6198
Langmuir 2007, 23, 6198-6204
Electrophoresis of a Sphere along the Axis of a Cylindrical Pore: Effects of Double-Layer Polarization and Electroosmotic Flow Jyh-Ping Hsu* and Zheng-Syun Chen Department of Chemical Engineering, National Taiwan UniVersity, Taipei, Taiwan 10617 ReceiVed January 11, 2007. In Final Form: March 9, 2007 The electrophoresis of a rigid sphere along the axis of a cylindrical pore is investigated theoretically. Previous analysis is extended to the case where the effects of double-layer polarization and electroosmotic flow can be significant. The influences of the surface potential, the thickness of the double layer, and the relative size of a pore on the electrophoretic behavior of a sphere are discussed. Some interesting results are observed. For example, if both a sphere and a pore are positively charged, then the mobility of the sphere has a local minimum as the thickness of its double layer varies. Depending upon the level of the surface potential of a sphere and the degree of significance of the boundary effect, the mobility of the sphere may change its sign twice as the thickness of its double layer varies. This result can play a significant role in electrophoresis measurements.
1. Introduction Electrophoresis has many applications in various fields. It is not only adopted often to characterize the charged conditions of entities of colloidal size but also used widely in both conventional and modern operations. One of the key effects in conducting electrophoresis and electrophoresis-related operations is the presence of a boundary. Electrophoresis conducted in a narrow space such as capillary electrophoresis is a typical example of the former, and electrodeposition where charged particles are driven by an applied electric field toward electrodes is an example of the latter. The charged conditions on a boundary can make things even more complicated because the associated electroosmotic flow will influence not only the magnitude of the electrophoretic velocity of a particle but also its direction. The boundary effect on electrophoresis has been studied extensively in the last few decades, and various types of problems have been investigated. These include the electrophoresis of a sphere parallel to a plate,1-4 along the axis of a cylindrical pore,2-6 positioned eccentrically in a cylindrical pore,7-9 through a converging-diverging pore,10 along the center line between two parallel plates,2,4 and in a spherical cavity.11-14 Although the presence of a boundary simply yields an extra viscous force on a particle arising from the nonslip condition on the boundary, the influences of other effects such as the level of surface potential, the thickness of the electrical double layer, and the charged conditions on the boundary surface can be profound. For instance, if a boundary is charged, an electroosmotic flow is generated that will influence both qualitatively and quantitatively the electrophoretic behavior of a particle.2 The interaction between * To whom correspondence should be addressed. E-mail: jphsu@ ntu.edu.tw. Fax: 886-2-23623040. (1) Keh, H. J.; Chen, S. B. J. Fluid Mech. 1988, 194, 377. (2) Keh, H. J.; Anderson, J. L. J. Fluid Mech. 1985, 153, 417. (3) Shugai, A. A.; Carnie, S. L. J. Colloid Interface Sci. 1999, 213, 298. (4) Ennis, J.; Anderson, J. L. J. Colloid Interface Sci. 1997, 185, 497. (5) Keh, H. J.; Chiou, L. C. Am. Inst. Chem. Eng. J. 1996, 42, 1397. (6) Hsu, J. P.; Ku, M. H. J. Colloid Interface Sci. 2005, 283, 592. (7) Yariv, E.; Brenner, H. Phys. Fluids 2002, 14, 3354. (8) Ye, C.; Xuan, X.; Li, D. Microfluid Nanofluid 2005, 1, 234. (9) Hsu, J. P.; Kuo, C. C. J. Phys. Chem. B 2006, 110, 17607. (10) Davison, S. M.; Sharp, K. V. J. Colloid Interface Sci. 2006, 303, 288. (11) Zydney, A. L. J. Colloid Interface Sci. 1995, 169, 476. (12) Lee, E.; Chu, J. W.; Hsu, J. P. J. Colloid Interface Sci. 1997, 196, 316. (13) Lee, E.; Chu, J. W.; Hsu, J. P. J. Colloid Interface Sci. 1998, 205, 65. (14) Hsu, J. P.; Hung, S. H.; Kao, C. Y. Langmuir 2002, 18, 8897.
Figure 1. Schematic representation of the problem considered where a sphere of radius a is placed on the axis of a long cylindrical pore of radius b. A uniform electric field E parallel to the axis of the pore is applied. The cylindrical coordinates (r, θ, z) with the origin located at the center of the sphere are adopted.
the double layer of a particle with a boundary can also lead to complicated results.6 In this study, previous analysis of the electrophoresis of a sphere along the axis of a cylindrical pore under the conditions of low surface potential and weak applied electric field6 is extended to the case of arbitrary surface potential and doublelayer thickness. Also, the surface of a sphere and that of a pore can be charged. This implies that the effects of double-layer polarization and electroosmostic flow, both of practical significance,15-16 can be important and should be taken into account.
10.1021/la070079m CCC: $37.00 © 2007 American Chemical Society Published on Web 05/01/2007
Electrophoresis of a Sphere
Langmuir, Vol. 23, No. 11, 2007 6199
Figure 2. Variation of the scaled transition length (Ltc/a) (a) as a function of κa at φr ) ζaz1e/kBT ) 1, ζ*b ) 0.2ζ*a, and λ ) 0.5 and (b) as a function of λ () a/b) at φr ) 1, ζ*b ) 0.2ζ*a, and κa ) 1.
2. Theory Let us consider the electrophoresis of a rigid sphere of radius a along the axis of a long cylindrical pore of radius b shown in Figure 1. The liquid phase is an incompressible Newtonian fluid with constant physical properties. The cylindrical coordinates (r, θ, z) are chosen with the origin located at the center of the sphere. A uniform electric field E is applied in the z direction, and U is the corresponding electrophoretic velocity of the sphere. The symmetric nature of the system under consideration suggests that only the (r, z) domain has to be considered. For the present case, the electrical potential φ can be described by the Poisson equation
∇ 2φ ) -
F
2
)-
∑j
zj enj (1)
Here, ∇2 is the Laplace operator, is the permittivity of the liquid phase, F is the space charge density, e is the elementary charge, and nj and zj are the number concentration and the valence of ionic species j, respectively. If we let Dj, kB, (15) Wong, P. K.; Wang, T. H.; Deval, J. H.; Ho, C. M. IEEE-ASME Trans. Mech. 2004, 9, 366. (16) Stone, H. A.; Stroock, A. D.; Ajdari, A. Annu. ReV. Fluid Mech. 2004, 36, 381. (17) Wiersema, P. H.; Loeb, A. L.; Overbeek, J. Th. G. J. Colloid Interface Sci. 1966, 22, 78.
Figure 3. Variation of the scaled electrophoretic mobility µE (defined in eq 28) as a function of κa at various levels of φr () ζaz1e/kBT) for the case of ζ*b ) 0. (a) λ ) 0.3, (b) λ ) 0.5, and (c) λ ) 0.7.
T, and v be the diffusivity of ionic species j, the Boltzmann constant, the absolute temperature, and the liquid velocity, respectively, then the conservation of ionic species at steady state leads to
[ (
∇‚ -Dj ∇nj +
) ]
zje n ∇φ + njv ) 0 kBT j
(2)
For convenience, φ is decomposed into an equilibrium potential or the potential in the absence of E, φ1, and a perturbed potential
6200 Langmuir, Vol. 23, No. 11, 2007
Hsu and Chen
described by 2
∇*2φ*1 ) -
(κa) 1 [exp(-φrφ*1) - exp(Rφrφ*1)] (1 + R) φr (6)
2 where ∇*2 ) ∇2/a2 is the scaled Laplace operator and κ ) [∑j)1 2 1/2 nj0(ezj) /kBT] is the reciprocal Debye length. It can be shown that eqs 1, 3, 4, 5, and 6 lead to
∇*2φ/2 -
(κa)2 [exp(-φrφ*1) + R exp(Rφrφ*1)]φ/2 ) (1 + R) (κa)2 [exp(-φrφ*1)g/1 + exp(Rφrφ*1)Rg/2] (7) (1 + R)
after neglecting terms involving products of small quantities such as g/1, g/2, and φ/2. Similarly, the variation of g/j can be approximated by employing eqs 3-5 to obtain
Figure 4. (a) Variation of β (defined in eq 27) as a function of κa at various levels of φr () ζaz1e/kBT) and (b) that of F*ez (defined in eq 31) as a function of κa for the case when ζ*b ) 0 and λ ) 0.5.
arising from E, φ2, that is, φ ) φ1 + φ2 .18 Also, nj is expressed as18
(
nj ) nj0 exp -
)
zje(φ1 + φ2 + gj) kBT
(3)
where gj is a perturbed potential that takes account of the deformation of the double layer surrounding a particle as it moves. Note that the form of eq 3 is assumed for convenience. Until now, gj has been considered to be an arbitrary function. Under conditions of practical significance, the applied electric field is weak compared with that established by the sphere and/or by the cavity so that the expressions for the distortion of the double layer, the electric potential, and the flow field near the sphere can be linearized. For example, the scaled number concentrations of ions, n*1 and n*2, can be approximated respectively by
n*1 ) exp(φrφ*1)[1 - φr(φ*2 + g*1)]
(4)
n*2 ) exp(Rφrφ*1)[1 + Rφr(φ*2 + g*2)]
(5)
where n*j ) nj/n10, φ/j ) φj/ζa, g/j ) gj/ζa, j ) 1, 2, R ) -z2/z1, φr ) ζaz1e/kBT, and ζa is the surface potential of the sphere. In scaled quantities, the equilibrium electrical potential can be (18) O’Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1978, 74, 1607.
∇*2g/1 - φr∇*φ*1‚∇*g*1 ) φr2Pe1v*‚∇*φ*1
(8)
∇*2g/2 - Rφr∇*φ*1‚∇*g*2 ) φr2Pe2v*‚∇*φ*1
(9)
Here, ∇* ) ∇/a is the scaled gradient operator, v* ) v/UE, UE ) ζa2/ηa is the magnitude of the velocity based on Smoluchowski’s theory when an electric field of strength ζa/a is applied, η is the viscosity of the liquid phase, and Pej ) (zje/kBT)2/ηDj for j )1, 2 is the electric Peclet number of ionic species j. We assume that both the sphere and the pore are nonconductive, impermeable to ionic species, and remain at constant surface potential. Also, the concentrations of ionic species reach the bulk values at both the inlet and the outlet of a pore. These lead to the following boundary conditions:
φ*1 )
ζa on the sphere surface ζa
(10)
ζb on the pore surface ζa
(11)
φ*1 ) φ*1 )
ζb I0(κr) ζa I0(κb)
|z| f ∞
r
