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김 재 영, 윤 병 준

포항공과대학교 화학공학과

Electrophoresis of Nonspherical Particles with Nonuniform Zeta Potential

Jae Young Kim and Byung Jun Yoon

Department of Chemical Engineering and

Division of Mechanical and Industrial Engineering, POSTECH

Introduction

The electrokinetic or zeta potential of a charged colloidal particle is commonly determined by using electrophoresis measurement.  In capillary electrophoresis the velocity of a charged particle is measured either by direct observation through microscope or by using other optical techniques such as laser light scattering. The zeta potential is then estimated using an appropriate relationship between the zeta potential and the electrophoretic velocity of the particle. The separation of the electrophoretic contribution and the estimation of the zeta potential therefrom require the theoretical analysis for a mathematical model which describes the motion of a charged particle in electrophoresis experiment.

The electrophoretic motion depends on many factors: the particle shape, the zeta potential distribution, the concentration of ions, and the electric field condition around the particle. For particles of simple shape there exists several theories for their electrophoretic motion under various conditions. But, when the particle shape is complex, the electrophoresis theories require more complicated analysis. In this study, we develope the electrophoresis theory for a slightly deformed sphere. The zeta potential distribution of the particle is nonuniform and the thickness of the electrical double layer is arbitrary.

Theory

The surface of a slightly deformed sphere may be represented by [eqn: r=a[1+beta f(theta, phi)]]

(1)

using a small parameter[eqn: beta]. Here,[eqn: (r, theta, phi)] are the spherical polar coordinates having their origin at the center of the undeformed sphere with radius a. The shape function [eqn: f] is an arbitrary function of[eqn: theta] and[eqn: phi]. The shape function is typically expressed in terms of the spherical harmonics [1]. However, in this work we use the multipole expansion form,

[eqn: f(theta, phi) = {rm M}^f + {rm boldD}^f cdot {rm boldnn} + {rm boldQ}^f bold: {rm boldnn} ,]

(2)

where

[eqn: {rm M}^f =&{1 over 4pi } int f~ \d{Omega} ,~~ {rm boldD}^f ={3 over 4pi} int {rm boldn} f~ \d{Omega}]

(3a,b)

[eqn: {rm boldQ}^f ={5 over 8pi} int (3 {rm boldnn} -bolddelta ) f~ \d{Omega}.]

(3c)

Here, [eqn: Omega] denotes the solid angle and[eqn: rmboldn] is the outward unit normal to the sphere. Note that the quadrupole moment[eqn: {rm boldQ}^f] is symmetric and traceless. By varying the components of[eqn: {rm boldD}^f] and[eqn: {rm boldQ}^f] in Eq. (2), one can get a wide variety of particle shapes. The zeta potential of the particle is assumed to be an arbitrary function [eqn: zeta(theta, phi)] of position on the particle surface. Following Eq. (2), the function [eqn: zeta] is expressed as

[eqn: zeta(theta, phi) = rm M^{zeta} + {boldD}^{zeta} cdot boldn + {boldQ}^{zeta} bold: boldnn .]

(4)

Following Eqs. (3a-c), the multipole moments of [eqn: zeta] are defined accordingly.

For a freely moving particle, at steady state, there is no force and torque on the particle. Ignoring the inertia effects, the force and torque balance equations are given by [2]

[eqn: rm bold A cdot bold U + boldB cdot bold omega =-epsilon int nabla Phi ~nabla^{2}psi ~\dV+ epsilon int boldV cdot nabla Phi~nabla^{2}psi~ \dV ,]

(5)

[eqn: rm {bold B}^{t}cdot boldU + boldC cdot boldomega =-&epsilon int (boldx - {boldx}^\c)boldtimes{nabla Phi}nabla^2 psi~ \dV # & + epsilon int rm boldW cdot {nabla Phi} nabla^2 psi~ \dV.]

(6)

Here,[eqn: rm {boldx} ^c] is a center of the particle, [eqn: epsilon] is the dielectric constant of the electrolyte solution. The second order tensors A, B and C are the hydrodynamic resistance tensors and[eqn: rm{bold B}^t] is the transpose of B. The tensorial velocities V and W are the Stokes flow fields around the particle. The electric potential[eqn: Phi] is due to the externally applied electric field with uncharged particle and satisfies the Laplace equation. The potential[eqn: psi], which is called by the electrical double layer potential, is due to the charged particle without external field. It is assumed that the electrical double layer potential is sufficiently low to satisfy the linearized Poisson-Boltzmann equation.

When b is small, a field variable such as the fluid velocity, the electrical potential[eqn: Phi], and the double layer potential [eqn: psi], can be expanded by the power series of b. Then the zeroth order field variables are for the spherical particle [3]. The boundary conditions of the first order problems can be obtained from the Taylor expansion of the field variable about [eqn: r=a], and we can set the first order problems within the spherical geometry. Furthermore, the integrals in Eqs. (5-6) can be approximated to the integrals over outside the undeformed sphere in the accuracy of the order of [eqn: beta]. Brenner [1] solved the Stokes problems around a deformed sphere upto the order of [eqn: beta] using Lamb's general solution and obtained the hydrodynamic resistance tensors and the tensorial velocities. We transform the Brenner's solution into the multipole expansion form that is suitable to our purpose. The first order problems for [eqn: Phi~ \and~ psi] may be solved analytically in the form of multipole expansion. After substitution of all the solutions into the Eqn. (5-6), one can determine the  electrophoretic motion of the particle.

Results and Discussion

Generally, the translational and rotational motion of a particle are coupled, Eqs. (5-6). But for a slightly deformed sphere, there is a point[eqn: rm {boldx} ^c] that makes[eqn: rm {boldB} = 0] and decouples the Eqs. (5-6) in the accuracy of the order of b. Since the integrals in Eqs. (5-6) are linear to the applied electric field[eqn: rm {boldE}^{infty}], the electrophoretic motions can be rewritten by

[eqn: {rm boldU ={epsilon over mu} psi^c boldM cdot {boldE}^{infty}},~~~ bold omega={epsilon over {mu a}}psi^c {rm boldN cdot {boldE}^{infty}, }]

(7a,b)

where M and N are the electrophoretic mobility tensors, and [eqn: psi^c] is a characteristic potential, and [eqn: mu] is the viscosity of the electrolyte solution. The angular integrations in the volume integrals are performed analytically over the solid angle. The resulting expressions of the mobility tensors contain the moments of [eqn: f], the moments of [eqn: zeta], and functions of ka, where 1/k is the Debye length. The dimensionless number ka represents the thickness of the electrical double layer. Large value of ka means that the thickness of the electrical double layer is thin relative to the particle dimension.

For simplicity, consider an axisymmetric problem where the electrophoretic mobility tensor M can be expressed by

[eqn: rm boldM = M_{||} boldd boldd - M_{BOT} (bolddelta - boldd boldd ) ,]

(8)

where[eqn: rmboldd] denotes the unit directional vector along the axis of revolution. When the particle shape is axisymmetric, the dipole moment of[eqn: f] is parallel to the axis of revolution and the quadrupole moment of[eqn: f] is diagonal and traceless. The moments of zeta potential follow the same characteristics. A typical trajectory of an axisymmetric particle with[eqn: beta=0.2] and ka = 1 is shown in Fig. 1. The particle shape is given by[eqn: f = -2 \cos theta + \cos^2 theta] and the zeta potential is [eqn: zeta/psi^c = 4/3 + \cos theta - \cos^2 theta]. The alignment of the particle along the applied electric field occurs over a relatively short time. The electrophoretic mobility components [eqn: rm M_{||}~ \and~ M_{BOT}] for the same particle at arbitrary ka are shown in Fig. 2. The results for the spherical particles with the same zeta potential distribution are also shown. As ka increases, the variation of [eqn: zeta] affects the electrophoretic mobility more strongly and thus the difference between [eqn: rmM_{||}~ \and~ M_{BOT}] becomes more pronounced.

From the trajectory analysis we can see that, when random Brownian rotation is not strong enough, axisymmetric particles including spherical particles with non uniform zeta potential may prefer to align along the electric field. For such particles the electrophoresis measurement only furnishes the value of [eqn: rmM_{||}]. Thus proper interpretation of the electrophoresis measurement of such particles requires a careful analysis.

References

1.  H. Brenner, The Stokes resistance of slightly deformed sphere, Chem. Eng. Sci.,      19, 519 (1964).

2.  M. Teubner, The motion of charged colloidal particles in electric fields, J. Chem.      Phys., 76, 5564 (1982).

3. B. J. Yoon, Electrophoretic motion of spherical particles with a nonuniform           charge distribution, J. Colloid Interface Sci., 142, 575 (1991).

Figure 1.  Trajectory and snapshots for the electrophoretic motion of an axisymmetric particle which lacks fore-aft symmetry.

Figure 2.  Normalized electtrophoretic mobility components for spheres and the axisymmetric particles considered in Figure 1.