Diffusiophoresis and electrophoresis of a charged sphere perpendicular to
two plane walls
Yu C. Chang, Huan J. Keh
∗Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan, Republic of China Received 24 January 2008; accepted 4 March 2008
Available online 16 April 2008
Abstract
The problem of diffusiophoretic and electrophoretic motions of a dielectric spherical particle in an electrolyte solution situated at an arbitrary position between two infinite parallel plane walls is studied theoretically in the quasisteady limit of negligible Peclet and Reynolds numbers. The applied electrolyte concentration gradient or electric field is uniform and perpendicular to the plane walls. The electric double layer at the particle surface is assumed to be thin relative to the particle radius and to the particle–wall gap widths, but the polarization effect of the diffuse ions in the double layer is incorporated. To solve the conservative equations, the general solution is constructed from the fundamental solutions in both cylindrical and spherical coordinates. The boundary conditions are enforced first at the plane walls by the Hankel transforms and then on the particle surface by a collocation technique. Numerical results for the diffusiophoretic and electrophoretic velocities of the particle relative to those of a particle under identical conditions in an unbounded solution are presented for various cases. The collocation results agree well with the approximate analytical solutions obtained by using a method of reflections. The presence of the walls can reduce or enhance the particle velocity, depending on the properties of the particle–solution system and the relative particle–wall separation distances. The boundary effects on diffusiophoresis and electrophoresis of a particle normal to two plane walls are found to be quite significant and complicated, and generally stronger than those parallel to the confining walls.
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Keywords: Diffusiophoresis; Electrophoresis; Boundary effects; Thin but polarized double layer
1. Introduction
The electrokinetic transport phenomena of charged colloidal particles suspended in electrolyte solutions are of much funda-mental and practical interest in the fields of chemical, biolog-ical, and electronic engineering and science. Electrophoresis, which results from the interaction between an applied elec-tric field and the elecelec-tric double layer surrounding a charged particle, is the most familiar example of various electrokinetic transport phenomena. The electrophoretic velocity U(0) of a uniformly charged particle in an unbounded ionic solution is simply related to the uniformly imposed electric field E∞ by the Smoluchowski equation[1–4],
(1)
U(0)=εζ
η E
∞.
* Corresponding author.
E-mail address:huan@ccms.ntu.edu.tw(H.J. Keh).
In this equation, ε and η are the dielectric permittivity and vis-cosity, respectively, of the fluid, and ζ is the zeta potential at the particle surface.
Diffusiophoresis, which is the movement of a particle in re-sponse to the macroscopic concentration gradient of a solute, can be another example of electrokinetic motions. In an un-bounded solution of a symmetrically charged electrolyte with a constant concentration gradient∇n∞, the diffusiophoretic ve-locity of a dielectric particle is[5–7]
(2) U(0)=εζ η kT Ze ∇n∞ n∞(0) α+ln cosh ¯ζ ¯ζ ,
with the dimensionless parameters
(3) α=D¯2− ¯D1 ¯ D2+ ¯D1 , (4) ¯ζ =Zeζ 4kT.
0021-9797/$ – see front matter © 2008 Elsevier Inc. All rights reserved.
Here n∞(0)is the macroscopic electrolyte concentration mea-sured at the particle center 0 in the absence of the particle, ¯D1
and ¯D2are the diffusion coefficients of the anion and cation,
re-spectively, Z is the absolute value of valences of ions, e is the charge of a proton, k is the Boltzmann constant, and T is the absolute temperature.
Diffusiophoresis of a charged particle in an electrolyte solu-tion can be considered as a linear combinasolu-tion of two effects: chemiphoresis due to the nonuniform adsorption of counteri-ons and depletion of co-icounteri-ons over the surface of the particle and electrophoresis driven by the macroscopic electric field gener-ated by the electrolyte concentration gradient. For the special case of ¯D2= ¯D1 or α= 0, the induced macroscopic electric
field disappears and Eq.(2)predicts that the particle movement (due to chemiphoresis only) is in the direction of increasing electrolyte concentration regardless of the sign of ζ and the par-ticle velocity increases monotonically with an increase in the magnitude of ζ .
As shown in Eqs.(1) and (2), the electrophoretic and diffu-siophoretic velocities of a dielectric particle having a uniform zeta potential on its surface are independent of the particle size and shape. However, their validity is based on the assumptions that the local radii of curvature of the particle are much larger than the thickness of the electric double layer surrounding the particle surface and that the effect of polarization (relaxation ef-fect) of the diffuse ions in the double layer due to nonuniform “osmotic” flow is negligible. In the past decades, important ad-vances have been made in the evaluation of the electrophoretic and diffusiophoretic velocities of dielectric particles relaxing these assumptions.
O’Brien and White[8]took the double-layer distortion from equilibrium as a perturbation to obtain numerical solutions for the electrophoretic velocity of a spherical particle of radius
a in a KCl solution which were applicable to arbitrary val-ues of ζ and κa, where κ−1 is the Debye screening length equal to (εkT /2Z2e2n∞)1/2. On the other hand, Dukhin and Derjaguin[3] obtained an analytical expression for the elec-trophoretic mobility of a dielectric sphere surrounded by a thin but polarized double layer in the solution of a symmetric electrolyte. Later, O’Brien[9] generalized this analysis to the case of electrophoretic motion of a charged sphere in the so-lution containing an arbitrary combination of electrolytes. The essence of this thin-layer polarization approach is that a thin dif-fuse layer can still transport a significant amount of electrolyte ions so as to affect the ionic transport outside the diffuse layer. The result for the electrophoretic velocity of a dielectric sphere with a thin but polarized double layer in a symmetric electrolyte solution can be expressed as[10]
(5) U0= εζ 3ηE ∞2+ c 1+ c2+ (c1− c2) ln cosh ¯ζ ¯ζ ,
where coefficients c1 and c2 are defined by Eqs. (A.6a) and
(A.6b). A comparison of Eq. (5) with the numerical results for the KCl solution[8]shows that the thin-layer polarization model is quite good over a wide range of zeta potentials when
κa >20. If|ζ | is small and κa is large, the interaction between the diffuse counterions and the particle surface is weak and the
polarization of the double layer is also weak. In the limit of (6) e2|¯ζ|/κa→ 0,
c1= c2= 1/2 and Eq.(5)reduces to the Smoluchowski
equa-tion (1). Unlike the prediction of Eq. (1), the electrophoretic velocity given by Eq.(5)in general is not a monotonic function of ¯ζ for a finite value of κa.
On the other hand, Prieve and Roman[11]obtained numeri-cal solutions of the diffusiophoretic velocity over a broad range of ¯ζ and κa for a charged sphere in concentration gradients of symmetric electrolytes (KCl or NaCl) using the method of O’Brien and White [8]. Analytical expressions for the veloc-ity of a dielectric sphere with a thin but polarized double layer undergoing diffusiophoresis in ionic solutions have also been derived[12,13], and the result for this diffusiophoretic velocity in a symmetric electrolyte solution is
U0= εζ 3η kT Ze ∇n∞ n∞(0) c1− c2 + α(2 + c1+ c2) (7) +2+ c1+ c2+ α(c1− c2) ln cosh ¯ζ ¯ζ ,
where coefficients c1 and c2 are defined by Eqs. (A.6c) and (A.6d). When κa > 20, the agreement between Eq.(7)and the numerical solutions[11]is excellent for all reasonable values of the zeta potential. In the limiting situation given by Eq.(6), the effect of double-layer polarization disappears, c1= c2= c1=
c2= 1/2, and Eq.(7)reduces to Eq.(2). Even for the case of ¯
D2= ¯D1, the particle velocity given by Eq.(7)for a finite value
of κa may not be a monotonic function of the magnitude of ¯ζ and its direction can reverse (toward lower electrolyte concen-tration) when|¯ζ| becomes large.
An examination of Eqs.(5) and (7)indicates that the effect of polarization of the diffuse layer is to decrease the particle velocity. The reason for this outcome is that the transport of the diffuse ions within the double layer reduces the local electrolyte gradient or electric field along the particle surface. Numerical calculations of Eqs.(5) and (7)show that, even when κa is as large as 300, the effect of ionic transport inside the diffuse layer cannot be ignored if|ζ| equals several kT /e.
In real situations of electrophoresis and diffusiophoresis, particles are not isolated and the surrounding fluid is exter-nally bounded by solid walls [14–17]. Thus, it is important to determine if the presence of neighboring boundaries signif-icantly affects the movement of particles. In the limiting case that Eqs. (1) and (2)are applicable, the normalized velocity field of the immense fluid that is dragged by a particle during diffusiophoresis is the same as for electrophoresis of the particle
[8]; thus, the boundary effects on electrophoresis under the sit-uation of infinitesimally thin double layer (satisfying Eq.(6)), which have been studied extensively in the past[17–24], can be taken to interpret those in diffusiophoresis.
When the polarization effect of diffuse ions in the electric double layer surrounding the particle is considered, the bound-ary effects on diffusiophoresis can be quite different from those on electrophoresis. Through the use of a boundary collocation technique, the diffusiophoretic and electrophoretic motions of a
Fig. 1. Geometrical sketch for the diffusiophoretic motion of a spherical particle perpendicular to two plane walls at an arbitrary position between them.
colloidal sphere with a thin but polarized diffuse layer perpen-dicular to a plane wall have been examined[25]. The wall effect in each transport mechanism was found to be a complicated function of the properties of the particle and ions. Recently, the diffusiophoretic and electrophoretic motions of a colloidal sphere with a thin polarized diffuse layer parallel to two plane walls at an arbitrary position between them were also inves-tigated by Chen and Keh[26]using the boundary collocation method. Numerical results of the wall correction to Eqs.(5) and (7)for the particle mobilities were presented for various values of the relative separation distances and other relevant parame-ters.
This paper is an extension of the previous works[25,26]to the situation of the diffusiophoretic and electrophoretic motions of a dielectric sphere with a thin but polarized double layer per-pendicular to two plane walls at an arbitrary position between them. The quasisteady equations of conservation applicable to each system are solved by using both the boundary-collocation technique numerically and a method of reflections analytically. The numerical solutions of the particle velocities are obtained with good convergence for various cases and agree well with the approximate analytical solutions. For the case of motions normal to a single plane wall, our results are in excellent agree-ment with those available in the literature[18,25]. Because the governing equations and boundary conditions concerning the general problem of diffusiophoresis and electrophoresis of a particle at an arbitrary position between two parallel plane walls in an arbitrary direction are linear, its solution can be obtained as a superposition of the solutions for its two subproblems: mo-tion parallel to the plane walls, which was previously examined
[26], and motion normal to the confining walls, which is con-sidered in this paper.
2. Analysis for diffusiophoresis
We consider, in this section, the quasisteady diffusiophoretic motion of a dielectric spherical particle of radius a in a solution of a symmetrically charged electrolyte perpendicular to two in-finite plane walls whose distances from the center of the particle are b and c, as shown inFig. 1. Here (ρ, φ, z) and (r, θ, φ) de-note the circular cylindrical and spherical coordinate systems, respectively, and the origin of coordinates is chosen at the par-ticle center. A linear electrolyte concentration field n∞(z) with a uniform gradient in the z direction is imposed in the ambient fluid far removed from the particle. The particle is charged uni-formly on the surface, and the thickness of the electric double
layers surrounding the particle and adjacent to the plane walls is assumed to be small in comparison with the radius of the par-ticle and the spacing between the parpar-ticle and each plane wall [i.e., κa 1, κ(b − a) 1, and κ(c − a) 1, which is valid for most practical situations where the electrolyte solution has a relatively strong ionic strength]. Hence, the fluid phase can be divided into two regions: an “inner” region defined as the thin double layer adjacent to each solid surface and an “outer” re-gion defined as the remainder of the fluid, which is electrically neutral. The purpose is to obtain the correction to Eq.(7) for the particle velocity due to the presence of the plane walls.
To determine the diffusiophoretic velocity of the particle, it is necessary to ascertain the electrochemical potential and ve-locity fields in the fluid phase.
2.1. Electrochemical potential distribution
The diffusiophoretic motion of a particle can be considered quasisteady if the Peclet and Reynolds numbers of the system are small. Hence, the conservation of each ionic species for the outer region of the fluid solution is governed by[9,12]
(8) ∇2μ
m= 0, m = 1, 2.
In the above Laplace equation, μm is the electrochemical
po-tential energy of ionic species m defined by
(9)
μm= μ0m+ kT ln nm+ zmeΦ,
where μ0
m is a constant, nm and zm are the concentration and
valence, respectively, of type m ions, and Φ is the electric potential. m equal to 1 and 2 refers to the anion and cation, re-spectively, so−z1= z2= Z > 0. Note that, in the outer region,
n1= n2= n, and both n and Φ also satisfy Laplace’s equation.
The boundary condition at the particle “surface” (outer limit of the thin double layer) can be obtained by solving for the elec-trochemical potentials in the inner region and using a matching procedure to ensure a continuous solution in the whole fluid phase, with the result[9,12]
r= a+: ∂μm ∂r = − 2 i=1 βmi 1 r2sin θ ∂ ∂θ sin θ∂μm ∂θ , (10) m= 1, 2,
where the relaxation coefficients
(11a) β11= 1 κ 4 1+3f1 Z2 e¯ζsinh ¯ζ−12f1 Z2 ( ¯ζ+ ln cosh ¯ζ) , (11b) β12= − 1 κ 12f1 Z2 ln cosh ¯ζ , (11c) β21= − 1 κ 12f2 Z2 ln cosh ¯ζ , β22= 1 κ −4 1+3f2 Z2 e−¯ζsinh ¯ζ (11d) +12f2 Z2 ( ¯ζ− ln cosh ¯ζ) .
In the above equation, fm= 2εk2T2/3ηe2D¯m and ¯ζ was
de-fined by Eq.(4). To obtain Eqs.(10) and (11), it was assumed that the concentration of each ionic species within the elec-tric double layer is related to the elecelec-tric potential energy by a Boltzmann distribution. Evidently, in the limit of Eq. (6),
β11= β12= β21= β22= 0.
Since the ionic electrochemical potentials far away from the particle approach the undisturbed distributions, we can write
(12) z= c, −b: μm= μm∞, (13) ρ→ ∞, −b < z < c: μm→ μm∞, where (14) μm∞= μ0m+ kT 1− (−1)mαln n∞
and α was defined by Eq.(3). The second term in the brackets of Eq.(14)represents the contribution from the macroscopic elec-tric field induced by the difference between cation and anion diffusion rates[6]. Note that the ionic electrochemical poten-tials have been set equal to different constants at the two parallel plane walls (which must be electrically conducting ones) to al-low a uniform electrolyte concentration gradient in their normal direction far from the particle. This situation can be gener-ated experimentally, as an example, from employing two planar electrodes on which some electrochemical reactions occur.
The electrochemical potential distributions, which are gov-erned by the linear Laplace equation, can be expressed as the superposition
(15)
μm= μm∞+ μmw+ μmp, m= 1, 2.
Here μmw is a separable solution of Eq.(8)in cylindrical
co-ordinates that represents the disturbance produced by the plane walls and is given by a Fourier–Bessel integral,
(16) μmw= ∞ 0 X(ω)eωz+ Y (ω)e−ωzωJ0(ωρ) dω,
where Jnis the Bessel function of the first kind of order n, and
X(ω)and Y (ω) are unknown functions of the separation vari-able ω. The last term on the right-hand side of Eq.(15), μmp,
is a separable solution of Eq.(8)in spherical coordinates rep-resenting the disturbance generated by the particle and is given by an infinite series in harmonics,
(17) μmp= ∞ n=0 Rmnr−n−1Pn(cos θ ),
where Pn is the Legendre polynomial of order n and Rmn are
unknown constants. Note that a solution for μm of the form
given by Eqs. (15)–(17) immediately satisfies the boundary condition at infinity in Eq.(13).
Substituting the electrochemical potential distribution μm
given by Eqs.(15)–(17)into the boundary condition (12)and applying the Hankel transform on the variable ρ lead to a so-lution for the functions X(ω) and Y (ω) in terms of the coef-ficients Rmn. After the substitution of this solution into Eqs.
(15)–(17), μmcan be expressed as (18) μm= μm∞+ ∞ n=0 Rmnδ(n1)(r, θ ), m= 1, 2,
where the function δ(n1)(r, θ )is defined by Eq.(B.1)in
Appen-dix B(in which the definite integral must be performed numer-ically).
Application of the boundary condition given by Eq.(10)to Eq.(18)yields ∞ n=0 2 m=1 RmnRm(θ )= kT |∇n∞| n∞(0) 1−2β11 a (1+ α) (19a) −2β12 a (1− α) cos θ, ∞ n=0 2 m=1 RmnRm(θ )= kT |∇n∞| n∞(0) 1−2β22 a (1− α) (19b) −2β21 a (1+ α) cos θ, where (20a) R1(θ )= 2β11 a − 1 δn(2)(a, θ )+ β11δn(4)(a, θ ), (20b) R2(θ )= β12 2 aδ (2) n (a, θ )+ δ(n4)(a, θ ) , (20c) R1(θ )= β21 2 aδ (2) n (a, θ )+ δ(n4)(a, θ ) , (20d) R2(θ )= 2β22 a − 1 δn(2)(a, θ )+ β22δ(n4)(a, θ ),
and the definitions of functions δn(2)(r, θ ) and δn(4)(r, θ ) are
given by Eqs. (B.2) and (B.4)(in which the integration must be performed numerically).
To satisfy the condition (19) exactly along the entire sur-face of the particle would require the solution of the entire infinite array of unknown constants Rmn. However, the
collo-cation method [25–27] enforces the boundary condition at a finite number of discrete points on any semicircular longitudi-nal generating arc of the sphere (from θ= 0 to π) and truncates the infinite series in Eq. (18)into a finite one. If the spheri-cal boundary is approximated by satisfying the condition(10)
at M discrete points on its generating arc, the infinite series in Eq.(18)is truncated after M terms, resulting in a system of 2M simultaneous linear algebraic equations in the truncated form of Eq.(19). This matrix equation can be numerically solved to yield the 2M unknown constants R1nand R2n required in the
truncated form of Eq.(18)for the electrochemical potential dis-tributions. The accuracy of the boundary-collocation/truncation technique can be improved to any degree by taking a sufficiently large value of M. Naturally, as M→ ∞ the truncation error vanishes and the overall accuracy of the solution depends only on the numerical integration required in evaluating the func-tions δn(2)(r, θ )and δn(4)(r, θ )in Eq.(20).
2.2. Fluid velocity distribution
Having obtained the solution for the ionic electrochemical potential distributions on the particle surface that drive the dif-fusiophoretic migration, we can now proceed to find the flow field. The fluid solution is assumed to be incompressible and Newtonian. Owing to the low Reynolds number encountered in diffusiophoresis, the fluid motion in the outer region is gov-erned by the quasisteady fourth-order differential equation for viscous axisymmetric creeping flows,
(21)
E2E2Ψ= 0,
in which the Stokes stream function Ψ is related to the compo-nents of fluid velocity v in cylindrical coordinates by (vφ= 0)
(22a) vρ= 1 ρ ∂Ψ ∂z, (22b) vz= − 1 ρ ∂Ψ ∂ρ ,
and the Stokes operator E2has the form
(23) E2= ρ ∂ ∂ρ 1 ρ ∂ ∂ρ + ∂2 ∂z2.
The boundary conditions for the fluid velocity at the particle surface[9,12], on the plane walls, and far from the particle are
(24) r= a+: v= Uez+ 2 m=1 Wm ∂μm r∂θ eθ, (25) z= c, −b: v= 0, (26) ρ→ ∞: v→ 0,
where Wmare coefficients for the apparent slip velocity (across
the thin electric double layer) at the particle surface due to the diffusioosmotic effect (a linear combination of both chemios-motic and electrooschemios-motic effects),
(27) Wm= 2ε η kT (Ze)2 (−1)m¯ζ − lncosh ¯ζ,
ezand eρare the unit vectors in cylindrical coordinates, er and eθ are the unit vectors in spherical coordinates, and U is the
dif-fusiophoretic velocity of the particle to be determined. Since the ionic electrochemical potentials are constants at the two con-ducting plane walls as set in Eq.(12), no apparent slip velocity occurs there as indicated by Eq.(25).
To solve the flow field, we express the stream function in the form
(28)
Ψ = Ψw+ Ψp.
Here Ψw is a Fourier–Bessel integral solution of Eq. (21) in
cylindrical coordinates that represents the disturbance produced by the plane walls and is given by
Ψw= ∞
0
A(ω)eωz+ B(ω)e−ωz+ C(ω)ωzeωz
(29) + D(ω)ωze−ωzρJ1(ωρ) dω,
where A(ω), B(ω), C(ω), and D(ω) are unknown functions of ω. The second part of Ψ , denoted by Ψp, is a separable
solution of Eq. (21)in spherical coordinates representing the disturbance generated by the charged sphere and is given by
(30) Ψp= ∞ n=2 Bnr−n+1+ Dnr−n+3 G−1/2n (cos θ ),
where G−1/2n is the Gegenbauer polynomial of the first kind of
order n and degree−1/2; Bn and Dnare unknown constants.
Note that the boundary condition in Eq. (26)is immediately satisfied by a solution of the form given by Eqs.(28)–(30).
Substituting the stream function Ψ given by Eqs. (28)– (30) into the boundary conditions in Eq. (25) and applying the Hankel transform on the variable ρ lead to a solution for
A(ω), B(ω), C(ω), and D(ω) in terms of the coefficients Bn
and Dn. After the substitution of this solution into Eqs.(28)–
(30), the fluid velocity components can be expressed as
(31a) vρ= ∞ n=2 Bnγ1n(1)(r, θ )+ Dnγ2n(1)(r, θ ) , (31b) vz= ∞ n=2 Bnγ1n(2)(r, θ )+ Dnγ2n(2)(r, θ ) ,
where the definitions of functions γin(j )(r, θ )for i and j equal to 1 or 2 are given by Eqs.(B.5) and (B.6)(in which the integration must be performed numerically).
The only boundary condition that remains to be satisfied is that on the particle surface. Substituting Eqs.(18) and (31)into Eq.(24), one obtains
(32a) ∞ n=2 Bnγ1n(1)(a, θ )+ Dnγ2n(1)(a, θ ) = 2 m=1 WmFm(θ )cos θ, (32b) ∞ n=2 Bnγ1n(2)(a, θ )+ Dnγ2n(2)(a, θ ) = U − 2 m=1 WmFm(θ )sin θ, where (33) Fm(θ )= −kT 1− (−1)mα|∇n ∞| n∞(0)sin θ+ ∞ n=0 Rmnδ(n3)(a, θ )
and the definition of the function δn(3)(r, θ )is given by Eq.(B.3).
The first 2M coefficients Rmn have been determined through
the procedure given in the previous subsection.
Equation (32) can be satisfied by utilizing the boundary-collocation technique presented for the solution of the electro-chemical potential field. Along a longitudinal generating arc at the particle surface, Eq.(32)is applied at N discrete points (val-ues of θ between 0 and π ) and the infinite series in Eq.(31)are truncated after N terms. This generates a set of 2N linear al-gebraic equations for the 2N unknown coefficients Bnand Dn.
The fluid velocity field is completely obtained once these coef-ficients are solved for a sufficiently large number of N .
2.3. Derivation of the particle velocity
The hydrodynamic force acting on the spherical particle can be determined from[27,28]
(34)
F= 4πηD2.
This expression shows that only the lowest-order coefficient D2
contributes to the drag force exerted on the particle by the fluid. Since the particle is freely suspended in the surrounding fluid, the net force acting on the particle must vanish. Apply-ing this constraint to Eq.(34), one has
(35)
D2= 0.
To determine the diffusiophoretic velocity U of the particle, Eq.(35)and the 2N algebraic equations resulting from Eq.(32)
are to be solved simultaneously.
3. Results and discussion for diffusiophoresis
The numerical results for the diffusiophoretic velocity of a charged spherical particle perpendicular to two plane walls at an arbitrary position between them, obtained by using the boundary-collocation method described in the previous section, are presented in this section. The system of linear algebraic equations to be solved for the coefficients R1nand R2nis
con-structed from Eq.(19), while that for Bnand Dnis composed
of Eq.(32). All the numerical integrations to evaluate the func-tions δ(i)n and γin(j )were done by the 180-point Gauss–Laguerre
guadrature.
When specifying the points along the half-circular generat-ing arc of the spherical particle where the boundary conditions are to be exactly satisfied, the first points that should be chosen are θ= 0 and π, since these stagnation points control the gaps between the particle and the plane walls. In addition, the point
θ= π/2 which defines the projected area of the particle normal
to the direction of motion is also important. However, an exami-nation of the systems of linear algebraic equations(19) and (32)
shows that the matrix equations become singular if these points are used. To overcome this difficulty, these points are replaced by closely adjacent basic points, i.e., θ= δ, π/2 − δ, π/2 + δ, and π− δ [25–27]. Additional points along the generating arc are selected as mirror-image pairs about the equatorial plane
θ= π/2 to divide the two quarter-circular arcs of the
parti-cle into equal segments. The optimum value of δ in this work is found to be 0.01◦, with which the numerical results of the particle velocity for various values of the parameters α, f1, Z,
ζ e/kT, κa, b/(b+ c), and a/b converge satisfactorily.
3.1. Motion normal to a single plane wall
Some of the typical boundary-collocation solutions for the diffusiophoretic velocity of a dielectric sphere normal to a sin-gle plane wall (with c→ ∞) for different values of the parame-ters α, Z, ζ e/kT , and a/b are presented inTable 1. The diffu-siophoretic velocity of an identical particle in an infinite fluid,
U0, given by Eq.(7), is used to normalize the wall-corrected
Fig. 2. Plots of the normalized diffusiophoretic velocity U/U0of a spherical particle perpendicular to a plane wall (with c→ ∞) versus the separation pa-rameter a/b with Z= 1, f1= 0.2, and ζ e/kT = −5 for various values of κa. The solid curves represent the case of α= 0 and the dashed curves denote the case of α= −0.2.
values. All of the results obtained under the collocation scheme converge satisfactorily to at least the significant figures shown in the table. The accuracy and convergence behavior of the trun-cation technique is principally a function of the ratio a/b. For the difficult case with a/b= 0.999, the numbers of colloca-tion points M= N = 200 are sufficiently large to achieve this convergence. Our collocation results inTable 1can be found to agree excellently with the existing numerical solutions obtained by using a similar boundary-collocation method[25]. Note that the electrolytes associated with α= 0 and −0.2 (taking Z = 1 and f1= 0.2) in the table are very close to the aqueous solutions
of KCl and NaCl, respectively. As shown inTable 1, the particle migrates with the velocity that would exist in the absence of the wall, given by Eq.(7), as a/b→ 0. The diffusiophoretic veloc-ity then may decrease or increase, or even reverse its direction, as the particle approaches the wall (with increasing a/b), de-pending on the combination of the relevant parameters. In any case, the particle velocity vanishes at the limit a/b→ 1.
In Appendix A, an approximate analytical solution for the same diffusiophoretic motion as that considered here is also obtained by using a method of reflections. The particle veloc-ity is given by Eq.(A.28), which is a power series expansion in λ (= a/b). The values of the wall-corrected normalized par-ticle velocity calculated from this asymptotic solution, with the
O(λ9)term neglected, are also listed in Table 1for compar-ison. It can be seen that the asymptotic formula (A.28) from the method of reflections for the normalized particle mobility
U/U0agrees quite well with the collocation results as long as
λ 0.6; the errors in most cases are less than 2%. However, the
accuracy of Eq.(A.28)deteriorates rapidly, as expected, when the relative spacing between the particle and the plane wall be-comes small (say, λ 0.8). The prediction of Eq.(A.28)may overestimate or underestimate the diffusiophoretic velocity of the particle, depending on the combination of the relevant para-meters.
Table 1
Normalized diffusiophoretic velocity of a dielectric sphere perpendicular to a single plane wall (with c→ ∞) obtained from the boundary-collocation method for the case f1= 0.2 and κa = 100
a/b U/U0 α= 0 α= −0.2 Z= 1 Z= 2 Z= 1 Z= 2 ζ e/kT= 2 0.2 0.99542 (0.99542) 0.99575 (0.99575) 0.99900 (0.99899) 0.99629 (0.99629) 0.4 0.96340 (0.96333) 0.96616 (0.96604) 0.99363 (0.99297) 0.97066 (0.97047) 0.6 0.87088 (0.87069) 0.88114 (0.87986) 0.98667 (0.97092) 0.89819 (0.89482) 0.8 0.65020 (0.66431) 0.67774 (0.68129) 0.97061 (0.83775) 0.72431 (0.70793) 0.9 0.43157 (0.49515) 0.46989 (0.50975) 0.86353 (0.61297) 0.53351 (0.52998) 0.95 0.26107 (0.38991) 0.29872 (0.39853) 0.64904 (0.41310) 0.35809 (0.40651) 0.99 0.06415 (0.29505) 0.08142 (0.29507) 0.20075 (0.18983) 0.10512 (0.28608) 0.995 0.03314 0.04341 0.10741 0.05692 0.999 0.0068 0.0093 0.0228 0.0124 ζ e/kT= 5 0.2 0.99628 (0.99628) 1.00286 (1.00286) 0.99697 (0.99697) 0.94469 (0.94471) 0.4 0.97053 (0.97037) 1.02472 (1.02412) 0.97634 (0.97609) 0.54463 (0.54846) 0.6 0.89716 (0.89450) 1.09373 (1.07872) 0.91883 (0.91384) −0.65639 (−0.54796) 0.8 0.72012 (0.70992) 1.24760 (1.10887) 0.77856 (0.74581) −3.45318 (−2.33078) 0.9 0.53023 (0.53827) 1.32572 (1.01832) 0.61143 (0.56930) −5.66735 (−2.95219) 0.95 0.36136 (0.42113) 1.27559 (0.90452) 0.44068 (0.43969) −6.56694 (−2.91781) 0.99 0.11668 (0.30816) 0.89325 (0.76140) 0.15405 (0.30872) −5.35464 (−2.58919) 0.995 0.06661 0.71943 0.08961 −4.41782 0.999 0.0164 0.4096 0.0226 −2.5812
Note. The figures in parentheses are asymptotic solutions calculated using Eq.(A.28). Table 2
Normalized diffusiophoretic velocity of a dielectric sphere perpendicular to two equally distant plane walls (with c= b) obtained from the boundary-collocation method for the case f1= 0.2 and κa = 100
a/b U/U0 α= 0 α= −0.2 Z= 1 Z= 2 Z= 1 Z= 2 ζ e/kT= 2 0.2 0.99232 (0.99233) 0.99310 (0.99311) 1.00076 (1.00076) 0.99437 (0.99437) 0.4 0.94181 (0.94257) 0.94775 (0.94842) 1.00582 (1.00514) 0.95737 (0.95784) 0.6 0.81559 (0.82975) 0.83379 (0.84624) 1.01279 (0.99996) 0.86334 (0.87227) 0.8 0.57210 (0.67553) 0.60930 (0.70219) 0.98474 (0.91849) 0.67050 (0.74153) 0.9 0.36698 (0.60439) 0.41151 (0.63192) 0.85040 (0.81012) 0.48386 (0.66867) 0.95 0.21849 (0.57587) 0.25896 (0.60203) 0.62403 (0.73151) 0.32181 (0.63354) 0.99 0.05313 (0.55808) 0.07050 (0.58216) 0.18838 (0.65559) 0.09416 (0.60726) 0.995 0.02742 0.03765 0.10045 0.05103 0.999 0.00566 0.00813 0.02130 0.01120 ζ e/kT= 5 0.2 0.99436 (0.99436) 1.01004 (1.01004) 0.99600 (0.99600) 0.87160 (0.87164) 0.4 0.95735 (0.95790) 1.07979 (1.07948) 0.96988 (0.97024) 0.00308 (0.01193) 0.6 0.86336 (0.87387) 1.26036 (1.25681) 0.90193 (0.90874) −2.20226 (−2.05070) 0.8 0.66984 (0.75174) 1.56955 (1.58573) 0.74932 (0.80875) −6.16959 (−5.35636) 0.9 0.48527 (0.68985) 1.70590 (1.85084) 0.58045 (0.74960) −8.61952 (−7.54210) 0.95 0.32920 (0.66306) 1.64942 (2.03081) 0.41533 (0.72061) −9.28579 (−8.85757) 0.99 0.10698 (0.64524) 1.16253 (2.20893) 0.14498 (0.69912) −7.15545 (−10.0748) 0.995 0.06133 0.93865 0.08451 −5.85974 0.999 0.01523 0.53734 0.02147 −3.4055
Note. The figures in parentheses are asymptotic solutions calculated using Eq.(A.38).
Some numerical solutions for the normalized diffusio-phoretic velocity U/U0of a spherical particle normal to a plane
wall as functions of a/b are depicted inFig. 2for various val-ues of the parameters α and κa. It can be seen that U/U0
decreases with an increase in κa for the otherwise specified
condition. This behavior is expected knowing that the elec-trochemical potential gradients on the particle surface near a normal plane wall decrease as κa (or G) increases (see the analysis inAppendix A). Under the situation of large κa, the diffusiophoretic mobility of the particle near the plane wall is
(a) (b)
Fig. 3. Plots of the normalized diffusiophoretic velocity U/U0of a spherical particle perpendicular to two plane walls with b/(b+ c) = 0.5 (solid curves) and b/(b+c) = 0.25 (dashed curves) versus the separation parameter a/b with f1= 0.2 and κa = 100 for various values of Z: (a) α = 0 and ζ e/kT = ±5; (b) α = −0.2 and ζ e/kT= −5.
a monotonic decreasing function of a/b. In the limiting case of κa→ ∞, the diffusiophoretic mobility of the particle is identical to the prediction by ignoring the polarization effect of the diffuse layer[20]. On the contrary, under the situation of relatively small κa (e.g., with κa= 20), the diffusiophoretic mobility of the particle increases with an increase in a/b as a/b is small or moderate, but decreases from a maximum with in-creasing a/b as a/b is sufficiently large. This interesting feature that U/U0may not be a monotonic decreasing function of a/b
and can even be greater than unity is understood because the wall effect of hydrodynamic resistance on the particle is in com-petition with the wall effect of electrochemical enhancement when a particle with small κa is undergoing diffusiophoretic motion normal to a plane wall.
3.2. Motion perpendicular to two plane walls
The diffusiophoretic velocity of a dielectric sphere normal to two plane wall is a complicated function of the properties of the particle and suspending solution as well as the relative separation distances. InTable 2, the collocation solutions for the normalized velocity U/U0 of a spherical particle located
between two parallel plane walls whose distance to one wall is the same as to the other (with c= b) undergoing diffusio-phoresis perpendicularly are presented for various values of the parameters α, Z, ζ e/kT , and a/b. The corresponding method-of-reflection solutions, given by Eq.(A.38)inAppendix Aas a power series expansion in λ (= a/b) correct to O(λ8), are also
listed in this table for comparison. Similar to the case of diffu-siophoresis of a spherical particle normal to a single plane wall considered in the previous subsection, the approximate analyti-cal formula(A.38)agrees quite well with the collocation results as long as λ < 0.6, but can have significant errors when λ > 0.6. A comparison betweenTable 2for the case of a slit andTable 1
for the case of a single normal plane indicates that the assump-tion that the boundary effect for two walls can be obtained by simple addition of single-wall effects can lead to a greater or smaller correction to diffusiophoretic motion, depending on the combination of the relevant parameters.
Numerical results of the normalized diffusiophoretic ve-locity U/U0 of a spherical particle perpendicular to two
plane walls at two particular positions between them (with
b/(b + c) = 0.25 and 0.5) obtained using the
boundary-collocation method are plotted as functions of a/b and Z in
Fig. 3 with f1= 0.2 and κa = 100. For the case of α = 0
( ¯D1= ¯D2), the macroscopic electric field induced by the
elec-trolyte concentration gradient vanishes and the particle velocity, which is due to the chemiphoretic effect only, is an even func-tion of zeta potential. It can be seen that, for a specified value of b/(b+ c), U/U0 is not necessarily a monotonic function
of a/b. For some cases, the particle will reverse the direction of diffusiophoresis and the magnitude of its normalized velocity can be dramatically varied when the separation distance be-tween the particle and the plane walls is changed. In general, no simple rule could appropriately predict the boundary ef-fect on diffusiophoresis. As expected, for a given value of a/b, the boundary effect on the diffusiophoretic particle is stronger for the case of b/(b+ c) = 0.5 than that of b/(b + c) = 0.25. Although it is difficult to obtain convergent results of the par-ticle velocity for situations of very small separation distances (a/b > 0.999), both physical reasoning and numerical tendency indicate that U/U0→ 0 as a/b → 1.
In Figs. 4 and 5, the collocation solutions for the normal-ized velocity U/U0 of a spherical particle situated midway
between two parallel plane walls (with c= b) undergoing dif-fusiophoresis perpendicularly are plotted versus ζ e/kT in the range from−8 to 8 for cases of α = 0 and −0.2, respectively, with a/b= 0.6 and various values of κa. Note that U = U0= 0
(a) (b)
(c)
Fig. 4. Plots of the normalized diffusiophoretic velocity U/U0of a spherical particle perpendicular to two equally distant plane walls (c= b) versus ζ e/kT with a/b= 0.6 and f1= f2= 0.2 (α = 0) for various values of κa: (a) Z = 1; (b) Z = 2; (c) Z = 3.
at ζ e/kT = 0. As expected, when the value of κa is large and the magnitude of ζ e/kT is small to satisfy Eq.(6), the diffusio-phoretic mobility of the particle approaches the value calculated by ignoring the polarization effect of the double layers (shown by a dashed line labeled with κa→ ∞ in each figure). These figures illustrate that the difference of the results obtained from the presence and absence of the polarization effect can be quite substantial and needs to be considered. For given values of Z and κa, there can be a maximum and a minimum of the nor-malized particle mobility occurring at an associated value of
ζ e/kT (in addition to the one with small magnitude appear-ing for the case of α = 0). If the value of κa is smaller or the counterions have a larger magnitude of valence, the locations of these extremes will shift toward a smaller magnitude of ζ e/kT . Because the direction of diffusiophoresis of an isolated spher-ical particle can reverse with the variation of Z, ζ e/kT , or κa
[11–13], these extremes arise at combinations of Z, ζ e/kT , and
κain which U0→ 0.
A careful comparison of the results inFigs. 3–5(orTable 2) for the case of a slit with those inFig. 2(orTable 1) for the case of a single wall reveals an interesting feature of the boundary ef-fect on diffusiophoresis of a colloidal sphere. The presence of a second, identical, normal plane wall, even at a symmetric posi-tion with respect to the sphere against the first, does not always enhance the wall effect on the diffusiophoretic particle induced by the first plane only. This outcome reflects again the fact that the confining wall can affect the electrochemical driving force and the viscous drag force on a particle in opposite directions. Each force is increased in its own direction as the value of a/b turns large, but to a different degree, for the case of diffusio-phoretic motion of a particle in a slit relative to that for the case of migration normal to a single plate. Thus, the net effect
(a) (b)
(c)
Fig. 5. Plots of the normalized diffusiophoretic velocity U/U0of a spherical particle perpendicular to two equally distant plant walls (c= b) versus ζ e/kT with a/b= 0.6, f1= 0.2 and α = −0.2 for various values of κa: (a) Z = 1; (b) Z = 2; (c) Z = 3.
composed of these two opposite forces for the slit case is not necessarily to enhance that for the case of a single wall.
InFig. 6, the collocation results for the normalized diffusio-phoretic velocity U/U0of a charged sphere normal to two plane
walls at various positions between them are plotted for some cases. The dashed curves (with a/b= constant) illustrate the effect of the position of the second wall (at z= c) on the parti-cle velocity for various values of the relative sphere-to-first-wall spacing b/a. The solid curves [with 2a/(b+ c) = constant] indicate the variation of the particle velocity as a function of the sphere position at various values of the relative wall-to-wall spacing (b+ c)/2a. It can be seen that the net wall effect can reduce or enhance the diffusiophoretic mobility U/U0 of the
particle. At a constant value of 2a/(b+ c), the particle expe-riences a minimum viscous drag force and in general has a
greatest velocity when it is located midway between the two walls (with c= b), as indicated in the cases ofFigs. 6a and 6c; the hydrodynamic drag increases and the diffusiophoretic ve-locity decreases as the particle approaches either of the walls [or the ratio b/(b+ c) decreases]. Interestingly, as illustrated in
Fig. 6a, at some specified values of a/b for the diffusiophoretic particle near a first wall, the presence and approach of a second plate can increase the velocities of the particle when it is far from the particle (c is large), and then reduce the particle ve-locity when it is close to the particle. On the other hand, for the case shown inFig. 6b, the net wall effect is to increase the dif-fusiophoretic mobility U/U0of the particle; at a constant value
of 2a/(b+ c), the normalized particle mobility has a smallest value when it is located midway between the two walls, where the particle experiences a minimum effect of electrochemical
(a) (b)
(c)
Fig. 6. Plots of the normalized diffusiophoretic velocity U/U0of a spherical particle perpendicular to two plane walls versus b/(b+ c) with a/b and 2a/(b + c) as parameters at f1= f2= 0.2 (α = 0), ζ e/kT = ±5, and κa = 100: (a) Z = 1; (b) Z = 2; (c) Z = 3.
enhancement, and becomes larger when it approaches either of the walls.
4. Electrophoresis
Considered in this section is the quasisteady electrophoretic motion of a dielectric sphere in a uniformly applied electric field E∞= E∞ezperpendicular to two large plane walls at an
arbitrary position between them. The velocity of the particle caused by the field is U= Uez. The bulk concentration n∞of
the symmetric electrolyte in the fluid beyond the electric double layer is constant now. Like the analysis in Section2, the thick-ness of the double layer is assumed to be much smaller than the radius of the particle and the surface-to-surface distance be-tween the particle and each wall, but the polarization effect in the thin diffuse layer is incorporated.
Outside the double layer, the electrochemical potentials μm
of the ions still satisfy Laplace’s equation(8)and boundary con-ditions(10)–(13), but their undisturbed values in Eq. (14)are replaced by
(36)
μm∞= μ0m+ kT ln n∞− (−1)mZeE∞z.
The solution for μm in this case can still be expressed as
Eq.(18)with coefficients Rmndetermined by
(37a) ∞ n=0 2 m=1 RmnRm(θ )= ZeE∞ 1−2β11 a + 2β12 a cos θ, (37b) ∞ n=0 2 m=1 RmnRm(θ )= −ZeE∞ 1+2β21 a − 2β22 a cos θ,
Table 3
Normalized electrophoretic velocity of a dielectric sphere perpendicular to a single plane wall (with c→ ∞) obtained from the boundary-collocation method for the case f1= f2= 0.2 and κa = 100
a/b U/U0 Z= 1 Z= 2 ζ e/kT= 2 0.2 0.99511 (0.99511) 0.99528 (0.99528) 0.4 0.96076 (0.96073) 0.96224 (0.96220) 0.6 0.86075 (0.86192) 0.86633 (0.86686) 0.8 0.62218 (0.64915) 0.63725 (0.65814) 0.9 0.39379 (0.48486) 0.41460 (0.49218) 0.95 0.22716 (0.38791) 0.24713 (0.39164) 0.99 0.05222 (0.30429) 0.06084 (0.30294) 0.995 0.02666 0.03169 0.999 0.00540 0.00660 ζ e/kT= 5 0.2 0.99557 (0.99557) 0.99639 (0.99639) 0.4 0.96460 (0.96453) 0.97135 (0.97124) 0.6 0.87502 (0.87475) 0.89916 (0.89788) 0.8 0.66040 (0.67333) 0.72497 (0.72654) 0.9 0.44731 (0.50674) 0.54830 (0.57710) 0.95 0.28047 (0.40243) 0.40387 (0.47989) 0.99 0.07873 (0.30794) 0.19900 (0.38934) 0.995 0.04327 0.14864 0.999 0.0100 0.0774
Note. The figures in parentheses are asymptotic solutions calculated using Eq.(A.28).
to replace Eq.(19). The governing equation, boundary condi-tions, and solution for the fluid velocity field have the same forms as those given by Eqs.(21)–(31). The final results for the electrophoretic velocity of the particle can be determined by the simultaneous solution of Eqs.(32) and (35), with
(38) Fm(θ )= (−1)mZeE∞sin θ+ ∞ n=0 Rmnδn(3)(a, θ ), in replacement of Eq.(33).
Some converged collocation solutions for the electrophoretic velocity of a dielectric sphere (which is an odd function of the parameter ζ e/kT ) normal to a plane wall (with c→ ∞) for various values of a/b are presented inTable 3, while the corre-sponding results for the electrophoretic velocity of the particle located at the median plane between two parallel plates (with
c= b) are given in Table 4. Now, the electrophoretic veloc-ity of the particle in an unbounded fluid given by Eq. (5) is used to normalize the wall-corrected values. The corresponding method-of-reflection solutions, given by Eqs.(A.28) and (A.38)
with the parameters defined by Eqs.(A.29) and (A.30)in Ap-pendix Aas power series expansions in λ (= a/b) correct to
O(λ8), are also listed in these tables for comparison. Analo-gous to the cases of diffusiophoresis considered in the previous section, the asymptotic formulas(A.28) and (A.38)for U/U0
agree quite well with the collocation results as long as λ 0.6, but can have significant errors for greater values of λ.
The collocation results for the normalized electrophoretic mobility U/U0 of a spherical particle normal to a plane wall
and the corresponding results for the particle undergoing elec-trophoresis at the median plane between two parallel plane
Table 4
Normalized electrophoretic velocity of a dielectric sphere perpendicular to two equally distant plane walls (with c= b) obtained from the boundary-collocation method for the case f1= f2= 0.2 and κa = 100
a/b U/U0 Z= 1 Z= 2 ζ e/kT= 2 0.2 0.99158 (0.99159) 0.99200 (0.99201) 0.4 0.93621 (0.93710) 0.93940 (0.94023) 0.6 0.79834 (0.81487) 0.80813 (0.82362) 0.8 0.53601 (0.65430) 0.55612 (0.66803) 0.9 0.32471 (0.58644) 0.34864 (0.60002) 0.95 0.18305 (0.56232) 0.20437 (0.57471) 0.99 0.04132 (0.54962) 0.04997 (0.56043) 0.995 0.02105 0.02605 0.999 0.00429 0.00547 ζ e/kT= 5 0.2 0.99268 (0.99268) 0.99465 (0.99465) 0.4 0.94456 (0.94532) 0.96010 (0.96081) 0.6 0.82400 (0.83831) 0.87544 (0.88916) 0.8 0.58870 (0.69370) 0.70927 (0.81423) 0.9 0.38813 (0.62920) 0.55821 (0.80712) 0.95 0.24141 (0.60478) 0.43411 (0.82104) 0.99 0.06839 (0.59083) 0.23830 (0.84422) 0.995 0.03783 0.18331 0.999 0.00892 0.09930
Note. The figures in parentheses are asymptotic solutions calculated using Eq.(A.38).
walls in the normal direction as functions of a/b are plotted in Fig. 7 for various values of κa and ζ e/kT (U/U0 is an
even function of ζ e/kT ). In either case, the value of U/U0is
a monotonic decreasing function of a/b. It appears that U/U0
decreases with an increase in κa and with a decrease in the magnitude of ζ e/kT . As expected, the presence of a second normal plane wall in general enhances the wall effect on the electrophoretic particle induced by the first plate only.
InFig. 8, the normalized electrophoretic mobility U/U0of
a spherical particle at the median plane between two parallel plane walls in the normal direction is plotted versus the parti-cle’s zeta potential for three cases of κa when the separation pa-rameter a/b is kept constant. When Z= 1, as shown inFig. 8a,
U/U0 is a monotonic increasing function of the magnitude of
the nondimensional zeta potential ζ e/kT ranging from 0 to 8 and the boundary effect is weakened (U/U0 is larger) as the
value of κa becomes smaller. However, when Z= 2 or 3 as il-lustrated in Figs. 8b and 8c, a maximum of U/U0 may exist
at a magnitude of ζ e/kT for a specified κa or at a value of
κafor a given ζ e/kT . If the specified κa increases, the maxi-mum occurs at a larger magnitude of ζ e/kT ; if the magnitude of the given ζ e/kT increases, the maximum occurs at a larger
κa. If the counterions have a larger magnitude of valence, the location of this maximum will shift toward a smaller magnitude of ζ e/kT or larger κa. Similar to the case of diffusiophoresis considered in the previous section, no simple rule could ap-propriately describe the boundary effects on the electrophoretic mobility of the particle, which are dependent on the combina-tion of ζ e/kT , κa, Z, f1, f2, and a/b.
In Fig. 9, collocation results for the normalized elec-trophoretic velocity U/U0of a dielectric sphere perpendicular
(a) (b)
Fig. 7. Plots of the normalized electrophoretic velocity U/U0of a spherical particle perpendicular to two plane walls (solid curves) and to a single plane wall (dashed curves) versus the separation parameter a/b with Z= 1 and f1= f2= 0.2: (a) ζ e/kT = ±5; (b) κa = 100.
to two plane walls at various positions between them are dis-played. It can be seen that the particle experiences a minimum viscous drag and has a greatest electrophoretic velocity when it is located midway between the two walls (with c= b). At a rel-atively small value of a/b for the electrophoretic particle near a first normal wall, the presence of a second plate is to further reduce the velocity of the particle, and the degree of this re-duction increases monotonically with a decrease in the relative distance between the particle and the second plate [or with an increase in b/(b+ c)]. For a specified value of a/b greater than about 0.8, however, the presence of a second plane wall is not necessarily to further reduce the particle velocity.
5. Conclusions
The numerical solutions and approximate analytical solu-tions for the quasisteady diffusiophoretic and electrophoretic motions of a charged sphere perpendicular to two infinite plane walls at an arbitrary position between them have been obtained in this work by using the boundary-collocation technique and the method of reflections, respectively, in the limit of vanish-ingly small Reynolds and Peclet numbers. It has been found that the boundary effects on these phoretic motions of a particle are quite significant and are complicated functions of the prop-erties of the particle and surrounding ions (ζ e/kT , κa, Z, f1,
and f2) and the separation distances (a/b and a/c). The
diffu-siophoretic or electrophoretic mobility of a particle near a wall is generally, but not necessarily, a monotonic decreasing func-tion of the separafunc-tion parameter a/b. When the value of a/b is sufficiently large, the effect of a confining wall can speed up or slow down the particle velocity relative to its isolated value depending on the values of the relevant parameters of the particle–electrolyte system. This behavior reflects the competi-tion between the relatively weak hydrodynamic retardacompeti-tion ex-erted by the neighboring wall on the particle migration and the
possible, strong phoretic enhancement due to the electrochem-ical interaction between the particle and the wall. No general rule can make an adequate prediction for such complicated phe-nomena present in the boundary effects on diffusiophoresis and electrophoresis.
The diffusiophoretic and electrophoretic mobilities of a spherical particle parallel to two infinite plane walls at an ar-bitrary position between them were calculated in a previous work[26]for various values of the parameters α, ζ e/kT , κa,
a/b, and b/(b+ c). It was also found that the particle mo-bilities may decrease or increase with increasing a/b and the particle can migrate faster than it would as a/b= 0, depend-ing the electrochemical boundary condition at the walls and the combination of the relevant parameters. The effect of viscous retardation is stronger and the effect of possible electrochem-ical enhancement is weaker in a transverse diffusiophoresis or electrophoresis than in a parallel motion. In general, the net boundary effects on diffusiophoresis and electrophoresis of a particle are stronger for the perpendicular migration. For the general problem of a particle undergoing diffusiophoresis or electrophoresis in an arbitrary direction with respect to the two parallel plane walls, the solution can be obtained by adding both the parallel and the transverse results vectorially.
Acknowledgment
Part of this research was supported by the National Science Council of the Republic of China.
Appendix A. Analysis of the diffusiophoresis and
electrophoresis of a spherical particle perpendicular to one or two plane walls by a method of reflections
In this appendix, the quasisteady diffusiophoretic and elec-trophoretic motions of a dielectric sphere in the solution of a
(a) (b)
(c)
Fig. 8. Plots of the normalized electrophoretic velocity U/U0of a spherical particle perpendicular to two equally distant plane walls (c= b) versus ζ e/kT with a/b= 0.6 and f1= f2= 0.2 for various values of κa: (a) Z = 1; (b) Z = 2; (c) Z = 3.
symmetric electrolyte perpendicular either to an infinite plane wall (with c→ ∞) or to two parallel plane walls with equal distances from the particle (c= b), as shown inFig. 1, will be analyzed using a method of reflections. The effect of the walls on the particle velocity U is sought in expansions of λ, which equals a/b, the ratio of the particle radius to the distance be-tween the particle center and the walls.
A.1. Motion normal to a single plane wall
For the problem of diffusiophoretic motion of a spherical particle driven by a uniform electrolyte concentration gradient ∇n∞ normal to an infinite plane wall, the governing
equa-tions(8) and (21)must be solved by satisfying the boundary conditions (10), (12), (13), and (24)–(26) with c→ ∞. The method-of-reflection solution for the ionic electrochemical
po-tential and velocity fields in the fluid phase consists of the fol-lowing series, whose terms depend on increasing powers of λ,
(A.1a)
μm= μm∞+ μ(m1)p+ μm(1)w+ μ(m2)p+ μ(m2)w+ · · · ,
(A.1b)
v= v(p1)+ v(w1)+ v(p2)+ v(w2)+ · · · ,
where the subscripts p and w represent the reflections from the particle and wall, respectively, and the superscript (i) denotes the ith reflection from that surface. In these series, all the ex-pansion sets of the electrochemical potential and velocity fields must satisfy Eqs.(8) and (21).
According to Eq.(A.1), the diffusiophoretic velocity of the particle can also be expressed in the series form,
(A.2)
(a) (b)
Fig. 9. Plots of the normalized electrophoretic velocity U/U0of a spherical particle perpendicular to two plane walls versus b/(b+ c) with a/b and 2a/(b + c) as parameters at f1= f2= 0.2, ζ e/kT = ±5, and κa = 100: (a) Z = 1; (b) Z = 2.
In this expression, U0is the diffusiophoretic velocity of an
iden-tical particle in the corresponding unbounded continuous phase given by Eq.(7); U(i)is related to∇μ(i)mwand vw(i)by the Faxen
law[28,29], (A.3) U(i)= 2 m=1 Gm ∇μ(i) mw 0+ v(i)w0+a 2 6 ∇2v(i) w 0,
where the subscript 0 to variables inside brackets denotes eval-uation at the position of the particle center. In Eq.(A.3),
(A.4a) G1= 4εkT 3η(Ze)2 (1+ g11− g21) ¯ζ+ (1 + g11+ g21)ln cosh ¯ζ , (A.4b) G2= 4εkT 3η(Ze)2 (−1 + g12− g22) ¯ζ+ (1 + g12+ g22)ln cosh ¯ζ ,
where ¯ζ is defined by Eq.(4), and
(A.5a) g11= 1 2 c1 + c1 , (A.5b) g12= 1 2 c1 − c1 , (A.5c) g21= 1 2 c2 − c2 , (A.5d) g22= 1 2 c2 + c2 . In Eq.(A.5), c1= 1 2a2 1
a2− 2aβ11+ 3aβ12+ aβ22+ 2β12β21
(A.6a) − 2β11β22 , c2= 1 2a2 1
a2− 2aβ22+ 3aβ21+ aβ11+ 2β12β21
(A.6b) − 2β11β22 , (A.6c) c1= c1− 3 β12 a1 , (A.6d) c2= c2− 3 β21 a1 , where (A.7) 1= 1 a2 a2+ aβ11+ aβ22− β12β21+ β11β22 ,
and the relaxation coefficients β11, β12, β21, and β22 were
de-fined by Eq. (11). In the limit of Eq. (6) or zero relaxation coefficients, Eqs. (A.5) and (A.6) reduce to c1= c2= c1 =
c2= g11= g22 = 1/2 and g12= g21= 0. In the other limit
of very large relaxation coefficients, these equations become
c1= c2= c1= c2 = g11= g22= −1 and g12= g21= 0.
The solution for the first reflected fields from the particle is
(A.8a) μ(m1)p= a3r−2cos θ 2 i=1 gmi|∇μi∞|, (A.8b) v(p1)=1 2U0a 3r−3(
2 cos θ er+ sin θeθ).
The velocity distribution shown in Eq. (A.8b) is identical to the irrotational flow surrounding a rigid sphere moving with velocity U0ez.
The boundary conditions for the ith reflected fields from the wall are derived from Eqs.(12), (13), (25), and (26),
(A.9a)
z= −b: μ(i)mw= −μ(i)mp,
(A.9b)
(A.9c)
r→ ∞, z > −b: μ(i)mw→ 0,
(A.9d)
v(i)w → 0.
The solution of μ(m1)wis obtained by applying Hankel transforms
on variable ρ in Eqs.(8) and (A.9a and A.9c)(taking i= 1), with the result
(A.10) μ(m1)w= a3(2b+ z)(2b+ z)2+ ρ2−3/2 2 i=1 gmi|∇μi∞|.
This reflected electrochemical potential field may be interpreted as arising from the reflection of the imposed field from a ficti-tious particle identical to the actual particle, its location being at the mirror-image position of the actual particle with respect to the plane z= −b (i.e., at ρ = 0, z = −2b). The solution of v(w1) can also be obtained by applying Hankel transforms to
the Stokes equation(21)twice and to the boundary conditions
(A.9b and A.9d), which results in
v(w1)= −1 2U0a 3 ∞ 0 ω2E(ω, z)J1(ωρ)eρ (A.11) + F (ω, z)J0(ωρ)ez dω, where (A.12a) E(ω, z)=2(b+ z)ω − 1e−ω(z+2b), (A.12b) F (ω, z)=2(b+ z)ω + 1e−ω(z+2b).
The contributions of μ(m1)w and v(w1) to the particle velocity are
determined using Eq.(A.3),
(A.13a) U(s1)= 2 m=1 Gm ∇μ(1) mw r=0= − 1 4Gλ 3U 0ez, (A.13b) U(h1)= v(w1)+a 2 6 ∇ 2 v(w1) r=0 = −1 4 2λ3− λ5U0ez, (A.13c) U(1)= U(s1)+ U(h1)=1 4 −(2 + G)λ3+ λ5U 0ez, where (A.14) G= 1 U0 2 m=1 2 i=1 Gmgmi|∇μi∞|.
After the substitution of Eq.(14)for μi∞and Eq.(A.5)for gmi,
Eq.(A.14)becomes
(A.15) G=kT|∇n ∞| U0n∞(0) 2 m=1 Gmbm, where (A.16) bm= cm− (−1)mcmα.
Equation(A.13a)shows that the reflected electrochemical po-tential field from the plane wall can decrease (if G > 0) or increase (if G < 0) the diffusiophoretic velocity of the parti-cle from its undisturbed value, while Eq.(A.13b)indicates that
the reflected velocity field is to decrease this velocity; the net effect of the reflected fields is expressed by Eq.(A.13c), which can enhance or retard the movement of the particle, depending on the combination of the values of G and λ. When G= 0, the reflected electrochemical potential field makes no contribution to the diffusiophoretic velocity. Eq. (A.13c)indicates that the wall correction to the velocity of the diffusiophoretic particle is O(λ3), which is weaker than that obtained for the corre-sponding sedimentation problem, in which the leading bound-ary effect is O(λ). Note that the necessbound-ary condition for the wall enhancement on the diffusiophoretic motion to occur is a small value of G and/or a value of λ close to unity such that the relation λ5> (2+ G)λ3is warranted.
The solution for the second reflected fields from the particle is μ(m2)p= −1 4λ 3a3r−2cos θ 2 i=1 2 j=1 gmjgj i|∇μi∞| + 3 16λ 4a4r−33 cos2θ− 1 2 i=1 2 j=1 hmjgj i|∇μi∞| (A.17a) + Oλ5a5, v(p2)= U0 −1 8Gλ 3a3r−3(
2 cos θ er+ sin θeθ)
− 9 64(5− 4H )λ 4a2r−2 3 cos2θ− 1er − 9 16H λ 4a4r−43 cos2θ− 1e
r+ 2 sin θ cos θeθ
+ 9
64λ
4a4r−4
9 cos2θ− 4er+ 54 sin θ cos θeθ
(A.17b) + Oλ6a4 , where (A.18) H= 1 U0 2 m=1 2 i=1 Hmgmi|∇μi∞|, with H1= − εkT η(Ze)2 (1+ 2h11− 2h21) ¯ζ (A.19a) + (1 + 2h11+ 2h21)ln cosh ¯ζ , H2= − εkT η(Ze)2 (−1 + 2h12− 2h22) ¯ζ (A.19b) + (1 + 2h12+ 2h22)ln cosh ¯ζ .
In Eqs.(A.17a) and (A.19),
(A.20a) h11= 1 2 d1+ d1 , (A.20b) h12= 1 2 d1− d1 , (A.20c) h21= 1 2 d2− d2 ,
(A.20d) h22= 1 2(d 2+ d2), where d1= 1 3a2 2
a2− 3aβ11+ 5aβ12+ 2aβ22+ 6β12β21
(A.21a) − 6β11β22 , d2= 1 3a2 2
a2− 3aβ22+ 5aβ21+ 2aβ11+ 6β12β21
(A.21b) − 6β11β22 , (A.21c) d1 = d1− 10β12 3a2 , (A.21d) d2 = d2− 10β21 3a2 , and (A.22) 2= 1 a2 a2+ 2aβ11+ 2aβ22− 4β12β21+ 4β11β22 .
The boundary conditions for the second reflected fields from the wall are obtained by substituting the result of μ(m2)pand v(p2)
into Eq.(A.9), with which Eqs.(8) and (21)can be solved as before to yield ∇μ(2) mw r=0= 1 256 16λ6 2 i=1 2 j=1 gmjgj i|∇μi∞| (A.23a) + 18λ8 2 i=1 2 j=1 hmjgj i|∇μi∞| + O λ9 ez, v(w2)+ a2 6 ∇ 2 v(w2) r=0 = U0 1 256 32G− 27(5 − 4H)λ6 + 1 512 27(5− 4H) − 32G − 180H (A.23b) − 99λ8+ Oλ9 ez.
The contribution of the second reflected fields to the particle velocity is obtained by combining Eqs.(A.3) and (A.23), which gives U(2)= U0 1 256 −27(5 − 4H) + 16(2G + G)λ6 (A.24) + 1 128(9− 72H + 9H − 8G)λ8+ Oλ9 ez, where (A.25a) G= 1 U0 2 m=1 2 i=1 2 j=1 Gmgmjgj i|∇μi∞|, (A.25b) H= 1 U0 2 m=1 2 i=1 2 j=1 Gmhmjgj i|∇μi∞|.
After the substitution of Eq.(14)for μi∞and Eq.(A.5)for gmi,
Eqs.(A.18) and (A.25)become
(A.26) H=kT|∇n ∞| U0n∞(0) 2 m=1 Hmbm, (A.27a) G=kT|∇n ∞| U0n∞(0) 2 m=1 2 j=1 Gmgmjbj, (A.27b) H=kT|∇n ∞| U0n∞(0) 2 m=1 2 j=1 Gmhmjbj.
Evidently, U(3)will be O(λ9). With the substitution of Eqs.
(A.13c) and (A.24)into Eq.(A.2), the particle velocity can be expressed as U= Uezwith U= U0 1−1 4(2+ G)λ 3+1 4λ 5 − 1 256 27(5− 4H) − 16(2G + G)λ6 (A.28) + 1 128(9− 72H + 9H − 8G)λ8+ Oλ9 .
The particle migrates along the imposed electrolyte concentra-tion gradient at a rate that can increase or decrease as the parti-cle approaches the wall. Owing to the linearity of the problem, the above analysis is valid when the particle is either approach-ing the plane wall or recedapproach-ing from it.
For the problem of electrophoretic motion of a spherical par-ticle caused by a constant external electric field E∞= E∞ez
perpendicular to a conducting plane, the above analysis still ap-plies, but now U0is the electrophoretic velocity of an isolated
sphere given by Eq.(5)and the expressions for G, H , G, and
Hgiven by Eqs.(A.15), (A.26), and (A.27)become
(A.29a) G= −ZeE ∞ U0 2 m=1 (−1)mGmcm, (A.29b) H= −ZeE ∞ U0 2 m=1 (−1)mHmcm, (A.30a) G= −ZeE ∞ U0 2 m=1 2 j=1 (−1)jGmgmjcj, (A.30b) H= −ZeE ∞ U0 2 m=1 2 j=1 (−1)jGmhmjcj.
Using the same method of reflections, Keh and Anderson[18]
obtained an asymptotic solution for the electrophoretic mobil-ity of a dielectric sphere surrounded by an infinitesimally thin electric double layer normal to a conducting plane wall correct to O(λ6). In the limit of Eq.(6), the formula(A.28)with coef-ficients given by Eqs.(A.29) and (A.30)is consistent with this earlier solution.
