帶電多孔粒子在表面帶電孔洞中之沉降運動

Abstract

本論文探討一個帶電球形多孔粒子，在一個表面帶電球形孔洞的中心位置進行擬穩態的沉降運動，粒子周圍及孔洞表面的電雙層可為任意厚度，並且考慮電雙層的鬆弛(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，以多孔粒子的固定(空間)電荷密度及孔壁的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，就可以求得多孔粒子內外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、及流體阻力三力平衡，可以求得多孔粒子沉降速度的解析形式表示式。 本研究發現，當帶電粒子於不帶電的孔洞中沉降時，粒子所帶電荷會降低粒子的沉降速度；而當不帶電粒子於帶電孔洞中沉降時，孔洞所帶電荷反而會加速粒子的沉降。當粒子與孔洞皆帶有固定電荷時，在粒子空間電荷與孔壁表面電荷電性相同的情況下，一般會加速粒子的沉降；在二者電荷電性相反的情況下，粒子沉降速度的變化情況則隨此二電荷之相對量而定。此外，在適當的情況下，孔壁表面電荷對多孔粒子沉降速度的影響效應，會隨著流體對粒子的滲透性增加以及粒子與孔洞半徑比值的減少而增加。

An analytical study is presented for the quasisteady sedimentation of a charged porous spherical particle located at the center of a charged spherical cavity with an arbitrary thickness of the electric double layers, in which the relaxation effect 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 fixed space charge density of the porous particle and surface charge density of the cavity as the small perturbation parameters. An analytical expression for the settling velocity of the confined charged porous 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 sedimentation velocity of the particle in an uncharged cavity and, due to the electroosmotic enhancement on the fluid recirculation in the cavity caused by the sedimentation-induced electric field, the presence of the fixed charge on the cavity wall increases the sedimentation velocity of an uncharged particle. For the case of a charged porous sphere settling in a charged cavity with their fixed charges in the same sign, the effect of electric interactions between the particle and cavity wall in general increases the sedimentation velocity of the particle. For the case of their fixed charges in opposite signs, the net effect of the fixed charges increases/reduces the sedimentation velocity if the magnitude of the fixed charge density of the cavity wall is relatively large/small. The effect of the surface charge at the cavity wall on the sedimentation of a porous particle is found to increase with an increase in the permeability for fluid flow inside the particle and with a decrease in the particle-to-cavity size ratio (i.e., an increase in the surface area of the cavity relative to that of the particle, which enhances the fluid recirculation effect).