doi:10.1016/j.jcis.2007.01.030
Copyright © 2007 Elsevier Inc. All rights reserved.
Effect of a charged boundary on electrophoresis: A sphere at an arbitrary position in a spherical cavity
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Jyh-Ping Hsu
, a, 
, Li-Hsien Yeha and Zheng-Syun Chena
aDepartment of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan
Received 7 December 2006;
accepted 11 January 2007.
Available online 15 February 2007.
Abstract
The effect of the presence of a charged boundary on the electrophoretic behavior of a particle is investigated by considering a sphere at an arbitrary position in a spherical cavity under conditions of low surface potential and weak applied electric field. Previous analyses are modified by using a more realistic electrostatic force formula and several interesting results, which are not reported in the literature, are observed. We show that the qualitative behavior of a particle depends largely on its position, its size relative to that of a cavity, and the thickness of the electric double layer. In general, the presence of a cavity has the effect of increasing the conventional hydrodynamic drag on a particle through a nonslip condition on the former. Also, a decrease in the thickness of the double layer surrounding a sphere has the effect of increasing the electrostatic force acting on its surface so that its mobility increases. However, this may not be the case when an uncharged particle in placed in a positively charged cavity, where the electroosmotic flow plays a role; for example, the mobility can exhibit a local maximum and the direction of electrophoresis can change.
Graphical abstract
Electrophoresis of a sphere at an arbitrary position in a spherical cavity.
Keywords: Electrophoresis; Effect of charged boundary; Sphere in spherical cavity
Fig. 1. The problem considered where a spherical particle of radius a is placed at an arbitrary position in a spherical cavity of radius b. A uniform electric field E0 parallel to the z-direction is applied. The centers of the particle and the cavity are at z=m and z=0, respectively, and θ is the solid angle.
Fig. 2. Variation of scaled electrophoretic mobility U* (a) as a function of λ (=a/b) at κa=1 and (b) as a function of κa at λ=0.4 for the case where an uncharged sphere is placed at the center of a positively charged spherical cavity. Solid curve, present result; discrete symbols, result of Zydney [2]. Key: 
, 
.
Fig. 3. Variation of (a) scaled hydrodynamic force coefficient D* and (b) scaled electrophoretic mobility U* as a function of P at various values of λ for the case of a positively charged sphere in an uncharged spherical cavity at 
, 
, and κa=1. Solid curves, present result; discrete symbols, results of Hsu et al. [6].
Fig. 4. Variation of (a) scaled electrostatic force 
and scaled excess hydrodynamic force 
and (b) scaled net driving force (
) as functions of P at various values of λ for the case of Fig. 3.
Fig. 5. Variation of (a) scaled electrophoretic mobility U* as a function of P at various values of κa and (b) as a function of κa at various values of P. Key: 
, 
, and λ=0.4.
Fig. 6. Variation of (a) scaled hydrodynamic force coefficient D* and (b) scaled electrophoretic mobility U* as functions of λ at various values of P. Key: 
, 
, and κa=1.
Fig. 7. Variation of scaled electrophoretic mobility U* as a function of P at various values of λ. Key: 
, 
, and κa=1.
Fig. 8. Variation of (a) scaled electrostatic force 
and scaled excess hydrodynamic force 
and (b) scaled net driving force (
) as a function of P at various values of λ for the case of Fig. 7.
Fig. 9. Some typical flow fields for the case of Fig. 7. (a) P=70% and λ=0.2; (b) P=95% and λ=0.2; (c) P=70% and λ=0.7; (d) P=95% and λ=0.7.
Fig. 11. Variation of (a) scaled electrophoretic mobility U* and (b) scaled electrostatic force 
and scaled excess hydrodynamic force 
as functions of κa at various values of P. Key: same as in Fig. 10.
Fig. 12. Variation of (a) scaled electrophoretic mobility U* and (b) scaled electrostatic force 
and scaled excess hydrodynamic force 
as functions of λ at various values of P. Key: Same as in Fig. 7.
Corresponding author. Fax: +886 2 23623040.
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