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  • 學位論文

球形膠體粒子於球形孔洞中之泳動

Phoretic Motions of Colloidal Spheres within a Spherical Cavity

指導教授 : 葛煥彰

摘要


膠體粒子於一連續相中受到外界所施加的溫度、電位、或溶質濃度梯度的驅動,所產生的輸送行為,稱為泳動。本研究考慮單一球形膠體粒子於一球形孔洞中之任意位置進行平行或垂直於它們中心連線之泳動,以邊界取點法半解析半數值計算粒子之擬穩態各種泳動速度。 首先,於第二章中,吾人考慮單一球形液滴於一充滿不互溶流體之球形孔洞中在Marangoni數和Reynolds數很小的情況下之熱毛細泳運動。施予平行或垂直於它們中心連線之定值溫度梯度,作為其驅動力。為求解溫度和流速分佈之主導方程式,建立一個包括與液滴和孔洞相關的兩個球座標系統之通解。以邊界取點法使通解滿足液滴表面和孔壁之邊界條件,可計算出在各種不同之內外流體黏度比、液滴相對熱傳導度、孔洞相對熱傳導度、粒子與孔洞半徑比,以及粒子與孔洞相對位置情形下之熱毛細泳速度,並與液滴在同心孔洞之解析結果相符。正規化之熱毛細泳速度會隨著粒子與孔洞半徑比的增加而減少,當粒子越靠近孔壁時其速度會越慢,在接觸孔壁時粒子則不會運動。當粒子與孔洞之半徑比和相對位置固定時,熱毛細泳速度會隨著流體黏度比或孔洞熱傳導度增加而增快。一般而言,孔洞邊界效應對於粒子垂直於其中心連線之運動會比平行於此連線時弱一些。 於第三章中,吾人探討單一介電球形膠體粒子在一個充滿電解質溶液之帶電球形孔洞中的電泳運動。外加電場為定值且可為平行或垂直於粒子與孔洞之連心線,而固體表面電雙層的厚度則假設為遠小於粒子半徑及粒子到邊界之間距。使用Laplace方程式和Stokes方程式之通解分別求解電位場以及流速分佈,並以邊界取點法使通解滿足粒子和孔壁之邊界條件,可求得在各種不同條件下粒子電泳之移動與轉動速度。孔壁的存在對於粒子電泳的影響相當複雜有趣而且重要。一般來說,粒子之電泳速度會隨著粒子與孔洞半徑比增加或粒子越靠近孔壁而減慢,而轉動速度則會隨著粒子與孔洞半徑比增加或粒子越靠近孔壁而增快,轉動方向通常與粒子作沉降運動時相反,但有些例外的情況。孔壁所產生的電滲透流能增強或減弱粒子電泳之移動和轉動速度,甚至可以改變運動之方向,取決於孔洞與粒子之zeta電位比值以及幾何參數。孔洞的存在對於粒子垂直於其中心連線之電泳速度的影響與平行此連線時之結果相近。 於第四章中,吾人考慮單一具有半透膜之球形胞囊粒子於一充滿溶液之球形孔洞中,受到任意方向之定值溶質濃度梯度驅動,所進行之滲透泳運動。為求解濃度以及流速分佈之主導方程式,需建立一個包括與胞囊粒子和孔洞相關的兩個球座標系統之通解,而以邊界取點法使通解滿足邊界條件,可計算出不同狀況下胞囊粒子滲透泳之移動和轉動速度;當胞囊粒子與孔洞同心時,其數值結果與解析結果一致。孔壁的存在對於滲透泳的影響相當重要而且有趣。一般來說,正規化之滲透泳移動和轉動速度會隨著粒子與孔洞半徑比增加或粒子越偏離孔洞中心而增快,且移動方向僅會稍微偏離外加濃度梯度,而轉動方向則與粒子作沉降運動時相反。 於第五章中,吾人探討單一球形膠體粒子於一充滿非電解質溶液之球形孔洞中,受到任意方向之定值溶質濃度梯度驅動,所進行之擴散泳運動。假設粒子與溶質之交互作用層厚度遠小於粒子的半徑及粒子到邊界之間距,但強吸附性溶質所造成之界面層中擴散溶質的極化效應則有考慮。可求出不同情況下粒子擴散泳之移動和轉動速度,當粒子與孔洞同心時其數值結果與解析結果一致。一般來說,正規化之擴散泳移動和轉動速度會隨著粒子與孔洞半徑比增加或粒子越偏離孔洞中心而分別減慢和加快,而轉動方向則與粒子作沉降運動時相反。

並列摘要


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.

參考文獻


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