* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download The Effect of Axial Concentration Gradient on
Lagrangian mechanics wikipedia , lookup
Introduction to gauge theory wikipedia , lookup
Speed of gravity wikipedia , lookup
Path integral formulation wikipedia , lookup
Time in physics wikipedia , lookup
Renormalization wikipedia , lookup
Relational approach to quantum physics wikipedia , lookup
Fundamental interaction wikipedia , lookup
Bohr–Einstein debates wikipedia , lookup
Electric charge wikipedia , lookup
Field (physics) wikipedia , lookup
Centripetal force wikipedia , lookup
Lorentz force wikipedia , lookup
Classical mechanics wikipedia , lookup
Standard Model wikipedia , lookup
Aharonov–Bohm effect wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Electrostatics wikipedia , lookup
Work (physics) wikipedia , lookup
Chien-Shiung Wu wikipedia , lookup
Equations of motion wikipedia , lookup
Newton's theorem of revolving orbits wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Elementary particle wikipedia , lookup
Classical central-force problem wikipedia , lookup
Microgravity Sci. Technol. (2010) 22:329–338 DOI 10.1007/s12217-010-9195-8 ORIGINAL ARTICLE The Effect of Axial Concentration Gradient on Electrophoretic Motion of a Charged Spherical Particle in a Nanopore Sang Yoon Lee · Sinan E. Yalcin · Sang W. Joo · Ashutosh Sharma · Oktay Baysal · Shizhi Qian Received: 30 November 2009 / Accepted: 3 April 2010 / Published online: 24 April 2010 © Springer Science+Business Media B.V. 2010 Abstract The electrophoretic motion of a charged spherical nanoparticle along the axis of a nanopore connecting two fluid reservoirs, subjected to an axial electric field and electrolyte concentration gradient, has been investigated using a continuum model. The model consists of the Poisson and Nernst–Planck equations for the electric potential and ionic concentrations and the Stokes equations for the hydrodynamic field with zero gravity. In addition to the electrophoresis generated by the externally imposed electric field, the particle also experiences diffusiophoresis arising from the externally imposed concentration gradient. The effects of the diffusiophoresis on the axial electrophoretic motion are examined with changes in the ratio of the particle size to the thickness of the electric double layer (EDL), and the imposed concentration gradient. Since the EDL thickness, the particle size, and the nanopore size are of the same order of magnitude, the diffusiophoresis is dominated by the induced electrophoresis driven by the generated electric field arising from the doublelayer polarization (DLP). For a relatively small κa p , the ratio of the particle size to the EDL thickness, the diffusiophoresis is dominated by the induced elec- S. Y. Lee · S. W. Joo · A. Sharma · S. Qian School of Mechanical Engineering, Yeungnam University, Gyongsan 712-749, South Korea S. E. Yalcin · O. Baysal · S. Qian (B) Department of Aerospace Engineering, Old Dominion University, Norfolk, VA 23529, USA e-mail: [email protected] A. Sharma Department of Chemical Engineering, Indian Institute of Technology, Kanpur 208016, India trophoresis from the type II DLP, which propels the particle toward regions with lower salt concentration. Depending on the magnitude and direction of the externally imposed concentration gradient, the electrophoretic motion can be accelerated, decelerated, and even reversed by the diffusiophoresis. Keywords Electrophoresis · Diffusiophoresis · Electrical double layer · Nanopore Introduction In recent years, there has been a growing interest in developing nanopore-based nanofluidic devices with features comparable in size to DNA, proteins and other biological molecules for biological and chemical analysis (Branton et al. 2008; Lemay 2009; Clarke et al. 2009). In nanofluidic devices, it is necessary to manipulate fluids or nanoparticles such as DNA for various applications. Since the sample volume is extremely small, the gravitational body force is negligible. The interfacial electrokinetic phenomena such as electroosmosis and electrophoresis are thus widely used to manipulate fluids and/or particles in microfluidic and nanofluidic applications (Li 2004; Masliyah and Bhattacharjee 2006; Branton et al. 2008; Lemay 2009; Clarke et al. 2009; Ai et al. 2009a, b, 2010a, b). 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 EDL. In the presence of an external electric field, both the charges on the particle and the ions in the EDL interact with the overall electric field near the particle, resulting in electrostatic forces acting on both 330 the particle and the fluid and resulting in simultaneous electrophoretic and electroosmotic motions. The electrophoresis of charged and non-charged particles that acquire charge by polarization has been widely utilized in characterizing, separating, and purifying colloidal particles and macromolecules, such as DNA fragments, proteins, drugs, viruses, and biological cells (Li 2004). For example, when a DNA molecule is electrophoretically driven through a nanopore, nucleobases would modify the ionic current through the nanopore, and thus the sequence of bases in DNA might be recorded by monitoring the current modulations (Branton et al. 2008; Lemay 2009; Clarke et al. 2009). This nanoporebased DNA sequencing method is called the third generation DNA sequencing, and its cost is believed to be sufficiently low. Therefore, this new technology might potentially revolutionize genomic medicine (Branton et al. 2008; Clarke et al. 2009). In the existing study on nanoparticle translocation through a nanopore, the particle motion is driven by the externally imposed electric field, while the electrolyte concentrations in the two fluid reservoirs 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 is studied for the first time. Since the electrolyte concentrations on both sides of the nanopore are different, diffusiophoretic motion is also induced in addition to the electrophoresis (Anderson and Prieve 1984; Keh and Li 2007; Qian et al. 2007; Lou and Lee 2008a, b; Keh and Wan 2008; Abecassis et al. 2008, 2009; Prieve 2008; Lou et al. 2009; Hsu et al. 2009, 2010; Hsu and Keh 2009; Zhang et al. 2009). Depending on the magnitude and direction of the imposed concentration gradient, the induced diffusiophoretic motion can enhance the particle’s electrophoretic motion or slow down nanoparticle translocation in a nanopore, and thus can be used to regulate the nanoparticles translocation process to achieve a nanometer-scale spa- Fig. 1 Schematic of a nanopore of length L and radius a connecting two identical reservoirs on either side. A concentration gradient of electrolyte solution and an electric field are applied across the two reservoirs. A charged spherical particle of radius ap bearing uniform surface charge density, σ p , is positioned at the center of the nanopore Microgravity Sci. Technol. (2010) 22:329–338 tial accuracy for DNA sequencing. In particular, it is conceivable that slowing down the motion of DNA in a nanopore by diffusiophoretic control seems especially attractive as it would offer a greater window of opportunity for enhanced spatio-temporal resolution. In the following section, a “Mathematical Model” is introduced based on the continuum hypothesis for the fluid motion and the ionic mass transport. The former is induced by the externally imposed electric field and concentration gradient and the latter accounts for the polarization of the EDL and is valid for any thickness of the EDL. The effect of the imposed concentration gradient on the electrophoretic motion of a nanoparticle along the axis of a nanopore is presented in “Results and Discussion”, followed by concluding remarks in “Conclusions”. Mathematical Model We consider an uncharged nanopore of length L and radius a connecting two identical reservoirs, as shown in Fig. 1. An axisymmetric cylindrical coordinate system (r, z) with the origin located at the center of the nanopore is used. A charged spherical nanoparticle of radius ap and surface charge density σ p is submerged in an electrolyte solution in 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 axisymmetrical 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 lengths L R 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 Microgravity Sci. Technol. (2010) 22:329–338 331 nanoparticle. We assume that the rigid 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, C L and C R . 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. Considering its good agreement with experimental observations for nanoparticle’s electrophoretic motion through a nanopore (Qian et al. 2006, 2008; Liu et al. 2007; Qian and Joo 2008) and with the molecular dynamics simulation of a liquid flowing through a nanopore of 2.2 nm in diameter and 6 nm in length (Huang et al. 2007), we adopt a continuum model that consists of the Poisson and Nernst-Planck (PNP) equations for the electrical potential and ionic concentrations, and the Navier–Stokes equations for the flow field to study the combined electrophoresis and diffusiophoresis in a nanopore. The Reynolds number of electrokinetic flows in nanopores is typically very small, and so we neglect the inertial terms in the Navier–Stokes equation, and model the fluid motion with the Stokes equations. The flow of an incompressible binary electrolyte solution then is described by ∇ •u=0 (1) and −∇ p + μ∇ 2 u − F(z1 c1 + z2 c2 )∇V = 0, (2) where u = uer + vez is the velocity (er and ez are unit vectors in the radial and axial directions, respectively), p is the pressure, V is the electric potential in the electrolyte solution, c1 and c2 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, F is the Faraday constant, and μ is the electrolyte solution’s dynamic viscosity. The last term on the left hand side of Eq. 2 represents the electrostatic force acting on the fluid through the interactions between the electric field and the net charge density in the electrolyte solution. The infinitesimal contribution of the body force due to gravitational acceleration is neglected. A non-slip boundary condition (i.e., u = v = 0) is specified at the rigid walls of the nanopore and the reservoirs (line segments CD, DE, and EF in Fig. 1). On 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 used. Symmetric boundary condition is used along the lines of symmetry, HI and KA. Slip boundary conditions are used on the segments BC and FG which represent the regions in the reservoirs and are far away from the entrances of the nanopore. Finally, along the surface of the particle (arc segment IJK in Fig. 1) translating with an electrokinetic velocity u p , we neglect the thickness of the adjacent Stern layer, and impose the no-slip condition as u(r, z) = u p ez , on IJK. (3) The particle’s electrokinetic velocity u p is determined by requiring the total force in the z direction (FT ) acting on the particle FT = F E + F D = 0, (4) where E FE = T · n · ez dS (5) S and FD = D T · n · ez dS (6) S are, respectively, the electrostatic and hydrodynamic forces acting on the particle. S is the particle’s surface; T E = εEE − 12 ε(E • E)I and T D = − pI+ μ(∇u + ∇uT ) are, the Maxwell stress tensor and the hydrodynamic stress tensor, respectively; ε is the permittivity of the electrolyte solution; E = −∇V is the electric field; and I is the unit tensor. The flux density of each aqueous species due to convection, diffusion, and migration is given by Dk Fck ∇V, k = 1 and 2. (7) RT In the above, Dk is the diffusion coefficient of the kth ionic species; T is the absolute temperature of the electrolyte solution; and R is the universal gas constant. Under steady state, the concentration of each species is governed by the Nernst–Planck equation Nk = uck − Dk ∇ck − zk ∇ • Nk = 0 (k = 1 and 2) (8) and the electric potential, V, in the electrolyte solution is governed by the Poisson equation −ε∇ 2 V = F(z1 c1 + z2 c2 ). (9) On the plane AB which is sufficiently far away from the nanopore, the ionic concentrations are the same as the bulk concentration of the electrolyte solution present in the left reservoir: c1 = c2 = C L on AB (10) 332 Microgravity Sci. Technol. (2010) 22:329–338 Similarly, the ionic concentrations on the plane GH are the same as the bulk concentration of the electrolyte solution in the right reservoir: c1 = c2 = C R on GH (11) Along the rigid walls of the reservoirs and the nanopore, and the surface of the nanoparticle, since the solid surfaces are impervious to ions, the net ionic fluxes normal to the rigid surfaces satisfy: n • Nk = n • (uck ) (k = 1 and 2) on CD, DE, EF, and IJK. (12) 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 Nernst–Planck equations: n • N1 = n • N2 = 0 on BC and FG. (13) Along the segments HI and KA, symmetric boundary condition is used for the Nernst–Planck equations: n • N1 = n • N2 = 0 on HI and KA. (14) Symmetric boundary condition for the electric potential in the electrolyte solution is used on the planes HI and KA: n • ∇V = 0 on HI and KA. (15) An external potential, φ0 , is applied along the plane AB: V = φ0 on AB. (16) Along the plane GH, the boundary condition for the electric potential is V=0 on GH. (17) 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 on BC and FG. (18) Since the rigid walls of the reservoirs (planes CD and EF) and the nanopore (plane DE) do not carry fixed charge, we use n • ∇V = 0 on CD, DE, and EF (19) Along the particle’s surface (line segment IJK), surface charge boundary condition is used: n • (−ε∇V) = σ p on IJK (20) Results and Discussion Note that the above model neglects the particle Brownian motion and assumes that the particle only translates along the axis of the nanopore. When the particle is off the centerline of the pore, particle translation and rotation should be considered. The model also neglects the effect of the finite ion size on the ionic mass transport. However, a good agreement of the predicted DNA translocation velocity with experimental measurements (Liu et al. 2007) suggests that the above model is able to capture the essential physics of the nanoparticle translocation process. The commercial package COMSOL version 3.5a (http://www.comsol.com), installed in a workstation with 96 GB RAM, is chosen to integrate the system with a finite-element method. Quadratic triangular elements with variable sizes are used to accommodate finer resolutions near the particle surface IJK where EDL is 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, the predictions of the electrophoretic motion of a particle through a nanopore driven by an imposed electric field are in good agreement with the approximate analytical solution and experimental results obtained from the literature (Qian et al. 2006, 2008; Liu et al. 2007; Qian and Joo 2008). The predictions of the diffusio-osmotic flow in a slit nanochannel, driven by a concentration gradient, agree with the results obtained from the literature (Qian et al. 2007). These good agreements of various benchmark problems under either electric field or concentration gradient make us confident of our following computational results. In this section, we present a few numerical results of the electrokinetic motion of a charged spherical nanoparticle along the axis of a nanopore filled with KCl electrolyte. We focus on the effects of the induced diffusiophoresis on the particle’s electrophoretic motion by varying the concentration ratio α = C R /C L for different values of κa p , ratio of particle radius to the EDL thickness (κ −1 = λ D = ε RT/2F 2 C0 is the dimensional EDL thickness with C0 = (C L + C R )/2). The diffusion coefficients of the ions K+ and Cl − are, respectively, 1.95 × 10−9 m2 /s and 2.03 × 10−9 m2 /s (page 195 in Masliyah and Bhattacharjee 2006). 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, a p = 5 nm, a/a p = 4, L R = 0.15 μm, b = 0.15 μm, φ0 = 100 mV, and σ p = −0.1 C/m2 . Microgravity Sci. Technol. (2010) 22:329–338 333 Figures 2 and 3 depict, respectively, the dimensionless particle velocity u p *, normalized by U 0 = ε R2 T 2 /(μa p F 2 ), as a function of the imposed concentration ratio, α, for κa p = 1 and 3. When α = 1, there is no external concentration gradient imposed, and the negatively charged particle moves along the opposite direction of the applied electric field. For the case of α = 1, a negative axial concentration gradient is imposed for α < 1 and vice-versa. For κa p = 1 and α < 1 (Fig. 2), as α gradually decreases from α = 1, the magnitude of the particle’s velocity decreases. When α is smaller than a threshold value, αc , at which the particle velocity is zero, the particle’s motion reverses, and the negatively charged particle migrates along the same direction of the imposed electric field. The particle moves towards lower salt concentration, and its motion is dominated by the diffusiophoretic motion. As α further decreases, the particle velocity first increases and eventually saturates and becomes independent of α as α → 0. On the contrary, as α gradually increases from α = 1, the magnitude of the particle’s velocity increases. When α is above a certain critical value, the particle’s velocity also becomes independent of the imposed concentration ratio. Therefore, one can accelerate, slow down or even reverse the particle’s electrophoretic motion by controlling the magnitude and direction of the imposed concentration gradient. For the case of κa p = 3 (Fig. 3), the induced diffusiophoresis basically slows down the particle’s electrophoretic motion for both α < 1 and α > 1, and the particle’s motion does not reverse in the range of 10−3 ≤ α ≤ 103 . Obviously, the diverse effects of the induced diffusiophoresis on the particle’s elec- 0.015 up* 0.01 0.005 0 -0.005 -0.01 -0.015 -4 10 10 -2 10 0 α 10 2 10 4 Fig. 2 Dimensionless particle velocity as a function of the concentration ratio, α, when κa p = 1 0 x 10 -3 up* -0.5 -1 -1.5 -2 -2.5 -4 10 10 -2 10 0 α 10 2 10 4 Fig. 3 Dimensionless particle velocity as a function of the concentration ratio, α, when κa p = 3 trophoretic motion depend on the ratio of the particle size to the EDL thickness, κa p , which will be elaborated later. Figure 4 depicts the fluid flow field near the negatively charged particle when κa p = 1 under the conditions of α = 0.3, 1, and 5. There is an obvious recirculating flow in the gap between the particle and the nanopore wall. A clockwise circulation is generated when the particle migrates upwards (i.e., the case of α = 0.3), and a counter-clockwise eddy is generated when the particle moves downwards (i.e., the cases of α = 1 and 5). Since the wall of the nanopore is assumed uncharged in this study, the fluid motion is very weak in the region far away from the charged particle. In the region closed to the charged particle, the fluid’s motion is induced by the particle’s motion and the electroosmotic flow (EOF) in the EDL surrounding the negatively charged particle. The negatively charged particle attracts positive ions and repels negative ions, resulting in higher concentration of the positive ions (Fig. 5) and lower concentration of the negative ions (Fig. 6) within the EDL surrounding the negatively charged particle. The net charge in gap between the particle and the nanopore wall is positive for different values of α. For α = 1, the interactions between the externally imposed, positive axial electric field and the net charge induces EOF directed towards the cathode, and drives the fluids in the gap upwards; The fluids near the particle moves downwards due to the particle’s motion. Consequently, a counter-clockwise motion is induced in the gap for the case of α = 1. For the case of α = 1, the fluid motion in the gap results from the interactions between the positive net charge and the overall electric field which includes both the externally 334 Microgravity Sci. Technol. (2010) 22:329–338 Fig. 4 Flow field in the gap between the particle and the wall of the nanopore for different concentration ratios when κa p = 1 imposed electric field and the generated electric field by the imposed concentration gradient. In the absence of the imposed electric field, the diffusiophoretic motion is generated by two mechanisms: one is induced electrophoresis generated by the induced electric fields arising from the difference of ionic diffusivities and DLP, and the other is Fig. 5 Distribution of the dimensionless ionic concentration of K+ near the negatively charged particle for different concentration ratios when κa p = 1 chemiphoresis generated by the induced osmotic pressure gradient around the particle (Lou and Lee 2008a, b; Hsu et al. 2009, 2010; Zhang et al. 2009; Dukhin 1993, 1995). The induced chemiphoresis always propels the particle towards higher salt concentration, regardless of the sign of charge on the particle. Figure 2 depicts that the resulting diffusiophoretic motion drags the α=1 α=0.1 α=5 1 1.1 20 5 1.1 1.1 20 2 1.2 5 20 2 1.2 5 2 1.2 1.1 1.1 1.1 1 Microgravity Sci. Technol. (2010) 22:329–338 Fig. 6 Distribution of the dimensionless ionic concentration of Cl− near the negatively charged particle for different concentration ratios when κa p = 1 335 α=1 α=0.1 0.9 α=5 1 0.9 0.8 0.9 0.5 0.8 0.1 0.1 0.5 0.1 0.5 0.8 0.9 0.9 1 particle towards lower salt concentration; therefore the diffusiophoresis is mainly controlled by the induced electrophoresis instead of chemiphoresis under the considered conditions. Since the diffusion mobilities of the anions and cations in KCl electrolyte are almost the same order of magnitude, the induced electric field arising from the difference in the ionic diffusivities, Ediffusivity , is very small, the induced electrophoresis driven by the generated electric field, Ediffusivity , thus is negligible. In addition to the generated electric field due to the difference in the ionic diffusivities, an electric field, leading to electrophoresis, is induced by the induced dipole moment, a consequence of DLP (Dukhin 1993, 1995). Due to the imposed concentration gradient, the double layer is polarization resulting in higher concentrations of counterions and coions and thinner EDL on the high-concentration side and lower concentrations of counterions and coions and thicker EDL on the low-concentration side of the particle. Note that the DLP stated here refers to the concentration polarization only by the imposed concentration gradient, and did not include the relaxation effect arising from the imposed and generated electric field, movement of the particle, and the fluid convection. Due to the DLP, the center of charge inside the EDL shifts away from the particle’s center. Together with the charge of the particle and the charge of the EDL, a dipole moment is induced, which induces an electric field reaching 0.9 beyond the limits of the EDL (Dukhin 1993, 1995). The concentration polarization inside the EDL by the imposed concentration gradient is named as the type I DLP, and its resulting electric field is named as EI−DLP , the direction of which is opposite to that of the applied concentration gradient when the particle is negatively charged (Lou and Lee 2008a, b; Hsu et al. 2009, 2010; Zhang et al. 2009). For example, the induced electric field due to the type I DLP, EI−DLP , is directed from higher salt concentration towards lower salt concentration (i.e., upward for α < 1 and downward for α > 1). Meanwhile, the concentration of the coions near the outer boundary of the EDL on the high-concentration side is higher than that on the low-concentration side of the particle (i.e., the cases of α = 0.1 and 5 in Fig. 6), which is called the type II DLP, generating an electric field, EII−DLP , the direction of which is opposite to that established by the counterions inside the EDL, EI−DLP (Lou and Lee 2008a, b; Hsu et al. 2009, 2010; Zhang et al. 2009). 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 the EDL; and the third one,EII−DLP , is generated by the type II DLP near the outer boundary of the EDL. The induced EI−DLP by the type I DLP always propels the particle towards higher salt concentration, while the 336 EII−DLP generated by the type II DLP always drags the particle towards lower salt concentration, regardless of the sign of charge on the particle. For thin EDL (i.e., κa p is large), 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 ratio of the nanopore size to the particle size, a/ap , is very large (the boundary effect arising from the nanopore wall is insignificant), and the magnitude of the particle’s surface charge is relatively low (Hsu et al. 2010); 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 (Hsu et al. 2010). If κa p is small (thick EDL), usually the electric field induced by the type II DLP dominates over that by the type I DLP. Depending on the difference of the diffusivities of the cations and anions which depends on the type of salt used, and the sign of the surface charge of the particle, the induced electrophoretic motion generated by the induced Ediffusivity might drive the particle towards either low- or high-concentration side. Usually, the induced electrophoretic effect by Ediffusivity is more significant than that driven by EI−DLP and EII−DLP if the boundary effect is insignificant (i.e., large a/ap and κa p ). In the present study, since Ediffusivity is very small due to the almost identical diffusivities of ions K+ and Cl− , the induced electrophoresis driven by the electric fields EI−DLP and EII−DLP arising from DLP dominates since the EDL thickness, the particle size and the pore size are of the same order of magnitude. Since the particle’s surface charge density is relatively high (i.e., σ p = −0.1 C/m2 ), the diffusiophoresis is dominated by the induced electrophoresis driven by the electric field generated by the type II DLP, EII−DLP , which propels the particle towards lower salt concentration. For α < 1, the salt concentration beneath the particle is higher than that above the particle, therefore, the generated electric field, EII−DLP , by the type II DLP is opposite to the imposed electric field, consequently, the induced electrophoretic motion generated by EII−DLP is opposite to the electrophoretic motion driven by the externally imposed electric field. Since the generated EII−DLP increases with the imposed concentration gradient, EII−DLP increases as α decreases (or 1 − α increases). Therefore, as α decreases from α = 1, the particle’s electrophoretic motion is slowed down by the opposite diffusiophoretic motion, leading to the decrease in the particle velocity, as shown in Fig. 2. As α further decreases, the induced EII−DLP exceeds the imposed electric field, and the particle motion is then reversed, as shown in Fig. 2. After the reversion, as α further decreases, the particle velocity increases due to the increase in the generated electric Microgravity Sci. Technol. (2010) 22:329–338 field EII−DLP . When the concentration ratio α is less than a certain value, the dimensionless ionic concentrations, normalized by C0 , in the left and right reservoirs shown in Fig. 1 are, respectively, c∗1 = c∗2 = 2/(1 + α) → 2 and c∗1 = c∗2 = 2α/(1 + α) → 0, and the concentration gradient saturates as α further decreases, resulting in a saturated particle velocity, as shown in Fig. 2. Since the magnitude of the generated electric field EII−DLP dominates over the imposed electric field, the overall electric field in the gap between the particle and the nanopore wall is directed opposite to that of the imposed one, and drags the positive net charges in the gap downward, as shown in Fig. 4 for α = 0.3. For α > 1, the salt concentration above the particle is higher than that below the particle, resulting in EII−DLP , and the direction of which is the same as the imposed electric field, leading to the enhancement of the electrophoretic motion as α increases. When α is above a certain value, the dimensionless concentrations in the left and right reservoirs shown in Fig. 1 are, respectively, c∗1 = c∗2 = 2/(1 + α) → 0 and c∗1 = c∗2 = 2α/(1 + α) → 2, and the concentration gradient saturates as α further increases leading to the saturation of the particle’s velocity shown in Fig. 2. Since the EDL surrounding the particle is relatively thick under the condition of κa p = 1, the induced electrophoresis driven by the generated EI−DLP is not significant. Since the overall electric field is directed upward, the positive net charge in the gap between the particle and the nanopore wall thus moves upward, shown in Fig. 4 for the case of α = 5. For κa p = 3 (Fig. 3), the effect of the diffusiophoretic motion on the electrophoretic motion becomes more complicated. Comparing to the case of κa p = 1, the EDL is thinner, and the induced electrophoresis driven by EII−DLP becomes smaller. The net particle motion will be the result of the competing three driving forces generated by the imposed electric field and the generated electric fields including EI−DLP and EII−DLP . For α < 1, the decrease of the particle’s velocity is primarily due to the induced electrophoresis driven by EII−DLP , which slows down the particle motion. The particle’s motion is not reversed since the opposite driving force from EII−DLP is smaller comparing to that of κa p = 1. For α > 1, the decrease of the particle’s velocity is mainly due to the induced electrophoresis driven by EI−DLP , which propels the particle towards higher salt concentration. In the absence of the imposed electric field, the same magnitude but opposite sign of the diffusiophoretic velocity is obtained when the direction of the imposed concentration gradient is reversed. Similarly, in the absence of the imposed concentration gradient, we obtained the same magnitude with opposite sign of electrophoretic velocity when the direction of Microgravity Sci. Technol. (2010) 22:329–338 the imposed electric field is reversed. However, since the hydrodynamic field, electric field and ionic mass transport are strongly coupled, as shown in the mathematical model described in “Mathematical Model”, the net electrokinetic motion driven by the combined electric field and concentration gradient is not simply the superposition of the electrophoresis solely driven by the imposed electric field and the diffusiophoresis solely driven by the imposed concentration gradient. The equilibrium double layer surrounding a spherical particle is spherically symmetrical. The double layer changes to a non-equilibrium state under the effect of any action which results in its deformation (Dukhin 1993, 1995). For example, in diffusiophoresis, the thickness of the EDL near the high-concentration side of the particle is thinner while the EDL is thicker near the low-concentration side of the particle. In electrophoresis, the externally imposed electric field directed from bottom to top (i.e., the positive axial direction) displaces the mobile positive counter-ions within the double layer to the top surface of the particle resulting in thinner EDL with higher concentration of counterions near the top and thicker EDL with lower concentration of counterions near the bottom of the particle. For α > 1, the salt concentration near the top surface of the particle is higher leading to higher concentration of the counterions within the EDL near the top surface of the particle. When an electric field directed from bottom to top is imposed, since the counterions are displaced along the double layer from bottom to top by the imposed electric field, more counterions appear on the top hemisphere, resulting in the enhancement of the type I DLP effect. Using the similar analysis, the type II DLP effect is reduced by the DLP arising from the imposed electric field. Therefore, the driving force arising from the type I DLP is stronger than that of the type II DLP, which reduces the particle’s velocity, as shown in Fig. 3 in the range of α > 1. For α < 1, the salt concentration near the bottom of the negatively charged particle is higher than that near the top surface of the particle, leading to higher concentration of counterions near the bottom and lower concentration of counterions near the top of the particle. The imposed electric field displaces the positive, counterions within the EDL towards the top surface of the particle, thus reduces the counterions’ concentration difference between bottom and top EDL and consequently reduces the type I DLP effect. Meanwhile, the type II DLP effect is enhanced by the DLP arising from the imposed electric field. Therefore, the induced electrophoresis driven by EII−DLP is stronger than that driven by EI−DLP for α < 1, leading to the reduction of the particle’s velocity, as shown in Fig. 3. 337 Conclusions The effects of the induced diffusiophoresis by an imposed concentration gradient on the electrophoretic motion of a charged particle along the axis of an uncharged nanopore has been numerically investigated using a continuum model, which consists of Poisson and Nernst-Planck equations for the potential and ionic concentrations, and Stokes equations for the flow field. The model accounts for double layer polarization induced by the imposed electric field and concentration gradient and the compression of the EDL by the impervious nanopore wall, and is valid for any thickness of the EDL and the imposed external fields. 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 induced electric field arising from the double layer polarization. When the EDL is thick, the diffusiophoretic motion is dominated by the induced electrophoresis driven by EII−DLP , which propels the particle towards lower salt 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 depending on the magnitude and direction of the imposed concentration gradient. When the EDL is relatively thin, the induced diffusiophoretic motion slows down the electrophoretic motion, regardless of the direction of the imposed concentration gradient. For a negatively charged particle, the imposed electric field enhances the type I DLP and reduces the type II DLP effects if the imposed concentration gradient and electric field are in the same direction. Otherwise, the imposed electric field enhances the type II DLP effect and reduces the type I DLP effect when the imposed concentration gradient is opposite to the direction of the imposed electric field. It is conceivable that diffusiophoresis can be used to regulate the nanoparticles translocation process to achieve a nanometer-scale spatial accuracy for DNA sequencing by controlling both the electric field and the concentration gradient. Acknowledgement 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 Abecassis, B., Cottin-Bizonne, C., Ybert, C., Adjari, A., Bocquet, L.: Boosting migration of large particles by solute contrasts. Nat. Mater. 7, 785–789 (2008) 338 Abecassis, B., Cottin-Bizonne, C., Ybert, C., Ajdari, A., Bocquet, L.: Osmotic manipulation of particles for microfluidic applications. New J. Phys. 11, 075022 (2009) Ai, Y., Joo, S.W., Jiang, Y., Xuan, X., Qian, S.: Transient electrophoretic motion of a charged particle through a converging-diverging microchannel: effect of direct current– dielectrophoretic force. Electrophoresis 30, 2499–2506 (2009a) Ai, Y., Beskok, A., Gauthier, D.T., Joo, S.W., Qian, S.: DC electrokinetic transport of cylindrical cells in straight microchannels. Biomicrofluidics 3, 044110 (2009b) Ai, Y., Park, S., Zhu, J., Xuan, X., Beskok, A., Qian, S.: DC electrokinetic particle transport in an L-shaped microchannel. Langmuir 26, 2937–2944 (2010a) Ai, Y., Qian, S., Liu, S., Joo, S.W.: Dielectrophoretic choking phenomenon in a converging-diverging microchannel. Biomicrofluidics 4, 013201 (2010b) Anderson, J.L., Prieve, D.C.: Diffusiophoresis-migration of colloidal particles in gradients of solute concentration. Sep. Purif. Methods 13, 67–103 (1984) Branton, D., Deamer, D.W., Marziali, A., Bayley, H., Benner, S.A., Butler, T., Di Ventra, M., Garaj, S., Hibbs, A., Huang, X.H., Jovanovich, S.B., Krstic, P.S., Lindsay, S., Ling, X.S.S., Mastrangelo, C.H., Meller, A., Oliver, J.S., Pershin, Y.V., Ramsey, J.M., Riehn, R., Soni, G.V., Tabard-Cossa, V., Wanunu, M., Wiggin, M., Schloss, J.A.: The potential and challenges of nanopore sequencing. Nat. Biotechnol. 26(10), 1146–1153 (2008) Clarke, J., Wu, H.C., Jayasinghe, L., Patel, L., Reid, S., Bayley, H.: Continuous base identification for single— molecule nanopre DNA sequencing. Nat. Nanotechnol. 4, 265–270 (2009) Dukhin, S.S.: Non-equilibrium electric surface phenomena. Adv. Colloid Interface Sci. 44, 1–134 (1993) Dukhin, S.S.: Electrochemical characterization of the surface of a small particle and nonequilibrium electric surface phenomena. Adv. Colloid Interface Sci. 61, 17–49 (1995) Hsu, L.Y., Keh, H.J.: Diffusioosmosis of electrolyte solutions around a circular cylinder at arbitrary zeta potential and double-layer thickness. Ind. Eng. Chem. Res. 48, 2443–2450 (2009) Hsu, J.P., Hsu, W.L., Chen, Z.S.: Boundary effect on diffusiophoresis: spherical particle in a spherical cavity. Langmuir 25, 1772–1784 (2009) Microgravity Sci. Technol. (2010) 22:329–338 Hsu, J.P., Hsu, W.L., Ku, M.H., Chen, Z.S., Tseng, S.: Diffusiophoresis of a sphere along the axis of a cylindrical pore. J. Colloid Interface Sci. 342, 598–606 (2010) Huang, C., Choi, P.Y.K., Nandakumar, K., Kostiuk, L.W.: Comparable study between continuum and atomistic approaches of liquid flow through a finite length cylindrical nanopore. J. Chem. Phys. 126, 224–702 (2007) Keh, H.J., Li, Y.L.: Diffusiophoresis in a suspension of chargeregulating colloidal spheres. Langmuir 23, 1061–1072 (2007) Keh, H.J., Wan, Y.W.: Diffusiophoresis of a colloidal sphere in nonelectrolyte gradients perpendicular to two plane walls. Chem. Eng. Sci. 63, 1612–1625 (2008) Lemay, S.G.: Nanopore-based biosensors: the interface between ionics and electronics. ACS Nano. 3(4), 775–779 (2009) Li, D.: Electrokinetics in Microfluidics. Elsevier Academic Press, London, UK (2004) Liu, H., Qian, S., Bau, H.H.: The effect of translocating cylindrical particles on the ionic current through a nanopore. Biophys. J. 92, 1164–1177 (2007) Lou, J., Lee, E.: Diffusiophoresis of a spherical particle normal to a plane. J. Phys. Chem. C 112, 2584–2592 (2008a) Lou, J., Lee, E.: Diffusiophoresis of concentrated suspensions of liquid drops. J. Phys. Chem. C 112, 12455–12462 (2008b) Lou, J., Shih, C.Y., Lee, E.: Diffusiophoresis of a spherical particle normal to an air–water interface. J. Colloid Interface Sci. 331, 227–235 (2009) Masliyah, J.H., Bhattacharjee, S.: Electrokinetics and Colloid Transport Phenomena. Wiley, New York (2006) Prieve, D.C.: Salt and migrate. Nat. Mater. 7, 769–770 (2008) Qian, S., Joo, S.W.: Analysis of self-electrophoretic motion of a spherical particle in a nanotube: effect of nonuniform surface charge density. Langmuir 24, 4778–4784 (2008) Qian, S., Wang, A., Afonien, J.K.: Electrophoretic motion of a spherical particle in a converging–diverging nanotube. J. Colloid Interface Sci. 303, 579–592 (2006) Qian, S., Das, B., Luo, X.: Diffusioosmotic flows in slit nanochannels. J. Colloid Interface Sci. 315, 721–730 (2007) Qian, S., Joo, S.W., Hou, W., Zhao, X.: Electrophoretic motion of a spherical particle with a symmetric nonuniform surface charge distribution in a nanotube. Langmuir 24, 5332–5340 (2008) Zhang, X.G., Hsu, W.L., Hsu, J.P., Tseng, S.: Diffusiophoresis of a soft spherical particle in a spherical cavity. J. Phys. Chem. B 113, 8646–8656 (2009)