Electrodiffusiophoretic Motion of a Charged Spherical Particle in a

Mar 2, 2010 - Comparing to the case of the uncharged nanopore (circles in Figure 2), the positive net charge surrounding the particle is increased due...
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Electrodiffusiophoretic Motion of a Charged Spherical Particle in a Nanopore Sinan E. Yalcin,† Sang Yoon Lee,‡ Sang W. Joo,‡ Oktay Baysal,† and Shizhi Qian*,†,‡ Department of Aerospace Engineering, Old Dominion UniVersity, Norfolk, Virginia 23529, and School of Mechanical Engineering, Yeungnam UniVersity, Gyongsan 712-749, South Korea ReceiVed: January 26, 2010; ReVised Manuscript ReceiVed: February 15, 2010

The electrodiffusiophoretic motion of a charged spherical nanoparticle in a nanopore subjected to an axial electric field and electrolyte concentration gradient has been investigated using a continuum model, composed of the Poisson-Nernst-Planck equations for the ionic mass transport and the Navier-Stokes equations for the flow field. The charged particle experiences electrophoresis in response to the imposed electric field and diffusiophoresis caused solely by the imposed concentration gradient. The diffusiophoretic motion is induced by two different mechanisms, an electrophoresis driven by the generated electric field arising from the difference of ionic diffusivities and the double layer polarization and a chemiphoresis due to the induced osmotic pressure gradient around the charged nanoparticle. The electrodiffusiophoretic motion along the axis of a nanopore is investigated as a function of the ratio of the particle size to the thickness of the electrical double layer, the imposed concentration gradient, the ratio of the surface charge density of the nanopore to that of the particle, and the type of electrolyte. Depending on the magnitude and direction of the imposed concentration gradient, one can accelerate, decelerate, and even reverse the particle’s electrophoretic motion in a nanopore by the superimposed diffusiophoresis. The induced electroosmotic flow in the vicinity of the charged nanopore wall driven by both the imposed and the generated electric fields also significantly affects the electrodiffusiophoretic motion. 1. Introduction When a charged particle is immersed in an electrolyte solution, the accumulation of a net electric charge near its surface leads to the formation of an electrical double layer (EDL) in the vicinity of the charged particle. Inside of the EDL, the concentration of the counterions is higher than that of the coions, resulting in a net charge within the EDL. In the presence of an electric field, both the charge on the particle and the net charge inside of the EDL interact with the electric field, resulting in electrostatic forces acting on both the particle and the fluid leading to simultaneous electrophoretic and electroosmotic flows. The electrophoresis of particles has been widely utilized in characterizing, separating, and purifying colloidal particles and macromolecules, such as DNA fragments, proteins, drugs, viruses, and biological cells.1-29 For example, a DNA molecule can be electrophoretically driven through a nanopore, nucleobases would modify the ionic current through the nanopore, and thus the sequence of bases in DNA can be recorded by monitoring the current modulations.21-29 This nanopore-based DNA sequencing method is called the third-generation DNA sequencing, and its cost is believed to be sufficiently low to revolutionize genomic medicine.14,23,24 The key to nanopore-based DNA sequencing is the ability to control the nanoparticle electrophoretic translocation process at a level that allows spatial information to be resolved at the nanometer scale, within the finite time resolution imposed by instrumental bandwidth.24,28 However, this is not attainable under the current resolution of the instrument since the DNA translocation velocity is too fast to detect the small ionic currents generated.24 Thus, a technique to slow down DNA translocation * To whom correspondence should be addressed. E-mail: [email protected]. † Old Dominion University. ‡ Yeungnam University.

through a nanopore is required. To date, many methods, including modifying viscosity, temperature, and imposed voltage bias, have been used to slow DNA translocation through the nanopore.26-28 For example, the combination of increasing the viscosity, decreasing the voltage, and lowering the temperature can reduce the DNA velocity by a factor of 10.27 However, an increase in the bulk viscosity or a decrease in the driving voltage reduces the possibility of DNA entering into the nanopore from the fluid reservoir and thus lowering the overall throughput.24,29 In the existing study on nanoparticle electrophoretic translocation through a nanopore, the particle is driven by the imposed electric field, while the electrolyte concentrations in the two fluid reservoirs connecting to the nanopore are the same. In the present study, electrophoretic motion of a charged nanoparticle in a nanopore connecting two fluid reservoirs filled with different electrolyte concentrations are studied, for the first time. In addition to the electrophoretic and electroosmotic flows arising from the imposed electric field, diffusioosmotic and diffusiophoretic flows are generated in response to the imposed electrolyte concentration gradient.30-54 The diffusiophoretic motion is induced by the generated electric field and osmotic pressure gradient arising from the imposed concentration gradient. The generated particle motion by the imposed electric field and concentration gradient is called electrodiffusiophoresis. One might slow down the nanoparticle translocation process in a nanopore by adjusting the solute concentrations in the two fluid reservoirs when the induced diffusiophoretic motion is opposite to the particle’s electrophoretic motion. Therefore, the possibility of slowing down the motion of DNA in a nanopore by diffusiophoretic control seems especially attractive. In the following section, a mathematical model is introduced based on the continuum hypothesis for the fluid motion, induced by the externally imposed electric field and concentration gradient, and the ionic mass transport, which accounts for the

10.1021/jp100784p  2010 American Chemical Society Published on Web 03/02/2010

Electrodiffusiophoretic Motion in a Nanopore

Figure 1. Schematics of a nanopore of length L and radius a connecting two identical reservoirs on either side. The surface charge density along the wall of the nanopore is σw. A charged spherical particle of radius ap bearing uniform surface charge density σp is positioned at the center of the nanopore. A concentration gradient of electrolyte solution and an electric field are applied across the two reservoirs.

polarization of the EDL and is valid for any EDL thickness and arbitrary magnitude of the imposed concentration gradient. The electrodiffusiophoretic motion in a nanopore under various conditions is presented in section 3, followed by concluding remarks in section 4. 2. Mathematical Model We consider a charged, spherical nanoparticle of radius ap and surface charge density σp submerged in a binary electrolyte solution, which is confined in a nanopore of length L and radius a connecting two identical fluid reservoirs on either side, as shown in Figure 1. We assume that the wall of the nanopore has a uniform surface charge density, σw. The two-dimensional cylindrical coordinate system (r, z) with origin located at the center of the nanopore is used. The z and r coordinates are, respectively, parallel and perpendicular to the axis of the nanopore. We assume that the nanoparticle is initially positioned with axis coinciding with the nanopore’s axis, and the location of the particle’s center of mass coincides with the origin. The symmetrical model geometry is represented by the region bounded by the outer boundary ABCDEFGH, the line of symmetry HI, the particle’s surfaces IJ and JK, and the symmetry line KA. The dashed line segments, AB, BC, FG, and GH, represent the regions in the reservoirs. The length LR and radius b of the reservoirs are sufficiently large to ensure that the electrochemical properties at the locations of AB, BC, FG, and GH are not influenced by the charged nanopore and nanoparticle. We also assume that the walls of the two reservoirs (line segments CD and EF) are electrically neutral surfaces. The left and right reservoirs are filled with two identical electrolyte solutions with different bulk concentrations, CL and CR. The segments AB and GH are borders with the reservoirs, between which a potential difference φ0 is applied. We also assume that there is no externally applied pressure gradient across the two reservoirs. Recently, a continuum mathematical model based on the Poisson-Nernst-Planck (PNP) equations for the ionic mass transport and the Navier-Stokes equations for the hydrodynamic field has been developed and used to predict a nanoparticle’s translocation process in a nanopore.55-60 Remarkable agreements between theoretical predictions and experimental data obtained from the literature and the predictions obtained from the molecular dynamics simulations pertaining to the translocation

J. Phys. Chem. B, Vol. 114, No. 11, 2010 4083 of DNA molecules in nanopores suggest that the continuum model, adopted in the current study, provides a reasonable description of the physics associated with the translocation process subjected to external fields including both an electric field and concentration gradient. In the following sections, we present the dimensionless mathematical models for the fluid motion, the ionic mass transport, and the particle’s motion in a nanopore. 2.1. Mathematical Model for the Fluid Motion. We consider a binary electrolyte solution of viscosity µ and permittivity ε maintained at a constant temperature T in the fluid reservoirs and inside of the nanopore. We use the macroscopic electrolyte concentration measured at the particle’s center in the absence of the particle and nanopore wall, C0 ) (CL + CR)/ 2, as the ionic concentration scale, RT/F as the potential scale, where R is the universal gas constant and F is the Faraday constant, the nanoparticle’s radius ap as the length scale, U0 ) εR2T2/(µapF2) as the velocity scale, and µU0/ap as the pressure scale. Due to the extremely low Reynolds numbers of electrokinetic flows in nanopores, the motion of the incompressible binary electrolyte solution is described by the modified Stokes equations

∇ · u* ) 0

(1)

1 -∇p* + ∇2u* - (κap)2(z1c*1 + z2c*2)∇V* ) 0 2

(2)

In the above, variables with * are dimensionless. u* ) u*er + V*ez is the fluid’s velocity vector. Hereafter, bold letters denote vectors; er and ez are, respectively, unit vectors in the r and z directions; u* and V* are, respectively, the velocity components in the r and z directions; p* is the pressure; V* is the electric potential in the electrolyte solution; c*1 and c*2 are, respectively, the molar concentrations of the positive and negative ions in the electrolyte solution; z1 and z2 are, respectively, the valences of the positive and negative ions; and κ-1 ) λD ) (εRT/2F2C0)1/2 is the dimensional EDL thickness. The last term on the lefthand side (LHS) of eq 2 represents the electrostatic force acting on the fluid through the interaction between the overall electric field and the net charge density in the electrolyte solution. In order to solve eqs 1 and 2, appropriate boundary conditions are required. A nonslip boundary condition (i.e., u* ) V* ) 0) is specified at the solid walls of the nanopore and the reservoirs (line segments CD, DE, and EF in Figure 1). At the planes AB and GH of the reservoirs, since they are far away from the nanopore and there is no externally applied pressure gradient across the two reservoirs, normal flow with pressure p* ) 0 is imposed. A symmetric boundary condition is used along the lines of symmetry, HI and KA. Slip boundary conditions are used on the segments BC and FG since they are far away from the entrances of the nanopore. Finally, along the surface of the particle (arc segment IJK in Figure 1) translating with an electrodiffusiophoretic velocity u*, p we neglect the thickness of the adjacent Stern layer and impose the no-slip condition as

u* ) u*pez

on IJK

(3)

The particle’s electrodiffusiophoretic velocity u*p is determined by requiring the total force in the z direction (F*T ) acting on the particle to vanish

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F*T ) F*E + F*D ) 0

(4)

F*E and F*D are, respectively, the z component of the electrostatic and hydrodynamic forces acting on the particle

F*E )

∫ (TE* · n) · ezds*

(5)

F*D )

∫ (TD* · n) · ezds*

(6)

and

E*

In the above, n is the unit vector normal to the corresponding surface. The boundary conditions on the segments BC and FG are defined with the assumption that these surfaces are in the bulk electrolyte reservoirs. Accordingly, zero normal flux is used for the NP equations on BC and FG. Along the segments HI and KA, the symmetric boundary condition is used for the NP equations. The symmetric boundary condition for the electric potential in the electrolyte solution is used on planes HI and KA. A j 0, is imposed across the planes AB and potential difference, φ GH; therefore

V* ) φ¯ 0

on AB

(13)

V* ) 0

on GH

(14)

D*

T and T are the dimensionless Maxwell stress tensor and the hydrodynamic stress tensor, respectively, and s* represents the surface of the particle. 2.2. Mathematical Model for the Ionic Mass Transport. A general multi-ion mass-transport model includes the NernstPlanck (NP) equations for the ionic concentrations and the Poisson equation for the electric potential in the electrolyte solution. The dimensionless flux density, normalized by U0C0, of each aqueous species due to convection, diffusion, and migration is given by

N*i ) u*c*i - Λi∇c*i - ziΛic*∇V* i

Since the surfaces of BC and FG are far away from the nanopore and are in the bulk electrolyte reservoirs, no charge boundary condition for the potential is used

n · ∇V* ) 0

i ) 1 and 2

(8)

(9)

On the plane AB, which is sufficiently far away from the nanopore, the concentrations of the positive and negative ions are the same as the bulk concentration of the electrolyte solution present in the left reservoir

c*1 ) c*2 ) 2/(1 + R)

on AB

(10)

where R ) CR/CL is the concentration ratio between the right and left reservoirs. Similarly, the ionic concentrations on the plane GH are the same as the bulk concentration of the electrolyte solution in the right reservoir

c*1 ) c*2 ) 2R/(1 + R)

on GH

(11)

At the walls of the reservoirs, the wall of the nanopore, and the surface of the nanoparticle (line segments CD, DE, EF, and IJK in Figure 1), since the solid surfaces are impervious to ions, the net ionic fluxes normal to the rigid walls are 0

n · N*1 ) n · N*2 ) 0

(15)

Since the walls of the reservoirs (planes CD and EF) do not carry fixed charge, the zero surface charge boundary condition for the Poisson equation is then used

n · ∇V* ) 0

on CD, DE, EF, and IJK

(12)

on CD and EF

(16)

Along the wall of the nanopore and the particle’s surface, the surface charge boundary condition is used

The electric potential is described by the Poisson equation

1 -∇2V* ) (κap)2(z1c*1 + z2c*) 2 2

on BC and FG

(7)

where Λi ) Di/D0 with D0 ) εR2T2/(µF2) and Di is the diffusion coefficient of the ith ionic species. Under steady state, the concentration of each species is governed by the NP equation

∇ · N*i ) 0

and

n · (-∇V*) ) σ¯ w

on DE

(17)

n · (-∇V*) ) σ¯ p

on IJK

(18)

and

The scale of the surface charge density is εRT/(apF). Note that the flow, concentration, and electric fields described above are strongly coupled. The fluid flow affects the ionic mass transport through the convection, and the concentration and electric fields affect the electrostatic forces acting on the net charge in the electrolyte solution and the charged particle. The fluid flow and the particle’s motion are coupled through the velocity boundary condition (eq 3) along the particle’s surface. Therefore, one has to simultaneously solve the coupled system including the Stokes eqs 1 and 2 and the Poisson-Nernst-Planck eqs 8 and 9 subjected to the aforementioned boundary conditions. 3. Results and Discussion The particle’s velocity up* is also unknown a priori and is determined iteratively using the Newton-Raphson method to estimate the offset correction starting from an appropriate initial guess.55-58 With an initial guess of the particle’s velocity, we simultaneously solve the coupled system using a commercial finite element package, COMSOL version 3.5a (www.comsol.com), installed in a workstation with 96 GB of RAM. On the basis of the obtained solution of the various fields, the net axial

Electrodiffusiophoretic Motion in a Nanopore force acting on the particle is evaluated using eqs 5 and 6. The resulting net force acting on the particle is not likely to satisfy the force balance eq 4, and the particle’s velocity is then corrected using the Newton-Raphson method. This iterative process is repeated until the magnitude of the net axial force becomes smaller than the absolute error bound, 10-6. This process typically converges within less than five iterations. Quadratic triangular elements with variable sizes are used to accommodate finer resolutions near the charged particle surface IJK and the wall of the nanopore DE where EDLs are present. Solution convergence is guaranteed through mesh-refinement tests on conservation laws. The mathematical model and its implementation with COMSOL have been validated by many benchmark tests. For example, we simulated the ionic mass transport near a charged planar surface in the absence of any external field using the Poisson-Nernst-Planck (PNP) equations without convection, and the numerical results are in good agreement with the analytical solution (Figure 2 in ref 61). The numerical results of the electroosmotic flow in a cylindrical nanopore driven by an externally imposed electric field obtained from the coupled PNP and Navier-Stokes equations are also in good agreement with the analytical solution (Figure 3 in ref 61). The electrophoretic motion driven by the externally imposed electric field (e.g., R ) 1) is also simulated using the model described in section 2, and our numerical results agree with the corresponding approximate analytical solution and experimental results obtained from the literature.55-60 We also simulated the diffusioosmostic flow in a slit nanochannel connecting to fluid reservoirs using the coupled system, and our numerical results40 agree with the results obtained by Pivonka and Smith.62 These good agreements of various benchmark problems under either an electric field or concentration gradient make us confident of our following computational results. In this section, we present a few numerical results of the electrodiffusiophoretic motion of a charged spherical nanoparticle in a nanopore under various conditions. We focus on the effects of the EDL thickness, κap, the ratio of the nanopore’s j p, and surface charge density to that of the particle, β ) σ j w/σ the bulk concentration ratio, R, on the particle’s translocation velocity in three different electrolytes (NaCl, KCl, and HCl). The diffusion coefficients of the ions K+, Na+, H+, and Clare, respectively, 1.95 × 10-9, 1.33 × 10-9, 9.53 × 10-9, and 2.03 × 10-9 m2/s (page 195 in ref 17). The temperature of the electrolyte solution in the reservoirs and the nanopore is maintained at 300 K. In the numerical simulations, the following parameters are used: L ) 0.5 µm, ap ) 5 nm, a/ap ) 4, LR ) 0.15 µm, b ) 0.15 µm, φ0 ) 100 mV, and σp ) -0.1 C/m2. Figure 2 depicts the particle’s electrophoretic velocity in NaCl, KCl, and HCl solutions in the absence of the axial concentration gradient (i.e., R ) 1) for β ) 1 (lines with triangles), 0 (lines with circles), and -1 (lines with squares) under the conditions of κap ) 1 (open symbols) and κap ) 3 (solid symbols). When the wall of the nanopore is not charged (e.g., β ) 0), as expected, the negatively charged particle electrophoretically migrates toward the anode. The electrophoretic velocity in HCl solution is lower than that of KCl, which is lower than that of NaCl, which arises from the difference in the diffusivities of the cations. The diffusion coefficient of H+ is higher than that of K+, which is higher than that of Na+. Cations are accumulated in the EDL surrounding the negatively charged spherical particle (Figure 3b), while the anions are repelled away from the negatively charged particle (Figure 4b), and the resulting EDL is spherically symmetrical without an external field. Under the effect of the

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Figure 2. Particle’s electrophoretic velocity in the absence of the concentration gradient (i.e., R ) 1) as a function of the type of electrolytes when κap ) 1 (lines with open symbols) and κap ) 3 (lines with solid symbols).

external electric field directed along the positive z direction, the mobile cations in the EDL move along the particle surface from the bottom hemisphere to the top hemisphere. The electrolyte concentration slightly decreases at the bottom hemisphere since the counterions (cations) migrate from the bulk to the diffuse layer (weak lateral flow of co-ions can be neglected) and increases at the top hemisphere since cations come from the diffuse layer to the adjoining bulk of the electrolyte solution.53 The concentration polarization in the EDL results in ion diffusion flows directed to the surface of the bottom hemisphere where the concentration is decreased and from the surface of the top hemisphere where the concentration is increased (Figure 3 in ref 53). The induced ion diffusion flow is proportional to D+/D-, where D+ and D- are the diffusivities of the cations and anions, respectively. The diffusion flow is opposite to the direction of the particle motion and thus retards the movement of the particle. Since D(H+) > D(K+) > D(Na+), the retardation force by the induced diffusion flow arising from the double layer polarization in HCl is higher than that in KCl, which is higher than that in NaCl. Consequently, the magnitude of the electrophoretic velocity in NaCl is higher than that in KCl, which is higher than that in HCl. As κap increases (i.e., the EDL thickness decreases), the EDL becomes more symmetrical, and the retardation effect by the induced ion diffusion flow decreases, resulting in an almost identical particle velocity for the three different electrolytes (i.e., the case of κap ) 3, solid circles in Figure 2). For the same electrolyte, the particle velocity decreases as κap increases. This behavior is attributed to the wall effects, which become more significant with the increase of the double layer thickness.55 When the nanopore wall is charged (i.e., β * 0), EDLs form in the vicinity of the charged nanopore wall and the charged nanoparticle. The net charge, which is proportional to the concentration difference between the cations and anions, surrounding the negatively charged nanoparticle will be increased (Figures 3a and 4a) or decreased (Figures 3c and 4c) if the nanoparticle and the nanopore carry the same or opposite surface charges. For β ) 1 (lines with triangles in Figure 2), both the nanoparticle and the nanopore wall are negatively charged. Comparing to the case of the uncharged nanopore (circles in

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Figure 3. Spatial distribution of the dimensionless concentration of the cations around the particle when β ) 1 (a), 0 (b), and -1 (c). κap ) 1 and R ) 1 in KCl solution.

Figure 4. Spatial distribution of the dimensionless concentration of the anions around the particle when β ) 1 (a), 0 (b), and -1 (c). κap ) 1 and R ) 1 in KCl solution.

Figure 2), the positive net charge surrounding the particle is increased due to the presence of the excess cations in the EDL near the nanopore wall (Figure 3a). The direction of the particle’s motion is reversed, and the negatively charged particle moves in the same direction as that of the applied electric field. Under this condition, the interaction between the applied axial electric field and the enriched net charge surrounding the particle and in the gap between the particle and the nanopore wall induces a strong electroosmotic flow (EOF) directed toward cathode. The induced strong EOF drags the particle toward the

cathode, resulting in a positive electrophoretic velocity. Under all other same conditions, the net charge surrounding the particle and in the gap between the particle and the nanopore wall decreases as κap increases, resulting in lower hydrodynamic driving force from the EOF and thus lower particle velocity for κap ) 3 compared to the case of κap ) 1. For β ) -1 (lines with squares in Figure 2), the particle is negatively charged, while the nanopore wall is positively charged. Compared to the case of β ) 0, the concentration of the cations (anions) surrounding the particle and in the gap between the particle and

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Figure 5. Electrodiffusiophoretic velocity as a function of the concentration ratio, R, in KCl solution when κap ) 1 (a) and 3 (b). The solid lines with circles, dashed lines with triangles, and dash-dotted lines with squares represent, respectively, the results of β ) 0, 1, and -1.

the nanopore wall is decreased (increased) due to the opposite charges in the two EDLs, as shown in Figures 3c and 4c, which decreases the positive net charge surrounding the negatively charged particle and thus reduces the retardation force arising from the electroosmotic flow, which is opposite to the direction of the particle motion. In addition, the net charge near the positively charged nanopore wall is negative because the concentration of the anions is higher than that of the cations, shown in Figures 3c and 4c, resulting in an EOF in the gap, the direction of which is the same as that of the particle electrophoretic motion. In addition to the electrical driving force arising from the interaction between the electric field and the charge on the particle, the generated EOF also provides a driving force for the particle to move toward the anode. Therefore, the magnitude of the negative electrophoretic velocity is higher for β ) -1 compared to that for β ) 0 due to the reduction of the retardation force and the additional driving force arising from the EOF in the gap between the particle and the nanopore wall. For the same electrolyte, as κap increases, the magnitude of the negative net charge near the positively charged nanopore wall increases due to the decrease in the effect of the opposite charges in the EDL of the nanoparticle, leading to an increase in the EOF in the gap between the particle and the nanopore wall and thus an increase in the magnitude of the electrophoretic velocity. Therefore, when the nanopore wall is uncharged, the electrostatic force acting on the particle is the driving force for the electrophoretic motion. The particle is driven by the generated EOF if the surface charge density of the nanopore wall is relatively high. Figure 5 depicts the particle’s translocation velocity as a function of the imposed concentration ratio, R, for a KCl electrolyte solution when κap ) 1 (a) and 3 (b). The solid lines with circles, dashed lines with triangles, and dash-dotted lines with squares represent, respectively, the results when β ) 0, 1, and -1. In Figure 5b, since the particle velocity at β ) 0 is very small compared to those at β ) (1, its magnitude is multiplied by a factor of 10 to enhance visibility. Similar results

are obtained for NaCl and HCl solutions, and their results thus are not shown. For the case of R * 1, a negative axial concentration gradient is imposed for R < 1 and vice versa. To clearly illustrate the diffusiophoretic effect, Figure 6 depicts the deviation of the particle electrodiffusiophoretic velocity from the electrophoretic velocity without imposing a concentration - u*(R ) 1), as a function of the imposed gradient, χ ) u*(R) p p concentration ratio, R, for KCl electrolyte solution when κap ) 1 (a) and 3 (b). The conditions are the same as those in Figure 5. Since the imposed electric field is fixed, the resulting electrodiffusiophphoretic velocity versus R, shown in Figure 5, is similar to the diffusiophoretic velocity as a function of the imposed concentration ratio, shown in Figure 6. When the nanopore wall is uncharged and κap ) 1 (solid line with circles in Figure 5a), as R decreases from R ) 1, the magnitude of the particle’s velocity decreases, and the particle translocation process is slowed down by imposing a negative axial concentration gradient. The direction of the particle’s motion is even reversed when R is smaller than a critical value, Rc, at which the particle’s velocity is 0. As the concentration ratio further decreases, the negatively charged particle migrates toward the cathode, and its velocity increases due to the increase in the imposed concentration gradient. When the concentration ratio is smaller than a certain value, the ionic concentration in the bottom reservoir is c*1 ) c*2 ) 2/(1 + R) f 2, and that in the top reservoir is c*1 ) c*2 ) 2R/(1 + R) f 0, leading to the saturation of the concentration gradient and thus the particle diffusiophoretic velocity. On the contrary, the electrophoretic velocity is accelerated by the diffusiophoresis when R > 1. As R increases from R ) 1, the particle velocity increases and becomes independent of R when the latter is large enough. When R is large enough, the concentration in the bottom reservoir is c*1 ) c*2 ) 2/(1 + R) f 0, and that in the top reservoir is c*1 ) c*2 ) 2R/(1 + R) f 2, resulting in the saturation of the imposed concentration gradient and consequently the particle motion, which is dominated by the generated diffusiophoresis. When the nanopore wall is uncharged and κap ) 3 (solid line with

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Figure 6. Particle velocity deviation χ as a function of the concentration ratio, R, in KCl solution when κap ) 1 (a) and 3 (b). The solid lines with circles, dashed lines with triangles, and dash-dotted lines with squares represent, respectively, the results of β ) 0, 1, and -1.

circles in Figure 5b), the EDL thickness is smaller than that of κap ) 1, and the particle translocation process is slowed down by applying a concentration gradient, regardless of the direction of the imposed concentration gradient. Under this condition, the direction of the particle’s motion is not reversed. Obviously, two diverse effects are obtained for κap ) 1 (relatively thick EDL) and κap ) 3 (relatively thin EDL) when a positive axial concentration gradient is applied (i.e., R > 1), which arises from the opposite directions of the generated diffusiophoretic motion at κap ) 1 and 3, shown in Figure 6, and will be elaborated in the following. The diffusiophoresis is induced by the chemiphoresis and the induced electrophoresis.42,45,48-50,53,54 The former one is generated by the induced osmotic pressure gradient around the charged particle, which propels the particle toward a higher salt concentration region, regardless of the sense of the particle’s charge. The latter one is induced by the generated electric fields arising from the difference of ionic diffusivities and from the double layer polarization (DLP) under the effect of the imposed concentration gradient. Usually, the electrophoretic effect dominates over the chemiphoretic effect. When the diffusive mobilities of anions and cations are unequal, an electric field, Ediffusivity, is induced so that the diffusion of high-mobility (lowmobility) ions is decelerated (accelerated). For example, since the diffusion mobilities of the anions and cations in KCl electrolyte are almost identical, the induced electric field arising from the difference in the ionic diffusivities is very small in KCl solution. Since the anions have a higher diffusion coefficient than the cations in NaCl solution, an electric field, Ediffusivity, is induced and directed from higher NaCl concentration to lower concentration. In contrast, an electric field, Ediffusivity, directed from lower electrolyte concentration to higher electrolyte concentration is generated in HCl solution because the cations have a higher diffusion coefficient than the anions. Similar to the externally imposed electric field, the electric field, Ediffusivity,

generated on account of the difference in the ionic diffusivities generates electrophoretic motion. With the imposed concentration gradient, inside of the EDL, the amount of the counterions on the high concentration side is higher than that on the low concentration side of the particle, resulting in an electric field, EI-DLP, the direction of which is opposite to (the same as) that of the applied concentration gradient when the particle is negatively (positively) charged. This kind of DLP is called the type I DLP, and its resulting electric field is named as EI-DLP. On the other hand, the concentration of the co-ions near the outer boundary of the EDL on the high concentration side is higher than that on the low concentration side of the particle, which is called the type II DLP. This generates an electric field, EII-DLP, the direction of which is opposite to that established by the counterions inside of the EDL, EI-DLP. Therefore, when a concentration gradient is imposed, an electric field is generated by three mechanisms; the first one, Ediffusivity, is established by the difference in the ionic diffusivities; the second one, EI-DLP, is established by the type I DLP inside of the EDL; and the third one, EII-DLP, is generated by the type II DLP near the outer boundary of the EDL. The induced electrophoresis stems from the interaction between the particle’s charge and the generated electric fields, Ediffusivity, EI-DLP, and EII-DLP. The induced EIDLP from the type I DLP always propels the particle toward higher electrolyte concentration, while EII-DLP generated by the type II DLP always drags the particle toward lower electrolyte concentration, regardless of the sign of charge on the particle. Depending on the difference of the diffusivities of the cations and anions, which depends on the type of electrolyte used, and the polarity of the particle’s surface charge, the electrophoretic motion generated by the induced Ediffusivity might drive the particle toward either the low or high concentration side. Usually, the induced electrophoretic effect generated by Ediffusivity is more significant if the boundary effect is insignificant (i.e., large a/ap and κap) and the ionic diffusivities are unequal. In

Electrodiffusiophoretic Motion in a Nanopore the current study, the electrophoretic effect arising from the DLP-induced electric fields dominates since the EDL thickness, the particle size, and the nanopore size are of the same order of magnitude. For thin EDL (i.e., relatively large κap), usually, the electric field generated by the type I DLP, EI-DLP, is stronger than that from the type II DLP, EII-DLP, if the boundary effect arising from the nanopore wall is insignificant and the magnitude of the particle’s surface charge is relatively low; however, the electric field arising from the type II DLP dominates over that from the type I DLP if the surface charge or surface potential of the particle is relatively high.52 For a thick EDL (i.e., relatively small value of κap), usually, the induced electrophoretic effect from EII-DLP dominates over that arising from the EI-DLP. Note that the degree of the aforementioned type I and II DLPs arising solely by the imposed concentration gradient will be further affected by the externally imposed electric field. For R < 1, the concentration near the bottom hemisphere is higher than that near the top hemisphere. Inside of the EDL, the concentration of the counterions (cations) near the bottom hemisphere is higher than that near the top hemisphere. Under the externally imposed axial electric field, the cations are displaced from the bottom hemisphere toward the top hemisphere, which reduces the counterions’ concentration difference between the bottom and top hemispheres and consequently reduces the electric field generated by the type I DLP effect. On the contrary, the co-ions at the outer boundary of the EDL are displaced from the top hemisphere toward the bottom hemisphere, which increases the co-ions’ concentration difference at the outer boundary of the EDL near the bottom and top hemispheres and thus increases the type II DLP effect. Therefore, for R < 1, the DLP arising from the imposed electric field reduces the degree of the type I DLP and increases the type II DLP. On the contrary, the imposed axial electric field reduces the degree of the type II DLP and increases the degree of the type I DLP when R > 1. For KCl, the generated Ediffusivity is very small due to the small difference in the diffusion coefficients of the cations and anions. For an uncharged nanopore (i.e., β ) 0) and κap ) 1 (solid line with circles in Figure 6a), the electrophoretic effect arising from EII-DLP dominates over that from EI-DLP. The generated EII-DLP propels the particle toward lower salt concentration, resulting in a positive diffusiophoretic velocity for R < 1 and negative diffusiophoretic velocity for R > 1, as shown in Figure 6a (solid line with circles). The results also suggest that the chemiphoretic effect is insignificant compared to the induced electrophoretic effect since the former one propels the particle toward higher salt concentration. As κap increases, the electric field generated by the type I DLP, EI-DLP, becomes stronger, and the generated electric field by the type II DLP, EII-DLP, becomes weaker. For κap ) 3 (solid line with circles in Figure 6b), the diffusiophoretic velocity results from the competition of the two opposite DLP effects. For R < 1, the electrophoretic effect by EII-DLP dominates over that by EI-DLP due to the increase in the type II DLP and decrease in the type I DLP by the DLP arising from the imposed electric field; thus, a positive diffusiophoretic velocity results, as shown in Figure 6b. When R > 1, the degree of the type II DLP is reduced, and that of the type I DLP is increased by the DLP arising from the imposed electric field, and the induced electrophoretic effect by EI-DLP dominates over that by EII-DLP, resulting in a positive diffusiophoretic velocity, as shown in Figure 6b. When the nanopore wall becomes charged, the electrodiffusiophoretic velocity versus R, shown in Figure 5, becomes more

J. Phys. Chem. B, Vol. 114, No. 11, 2010 4089 complicated, which mainly arises from the complicated variation of the diffusiophoretic velocity as a function of R, shown in Figure 6. Regardless of the polarity of the nanopore’s charge, as R decreases from R ) 1, the generated diffusiophoretic motion is directed from higher salt concentration toward lower salt concentration, and the magnitude of its velocity increases, obtains a local maximum, and then declines. As R further decreases, the direction of the diffusiophoretic motion is reversed and is directed toward higher salt concentration. When the direction of the imposed concentration gradient is reversed (i.e., R > 1), an opposite diffusiophoretic motion is induced. As R increases from R ) 1, the diffusiophoretic motion is directed from the higher concentration side to the lower concentration side, and its magnitude increases, obtains a local maximum, and then decreases as the imposed concentration gradient increases. When R exceeds a certain value, the diffusiophoretic motion is reversed, and its magnitude increases as R further increases. When the nanopore wall is charged, in addition to the EDL surrounding the charged nanoparticle, an EDL forms in the vicinity of the charged nanopore wall. Figures 7 and 8 depict, respectively, the spatial distribution of the dimensionless concentrations of the cations and anions around the negatively charged particle for β ) 1(a), 0 (b), and -1 (c) in KCl solution with R ) 0.001 and κap ) 1. Figure 9 depicts the concentration difference between the cations and anions, c*1 - c*2 , under the same conditions as those shown in Figures 7 and 8. Similar spatial concentration distributions are obtained for R ) 0.1 and thus are not shown. Compared to the case of β ) 0, which corresponds to an uncharged nanopore wall (Figures 7b, 8b, and 9b), both the nanoparticle and the nanopore wall are negatively charged for β ) 1, and the presence of the EDL of the nanopore wall enriches the concentration of the cations (Figure 7a) and depletes the concentration of the anions (Figure 8a), resulting in higher net charge surrounding the particle and in the gap between the particle and the nanopore wall, shown in Figure 9a. For β ) -1, the concentration of the cations (anions) is decreased (increased), arising from the opposite charges of the nanoparticle and the nanopore wall. Compared to the case of R ) 1 in the absence of the imposed concentration gradient (Figures 3 and 4), the contour lines of the ionic concentrations shown in Figures 7 and 8 become more asymmetric around the particle, implying significant DLP arising from the imposed concentration gradient. Since the electric field generated by the type II DLP dominates over that arising from the type I DLP, the overall electric field is dominated by EIIDLP, which is directed toward higher salt concentration. For β ) 1 and κap ) 1 (dashed line with triangles in Figure 6a), as R decreases from R ) 1, the electrical driving force arising from the interaction between the negative charge of the particle and the generated EII-DLP drives the particle toward lower salt concentration, resulting in a positive diffusiophoretic velocity. The interaction between EII-DLP and the positive net charge surrounding the particle and in the gap between the particle and the wall generates electroosmotic flow, which is directed toward higher salt concentration. The direction of the generated electroosmotic flow is opposite to that of the diffusiophoretic motion. As R decreases, the concentration gradient increases, and the magnitude of the generated electric field EII-DLP and the net charge, c*1 - c*2 , increase, which simultaneously increases the electrical driving force acting on the particle and the body force in the fluid, and the latter increases the opposite electroosmotic flow. The diffusiophoretic velocity attains a maximum due to the competition between the

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Figure 7. Spatial distribution of the dimensionless concentration of the cations around the negatively charged particle when β ) 1 (a), 0 (b), and -1 (c). κap ) 1 and R ) 0.001 in KCl solution.

Figure 8. Spatial distribution of the dimensionless concentration of the anions around the negatively charged particle when β ) 1 (a), 0 (b), and -1 (c). κap ) 1 and R ) 0.001 in KCl solution.

hydrodynamic force arising from the generated opposite electroosmotic flow and the electrical driving force acting on the particle. As R further decreases, the rate of increase in the hydrodynamic force is faster than the rate of increase in the corresponding electrical driving force, resulting in the decrease in the diffusiophoretic velocity. When R is smaller than a certain

value (or the imposed concentration gradient exceeds a certain value), the hydrodynamic force arising from the induced electroosmotic flow becomes the driving force for the particle motion, and the particle is driven by the generated electroosmotic flow, leading to the reverse of the diffusiophoretic velocity, shown in Figure 6a. On the contrary, the generated

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Figure 9. Spatial distribution of the concentration difference, c1* - c2*, around the negatively charged particle when β ) 1 (a), 0 (b), and -1 (c). κap ) 1 and R ) 0.001 in KCl solution.

EII-DLP is directed in the positive axial direction for R > 1. As R increases from R ) 1, the electrical driving force stemming from the interaction between EII-DLP and the negative charge on the particle induces negative diffusiophoretic velocity. The generated electroosmotic flow arising from the interaction between EII-DLP and the positive net charge is directed along the positive axial direction. As R increases, both the magnitude of EII-DLP and the net charge surrounding the particle and in the gap increase, which leads to the increase in the electrical driving force and the hydrodynamic retardation force acting on the particle. When the rate of increase in the retardation force is faster than that of the electrical driving force, the magnitude of the diffusiophoretic velocity attains the maximum and then declines. As R further increases, the induced electroosmotic flow provides the driving force and drags the particle moving toward higher salt concentration. For β ) 1 and κap ) 3 (dashed line with triangles in Figure 6b), the variation of the diffusiophoretic velocity with R is similar to that under κap ) 1, except that the direction of the diffusiophoretic motion is not reversed. In addition, the magnitude of the diffusiophoretic velocity for κap ) 3 is higher than that for κap ) 1. As κap increases, the EDL thickness decreases, and the enrichment of the cations and the depletion of the anions surrounding the particle and in the gap become less significant. The net charge surrounding the particle and in the gap for κap ) 3 is lower than that under κap ) 1, leading to relatively weaker electroosmotic flow for κap ) 3. Consequently, the retardation force arising from the electroosmotic flow for κap ) 3 is lower than that of κap ) 1, resulting in higher magnitude of the diffusiophoretic velocity. Since the generated electroosmotic flow is not strong enough, the particle is driven by the electrical driving force arising from the interaction between EII-DLP and the negative charge of the particle, and the direction of the diffusiophoretic motion is not reversed for κap ) 3. For β ) -1 and κap ) 1 (dash-dotted line with squares in Figure 6a), similar to the cases of β ) 0 and 1, EII-DLP generated by the type II DLP dominates. The net charge is positive in the region surrounding the negatively charged particle and is

negative in the region near the positively charged nanopore wall, as shown in Figure 9c. The magnitude of the net charge increases as the magnitude of the imposed concentration gradient increases. For R < 1, EII-DLP is directed in the negative axial direction. As R decreases from R ) 1, the electrical driving force arising from the interaction between EII-DLP and the negative charge on the particle generates positive diffusiophoretic motion. The interaction between the generated EII-DLP and the positive net charge surrounding the particle generates electroosmotic flow which is opposite to the particle motion. In addition to the fluid motion, which tends to retard the particle motion, the interaction between EII-DLP and the negative net charge near the positively charged nanopore wall induces electroosmotic flow, the direction of which is the same as that of the particle diffusiophoretic motion, and thus enhances the particle motion. Therefore, the diffusiophoretic velocity for β ) -1 is higher than that of β ) 1 since the electroosmotic flow near the positively charged nanopore wall provides an additional driving force while the electroosmotic flow near the negatively charged nanopore wall retards the particle motion. As R decreases, the generated electric field EII-DLP and the magnitude of the net charge increase, leading to the increase in the electrical driving force, the driving force from the electroosmotic flow near the nanopore wall, and the retardation force from the opposite electroosmotic flow near the particle. Similar to the case of β ) 1, as R decreases, the diffusiophoretic velocity increases, peaks, and then declines. The particle motion is reversed when the hydrodynamic drag arising from the fluid motion in the vicinity of the particle becomes the driving force. On the country, an opposite variation of the diffusiophoretic velocity with R is obtained when the direction of the imposed concentration gradient is reversed. As κap increases, the influence of the EDL formed in the vicinity of the nanopore wall on the ionic concentration in the EDL surrounding the nanoparticle decreases, resulting in higher net charge and consequently stronger fluid motion surrounding the particle. Therefore, the magnitudes of the particle’s velocity at R ) 0.001 and 1000

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under κap ) 3 are higher than those at κap ) 1, under which the particle motion is driven by the fluid motion surrounding the particle. 4. Conclusions The electrodiffusiophoretic motion of a charged particle in a nanopore simultaneously subjected to an axial electric field and solute concentration gradient has been numerically investigated using a mathematical model, composed of Poisson-NernstPlanck equations for the ionic mass transport and Stokes equations for the hydrodynamic field. The model takes into account the full interactions of the particle, fluid, electric field, and ionic mass transport, under conditions of an arbitrary level of surface potential on the particle and the nanopore wall, the EDL thickness, and the magnitude of the imposed concentration gradient and electric field. The charged particle experiences both electrophoresis driven by the imposed electric field and diffusiophoresis arising from the imposed concentration gradient. Since the EDL thickness, the nanoparticle size, and the nanopore size are of the same order of magnitude, the induced diffusiophoresis is dominated by the induced electrophoresis driven by the generated electric field arising from the type II double layer polarization, which drives the particle toward lower electrolyte concentration, regardless of the sign of the particle’s surface charge. The diffusiophoretic motion can be used to enhance, slow down, or even reverse the electrophoretic motion in a nanopore depending on the magnitude and direction of the imposed concentration gradient and the surface charge of the nanopore wall. When the wall is not charged, the induced diffusiophoretic motion slows down the electrophoretic motion, regardless of the direction of the imposed concentration gradient if the thickness of the EDL is relatively small; the electrophoretic motion is accelerated (decelerated) under a concentration gradient which is in the same (opposite) direction of the applied electric field if the EDL thickness is relatively large. When the nanopore wall is charged, induced electroosmotic flow arising from the interaction between the generated electric field and the net charge near the nanopore wall will also influence the diffusiophoretic motion. Regardless of the sign of the nanopore’s surface charge, the diffusiophoretic motion is driven by the electrical driving force stemming from the interaction between the generated electric field EII-DLP from the type II DLP and the charge of the particle, and the particle migrates from higher electrolyte concentration toward lower electrolyte concentration if the magnitude of the imposed concentration gradient is relatively small; the hydrodynamic force arising from the induced electroosmotic flow surrounding the particle provides the driving force for the particle motion, and the particle moves toward higher electrolyte concentration if the magnitude of the imposed concentration gradient is relatively high. One might use electrodiffusiophoresis to regulate the nanoparticle’s translocation process in a nanopore to achieve a nanometer-scale spatial accuracy for DNA sequencing by simultaneously controlling the electric field and the concentration gradient. Acknowledgment. This work is supported by the World Class University Grant No. R32-2008-000-20082-0 of the Ministry of Education, Science and Technology of Korea. References and Notes (1) Angelova, A.; Angejov, B.; Lesieur, S.; Mutafchieva, R.; Ollivon, M.; Bourgaux, C.; Willumeit, R.; Couvreur, P. J. Drug DeliVery Sci. Technol. 2008, 18, 41–45. (2) Kim, P.; Baik, S.; Suh, K. Y. Small 2008, 4, 92–95.

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