Phoretic Motions of Colloidal Spheres within a Spherical Cavity
Date Issued
2014
Date
2014
Author(s)
Lee, Tai-Cheng
Abstract
Driven by a temperature, electric potential, or solute concentration gradient, the transport of colloidal particles in a continuous medium is known as the “phoretic motion”. In this work, a boundary collocation method is used to calculate semi-analytically the various phoretic velocities of a small spherical particle at an arbitrary position within a spherical cavity in the direction parallel or perpendicular to the line connecting the particle and cavity centers at the quasi-steady state.
First, in Chapter 2, the thermocapillary migration of a spherical fluid drop situated at an arbitrary position in a second fluid within a spherical cavity is studied in the limit of negligible Marangoni and Reynolds numbers. The imposed temperature gradient is parallel or perpendicular to the line through the drop and cavity centers. To solve the thermal and hydrodynamic governing equations, the general solutions are constructed from the fundamental solutions in the two spherical coordinate systems based on the drop and cavity. The boundary conditions at the drop surface and cavity wall are satisfied by the collocation technique. Numerical results for the thermocapillary migration velocity of the drop normalized by its value in an unbounded medium are presented for various values of the relative viscosity and thermal conductivity of the drop, the relative conductivity of the cavity phase, the drop-to-cavity radius ratio, and the relative distance between the drop and cavity centers. In the particular case of the migration of a spherical drop in a concentric cavity, these results agree excellently with the exact solution derived analytically. The normalized thermocapillary migration velocity decreases with increases in the drop-to-cavity radius ratio and in the relative distance between the drop and cavity centers, vanishing as the drop surface touches the cavity wall. For a given configuration, this velocity augments with increases in the relative viscosity of the drop and thermal conductivity of the cavity phase. In general, the boundary effects on the thermocapillary motion perpendicular to the line connecting the drop and cavity centers is weaker than that parallel to this line.
An investigation is presented in Chapter 3 for the electrophoretic motion of a dielectric colloidal sphere located at an arbitrary position inside a charged spherical cavity filled with an ionic fluid. The applied electric field is parallel or perpendicular to the line through the centers of the particle and cavity, and the electric double layers adjacent to the solid surfaces are assumed to be much thinner than the particle radius and any gap width between the surfaces. The general solutions to the Laplace and Stokes equations governing the electric potential and fluid velocity fields, respectively, are established from the superposition of their basic solutions in the two spherical coordinate systems about the two centers, and the boundary conditions are satisfied by the collocation method. Results for the translational and angular velocities of the confined particle are obtained for various cases. When the particle is positioned at the center of the cavity, these results are in excellent agreement with the available analytical solution. The effects of the cavity wall on the electrokinetic motion of the particle are interesting, complicated, and significant. In general, the electrophoretic translational/rotational mobility of the particle decreases/increases with increases in the particle-to-cavity radius ratio and in the relative distance between the particle and cavity centers (the direction of rotation is opposite to that of a corresponding settling particle), but there exist some exceptions. The direct and recirculating cavity-induced electroosmotic flows can strengthen or weaken the electrophoretic translation and rotation of the particle and even reverse their directions, depending on the cavity-to-particle zeta potential ratio and geometric parameters. The effect of the cavity wall on the electrokinetic translation of a particle perpendicular to the line connecting their centers only slightly differs from that parallel to this line.
In Chapter 4, the osmophoretic motion of a spherical vesicle with a semipermeable membrane located at an arbitrary position within a spherical cavity filled with a fluid solution is studied, where a constant solute concentration gradient is imposed in an arbitrary direction with respect to the line connecting the centers of the vesicle and cavity. The general solutions of conservation equations for the solute species and fluid momentum are constructed from the superposition of fundamental solutions in the two spherical coordinate systems based on the vesicle and cavity, and the boundary conditions are satisfied by the collocation method. The translational and rotational velocities of the osmophoretic vesicle are calculated for various cases. In the particular case of the osmophoresis of a vesicle in a concentric cavity, the result agrees excellently with the available exact solution. The effects of the cavity wall on osmophoresis are significant and interesting. In general, the normalized translational and rotational velocities of the osmophoretic vesicle increase with increases in the vesicle-to-cavity radius ratio and its relative distance from the cavity center, and the translational velocity deflects little from the imposed solute concentration gradient. The direction of rotation of a confined vesicle undergoing osmophoresis is opposite to that of a corresponding settling particle.
In Chapter 5, an investigation is presented for the diffusiophoresis of a spherical particle situated at an arbitrary position inside a spherical cavity filled with a nonionic solution, where a uniform solute concentration gradient is prescribed in an arbitrary direction relative to the line through the particle and cavity centers. The interfacial layer of particle-solute interaction is assumed to be thin relative to the particle radius and any gap width between the particle and cavity surfaces, but the polarization effect of the diffuse solute in the interfacial layer caused by the strong adsorption of the solute is incorporated. The solutions of the solute concentration and fluid velocity are constructed from the superposition of general solutions in the two spherical coordinate systems based on the particle and cavity, and the boundary conditions are satisfied by the collocation method. The translational and rotational velocities of the diffusiophoretic particle are determined for various cases. In the special case of a particle in a concentric cavity, these results are in excellent agreement with the exact solution. The effects of the cavity wall on diffusiophoresis are significant and interesting. In general, the normalized translational velocity decreases and rotational velocity increases with increases in the particle-to-cavity radius ratio and the normalized distance between the particle and cavity centers, and the translational velocity deflects little from the applied solute concentration gradient. The direction of rotation of a confined particle undergoing diffusiophoresis is opposite to that of a corresponding sedimenting particle.
Subjects
邊界效應
球形粒子
球形孔洞
熱毛細運動
電泳
電滲透
滲透泳
擴散泳
粒子移動和轉動
Type
thesis
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