帶電軟質粒子在表面帶電孔洞中之電泳運動

Abstract

本論文探討一個具有多孔層吸附於固體核心表面之軟質粒子，在一個充滿任意電解質溶液之帶電球形孔洞的中心進行擬穩態的電泳運動，其中的多孔層具有溶劑與離子皆可穿透性，且其固定電荷密度及對流體之摩擦阻力皆為均勻分布，而粒子周圍與孔洞表面的電雙層相對於粒子半徑可為任意厚度。透過線性化之Poisson-Boltzmann方程式和Laplace方程式，可分開求解平衡電位分佈與外加電場所造成的偏離平衡之微擾項，再利用靜電力修正之Stokes方程式和Brinkman方程式可求得在多孔層內外之流場分佈。最後藉由作用於軟質粒子的靜電作用力與流體阻力之平衡，可以求得以固體核心、多孔層和孔洞三者固定電荷密度為分項之電泳速度的線性解析形式表示式。此解析解為粒子電動力半徑,多孔層流體阻力參數, 粒子固體核心半徑比值和粒子與孔洞半徑比的函數。 本研究發現，帶電孔洞所造成的邊界效應會對粒子的運動情形產生明顯的影響。此外，針對粒子固體核心半徑比值的兩種極限情況，可將軟質粒子電泳速度之解析解分別衍伸成固體粒子和多孔粒子在帶電球形孔洞中之電泳運動情形。

An analytical study of the quasi-steady electrophoretic motion of a soft particle composed of a charged spherical rigid core and a surrounding porous shell positioned at the center of a charged spherical cavity filled with an arbitrary electrolyte solution is presented. In the solvent-permeable and ion-penetrable porous surface layer of the particle, idealized frictional segments with fixed charges are assumed to distribute at a constant density. The thickness of the electric double layers around the particle and adjacent to the cavity wall is arbitrary relative to the particle radius. Through the use of the linearized Poisson-Boltzmann equation and the Laplace equation, the equilibrium double-layer potential distribution and its perturbation caused by the applied electric field are separately solved. The modified Stokes and Brinkman equations governing the fluid flow fields outside and inside the porous surface layer, respectively, are solved subsequently. An explicit formula for the electrokinetic migration velocity of the soft particle in terms of the fixed charge densities on the rigid core surface, in the porous shell, and on the cavity wall is obtained from a balance between its electrostatic and hydrodynamic forces. This formula is valid for arbitrary values of the Debye screening parameter, the reciprocal of the length characterizing the extent of flow penetration inside the porous layer, the radius of the rigid core of the particle, the radius of the soft sphere, and the radius of the cavity. The effect of the surface charge at the cavity wall on the particle migration can be significant and the particle may reverse the direction of its migration. In the limiting cases of large and small values of the radius of the rigid core of the particle, the analytical solution describing the migration velocity for a charged soft sphere reduces to that for a charged impermeable sphere and for a charged porous sphere, respectively, in a concentric spherical cavity.