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Diffusiophoresis and electrophoresis of a charged sphere parallel to one
or two plane walls
Po Y. Chen, Huan J. Keh
∗Department of Chemical Engineering, National Taiwan University, Taipei 106-17, Taiwan, Republic of China
Received 7 September 2004; accepted 20 January 2005 Available online 2 March 2005
Abstract
The diffusiophoretic and electrophoretic motions of a dielectric spherical particle in an electrolyte solution located between two infinite parallel plane walls are studied theoretically. The imposed electrolyte concentration gradient or electric field is constant and parallel to the two plates, which may be either impermeable to the ions/charges or prescribed with the far-field concentration/potential distribution. The electrical 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 mobile ions in the diffuse layer is incorporated. The presence of the neighboring walls causes two basic effects on the particle velocity: first, the local electrolyte concentration gradient or electric field on the particle surface is enhanced or reduced by the walls, thereby speeding up or slowing down the particle; second, the walls increase the viscous retardation of the moving particle. To solve the conservative equations, the general solution is constructed from the fundamental solutions in both rectangular and spherical coordinates. The boundary conditions are enforced first at the plane walls by the Fourier 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 values of the relevant parameters including the relative separation distances between the particle and the two plates. For the special case of motions of a spherical particle parallel to a single plate and in the central plane of a slit, the collocation results agree well with the approximate analytical solutions obtained by using a method of reflections. The presence of the lateral walls can reduce or enhance the particle velocity, depending on the properties of the particle–solution system, the relative particle– wall separation distances, and the electrochemical boundary condition at the walls. In general, the boundary effects on diffusiophoresis and electrophoresis are quite significant and complicated, and they no longer vary monotonically with the separation distances for some situations. 2005 Elsevier Inc. All rights reserved.
Keywords: Diffusiophoresis; Electrophoresis; Boundary effects; Thin but polarized double layer
1. Introduction
The transport behavior of small particles in a continuous medium at low Reynolds number is of much fundamental and practical interest. In general, driving forces for motions of colloidal particles include concentration gradients of the particles themselves (diffusion), bulk velocities of the dis-perse medium (convection), and gravitational fields (sedi-mentation). Problems of the colloidal transport induced by these well-known driving forces have been treated
exten-*Corresponding author.
E-mail address:huan@ntu.edu.tw(H.J. Keh).
sively in the past. Another category of driving forces for the motions of colloidal particles involves a nonuniform im-posed field (such as electric potential, temperature, or solute concentration) that interacts with the surface of each parti-cle. The particle motions associated with this mechanism, known as “phoretic motions,” have also received a consider-able amount of attention recently[1–4].
Perhaps the most familiar example of phoretic motion is electrophoresis, which results from the interaction between an external electric field and the electric double layer sur-rounding a charged particle and is widely used for parti-cle characterization and separation in a variety of colloidal and biological systems [5–7]. The electrophoretic veloc-0021-9797/$ – see front matter 2005 Elsevier Inc. All rights reserved.
ity U(0) of a single dielectric particle suspended in an un-bounded electrolyte solution is simply related to the uni-formly applied electric field E∞by the Smoluchowski equa-tion[8–10],
(1) U(0)= εζ
4π ηE
∞.
Here, ε/4π is the fluid permittivity, η is the fluid viscosity, and ζ is the zeta potential of the particle surface.
Another example of phoretic motions is diffusiophore-sis, which is the migration of a particle in response to the macroscopic concentration gradient of a solute and can be applied to certain latex-particle coating processes [1,11]. The particle moves toward or away from a region of higher solute concentration, depending on long-range interactions between the solute molecules and the particle. In an un-bounded solution of a symmetric electrolyte with a constant concentration gradient∇n∞, the diffusiophoretic velocity of a charged particle is[12,13] (2) U(0)= εζ 4π η kT Ze ∇n∞ n∞(0)(α+ ¯ζ −1ln cosh ¯ζ ),
with the dimensionless parameters
(3) α=D2− D1 D2+ D1 , (4) ¯ζ =Zeζ 4kT.
Here, n∞(0) is the macroscopic electrolyte concentration
measured at the particle center 0 in the absence of the parti-cle, D1and D2are the diffusion coefficients of the anion and cation, respectively, Z is the absolute value of the valences of the ions, e is the charge of a proton, k is the Boltzmann constant, and T is the absolute temperature. For the special case D2= D1 or α= 0, 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 particle velocity is a monotonic increasing
func-tion of the magnitude of ζ .
Equations(1) and (2)indicate that the electrophoretic and diffusiophoretic velocities of a dielectric particle having a uniform zeta potential on its surface are independent of the particle size and shape (and there is no rotational motion of the particle). However, their validity is based on the as-sumptions that the local radii of curvature of the particle are much larger than the thickness of the electric double layer at the particle surface and that the effect of polarization (re-laxation effect) of the diffuse ions in the double layer due to nonuniform “osmotic” flow is negligible. In fact, impor-tant advances were made in the past in the evaluation of the phoretic velocities of colloidal particles relaxing these as-sumptions.
Taking the double-layer distortion from equilibrium as a perturbation, O’Brien and White[14]obtained a numeri-cal numeri-calculation for the electrophoretic velocity of a dielectric
sphere of radius a in a KCl solution which was applicable to arbitrary values of ζ and κa, where κ−1is the Debye screen-ing length equal to (8π Z2e2n∞/εkT )1/2. On the other hand, Dukhin and Derjaguin [1] obtained an analytical expres-sion for the electrophoretic mobility of a spherical particle surrounded by a thin but polarized double layer in the solu-tion of a symmetric electrolyte. Later, O’Brien[15] general-ized this analysis to the case of electrophoretic motion of a charged sphere in the solution containing an arbitrary com-bination of electrolytes. The essence of this thin-layer polar-ization approach is that a thin diffuse layer can still transport a significant amount of electrolyte ions in such a way 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[16]
(5) U0= εζ 12π ηE ∞2+ c 1+ c2+ (c1− c2) ¯ζ−1ln cosh ¯ζ ,
where coefficients c1and c2are defined by Eqs.(A.6a) and
(A.6b). A comparison of Eq.(5)with the numerical results for the KCl solution[14]shows that the thin-layer polariza-tion model is quite good over a wide range of zeta potentials when κa > 20. If|ζ | is small and κa is large, the interac-tion 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)
(κa)−1exp2|¯ζ|→ 0,
c1= c2= 1/2 and Eq. (5) reduces to the Smoluchowski equation(1). In general, the electrophoretic velocity given by Eq.(5)is not a monotonic function of ¯ζ for a finite value of κa, unlike the prediction of Eq.(1).
In contrast, Prieve and Roman [17]obtained a numeri-cal solution for 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[14]. On the other hand, ana-lytical expressions for the velocity of a dielectric sphere with a thin but polarized double layer undergoing diffusiophore-sis in electrolyte solutions have also been derived[4,18]. The result for this diffusiophoretic velocity in a symmetric elec-trolyte solution is U0= εζ 12π η kT Ze ∇n∞ n∞(0) c1− c2+ α(2 + c1+ c2) (7) +2+ c1 + c2+ α(c1− c2)¯ζ−1ln cosh ¯ζ ,
where coefficients c1and c2are defined by Eqs.(A.6c) and (A.6d). When κa > 20, the agreement between Eq.(7)and the numerical solution [17] 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 concentration) when|¯ζ| becomes large.
It could be found from Eqs.(5) and (7)that the effect of polarization of the diffuse layer is to decrease the particle ve-locity. The reason for this outcome is that the transport of the diffuse ions within the double layer reduces the local elec-trolyte 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 in-side the diffuse layer cannot be ignored if|ζ | equals several
kT /e.
In practical applications of diffusiophoresis and elec-trophoresis, such as transport in channels of microfluidic de-vices[19–22], particles are not isolated and the surrounding fluid is externally bounded by solid walls. Thus, it is im-portant to determine if the presence of neighboring bound-aries significantly affects the movement of particles. In the limiting case where Eqs.(1) and (2)are applicable, the nor-malized velocity field of the immense fluid that is dragged by a particle during diffusiophoresis is the same as for the electrophoresis of the particle[2]; thus, the boundary effects on electrophoresis under the situation of an infinitesimally thin double layer [satisfying Eq.(6)], which have been stud-ied extensively in the past[23–29], can be taken to interpret those in diffusiophoresis.
When the polarization effect of diffuse ions in the double layer surrounding the particle is considered, the boundary ef-fects 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 colloidal sphere with a thin but polarized diffuse layer perpendicular to a plane wall were examined[30]. The wall effect on each transport mechanism was found to be a com-plicated function of the properties of the particle and ions. In this work we present a theoretical investigation of the diffusiophoretic and electrophoretic motions of a charged sphere with a thin but polarized double layer parallel to a single plane wall and to two plane walls at an arbitrary posi-tion between them. The quasi-steady equaposi-tions of conserva-tion applicable to each system are solved by using both the boundary collocation technique numerically and a method of reflections analytically. The exact numerical solutions for the particle velocities are obtained with good convergence for various cases and agree well with the approximate ana-lytical solutions. In the limiting case of Eq.(6), our results are in excellent agreement with those available in the litera-ture for electrophoresis[23,24].
2. Analysis for diffusiophoresis
In this section we consider the quasi-steady diffusio-phoretic motion of a dielectric spherical particle of radius
a in a solution of a symmetrically charged electrolyte
par-allel to two infinite plane walls whose distances from the center of the particle are b and c, as shown inFig. 1. Here
Fig. 1. Geometrical sketch for the diffusiophoresis of a spherical particle parallel to two plane walls at an arbitrary position between them.
(x, y, z), (ρ, φ, z), and (r, θ, φ) denote the rectangular,
cir-cular cylindrical, and spherical coordinate systems, respec-tively, and the origin of coordinates is chosen at the particle center. A linear electrolyte concentration field n∞(x) with a
uniform gradient in the x direction is imposed in the sur-rounding fluid far away from the particle. The particle is charged uniformly on the surface, and the thickness of the electrical double layer is assumed to be small in compari-son with the radius of the particle and the spacing between the particle and each wall. Hence, the fluid phase can be di-vided into two regions: an “inner” region defined as the thin double layer adjacent to each solid surface and an “outer” region defined as the remainder of the fluid, which is elec-trically neutral. The objective is to determine the correction to Eq.(7)for the particle velocity due to the presence of the plane walls.
Before the diffusiophoretic velocity of the particle is de-termined, the electrochemical potential and velocity fields in the fluid phase need to be found.
2.1. Electrochemical potential distribution
The Peclet number of the system is assumed to be small. Hence, the equation of conservation of each ionic species for the outer region of the fluid solution is the Laplace equation
[4,15],
(8) ∇2µ
m= 0, m = 1, 2.
In Eq. (8), µm is the electrochemical potential energy of
ionic species m, defined by
(9)
µm= µ0m+ kT ln nm+ zmeΦ,
where µ0mis a constant, nmand zmare 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, respectively, so−z1= z2= Z > 0. Note that, in the outer region, n1= n2= n, and both n and Φ also satisfy Laplace’s equation.
Equation(8)satisfies the boundary condition at the parti-cle “surface” (outer limit of the thin double layer) obtained
by solving for the electrochemical potentials in the inner re-gion and using a matching procedure to ensure a continuous solution in the whole fluid phase[4,15],
(10) r= a+: er· ∇µm= − 2 i=1 βmiI: ∇s∇sµi, m= 1, 2,
where the relaxation coefficients are
β11= 1 κ 4 1+3f1 Z2 exp( ¯ζ ) sinh ¯ζ (11a) −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 exp(−¯ζ) sinh ¯ζ (11d) +12f2 Z2 ( ¯ζ− ln cosh ¯ζ) .
In the above equation, fm= εk2T2/6π ηe2Dm, ¯ζ was
de-fined by Eq.(4), er is the unit vector in the direction of r,
I is the unit dyadic,∇s= (I − erer)· ∇ denotes the
gradi-ent operator along the particle surface, and U and are the instantaneous translational and angular velocities, respec-tively, of the particle to be determined. To obtain Eqs.(10) and (11), it was assumed that the concentration of each ionic species within the electric double layer is related to the elec-tric potential energy by a Boltzmann distribution. Evidently, in the limit of Eq.(6), β11= β12= β21= β22= 0.
The ionic electrochemical potentials far away from the particle approach the undisturbed values. Thus,
(12) z= c, −b: ∂µm ∂z = 0, (13) ρ→ ∞: µm→ µm∞, where (14) µm∞= µ0m+ kT 1− (−1)mαln n∞
and α was defined by Eq.(3). The second term in the brack-ets of Eq.(14)represents the contribution from the macro-scopic electric field induced by the difference of ion diffu-sion rates[13,31]. Note that the boundary conditions given by Eq.(12)apply for the case of two impermeable and non-conducting plane walls (which can appear in practice) with a negligible relaxation effect in their thin interfacial diffuse layers. For the case of diffusiophoretic motion of a parti-cle parallel to two plane walls prescribed with linear elec-trochemical potential profiles consistent with the far-field
distributions (which is less practical), Eq.(12)should be re-placed by
(15)
z= c, −b: µm= µm∞.
Since the governing equation and boundary conditions are linear, one can express the electrochemical potential dis-tribution µm, which is symmetric with respect to y and
anti-symmetric with respect to x, as the superposition
(16)
µm= µmw+ µmp, m= 1, 2.
Here, µmwis a double Fourier integral solution of Eq.(8)in
rectangular coordinates that represents the disturbance pro-duced by the plane walls plus the undisturbed field and is given by µmw= µm∞+ ∞ 0 ∞ 0 (Xmeκz+ Yme−κz) (17) × sin( ˆαx) cos( ˆβy) d ˆα d ˆβ,
where Xmand Ymare unknown functions of separation
vari-ablesˆα and ˆβ, and κ = ( ˆα2+ ˆβ2)1/2. The second term on the right-hand side of Eq.(16), µmp, is a solution of Eq.(8)in
spherical coordinates representing the disturbance generated by the spherical particle and is given by an infinite series in harmonics, (18) µmp= ∞ n=1 Rmnr−n−1Pn1(µ) cos φ,
where Pn1is the associated Legendre function of order n and degree one, µ is used to denote cos θ for brevity, and Rmnare
unknown constants. Note that a solution for µmof the form
given by Eqs.(16)–(18)immediately satisfies the boundary condition at infinity in Eq.(13).
Substituting the electrochemical potential distribution µm
given by Eqs. (16)–(18) into the boundary conditions in Eq. (12) or (15)and applying the Fourier sine and cosine transforms on the variables x and y, respectively, lead to a solution for the functions Xm and Ym in terms of the
co-efficients Rmn. After the substitution of this solution into
Eq.(17)and utilization of the integral representations of the modified Bessel functions of the second kind, the distribu-tion µmcan be expressed as
(19) µm= µm∞+ ∞ n=1 Rmnδn(1)(r, µ) cos φ, m= 1, 2,
and the function δn(1)(r, µ) is defined by Eq.(B.1)in Appen-dix B.
Applying the boundary condition given by Eq. (10) to Eq.(19)yields ∞ n=1 2 m=1 RmnRm(µ)= kT 1−2β11 a (1+ α) (20a) −2β12 a (1− α) (1− µ2)1/2,
∞ n=1 2 m=1 RmnRm(µ)= kT 1−2β22 a (1− α) (20b) −2β21 a (1+ α) (1− µ2)1/2, where (21a)
R1(µ)= (2β11− a)δn(2)(a, µ)+ aβ11δn(4)(a, µ),
(21b)
R2(µ)= β12
2δ(2)n (a, µ)+ aδ(4)n (a, µ),
(21c)
R1(µ)= β21
2δ(2)n (a, µ)+ aδ(4)n (a, µ),
(21d)
R2(µ)= (2β22− a)δn(2)(a, µ)+ aβ22δn(4)(a, µ),
and the definitions of functions δn(2)(r, µ) and δ(4)n (r, µ) are
given by Eqs.(B.2) and (B.4). Note that the dependence of φ factors out in Eq.(20)and the definite integrals in δn(1), δn(2),
and δn(4)must be performed numerically.
To satisfy the condition in Eq. (20) exactly along the entire surface of the particle would require the solution of the entire infinite array of unknown constants Rmn.
How-ever, the collocation method[30,32]enforces the boundary condition at a finite number of discrete points on the half-circular generating arc of the sphere (from θ= 0 to θ = π) and truncates the infinite series in Eq.(19)into finite ones. If the spherical boundary is approximated by satisfying the condition of Eq. (10)at M discrete points on its generat-ing arc, the infinite series in Eq. (19)is truncated after M terms, resulting in a system of 2M simultaneous linear al-gebraic equations in the truncated form of Eq. (20). This matrix equation can be numerically solved to yield the 2M unknown constants Rmn required in the truncated form of
Eq.(19)for the electrochemical potential distribution. 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 van-ishes and the overall accuracy of the solution depends only on the numerical integration required in evaluating the ma-trix elements.
2.2. Fluid velocity distribution
With knowledge of the solution for the ionic electro-chemical potential distribution on the particle surface which drives the 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, the fluid motion in the outer region caused by the diffusiophoretic motion of the particle is governed by the Stokes equations,
(22a)
η∇2v− ∇p = 0,
(22b) ∇ · v = 0,
where v is the velocity field for the fluid flow and p is the dynamic pressure distribution.
The boundary conditions for the fluid velocity at the par-ticle surface[4,15], on the plane walls, and far removed from the particle are
(23) r= a+: v = vs≡ U + × r + v(s), (24) z= c, −b: v = 0, (25) ρ→ ∞: v = 0.
In Eq.(23), the apparent slip velocity due to diffusioosmotic effect is (26) v(s)= 2 m=1 Wm∇sµm, where (27) Wm= ε 2π η kT (Ze)2 −(−1)m¯ζ + lncosh ¯ζ,
and U= Uexand = Ωeyare the translational and angular
velocities of the particle undergoing diffusiophoresis, to be determined. For the asymmetric problem, b= c, the assump-tion that the sphere would migrate in a direcassump-tion parallel to the solute concentration gradient is justified in the absence of fluid inertia. Note that the possible osmotic flow caused by the plane walls is ignored for simplicity.
A fundamental solution to Eq. (22) which satisfies Eqs.(24) and (25)can be obtained in a way similar to that for Eq.(19), with the form
(28) v= vxex+ vyey+ vzez, where (29a) vx= ∞ n=1 An(An+ αn)+ Bn(Bn + βn)+ Cn(Cn + γn) , (29b) vy= ∞ n=1 An(An+ αn)+ Bn(Bn+ βn)+ Cn(Cn+ γn) , (29c) vz= ∞ n=1 An(An + αn)+ Bn(Bn+ βn)+ Cn(Cn+ γn) .
Here, the primed An, Bn, Cn, αn, βn, and γn are functions
of position involving associated Legendre functions of µ or cos θ defined by Eq. (2.6) and in the form of integra-tion (which need to be performed numerically) defined by Eq. (C.1) of Ganatos et al.[32], and An, Bn, and Cnare
un-known constants.
The boundary condition that remains to be satisfied is that on the particle surface. Substituting Eqs.(19) and (28)into Eq.(23), one obtains
∞ n=1 An(An+ αn)+ Bn(Bn + βn)+ Cn(Cn + γn) r=a = U + aΩµ (30a) − 2 m=1 Wm H1m(µ)µ cos2φ+ H2m(µ) sin2φ , ∞ n=1 An(An+ αn)+ Bn(Bn+ βn)+ Cn(Cn+ γn) r=a (30b) = − 2 m=1 Wm H1m(µ)µ− H2m(µ) sin φ cos φ, ∞ n=1 An(An + αn)+ Bn(Bn+ βn)+ Cn(Cn+ γn) r=a = −aΩ(1 − µ2)1/2cos φ (30c) + 2 m=1 WmH1m(µ)(1− µ2)1/2cos φ, where H1m(µ)= kT 1− (−1)mα|∇n ∞| n∞(0)µ (31a) +1 a ∞ n=1 Rmnδn(3)(a, µ), H2m(µ)= kT 1− (−1)mα|∇n ∞| n∞(0) (31b) + 1 a(1− µ2)1/2 ∞ n=1 Rmnδn(1)(a, µ),
and the function δ(3)n (r, µ) is defined by Eq.(B.3). The first
2M coefficients Rmnhave been determined through the
pro-cedure given in the previous subsection.
Careful examination of Eqs.(30a)–(30c)shows that the solution of the coefficient matrix generated is independent of the φ coordinate of the boundary points on the surface of the sphere r= a. Thus, these relations can be satisfied by utilizing the collocation technique presented for the solution of the electrochemical potential field. At the particle surface, Eqs.(30a)–(30c)are applied at N discrete points (values of
θ between 0 and π ) and the infinite series in Eq.(29)are truncated after N terms. This generates a set of 3N linear al-gebraic equations for the 3N unknown coefficients An, Bn,
and Cn. The fluid velocity field is completely obtained once
these coefficients are solved for a sufficiently large value of N .
2.3. Derivation of the particle velocities
The drag force and torque exerted by the fluid on the spherical particle can be determined from[32]
(32a) F= −8πηA1ex,
(32b) T= −8πηC1ey.
These expressions show that only the lowest-order coeffi-cients A1and C1in Eq.(29)contribute to the hydrodynamic force and couple acting on the particle.
Because the particle is freely suspended in the surround-ing fluid, the net force and torque exerted on the particle must vanish. Applying this constraint to Eq.(32), one has
(33)
A1= C1= 0.
To determine the translational and angular velocities U and
Ω of the particle, Eq.(33)and the 3N algebraic equations resulting from Eq.(30)are to be solved simultaneously.
3. Results and discussion for diffusiophoresis
The solution for the diffusiophoretic motion of a charged spherical particle parallel to two plane walls at an arbitrary position between them, obtained using the boundary collo-cation method described in the previous section, is presented in this section. The system of linear algebraic equations to be solved for the coefficients R1n and R2n is constructed from Eq. (20), while that for An, Bn, and Cn is derived
from Eq.(30). All the numerical integrations to evaluate the primed αn, βn, and γnas well as δ(i)n functions were done by
80-point Gauss–Laguerre quadrature.
When the points along the semicircular generating arc of the sphere (with a constant value of φ) where the boundary conditions are to be exactly satisfied are being specified, the first points that should be chosen are θ= 0 and π, since these points define the projected area of the particle normal to the direction of motion and control the gaps between the particle and the neighboring plates. In addition, the point θ= π/2 is also important. However, an examination of the systems of linear algebraic equations(20) and (30)shows that the ma-trix equations become singular if these points are used. To overcome this difficulty, these points are replaced by closely adjacent points, i.e., θ = δ, π/2 − δ, π/2 + δ, and π − δ
[30,32]. Additional points along the boundary are selected as mirror-image pairs about the plane θ = π/2 to divide the two quarter-circular arcs of the particle into equal seg-ments. The optimum value of δ in this work is found to be 0.1◦, for which the numerical results of the particle veloci-ties converge satisfactorily. In selecting the boundary points, any value of φ may be used except for φ= 0, π/2, and π since the matrix equation(30)is singular for these values. 3.1. Motion parallel to a single plane wall
The boundary effects on diffusiophoresis are complicated functions of the properties of the particle and suspending so-lution, the electrochemical condition of the boundary, and the particle–boundary separation distance. Some typical col-location solutions for the translational and rotational ve-locities of a dielectric sphere undergoing diffusiophoresis
Table 1
Normalized translational and rotational velocities of a spherical particle un-dergoing diffusiophoresis parallel to a single impermeable and nonconduct-ing plane wall computed from the exact boundary-collocation solution and the asymptotic method-of-reflection solution for the case Z= 1, f1= 0.2,
κa= 100, and ζ e/kT = 2
a/b U/U0 −aΩ/U0
Exact solution Asymptotic solution Exact solution Asymptotic solution α= 0 0.1 0.99992 0.99992 0.00002 0.00002 0.2 0.99934 0.99935 0.00030 0.00030 0.3 0.99788 0.99798 0.00154 0.00152 0.4 0.99521 0.99578 0.00494 0.00480 0.5 0.99103 0.99316 0.01242 0.01172 0.6 0.98490 0.99115 0.02702 0.02430 0.7 0.97580 0.99152 0.05412 0.04502 0.8 0.96060 0.99694 0.10518 0.07680 0.9 0.92322 1.01113 0.21643 0.12302 0.95 0.85475 0.34835 0.99 0.2493 0.7542 α= −0.2 0.1 0.99969 0.99969 0.00002 0.00002 0.2 0.99755 0.99758 0.00030 0.00030 0.3 0.99169 0.99200 0.00156 0.00152 0.4 0.97982 0.98159 0.00510 0.00480 0.5 0.95838 0.96544 0.01323 0.01172 0.6 0.92042 0.94326 0.03018 0.02430 0.7 0.84954 0.91547 0.06473 0.04502 0.8 0.69592 0.88342 0.13897 0.07680 0.9 0.21212 0.84950 0.33712 0.12302 0.95 −0.75566 0.63714 0.99 −8.8227 2.2435
parallel to a plane wall (with c→ ∞) for different val-ues of the parameters α and a/b are presented inTables 1 and 2 for the cases of an impermeable and nonconducting wall and a wall with the imposed far-field electrolyte con-centration gradient, respectively. The corresponding velocity for the diffusiophoretic motion of the particle in an infi-nite fluid, U0, given by Eq. (7), is used to normalize the boundary-corrected values. All of the results obtained un-der the collocation scheme converge satisfactorily to at least the significant figures shown in the tables. The accuracy and convergence behavior of the truncation technique are princi-pally a function of the ratio a/b. For the difficult case with
a/b= 0.99, the numbers of collocation points M = 40 and N = 40 are sufficiently large to achieve this convergence.
For some cases, the particle will reverse the direction of dif-fusiophoresis and the magnitude of its normalized velocity can be dramatically varied when the separation distance is increased. Note that the situations associated with α= 0 and
α= −0.2 (taking Z = 1 and f1= 0.2) in the tables are close to the particle migration in the aqueous solutions of KCl and NaCl, respectively.
Through the use of spherical bipolar coordinates, Keh and Chen [24] obtained numerical solutions for the nor-malized translational and rotational velocities of a dielectric sphere surrounded by an infinitesimally thin electrical
dou-Table 2
Normalized translational and rotational velocities of a spherical particle un-dergoing diffusiophoresis parallel to a single plane wall prescribed with the far-field electrolyte concentration profile computed from the exact boundary-collocation solution and the asymptotic method-of-reflection so-lution for the case Z= 1, f1= 0.2, κa = 100, and ζ e/kT = 2
a/b U/U0 −aΩ/U0
Exact solution Asymptotic solution Exact solution Asymptotic solution α= 0 0.1 0.99984 0.99984 0.00002 0.00002 0.2 0.99872 0.99873 0.00030 0.00030 0.3 0.99573 0.99588 0.00154 0.00152 0.4 0.99000 0.99078 0.00496 0.00480 0.5 0.98046 0.98340 0.01252 0.01172 0.6 0.96550 0.97430 0.02736 0.02430 0.7 0.94215 0.96475 0.05498 0.04502 0.8 0.90370 0.95699 0.10651 0.07680 0.9 0.82932 0.95425 0.21260 0.12302 0.95 0.75204 0.32114 0.99 0.5976 0.5205 α= −0.2 0.1 1.00006 1.00006 0.00002 0.00002 0.2 1.00050 1.00050 0.00030 0.00030 0.3 1.00188 1.00186 0.00152 0.00152 0.4 1.00511 1.00497 0.00480 0.00480 0.5 1.01179 1.01112 0.01174 0.01172 0.6 1.02480 1.02218 0.02447 0.02430 0.7 1.04930 1.04080 0.04620 0.04502 0.8 1.09390 1.07050 0.08474 0.07680 0.9 1.16276 1.11587 0.18095 0.12302 0.95 1.17697 0.33196 0.99 1.0353 0.7646
ble layer undergoing electrophoresis parallel to a noncon-ducting plane wall. These solutions, which can apply to the case of diffusiophoresis of a sphere in the limit of Eq.(6) par-allel to an impermeable plane wall, can be used to check the correctness of our collocation solutions. It has been found that our solutions for the particle velocities in this limiting case agree excellently with the bipolar-coordinate solutions, although the comparison is not shown here for conciseness.
InAppendix A, an approximate analytical solution for the same diffusiophoretic motion as that considered here is also obtained using a method of reflections. The translational and angular velocities of a spherical particle near a lateral plate are given by Eqs. (A.14a) and (A.14b), which are power series expansions in λ (= a/b). The values of the wall-corrected normalized particle velocities calculated from this asymptotic solution, with the O(λ6) term neglected, are also
listed in Tables 1 and 2 for comparison. It can be seen that the asymptotic formula of Eq. (A.14a)resulting from the method of reflections for U/U0 agrees very well with the exact results as long as λ 0.5; the errors in all cases are less than 0.8%. However, the accuracy of Eqs.(A.14a) and (A.14b)[especially Eq. (A.14b)for aΩ/U0, in which the leading term is O(λ4)] deteriorates rapidly, as expected,
when the relative spacing between the particle and the plane wall becomes small.
(a)
(b)
Fig. 2. Plots of the normalized diffusiophoretic mobility U/U0of a
spher-ical particle parallel to a plane wall versus the separation parameter a/b with Z= 1, f1= 0.2, and ζ e/kT = −5 for various values of κa: (a) α = 0;
(b) α= −0.2. The solid curves represent the case of an impermeable and nonconducting wall, and the dashed curves denote the case of a wall on which the far-field electrolyte concentration gradient is imposed.
Some exact numerical solutions for the normalized dif-fusiophoretic velocity U/U0of a spherical particle parallel to a plane wall as functions of a/b are depicted in Fig. 2
for various values of κa. It can be seen that, under the speci-fied condition, U/U0increases with an increase in κa for the case of an impermeable and nonconducting wall [the bound-ary condition(12)is used], but decreases with an increase in
κa for the case of a plane wall prescribed with the far-field
electrolyte concentration distribution [the boundary condi-tion(15)is used] for the otherwise specified condition. This increase and decrease in particle mobility become more pro-nounced as a/b increases. This behavior is expected, know-ing that the electrochemical potential gradients on the
par-ticle surface near an impermeable and nonconducting wall increase as κa (or G) increases and these gradients near a wall with the imposed far-field concentration gradient de-crease as κa (or G) inde-creases for the present case (see the analysis inAppendix A).
Examination of the results inFig. 2 reveals an interest-ing feature. For the case of a plane wall impermeable to the ions/charges under the situation of large κa (e.g., with
κa→ ∞), the diffusiophoretic mobility of the particle
de-creases with increased a/b when a/b is small, but inde-creases from a minimum with increased a/b when a/b is suffi-ciently large. When the gap between the particle and the wall turns thin, the particle can even move faster than it would at
a/b= 0. For example, as κa → ∞ and when a/b = 0.99,
the diffusiophoretic velocity for a typical case can be as much as 15% higher than the value with the wall far away from the particle. Under the situation of relatively small κa, the diffusiophoretic mobility of the particle near the imper-meable and nonconducting wall is a monotonic decreasing function of a/b. For a case where a linear electrochemi-cal potential profile is prescribed on the plane wall, which is consistent with the far-field distribution under the situa-tion of small κa (e.g., with κa= 20), the diffusiophoretic mobility of the particle may increase monotonically with the increase of a/b from a/b= 0. Under the situation of relatively large κa, the diffusiophoretic mobility of the par-ticle near the wall prescribed with the far-field electrolyte concentration distribution becomes a monotonic decreasing function of a/b. This interesting feature that U/U0may not be a monotonic decreasing function of a/b and can even be greater than unity is understandable because the wall effect of hydrodynamic resistance on the particle is in competition with the wall effect of electrochemical enhancement when a particle with large κa is undergoing diffusiophoretic mo-tion parallel to an impermeable and nonconducting plate or when a particle with small κa is moving near a lateral plate with the imposed far-field electrolyte concentration gradient. A careful examination of the asymptotic formula for U/U0 given by Eq.(A.14a)shows good agreement of the numeri-cal outcome inFig. 2with the analytical solution.
The results inTables 1 and 2indicate that the dielectric sphere undergoing diffusiophoresis parallel to a plane wall rotates in the direction opposite to that for a sphere migrat-ing in the same direction but under a body-force field (e.g., a gravitational field). The explanation for this behavior is anal-ogous to the case of electrophoresis of a charged sphere with a thin double layer parallel to a nonconducting plate[24]. For an otherwise specified condition, the magnitude of the normalized rotational velocity of the diffusiophoretic sphere near a given plane wall is a monotonically increasing func-tion of a/b.
3.2. Motion parallel to two plane walls
Some converged collocation solutions for the normalized velocity U/U0 of a charged spherical particle undergoing
Table 3
Normalized diffusiophoretic velocity of a spherical particle along the me-dian plane between two parallel plane walls computed from the exact boundary-collocation solution and the asymptotic method-of-reflection so-lution for the case Z= 1, f1= 0.2, κa = 100, and ζ e/kT = 2
a/b U/U0 α= 0 α= −0.2 Exact solution Asymptotic solution Exact solution Asymptotic solution For impermeable plane walls
0.1 0.99968 0.99968 0.99915 0.99915 0.2 0.99751 0.99751 0.99325 0.99325 0.3 0.99202 0.99207 0.97760 0.97768 0.4 0.98232 0.98272 0.94781 0.94860 0.5 0.96820 0.97005 0.89890 0.90342 0.6 0.94992 0.95629 0.82267 0.84166 0.7 0.92794 0.94567 0.70053 0.76285 0.8 0.90191 0.94489 0.47632 0.67198 0.9 0.86257 0.96346 −0.12763 0.57489 0.95 0.80313 −1.24283 0.99 0.1946 −10.226
For plane walls prescribed with the far-field concentration profile
0.1 0.99952 0.99952 0.99991 0.99991 0.2 0.99620 0.99620 0.99939 0.99940 0.3 0.98762 0.98764 0.99829 0.99843 0.4 0.97197 0.97221 0.99712 0.99780 0.5 0.94811 0.94953 0.99731 0.99951 0.6 0.91521 0.92083 1.00195 1.00719 0.7 0.87210 0.88937 1.01684 1.02650 0.8 0.81568 0.86085 1.05193 1.06553 0.9 0.73429 0.84380 1.11612 1.13524 0.95 0.66825 1.13957 0.99 0.5615 1.0560
diffusiophoresis on the median plane between two parallel plane walls (with c= b and Ω = 0) for various values of the parameters α and a/b are presented inTable 3for the two cases of impermeable and nonconducting walls and walls prescribed with the far-field electrolyte concentration dis-tribution. The corresponding method-of-reflection solutions, given by Eq.(A.21)inAppendix Aas a power series expan-sion in λ (= a/b) correct to O(λ5), are also listed in this
table for comparison. Similarly to the case of migration of a spherical particle parallel to a single plane wall considered in the previous section, the approximate analytical formula of Eq.(A.21)agrees very well with the exact results as long as λ 0.5, but can have significant errors when λ 0.6. In general, Eq.(A.21)overestimates the diffusiophoretic ve-locity of the particle. A comparison betweenTable 3for the case of a slit andTables 1 and 2for the case of a single par-allel plane indicates that the assumption that the boundary effect for two walls can be obtained by simple addition of single-wall effects in general leads to a smaller correction to diffusiophoretic motion when a/b is small but can give a greater correction as a/b becomes large.
In Fig. 3, typical collocation results for the normalized diffusiophoretic mobility U/U0 of a dielectric sphere mi-grating on the median plane between two parallel plane walls are plotted as functions of a/b for several values of α
(a)
(b)
Fig. 3. Plots of the normalized diffusiophoretic mobility U/U0of a
spheri-cal particle migrating on the median plane between two parallel plane walls (with c= b) versus the separation parameter a/b with Z = 1, f1= 0.2,
and ζ e/ kT= −5 for several values of κa: (a) α = 0; (b) α = −0.2. The solid curves represent the case of impermeable and nonconducting walls, and the dashed curves denote the case of walls prescribed with the far-field electrolyte concentration distribution.
and κa. Analogously to the corresponding motion of a par-ticle parallel to a single plane wall under the same condition illustrated inFig. 2, for a specified value of a/b, U/U0 de-creases with an increase in κa for the case of walls with the imposed far-field electrolyte concentration gradient and in-creases with an increase in κa for the case of impermeable and nonconducting walls. Again, for the case of imperme-able and nonconducting walls in the situation of large κa, the diffusiophoretic mobility of the particle may first go through a minimum with the increase of a/b from a/b= 0 and then increase monotonically, and the particle can even move faster than it would at a/b= 0. Also, for the case
of walls prescribed with the far-field electrolyte concentra-tion distribuconcentra-tion under the situaconcentra-tion of small κa, the particle mobility may increase monotonically with an increase in
a/b from a/b= 0. This result indicates that the effect of
electrochemical enhancement, rather than that of hydrody-namic resistance, can be overriding when the particle–wall gap thickness is small. An examination of the asymptotic formula for U/U0in Eq.(A.21)also shows good agreement of the trend inFig. 3with the analytical solution.
A careful comparison of the curves inFig. 3for the case of a slit with the corresponding curves inFig. 2for the case of a single wall reveals an interesting feature of the boundary effect on diffusiophoresis of a colloidal sphere. The presence of a second, identical, lateral plane wall, even at a symmetric position with respect to the sphere against the first, does not always enhance the wall effect on the diffusiophoretic parti-cle induced by the first plate only. This result reflects again the fact that the lateral wall can affect the electrochemical driving force and the viscous drag force on a particle in op-posite directions. Each force is increased in its own direction as the value of a/b turns small, but to a different degree, for the case of diffusiophoretic motion of a particle in a slit rela-tive to that for the case of migration parallel to a single plate. Thus, the net effect composed of these two opposite forces for the slit case is not necessarily to enhance that for the case of a single wall.
Fig. 4shows the collocation results for the normalized translational velocity U/U0and rotational velocity aΩ/U0 of a charged sphere undergoing diffusiophoresis parallel to two impermeable and nonconducting plane walls at various positions between them for a typical case. The dashed curves (with a/b= constant) illustrate the effect of the position of the second wall (at z= c) on the particle velocities for var-ious 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 velocities as functions of the sphere position at various values of the relative wall-to-wall spac-ing (b+ c)/2a. As illustrated inFig. 4a, the net wall effect for the given case is to reduce the diffusiophoretic mobility
U/U0of the particle. At a constant value of 2a/(b+ c), the particle in general (with exceptions) experiences a minimum viscous drag force and has a greatest translational velocity (without rotation) when it is located midway between the two walls (with c= b); the hydrodynamic drag increases, the translational velocity decreases, and the rotational veloc-ity increases as the particle approaches either of the walls [or the ratio b/(b+c) decreases]. At a specified value of a/b for the diffusiophoretic particle near a first lateral wall, the pres-ence of a second plate is to further reduce the translational and rotational velocities of the particle, and the degree of this reduction increases monotonically with a decrease in the rel-ative distance between the particle and the second plate [or with an increase in b/(b+ c)].
On the other hand, for some cases such as the diffusio-phoresis of a dielectric sphere with a large value of κa par-allel to two impermeable and nonconducting plane walls or
(a)
(b)
Fig. 4. Plots of the normalized velocities of a spherical particle undergo-ing diffusiophoresis parallel to two impermeable and nonconductundergo-ing plane walls versus the ratio b/(b+ c) for the case of Z = 1, f1= f2= 0.2,
κa= 1000, and |ζ e/kT | = 5 with a/b and 2a/(b + c) as parameters: (a)
translational velocity U/U0; (b) rotational velocity aΩ/U0.
with a small value of κa parallel to two plates prescribed with the far-field electrolyte concentration distribution, the net wall effect can increase the diffusiophoretic mobility of the particle relative to its isolated value. At a fixed value of 2a/(b+ c) in these cases, the normalized particle mobility has a relatively small value as it is located midway between the two walls, where the particle experiences a minimum effect of electrochemical enhancement, and becomes rela-tively large when it approaches either of the walls. At a given value of a/b for the diffusiophoretic particle and the first lat-eral plate, the effect induced by the presence of the second plate on the particle mobility is not necessarily a monotonic function of its distance from the particle. This dependence is quite complicated and is not graphically presented here, for conciseness.
4. Electrophoresis
In this section, we consider the steady electrophoretic motion of a dielectric sphere in a uniformly applied electric field E∞= E∞ex parallel to two plane walls. The
transla-tional and angular velocities of the particle caused by the field are U= Uex and = Ωey, respectively. 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 thickness of the double layer is as-sumed to be much smaller than the radius of the particle and the surface-to-surface distance between the particle and each wall, but the polarization effect in the thin diffuse layer is in-corporated.
Outside the double layer, the electrochemical potentials
µmof the ions still satisfy Laplace’s equation(8)and
bound-ary conditions (10)–(15), but their undisturbed values in Eq.(14)are replaced by
(34)
µm∞= µ0m+ kT ln n∞− (−1)mZeE∞x.
The solution for µm in this case can still be expressed as
Eq.(19)with coefficients Rmndetermined by
∞ n=1 2 m=1 RmnRm(µ) (35a) = ZeE∞(a− 2β 11+ 2β12)(1− µ2)1/2, ∞ n=1 2 m=1 RmnRm(µ) (35b) = −ZeE∞(a− 2β22+ 2β21)(1− µ2)1/2,
to replace Eq.(20). The governing equations, boundary con-ditions, and solution for the fluid velocity field have the same forms as those given by Eqs.(22)–(29). The final results for the translational and angular velocities of the particle can be determined by the simultaneous solution of Eqs.(30) and (33), with (36a) H1m(µ)= −(−1)mZeE∞µ+ 1 a ∞ n=1 Rmnδn(3)(a, µ), H2m(µ)= −(−1)mZeE∞ (36b) + 1 a(1− µ2)1/2 ∞ n=1 Rmnδ(1)n (a, µ),
in replacement for Eq.(31).
Some converged collocation solutions for the transla-tional and rotatransla-tional velocities of a dielectric sphere under-going electrophoresis parallel to a plane wall (with c→ ∞) for various values of a/b are presented in Table 4, while the corresponding results for the electrophoretic velocity of the particle on the median plane between two parallel plates (with c= b) are given in Table 5. Now, the elec-trophoretic velocity of the particle in an unbounded fluid
Table 4
Normalized translational and rotational velocities of a spherical particle undergoing electrophoresis parallel to a single plane wall computed from the exact boundary-collocation solution and the asymptotic method-of-reflection solution for the case Z= 1, f1= f2= 0.2, κa = 100, and
ζ e/kT= 2
a/b U/U0 −aΩ/U0
Exact solution Asymptotic solution Exact solution Asymptotic solution For a nonconducting plane wall
0.1 0.99994 0.99994 0.00002 0.00002 0.2 0.99950 0.99952 0.00030 0.00030 0.3 0.99842 0.99853 0.00154 0.00152 0.4 0.99655 0.99709 0.00493 0.00480 0.5 0.99389 0.99571 0.01235 0.01172 0.6 0.99054 0.99556 0.02675 0.02430 0.7 0.98685 0.99853 0.05320 0.04502 0.8 0.98377 1.00740 0.10222 0.07680 0.9 0.98557 1.02603 0.20582 0.12302 0.95 0.99640 0.32277 0.99 1.0898 0.6057
For a plane wall prescribed with the far-field potential profile
0.1 0.99982 0.99982 0.00002 0.00002 0.2 0.99856 0.99856 0.00030 0.00030 0.3 0.99520 0.99532 0.00154 0.00152 0.4 0.98868 0.98948 0.00498 0.00480 0.5 0.97772 0.98085 0.01259 0.01172 0.6 0.96031 0.96988 0.02762 0.02430 0.7 0.93278 0.95774 0.05574 0.04502 0.8 0.88706 0.94652 0.10842 0.07680 0.9 0.80016 0.93935 0.21538 0.12302 0.95 0.71488 0.32020 0.99 0.5591 0.4984 Table 5
Normalized electrophoretic velocity of a spherical particle along the me-dian plane between two plane walls computed from the exact boundary-collocation solution and the asymptotic method-of-reflection solution for the case Z= 1, f1= f2= 0.2, κa = 100, and ζ e/kT = 2
a/b U/U0
For nonconducting plane walls
For plane walls prescribed with the far-field potential profile Exact solution Asymptotic solution Exact solution Asymptotic solution 0.1 0.99973 0.99973 0.99948 0.99948 0.2 0.99788 0.99791 0.99592 0.99591 0.3 0.99328 0.99340 0.98669 0.98664 0.4 0.98534 0.98586 0.96977 0.96985 0.5 0.97426 0.97619 0.94381 0.94493 0.6 0.96105 0.96690 0.90762 0.91288 0.7 0.94784 0.96253 0.85944 0.87674 0.8 0.93916 0.97005 0.79502 0.84199 0.9 0.94937 0.99928 0.70090 0.81694 0.95 0.98303 0.62704 0.99 1.1572 0.5167
given by Eq. (5) is used to normalize the wall-corrected values. The corresponding method-of-reflection solutions, given by Eqs.(A.14a), (A.14b), and (A.21)with the parame-ter G defined by Eq.(A.16)inAppendix Aas power series expansions in λ (= a/b) correct to O(λ5), are also listed
(a)
(b)
Fig. 5. Plots of the normalized electrophoretic velocity of a dielectric sphere parallel to a plane wall versus the separation parameter a/b with Z= 1 and
f1= f2= 0.2: (a) |ζ e/kT | = 5; (b) κa = 100. The solid curves represent
the case of an impermeable and nonconducting wall, and the dashed curves denote the case of a wall on which the far-field electrolyte concentration gradient is imposed.
in these tables for comparison. Analogously to the cases of diffusiophoresis considered in the previous section, the as-ymptotic formulas of Eqs. (A.14a) and (A.21) for U/U0 agree quite well with the exact results as long as λ 0.6, but can have significant errors for greater values of λ.
In Fig. 5, typical collocation results for the normalized electrophoretic mobility U/U0of a spherical particle paral-lel a plane wall as functions of a/b are plotted for various values of κa and ζ e/ kT . The corresponding results for the particle undergoing electrophoresis on the median plane be-tween two parallel plane walls are depicted inFig. 6. For the case of nonconducting plane walls under the specified con-dition, the value of U/U0decreases with an increase in a/b as a/b is small, but increases from a minimum (to a
max-(a)
(b)
Fig. 6. Plots of the normalized electrophoretic velocity of a dielectric sphere migrating on the medium plane between two parallel plane walls versus the separation parameter a/b with Z= 1 and f1= f2= 0.2: (a) |ζ e/kT | = 5;
(b) κa= 100. The solid curves represent the case of an impermeable and nonconducting wall, and the dashed curves denote the case of a wall on which the far-field electrolyte concentration gradient is imposed. imum if the value of κa is small) with increasing a/b and can be greater than unity as a/b is sufficiently large. For the case of plane walls prescribed with the undisturbed elec-tric potential distribution, U/U0 is a monotonic decreasing function of a/b. It appears that U/U0 in general increases with an increase in κa and with a decrease in ζ e/ kT for the case of nonconducting walls, but decreases with an increase in κa and with a decrease in ζ e/ kT for the case of plane walls prescribed with the undisturbed potential distribution. However, similar to the case of diffusiophoresis considered in the previous section, no simple rule could appropriately describe the boundary effects on the electrophoretic mobil-ity of the particle, which are dependent on the combination of ζ e/ kT , κa, Z, f1, f2, and a/b.
(a)
(b)
Fig. 7. Plots of the normalized velocities of a dielectric sphere undergoing electrophoresis parallel to two impermeable and nonconducting plane walls versus the ratio b/(b+ c) for the case of Z = 1, f1= f2= 0.2, κa = 1000,
and|ζ e/kT | = 5 with a/b and 2a/(b + c) as parameters: (a) translational velocity U/U0; (b) rotational velocity aΩ/U0.
Some collocation results for the normalized translational velocity U/U0and rotational velocity aΩ/U0of a dielectric sphere undergoing electrophoresis parallel to two noncon-ducting plane walls at various positions between them are displayed inFig. 7. Similar to the cases of diffusiophoresis, the particle experiences a minimum viscous drag and has a greatest translational velocity (without rotation) when it is located midway between the two walls (with c= b). At a specific value of a/b for the electrophoretic particle near a first lateral wall, the presence of a second plate is to further reduce the translational and rotational velocities of the parti-cle, and the degree of this reduction increases monotonically with a decrease in the relative distance between the particle and the second plate [or with an increase in b/(b+ c)].
5. Conclusions
In this work, the exact numerical solutions and approx-imate analytical solutions for the quasi-steady diffusio-phoretic and electrodiffusio-phoretic motions of a charged sphere parallel to two infinite plane walls at an arbitrary position between them have been obtained by using the boundary-collocation technique and the method of reflections, respec-tively. Both the cases of impermeable and nonconducting walls and of walls with the imposed far-field electrolyte concentration gradient were examined in the limit of van-ishingly 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 func-tions of the properties of the particle and surrounding ions (ζ e/ kT , κa, Z, f1, and f2), the electrochemical condi-tions of the boundaries, and the separation distances (a/b and a/c). The diffusiophoretic or electrophoretic mobility of a particle near a wall is generally, but not necessarily, a monotonic decreasing function of the separation parameter
a/b. When the value of a/b is sufficiently large, the effect
of a lateral wall can speed up or slow down the particle ve-locity relative to its isolated value depending on the values of the relevant parameters of the particle–electrolyte sys-tem and the electrochemical boundary condition at the wall. This behavior reflects the competition between the relatively weak hydrodynamic retardation exerted by the neighbor-ing wall on the particle migration and the possible, strong phoretic enhancement due to the electrochemical interaction between the particle and the lateral wall. No general rule can make an adequate prediction for such complicated phenom-ena present in the boundary effects on diffusiophoresis and electrophoresis.
Acknowledgment
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 parallel to one or two plane walls by a method of reflections
In this Appendix, we analyze the quasi-steady diffusio-phoretic and electrodiffusio-phoretic motions of a dielectric sphere of radius a with uniform zeta potential ζ in the solution of a symmetric electrolyte either parallel to an infinite flat wall (with c→ ∞) or on the median plane between two paral-lel plates (with c= b), as shown inFig. 1, by a method of reflections. The effect of the walls on the translational ve-locity U and angular veve-locity of the particle is sought in expansions of λ, which equals a/b, the ratio of the particle radius to the distance between the wall and the center of the particle.
A.1. Motion parallel to a plane wall
For the problem of diffusiophoretic motion of a spher-ical particle driven by a uniform electrolyte concentration gradient∇n∞parallel to an impermeable and nonconduct-ing plane wall, the governnonconduct-ing equations (8) and (22) for the electrochemical potentials of the ions and the fluid flow field must be solved by satisfying the boundary con-ditions (10), (12), (13), and (23)–(25) with c→ ∞. The method-of-reflection solution consists of the series, whose terms depend on increasing powers of λ,
(A.1a)
µm= µm∞+ µ(1)mp+ µ(1)mw+ µ(2)mp+ µ(2)mw+ · · · ,
(A.1b) v= v(1)p + v(1)w + v(2)p + v(2)w + · · · ,
where subscripts p and w represent the reflections from the particle and the wall, respectively, and the superscript (i) de-notes the ith reflection from that surface. In these series, all the expansion sets of the corresponding electrochemical po-tential and velocity fields for the fluid solution must satisfy Eqs.(8) and (22). The advantage of this method is that it is necessary to consider boundary conditions associated with only one surface at a time.
According to Eq.(A.1), the translational and angular ve-locities of the particle can also be expressed in the series forms
(A.2a) U= U0ex+ U(1)+ U(2)+ · · · ,
(A.2b)
= (1)+ (2)+ · · · .
In these expressions, U0is the diffusiophoretic velocity of an identical particle suspended freely in the continuous phase far from the wall given by Eq.(7); U(i)and (i)are related to∇µ(i)mwand v(i)w by[33]
(A.3a) U(i)= 2 m=1 Gm ∇µ(i) mw 0+ v(i)w0+a 2 6 ∇2v(i) w 0, (A.3b) (i)=1 2 ∇ × v(i) w 0,
where the subscript 0 to variables inside brackets de-notes evaluation at the position of the particle center. In Eq.(A.3a), G1= εkT 3π η(Ze)2 (1+ g11− g21) ¯ζ (A.4a) + (1 + g11+ g21) ln cosh ¯ζ , G2= εkT 3π η(Ze)2 (−1 + g12− g22) ¯ζ (A.4b) + (1 + g12+ g22) ln cosh ¯ζ ,
where ¯ζ is defined by Eq.(4),
(A.5a) g11= 1 2(c 1+ c1), (A.5b) g12= 1 2(c 1− c1), (A.5c) g21= 1 2(c 2− c2), (A.5d) g22= 1 2(c 2+ c2). In Eq.(A.5), c1= 1 2a2 1 (a2− 2aβ11+ 3aβ12 (A.6a) + aβ22+ 2β12β21− 2β11β22), c2= 1 2a2 1
(a2− 2aβ22+ 3aβ21+ aβ11
(A.6b) + 2β12β21− 2β11β22), (A.6c) c1= c1− 3 β12 a1 , (A.6d) c2= c2− 3 β21 a1 , where (A.7) 1= 1 a2(a 2+ aβ 11+ aβ22− β12β21+ β11β22),
and the relaxation coefficients β11, β12, β21, and β22are 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(1)mp= a3r−2sin θ cos φ 2 i=1 gmi|∇µi∞|, v(1)p =1 2U0a 3r−3 (2 sin θ cos φ er (A.8b) − cos θ cos φ eθ+ sin φ eφ).
The velocity distribution shown in Eq.(A.8b)is identical to the irrotational flow surrounding a rigid sphere moving with velocity U0ex.
The boundary conditions for the first reflected fields from the wall are derived from Eqs.(12), (13), (24), and (25):
(A.9a) z= −b: ∂µ (1) mw ∂z = − ∂µ(1)mp ∂z , (A.9b) v(1)w = −v(1)p ; (A.9c) r→ ∞, z > −b: µ(1)mw→ 0, (A.9d) v(1)w → 0.
