(1)延遲時間的特徵值表示法 (2)以擴散過程延遲時間及平均首度經過時間探討成核誘導期及膠體穩定性

Abstract

本文先以利用理論推導延遲時間及其二階時距的格林函數表示式。其中格林函數的邊界條件為兩端皆為吸收性。其次，文中利用擴散的觀點討論均相成核以及膠體凝結。以延遲時間及平均首度經過時間探討成核誘導期及膠體穩定性。 透過Sturm-Liouville運算子的性質，可以發現當原本無化學反應的擴散過程，在導入了一階化學反應之後，其特徵值會增加，而延遲時間及其二階時距會減少。在均相成核的部分，與擴散最大的不同在於均相成核是離散座標下的隨機過程，而後者是在連續座標下。有關成核系統的動力方程式，皆以拉普拉斯轉換的方式求解之。如此便能求得延遲時間以及平均首度經過時間與兩者之二階時距在離散座標下的表示式。利用推導得出的公式計算水蒸氣凝結的延遲時間以及平均首度經過時間；計算結果顯示延遲時間以及平均首度經過時間隨著蒸氣壓越大而變小。若分別以上述兩者代表水蒸氣凝結所需的誘導時間，則兩者皆顯示蒸氣壓越大誘導時間越短。 穩定性在傳統上是用穩定比來作為定量的評估。本文試圖以擴散的觀點來解釋膠體的穩定性，並選用了延遲時間的相對比或平均首度經過時間的相對比來取代穩定比。計算結果顯示後者與穩定比之值相當接近。能障高度與穩定比則可利用最陡下降斜率法推導出兩者的近似關係。文末提出一條線性方程式，可由給定的參數估計臨界凝結電解質濃度。

In this thesis, the eigenvalue and Green’s function representations for the time lag of first and second moments were formulated. The Green’s function mentioned above is the one subject to the boundary conditions on both ends being absorbing. The homogeneous nucleation and the coagulation of colloids were discussed with the help of diffusion. Time lag and mean first passage time were employed to interpret the induction time in nucleation and the stability of colloids. The time lag of the first and of second moments will decrease as a result of the properties of Sturm-Liouville operator. We have derived the kinetic equations of homogeneous nucleation in the discrete number of particle coordinate, followed by solving in the Laplace domain. In this way, time lag, mean first passage time, and their corresponding second moments can be obtained. The formulas were tested in the problem of condensing water vapor. The results show that induction time for vapor condensation decreases with increasing vapor pressure. The stability of colloids is commonly expressed by stability ratio. We attempted to interpret the stability with the viewpoint of diffusion via the parameters, relative time lag and relative mean first passage time. It is indicated that relative mean first passage time matches with stability ratio quite well. The relation between the barrier height and the stability ratio is also discussed by applying the method of steepest descent, to obtain an approximate formula. Furthermore, a linear equation was proposed to calculate the critical coagulation concentration from known parameters.