Electrophoresis of a Sphere Normal to a Plane at Arbitrary Electrical
Potential and Double Layer Thickness
Ming-Hui Chih, Eric Lee, and Jyh-Ping Hsu1
Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617, Republic of China
Received June 15, 2001; accepted January 18, 2002; published online March 13, 2002
The electrophoretic movement of a sphere normal to an un-charged, planar surface is analyzed theoretically, taking the effect of double layer polarization into account. Here, both the surface potential of the particle and the thickness of the double layer sur-rounding it can be arbitrary. We show that if double layer polar-ization is neglected, the effect of the surface potential of a particle on its electrophoretic velocity is inappreciable. On the contrary, it becomes significant if double layer polarization is present. However, if the distance between the particle and the surface is sufficiently close, since the hydrodynamic effect dominates, the influence of the surface potential and double layer polarization becomes insignifi-cant. C2002 Elsevier Science (USA)
INTRODUCTION
The presence of a boundary on the electrophoretic behavior of a charged entity is of fundamental significance and draws the attention of many researches in the past two decades. Typi-cal examples include the electrophoresis of a particle through a pore or in a porous medium and that of a concentrated disper-sion. Zydeny (1) and Lee et al. (2, 3), for instance, considered the electrophoretic behavior of a sphere in a spherical cavity. Morrison and Stukel (4) investigated the electrophoresis of a sphere normal to a plane under the conditions of thin double layer and low surface potential. Bispherical coordinates were chosen for the description of the electrokinetic equations. They found that the plane has the effect of retarding the movement of the particle, and the closer the particle to the plane the more significant it is. Pointing out that the analysis of Morrison and Stukel is incorrect, Keh and Lien (5) reanalyzed the problem. Keh and Anderson (6) considered the electrophoretic behavior of a sphere for the cases in which it moves normal to a plane, parallel to a plane, and along the axis of a cylindrical cavity. Analytical solution was derived based upon a reflection method under the conditions of low electrical potential and thin double layer. It was found that an electroosmotic effect will be induced by a charged boundary. The same problem was discussed by
1To whom correspondence should be addressed. E-mail: jphsu@ccms.ntu.
edu.tw.
Ennis and Anderson (7) through considering a double layer of finite thickness, but it cannot reach the boundary. The problem that an arbitrary prolate body of revolution moves normal to a conducting plane for the case of thin double layer was an-alyzed by Feng and Wu (8). The electrophoresis of a particle normal to a plane at low electrical potential but arbitrary dou-ble layer thickness was considered by Shugai and Carnie (9). Recently, Tang et al. (10) analyzed the electrophoretic behavior of a charge-regulated sphere normal to a plane at low electrical potential.
The electrophoresis of spherical particles has been studied ex-tensively (see, e.g., (11–24)). Most of the results reported, how-ever, are limited to special cases such as low electrical potential and thin double layer. This is mainly due to the nonlinear nature of the governing electrokinetic equations, and solving them an-alytically becomes almost impossible. For the case of a sphere which moves normal to a planar surface, previous analyses are restricted to low surface potential and the effect of double layer polarization (DLP) is negligible. In this work, these limitations are removed, and the general electrokinetic equations are solved numerically.
THEORY
Referring to Fig. 1, we consider the electrophoresis of a sphere of radius a normal to an uncharged plane subject to a uniform electric field Ez with strength Ez in the Z direction. Let h be
the distance between the center of the particle and the surface. The liquid phase contains z1: z2 electrolyte solution, z1 and z2 being respectively the valences of cations and anions, with n20= n10/α, n10 and n20 being respectively the bulk
concen-trations of cations and anions. Suppose that the velocity of the particle U is slow so that a quasi-steady state can be assumed. The bispherical coordinates (ξ, η, φ) are adopted with η = 0 and
η = η0represent respectively the planar surface and the surface
of the particle. The symmetric nature of the problem suggests that only half of the (ξ, η) domain needs to be considered. The bispherical coordinates used are related to the cylindrical coor-dinates (r, θ, Z) by (25)
Z = c sinhη
coshη − cos ξ [1]
383 0021-9797/02 $35.00
C
2002 Elsevier Science (USA) All rights reserved.
FIG. 1. The bispherical coordinates (ξ, η, φ) adopted where η = 0 and η =
η0represent respectively the plane and the surface of particle, a is particle radius,
and h is the distance between the center of the particle and the plane. A uniform electric field Ezis applied in the Z direction, and U is the velocity of particle.
ρ = c sinξ
coshη − cos ξ, [2]
where c is the focal length and 0≤ η ≤ η0, 0 ≤ ξ ≤ π, η0=
cosh−1(h/a).
The elctrokinetic equations of the present problem comprise the governing equations for ion conservation, electrical field, and flow field. The conservation of ions in the liquid phase yields
∇ · Dj ∇nj+ zjenj kBT ∇φ − njv = 0 j= 1, 2, [3]
where∇ is the gradient operator, nj, Dj, and zjare respectively
the number concentration, the diffusivity, and the valence of ion species j, e, and φ are the elementary charge and the electrical potential respectively, v is the fluid velocity, and kB and T are
respectively the Boltzmann constant and the absolute temper-ature. Suppose that the electrical field can be described by the Poission equation ∇2φ = −ρe ε = − 2 j=1 zjenj ε , [4]
whereρeis the space charge density, andε is the permittivity of the liquid phase. We consider incompressible fluid with constant physical properties. Assume that the flow field can be described by the Navier–Stokes equation
∇ · v = 0 [5]
µ∇2v− ∇ p − ρ
e∇φ = 0. [6]
In these expressions p is the pressure andµ is the viscosity. For convenience,φ is decomposed into the sum of the elec-trical potential in the absence of the applied electric field (or the equilibrium electrical potential)φ1, and a perturbed electrical
potential φ2, which arises from the applied electric field (16).
The distortion of the double layer surrounding the particle can be simulated by nj= nj 0exp −zje(φ1+ φ2+ gj) kBT , j = 1, 2, [7]
where gjis a perturbed function. Suppose that the applied
elec-tric field is weak, andφ2and gj are much smaller thanφ1. In
this case Eq. [7] can be approximated by
nj = nj 0exp − zje kBTφ1 1−zje kBT(φ2+ gj) , j = 1, 2 [8]
and the space charge density can be approximated by
ρ ∼= 2 j=1 zjenj 0exp −zje kBTφ1 1− zje kBT(φ2+ gj) . [9]
For a simpler mathematical manipulation, we define the scaled quantitiesφ∗j = φj/ζa, g∗j = gj/ζa, and n∗j= nj/n10, whereζa
is the electrical potential on particle surface. In scaled quantities the equilibrium electrical potential can be expressed as
∇∗2φ∗ 1 = − 1 (1+ α) (κa)2 φr [exp(−φrφ1∗)− exp(αφrφ1∗)], [10]
whereφr = ζa/(z1e/kBT ) is the scaled surface potential of the
particle, and the reciprocal Debye lengthκ and the scaled gra-dient operator∇∗2are defined by
κ = 2 j=1 nj 0(ezj)2/εkBT 1/2 [11] ∇∗2= x2 c∗2 ∂2 ∂η2 + ∂2 ∂ξ2 − sinhη x ∂ ∂η+ cosξ cosh η − 1 x sinξ ∂ ∂ξ . [12] In these expression c∗ = c/a, and x = cosh η − cos ξ.
Suppose that the equilibrium electrical potential vanishes at a point far away from particle surface. Then
φ∗
1 = 0, η = 0 and ξ = 0, or η = 0. [13]
The symmetric nature of the problem implies that
∂φ∗ 1
∂ξ = 0, ξ = 0 or ξ = π. [14]
Assuming constant electrical potential on particle surface, we have
φ∗
Employing the relationφ = φ1+ φ2, Eqs. [4] and [10] yield ∇∗2φ∗ 2− (κa)2 (1+ α)[exp(−φrφ ∗ 1)+ α exp(αφrφ1∗)]φ2∗ = (κa)2 (1+ α)[exp(−φrφ ∗ 1)g∗1+ exp(αφrφ1∗)αg∗2]. [16]
The applied electric field is in the Z direction, therefore, we have, for the scaled perturbed potentialφ2∗,
φ∗
2 = −E∗z· r∗, η = 0 and ξ = 0, or η = 0 [17]
∂φ∗ 2
∂η = 0, η = η0. [18]
The symmetric nature of the problem suggests that
∂φ∗ 2
∂ξ = 0, ξ = 0 or ξ = π. [19]
In these expressions, E∗z and r= r∗· ir are respectively the
scaled electric field and the direction vector where
r∗= c∗ coshη + cos ξ x 1/2 . [20]
Substituting Eq. [8] into Eq. [3] gives ∇∗2g∗
1− φr∇∗φ1∗· ∇∗g1∗ = φr2Pe1v∗· ∇∗φ1∗ [21]
∇∗2g∗
2+ αφr∇∗φ1∗· ∇∗g2∗ = φr2Pe2v∗· ∇∗φ1∗. [22]
In these expressions Pej = ε(Z1e/kBT )2/ηDj, j = 1, 2, is the
electric Peclet number of ionic species j , and v∗= v/UE, UE = (εζk2/ηa) is the magnitude of the velocity of an isolated
particle based on Smoluchowski’s theory when an electric field of strength (ζa/a) is applied,
Suppose that the perturbed electrical potential arises from the induced electric field vanishes at a point far away from particle and plane surface. Then
g∗j = −φ2∗, η = 0 and ξ = 0, or η = 0. [23]
The surface of particle is impenetrable to ionic species, which implies that
∂g∗ j
∂η = 0, η = η0. [24]
The symmetric nature of the problem requires that
∂g∗ j
∂ξ = 0, ξ = 0 or ξ = π. [25]
In terms of stream function, Eq. [5] becomes, after taking curl,
µiϕ
ωE4ψ − ∇ × (ρ∇φ) = 0, [26]
whereω = c sin ξ/x is the distance from the plane ξ = π. If we let vη and vξ be the η and ξ components of velocity v, then vη = x 2 c2sinξ ∂ψ ∂ξ [27] vξ = −x 2 c2sinξ ∂ψ ∂η. [28]
Substituting Eq. [9] into Eq. [26] and rewriting the resultant expression in scaled form, we obtain
E4ψ∗ = −(κa)2x sinξ c∗ ∂φ∗ 1 ∂ξ ∂φ∗ 2 ∂η − ∂φ∗ 1 ∂η ∂φ∗ 2 ∂ξ . [29]
Here,ψ∗= ψ/UEa is the scaled stream function, and E4 = E2E2, with E2defined by E2= x 2 c∗2 ∂2 ∂η2 + ∂2 ∂ξ2 + sinhη x ∂ ∂η + 1− cos ξ cosh η x sinξ ∂ ∂ξ . [30] The following boundary conditions are assumed for the flow field: ψ∗ =−1 2 ω ∗2U∗ and ∂ψ∗ ∂η = c∗2
x3 sinξ sinh ηU
∗, η = η 0 [31] ψ∗ = 0 and ∂ψ∗ ∂ξ = 0, ξ = 0 or ξ = π [32] ψ∗= 0 and ∂ψ∗ ∂η = 0, η = 0 [33] ψ∗ r∗2 → 0, η = 0 and ξ = 0. [34]
In these expressions, ω∗= c∗sinξ/x and U∗ = U/UE.
Equations [31]–[34] imply that the velocity of the particle is U, and the fluid at infinity is stationary.
The forces acting on the particle include the electric force
FE zand the hydrodynamic force FDz, which can be evaluated
respectively by FE z= 2π π 0 σ(−∇φ) δs [35]
FIG. 2. Variation of scaled mobility (U∗/E∗) as a function ofκa at various scaled surface potentialφr without considering DLP for the caseα = 1.0 and
η0= 2.0. The dashed line represents the result based on the linearized Eq. [4].
and FDz= µπ π 0 ω3 ∂ ∂n E2ψ ω2 δs − π π 0 ω2ρe∂φ ∂sδs. [36]
In these expressions s represents the tangential surface, and the charge density on particle surfaceσ can be evaluated by
σ = εx
c ∂φ
∂η. [37]
Substituting Eq. [37] into Eq. [35] yields
FE z∗ = 2π π 0 −sinξ x ∂φ∗ ∂η × (1− cosh η cos ξ)∂φ∗ ∂η− sinh η sin ξ ∂φ∗ ∂ξ dξ. [38]
Substituting Eq. [7a] into Eq. [41] gives
FDz∗ = π π 0 −ω∗3 ∂ ∂η E2ψ∗ ω∗2 dξ − (κa)2 (1+ α)φr π × π 0 ω∗2[exp(−φ rφ1∗)(1− φr(φ∗2+ g∗1)) + exp(αφrφ∗1(1+ αφr(φ2∗+ g2∗)) ∂φ∗ ∂ξ dξ. [39]
The problem under consideration can be decomposed into two
subproblems (6). In the first, the particle moves at constant ve-locity in the absence of the applied electric field, and in the second it is held constant under the external electric field. Let f1
and f2be respectively the net forces acting on the particle in the
first and second problems. We have f1= χU∗and f2= β E∗.
At steady state the net force acting on the particle vanishes, and, therefore, its mobility can be expressed as U∗/E∗= −β/χ.
RESULTS AND DISCUSSION
The behavior of the system under consideration is examined through numerical simulation. A pseudo-spectral method based on Chebyshev polynomial (26), which is readily applicable to the present problem, is adopted for the resolution of the gov-erning equations subject to the boundary conditions assumed.
FIG. 3. Variation of scaled mobility (U∗/E∗) as function ofκa at various scaled surface potentialφrfor the case when DLP is considered with Pe1= 0.01,
The numerical procedure is introduced briefly in the Appendix. The method adopted has a fast rate of convergence, and the convergent properties are independent of the associated bound-ary conditions. Also, the mini-max property typically associated with the Chebyshev polynomial is maintained.
Figure 2 shows the variations of the scaled mobility (U∗/E∗) as a function ofκa at various scaled surface potential φr with-out considering DLP, which can be obtained by letting gj= 0
in Eq. [7], and the corresponding result when DLP is taken into account is illustrated in Fig. 3. For comparison, the results based on linearized Eq. [4], that is, those at low electrical potentials, are also presented in these figures. Figure 2 reveals that ifκa is small (i.e., thick double layer), (U∗/E∗) becomes indepen-dent of the surface potential of particle. This is because if the double layer is sufficiently thick, it will reach the planar sur-face. In this case, the movement of the particle is dominated by hydrodynamic conditions, and the effect of the electric field is less significant. Asκa increases, the presence of the planar sur-face on the movement of the particle becomes less significant, and (U∗/E∗) increases accordingly. Figure 2 also suggests that (U∗/E∗) increases with the increase in the surface potential of particle, but the degree of increase is inappreciable. As can be seen in Fig. 3, the behavior of (U∗/E∗) at smallκa when the effect of DLP is taken into account is similar to that when it is ne-glected. The situation is different, however, ifκa becomes large. This figure reveals that (U∗/E∗) decreases with the increase in the surface potential of particle, and the degree of increase is appreciable. This is because the polarization of the double layer has the effect of inducing an internal electric field, which is in the inverse direction as that of the applied electric field. Since the strength of the induced electric field increases with the surface potential of the particle, (U∗/E∗) decreases with the increase in
φr. This phenomenon can also be explained by the variation of
β as a function of κa, as illustrated in Fig. 4. According to its
FIG. 4. Variation ofβ as function of κa at various scaled surface potential
φrfor the case when DLP is considered with Pe1= 0.01, Pe2= 0.01, α = 1.0,
andη0= 2.0.
FIG. 5. Variation of scaled mobility (U∗/E∗) as function ofη0at various
scaled surface potentialφrfor the case when DLP is considered with Pe1= 0.01,
Pe2= 0.01, α = 1.0, and κa = 1.0.
definition,β is a measure for the net force acting on a particle in problem 2. Figure 4 shows that ifφr is fixed,β increases with the increase inκa. This is because the thinner the double layer, the smaller the resistant force experienced by a particle. Again, the presence of DLP has the effect of reducing the elec-tric force arising from the applied elecelec-tric field, and the larger theφr the more significant the effect DLP.
Figure 5 illustrates the variation of the scaled mobility (U∗/E∗) as a function ofη0at various scaled surface potential φrfor the case when DLP is considered. According to its defini-tion (h/a = cosh η0),η0is a measure for the distance between
particle and surface, the smaller theη0the closer the distance.
As can be seen from Fig. 5, (U∗/E∗) decreases with the decrease inη0. This is expected since the closer the distance between the
particle and the surface the more significant the hydrodynamic hindrance of the movement of the former by the latter. Figure 5 suggests that ifη0 < 1.0, the mobility of a particle becomes
in-sensitive to its surface potential. This is because if the distance between the particle and the surface is sufficiently close, the hydrodynamic effect dominates the electrical effect.
CONCLUSION
We conclude that under the condition in which a particle and a planar surface are not close, the surface potential of the farmer on its electrophoretic mobility is negligible when double layer polarization is neglected, but it becomes crucial if double layer polarization is taken into account. However, if the distance be-tween the particle and the planar surface is sufficiently close, the hydrodynamic effect dominates, and the electric effects are insignificant.
APPENDIX
In the present case, the computational domain is two dimen-sional, and the pseudo-spectral method is applied in both the
η and ξ directions. For instance, the Nth-order × Mth-order
approximation to the function fN M(η, ξ) is expressed by
fN M(η, ξ) = N i=0 M j=0 fN M(ηi, ξj)hi(η)hj(ξ), [A-1]
where fN M(ηi, ξj) is the value of fN M at the kth collocation
point, where k= [(N − 1)i + j]. The interpolation polynomials
hi(η) and hj(ξ) depend on the collocation points and these points
are determined by mapping the computational domain onto the square [−1, 1] × [−1, 1] through η = η0 2 y+ η0 2 [A-2] ξ =π 2(x+ 1). [A-3]
The N+ 1 interpolation points in the interval [−1, 1] are chosen to be the extreme values of an N th-order Chebyshev polynomial
TN(y) yj = cos π j N , j = 0, 1, . . . , N. [A-4]
The corresponding interpolation polynomial hi(y) is hj(y)= (−1)j+1 (1− y2)(d TN(y)/dy) cjN2(y− yj) j = 0, 1, . . . , N, [A-5] where cjis defined by cj = 2, j = 0, N 1, 1 ≤ j ≤ N − 1. [A-6]
Both the partial derivatives and the integration of fN M(η, ξ)
are evaluated based on Eq. [A-1]. The corresponding nonlinear system is then solved with a Newton–Raphson iteration scheme.
Double precision is used throughout the computation, and grid independence is checked to ensure that the mesh used is fine enough. For eachκa the calculations consist of 41 × 41 nodal points for (Nη× Mξ).
ACKNOWLEDGMENT
This work is supported by the National Science Council of the Republic of China.
REFERENCES
1. Zydney, A. L., J. Colloid Interface Sci. 169, 476 (1995).
2. Lee, E., Chu, J. W., and Hsu, J. P., J. Colloid Interface Sci. 196, 316 (1997). 3. Lee, E., Chu, J. W., and Hsu, J. P., J. Colloid Interface Sci. 205, 65 (1998). 4. Morrison, F. A., and Stukel, J. J., J. Colloid Interface Sci. 33, 88 (1969). 5. Keh, H. J., and Lien, L. C., J. Chin. Inst. Chem. Eng. 20, 283 (1989). 6. Keh, H. J., and Anderson, J. L., J. Fluid Mech. 153, 417 (1985). 7. Ennis, J., and Anderson, J. L., J. Colloid Interface Sci. 185, 497 (1997). 8. Feng, J. J., and Wu, W. Y., J. Fluid Mech. 264, 41 (1994).
9. Shugai, A. A., and Carnie, S. L., J. Colloid Interface Sci. 213, 298 (1999). 10. Tang, Y. P., Chih, M. H., Lee, E., and Hsu, J. P., J. Colloid Interface Sci.
242, 121 (2001).
11. Von Smoluchowski, M., Z. Phys. Chem. 92, 129 (1918). 12. Huckel, E., Phys. Z. 25, 204 (1924).
13. Henry, D. C., Proc. R. Soc. London Ser. A 133, 106 (1931). 14. Booth, F., Proc. R. Soc. London Ser. A 203, 514 (1950).
15. Wiersema, P. H., Loeb, A. L., and Overbeek, J. Th. G., J. Colloid Interface
Sci. 22, 78 (1966).
16. O’Brien, R. W., and White, L. R., J. Chem. Soc. Faraday Trans. 2 74, 1607 (1978).
17. O’Brien, R. W., and Hunter, R. J., Can. J. Chem. 59, 1878 (1981). 18. Ohshima, H., Healy, T. W., and White, L. R., J. Chem. Soc. Faraday Trans.
2 79, 1613 (1983).
19. Ohshima, H., Adv. Colloid Interface Sci. 62, 189 (1995).
20. Levine, S., and Neale, G. H., J. Colloid Interface Sci. 47, 520 (1974). 21. Kozak, M. W., and Davis, E. J., J. Colloid Interface Sci. 127, 497 (1989). 22. Kozak, M. W., and Davis, E. J., J. Colloid Interface Sci. 129, 166 (1989). 23. Ohshima, H., J. Colloid Interface Sci. 188, 481 (1997).
24. Lee, E., Chu, J. W., and Hsu, J. P., J. Colloid Interface Sci. 209, 240 (1999).
25. Happel, J., “Low Reynolds Number Hydrodynamics,” Noordhoff, Leyden, The Netherlands, 1973.
26. Canuto, C., Hussaini, M. Y., Quarteroni, A., and Zang, T. A., “Spectral Methods in Fluid Dynamics,” Springer-Verlag, Berlin/ New York, 1986.
