ai et al. [18] for analytic solution. In grid 1 and grid 2, we considered a non-uniform grid size where δ r is assumed to vary between 0.0025 to 0.01 with δ z is either 0.0125 (for grid 1) or δ z = 0.008 (for grid 2). In grid 2, we consider δ z = 0.01 and 0.0025 ≤ δ r ≤ 0.002. To validity of our numerical method we have compared our numerical solution with analytic solution Ai et al. [18] for the axial velocity u of an EOF in a cylindrical channel when there is no PEL layer (i.e.,h = 0). The bulk electrolyte is 10 mM in KCl solution and surface charge density (σ ) = −1 mC/m2 (i.e., corresponding ζ = −0.11) and the exter-

′

pi, j+1 − 2pi, j + pi, j−1

u∗i, j − u∗i−1, j ∆z

′

tum equations. At each iteration, the Poisson equation for electric potential is solved through a central difference scheme. Thus, at any time step a single iteration in this algorithm follows these sequential steps:

=

(11)

′

The variable p denotes the pressure correction, and u∗ and v∗ denote the velocity components obtained by solving the momen-

5



nal imposed electric field is −50 KV/m. Fig. 3 suggests that the results obtained by grid 2 and grid 3 agree fairly well each other and these results are in close agreement with the result due to Ai et al. [18]. Thus, we find grid 2 is optimal.

wall surface. Thus, the screening of EOF by PEL is enhanced with decreasing the charged density of PEL. The slow decrease of axial EOF velocity at high charge densities in PEL can be described to an increased fraction of counterions outside the PEL. As a result, the EOF velocity decrease with the increase of scaled charge density of PEL as shown in Fig. 4a. Fig. 4(b) and (c) described the corresponding distribution of induced potential φ and ions g, f respectively. Two terms ((κ a)2 (g − f )/2 − hQ f ix) arise in R.H.S of Poisson Eq. 2, first term consists the net charge density multiple with the factor (κ a)2 /2 and second term scaled charge density Q f ix of PEL. For low ionic concentration, the net charged density is non-zero and factor (κ a)2 /2 is very small. Thus, the second term, scaled charge density Q f ix in PEL lead to significant effect of induced potential as illustrated in Fig. 4b. Since, nanopore wall is negatively charged and PEL is positively charged, more anions are electrostatically attracted into the nanopore, whereas more cations are repelled out. Therefore, the magnitude of the anions flux inside the nanopore is significantly larger than that in the reservoirs, resulting in an enrichment of ionic concentrations on the anode side of the nanopore. The increases of scaled charged density in PEL actually increase the net charge density of ions ρe (= g − f ) in Fig. 4c. Fig. 5a depicts the the distribution of induced potential φ for different values of bulk ionic concentration when the scaled charged density Q f ix = 10 and PEL thickness as = 4nm and the corresponding distribution of ions are presented in Fig. 5b. The variation of softness parameter on axial velocity for various values of bulk ionic concentration are shown in Fig. 5c when the scaled charge density of the nanopore Q f ix = 10. The increasing of bulk ionic concentration leads the net charge density tend to zero and hence, increase the EOF velocity. The softness parameter mainly increase the EOF velocity in the PEL. The cross-sectional non-dimensional average flow EOF (Um ) is given by

Results and Discussions The geometry of the nanopore corresponds to the typical experimental design of nanopore based nanofluidic devices, where the radius of the nanopore is 3 − 30nm [25, 26, 27]. Since, the diameter of a water molecule and common univalent ions is about three angstroms; thus the liquid Knudsen number defined as the ratio of a molecular diameter to the channel height is about 0.06. This help to compare these continuum results which treat the ions as point charges with molecular dynamics (MD) simulations and this has already been done in one dimension Zhu et al. [28] where it is shown that wall exclusion effects become important at about channel height 6nm or about twenty molecular diameters. The typical values of softness degree of PEL λs−1 lies between 0.1 to 10 nm (Yeh et al. [16], Drop et al. [24]) and the corresponding softness parameter (β 2 ) varies from 1 to 100 when radius of the cylindrical nanopore r is 10nm. The value of surface density(σs of PE chains grafted to the membranae typically ranges from 0.1 to 0.6nm2. The σs = 0.125nm2 corresponding Q f ix = 28.5 when φ0 = 0.02525V, as = 4nm and r = 10nm. We considered the surface potential ζ of the nanopore wall is 1 and net charged density of the PEL is positive varying from 0 to 50. For illustration, we consider a PE-modified nanopore with length l = 80 nm, radius a = 10 nm, and thickness of PEL rs varies form 19nm. it is assumed that the channel is filled with an incompressible Newtonian electrolyte (e.g., NaCl with water) with uniform permittivity εe = 695.4074 × 10−12C/V m, viscosity µ = 0.001 Kg/m sec, density ρ = 1000Kg/m3, Boltzmann constant kB = 1.381 × 10−23 J/K, elementary electric charge e = 1.602 × 10−19 C, gas constant R = 8.315 J/mol K at temperature T = 300K and diffusion coefficient for both the ions is assumed to be same as Di = 1.3 × 10−10 m2 /sec. Solutions are presented for various values of the ionic concentration of electrolyte, external electric field, surface potential, thickness of PEL and surface charged density of polyelectrolyte chains grafted to the membranae. The influence of PEL scaled charge density on the axial velocity, induced potential and ion distribution are presented in Figs.4(a-c) for fixed values of softness parameter β 2 = 10 and bulk ion concentration C = 1mM. Since, the nanopore wall is negatively charged (ζ = −1), the increase of positive charge density in PEL actually reduces the electroosmotic velocity. When scaled charge density in PEL decreases, more counterions penetrate into the PEL. It is due to the fact that the increased distance between neighboring PEL points weakens the excludedvolume interaction between PEL or leads to more free volume in the PEL, which makes more counterions to be adsorbed on the

Um = UHS

Z

s

u · nds π a2

(12)

where n is the unit normal vector outward the surface and s denotes the cross sectional area. The distribution of the ratio of cross-sectional average flow Um /UPlane with scaled charged density Q f ix is illustrated in Fig. 6 for two different ionic concentration 1mM and 1000mM. Here, UPlane is the average flow in a plane cylindrical channel without PEL. The average flow rate decreases with the increase of scaled charged density in PEL for both low and high ionic concentration cases but it is strictly decreasing in nature for high ionic concentration. We considered the thickness of the PEL as = 4nm, softness parameter β 2 = 10 and the surface potential of the nanopore wall ζ = 1. Since the surface of the nanopore is negatively charged, the increase of positive scaled charged density of PEL are actually reduces the

6



φ

(a)

(b)

(b)

φ

(a)

Qfix

β β

β

(c)

(c)

FIGURE 4. DISTRIBUTION OF (a) AXIAL VELOCITY u, (b) INDUCED POTENTIAL φ AND (c) IONIC SPECIES g, f , FOR DIFFERENT VALUES OF SCALED CHARGED DENSITY IN PEL Q f ix . HERE, BULK CONCENTRATION C = 1mM, SOFTNESS PARAMETER β 2 = 10, NANOPORE RADIUS a = 10nm, PEL THICKNESS as = 4nm, SURFACE POTENTIAL ζ = −1 AND EXTERNAL IMPOSED ELECTRIC FIELD E0 = 106 V/m. ALLOW INDICATES INCREASING VALUES OF Q f ix AS 0,1,10,20,30,40 and 50.

FIGURE 5. DISTRIBUTION OF (a) INDUCED POTENTIAL POTENTIAL φ WHEN β 2 = 1 (b) IONIC SPECIES g, f FOR DIFFERENT VALUES OF IONIC CONCENTRATION WHEN SOFTNESS PARAMETER β 2 = 1 AND (c) AXIAL VELOCITY u FOR DIFFERENT VALUES OF THE SOFTNESS PARAMETER β 2 = 1, 10, 100. HERE, SCALED CHARGED DENSITY IN PEL Q f ix = 10, NANOPR RADIUS a = 10nm, PEL THICKNESS as = 4nm, SURFACE POTENTIAL ζ = 1 AND EXTERNAL IMPOSED ELECTRIC FIELD E0 = 106 V/m. SOLID; DASH; DASH DOT AND DASH DOT DOT LINES REPRESENT BULK IONIC CONCENTRATION AS C = 1mM; 10mM; 100mM; 1000mM.

EOF velocity and hence decrease the average flow rate. The driving force for EOF is the product of the net charge density of ions ρe and the local electric field E. The increase of ionic concentration, net charge density of the ion becomes very small and flow is governed mainly by high positive charged density of PEL. Fig. 7(a) and (b) show the distribution of dimensional average flow rate with scaled charge density of PEL for different values of the softness parameter β 2 = 1, 10, 100 when bulk ionic concentration C = 1 mM (κ a = 1) and C = 1000mM (κ a = 33) respectively. The average flow rate decreases with the increase of PEL scaled charge density for both low and high ionic concentration cases, but the amplitude of the average flow rate is much more in high ionic concentration. Since, the softness parame-

ter mainly affects the hydrodynamic field inside the nanopore and it is inversely proportional to the hydrodynamics frictional coefficient of PEL. The increase of softness parameter actually increase the EOF velocity in PEL. The influence of the nanopore radius a and PEL thickness as on average flow rate are shown in Fig. 8(a) and (b) respectively. Here, EOF velocity is a function of bulk ionic concentration. Fig. 8a shows the distribution of cross-sectional average flow rate with nanopore radius for different bulk ionic concentration C as 1mM, 10mM, 100mM and 1000mM for fixed PEL thick-

7



1000

1 0.5

Um(µm/sec)

0

Um/UPlane

0

-1000

-0.5 -1

-2000

κa=1 κa=33

-1.5 -2 -2.5 0

β 2=100 β 2=10 β 2=1

-3000 10

20

Qfix

30

40

10

50

50

40

50

Um(µm/sec)

-5000 β =100 β 2=10 2 β =1 2

10

20

Qfix

30

(b)

FIGURE 7. DISTRIBUTION OF THE DIMENSIONAL CROSSSECTIONAL AVERAGE FLOW Um WITH SCALED CHARGED DENSITY WITH SCALED CHARGED DENSITY Q f ix IN DIFFERENT VALUES OF SOFTNESS PARAMETER β 2 = 1, 10, 100 WHEN (a) κ a = 1 (I = 1 mM) AND (b) κ a = 33 (I = 1000mM). HERE, NANOPORE RADIUS a = 10nm, PEL THICKNESS as = 4nm, SURFACE POTENTIAL ζ = −1 AND EXTERNAL IMPOSED ELECTRIC FIELD E0 = 106 V/m.

Q f ix = 1, 10, 20 and 30. But there is no difference of average current density for different scaled charge in high ionic concentration cases. The increase of net charge density increase of axial velocity for both low and high ionic concentration cases. But, for high ionic concentration, net charge density ρe (= g − f ) is very small. Therefore, the contribution of the last term of equation Eq. 13 (i.e., Pe × u × (g − f )) is negligible for high ionic concentration cases and average current density basically is governed by applied electric potential.

(13) Conclusions The motivation of the present study is to study the behavior of electrokinetic flow in a polyelectrolyte coated conical nanopore. The current study presents a numerical investigation on electrokinetic flow through a polyelectrolyte coated conical nanopore. The characteristic of electrokinetic flow is based on the non-linear Nernst-Planck equations for the ion transport coupled with the Brinkman extended Navier-Stokes equations for fluid flow and the Poisson equation for induced electric potential in the exterior and interior of the PEL. The ratio of the average flow with plane cylindrical channel, decreases with the increase

where j0 (= Di en0 /a) is the characteristic electric current density. The cross-sectional average current density (Iz ) along the z−axis is defined as j · nds

40

0

-10000

ness as = 4nm and softness parameter β 2 = 10. The averaged flow rate increases with the increase of bulk ionic concentration and nanopore radius and it becomes constant for high ionic concentration. Similarly, Fig. 8b depicts the cross-sectional average flow rate with the PEL thickness for different bulk ionic concentration for fixed nanopore radius a = 10nm and softness parameter β 2 = 10. The average flow rate decreases with the increase of PEL thickness. We defined critical values of scaled charged density in such a manner so that there will be no fluid flow through PEL coated nanopore. Fig. 9a shows the distribution of critical values of ¨ scaled charged density of PEL with Debye-Huckel parameter κ a when softness parameter, thickness of PEL and nanopore radius are fixed. Fig.9(b) describes the corresponding distribution of the ions concentration g and f for scaled charge density of PEL Q f ix = 20, 30, 40 and 50. The variation of cross-sectional average current density Iz for various values of scaled charge density of PEL is presented in Fig. 10(a) and (b) for both low ionic concentration C = 1mM and high ionic concentration C = 1000mM respectively. The current density is defined by

Z

30

5000

-15000

Iz =

Qfix

(a)

FIGURE 6. THE RATIO OF CROSS-SECTIONAL AVERAGE FLOW Um /UPlane WITH SCALED CHARGED DENSITY Q f ix BETWEEN TWO DIFFERENT IONIC CONCENTRATION (1mM and 1000mM). HERE, NANOPORE RADIUS a = 10nm, PEL THICKNESS as = 4 nm, SOFTNESS PARAMETER β 2 = 10, SURFACE POTENTIAL ζ = −1 AND EXTERNAL IMPOSED ELECTRIC FIELD E0 =106 V/m.

j = eΣzi Ni = j0 [−Σzi ∇ni − Σz2i ni ∇Φ + PeqΣzi ni ]

20

(14)

s

The average current density increases with the increase of applied electric field for different scaled charged density of PEL as

8



0.0035

6000

1000mM 0.003

100mM 0.0025

4000

10mM

κa

Um(µm/sec)

5000

0.002

3000

0.0015

1mM

2000

0.001

1000

0.0005 10

20

a

30

40

50

20

25

30

35

40

45

50

Qfix

(a)

(a)

10000 12 10

1000mM

6000

8

f,g

Um(µm/sec)

8000

100mM

6

4000

10mM

4

2000 2

1mM 0.2

0.4

h

0.6

0 0

0.8

0.2

0.4

r

0.6

0.8

1

(b)

(b)

FIGURE 8. DISTRIBUTION OF THE DIMENSIONAL CROSSSECTIONAL AVERAGE FLOW Um WITH (a) NANOPORE RADIUS a WHEN PEL THICKNESS as = 4nm AND (b) PEL THICKNESS as WHEN NANOPORE RADIUS a = 10nm FOR DIFFERENT VALUES OF IONIC CONCENTRATION. HERE, SCALED CHARGED DENSITY Q f ix = 1, SOFTNESS PARAMETER β 2 = 10, SURFACE POTENTIAL ζ = −1 AND EXTERNAL IMPOSED ELECTRIC FIELD E0 =106 V/m.

FIGURE 9. (a) CRITICAL VALUES OF SCALED CHARGED DENSITY Q f ix WITH κ a FOR NO FLUID FLOW SITUATION AND (b) THE CORRESPONDING IONS DISTRIBUTION (g, f ) when Q f ix = 20, 30, 40, 50. HERE, NANOPORE RADIUS a = 10nm, PEL THICKNESS as = 4nm, SOFTNESS PARAMETER β 2 = 10, SURFACE POTENTIAL ζ = −1 AND AND EXTERNAL IMPOSED ELECTRIC FIELD E0 =106 V/m.

[2] Kang, Y., Yang, C., and Huang, X., 2002, ”Electroosmotic flow in a capillary annulus with high zeta potentials”, Journal of Colloid and Interface Science, Vol. 253, pp. 285-294. [3] Berg, P., Findlay, J., 2011, ”Analytical solution of the PoissonNernstPlanckStokes equations in a cylindrical channel”, Proceedings of the Royal Society: A, Vol. 467, pp. 31573169. [4] Ohshima, H., 2016, ”A simple algorithm for the calculation of the electric double layer potential distribution in a charged cylindrical narrow pore”, Colloid and Polymer Science, Vol. 294, pp. 1871-1875. [5] Bhattacharyya, S. and Bera, S., 2015, ”Combined electroosmosis-pressure driven flow and mixing in a microchannel with surface heterogeneity”, Applied Mathematical Modelling, Vol. 39, pp. 4337-4350 . [6] Chen, X., Lam, Y.C., Chen, X.Y., Chai, J.C. and Yang, C., 2005, ”Numerical simulation of electroosmotic flow with step change in zeta potential”, Innovation in Manufacturing Systems and Technology (IMST), MIT Libraries. URI: http://hdl.handle.net/1721.1/7457 [7] Cao, Q., Zuo, C., Li, L., Ma, Y., Li, N., 2010, ”Electroosmotic flow in a nanofluidic channel coated with neutral polymers”, Microfluid Nanofluid, Vol. 9, pp. 1051-1062.

of the PEL scaled charge density for both low and high ionic concentration cases when softness parameter and thickness of PEL are fixed. The average flow rate decreases with the increase of the PEL scaled charge density in both low and high ionic concentration cases for fixed PEL thickness. The increase of nanopore radius increases the cross sectional averaged flow for fixed scaled charge density and thickness in PEL. The increase of PEL thickness decreases average flow rate for fixed scale charged density in PEL. The average current density increases with the increase of applied electric field.

Acknowledgement One of the authors (S.Bera) thanks the Science & Engineering Research Board (SERB), Department of Science and Technology, Government of India for providing the partial financial support through the project grant (File no: ECR/2016/000771).

REFERENCES [1] Probstein, R. F., 1994, ”Physicochemical Hydrodynamics: An Introduction”, 2nd ed., Wiley Interscience, New York.

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0.6 Qfix=1 Qfix=10 Qfix=20 Qfix=30

0.5

[15]

Iz

0.4 0.3 0.2 0.1 0 3 10

4

10

10

5

10

[16]

6

Ez

(a) 0.4

[17]

Qfix=1 Qfix=10 Qfix=20 Qfix=30

Iz

0.3

0.2

[18]

0.1

0 3 10

4

10

10

5

6

10

Ez

[19]

(b)

FIGURE 10. VARIATION OF AVERAGE CURRENT DENSITY Iz WITH EXTERNAL ELECTRIC FIELD Ex FOR DIFFERENT VALUES OD SCALED CHARGED DENSITY Q f ix WHEN (a) κ a = 1 (b) κ a = 33. HERE, NANOPORE RADIUS a = 10nm, PEL THICKNESS as = 4nm, SOFTNESS PARAMETER β 2 = 10 AND SURFACE POTENTIAL ζ = −1.

[20]

[21]

[8] Zuo, Y., Wang, G., Yu, Y., Zuo, C., Liu, Z., Hu, D., Wang, Y., 2014, ”Suppression of electroosmotic flow by polyampholyte brush”, Microfluid Nanofluid, Vol. 17, pp. 923-931. [9] Wu, R. C., Papadopoulos, K. D., 2000, ”Electroosmotic flow through porous media: cylindrical and annular models”, Colloids and Surfaces A: Physicochemical and Engineering Aspects, Vol. 161, pp. 469-476. [10] Cao, Q., Zuo, C. , Li, L., Ma, Y., Li, N., 2011, ”Controlling electroosmotic flow by polymer coating: a dissipative particle dynamics study”, Microfluid Nanofluid, Vol. 10, pp. 977-990. [11] Scales, N. and Tait, N., Modelling electroosmotic flow in porous media for microfluidic applications, Proc. Int. Conf. on MEMS, NANO and Smart Systems (ICMENS),IEEE Computer Society. [12] Kim, Y., Cha, M., Choi, Y., Joo, H., and Lee, J., 2013, ”Electrokinetic separation of biomolecules through multiple nano-pores on membrane”, Chemical Physics Letters, Vol. 561-562, pp. 63-67. [13] Mao, M., Sherwood, J. D., and Ghosal, S., 2014, ”Electroosmotic flow through a nanopore”, Journal of Fluid Mechanics, Vol. 749, pp. 167-183. [14] Berg, P., and Ladipo, K., 2009, ”Exact solution of an electro-osmotic flow problem in a cylindrical channel of

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