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Langmuir 2008, 24, 390-398
Electrophoresis of a Colloidal Sphere in a Spherical Cavity with Arbitrary Zeta Potential Distributions and Arbitrary Double-Layer Thickness Huan J. Keh* and Tzu H. Hsieh Department of Chemical Engineering, National Taiwan UniVersity, Taipei 10617, Taiwan, Republic of China ReceiVed August 4, 2007. In Final Form: September 25, 2007 The electrophoretic motion of a dielectric sphere situated at the center of a spherical cavity with an arbitrary thickness of the electric double layers adjacent to the particle and cavity surfaces is analyzed at the quasisteady state when the zeta potentials associated with the solid surfaces are arbitrarily nonuniform. Through the use of the multipole expansions of the zeta potentials and the linearized Poisson-Boltzmann equation, the equilibrium double-layer potential distribution and its perturbation caused by the applied electric field are separately solved. The modified Stokes equations governing the fluid velocity field are dealt with using a generalized reciprocal theorem, and explicit formulas for the electrophoretic and angular velocities of the particle valid for all values of the particle-to-cavity size ratio are obtained. To apply these formulas, one only has to calculate the monopole, dipole, and quadrupole moments of the zeta potential distributions at the particle and cavity surfaces. In some limiting cases, our result reduces to the analytical solutions available in the literature. In general, the boundary effect on the electrophoretic motion of the particle is a qualitatively and quantitatively sensible function of the thickness of the electric double layers relative to the radius of the cavity.
1. Introduction When a charged particle is suspended in an electrolyte solution, it is surrounded by a diffuse cloud of ions carrying a total charge equal and opposite in sign to that of the particle. This distribution of fixed surface charge and adjacent diffuse ions is known as an electric double layer. For a charged particle subjected to an external electric field, a force is exerted on both parts of the double layer. The particle is attracted toward the electrode of its opposite sign, while the ions in the diffuse layer migrate in the other direction. This particle motion is called electrophoresis and has long been applied to the particle characterization and separation in a variety of colloidal systems. On the other hand, the fluid is dragged to flow by the motions of the particle and of the ions. To evaluate the electrophoretic mobility of the particle, it is usually necessary to first solve for the electric potential and velocity fields in the fluid phase. Henry1 was the first person who took into account the finite thickness of the electric double layer to analyze the electrophoresis of a colloidal particle. Assuming that the double layer is not distorted from the equilibrium state, he derived an analytical formula for the electrophoretic mobility of a uniformly charged sphere of radius a with small zeta potential ζp for the entire range of κa, where κ-1 is the Debye screening length characterizing the thickness of the double layer. In the two limits κa f ∞ and κa f 0, this result reduces to the well-known Smoluchowski equation and Huckel expression, respectively. Taking the doublelayer distortion from equilibrium as a perturbation, O’Brien and White2 performed a numerical calculation for the electrophoretic mobility of a dielectric sphere in electrolyte solutions, which was applicable to arbitrary values of ζp and κa. Beyond that, analytical expressions for the electrophoretic mobility of a * To whom correspondence should be addressed. Fax: +886-223623040. E-mail:
[email protected]. (1) Henry, D. C. Proc. R. Soc. London, Ser. A 1931, 133, 106. (2) O’Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1978, 74, 1607.
spherical3-5 or nonspherical6-8 particle surrounded by a thin but polarized double layer in electrolyte solutions have also been obtained. These numerical and analytical results indicate that the polarization effect (or relaxation effect) of the double layer impedes the particle’s movement because an opposite electric field is induced in the distorted ion cloud, which acts against the motion of the particle. This polarization effect disappears for the two limiting cases κa f ∞ and κa f 0 and for the situation of low ζp. In practical applications of electrophoresis, colloidal particles are not isolated and will move in the presence of neighboring boundaries. For example, electrophoresis in porous gels or membranes could be applied to permit separations based on both the size and the charge of the particles.9 In capillary electrophoresis, gels in the capillary column can minimize the particle diffusion, prevent the particle adsorption to the capillary walls, and eliminate electroosmosis, while serving as the anticonvective media.10 Deep electrophoresis penetration and deposition of inert colloidal particles over the interstitial surfaces of porous composites has been suggested in the aerospace industry to protect the composites from burning or deterioration.11 Therefore, the boundary effects on electrophoresis are of great importance and have been studied extensively in the past for various geometries of uniformly charged spheres and boundaries in the cases of infinitesimally thin electric double layers,12-21 thin but polarized double layers,22-24 and thick double layers.25,26 In fact, many colloidal particles have heterogeneous surface structure or chemistry and are nonuniformly charged. For instance, (3) Dukhin, S. S.; Derjaguin, B. V. In Surface and Colloid Science; Matijevic, E., Ed.; Wiley: New York, 1974; Vol. 7. (4) O’Brien, R. W. J. Colloid Interface Sci. 1983, 92, 204. (5) Chen, S. B.; Keh, H. J. J. Fluid Mech. 1992, 238, 251. (6) O’Brien, R. W.; Ward, D. N. J. Colloid Interface Sci. 1988, 121, 402. (7) Keh, H. J.; Chen, S. B. Langmuir 1993, 9, 1142. (8) Keh, H. J.; Huang, T. Y. J. Colloid Interface Sci. 1993, 160, 354. (9) Jorgenson, J. W. Anal. Chem. 1986, 58, 743A. (10) Ewing, A. G.; Wallingford, R. A.; Olefirowicz, T. M. Anal. Chem. 1989, 61, 292A. (11) Haber, S.; Gal-Or, L. J. Electrochem. Soc. 1992, 139, 1071.
10.1021/la702399u CCC: $40.75 © 2008 American Chemical Society Published on Web 12/18/2007
Electrophoresis of a Sphere in a Spherical CaVity
elementary clay particles are flat disks with edges having a different surface charge density or zeta potential than the faces. Distributions of surface charge or potential for particles can also result from aggregation of different species of colloids. Even if a particle is homogeneously charged on its surface, an applied electric field could cause rearrangement of these charges if they are mobile.27 A distribution of zeta potential on particle surfaces has been found to lead to colloidal instability; even the average zeta potential should be sufficiently high to keep the suspension stable.28,29 The electrophoretic motion of a dielectric particle with nonuniform zeta potential and thin electric double layer was first analyzed thoroughly by Anderson,30,31 although it had also been discussed to some extent earlier.32 It was found that, in terms of the multipole moments of the zeta potential, the electrophoretic mobility of the particle depends not only on the monopole moment (area-averaged zeta potential) but also on the quadrupole moment, and the dipole moment contributes to the particle rotation, which tends to align the particle with the electric field. This analysis was later extended to the cases of a nonuniformly charged particle with a thin but polarized double layer33 and a thick double layer.34-37 On the other hand, the idea that particles can have random surface charge nonuniformity has also been demonstrated experimentally.38,39 The electrophoretic motion of nonuniformly charged particles in the proximity of confining walls could also be encountered in some real situations. In addition to the possible examples mentioned above, the translation and rotation of each of an array of nonuniformly charged bichromal spheres in its own elastomermade and solvent-filled cavity with either a monopole or a dipole on its wall controlled by imposing a voltage of either positive or negative polarity have been applied to a technology of electric paper displays (known as Gyricon displays).40,41 Also, an electrophoretic positioning process has been employed in electronic fabrication techniques for assembling individual microdevices, such as an InGaAs light-emitting diode or a nanowire, which are nonuniformly charged and must have all electric contacts available on one surface, onto the contact electrodes of a silicon circuit by biasing the contacts to control (12) Morrison, F. A.; Stukel, J. J. J. Colloid Interface Sci. 1970, 33, 88. (13) Keh, H. J.; Anderson, J. L. 1985 J. Fluid Mech. 1985, 153, 417. (14) Keh, H. J.; Chen, S. B. J. Fluid Mech. 1988, 194, 377. (15) Keh, H. J.; Lien, L. C. J. Fluid Mech. 1991, 224, 305. (16) Keh, H. J.; Horng, K. D.; Kuo, J. J. Fluid Mech. 1991, 231, 211. (17) Feng, J. J.; Wu, W. Y. J. Fluid Mech. 1994, 264, 41. (18) Loewenberg, M.; Davis, R. H. J. Fluid Mech. 1995, 288, 103. (19) Keh, H. J.; Chiou, J. Y. AIChE J. 1996, 42, 1397. (20) Hao, Y.; Haber, S. Int. J. Multiphase Flow 1998, 24, 793. (21) Yariv, E.; Brenner, H. J. Fluid Mech. 2003, 484, 85. (22) Venema, P. J. Fluid Mech. 1995, 282, 45. (23) Keh, H. J.; Jan, J. S. J. Colloid Interface Sci. 1996, 183, 458. (24) Chen, P. Y.; Keh, H. J. J. Colloid Interface Sci. 2005, 286, 774. (25) Zydney, A. L. J. Colloid Interface Sci. 1995, 169, 476. (26) Ennis, J.; Anderson, J. L. J. Colloid Interface Sci. 1997, 185, 497. (27) Bazant, M. Z.; Squires, T. M. Phys. ReV. Lett. 2004, 92, 066101. (28) Miklavic, S. J.; Chan, D. Y. C.; White, L. R.; Healy, T. W. J. Phys. Chem. 1994, 98, 9022. (29) Grant, M. L.; Saville, D. A. J. Colloid Interface Sci. 1995, 171, 35. (30) Anderson, J. L. J. Colloid Interface Sci. 1985, 105, 45. (31) Fair, M. C.; Anderson, J. L. J. Colloid Interface Sci. 1989, 127, 388. (32) Teubner, M. J. Phys. Chem. 1982, 76, 5564. (33) Solomentsev, Y. E.; Pawar, Y.; Anderson, J. L. J. Colloid Interface Sci. 1993, 158, 1. (34) Yoon, B. J. J. Colloid Interface Sci. 1991, 142, 575. (35) Velegol, D.; Feick, J. D.; Collins, L. R. J. Colloid Interface Sci. 2000, 230, 114. (36) Kim, J. Y.; Yoon, B. J. J. Colloid Interface Sci. 2002, 251, 318. (37) Kim, J. Y.; Yoon, B. J. J. Colloid Interface Sci. 2003, 262, 101. (38) Feick, J. D.; Velegol, D. Langmuir 2002, 18, 3454. (39) Feick, J. D.; Chukwumah, N.; Noel, A. E.; Velegol, D. Langmuir 2004, 20, 3090. (40) Crawford, G. P. IEEE Spectrum 2000, 37 (10), 40. (41) Crowley, J. M.; Sheridon, N. K.; Romano, L. J. Electrost. 2002, 55, 247.
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Figure 1. Geometric sketch for the electrophoresis of a colloidal sphere in a concentric spherical cavity.
the placement of these devices with the precision required.42,43 Recently, the electrophoretic motions of a dielectric sphere in a concentric spherical cavity with nonuniform zeta potential distributions at the solid surfaces44 and that of a nonuniformly charged circular cylinder near a plane wall parallel to its axis45 have been analytically investigated for the case of infinitesimally thin electric double layers, and explicit expressions for the translational and angular velocities of the particles in terms of the monopole, dipole, and quadrupole moments of the zeta potentials were obtained. The objective of this paper is to extend the previous work44 for the electrophoretic motion of a dielectric sphere in a concentric spherical cavity with arbitrary distributions of the zeta potentials at the solid surfaces but very thin electric double layers to the general case with an arbitrary thickness of the double layers. Through the use of the multipole expansions of the zeta potentials and a generalized reciprocal theorem, the linearized PoissonBoltzmann equation and the modified Stokes equations are solved for the equilibrium double-layer potential distribution and the velocity field, respectively, for the fluid phase. The geometric symmetry in this model system allows an analytical solution of the translational and angular velocities of the electrophoretic particle to be obtained (as given by eqs 28-31 and illustrated in Figures 2-7).
2. Analysis We consider the electrophoretic motion of a dielectric spherical particle of radius a and zeta potential ζp in a concentric spherical cavity (or pore) of radius b and zeta potential ζw filled with an electrolyte solution, as illustrated in Figure 1, at the quasisteady state. Both ζp and ζw can be nonuniform and are taken as arbitrary functions of the position over the particle and cavity surfaces. The applied electric field (or the external electric field in the absence of the particle) is constant and equals E∞ez, where ez is the unit vector in the positive z direction. The rectangular coordinates (x, y, z) and spherical coordinates (r, θ, φ) are established with their origin at the particle and cavity center. Gravitational effects are ignored. The purpose is to obtain the electrophoretic velocity of the particle when the thickness of the electric double layers adjacent to the particle and cavity surfaces is arbitrary relative to the particle or cavity radius. To determine the electrophoretic velocity of the particle inside the cavity, it is necessary to ascertain the electric potential distribution and velocity field in the fluid phase. (42) Edman, C. F.; Swint, R. B.; Gurtner, C.; Formosa, R. E.; Roh, S. D.; Lee, K. E.; Swanson, P. D.; Ackley, D. E.; Coleman, J. J.; Heller, M. J. IEEE Photonics Technol. Lett. 2000, 12, 1198. (43) Smith, P. A.; Nordquist, C. D.; Jackson, T. N.; Mayer, T. S.; Martin, B. R.; Mbindyo, J.; Mallouk, T. E. Appl. Phys. Lett. 2000, 77, 1399. (44) Keh, H. J.; Hsieh, T. H. Langmuir 2007, 23, 7928. (45) Hsieh, T. H.; Keh, H. J. J. Colloid Interface Sci. 2007, 315, 343.
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2.1. Electric Potential Distribution. Following previous analyses1,25,26,32,34-37 of electrophoresis, we assume that the total electric potential ψ(r, θ, φ) in the fluid phase between the particle and the cavity wall can be expressed as the linear superposition of the potential ψ1(r, θ, φ) that would exist in the absence of the imposed electric field E∞ez (induced by the fixed surface charges and mobile ions) and the potential ψ2(r, θ) that arises from this imposed electric field,
ψ ) ψ 1 + ψ2
(1)
This linear superposition is valid if E∞ is weak relative to the field associated with the electric double layers, so the distribution of the diffuse ions is hardly distorted from the equilibrium state. Using the Debye-Huckel approximation applicable for the case of low electric potentials, the equilibrium double-layer potential distribution ψ1 is governed by the linearized PoissonBoltzmann equation,
∇2ψ1 ) κ2ψ1
(2)
where κ is the Debye screen parameter. The boundary conditions for ψ1 at the particle surface and cavity wall are simply that
r ) a: ψ1 ) ζp
(3a)
r ) b: ψ1 ) ζw
(3b)
The general solution of eq 2 in spherical coordinates is34 ∞
ψ1 )
Qp,w )
1 Sp,w
θ)e
imφ
(4)
where
Kn(x) ) e
(-1) ∑ s)0
Ln(x) ) e
-x
(2x)sn!(2n - s)! (2n)!s!(n - s)!
(5a)
(2x)sn!(2n - s)!
n
∑ s)0
(2n)!s!(n - s)!
(5b)
Pmn is the associated Legendre function of order m and degree n, and anm and bnm are unknown coefficients to be determined from the boundary conditions. The zeta potentials ζp and ζw are arbitrary functions of (θ, φ) and can be expressed in terms of the multipole expansions,30,36
5 ζp ) Mp + 3Dp‚n + Qp:nn 2
(6a)
5 ζw ) Mw + 3Dw‚n + Qw:nn 2
(6b)
where Mp,w, Dp,w, and Qp,w are the monopole, dipole, and quadrupole moments of ζp,w, respectively, defined by the following integrals over the particle surface Sp and cavity surface Sw:
Mp,w ) D
p,w
1 Sp,w
1 ) p,w S
∫S
∫S
p,w
p,w
ζp,wdS
5 W1Dw)‚n + (P2Qp + W2Qw):nn (8) 2 where
Pn )
Kn(κr)Ln(κb) - Kn(κb)Ln(κr) a n+1 Kn(κa)Ln(κb) - Kn(κb)Ln(κa) r
(9a)
Wn )
Kn(κa)Ln(κr) - Kn(κr)Ln(κa) b n+1 Kn(κa)Ln(κb) - Kn(κb)Ln(κa) r
(9b)
()
()
∂ψ2 ) 0 at r ) a ∂r
(11)
The above equation follows from the continuity of electric current at the particle surface, although O’Brien and White2 have shown that the choice of this boundary condition does not affect the final result of the particle velocity. At the surface of the cavity wall, the electric potential distribution gives rise to the applied electric field E∞ez when the particle does not exist. Thus, a reasonable choice of the boundary condition there is
ψ2 ) -E∞r cos θ at r ) b
(12)
Here, we have set ψ2 ) 0 on the plane z ) 0 for convenience without the loss of generality. The solution of eq 10 subject to boundary conditions 11 and 12 is
(7a) (7b)
(10)
This Laplace equation instead of the Poisson-Boltzmann equation is used to obtain the electric potential field in the case where the particle and cavity walls are neutral (zeroth order of a perturbation in the zeta potentials ζp and ζw), which, together with the potential field ψ1 (zeroth order of a perturbation in the applied electric field E∞), will be sufficient to determine the particle velocity correct to first orders in ζp, ζw, and E∞.26 Since the particle is assumed to be perfectly insulating (surface conductance is neglected), the boundary condition for ψ2 at the surface of the particle is
ψ2 ) ζp,wndS
(7c)
ψ1 ) P0Mp + W0Mw + 3(P1Dp +
∇2ψ2 ) 0
bnmLn(κr)]Pmn (cos
s
ζp,w(3nn - I)dS
n is the unit normal vector on the particle surface pointing toward the fluid phase, and I is the unit dyadic. Various nonuniform zeta potentials ζp,w can result from eq 6, in which the higher-order moments (which make no contribution to the electrophoretic velocity of the particle, as will be discussed later) are neglected, with appropriate choices of the moments Mp,w (i.e., area-averaged zeta potentials), Dp,w, and Qp,w (which are symmetric and traceless). Applying eq 3 to eq 4 with the zeta potentials expressed by eq 6, we obtain the equilibrium electrostatic potential distribution ψ1 as an explicit function of the moments Mp,w, Dp,w, and Qp,w:
r-n-1[anmKn(κr) + ∑ ∑ n)0 m)-n
n
p,w
The electric potential distribution ψ2 developed in the presence of the external electric field is governed by
n
x
∫S
where λ ) a/b.
2E∞ 2 + λ3
(
r+
)
a3 cos θ 2r2
(13)
Electrophoresis of a Sphere in a Spherical CaVity
Langmuir, Vol. 24, No. 2, 2008 393
The boundary condition at the cavity wall may alternatively be taken as the local electric potential gradient is equal in magnitude to the prescribed electric field. In this case, the Dirichlet approach given by eq 12 becomes the following Neumann approach:25,46
∂ψ2 ) -E∞ cos θ at r ) b ∂r
(
)
a3 r + 2 cos θ ψ2 ) 3 1-λ 2r
(15)
In fact, eq 13 predicts that the electric potential at the particle surface (r ) a) is decreased by the presence of the cavity by a factor of (1 + λ3/2)-1, whereas eq 15 suggests that this potential is increased by a factor of (1 - λ3)-1. In the limit λ f 0, as expected, eqs 13 and 15 become identical and reduce to the potential distribution for a nonconducting sphere in an unbounded medium. As it was discussed previously44 and as the results in the following section illustrate, the potential distribution given by eq 13, which also satisfies the condition that its tangential gradient at the cavity wall is consistent with the applied electric field, might be a better choice in comparison with that in eq 15. 2.2. Fluid Velocity Field. With knowledge of the solution for the electric potential distribution in the fluid phase between the particle and the cavity wall, we can now proceed to deal with the fluid velocity field. Because the Reynolds numbers of electrokinetic flows are small, the fluid motion is governed by the Stokes equations modified with an additional electrostatic body force term,
η∇2v ) ∇p + F∇ψ
(16a)
∇‚v ) 0
(16b)
where v and p are the fluid velocity and dynamic pressure distributions, respectively, η is the fluid viscosity, and F is the space charge density. Using Poisson’s equation and eqs 1, 2, and 10, one obtains
F ) -κ2ψ1
(19a)
∇‚vj ) 0
(19b)
the no-slip constraint at the cavity wall,
vj ) 0 at r ) b (14)
Note that, although the normal component of the electric potential gradient at the cavity wall given by this boundary condition is consistent with the applied electric field, its tangential (angular) component is not specified. The solution of eq 10 subject to eq 11 and boundary condition 14 is given by
E∞
η∇2vj ) ∇pj
(17)
and some boundary condition at the particle surface r ) a. A generalization of the reciprocal theorem of Lorentz32,47 for the transformation between the charged and uncharged systems with the application of eqs 18b and 20 leads to
∫ ∫r)a v‚Πh ‚ndS ) ∫ ∫r)a vj‚Π‚ndS + ∫ ∫ ∫a
