多孔纖維狀介質中電解質溶液之擴散滲透

Abstract

本論文以理論探討在規則、平行排列且表面帶電的圓柱體所構成的纖維介質中，由於外加軸向固定濃度梯度所引起的電解質溶液擴散滲透流。相對於圓柱半徑而言，在圓柱表面的電雙層可以是任意厚度。我們利用unit cell model來解釋圓柱彼此之間的影響。 此研究包含了兩部份。第一部份是分析圓柱表面為低電位的纖維型介質中，電解質溶液的擴散滲透流。流體相中的電位分布，可由求解線性化後的Poisson-Boltzmann方程式而得到。在流體中，會因為電解質離子的擴散而引起電流，利用電解質離子的淨質傳通量及沒有淨電流存在的限制，可以求解出由外加電解質濃度梯度所誘導出的具有徑向分佈的軸向巨觀誘導電場。再利用modified Navier-Stokes方程式可以得到流體速度分佈精確至圓柱表面電位或表面電荷密度的二次項近似解析解，此流體速度的貢獻包含了電滲透和化學滲透兩部份，且速度分佈為此纖維型介質的孔隙度的函數。 此研究的另一部分是探討多孔纖維狀介質中擴散滲透流，但其中圓柱的表面電位可為任意值，不再受限為低電位，流體中的電位分佈可由求解Poisson-Boltzmann方程式而得到近似解析解。另外考慮流體中由電解質離子的擴散、電解質離子的電遷移以及電解質離子濃度梯度造成擴散滲透的對流等因素對電流的影響，利用沒有淨電流產生的要求，可以求得軸向巨觀誘導電場的徑向分佈。最後，使用半解析的方法求解modified Navier-Stokes方程式得到速度分佈。 流體擴散滲透速度方向可由圓柱的表面電位(或表面電荷密度)、電解質溶液的性質和其他相關的因素所決定。研究結果顯示，誘導電場隨徑向位置而變化的效應和電解質濃度梯度造成擴散滲透流的對流效應具有很顯著的影響。

The steady diffusioosmosis of electrolyte solutions in the fibrous medium constructed by a homogeneous array of parallel charged circular cylinders caused by constant concentration gradients imposed in the direction along the axes of the cylinders is theoretically investigated. The electric double layer surrounding each cylinder may have an arbitrary thickness relative to the radius of the cylinder. We employ a unit cell model to account for the effect of cylinders on each other. The study contains two parts. In the first part, the diffusioosmotic flow in the fibrous medium is analyzed for the case of low zeta potential. The electrostatic potential distribution in the fluid phase of a cell is obtained by solving the linearized Poisson-Boltzmann equation, and the macroscopic electric field induced by the imposed electrolyte concentration gradient is determined analytically as a function of the radial position with the constraint of no net electric current arising from the cocurrent diffusion of the electrolyte ions. A closed-form formula for the fluid velocity profile of the electrolyte solution due to the combination of electroosmotic and chemiosmotic contributions as a function of the porosity of the array of cylinders correct to the second order of their surface charge density or zeta potential is derived as the solution of a modified Navier-Stokes equation. In the other part, the diffusioosmotic flow in fibrous porous media with an arbitrary surface potential is examined, and the electrostatic potential distribution in the fluid phase is determined by an analytical approximation to the solution of the general Poisson-Boltzmann equation. Solving the modified Navier-Stokes equation with the constraint of no net electric current arising from the cocurrent diffusion, electric migration, and diffusioosmotic convection of the electrolyte ions, the macroscopic electric field and the fluid velocity along the axial direction are obtained semianalytically as functions of the radial position in a self-consistent way. The direction of the diffusioosmotic flow relative to the concentration gradient is determined by the combination of the zeta potential (or surface charge density) of the cylinder, the properties of the electrolyte solution, and other relevant factors. The effects of the radial distribution in the induced macroscopic electric field and of the ionic convection in the electric double layer on the diffusioosmotic flow are found to be quite significant in practical situations.