Electrophoresis of a Cylindrical Particle with a Nonuniform Zeta

12790

J. Phys. Chem. C 2009, 113, 12790–12798

Electrophoresis of a Cylindrical Particle with a Nonuniform Zeta Potential Distribution Parallel to a Charged Plane Wall Li J. Wang and Huan J. Keh* Department of Chemical Engineering, National Taiwan UniVersity, Taipei 10617, Taiwan, Republic of China ReceiVed: April 3, 2009; ReVised Manuscript ReceiVed: May 31, 2009

An analytical study is presented for the steady, transverse electrophoretic motion of a circular cylindrical particle with an arbitrary angular distribution of its surface potential parallel to a plane wall prescribed with the potential distribution consistent with the applied electric field. The electric double layers adjacent to the solid surfaces are assumed to be very thin with respect to the particle radius and spacing between the surfaces. The two-dimensional electrostatic and hydrodynamic governing equations are solved using cylindrical bipolar coordinates, and the typical electric field line, equipotential line, and streamline patterns are exhibited. The explicit formulas for the electrophoretic and angular velocities of the particle are obtained with the contribution from the electroosmotic flow produced by the interaction between the applied electric field and the thin double layer adjacent to the plane wall. To apply these formulas, one only has to calculate the leading multipole moments of the zeta potential distribution at the particle surface. The existence of a plane wall can cause the translation or rotation of the particle, which does not occur in an unbounded fluid with the same applied electric field. The boundary effects on the electrophoretic motion of a uniformly or nonuniformly charged particle resulting from the parallel plane wall prescribed with the far-field potential distribution are quite different from those produced by a corresponding insulating wall. 1. Introduction Electrophoresis refers to the motion of charged particles in an electrolyte solution subject to an imposed electric field, which has long been used as an effective technique for separation and identification of biologically active compounds in the biochemical and clinical fields. A simple expression for the electrophoretic velocity of a dielectric particle of arbitrary shape is the Smoluchowski equation1

U0 )

εζp E η ∞

(1)

where ζp is the zeta potential of the particle surface, η is the fluid viscosity, ε is the fluid permittivity, E∞ is the constant applied electric field, and the particle does not rotate. Equation 1 is valid on the basis of several assumptions: (i) the thickness of the electric double layer adjacent to the particle surface is much smaller than the local radii of curvature of the particle; (ii) the fluid surrounding the particle is unbounded; (iii) the zeta potential is uniform on the length scale of the particle. Although many colloidal particles undergoing electrophoretic motion fulfill the first restriction, electrophoresis of particles with finite thickness of the double layers is encountered in certain cases so that relevant corrections to the Smoluchowski equation were obtained in the past for spherical and nonspherical particles.2-7 In electrophoresis applications to particle analysis or separation, particles usually move in the vicinity of solid boundaries. For instance, electrophoresis in porous gels or membranes could be applied to permit separations on the basis of the size and charge of the particles.8 In capillary electrophoresis, gels in the capillary column can minimize particle diffusion, prevent * To whom correspondence should be addressed. E-mail: [email protected]. Telephone: 886-2-33663048. Fax: 886-2-23623040.

particle adsorption to the capillary walls, and eliminate electroosmosis, while serving as the anticonvective media.9 Deep electrophoresis penetration and deposition of inert colloidal particles over the interstitial surfaces of porous composites has been suggested in the aerospace industry to protect the composites from burning or deterioration.10 In electrokinetically driven microfluidic devices used for biological particle manipulation, the dimension of the microchannels is usually comparable with the size of the particles undergoing electrophoretic movement.11 Therefore, the boundary effects on electrophoresis are of great importance and have been studied extensively in the last four decades for various cases of uniformly charged colloidal particles and boundaries.12-26 In fact, many colloidal particles have a heterogeneous surface structure or chemistry and are nonuniformly charged in electrolyte solutions.27-33 The electrophoresis of a dielectric particle with an arbitrary zeta potential distribution and a very thin electric double layer has been analyzed by Teubner34 and Anderson.35,36 This pioneering analysis was later extended to the cases of nonuniformly charged particles with a thin but polarized double layer37 and a double layer of arbitrary thickness.38-41 The electrophoresis of a nonuniformly charged particle in the vicinity of a solid boundary could also occur in some practical situations. In addition to the possible examples mentioned above, the electrophoretic translation and rotation of an array of nonuniformly charged bichromal spheres in their individual solvent-filled cavities have been applied to a technology of electric paper displays (known as Gyricon displays).42,43 Also, an electrophoretic positioning process has been employed in electronic fabrication techniques for assembling nonuniformly charged microdevices onto the selected contact electrodes of a silicon circuit.44,45 Recently, the electrophoretic motions of a dielectric sphere in a concentric spherical cavity with arbitrary zeta potential distributions at the solid surfaces46 and of a

10.1021/jp903077e CCC: $40.75  2009 American Chemical Society Published on Web 06/26/2009

Nonuniform Zeta Potential Distribution

J. Phys. Chem. C, Vol. 113, No. 29, 2009 12791

y)

c sin ξ cosh ψ - cos ξ

(2b)

where 0 e ψ < ∞, 0 e ξ e 2π, and c is a characteristic length in the bipolar coordinate system. The curve ψ ) ψ0 > 0 represents the circle (or the cylinder) of radius a ) c csch ψ0, with its center at the point (x ) d ) ccoth ψ0, y ) 0). The limiting case ψ ) 0 generates a circle of infinite radius and corresponds to the entire y-axis (or the plane x ) 0). The ratio of the radius of the cylinder to the distance of the axis of the cylinder from the plane is related to ψ0 by Figure 1. Geometric sketch of the transverse electrophoresis of a circular cylindrical particle parallel to a plane wall.

nonuniformly charged circular cylinder near a conducting or insulating plane wall parallel to its axis47 were analytically investigated for the case of infinitesimally thin electric double layers. However, the electrophoresis of a colloidal sphere in a spherical cavity48 and in a nanotube49 have also been studied theoretically with nonuniform surface charge and potential distributions and arbitrary double-layer thicknesses. In this article, we analyze the steady electrophoretic motion of a dielectric circular cylinder with an arbitrary zeta potential distribution in the angular direction under a transversely and uniformly applied electric field parallel to a charged plane wall. The electric double layers are assumed to be very thin, and the electric potential distribution at the outer edge of the thin double layer adjacent to the plane wall is taken to be consistent with the applied electric field. The explicit solution for the wallcorrected electrophoretic velocity of the particle is obtained in eq 20a. Because the governing equations and boundary conditions concerning the general two-dimensional problem of electrophoresis of a circular cylindrical particle in an arbitrary direction with respect to a nearby plane wall prescribed with the far-field potential distribution are linear, its solution can be obtained as a superposition of the solutions for its two subproblems: motion normal to the plane wall, which was previously examined,47 and motion parallel to the confining wall, which is considered in this article. 2. Analysis We consider the steady electrophoresis of a long circular cylindrical particle of radius a driven by a constant electric field E∞ ) E∞ey applied perpendicular to its axis and parallel to a large plane wall, prescribed with a potential distribution consistent with the applied electric field and located at a distance d from the axis as shown in Figure 1, where ey together with ez and ex are the unit vectors in the rectangular coordinate system (x,y,z). The zeta potential on the particle surface can be an arbitrary function of its azimuth angle. The thickness of the electric double layers surrounding the particle and adjacent to the plane wall is assumed to be much smaller than the radius of the cylinder and the gap width between the solid surfaces. Our objective is to determine the electrophoretic velocity of the nonuniformly charged cylindrical particle in the presence of the charged plane wall. The cylindrical bipolar coordinate system (ξ,ψ,z) as illustrated in Figure 2 is also used to solve the problem. These coordinates and rectangular coordinates are related by50

x)

c sinh ψ cosh ψ - cos ξ

(2a)

λ ) a/d ) sech ψ0

(3)

To determine the electrophoretic velocity of the cylindrical particle near the plane wall, it is necessary to ascertain the electric potential and velocity fields in the fluid phase. 2.1. Electric Potential Distribution. Because the fluid outside the thin double layers is neutral and of constant conductivity, the electric potential distribution Φ(ξ,ψ) is governed by the Laplace equation

∇2Φ ) 0

(4)

where the expressions of the operator ∇2 in rectangular and bipolar coordinates are given by eq B2 in Appendix B. The cylindrical particle is assumed to be nonconducting, and the plane wall is prescribed with the far-field potential distribution. Thus, the boundary conditions are

ψ ) ψ0, ψ ) 0,

eψ · ∇Φ ) 0

Φ ) -cE∞ sin ξ(cosh ψ - cos ξ)-1

(5a)

(5b)

where the expressions of the operator ∇ in rectangular and bipolar coordinates are given by eq B1, and eξ and eψ are the unit vectors in bipolar coordinates, which relate to unit vectors ex and ey in rectangular coordinates by eqs B3 and B4. As shown in Appendix A, the solution to eq 4 subject to the boundary conditions in eq 5a is

Figure 2. Two-dimensional bipolar coordinates (ξ,ψ) and rectangular coordinates (x,y).

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J. Phys. Chem. C, Vol. 113, No. 29, 2009

Wang and Keh

∞

Φ ) -2cE∞

∑ e-nψ

0

ψ ) 0,

sech nψ0 sinh nψ sin nξ -

v ) V∞ey ) -

n)1

cE∞ sin ξ(cosh ψ - cos ξ)-1

(6)

The electric field function V is related to the electric potential by the formulas

∂Φ ∂V ) ∂ξ ∂ψ

(7a)

∂Φ ∂V )∂ψ ∂ξ

(7b)

The imaginary part of the complex potential satisfying the above Cauchy-Riemann equations (or eqs B5 and B6) and with its real part given by eq 6 is the electric field function ∞

V ) -2cE∞

∑ e-nψ

0

sech nψ0 cosh nψ cos nξ +

n)1

cE∞ sinh ψ(cosh ψ - cos ξ)-1

η∇2v - ∇p ) 0

(9a)

∇·v ) 0

(9b)

where v is the fluid velocity and p is the dynamic pressure. Taking the curl of eq 9a and introducing eq 9b, the stream function Ψ results in

∇4Ψ ) ∇2(∇2Ψ) ) 0

(10)

The stream function is related to the velocity components in bipolar coordinates by

1 ∂Ψ Vξ ) (cosh ψ - cos ξ) c ∂ψ

(11a)

1 ∂Ψ Vψ ) - (cosh ψ - cos ξ) c ∂ξ

(11b)

Because the electric field interacts with the thin double layer at each of the solid surfaces to produce a relative electroosmotic flow at the outer edge of the double layer, the boundary conditions for the fluid velocity are

ψ ) ψ0,

v ) Uxex + Uyey + aΩeξ +

εζp ∇Φ η

(12a)

(12b)

where ζp and ζw are, respectively, the zeta potentials associated with the particle and plane wall, Uxex + Uyey and Ωez are, respectively, the translational and angular velocities of the electrophoretic cylinder (which is force- and torque-free) to be determined, and Φ has been given by eq 6. The unit vectors ex and ey in rectangular coordinates in eq 12a can be related to the unit vectors eξ and eψ in bipolar coordinates by eqs B3 and B4. Note that because ζp can be a general function of the angular position on the particle surface, Ux appears in eq 12a. As both the governing equation and boundary conditions are linear, the total flow can be decomposed into two parts. First, we consider the fluid velocity field v1 about a circular cylinder (with its surface at ψ ) ψ0) moving with the translational velocity Uxex + Uyey and angular velocity Ωez near the plane wall (at x ) 0) but with no electrokinetic slip velocity at the particle surface, while the plane wall and fluid far away from the cylinder are moving with a velocity equal to V∞ey. The stream function for this creeping flow was obtained, and the hydrodynamic drag force F1 and torque T1 acting on the cylinder per unit length is16

(8)

2.2. Fluid Velocity Distribution. Having obtained the solution for the electric potential distribution, we can now proceed to find the fluid flow field. Because of the low Reynolds numbers encountered in electrokinetic flows, the fluid motion outside the thin electric double layers is governed by the Stokes equations

εζwE∞ e η y

(

F1 ) -4πη

)

Uy - V∞ Ux ex + ey ψ0 - tanh ψ0 ψ0

T1 ) -4πηa2Ω coth ψ0ez

(13a) (13b)

Next, we consider the fluid flow produced by the electrokinetic tangential velocity at the surface (outer edge of the electric double layer) of a stationary circular cylinder near a stationary, no-slip plane wall, which satisfies the boundary conditions

ψ ) ψ0, ψ ) 0,

v2 )

εζp ∇Φ η

v2 f 0

(14a)

(14b)

Superposing this velocity field v2 with v1 leads to the total velocity field caused by the electrophoresis of a circular cylinder under an applied electric field normal to its axis and parallel to a plane wall. By obtaining the hydrodynamic force F2 and torque T2 exerted on the stationary cylinder per unit length, adding them, respectively, to force F1 and torque T1 given by eq 13a, and equating the sums to zero, we find that the translational and angular velocities of the electrophoretic cylinder with wall corrections will result. The zeta potential of the cylinder is a general function of the azimuth angle θ and, for the convenience of mathematical derivation and physical interpretation, it can be expressed in terms of the multipole expansion35,47

ζp ) M + D · er + Q:erer

(15)

where er and eθ are the unit vectors in polar coordinates (r,θ), and the monopole, dipole, and quadrupole moments M, D, and Q, respectively, are defined by


Q)

∫02π ζpdθ

(16a)

∫02π ζperdθ

(16b)

∫02π ζp(erer - eθeθ)dθ

(16c)

M)

1 2π

D)

1 π

1 π

J. Phys. Chem. C, Vol. 113, No. 29, 2009 12793 Substituting eqs 13a and 18a into the above constraints, we obtain the wall-corrected translational velocities Ux and Uy and angular velocity Ω of the cylinder as Ux )

Uy )

with the higher-order moments being neglected. Various distributions of nonuniform zeta potential ζp can result from appropriate choices of the moments M (which is the areaaveraged zeta potential), D, and Q (which is symmetric). A general solution to eq 10 in bipolar coordinates is50

Ψ)

εE∞c (cosh ψ - cos ξ)-1[Aψ(cosh ψ - cos ξ) + η (B + Cψ)sinh ψ - Dψ sin ξ

∞

+

∑ {[a

n

cosh(n + 1)ψ + bn sinh(n + 1)ψ +

εE∞ tanh ψ0 sinh ψ0 sech 2ψ0{2Dy + Qxy[(sinh ψ0 2η sinh 3ψ0)P1 - 4 cosh ψ0]} (20a)

{

εE∞ sinh ψ0 2(M - ζw)sinh ψ0 - Dx tanh ψ0 + η cosh 2ψ0 1 Qyy 2 sinh3 ψ0 - (2RO1 + R1)cosh 2ψ0 4

]}

[

Ω)

εE∞ tanh ψ0{-(M - ζw)csch ψ0 sech 2ψ0 + ηa

Dx(3 + cosh 4ψ0)csch 4ψ0 - Qyy[-csch ψ0 1 + (sech 2ψ0 - 2)sinh ψ0 + (2RO0 + R0)cosh ψ0 + 2 1 (2RO1 + R1)]} 4

(20c) where On, Pn, Rn, and R are defined by eqs A24-A28 and A32. In the limiting case of λ f 0, eq 20b with ζw ) 0 reduces to

n)1

cn cosh(n - 1)ψ + dn sinh(n - 1)ψ]cos nξ + [a′n cosh(n + 1)ψ +

U0 )

b′n sinh(n + 1)ψ + c′ncosh(n - 1)ψ + d′n sinh(n - 1)ψ]sin nξ}]

(17)

where the coefficients A, B, C, D, an, bn, cn, dn, a′n, b′n, c′n, and d′n should be determined by the boundary conditions in eq 14a using eqs 6, 11a, and 15. After considerable algebraic manipulation, the explicit solution of these coefficients is obtained in Appendix A. It can be found that the infinite series in eq 17 converge quite rapidly. The hydrodynamic drag force and torque acting on the stationary cylinder per unit length due to the electrokinetic motion are16

(18a)

ε 1 ME∞ - Q · E∞ η 2

(21a)

ε D × E∞ ηa

(21b)

(

Ω0 )

)

which are the translational and angular velocities of a nonuniformly charged circular cylinder undergoing transverse electrophoresis in an unbounded fluid. For the electrophoretic motion of a uniformly charged circular cylinder (i.e., M ) ζp, D ) 0, and Q ) 0) in a transversely applied electric field parallel to a plane wall, eq 20c becomes Uy )

ε(ζp - ζw)E∞ 2 sinh2 ψ0 η cosh 2ψ0

(22a)

ε(ζp - ζw)E∞ sech ψ0 aη cosh 2ψ0

(22b)

Ω)-

F2 ) 4πεE∞(Dex + Cey)

(20b)

and Ux ) 0 as expected.

T2 ) -4πεE∞a(A sinh ψ0 + C cosh ψ0)ez

(18b)

where the coefficients A, C, and D are given by eqs A4, A6, and A7 in Appendix A. 2.3. Derivation of the Particle Velocities. Because the net force and torque exerted on the electrophoretic cylinder must vanish, one can write

F1 + F2 ) 0,

T1 + T2 ) 0

(19)

3. Results and Discussion 3.1. Electrophoresis of a Cylinder with a Uniform Zeta Potential. The formulas for the translational and angular velocities of a circular cylindrical particle with a uniform zeta potential undergoing transverse electrophoresis under the applied electric field E∞ ) E∞ey parallel to a plane wall prescribed with the far-field potential distribution are given in eq 22a. These velocities normalized by the Smoluchowski electrophoretic velocity with the correction of the electroosmosis caused by the presence of the plane wall are plotted versus the separation parameter λ (defined by eq 3) in Figure 3. It shows that the particle moves without rotation with the velocity that would exist in the absence of the wall as λ f 0. Opposite to the case of corresponding electrophoresis parallel to an insulating plane

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Figure 3. Plots of the normalized translational and rotational velocities of a circular cylinder with a uniform zeta potential undergoing electrophoresis parallel to a plane wall prescribed with the far-field potential distribution versus the separation parameter λ.

Wang and Keh

Figure 5. Streamlines around a circular cylinder undergoing electrophoresis parallel to a plane wall prescribed with the far-field potential distribution with λ ) 1/2.

TABLE 1: Variation of the Ratio s/d with respect to the Separation Parameter λ

Figure 4. Electric field lines (solid curves) and equipotential lines (dashed curves) for the electrophoretic motion of a circular cylinder parallel to a plane wall prescribed with the far-field potential distribution with λ ) 4/5.

wall,16 the electrophoretic velocity of the cylinder predicted by eq 22b is a monotonic decreasing function of λ and vanishes in the limit λ f 1, where the cylindrical particle touches the plane wall (or more precisely, the outer edges of the thin electric double layers are in contact). In Figure 4, the electric field lines and equipotential curves for the case of λ ) 4/5 are depicted using eqs 6 and 8. The local electric field at the particle surface on the near side to the plane wall is weakened in comparison with that on the far side. Therefore, the electrokinetic and hydrodynamic effects caused by the plane wall are to retard the translational movement of the cylinder, consistent with the prediction in Figure 3. At the same time as the translation, the electrophoretic cylinder rotates with an angular velocity in the direction of ez(ζw/ζp - 1) (opposite to the direction for a sphere sedimenting in the same direction parallel to a plane wall) whose magnitude increases monotonically with an increase in λ and becomes ε|ζp - ζw|E∞/ ηa in the limit λ f 1. The streamlines of the fluid flow around a circular cylinder undergoing transverse electrophoretic motion parallel to a neutral plane wall prescribed with the far-field potential distribution for the situation when the radius of the cylinder is equal to the surfaceto-surface distance between the cylinder and the wall (λ ) 1/2) are exhibited in Figure 5. The fluid flow contains two stagnation

λ

s/d

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.95 0.99

0.60404 0.50133 0.48495 0.49346 0.51729 0.53663 0.55308 0.66552 0.66669 0.54029 0.31471

points on the wall where two recirculation regions meet each other. The results of s/d versus λ, where s denotes the distance between either of the stagnation points and the origin of the coordinate system, are presented in Table 1. Different from the case of corresponding electrophoresis parallel to an insulating plane wall,16 s/d is not a monotonic function of λ in the current case. 3.2. Electrophoresis of a Cylinder with a Zeta Potential Distribution. The explicit formulas for the translational and angular velocities of a nonuniformly charged circular cylinder undergoing transverse electrophoresis in the applied electric field E∞ ) E∞ey parallel to a plane wall prescribed with the far-field potential distribution are given in eq 20a. For illustrative examples, we follow a previous work47 to consider four cases of the odd or even zeta potential distribution on the particle surface. Case I:

ζp ) ζ0 sin θ

(23a)

ζp ) ζ0 cos θ

(23b)

ζp ) ζ0 sin 2θ

(23c)

ζp ) ζ0 cos 2θ

(23d)

Case II:

Case III:

Case IV:

where ζ0 is a constant, and θ is the azimuth angle clockwise from the positive x axis in Figure 2. Note that the monopole



Figure 6. Plots of the normalized velocity ηUx/εζ0E∞ of a circular cylinder with a zeta potential distribution given by eq 23a in a transversely applied electric field parallel to a plane wall versus the separation parameter λ.

Figure 7. Plots of the normalized velocities -ηUy/εζ0E∞ and ηaΩ/ εζ0E∞ of a circular cylinder with a zeta potential distribution given by eq 23b in a transversely applied electric field parallel to a plane wall versus the separation parameter λ.

and quadrupole moments vanish in Cases I and II, whereas the monopole and dipole moments disappear in Cases III and IV. After calculating the multipole moments according to eq 16a and substituting them into eq 20a, we obtain the translational and angular velocities of the cylindrical particle for each of the four cases as functions of λ. For convenience in the following discussion, a neutral plane wall (ζw ) 0) will be taken in all cases. For Case I, the dipole moment D ) ζ0ey, and there is no translation in the direction of the imposed electric field (Uy ) 0) and no rotation (Ω ) 0) of the cylindrical particle due to the antisymmetry of the zeta potential distribution on the particle surface about the x axis (or the plane y ) 0), regardless of the value of λ. Figure 6 shows the result of the lateral velocity of the particle, Ux, as a function of λ. The existence of the plane wall generates a finite lateral velocity of the cylinder in the direction of exζ0/|ζ0| as long as 0 < λ < 1, but this velocity disappears in both limits of λ. It can be shown that the maximal magnitude of Ux equals about 0.238ε|ζ0|E∞/η, which occurs at

Figure 8. Plots of the normalized velocity -ηUx/εζ0E∞ of a circular cylinder with a zeta potential distribution given by eq 23c in a transversely applied electric field parallel to a plane wall versus the separation parameter λ.

λ ) √(5 - √17)/2 For Case II defined by eq 23b, the dipole moment D ) ζ0ex, and the lateral velocity Ux ) 0 due to the symmetry of the zeta potential distribution on the surface of the cylindrical particle about the x axis, irrespective of the value of λ. The results of the translational velocity Uy in the direction of the applied electric field and of the angular velocity Ω of the particle as functions of λ are plotted in Figure 7. For an isolated cylinder (with λ ) 0), the particle rotates with an angular velocity Ω ) εζ0E∞/ηa without translation as given by eq 21a. Different from the case of corresponding electrophoresis parallel to an insulating plane wall,47 the existence of the plane wall prescribed with the far-field potential distribution (with a finite value of λ) decreases this angular velocity and causes a finite translational velocity Uy of the cylinder in the direction of -eyζ0/|ζ0|. In the limit λ ) 1, the particle again rotates with an angular velocity Ω ) εζ0E∞/ηa without translation exactly as the case of λ ) 0. The minimum value of Ωηa/εζ0E∞ equals about 0.828, which takes places at λ ) [2 - (2)1/2]1/2. Interestingly, the λ dependence of the normalized electrophoretic velocity -ηUy/εζ0E∞ in this case is entirely the same as that of the normalized lateral velocity ηUx/εζ0E∞ in Case I.

For Case III defined by eq 23c, the quadrupole moment Q ) ζ0(exey + eyex), and Uy ) Ω ) 0 for any value of λ because of the antisymmetry of the zeta potential distribution on the cylinder surface about the x axis. Figure 8 illustrates the result of the lateral velocity Ux of the particle as a function of λ. For the case of an isolated cylinder (with λ ) 0), the particle translates with a lateral velocity Ux ) -εζ0E∞/2η as given by eq 21a. The approach of a plane wall prescribed with the farfield potential distribution (with an increase in λ) decreases the magnitude of this lateral velocity of the cylinder monotonically (because of the reduction of the local electric field at the particle surface on the side next to the wall and the viscous retardation caused by the wall) to zero in the limit λ ) 1. For Case IV defined by eq 23d, the quadrupole moment Q ) ζ0(exex - eyey), and there is no velocity of the circular cylinder in the direction normal to the imposed electric field (Ux ) 0) due to the symmetry of the zeta potential distribution on the particle surface about the x axis, irrespective of the value of λ. The results of the electrophoretic velocities Uy and Ω of the particle as functions of λ are plotted in Figure 9. For an isolated cylinder (with λ ) 0), the particle translates with a velocity Uy ) εζ0E∞/2η without rotation as predicted by eq 21a. The approach of a plane wall (with an increase in λ) first increases the magnitude of the translational velocity Uy of the cylinder

12796


Wang and Keh occur in an unbounded fluid. Interestingly, although the boundary effect of the confining plane wall on electrophoresis of a uniformly charged particle is much stronger for the perpendicular motion than for the parallel motion, this tendency is not necessarily true for the case of a nonuniformly charged cylinder. For the general two-dimensional problem of a circular cylindrical particle with an arbitrary zeta potential distribution in the angular direction undergoing electrophoresis in an arbitrary direction with respect to a nearby plane wall prescribed with the far-field potential distribution, the solution can be obtained by adding the parallel and perpendicular results vectorially. Appendix A: Detailed Derivation of the Solution in the Analysis

Figure 9. Plots of the normalized velocities ηUy/εζ0E∞ and -ηaΩ/ εζ0E∞ of a circular cylinder with a zeta potential distribution given by eq 23d in a transversely applied electric field parallel to a plane wall versus the separation parameter λ.

In the Analysis, a general solution to eq 4 suitable to satisfy eq 5a is ∞

to a maximum (with ηUy/εζ0E∞ = 0.528) at a finite value of λ (about 0.625) and then reduces it monotonically to zero in the limit λ ) 1, opposite to the case of corresponding electrophoresis parallel to an insulating plane wall.47 At the same time, the approach of the plane wall causes a finite angular velocity Ω of the cylinder, first increasing its magnitude in the direction of ezζ0/|ζ0| to a maximum (with ηaΩ/εζ0E∞ = 0.036) at a finite value of λ (about 0.663), then reducing this angular velocity, and finally reversing its direction (at λ = 0.813) and increasing its magnitude monotonically until Ω ) -εζ0E∞/ηa in the limit λ ) 1.

Φ ) G0 + G1ψ +

∑ [(Rn cosh nψ +

n)1

Sn sinh nψ)cos nξ

(A1)

+ (Rn′cosh nψ + Sn′sinh nψ)sin nξ] cE∞ sin ξ(cosh ψ - cos ξ)-1 in which the last term is the electric potential distribution that would exist in the absence of the cylinder. Applying the boundary conditions in eq 5b, one can easily obtain the coefficients in eq A1 as

4. Conclusion The steady transverse electrophoresis of a dielectric circular cylinder with a thin electric double layer and an arbitrary angular zeta potential distribution on its surface parallel to a large plane wall prescribed with the potential distribution consistent with the uniformly applied electric field has been analytically studied. Through the use of cylindrical bipolar coordinates, the explicit expressions for the wall-corrected translational and angular velocities of the electrophoretic cylindrical particle are obtained in eq 20a. Before using these expressions, one only has to evaluate the leading multipole moments of the zeta potential distribution at the particle surface defined by eqs 15 and 16a. Several illustrative examples of odd and even zeta potential distributions on the cylindrical particle are presented to discuss the boundary effects on the electrophoretic velocities of the nonuniformly charged particle in detail. The presence of the confining wall can produce translation or rotation of the nonuniformly charged particle that does not happen in an unbounded fluid with the same applied electric field. The boundary effects on the electrophoresis of a uniformly or nonuniformly charged particle caused by the parallel plane wall prescribed with the far-field potential distribution are quite different from those resulting from a corresponding insulating wall. The two-dimensional electrophoretic motion of a dielectric circular cylinder with an arbitrary zeta potential distribution in the angular direction normal to a nearby conducting plane wall (i.e., consistent with the applied electric field) was analyzed in a previous work.47 It was also found that the existence of a normal plane wall reduces the electrophoretic velocity of a uniformly charged particle and can causes the translation or rotation of a nonuniformly charged particle, which does not

G0 ) G1 ) Rn ) Sn ) Rn′ ) 0

(A2)

Sn′ ) -2cE∞e-nψ0 sech nψ0

(A3)

and

Substituting eqs A2 and A3 into eq A1, we obtain eq 6 for the solution of the electric potential. The coefficients in eq 17 for the stream function subject to the boundary conditions in eq 14a with eqs 6, 11a, and 15 are obtained as follows

csch2 ψ0 A) {4(M - ζw)(2ψ0 sech 2ψ0 8ψ0 sinh 2ψ0 + tanh 2ψ0) - 2Dx sech 2ψ0[(3 + cosh 4ψ0)ψ0 sech ψ0 4 sinh3 ψ0] - Qyy[8ψ0 cosh 2ψ0 + 2 sech ψ0sinh2 ψ0(sinh 4ψ0 2 sinh 2ψ0 - 4ψ0) - 2(2RO0 + R0)ψ0 sinh 2ψ0 - (2RO1 + R1)(2ψ0 + sinh 2ψ0)sinh ψ0]} (A4)



B ) -A C)

D)

(A5)

{

sinh ψ0 sinh ψ0 [2(M - ζw) - Dx sech ψ0 + ψ0 cosh 2ψ0 Qyy 2Qyy sinh2 ψ0] (A6) (2RO1 + R1) 4

}

sech 2ψ0 sinh ψ0 {2Dy + Qxy[(sinh ψ0 2(ψ0 coth ψ0 - 1) sinh 3ψ0)P1 - 4 cosh ψ0]} (A7) a1 ) -

Ln ) (M - ζw - Dx cosh ψ0 - Qyy cosh2 ψ0)(Fn+1 + Fn-1) + Qyy 2Fn Dx - (M - ζw) cosh ψ0 + (5 cosh ψ0 - cosh 3ψ0) + 4 Qyy (2ROn + Rn)sinh ψ0 (A22) 2

[

]

L′n ) [(Dy - 2Qxy cosh ψ0)(Fn+1 - Fn-1) 2QxyPn sinh ψ0]sinh ψ0 ∞

O0 )

On )

(A8) (A9)

n

∞

k)1

k)n+1

∑ Fkckn + ∑

(A24)

0

Fkcnk

∑

(n g 1)

(Fk-1 - Fk+1)skn +

∑

(Fk-1 - Fk+1)snk

k)n+1

(A26)

(A10)

n sinh ψ0 cosh nψ0 - cosh ψ0 sinh nψ0 n2(cosh 2ψ0 - 1) + 1 - cosh 2nψ0

∞

Ln

(n g 2)

∑ (Fk+2 + Fk-2)e-kψ

R0 )

0

+ (F1 - F-1)e-ψ0 + F2

k)1

(A27)

(A11) bn ) an

(1 - n)[cosh(n - 1)ψ0 - cosh(n + 1)ψ0] (1 - n)sinh(n + 1)ψ0 + (1 + n)sinh(n - 1)ψ0 (n g 2) (A12) cn ) -an

a′n )

n

∞

k)1

k)n+1

∑ (Fk+2 + Fk-2)ckn + ∑

(F1 - F-1)c1n + F2c0n

cnk ) 2e-kψ0 cosh nψ0

(A30)

snk ) 2e-kψ0 sinh nψ0

(A31)

R ) cosh 2ψ0 - 2

(A32)

(n g 2)

(A14)

a′1 )

2ψ0 - sinh 2ψ0 D 2(cosh 2ψ0 - 1)

(A15)

1 b′1 ) D 2

(A16)

c′1 ) -a′1

(A17)

L′n (n g 2)

(1 - n)[cosh(n - 1)ψ0 - cosh(n + 1)ψ0] (1 - n)sinh(n + 1)ψ0 + (1 + n)sinh(n - 1)ψ0 (n g 2) (A19)

c′n ) -a′n d′n )

1+n b′ 1-n n

In the above equations,

(n g 2)

(n g 1) (A28)

(A29)

Appendix B: Some Formulas Relating Bipolar and Rectangular Coordinates In eqs 4 and 5a, the operators are given by

∇ ) ex

(A18) b′n ) a′n

(Fk+2 + Fk-2)cnk +

Fn ) ne-nψ0(tanh nψ0 + 1)

1+n b 1-n n

n2(cosh 2ψ0 - 1) + 1 - cosh 2nψ0

Rn )

(A13)

dn )

n sinh ψ0 cosh nψ0 - cosh ψ0 sinh nψ0

(A25)

∞

n

Pn )

k)1

c1 ) -a1 an )

∑ Fke-kψ

k)1

2ψ0 tanh ψ0C + (sinh 2ψ0 - 2 tanh ψ0)A 2(cosh 2ψ0 - 1)

1 b1 ) A 2

(A23)

(A20)

∂ ∂ ∂ ∂ 1 + ey ) (cosh ψ - cos ξ) eξ + eψ ∂x ∂y c ∂ξ ∂ψ (B1)

(

)

and

∇2 )

(

)

2 ∂2 ∂2 1 ∂2 2 ∂ + ) (cosh ψ cos ξ) + ∂x2 ∂y2 c2 ∂ξ2 ∂ψ2 (B2)

(A21) The unit vectors ex and ey in rectangular coordinates are related to the unit vectors eξ and eψ in bipolar coordinates by

12798

ex )


Wang and Keh

1 [-sinh ψ sin ξeξ cosh ψ - cos ξ (cosh ψ cos ξ - 1)eψ] (B3)

and

ey )

1 [(cosh ψ cos ξ - 1)eξ cosh ψ - cos ξ sinh ψ sin ξeψ] (B4)

Using eq 2a and the chain rule, one can find that eq 7a is equivalent to

∂Φ ∂V ) ∂x ∂y

(B5)

∂Φ ∂V )∂y ∂x

(B6)

and

which are the Cauchy-Riemann equations in rectangular coordinates. Acknowledgment. This research was partly supported by the National Science Council of the Republic of China. References and Notes (1) Morrison, F. A. J. Colloid Interface Sci. 1970, 34, 210. (2) Henry, D. C. Proc. R. Soc. London, Ser. A 1931, 133, 106. (3) Booth, F. Proc. Roy. Soc. A 1950, 203, 514. (4) O’Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1978, 74, 1607. (5) O’Brien, R. W. J. Colloid Interface Sci. 1983, 92, 204. (6) Keh, H. J.; Chen, S. B. Langmuir 1993, 9, 1142. (7) Keh, H. J.; Huang, T. Y. J. Colloid Interface Sci. 1993, 160, 354. (8) Jorgenson, J. W. Anal. Chem. 1986, 58, 743A. (9) Ewing, A. G.; Wallingford, R. A.; Olefirowicz, T. M. Anal. Chem. 1989, 61, 292A. (10) Haber, S.; Gal-Or, L. J. Electrochem. Soc. 1992, 139, 1071. (11) Unni, H. N.; Keh, H. J.; Yang, C. Electrophoresis 2007, 28, 658.

(12) Morrison, F. A.; Stukel, J. J. J. Colloid Interface Sci. 1970, 33, 88. (13) Keh, H. J.; Anderson, J, L. J. Fluid Mech. 1985, 153, 417. (14) Keh, H. J.; Chen, S. B. J. Fluid Mech. 1988, 194, 377. (15) Keh, H. J.; Lien, L. C. J. Fluid Mech. 1991, 224, 305. (16) Keh, H. J.; Horng, K. D.; Kuo, J. J. Fluid Mech. 1991, 231, 211. (17) Loewenberg, M.; Davis, R. H. J. Fluid Mech. 1995, 288, 103. (18) Zydney, A. L. J. Colloid Interface Sci. 1995, 169, 476. (19) Keh, H. J.; Chiou, J. Y. AIChE J. 1996, 42, 1397. (20) Keh, H. J.; Jan, J. S. J. Colloid Interface Sci. 1996, 183, 458. (21) Ennis, J.; Anderson, J. L. J. Colloid Interface Sci. 1997, 185, 497. (22) Hao, Y.; Haber, S. Int. J. Multiphase Flow 1998, 24, 793. (23) Yariv, E.; Brenner, H. J. Fluid Mech. 2003, 484, 85. (24) Chen, P. Y.; Keh, H. J. J. Colloid Interface Sci. 2005, 286, 774. (25) Hsu, J. P.; Yeh, L. H. J. Phys. Chem. B 2007, 111, 2579. (26) Chang, Y. C.; Keh, H. J. J. Colloid Interface Sci. 2008, 322, 634. (27) Miklavic, S. J.; Chan, D. Y. C.; White, L. R.; Healy, T. W. J. Phys. Chem. 1994, 98, 9022. (28) Koopal, L. K. Electrochim. Acta 1996, 41, 2293. (29) Feick, J. D.; Velegol, D. Langmuir 2002, 18, 3454. (30) Feick, J. D.; Chukwumah, N.; Noel, A. E.; Velegol, D. Langmuir 2004, 20, 3090. (31) Bazant, M. Z.; Squires, T. M. Phys. ReV. Lett. 2004, 92, 066101. (32) Rao, S.; Zydney, A. L. Biotechnol. Bioeng. 2005, 91, 733. (33) Golestanian, R.; Liverpool, T. B.; Ajdari, A. New J. Phys. 2007, 9, 126. (34) Teubner, M. J. Phys. Chem. 1982, 76, 5564. (35) Anderson, J. L. J. Colloid Interface Sci. 1985, 105, 45. (36) Fair, M. C.; Anderson, J. L. J. Colloid Interface Sci. 1989, 127, 388. (37) Solomentsev, Y. E.; Pawar, Y.; Anderson, J. L. J. Colloid Interface Sci. 1993, 158, 1. (38) Yoon, B. J. J. Colloid Interface Sci. 1991, 142, 575. (39) Velegol, D.; Feick, J. D.; Collins, L. R. J. Colloid Interface Sci. 2000, 230, 114. (40) Kim, J. Y.; Yoon, B. J. J. Colloid Interface Sci. 2002, 251, 318. (41) Kim, J. Y.; Yoon, B. J. J. Colloid Interface Sci. 2003, 262, 101. (42) Crawford, G. P. IEEE Spectrum 2000, 37 (10), 40. (43) Crowley, J. M.; Sheridon, N. K.; Romano, L. J. Electrostatics 2002, 55, 247. (44) Edman, C. F.; Swint, R. B.; Gurtner, C.; Formosa, R. E.; Roh, S. D.; Lee, K. E.; Swanson, P. D.; Ackley, D. E.; Coleman, J. J.; Heller, M. J. IEEE Photonics Technol. Lett. 2000, 12, 1198. (45) Smith, P. A.; Nordquist, C. D.; Jackson, T. N.; Mayer, T. S.; Martin, B. R.; Mbindyo, J.; Mallouk, T. E. Appl. Phys. Lett. 2000, 77, 1399. (46) Keh, H. J.; Hsieh, T. H. Langmuir 2007, 23, 7928. (47) Hsieh, T. H.; Keh, H. J. J. Colloid Interface Sci. 2007, 315, 343. (48) Keh, H. J.; Hsieh, T. H. Langmuir 2008, 24, 390. (49) Qian, S.; Joo, S. W.; Hou, W. S.; Zhao, X. Langmuir 2008, 24, 5332. (50) Jeffery, G. B. Proc. R. Soc. London, Ser. A 1922, 101, 169.

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