Boundary effects on electrophoresis of a colloidal cylinder
with a nonuniform zeta potential distribution
Tzu H. Hsieh, Huan J. Keh
∗Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan, Republic of China Received 16 May 2007; accepted 22 June 2007
Available online 31 July 2007
Abstract
The electrophoretic motion of a long dielectric circular cylinder with a general angular distribution of its surface potential under a transversely imposed electric field in the vicinity of a large plane wall parallel to its axis is analyzed. The thickness of the electric double layers adjacent to the solid surfaces is assumed to be much smaller than the particle radius and the gap width between the surfaces, but the applied electric field can be either perpendicular or parallel to the plane wall. The presence of the confining wall causes three basic effects on the particle velocity: (1) the local electric field on the particle surface is enhanced or reduced by the wall; (2) the wall increases viscous retardation of the moving particle; (3) an electroosmotic flow of the suspending fluid may exist due to the interaction between the charged wall and the tangentially imposed electric field. Through the use of cylindrical bipolar coordinates, the Laplace and Stokes equations are solved analytically for the two-dimensional electric potential and velocity fields, respectively, in the fluid phase, and explicit formulas for the quasisteady electrophoretic and angular velocities of the cylindrical particle are obtained. To apply these formulas, one has only to calculate the multipole moments of the zeta potential distribution at the particle surface. It is found that the existence of a plane wall near a nonuniformly charged particle can cause its translation or rotation which does not occur in an unbounded fluid with the same applied electric field.
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Keywords: Electrophoresis; Nonuniform zeta potential distribution; Circular cylindrical particle; Plane wall; Boundary effect
1. Introduction
A charged particle suspended in an electrolyte solution is surrounded by a diffuse cloud of ions carrying a total charge equal and opposite in sign to that of the particle. This distrib-ution of fixed charge and diffuse ions is known as an electric double layer. When an electric field is imposed on the particle, a force is exerted on both parts of the double layer. The parti-cle is attracted toward the electrode of its opposite sign, while the ions in the diffuse layer migrate in the other direction. This particle motion is termed electrophoresis and has long been ap-plied to the particle characterization or separation in a variety of colloidal and biological systems.
The electrophoretic velocity U of an isolated particle is re-lated to the applied electric field E∞ by the Smoluchowski
* Corresponding author.
E-mail address:[email protected](H.J. Keh).
equation[1–3],
(1)
U=εζp
η E∞.
Here, η and ε represent the viscosity and permittivity, respec-tively, of the solution surrounding the particle, and ζp is the
zeta potential associated with the particle surface. This for-mula is valid on the basis of several assumptions: (i) the lo-cal radii of curvature of the particle are much larger than the thickness of its electric double layer; (ii) the ambient fluid is unbounded; (iii) the zeta potential is uniform on the length scale of the particle. The first restriction also implies that the double layer remains approximately in equilibrium despite the migration of the particle and diffuse ions. Even though many colloidal particles undergoing electrophoresis fulfill this condi-tion, electrophoresis of particles with thick or distorted double layers does occur in certain cases so that relevant corrections to the Smoluchowski prediction in Eq.(1)are necessary and have been obtained[4–8].
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In practical applications of electrophoresis, colloidal parti-cles are not isolated and will move in the presence of neigh-boring boundaries[9–11]. Therefore, the boundary effects on electrophoresis are of great importance and have been studied extensively in the past for various cases of uniformly charged colloidal spheres and boundaries in the limit of thin electric double layers. Using a method of reflections, Keh and Anderson
[12]analyzed the electrophoretic motions of a dielectric sphere normal to a large conducting plane, parallel to a large dielectric plane, along the axis of a long circular tube, and along the cen-tral plane between two large parallel plates. Through an exact representation in spherical bipolar coordinates or a lubrication theory, semianalytical solutions for the electrophoretic veloc-ity of a colloidal sphere in the vicinveloc-ity of an infinite plane wall have also been obtained in two principal cases: the migration perpendicular to a conducting plane [13–15] and the move-ment parallel to an insulating wall[16,17]. Subsequently, the boundary effects on electrophoresis of a charged sphere were investigated for geometries like migration along the axis of a circular orifice or disk[18], movement in a circular cylindrical pore at an axial[19]or eccentric[20,21]position, and motion in between two parallel plane walls[22–24]. The boundary ef-fects on electrophoresis have also been theoretically examined for the cases of spherical particles with thick or distorted double layers[24–28]and of nonspherical particles[29–31].
On the other hand, many colloidal particles have hetero-geneous surface structures or chemistry and are nonuniformly charged. For instance, elementary clay particles are flat disks with edges having a different charge density or zeta potential from the faces. Distributions of surface charge or potential for particles can also result from aggregation of different species of colloids. Even if a particle is homogeneously charged on its surface, an applied electric field could cause rearrangement of these charges if they are mobile[32]. A distribution of zeta po-tential on particle surfaces has been found to lead to colloidal instability, even the average zeta potential should be sufficiently high to keep the suspension stable[33,34]. The electrophoretic motion of a dielectric sphere with nonuniform zeta potential and thin electric double layer was first analyzed thoroughly by Anderson[35], although it had also been discussed to some ex-tent earlier [36]. It was found that, in terms of the multipole moments of the zeta potential, the electrophoretic mobility de-pends not only on the monopole moment (area-averaged zeta potential) but also on the quadrupole moment, and the dipole moment contributes to particle rotation which tends to align the particle with the electric field. This analysis was later extended to the cases of a nonuniformly charged spherical particle with a double layer of finite thickness[37–40]and a nonuniformly charged nonspherical particle[41–45]. Recently, that particles can have random charge nonuniformity has also been demon-strated experimentally[46,47].
The electrophoretic motion of nonuniformly charged parti-cles in the proximity of confining walls could also be encoun-tered in some real situations. For example, the translation and rotation of each of an array of nonuniformly charged bichro-mal spheres in its own elastomer-made and solvent-filled cavity controlled by imposing a voltage of either positive or negative
polarity have been applied to a technology of electric paper displays[48,49]. Also, an electrophoretic positioning process has been employed in electronic applications for assembling very small individual devices, such as an InGaAs light-emitting diode or a nanowire, which is nonuniformly charged and must have all electric contacts available on one surface, onto the tact electrodes of a silicon circuit by biasing the contacts to con-trol the placement of these devices with the precision required
[50,51]. Recently, the electrophoresis of a dielectric spherical particle in a concentric spherical cavity with nonuniform zeta potential distributions at the solid surfaces has been investigated and analytical expressions for the translational and angular ve-locities of the particle in terms of the monopole, dipole, and quadrupole moments of the zeta potentials were obtained[52].
The objective of this paper is to determine the electrophoretic velocity of a long dielectric circular cylinder with an a nonuni-form zeta potential distribution in the angular direction near a large plane wall parallel to its axis in transversely applied elec-tric fields. The elecelec-tric double layers are assumed to be thin compared with the radius of the cylindrical particle and with the surface-to-surface spacing between the particle and the wall. A cylindrical bipolar coordinate system is used to solve the qua-sisteady problem. In the next section, the electrophoresis of a circular cylinder caused by an imposed electric field in the di-rection perpendicular to its axis and to a conducting plane wall is examined. The analytical solution for the wall-corrected elec-trophoretic velocity of the particle is obtained in Eqs.(20a) and (20b). The analysis of a complementary problem to that treated in Section2, the electrophoretic motion of a circular cylinder driven by an applied electric field in the direction perpendicular to its axis and parallel to a dielectric plane wall, is presented in Section3. The general expressions for the electrophoretic ve-locity of the particle in this case are given in Eqs.(28a)–(28c).
2. Electrophoresis in an applied electric field perpendicular to a conducting plane wall
In this section we consider the quasisteady electrophoretic motion of a long circular cylindrical particle of radius a caused by a uniform electric field E∞= E∞eximposed normal to its
axis and to a large conducting plane wall located at a distance d from the axis, as illustrated inFig. 1a, where extogether with ey
and ezare the principal unit vectors in the Cartesian coordinate
system (x, y, z) with a right-handed screw. The zeta potential ζp
on the surface of the particle at r= a can be a general function of the azimuth angle θ , where (r, θ, z) are circular cylindrical coordinates. The thickness of the electric double layers sur-rounding the particle and adjacent to the plane wall is assumed to be much smaller than the radius of the cylinder and the spac-ing between the solid surfaces. Gravitational and end effects are neglected. Our purpose is to determine the electrophoretic ve-locity of the nonuniformly charged cylindrical particle in the presence of the plane wall.
For convenience in satisfying the boundary conditions at the solid surfaces, an orthogonal curvilinear coordinate system (ξ, ψ, z) known as cylindrical bipolar coordinates and shown in
(a)
(b)
Fig. 1. Geometric sketch for the transverse electrophoresis of a circular cylinder in the proximity of a plane wall: (a) electric field applied perpendicular to the wall; (b) electric field imposed parallel to the wall.
Fig. 2. The two-dimensional bipolar coordinates (ξ, ψ) and rectangular coordi-nates (x, y).
is related to rectangular coordinates in any plane z= constant by the relation[53,54] (2a) x= csinh ψ cosh ψ− cos ξ, (2b) y= csin ξ cosh ψ− cos ξ,
where−∞ < ψ < ∞, 0 ξ 2π, and c is a characteristic length in the bipolar coordinate system which is positive.
The curves ψ= constant correspond to a family of noninter-secting, coaxial circles (or cylinders) whose centers all lie along the x axis. The special case ψ= 0 generates a circle of infinite radius and corresponds to the entire y axis (or the plane x= 0).
ψ= ψ0>0 represents the circle (or the cylinder) of radius a= c csch ψ0, with its center at the point (x= d = c coth ψ0, y= 0). The ratio of the radius of the cylinder to the distance of
the axis of the cylinder from the plane is related to ψ0by
(3)
λ= a/d = sech ψ0.
Before determining the electrophoretic velocity of the cylin-drical particle near the plane wall, the electric potential and velocity fields in the fluid phase must be solved.
2.1. Electric potential distribution
The fluid outside the thin double layers is electrically neu-tral and of constant conductivity; hence the electric potential distribution Φ(ξ, ψ) is governed by the Laplace equation,
(4) ∇2Φ= 0.
Here, the operator∇2in bipolar coordinates has the form (5) ∇2= 1 c2(cosh ψ− cos ξ) 2 ∂2 ∂ξ2 + ∂2 ∂ψ2 .
The potential gradient far away from the cylinder approaches the applied electric field, and the cylindrical particle is assumed to be perfectly insulating. Also, the plane boundary is consid-ered as a perfectly conducting wall and its potential is taken to be zero for convenience. Thus, the boundary conditions for Φ are (6a) ψ= ψ0: eψ· ∇Φ = 0, (6b) ψ= 0: Φ→ −E∞x, where (7) ∇ = 1 c(cosh ψ− cos ξ) eξ ∂ ∂ξ + eψ ∂ ∂ψ ,
and eξ and eψ are the principal unit vectors in bipolar
coordi-nates. Note that
(8a)
ex=
1 cosh ψ− cos ξ
− sinh ψ sin ξeξ
− (cosh ψ cos ξ − 1)eψ
, (8b) ey= 1 cosh ψ− cos ξ
(cosh ψ cos ξ− 1)eξ− sinh ψ sin ξeψ
.
The solution to Eq. (4) subject to the boundary conditions in Eqs.(6a) and (6b)is[29]
(9) Φ= −2cE∞ ∞ n=1 e−nψ0sech nψ 0sinh nψ cos nξ
− cE∞cosh ψsinh ψ− cos ξ,
in which the last term is the electric potential distribution that would exist in the absence of the cylinder.
2.2. Fluid velocity distribution
With knowledge of the solution for the electric potential dis-tribution in the fluid phase, we can now proceed to find the fluid flow field. Because the Reynolds number associated with elec-trophoretic motions is small, the velocity distribution for the
fluid outside the thin electric double layers is governed by the Stokes equations, (10a) η∇2v− ∇p = 0, (10b) ∇ · v = 0,
where v is the fluid velocity distribution and p is the dynamic pressure. Taking the curl of both sides of Eq.(10a)and intro-ducing Eq.(10b)and the stream function Ψ result in a fourth-order linear partial differential equation,
(11) ∇4Ψ = ∇2∇2Ψ= 0.
The stream function is related to the velocity components in bipolar coordinates by the formulas
(12a) vξ= 1 c(cosh ψ− cos ξ) ∂Ψ ∂ψ, (12b) vψ= − 1 c(cosh ψ− cos ξ) ∂Ψ ∂ξ .
At the surface of the cylinder, the electric field acting on the diffuse ions within the double layer produces a relative tan-gential fluid velocity at the outer boundary of the double layer as given by the Helmholtz expression for the electroosmotic flow[55]. At a distance far away from the particle and on the conducting plane wall, the fluid is motionless. Therefore, the boundary conditions for the velocity field are
(13a) ψ= ψ0: v= Uxex+ Uyey+ aΩeξ+ εζp η ∇Φ, (13b) ψ= 0: v→ 0,
where Uxex+ Uyeyand Ωezare the translational and angular
velocities, respectively, of the electrophoretic cylinder to be de-termined and the expression for Φ is given by Eq.(9). Note that
Uyand Ω appear in Eq.(13a)since the zeta potential ζpcan be
a general function of the angular position on the particle sur-face. Because the cylinder is freely suspended in the fluid, the net force and net torque exerted by the fluid on the cylinder per unit length must vanish.
Since the governing equation and boundary conditions are linear, the total flow can be decomposed into two parts. First, we consider the fluid velocity field v1about a circular cylinder
(with its surface at ψ= ψ0) translating with a velocity Uxex+ Uyey and rotating with an angular velocity Ωez near a plane
wall (at x= 0), but with no electrokinetic slip velocity at the particle surface. The stream function for this creeping flow was obtained and the drag force F1 and torque T1 exerted by the
fluid on the cylinder per unit length is[29]
(14a) F1= −4πη Ux ψ0− tanh ψ0 ex+ Uy ψ0ey , (14b) T1= −4πηa2Ωcoth ψ0ez.
The above equations indicate that the translation and rotation for the two-dimensional creeping motion of a circular cylinder near a plane wall are not coupled with each other.
Next, we consider the fluid flow caused by the electrokinetic tangential velocity at the surface (outer edge of the electric dou-ble layer) of a stationary circular cylinder near a plane wall,
namely, the flow subject to the boundary conditions given by Eqs. (13a) and (13b) with Ux = Uy= Ω = 0. Superposing
this velocity field v2with v1 will yield the total velocity field
produced by the electrophoretic motion of a cylinder under an applied electric field normal to its axis and to a plane wall. By obtaining the hydrodynamic force F2and torque T2exerted on
the stationary cylinder, adding them respectively to the force
F1and torque T1given by Eqs.(14a) and (14b), and equating
the sums to zero, the translational and angular velocities of the electrophoretic cylinder with wall corrections will result.
The zeta potential ζp is a general function of the azimuth
angle θ on the cylinder surface r= a and can be expressed in terms of the multipole expansions,
(15)
ζp= M + D · er+ Q : erer.
Here er and eθ are the basic unit vectors in polar coordinates
(r, θ ), and the monopole, dipole, and quadrupole moments M,
D, and Q, respectively, are defined by the following integrals
over the particle surface,
(16a) M= 1 2π 2π 0 ζpdθ, (16b) D= 1 π 2π 0 ζperdθ, (16c) Q= 1 π 2π 0 ζp(erer− eθeθ)dθ,
and the higher-order moments are neglected. Various distribu-tions of nonuniform zeta potential ζpcan result from
appropri-ate choices of the moments M (i.e., area-averaged zeta poten-tial), D, and Q (which is symmetric and traceless).
A general solution to the biharmonic equation(11)in bipolar coordinates, suitable for satisfying boundary conditions on the cylindrical particle and plane wall, has been given by[53,56]
(17)
Ψ=εE∞c
η (cosh ψ− cos ξ) −1
Aψ (cosh ψ− cos ξ) + (B + Cψ) sinh ψ − Dψ sin ξ + ∞ n=1 ancosh(n+ 1)ψ + bnsinh(n+ 1)ψ + cncosh(n− 1)ψ + dnsinh(n− 1)ψ cos nξ +ancosh(n+ 1)ψ + bnsinh(n+ 1)ψ + cncosh(n− 1)ψ + dnsinh(n− 1)ψ sin nξ .
The coefficients A, B, C, D, an, bn, cn, dn, an, bn, cn, and dn (in which d1 and d1 are trivial) should be determined by
the boundary conditions given by Eqs.(13a) and (13b) with
Ux= Uy= Ω = 0 using Eqs.(9), (12a), (12b), and (15). After
considerable algebraic manipulation, analytical results of these coefficients are obtained and given inAppendix A.
The drag force and torque exerted on the stationary cylinder per unit length by the fluid due to the electrokinetic motion are
(18a)
F2= 4πεE∞(Dex+ Cey),
(18b)
T2= −4πεE∞a(Asinh ψ0+ C cosh ψ0)ez,
where the coefficients A, C, and D are given by Eqs. (A.1), (A.3), and (A.4).
2.3. Derivation of the particle velocities
Since the net hydrodynamic force and torque acting on the electrophoretic cylinder must vanish, we have
(19a)
F1+ F2= 0,
(19b)
T1+ T2= 0.
With the substitution of Eqs.(14a), (14b) and (18a), (18b)into the above constraints, the translational velocities Ux and Uyas
well as the angular velocity Ω of the cylinder near the conduct-ing plane wall are determined as
(20a)
Ux= εE∞
η sinh ψ0tanh ψ0sech 2ψ0
2M sinh ψ0− Dxtanh ψ0
+1 4Qxx
(cosh 4ψ0+ 1)O1+ cosh 2ψ0(R1− 4O1)
+ 6 sinh ψ0− 2 sinh 3ψ0 , (20b) Uy= εE∞ η tanh ψ0sech 2ψ0 −Dycosh ψ0 +1 4Qyx
6+ 2 cosh 4ψ0− (sinh 3ψ0+ sinh 5ψ0)P0
, (20c) Ω=εE∞ ηa −Dytanh 2ψ0+ 2Qxy (1− cosh 2ψ0)P0
+ (2 + sech 2ψ0)sinh ψ0− sech ψ0tanh ψ0
,
where On, Pn, and Rnare defined by Eqs.(A.21)–(A.24).
In the limit λ→ 0, Eqs.(20a)–(20c)reduce to
(21a) U0= ε η MI−1 2Q · E∞, (21b) Ω0= ε ηaD× E∞,
which are the translational and angular velocities of a nonuni-formly charged circular cylinder undergoing two-dimensional electrophoresis in an unbounded fluid.
For a uniformly charged circular cylinder undergoing elec-trophoretic motion in a transversely applied electric field nor-mal to a conducting plane wall, Eqs.(20a)–(20c)become
(22) Ux=2εζpE∞ η sinh2ψ0 cosh 2ψ0 tanh ψ0,
and Uy = Ω = 0. Equation (22) corrects an inadvertent
er-ror for the previous result obtained by Keh et al.[29]in their Eq. (4.10).
2.4. Results and discussion
The analytical solution for the translational and angular ve-locities of the cylindrical particle undergoing transverse elec-trophoresis under the applied electric field E∞= E∞ex
per-pendicular to a conducting plane wall is obtained in Eqs.(20a)– (20c). For illustrative examples, we consider four cases of the zeta potential distribution on the surface of the particle.
(23a) Case I: ζp= ζ0sin θ,
(23b) Case II: ζp= ζ0cos θ,
(23c) Case III: ζp= ζ0sin 2θ,
(23d) Case IV: ζp= ζ0cos 2θ,
where ζ0 is a constant and θ is the azimuth angle clockwise
from the positive x axis inFig. 2. Note that both the monopole moment (area-averaged zeta potential) and the quadrupole mo-ment disappear in Cases I and II, while both the monopole and the dipole moments vanish in Cases III and IV. After calculat-ing the multipole moments accordcalculat-ing to Eqs.(16a)–(16c)and substituting them into Eqs.(20a)–(20c), we obtain the transla-tional and angular velocities of the cylindrical particle for each of the four cases as functions of λ, the ratio of the radius of the cylinder to the distance of the axis of the cylinder from the plane wall.
For Case I, the dipole moment D= ζ0ey, and there is no
particle velocity in the direction of the imposed electric field (Ux= 0) owing to the antisymmetry of the zeta potential
distri-bution on the surface of the cylindrical particle about the x axis (or the plane y= 0), regardless of the value of the parameter λ. The results of the lateral velocity Uyand angular velocity Ω of
the particle as functions of λ are plotted inFig. 3. The existence of the conducting plane wall depresses the local electric field at the particle surface on the side next to the wall compared with that on the far side[29], and thus reduces the magnitude of the angular velocity of the particle from ε|ζ0|E∞/ηa(in the
direc-tion of−ezζ0/|ζ0|, as predicted by Eq.(21b)) at λ= 0 to zero at
Fig. 3. Plots of the normalized velocities−ηUy/εζ0E∞and−ηaΩ/εζ0E∞
of a circular cylinder with a zeta potential distribution given by Eq.(23a)in a transversely applied electric field perpendicular to a conducting plane wall versus the separation parameter λ.
Fig. 4. Plots of the normalized velocity−ηUx/εζ0E∞of a circular cylinder
with a zeta potential distribution given by Eq.(23b)in a transversely applied electric field perpendicular to a conducting plane wall versus the separation parameter λ.
λ= 1. Interestingly, there exists an accompanying finite lateral
velocity Uyof the cylinder in the direction of−eyζ0/|ζ0|
(pre-dictable from observing the strength and direction distributions of the tangential electroosmotic velocity at the particle surface given in Eq.(13a)) as long as 0 < λ < 1, but this velocity dis-appears in both limits of λ as expected. It can be shown that the maximal magnitude of Uy equals ε|ζ0|E∞/2
√
2η, which occurs at λ=√2/3.
For Case II defined by Eq.(23b), the dipole moment D=
ζ0ex, and Uy= Ω = 0 due to the symmetry of the zeta
poten-tial distribution on the surface of the cylindrical particle about the x axis, irrespective of the value of λ. As shown inFig. 4, the electrophoretic velocity of the particle, Ux, has a finite
magnitude in the direction of−exζ0/|ζ0| for all finite
separa-tions (0 < λ < 1) and, as expected, vanishes in both limits of
λ. It can be found that the maximal magnitude of Ux equals (3+√33 )3/2ε|ζ0|E∞/4(17+ 3
√
33 )η, which takes place at
λ= 2/(7 +√33 )1/2.
For Case III, the quadrupole moment Q= ζ0(exey+ eyex),
and there is no velocity of the cylinder in the direction of the applied electric field (Ux= 0) for any value of λ, owing to the
antisymmetry of the zeta potential distribution on the particle surface about the x axis. The results of the lateral velocity Uy
and angular velocity Ω of the particle as functions of λ are plotted inFig. 5. For an isolated cylinder (with λ= 0), the par-ticle translates with a lateral velocity Uy= −εζ0E∞/ηwithout
rotation, as given by Eqs. (21a)–(21b). The existence of the conducting plane wall (with a finite value of λ) reduces the magnitude of this lateral velocity and causes a finite angular velocity Ω of the cylinder in the direction of ezζ0/|ζ0|, and
both velocities vanish in the limit λ= 1. The maximal mag-nitude of Ω equals about 0.176ε|ζ0|E∞/ηa, which occurs near λ= 0.915.
For Case IV defined by Eq.(23d), the quadrupole moment
Q= ζ0(exex− eyey), and Uy= Ω = 0 for any value of λ due
to the symmetry of the zeta potential distribution on the
parti-Fig. 5. Plots of the normalized velocities−ηUy/εζ0E∞and ηaΩ/εζ0E∞of a
circular cylinder with a zeta potential distribution given by Eq.(23c)in a trans-versely applied electric field perpendicular to a conducting plane wall versus the separation parameter λ.
Fig. 6. Plots of the normalized velocity−ηUx/εζ0E∞of a circular cylinder
with a zeta potential distribution given by Eq.(23d)in a transversely applied electric field perpendicular to a conducting plane wall versus the separation parameter λ.
cle surface about the x axis. The result of the particle velocity in the direction of the imposed electric field, Ux, as a function
of λ is plotted inFig. 6. For the case of an unconfined cylin-der (with λ= 0), the particle translates with an electrophoretic velocity Ux= −εζ0E∞/2η, as predicted by Eq.(21a). When
the conducting plane wall exists, as expected, the magnitude of this electrophoretic velocity decreases with an increase in λ and vanishes in the limit λ= 1.
3. Electrophoresis in an applied electric field parallel to an insulating plane wall
We now consider the two-dimensional quasisteady elec-trophoretic motion of a long circular cylinder of radius a
(repre-sented by ψ= ψ0) under a uniform electric field E∞= E∞ey
imposed perpendicular to its axis (with coordinates x= d and
y= 0) and parallel to a large dielectric plane wall (located at x= 0), as shown inFig. 1b. The zeta potential distribution on the surface of the cylindrical particle can be a general function of the azimuth angle θ . As in the previous section, the assump-tion of thin electric double layers is employed. Our objective is to find the wall-corrected electrophoretic velocity of the parti-cle.
3.1. Electric potential distribution
The electrostatic equation governing the potential distribu-tion Φ(ξ, ψ) is the Laplace equadistribu-tion(4). Since the potential gradient far away from the cylinder approaches the applied electric field and both the cylinder and the wall are perfectly insulating, the electric potential is subject to the boundary con-ditions (24a) x= 0: eψ· ∇Φ = 0, (24b) ψ= ψ0: eψ· ∇Φ = 0, (24c) x2+ y21/2→ ∞ and x >0: Φ→ −E∞y.
The solution of Eq. (4)satisfying the above boundary condi-tions is given by[29] (25) Φ= −2cE∞ ∞ n=1 e−nψ0csch nψ 0cosh nψ sin nξ
− cE∞cosh ψsin ξ− cos ξ,
where the last term is the undisturbed potential distribution (in the absence of the particle).
3.2. Fluid velocity distribution
Having obtained the solution for the electric potential distri-bution in the fluid phase, we can now proceed to find the flow field. The fluid motion outside the thin electric double layers is governed by Eq.(11)and subject to the boundary conditions
(26a) x= 0: v=εζw η ∇Φ, (26b) ψ= ψ0: v= Uxex+ Uyey+ aΩeξ+ εζp η ∇Φ, (26c) x2+ y21/2→ ∞ and x > 0: v→ v∞ey= − εζwE∞ η ey,
where ζw is the zeta potential associated with the plane wall,
which is taken as a constant, Uxex+ Uyey and Ωez are the
translational and angular velocities, respectively, of the elec-trophoretic cylinder to be determined, and the expression for Φ is given by Eq.(25). Note that Uxexists in Eq.(26b)due to the
nonuniformity of ζpon the particle surface, and Eqs.(26a) and
(26c)allow an electroosmotic flow induced by the interaction of the applied electric field with the charged plane wall.
Similarly to the case dealt with in the previous section, the total flow can be decomposed into two parts. First, we consider the fluid velocity field v1about a circular cylinder moving near
the plane wall with the translational velocity Uxex+ Uyeyand
angular velocity Ωez, while the plane wall and the fluid far
away from the cylinder are moving with a velocity equal to
v∞ey, but with no electrokinetic slip velocity at either of the
solid surfaces. For this creeping flow, the drag force F1 and
torque T1per unit length exerted by the fluid on the cylinder is
still given by Eqs.(14a), (14b)with Uy− v∞to replace Uy.
Next, we consider the fluid flow caused by the electrokinetic tangential velocities at the solid surfaces (i.e., outer edges of the double layers) of a stationary circular cylinder and a nearby plane wall moving with a velocity equal to−v∞ey, which
sat-isfies the boundary conditions
(27a) x= 0: v2= εζw η ∇Φ − v∞ey, (27b) ψ= ψ0: v2= εζp η ∇Φ, (27c) x2+ y21/2→ ∞ and x 0: v2→ 0.
Here the electric potential distribution Φ is provided by Eq. (25). Superposing the velocity field v2 with v1 yields
the total fluid velocity field produced by the transverse elec-trophoretic motion of a circular cylinder subject to an im-posed electric field parallel to a plane wall and specified by Eqs.(26a)–(26c). By obtaining the drag force F2and torque T2
per unit length exerted by the fluid on the stationary cylinder, individually adding these to the force F1and torque T1given
by Eqs.(14a) and (14b)with Uy−v∞to replace Uy, and
equat-ing the results to zero, the translational and angular velocities of the cylinder will result.
The stream function Ψ2 associated with v2can also be
ex-pressed by Eq. (17), and the coefficients A, B, C, . . . , etc., should be determined by applying Eqs.(27a)–(27c)to Eq.(17)
and using Eqs.(12a), (12b) and (25). The procedure is straight-forward but tedious, and the result is given inAppendix B. The force F2and torque T2per unit length exerted on the
station-ary cylinder by the fluid due to the electrokinetic motion can be easily obtained by the substitution of the coefficients A, C, and
Dgiven by Eqs.(B.1), (B.3), and (B.4)into Eqs.(18a), (18b).
3.3. Derivation of the particle velocities
Using the constraints that the net force F1+ F2 and net
torque T1+ T2acting on the electrophoretic cylinder must
van-ish, we obtain the translational and rotational velocities of the cylinder near an insulating plane wall as
(28a) Ux= εE∞ η tanh ψ0 Dysech ψ0+ 2Qxy P1sinh2ψ0− 1 ,
(28b) Uy= εE∞ η (M− ζw)coth 2ψ0− 1 2Dxcsch ψ0 +1 4Qyy (cosh 4ψ0− 5) csch 2ψ0 + sinh 2ψ0 (cosh 2 ψ0− 2)O0+ R0 , (28c) Ω=εE∞ 2ηa csch ψ0 sech2ψ0 −(M − ζw) +1 2Dx(cosh ψ0+ cosh 3ψ0) + Qyy 1+ tanh2ψ0 + sinh2ψ 0(2+ (cosh 2ψ0− 2)O0+ R0) .
Here, On, Pn, and Rnare defined by Eqs.(B.21)–(B.25). Again,
in the limit λ→ 0, Eqs.(28a)–(28c)with ζw= 0 reduces to
Eqs. (21a) and (21b) for the electrophoresis of the circular cylinder in an unbounded fluid.
For a uniformly charged circular cylinder undergoing elec-trophoretic motion in a transversely applied electric field paral-lel to an insulating plane wall, Eqs.(28a)–(28c)become
(29a) Uy= εE∞ η (ζp− ζw)coth 2ψ0, (29b) Ω= −εE∞ ηa (ζp− ζw) sech ψ0 sinh 2ψ0 ,
and Ux= 0. Equations(29a) and (29b)are identical to the
pre-vious result obtained by Keh et al.[29].
3.4. Results and discussion
The analytical solution for the translational and angular ve-locities of the circular cylinder undergoing transverse elec-trophoresis under the applied electric field E∞= E∞eyparallel
to a dielectric plane wall is obtained in Eqs.(28a)–(28c). Again, we consider the four cases of the zeta potential distribution on the surface of the cylindrical particle defined by Eqs. (23a)– (23d) as illustrative examples. After calculating the multipole moments according to Eqs.(16a)–(16c)and substituting them into Eqs.(28a)–(28c), we obtain the translational and angular velocities of the particle as functions of the parameter λ defined by Eq.(3) for each of the four cases. For convenience in the following discussion, the condition ζw= 0 will be taken for the
plane wall in all cases.
For Case I defined by Eq. (23a), there is no translation in the direction of the applied electric field (Uy= 0) and no
rota-tion (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), irrespective of the value of λ. The result of the lateral velocity of the particle, Ux, as a
func-tion of λ is plotted inFig. 7. The existence of the insulating plane wall enhances the local electric field at the particle sur-face on the near side to the plane wall in comparison with that on the far side[29], and thus generates a finite lateral velocity of the cylinder in the direction of exζ0/|ζ0| (predictable from
observing the strength and direction distributions of the tan-gential electroosmotic velocity at the particle surface given in
Fig. 7. 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 conducting plane wall versus the separation parameter λ.
Fig. 8. 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 conducting plane wall versus the separation parameter λ.
Eq.(26b)) as long as 0 < λ < 1, but this velocity disappears in both limits of λ. It can be found that the maximal magnitude of
Uxequals ε|ζ0|E∞/4η, which occurs at λ= 1/
√ 2.
For Case II, the lateral velocity Ux= 0 for any value of λowing to the symmetry of the zeta potential distribution on the surface of the cylindrical particle about the x axis. The results of the translational velocity Uy in the direction of the
imposed electric field and of the angular velocity Ω of the par-ticle as functions of λ are plotted inFig. 8. For an isolated cylinder (with λ= 0), the particle rotates with an angular veloc-ity Ω= εζ0E∞/ηawithout translation, as given by Eqs.(21a)
and (21b). The existence of the dielectric plane wall (with a finite value of λ) increases this angular velocity and causes a finite translational velocity Uy of the cylinder in the direction
Fig. 9. 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 conducting plane wall versus the separation parame-ter λ.
increase in the parameter λ to infinity in the limit λ= 1 (in which the local tangential electric field at the contact point be-comes infinity). This behavior is predictable from observing the strength and direction distributions of the tangential electroos-motic velocity at the particle surface given in Eq.(26b).
For Case III defined by Eq.(23c), Uy= Ω = 0 due to the
an-tisymmetry of the zeta potential distribution on the particle sur-face about the x axis, regardless of the value of λ. The result of the lateral velocity Uxof the particle as a function of λ is
plot-ted inFig. 9. For the case of an isolated cylinder (with λ= 0), the particle translates with a lateral velocity Ux= −εζ0E∞/2η
as predicted by Eq.(21a). The approach of an insulating plane wall (with an increase in λ) first increases the magnitude of this lateral velocity of the cylinder (the influence of the enhance-ment of the local electric field at the particle surface is stronger than the effect of the viscous retardation caused by the wall) to a maximum at a finite value of λ, then reduces it (the effect of the viscous retardation dominates) to zero in the limit λ= 1. The maximal magnitude of Uxequals about 0.516ε|ζ0|E∞/η,
which takes place near λ= 0.500.
For Case IV, there is no velocity of the circular cylin-der in the direction perpendicular to the applied electric field (Ux = 0) for any value of λ, owing to the
symme-try of the zeta potential distribution on the particle surface about the x axis. The results of the electrophoretic veloci-ties Uy and Ω of the particle as functions of λ are plotted
in Fig. 10. For an isolated cylinder (with λ= 0), the par-ticle translates with a velocity Uy = εζ0E∞/2η without
ro-tation, as given by Eqs. (21a) and (21b). The existence of a nearby insulating plane wall (with a finite value of λ) in-creases this electrophoretic velocity and causes a finite angu-lar velocity Ω of the cylinder in the direction of−ezζ0/|ζ0|.
Both magnitudes of Uy and Ω increase with an increase in
the parameter λ to infinity in the limit λ= 1 (where the local tangential electric field at the contact point is infin-ity).
Fig. 10. 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 conducting plane wall versus the separation parameter λ.
4. Concluding remarks
In this work, the two-dimensional (transverse) electropho-retic motion of a dielectric circular cylinder with a general angular zeta potential distribution on its surface in the proxim-ity of a large plane wall has been analytically investigated at the quasisteady state. The thickness of the electric double layers ad-jacent to the solid surfaces is assumed to be much smaller than the particle radius and the gap width between the solid surfaces. A cylindrical bipolar coordinate system has been used to solve the Laplace and Stokes equations for the electric potential and velocity fields, respectively, in the fluid phase in two fundamen-tal cases: external electric fields applied normal to a conducting plane wall and parallel to a dielectric plane wall. The trans-lational and angular velocities of the cylindrical particle are obtained in explicit expressions (20a)–(20c) and (28a)–(28c)
for the two cases. Before using these equations, one has only to evaluate the multipole moments of the zeta potential distribu-tion at the particle surface defined by Eqs.(15) and (16a)–(16c). The contributions from the electroosmotic flow produced by the interaction of the tangentially applied electric field with the thin electric double layer adjacent to the plane wall and from the wall-corrected electrophoretic driving force to the particle ve-locities can be superimposed due to the linearity of the problem. Several illustrative examples of the cylindrical particle–plane wall system with odd and even zeta potential distributions, re-spectively, are given to discuss in detail the boundary effects on the electrophoretic velocities of the nonuniformly charged particle.
For the purpose of obtaining analytical solutions for the transverse electrophoretic motion of a circular cylinder in the proximity of a plane wall parallel to its axis, the effect of the ends of the cylinder has been ignored in our analyses. In order to investigate the electrophoresis of a cylinder having a relatively large but finite length with or without a confining plane wall, the use of a slender-body theory[42,57]or numerical calculations might be needed. However, our results in Eqs.(20a)–(20c) and
(28a)–(28c)demonstrate that the electrophoretic velocities of the cylinder are independent of its length (or the size of the par-ticle). Therefore, it is reasonable to expect that, unless the zeta potential or surface charge density at the ends of the cylinder is relatively high, the end effect on its electrophoretic velocities will not be significant.
Acknowledgment
This research was partly supported by the National Science Council of the Republic of China.
Appendix A. Coefficients in Eq.(17)satisfying boundary conditions given by Eq.(13)withUx= Uy= Ω = 0
In Section 2, the coefficients in Eq. (17) for the stream function determined using the boundary conditions given by Eq. (13)with Ux= Uy= Ω = 0 and Eqs.(9), (12), and (15)
are obtained as follows:
(A.1) A= Dy(1+ 2ψ0coth ψ0)sech 2ψ0 cosh ψ0 ψ0 − Qyx csch 4ψ0 2ψ0 16ψ0+ 5 sinh 2ψ0+ sinh 6ψ0 − 8 cosh2ψ 0 2ψ0+ cosh 2ψ0 P0sinh ψ0 × (4ψ0+ sinh 2ψ0)− 4ψ0 , (A.2) B= −A, (A.3) C=sech 2ψ0 ψ0 −Dysinh ψ0+ 1 4Qyxtanh ψ0 6+ 2 cosh 4ψ0 − P0(sinh 3ψ0+ sinh 5ψ0) , (A.4) D=sinh ψ0sech 2ψ0 ψ0coth ψ0− 1 2M sinh ψ0− Dxtanh ψ0 +1 4Qxx
(cosh 4ψ0+ 1)O1+ cosh 2ψ0(R1− 2O1)
+ 6 sinh ψ0− 2 sinh 3ψ0
,
(A.5)
a1= −2ψ0tanh ψ0C+ (sinh 2ψ0− 2 tanh ψ0)A
2(cosh 2ψ0− 1) , (A.6) b1= 1 2A, (A.7) c1= −a1, (A.8) an=
nsinh ψ0cosh nψ0− cosh ψ0sinh nψ0
[n2(cosh 2ψ 0− 1) + 1 − cosh 2nψ0] sinh ψ0 Hn (n 2), (A.9) 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 2), (A.10) cn= −an (n 2), (A.11) dn=1+ n 1− nbn (n 2), (A.12) a1= 2ψ0− sinh 2ψ0 2(cosh 2ψ0− 1) D, (A.13) b1=1 2D, (A.14) c1 = −a1, (A.15)
an = nsinh ψ0cosh nψ0− cosh ψ0sinh nψ0
[n2(cosh 2ψ 0− 1) + 1 − cosh 2nψ0] sinh ψ0 Hn (n 2), (A.16) 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 2), (A.17) cn = −an (n 2), (A.18) dn =1+ n 1− nb n (n 2).
In the above equations,
(A.19) Hn= sinh ψ0 (Dy− 2Qxycosh ψ0)(Fn+1− Fn−1) + 2Qxysinh ψ0Pn , (A.20) Hn=1 2 −2(M− Dxcosh ψ0)+ (1 + cosh 2ψ0)Qxx × (Fn+1+ Fn−1)+ 4(M cosh ψ0− Dx) + (5 cosh ψ0− cosh 3ψ0)Qxx Fn + Qxx
sinh ψ0(Rn− 5On)+ sinh 3ψ0On ,
(A.21) On= n k=1 Fkskn+ ∞ k=n+1 Fksnk, (A.22) P0= ∞ k=1 (Fk+1− Fk−1)e−kψ0+ F1, (A.23) Pn= n k=1 (Fk+1− Fk−1)ckn+ ∞ k=n+1 (Fk+1− Fk−1)cnk+ c0kF1 (n 1), (A.24) Rn= n k=1 (Fk+2− Fk−2)skn+ ∞ k=n+1 (Fk+2− Fk−2)snk − (F1+ F−1)s1n, (A.25) Fn= ne−nψ0(tanh ψ0+ 1), (A.26) cnk= 2e−kψ0cosh nψ0, (A.27) snk= 2e−kψ0sinh nψ0.
Appendix B. Coefficients in Eq.(17)satisfying boundary conditions given by Eqs.(27a)–(27c)
In Section 3, the coefficients in Eq. (17) for the stream function determined using the boundary conditions given by Eqs.(27a)–(27b)with Eqs.(12a), (12b), (15), and (25)are
ob-tained as follows: (B.1) A= 1 8ψ0 csc h3ψ0 Msec hψ0(4ψ0− sinh 4ψ0) + 2Dx(sinh 2ψ0− 2ψ0cosh 2ψ0) − 2Qyy 21+ cosh 2ψ0− csc h2ψ0 + ψ0 coth ψ0 5+ csc h2ψ0 − tanh ψ0 + coth ψ0(4ψ0+ sinh 2ψ0) ×2(cosh 2ψ0− 2)O0+ R0 (B.2) B= −A, (B.3) C= 1 2ψ0 (2M coth 2ψ0− Dxcsch ψ0) +Qyy 4ψ0 2 tanh ψ0− 2 coth ψ0 + sinh 2ψ0 2+ R0+ (cosh 2ψ0− 2)O0 , (B.4) D= 1 2(ψ0coth ψ0− 1) ×Dysech ψ0+ 2Qxy sinh2ψ0P1− 1 , (B.5)
a1= −2ψ0tanh ψ0C+ (sinh 2ψ0− 2 tanh ψ0)A
2(cosh 2ψ0− 1) , (B.6) b1=1 2A, (B.7) c1= −a1, (B.8) an=
nsinh ψ0cosh nψ0− cosh ψ0sinh nψ0
[n2(cosh 2ψ 0− 1) + 1 − cosh 2nψ0] sinh ψ0 Ln (n 2), (B.9) 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 2), (B.10) cn= −an (n 2), (B.11) dn= 1+ n 1− nbn (n 2), (B.12) a1 = 2ψ0− sinh 2ψ0 2(cosh 2ψ0− 1) D, (B.13) b1 =1 2D, (B.14) c1 = −a1 (B.15)
an = nsinh ψ0cosh nψ0− cosh ψ0sinh nψ0
[n2(cosh 2ψ 0− 1) + 1 − cosh 2nψ0] sinh ψ0 Ln (n 2), (B.16) 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 2), (B.17) cn = −an (n 2), (B.18) dn =1+ n 1− nb n (n 2).
In the above equations,
(B.19) Ln= 1 2 2(−M + cosh ψ0Dx)+ (1 + cosh 2ψ0)Qyy × (Gn+1+ Gn−1)+ 4(M cosh ψ0− Dx)
+ (cosh 3ψ0− 5 cosh ψ0)Qyy
Gn
− Qyy
sinh ψ0(Rn− 5On)+ sinh 3ψ0On ,
(B.20) Ln= sinh ψ0 (Dy− 2Qyxcosh ψ0)(Gn−1− Gn+1) + 2Qyxsinh ψ0Pn , (B.21) O0= ∞ k=1 Gke−kψ0, (B.22) On= n k=1 Gkckn+ ∞ k=n+1 Gkcnk (n 1), (B.23) Pn= n k=1 (Gk−1− Fk+1)skn+ ∞ k=n+1 (Gk−1− Gk+1)snk, (B.24) R0= ∞ k=1 (Gk+2+ Gk−2)e−kψ0+ (G1− G−1)e−ψ0+ G2, (B.25) Rn= n k=1 (Gk+2+ Gk−2)ckn+ ∞ k=n+1 (Gk+2+ Gk−2)cnk + (G1− G−1)c1n+ G2c0n (n 1), (B.26) Gn= −ne−nψ0(coth ψ0+ 1),
where cnk and snk are given by Eqs.(A.26) and (A.27).
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