帶電粒子於帶電孔洞中之沉降

Abstract

本論文探討一個帶電球形粒子，在一個帶電球形孔洞的中心位置進行擬穩態的沉降運動，粒子及孔洞表面的電雙層可為任意厚度，並且考慮電雙層的極化效應。在系統只有些微偏離平衡狀態的情況下，可利用正規微擾法，以粒子表面電荷密度和孔壁表面電荷密度為微小參數，將原本交互聯立的非線性電動力微分方程式轉換成線性的方程式，再結合其相對應的邊界條件，就可以求得孔洞內部電解質溶液的電化學位能分佈、電位分佈以及流場分佈。求解電動力方程式後，藉由作用於粒子的重力、電力、流體阻力三力平衡，可以求得粒子沉降速度的解析形式表示式。 本研究發現，在粒子表面電荷與孔壁表面電荷電性相同的情況下，會加速粒子的沉降；在二表面電荷電性相反的情況下，粒子沉降速度的變化情況則不一定。此外，在適當的情況下，孔壁表面電荷對粒子沉降速度的影響效應，隨著粒子與孔洞半徑比值的減少而顯著的增加。

An analytical study is presented for the quasisteady sedimentation of a charged spherical particle located at the center of a charged spherical cavity with an arbitrary thickness of the electric double layers. The overlap of the double layers is allowed, and the polarization (relaxation) effect in the double layers is considered. The electrokinetic equations that govern the ionic concentration distributions, electric potential profile, and fluid flow field in the electrolyte solution are linearized assuming that the system is only slightly distorted from equilibrium. Using a perturbation method, these linearized equations are solved for a symmetrically charged electrolyte with the surface charge densities of the particle and cavity as the small perturbation parameters. An analytical expression for the settling velocity of the charged sphere in closed form is obtained from a balance among the gravitational, electrostatic, and hydrodynamic forces acting on it. Our results indicate that the presence of the particle charge reduces the magnitude of the sedimentation velocity of the particle in an uncharged cavity and the presence of the fixed charge on the cavity surface increases the magnitude of the sedimentation velocity of an uncharged particle in a charged cavity. For the case of a charged sphere settling in a charged cavity with equivalent surface charge densities, the net effect of the fixed charges will increase the sedimentation velocity of the particle. For the case of a charged sphere settling in a charged cavity with their surface charge densities in opposite signs, the net effect of the fixed charges in general reduces/increases the sedimentation velocity of the particle if the surface charge density of the particle has a greater/smaller magnitude than that of the cavity. The effect of the surface charge at the cavity wall on the sedimentation of a colloidal particle is found to increase with a decrease in the particle-to-cavity size ratio and can be significant in appropriate situations.