圓柱形膠體粒子泳動之邊界效應

Abstract

本論文對圓柱形粒子在一連續相中的擬穩態緩慢運動、熱泳及電泳運動之邊界效應作理論探討。考慮單一圓柱形膠體粒子垂直或平行於單一平板之緩慢運動或泳動，利用雙圓柱座標系統求解二維主導方程式，計算粒子之運動速度。 第二章探討平板的存在對一圓柱形粒子緩慢運動所產生的邊界效應。針對單一表面滑移圓柱形粒子接近一固定不動的平板之擬穩態移動與轉動來分析，考慮粒子表面上的摩擦滑移效應，在低Reynolds數的條件下，可求解主導流速分佈的動量方程式，以半解析半數值之方式，計算單一圓柱形粒子之移動及轉動速度，得到粒子移動速度與轉動速度彼此不相互影響之結果。圓柱形粒子接近平板進行緩慢運動時，流體施加的拖曳力及扭力隨著粒子表面摩擦滑移係數增加而減少。當粒子垂直平板運動時所受到平板之阻礙效應最大，而當粒子平行平板運動時所受到平板之阻礙效應則最小。在粒子與平板間的相對距離相同時，平板存在而造成對圓柱形粒子的流力阻礙效應比球形粒子所受到的流力阻礙效應顯著。 第三章探討平板的存在，對一圓柱形氣膠粒子熱泳運動所產生的邊界效應。考慮單一圓柱形氣膠粒子在忽略流體慣性項與溫度對流效應下之擬穩態熱泳運動。由於Knudsen數較小，因此粒子周圍之流體可視為連續體，在粒子表面考慮有溫度躍差、熱滑移及摩擦滑移的現象。平板之邊界效應，其一來自於氣膠粒子與平板間產生的溫度梯度交互作用，另一為流體之黏滯阻礙作用。在低Peclet數與低Reynolds數的假設下，求解系統之能量及動量主導方程式，可求得溫度場與流場，以半解析半數值之方式，計算單一圓柱形氣膠粒子之熱泳速度。由於氣膠粒子與粒子周圍流體之熱傳導性差異、氣膠粒子之表面特性、氣膠粒子與平板的相對距離及外界所施加的溫度梯度方向之不同，平板之邊界效應可降低或增加氣膠粒子的運動速度。當粒子與平板的相對距離相同時，平板的存在對圓柱形氣膠粒子的熱泳運動影響比球形氣膠粒子熱泳運動所受到的邊界效應顯著。 第四章探討平板的存在，對一表面帶電不均勻之圓柱形膠體粒子電泳運動所產生的邊界效應。針對一表面存在隨角度變化之帶電分佈圓柱形粒子在外加一均勻電場下之擬穩態電泳，外加電場方向為平行平板，假設電雙層厚度相較於粒子半徑與粒子到邊界之距離皆為非常薄。平板之邊界效應來自於以下三個影響：(1) 膠體粒子表面之電場會因邊界存在而增強或減弱；(2)邊界存在會增強粒子運動時所受之黏滯阻礙力；(3)邊界表面上之電荷與邊界切線方向之電場作用使流體進行電滲透流動。求解二維主導方程式，可求得電場與流場，以半解析半數值之方式，計算粒子電泳移動及轉動速度。計算粒子所帶不均勻表面電位之multipole moments，即可利用本研究結果所得之關係式，求得圓柱形膠體粒子電泳移動與轉動速度。吾人發現當粒子鄰近邊界時，邊界效應可使得無邊界存在時不進行移動或轉動運動之粒子產生移動或轉動速度。 第五章探討平板的存在，對一表面均勻帶電之圓柱形膠體粒子之電泳運動所產生的邊界效應。考慮圓柱形膠體粒子表面電雙層存在極化現象時，粒子於電解質溶液中在外加一均勻電場下之擬穩態電泳，電雙層厚度相較於粒子半徑與粒子到邊界之距離皆為非常薄。平板之邊界效應，其一來自於膠體粒子與平板間產生的離子電化學電位梯度交互作用，另一為流體之黏滯阻礙作用。求解系統之能量及動量主導方程式，可求得電化學電位場與流場，以半解析半數值之方式，計算單一圓柱形膠體粒子之電泳移動速度及轉動速度。在不同極化參數下，由於膠體粒子之表面特性、電解質溶液性質、膠體粒子與平板的相對距離以及外界所施加的電場方向之不同，平板之邊界效應可降低粒子的運動速度。當粒子與平板間的相對距離相同時，平板的存在對圓柱形膠體粒子的電泳運動影響比球形膠體粒子電泳運動所受到的邊界效應顯著。

In this thesis, boundary effects on the two-dimensionalcreeping motion, thermophoresis and electrophoresis of cylindrical particles in a continuous medium are theoretically studied. Through the use of cylindrical bipolar coordinates, the transport governing equations are solved to calculate the various velocities of acylindrical particle migrating normal or parallel to a plane wall. In Chapter 2, an analytical study is presented for the creeping flow caused by a long circular cylindrical particletranslating and rotating in a viscous fluid near a large plane wall parallel to its axis. The fluid is allowed to slip at thesurface of the particle. The Stokes equations for the fluid velocity field are solved in the quasisteady limit. Semi-analytical solutions for the drag force and torque acting on the particle by thefluid are obtained for various values of the slip coefficient associated with the particle surface and of the relativeseparation distance between the particle and the wall. The results indicate that the translation and rotation of theconfined cylinder are not coupled with each other. For the motion of a no-slip cylinder near a plane wall, ourhydrodynamic drag force and torque results reduce to the closed-form solutions available in the literature. Theboundary-corrected drag force and torque acting on the particle decrease with an increase in the slip coefficient for anotherwise specified condition. The plane wall exerts the greatest drag on the particle when the migration occurs normalto it, and the least in the case of motion parallel to it. The enhancement in the hydrodynamic drag force and torque on atranslating and rotating particle caused by a nearby plane wall is much more significant for a cylinder than for a sphere. In Chapter 3, an analytical study is presented for the thermophoretic motion of a circular cylindrical particle in a gaseous medium with a transversely imposed temperature gradient near a large plane wall parallel to its axis in the quasisteady limit of negligible Peclet and Reynolds numbers. The Knudsen number is assumed to be small so that the fluid flow is described by a continuum model with a temperature jump, a thermal slip, and a frictional slip at the particle surface. The presence of the confining wall causes two basic effects on the particle velocity: first, the local temperature gradient on the particle surface is altered by the wall, thereby speeding up or slowing down the particle; secondly, the wall enhance the viscous retardation of the moving particle. The transport equations governing this problem are solved and the wall effects on the thermophoresis of the aerosol cylinder are computed for various cases. The presence of the plane wall can reduce or enhance the particle velocity, depending upon the relative thermal conductivity and surface properties of the particle, the relative particle-wall separation distance, and the direction of the applied temperature gradient. The direction of the thermophoretic motion of a cylindrical particle near a plane wall is different from that of the prescribed thermal gradient, except when it is oriented parallel or perpendicular to the wall. The effects of the plane wall on the thermophoresis of a cylinder are found to be much more significant than those for a sphere at the same separation. In Chapter 4, an analytical study is presented for the steady, transverse electrophoretic motion of a circular cylindrical particle with an arbitrary angular distribution of its surface potential parallel to a plane wall prescribed with the potential distribution consistent with the applied electric field. The electric double layers adjacent to the solid surfaces are assumed to be very thin with respect to the particle radius and spacing between the surfaces. The electrostatic and hydrodynamic governing equations are solved, and the typical electric field line, equipotential line, and streamline patterns are exhibited. The explicit formulas for the electrophoretic and angular velocities of the particle are obtained with the contribution from the electroosmotic flow produced by the interaction between the applied electric field and the thin double layer adjacent to the plane wall. To apply these formulas, one only has to calculate the leading multipole moments of the zeta potential distribution at the particle surface. The existence of a plane wall can cause the translation or rotation of the particle, which does not occur in an unbounded fluid with the same applied electric field. The boundary effects on the electrophoretic motion of a uniformly or nonuniformly charged particle resulting from the parallel plane wall prescribed with the far-field potential distribution are quite different from those produced by a corresponding insulating wall. In Chapter 5, an analytical study is presented for the electrophoretic motion of a circular cylindrical particle in an electrolyte solution with a transversely imposed electric field near a large plane wall parallel to its axis in the quasisteady limit. The electric double layers at the solid surfaces are assumed to be thin relative to the particle radius and to the particle-wall gap width, but the polarization effect of the diffuse ions in the double layer surrounding the particle is incorporated. The presence of the confining wall causes two basic effects on the particle velocity: first, the local ionic electrochemical potential gradients on the particle surface are altered by the wall, thereby affecting the motion of the particle; secondly, the wall enhances the viscous retardation of the moving particle. The transport equations governing this problem are solved and the wall effects on the electrophoresis of the cylinder are determined for various cases. The presence ofthe plane wall prescribed with the ionic electrochemical potentials consistent with the far-field distributionsreduces the electrophoretic mobility of the particle, which depends upon the properties of the particle–solutionsystem, the relative particle–wall separation distance, and the direction of the applied electric field relativeto the plane wall. The direction of the electrophoretic migration of a cylindrical particle near a plane wall isdifferent from that of the prescribed electric field, except when it is oriented parallel or perpendicular to thewall. The effects of the plane wall on the electrophoresis of a cylinder are found to be much more significantthan those for a sphere at the same separation.