Analysis of Self-Electrophoretic Motion of a Spherical Particle in a

NeVada 89154-4027, and School of Mechanical Engineering, Yeungnam UniVersity,. Gyongsan 712-749, South Korea. ReceiVed December 14, 2007. In Final ...
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Langmuir 2008, 24, 4778-4784

Analysis of Self-Electrophoretic Motion of a Spherical Particle in a Nanotube: Effect of Nonuniform Surface Charge Density Shizhi Qian*,† and Sang W. Joo†,‡ Department of Mechanical Engineering, UniVersity of NeVada, Las Vegas, Las Vegas, NeVada 89154-4027, and School of Mechanical Engineering, Yeungnam UniVersity, Gyongsan 712-749, South Korea ReceiVed December 14, 2007. In Final Form: January 21, 2008 Autonomous motions of a spherical nanoparticle in a nanotube filled with an electrolyte solution were investigated using a continuum theory, which consisted of the Nernst-Planck equations for the ionic concentrations, the Poisson equation for the electric potential in the solution, and the Stokes equation for the hydrodynamic field. Contrary to the usual electrophoresis, in which an external electric field is imposed to direct the motion of charged particles, the autonomous motion originates from the self-generated electric field due to the ionic concentration polarization of the liquid medium surrounding an asymmetrically charged particle. In addition to the particle motion, the interaction between the electric field generated and the free charges of the polarized solution induces electroosmotic flows. These autonomous motions of the fluid as well as the particle were examined with focus on the effects of the surface-charge distribution of the particle, the size of the nanotube, and the thickness of the electric double layer, which affected the direction and the speed of the particle significantly.

1. Introduction Electrophoresis is a well-known electrokinetic phenomenon, referring to the motion of a particle in a fluid medium induced by an externally applied electric field. The electrophoresis of charged and noncharged particles that acquire charge by polarization has been used widely for characterizing, separating, and purifying colloidal particles and macromolecules, such as DNA fragments, proteins, drugs, viruses, and biological cells.1-3 A large body of literature is available on the electrophoretic motion of rigid particles in an unbounded and in a confined geometry.2,4-21 More recently, it was conceived that particle motions driven by an electric field not from an external source but by selfgeneration can be very useful in transporting and manipulating * Corresponding author. E-mail: [email protected]. † University of Nevada, Las Vegas. ‡ Yeungnam University. (1) Li, D. Electrokinetics in Microfluidics; Elsevier Academic Press: New York, 2004. (2) Hunter, R. J. Foundations of Colloid Science, 2nd ed.; Oxford University Press: New York, 2001. (3) Masliyah, J. H.; Bhattacharjee, S. Electrokinetics and Colloid Transport Phenomena; John Wiley and Sons Inc.: New York, 2006. (4) Keh, H. J.; Anderson, J. L. J. Fluid Mech. 1985, 153, 417. (5) Keh, H. J.; Chiou, J. Y. AIChE J. 1996, 42, 1397. (6) Ennis, J.; Zhang, H.; Stevens, G.; Perera, J.; Scales, P.; Carnie, S. J. Membrane Sci. 1996, 119, 47. (7) Ennis, J.; Anderson, J. L. J. Colloid Interface Sci. 1997, 185, 497. (8) Yariv, E.; Brenner, H. Phys. Fluids 2002, 14, 3354. (9) Hsu, J. P.; Kao, C. Y. J. Phys. Chem. B 2002, 106, 10605. (10) Hsu, J. P.; Hung, S. H. Langmuir 2003, 19, 7469. (11) Hsu, J. P.; Hung, S. H.; Kao, C. Y.; Tseng, S. Chem. Eng. Sci. 2003, 58, 5339. (12) Hsu, J. P.; Ku, M. H.; Kao, C. Y. J. Colloid Interface Sci. 2004, 276, 248. (13) Liu, H.; Bau, H. H.; Hu, H. H. Langmuir 2004, 20, 2628. (14) Xuan, X.; Xu, B.; Li, D. Anal. Chem. 2005, 77, 4323. (15) Hsu, J. P.; Ku, M. H. J. Colloid Interface Sci. 2005, 283, 592. (16) Qian, S.; Wang, A.; Afonien, J. K. J. Colloid Interface Sci. 2006, 303, 579. (17) Liu, H.; Qian, S.; Bau, H. H. Biophys. J. 2007, 92, 1164. (18) Hsu, J. P.; Yeh, L. H. Langmuir 2007, 23, 8637. (19) Hsu, J. P.; Yeh, L. H.; Chen, Z. S. J. Colloid Interface Sci. 2007, 310, 281. (20) Anderson, J. L. J. Colloid Interface Sci. 1985, 105, 45. (21) Hsieh, T. H.; Keh, H. J. J. Colloid Interface Sci. 2007, 315, 343.

micro- and nanoscale objects for many applications, such as the self-assembly of superstructures, roving sensors, drug-delivery systems, and useful nanomachinery.22 The autonomous motion of a particle, or self-electrophoresis, has indeed been observed in microorganisms,23,24 bimetallic platinum-gold nanorods,25-27 vancomycin,28 cyclodextrin derivatives,29 carbon fibers with different enzymes coated on its two ends,30 and polystyrene colloidal particles coating platinum on one end while keeping the second half as the nonconducting polystyrene.31 As compared to the extensive investigations performed for conventional electrophoresis, analytical studies of the selfelectrophoretic motion of particles are very limited. Golestanian et al.32 gave generic considerations for the design of small phoretic swimmers in an unbounded medium. On the basis of a linearized formula for general phoretic motions of various external driving forces, explained in the review of Anderson33 and references therein, it was shown that an autonomous motion requires symmetry breaking in either or both of the patterns of activity and mobility, and the swimming velocity is independent of the size for a given geometrical shape and surface pattern. The energy for the self-propulsion stems from the conversion of chemical or electrical energy.33 Precise prediction of the self-electrophoretic speed based on first principles and quantitative analysis on the influence of a bounding surface has yet to be completed. (22) Paxton, W. F.; Sundararajan, S.; Mallouk, T. E.; Sen, A. Angew. Chem., Int. Ed. 2006, 45, 5420. (23) Mitchell, P. FEBS Lett. 1972, 28, 1. (24) Lammert, P. E.; Prost, J.; Bruinsma, R. J. Theor. Biol. 1996, 178, 387. (25) Fournier-Bidoz, S.; Arsenault, A. C.; Manners, I.; Ozin, G. A. Chem. Commun. (Cambridge, U.K.) 2005, 4, 441. (26) Paxton, W. F.; Baker, P. T.; Kline, T. R.; Wang, Y.; Mallouk, T. E.; Sen, A. J. Am. Chem. Soc. 2006, 128, 14881. (27) Wang, Y.; Hernandez, R. M.; Bartlett, D. J.; Bingham, J. M.; Kline, T. R.; Sen, A.; Mallouk, T. E. Langmuir 2006, 22, 10451. (28) Fanali, S.; Desiderio, C.; Schulte, G.; Heitmeier, S.; Strickmann, D.; Chankvetadze, B.; Blaschke, G. J. Chromatogr., A 1998, 800, 69. (29) Schulte, G.; Heitmeier, S.; Chankvetadze, B.; Blaschke, G. J. Chromatogr., A 1998, 800, 77. (30) Mano, N.; Heller, A. J. Am. Chem. Soc. 2005, 127, 11574. (31) Howse, J. R.; Jones, R. A. L.; Ryan, A. J.; Gough, T.; Vafabakhsh, R.; Golestanian, R. Phys. ReV. Lett. 2007, 99, 48102. (32) Golestanian, R.; Liverpool, T. B.; Ajdari, A. New J. Phys. 2007, 9, 126. (33) Anderson, J. L. Ann. ReV. Fluid Mech. 1989, 21, 61.

10.1021/la703924w CCC: $40.75 © 2008 American Chemical Society Published on Web 03/27/2008

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nanoparticle and its surrounding fluid in a nanotube under various conditions are presented in section 4, followed by concluding remarks in section 5.

2. Mathematical Model

Figure 1. Configuration of the system with a charged spherical particle of radius a translating along the axis of a nanotube filled with an electrolyte solution. The particle is divided into three parts (the top, the middle, and the bottom), which are charged on the surface with densities σT, σM, and σB, respectively. The nanotube carries no surface charge.

In more advanced applications of self-electrophoresis, a nanoparticle may need to be transported through a confined space with dimensions comparable to its diameter, such as a nanotube. The presence of the bounding surface, in such cases, can greatly alter the self-electrophoretic motion of the particle. In addition to the additional hydrodynamic shear stress it imposes, the bounding surface can deform the self-generated electric field and influence the electroosmotic flow in the electric double layer (EDL) adjacent to the charged particle. The charge on its surface also can complicate the entire dynamics through its EDL, which may reach the nanoparticle. For electrophoretic motions of microscale particles with an external electric potential gradient, it is reported that the enhancement of the driving force due to an adjacent rigid plane can sometimes exceed the viscous retardation, giving rise to an increase in the phoretic speed33,34 when a symmetry of the flow system and a slip boundary condition for the infinitely thin double layer are applied. For nanoparticles with a spontaneous driving force, the effect of a bounding surface can be substantially different. In the present study, we applied conservation laws based on continuum theory to predict the dynamics of a self-electrophoretic nanoparticle without resorting to empirical constants. For the first time, the effect of a bounding surface was incorporated. To delineate its geometrical effect without complications due to the interactive dynamics of its EDL, the nanotube, considered as the bounding surface, is assumed to be uncharged. In the following section, a mathematical model for the fluid motion, the selfgenerated electric field, and the ionic mass transport, which accounts for the polarization of the EDL, is described without any restriction on the thickness of the EDL. Numerical methods used for integrating the highly coupled system are explained in section 3. The autonomous motions of an asymmetrically charged (34) Keh, H. J.; Chen, S. B. J. Fluid Mech. 1988, 194, 377.

We considered a spherical nanoparticle of radius a translating with a constant speed along the center of a nanotube of length 2h and radius b, as shown in Figure 1. The nanotube was filled with a symmetric binary electrolyte solution with viscoity µ, valences z1 and z2, diffusion coefficients D1 and D2, and uniform temperature T. The electrical charge on the surface of the nanoparticle was nonuniform and is expressed for convenience by assigning three piece-wise uniform values, σT, σM, and σB, respectively, for the top (EF), the middle (FG), and the bottom (GH) part of the spherical surface, whose areas are controlled by the angles θT and θB, indicated in Figure 1. Considering previous success in analyzing liquid flow through a cylindrical nanopore 2.2 nm in diameter and 6 nm in length,35 we adopted a continuum approach in describing the system. Using a, U0 ) C0RTa/µ, µU0/a, and RT/F as the length, the velocity, the pressure, and the electric potential scale, respectively, where C0 is the far-field balanced concentration of both positive and negative ions, R is the universal gas constant, and F is Faraday’s constant, the mass and momentum conservation laws are described in nondimensional quantities by

∇‚u ) 0

(1)

-∇p + ∇2u - (z1c1 + z2c2)∇V ) 0

(2)

and

Here,u ) urer + uzez is the dimensionless velocity vector, in terms of the radial and axial unit base vectors er and ez, and p is the dimensionless pressure. The forcing term in eq 2 shows nonlinear cross-coupling of c1 and c2, the dimensionless molar concentrations of the positive and negative ions in the electrolyte solution, respectively, and V, the dimensionless electric potential, to the flow field. Without a loss of generality, we assigned subscripts 1 and 2 to positive and negative ions, respectively. An axisymmetric cylindrical coordinate system (r, z) was used with the origin fixed at the center of the translating particle and z-axis directly upward, as shown in Figure 1. In the momentum eq 2, the unsteady and the convective acceleration terms were ignored for the present problem of steady flow with an extremely low Reynolds number, and the electrostatic force through the interaction between the self-generated electric field and the net charge density in the electrolyte solution was modeled as the dominant body force. Conservation laws for the electric potential and the concentration of each species need to be invoked for closure of the system. For a steady state, these reduce to the Poisson equation

-∇2V )

A2 (z c + z2c2) 2 1 1

(3)

where A ) (2F2C0a2/RT)1/2 is a dimensionless parameter measuring the radius of the nanoparticle relative the Debye length with  denoting the permittivity of the electrolyte solution and the steady Nernst-Planck equation

∇‚Nk ) 0 (k ) 1 and 2)

(4)

(35) Huang, C.; Choi, P. Y. K.; Nandakumar, K.; Kostiuk, L. W. J. Chem. Phys. 2007, 126, 224702.

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Here, the dimensionless flux density vector Nk for each species due to convection, diffusion, and migration is given by

Nk ) Pekuck - ∇ck - zkck∇V (k ) 1 and 2)

(5)

where Pek ) U0a/Dk is the Peclet number for each species. The length of the nanotube was considered to be much larger than the radius of the particle (H ) h/a . 1), so that in the absence of any externally applied pressure gradient along the tube, the boundary conditions to be applied on AB and CD in Figure 1 are

(∇u + (∇u)T)‚n ) 0, p ) 0, c1 ) c2 ) 1, n∇V ) 0 on z ) (H (6) which express no viscous stress, undisturbed pressure, and bulk concentrations and no axial electric potential gradient. The unit outward normal vector n ) (nr, nz) is directed from the fluid into a bounding phase. On the solid wall BC of the uncharged nanotube, no slip and no penetration of momentum, ionic flux, and electric field exist

ur ) uz ) 0, n‚N1 ) n‚N2 ) 0, n‚∇V ) 0 on r ) B (7) where B ) b/a is the dimensionless radius of the nanotube. Along the line segments DE and HA, symmetry conditions, or no flux conditions for all quantities, are imposed. On the surface of the nanoparticle, translating with a dimensionless self-electrophoretic velocity Vp, we neglected the thickness of the adjacent Stern layer and imposed the no-slip and no-ionic flux conditions as

u(r, z) ) Vpez, n‚N1 ) n‚N2 ) 0 on EFGH

(8)

The boundary condition for the electric potential on EFGH is given as

{

CT (0 < θ < θT) n‚(-∇V) ) CM (θT < θ < π - θB) CB (π - θB < θ < π)

(9)

(10)

To determine the self-electrophoretic velocity Vp in eq 8, the axial force balance between the electrostatic and the hydrodynamic force acting on the particle must be imposed, where the integration is performed over the particle surface S

∫∫

( ) [∫∫ (

∂V C dS S i ∂z A2 2

S

) (

)]

∂uz ∂uz ∂ur + nr + 2 - p nz dS ) 0 (11) ∂r ∂z ∂z

Dynamics of the strongly coupled system described previously mainly was controlled by the parameters

P ) (A, B, CT, CM, CB, θT, θB)

3. Numerical Method The flow, concentration, and electric fields described previously are strongly coupled, while the inhomogeneous boundary conditions on the particle surface are piece-wise continuous. We thus resort to a direct numerical simulation to understand the self-electrophoretic motion and associated flow dynamics for a realistic choice of control parameters. The commercial package COMSOL version 3.3b (www.femlab.com), installed in a 64-bit dual-processor workstation with 32GB RAM, was chosen to integrate the system with a finite element method. Quadratic triangular elements with variable sizes were used to accommodate finer resolutions near the particle surface EFGH where EDL was present. Solution convergence was guaranteed through mesh refinement tests on conservation laws. The self-electrophoretic velocity of the particle, used in prescribing the boundary condition (eq 8), is in fact unknown a priori and must be determined iteratively in conjunction with the finite element solver. Starting from an appropriate initial guess V 0p, we obtained a solution set to eqs 1-4, from which the net axial force FT acting on the nanoparticle, or the left-handside of eq 11, was calculated. A subsequent task was to update the value of Vp until the condition (eq 11) was satisfied. We used the Newton-Raphson method16,17 and used the iteration algorithm

V n+1 ) V np - J-1(V np)FT(V np) (n ) 0, 1, ...) p

(13)

where the Jacobian matrix J(V np) was evaluated numerically with

where θ is measured from the top, and the three surface-charge parameters are defined as

σiFa (i ) T, M, and B) Ci ) RT

(2µDkF2) depends solely on the material properties of the solution, and are left out, along with the valences zk, from the primary control parameters. The tube length H is to be large enough to avoid its influence and so is not considered as a control parameter in this study. Here, we recall that A and B are measures of the nanoparticle (relative to the Debye length) and the nanotube (relative to the nanoparticle), respectively, while the three Ci values and the two θ values collectively describe the surfacecharge distribution on the nanoparticle. These parameters are of primary interest in the present study and will be examined further through numerical analysis.

(12)

where the Peclet numbers Pek, the valences zk, and the tube length H were excluded. The Peclet numbers, defined just below eq 5, can be rewritten as Pek ) A2Pek, where Pek ) R2T2/

J(V np) )

[ ] ∂FT(V np) ∂V np

(14)

This iterative process coupled with the COMSOL finite element procedure was terminated when FT was reduced to below the absolute error boundary.

4. Results and Discussion The direct numerical simulation summarized previously was performed rather exhaustively for a wide variety of parameter values. Here, we present a few select cases that were useful in delineating the effects of thickness of the EDL, tube size, and charge distribution on the particle. For convenience and consistency, we chose a reference case, represented by P0 ) (1.63, 4, -2.75, -2.75, 2.75, π/4, π/4), ( Pe1,Pe2) ) (0.1184, 0.1149), z1 ) 1, z2 ) -1, and H ) 20. A typical example of this case is a nanoparticle with a 5 nm radius, with approximately 85% area on the top charged negatively with -0.01 C/m2 and the other 15% on the bottom charged positively with 0.01 C/m2, in a 10 mM KCl aqueous solution with a viscosity of 0.001 kg/(m s) at 298 K contained in a nanotube with a 20 nm radius. While the primary parameters P may be varied in the cases discussed next, the values for the minor parameters were

Effect of Nonuniform Surface Charge Density

Figure 2. Self-electrophoretic velocity vs β ) CM/CT. CT ) -CB ) -2.75 (solid line) and CT ) -CB ) 2.75 (dashed line). (a) A ≈ 0.52; (b) A ≈ 1.63; and (c) A ≈ 5.15.

fixed as stated previously unless otherwise noted. The relative tube length fixed at H ) 20 for computational efficiency was determined to be a sufficiently large value through a series of convergence tests for various parameters chosen. It is understood further that an unspecified parameter value in the cases discussed next implies the corresponding reference case value. Figure 2 shows the self-electrophoretic velocity of a particle through a nanotube for P ) (A, 4, (2.75, CM, (2.75, π/4, π/4) where A ) 0.52 (Figure 2a), 1.63 (Figure 2b), and 5.15 (Figure 2c) are considered, and the surface charge density on the middle of the nanoparticle was varied, while that on the top and the bottom was fixed with equal magnitudes but opposite signs. The parameter A was adjusted by varying the thickness of the EDL through the change of the bulk concentration C0, and its value increased as the thickness of the EDL decreased under the same particle size a. The direction of the self-electrophoretic motion was reversed if the polarities of the top and the bottom were switched. It is to be noted, however, that due to the absence of external pressure or to an electric potential gradient that determines the directionality of the system, a positive axial motion of the particle is simply a mirror image of the corresponding negative axial motion. An upward motion of a particle for a prescribed surface-charge distribution is identical to a downward motion with the same speed for an exact mirror image surface-charge distribution about z ) 0, which is clearly shown in Figure 2 for all thicknesses of the EDL in the vicinity of the particle. The reference case by P0 is located in Figure 2b at β ) CM/CT ) 1, with the corresponding dimensionless self-electrophoretic velocity of approximately -0.2. If the surface charge on the middle of the particle bears the same value as that on the bottom (CM/CT ) -1) instead of the top, the direction of the self-electrophoretic motion is reversed with the speed unaltered. When the surface charge of a particle is divided unevenly into a positive and a negative part with equal charge densities, the self-electrophoretic motion occurs in the direction of the smaller part regardless of its polarity. The magnitude of the self-electrophoretic velocity increases with a decrease in the symmetry or antisymmetry of the charge-density distribution. For all thicknesses of the EDL considered, this change in velocity appears to be linearly proportional to the relative change of the charge density on the middle. It also can be deduced easily from the three images given in Figure 2 that an increase in the thickness of the EDL results in an increase in the self-electrophoretic velocity. The phenomenon associated with the autonomous motion of a particle can be understood in more detail by examining the local fields of quantities involved. Figure 3 presents the

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distributions of the ionic concentrations of K+ (Figure 3a) and Cl- (Figure 3b), net charge density (Figure 3c), electric potential (Figure 3d), induced electric field (Figure 3e), and velocity field and streamlines (Figure 3f) for the reference case P0. For clarity, only the region surrounding the particle is shown. Because of a mechanism analogous to the electroosmosis, more counterions are attracted to the charged particle and the co-ions are repelled. For example, positive K+ ions accumulate in the EDL next to the top and middle of the nanoparticle due to the presence of the negative surface charge, while the co-ions Cl- are repelled from the EDL. In the EDL adjacent to the bottom of the particle, the concentration of the K+ ions is depleted, and that of Cl- is enriched due to the positive surface charge along the bottom surface of the particle. Because of the presence of asymmetric surface charge around the particle, the concentration distributions for both ions are asymmetric, and a concentration gradient is thus generated. The net charge density is positive in the EDL adjacent to the top and the middle of the nanoparticle and is negative in the EDL adjacent to the bottom. As seen in the Poisson equation (eq 3), a small difference in the concentrations of the counterion and co-ion can generate an electric field. The generated electric field near the top and middle of the particle is directed from the fluid toward the particle, which is directed from the particle toward the fluid in the region near the bottom of the particle. In other words, the bottom of the particle is the source of the electric field, and the top and middle of the particle is the sink of the self-generated electric field. In the region near the top and the bottom of the particle, the axial component of the electric field is dominant. However, the negative radial electric field is dominant in the gap between the particle and the wall of the nanotube. The interaction between the generated electric field and the asymmetric surface charge around the particle induces an axial electrostatic force on the particle that is positive on the top semisphere and negative on the bottom semisphere of the particle. The net electrostatic force on the particle is directed in the negative z-direction for this case, resulting in a negative self-electrophoretic velocity, as seen in Figure 2. The interaction between the generated electric field and the net charge in the liquid medium induces a flow of the electroosmotic type, which is directed downward near the particle and has a counterclockwise recirculating flow in the gap between the particle and the wall of the nanotube. The fluid near the particle moves downward, which is mainly due to the downward particle motion. In addition, the fluid’s motion also is affected by the electrostatic force on the liquid. For example, near the top of the particle, the negative axial electric field interacts with the positive excessive ions, resulting in a negative axial electrostatic force on the liquid, which, in turn, induces downward fluid motion. The direction of the electrostatic force on the fluid is the same as that of the particle’s velocity, leading to an enhancement of the fluid’s velocity on the top of the particle. Near the bottom of the particle, the interaction between the negative axial electric field and the excessive negative ions generates a positive electrostatic force on the fluid that is opposite to the direction of the particle’s velocity. The fluid’s motion near the bottom is thus decelerated. In the gap between the particle and the tube wall, the interaction between the radial electric field and the excessive positive ions generates a counterclockwise circulating fluid flow. Away from the particle, the fluid’s motion is mainly driven by the electrostatic force on the liquid that is the product of the interaction between the generated electric field and the excessive ions. When the surface charge on the middle is positive, the excessive ions will be positive ions in the EDL next to the top and are negative ions in the EDL adjacent to the middle and bottom. The generated axial electric

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Figure 3. Distribution of (a) concentration of the positive ions (K+); (b) concentration of the negative ions (Cl-); (c) net charge density, c1 - c2; (d) electric potential; (e) generated electric field; and (f) velocity field and streamlines.

field is negative, inducing a positive axial electrostatic force on the particle, and results in a positive electrophoretic velocity as shown in Figure 2b. The upward particle motion would induce an upward fluid motion near the particle and a clockwise recirculating flow in the gap between the particle and the wall of the nanotube. The magnitudes of the generated electric field and the resulting axial electrostatic force increase with the increase in the nonuniformity of the surface charge, which explains the changes in the particle speed seen in Figure 2. Similarly, when the polarities of the surface charges on the top and bottom are reversed, the direction of the axial component of the electric field generated and of the electrostatic force on the particle would be reversed, resulting in a reversal of motion also seen in Figure 2. Figure 4 depicts the effect of the change in the surface areas for piece-wise uniform charges on the self-electrophoretic velocity. The cases shown are identical to those in Figure 2 for three different thicknesses of EDL, with additional lines plotted to include two additional surface-charge areas instead of the redundant switch in the polarity of the top and bottom in Figure 2. Three different surface-area combinations were determined according to θT ) θB ) π/3 (solid line in Figure 4), θT ) θB ) π/4 (dashed line in Figure 4), and θT ) θB ) π/6 (dasheddotted line in Figure 4). The actual surface areas of the top, middle, and bottom slices can be computed approximately by 2πa2(1 - cos θT), 2πa2(cos θT + cos θB), and 2πa2(1 - cos θB), respectively. Under identical conditions, the particle speed is the largest for the case θT ) θB ) π/4 and is the lowest for the case θT ) θB ) π/6. In other words, there is an optimal value for θT and θB, for which the self-electrophoretic speed reaches a maximum. Figure 5 shows variations in the self-electrophoretic velocity with a change in the angle θ (θT ) θB) for three different

Figure 4. Self-electrophoretic velocity vs β ) CM/CT for θT ) θB ) π/3 (solid line), θT ) θB ) π/4 (dashed line), and θT ) θB ) π/6 (dashed-dotted line). CT ) -CB ) -2.75. (a) A ≈ 0.52, (b) A ≈ 1.63, and (c) A ≈ 5.15.

values of the charge density on the middle CM. For the cases considered, the maximum in the self-electrophoretic speed was reached approximately at θT ) θB )50°. The dependence of the particle’s autonomous motion on the surface areas of the asymmetric surface charge originates from the variations of the self-generated electric field and the resulting electrostatic force on the particle with the surface areas of the three slices. When the middle part is uncharged (solid line in Figure 5), the particle with an antisymmetric surface charge (i.e., CB/CT ) -1) does not move due to zero net electrostatic force on the particle. In the cases considered previously, the surface-charges on the top and bottom of the nanoparticle are set to be equal in magnitude but opposite in polarity. In Figure 6, the effect of an unbalanced top and bottom charge is studied by gradually changing the relative surface-charge density of the bottom to

Effect of Nonuniform Surface Charge Density

Langmuir, Vol. 24, No. 9, 2008 4783

Figure 5. Self-electrophoretic velocity vs θ () θT ) θB) for CM/CT ) 0 (solid line), 1 (dashed line), and 2 (dashed-dotted line).

Figure 7. Contours of constant self-electrophoretic velocity on the (CB/CT, CM/CT) plane.

Figure 6. Self-electrophoretic velocity vs CB/CT for CM/CT ) -1 (solid line), 0 (dashed line), and 1 (dashed-dotted line).

the top. The reference case P0 is located at CB/CT ) -1 on the dashed-dotted line in Figure 6 and is in a downward motion. The direction and speed of the motion can be controlled by changing the ratio CB/CT. It is seen that a window of CB/CT exists for the self-electrophoretic motion reversal with a maximum from the downward to the upward direction. The window is bounded by the values for zero electrophoretic velocity, with the upper bound located at CB/CT ) 1, which corresponds to a uniform charge distribution on the entire particle surface. While this upper bound is fixed, the lower bound of the window varies with the charge on the middle. As the charge on the middle increases gradually from a negative to a positive value, the lower bound decreases, while the maximum reversal speed increases. The effect of surface-charge distribution can be more collectively understood by examining Figure 7, where contours of constant self-electrophoretic velocity are given on a (CB/CT, CM/CT) plane. Two branches for the stationary state are seen. When the top and the bottom charge are identical (CB/CT ) 1), no phoretic motion is induced regardless of the charge on the middle for the symmetric area distribution θT ) θB ) π/4 chosen. Another line for zero velocity corresponds to the lower bound for the window of reversal motion in Figure 6 and shows an almost linear dependence on the charge of the middle. For fixed CM/CT, self-electrophoretic motion can be controlled to a desired motion by an appropriate choice of the combination CB/CT. When CB/CT is fixed to a positive value (same polarity), however, controlling the direction of motion by changing the charge on

Figure 8. Self-electrophoretic velocity vs CB/CT for θT ) θB ) π/2. A ≈ 0.52 (dashed line) and 1.63 (solid line).

the middle is not realizable. The contour plot of Figure 7 can be useful as a self-electrophoretic motion map in designing the speed and direction of the particle motion desired. If we set θT ) θB ) π/2, the surface in the middle vanishes, and the surface-charge distribution is determined by the top and the bottom. Figure 8 shows one such case, where the selfelectrophoretic velocity is plotted against the ratio of the bottom to the top charge. For this particular case, the particle ceases to move if CB/CT )1, symmetric (or uniform) charge distribution, or CB/CT ) -1, antisymmetric charge distribution about z ) 0. The dashed line in Figure 8 represents the case with a thicker EDL and shows an enhanced phoretic speed and sensitivity to the charge ratio. Figure 9 shows the effect of the size of the nanotube on the self-electrophoretic velocity of the nanoparticle. If the middle of the particle is uncharged, due to the antisymmetric top and bottom charge distribution (CB/CT ) -1) and the symmetric surface distribution set by θT ) θB ) π/4, the self-electrophoretic velocity vanishes for all values of B. For other charge distributions accommodating self-electrophoretic motions, it is seen that the increase in the tube size B results in the increase in the speed. The rate of increase in the speed decreases with B until the speed asymptotes to a finite value for a large B. The maximum phoretic velocity reached for the cases shown is 0.2898. The wall effect

4784 Langmuir, Vol. 24, No. 9, 2008

Figure 9. Self-electrophoretic velocity vs B for CM/CT ) -1 (solid line), 0 (dashed line), and 1 (dashed-dotted line).

of the nanotube persists until the size of the particle and the tube have different orders of magnitude. As stated previously, the effect of confinement is studied only for the radial dimension in the present study, assuming that the axial length of the nanotube in practice will be much larger than the chosen value H ) 20 and consequently does not affect the phoretic motion. While the radial confinement can alter the phoretic motion quantitatively, as shown in Figure 9, a qualitative change, such as reversal of the self-phoresis direction, has not been observed for numerous simulations performed.

5. Conclusion The autonomous motions of an asymmetrically charged spherical particle and the electrolyte solution in a nanotube in

Qian and Joo

the absence of an externally applied electric field were studied theoretically with a general multi-ion mass transport model that takes into account the distortion and polarization of the EDL adjacent to the charged particle. Contrary to the conventional electrophoresis technique in which the particle’s motion is driven by an external electric field, the self-electrophoretic motion of the nonuniformly charged particle originates from the selfgenerated electric field by the asymmetric ionic concentration polarization. In addition to the autonomous particle motion, the interaction between the generated electric field and the excessive charges in the polarized liquid medium induces an autonomous fluid motion. A recirculating flow was induced in the gap between the particle and the wall of the nanotube, and the direction of the recirculating flow depends on the asymmetric surface charge distribution. The induced recirculating flow may be used to enhance mixing or to stretch, fold, or stall molecules in nanofluidic applications. The induced motions of the particle and the fluid decrease as the thickness of the EDL decreases. Therefore, one can enhance the autonomous motion by adjusting the bulk electrolyte concentration. The magnitude and direction of the induced particle motion highly depends on the polarity and magnitude of the asymmetric surface charge distribution of the nanoparticle. The self-electrophoretic velocity decreases consistently with a decrease in the relative size of the nanotube to the particle transported. The predicted increase in the phoretic speed due to an adjacent bounding surface in the presence of an external driving force has not been observed for the present case of selfelectrophoresis. The surface charge on the tube can greatly influence this phoretic speed and will be a subject for future investigations. Acknowledgment. This work was supported by a UNLV PRA grant and, in part (S.W.J.), by Yeungnam University. LA703924W