Electrostatic Interactions between Colloidal Particles in a Salt-free Medium

本論文中推導出了在不含鹽類的介質中，平面、圓柱、與圓球狀單顆膠體表面的空間電位分布近似解析解；根據單一面的結果，兩膠體表面間的空間電位分布也由類比的方式而導出。結果顯示曲率效應在粒徑超過100倍的粒子電雙層厚度時可以忽略，而當需要考慮曲率效應時，曲面系統的電位分布結果可由平面系統的結果乘上一曲率校正函數而來。由已推導出的粒子間電位分布可進一步求得粒子間每單位面積之靜電作用力，也就是滲透壓；而在同樣的參數之下，滲透壓的大小依序為平面系統>圓柱系統>圓球系統。最後，吾人求出了球型膠體分散系統在不含鹽類的介質中之臨界凝聚濃度。其中，傳統的 Schulze-Hardy 通則於此分散相中與反離子之臨界凝聚濃度偏差大小可由一校正因子來修正，而此校正因子同時為系統中反離子價數、粒子表面電位、以及凝聚發生時粒子間中點電位的函數。吾人發現此校正因子大致上會隨著表面電位與反離子濃度的增加而變大，但是當表面電位夠高時，校正因子會趨近於定值—0.8390。Approximate analytical expressions for the electrical potential of planar, cylindrical, and spherical surfaces are derived for the case when the dispersion medium contains counterions only. Based on the results for single surfaces, those for two identical surfaces can be derived. The curvature effect of a surface on the electrical potential distribution can be neglected when the order of its radius exceeds about 102 of the thickness of the corresponding double layer. If this effect needs be considered, it can be taken into account by multiplying a correction function to the electrical potential of a planar surface. The electrical potential at the center between two surfaces derived is readily applicable to the evaluation of the electrostatic force per unit area between two surfaces, or the osmotic pressure. For the same set of parameters, the magnitude of the osmotic pressure for various types of surfaces follows the order planar surface>cylindrical surfaces>spherical surfaces. Furthermore, the planar results obtained are used to evaluate the critical coagulation concentration of a spherical dispersion. A correction factor, which is a function of the valence of counterions, the surface potential of a particle, and the potential on the mid-plane between two particles at the onset of coagulation, is derived to modify the classic Schulze-Hardy rule for the dependence of the critical coagulation concentration on the valence of counterions. The correction factor is found to increase with the increase in the valence of counterions and/or with the increase in the surface potential. However, it approaches a constant value of 0.8390 if the surface potential is sufficiently high.中文摘要………………………………………………………………………………I 英文摘要……………………………………………………………………………...II 目錄…………………………………………………………………………………III 表目錄………………………………………………………………………………...V 圖目錄………………………………………………………………………………VII 第一章 緒論…………………………………………………………………………1 第二章 文獻回顧…………………………………………………………………....3 2-1 電雙層理論……………………………………………………………….....3 2-1.1 波松方程式…………………………………………………………..3 2-1.2 波茲曼分布…………………………………………………………..4 2-1.3 波松—波茲曼方程式………………………………………………..5 2-1.4 Debye-Hückel 理論……………………………………………….....6 2-1.5 單一電雙層的Gouy-Chapman 理論……………………………......7 2-2 兩膠體粒子間的交互作用……………………………………………….....8 2-2.1 凡得瓦爾作用能……………………………………………………..9 2-2.2 靜電斥力作用能……………………………………………………10 2-2.3 大型膠體粒子間的Derjaguin 近似………………………………10 2-2.4 臨界凝聚濃度………………………………………………………11 2-3 不含鹽類的膠體分散系統………………………………………………12 第三章 理論分析…………………………………………………………………..15 3-1 單顆膠體粒子周圍之空間電位分布……………………………………15 3-2 兩顆相同膠體粒子間的空間電位分布…………………………………19 3-3 兩顆相同膠體粒子間之作用力…………………………………………...24 3-4 球型膠體粒子的臨界凝聚濃度…………………………………………24 第四章 結果與討論………………………………………………………………27 4-1 粒徑大小對空間電位分布的影響………………………………………27 4-2 單顆膠體粒子周圍的空間電位分布……………………………………27 4-3 兩顆膠體之間的空間電位分布…………………………………………...28 4-4 靜電排斥作用……………………………………………………………...29 4-5 各參數對臨界凝聚的影響………………………………………………30 第五章 結論………………………………………………………………………..33 符號說明……………………………………………………………………………35 參考文獻……………………………………………………………………………39 附錄A PBE於曲面座標下解析解之誤差比………………………………………70 附錄B 數值方法之比較…………………………………………………………73 B-1 數值方法之Formulation…………………………………………………73 B-2 數值方法結果分析………………………………………………………..79 B-2.1有限差分法中網格點對精確度的影響…………………………....79 B-2.2 正交配位法中N對曲線分布的影響……………………………...80 B-2.3 綜合比較……………………………………………………….......81 B-2.4 收斂比較…………………………………………………………...82 B-2.5 計算時間與誤差比較……………………………………………...8