應用矩陣法研究薄膜反應–擴散

Abstract

A. 薄膜內催化劑隨位置做線性分佈，且伴隨著一次反應的擴散薄膜中嵌 入隨位置呈線性分佈的催化劑，討論其間伴隨一次反應發生的擴散。溶質 濃度變化同時與位置、時間相關，為了避免單位在運算過程中，帶來困擾 ，我們利用變數的特性值(characteristic value)，改寫約化後的反應－ 擴散偏微分方程式，再經過變數變換，其解是 Airy 函數的線性組合。再 依據 Siegel 對各參數定義，借重行列式將複雜的表示式加以簡化。此後 ，利用已知無反應的擴散參數值，印證表示式的正確性。 B. 薄膜內催化 劑隨位置做二次分佈，且伴隨著一次反應的擴散延續上一章的探討，我們 希望了解薄膜內催化劑隨位置做二次分佈，對伴隨著一次反應－擴散的影 響。經過二次變數變換，反應－擴散偏微分方程式的解是 Weber 函數的 線性組合。若曲線分佈中二次項係數趨近於0，則相當線性分佈情形，我 們已經印證了其正確性，可知表示式無誤。 C. 以矩陣方法研究經過薄膜 串－並聯，伴隨著反應的擴散在此，考慮不同的薄膜擴散路徑，經過由二 片並聯薄層、串聯另一薄層之組合薄膜，它的傳輸矩陣(transmission matrix)可以求得。根據 Siegel的理論，已知 Laplace定義域中的傳輸矩 陣後，即可利用第一列、第二行位置之矩陣元素，求出穿透率 (P)及延遲 時間(tL)。由於傳輸矩陣具有行列式值為 1 的特性，tL 的方向不變性可 以得證。 D. n片薄膜串聯的傳輸矩陣我們希望發展串聯薄膜之特徵參數 通式。矩陣本身具有相當多的特性可供運用，先求出 2×2 傳輸矩陣特徵 值 (eigenvalue)，依據 Cayley- Hamilton 理論，分別代入後求解，即 可得知 n 片薄膜串聯的傳輸矩陣，所以描述 n 片薄膜串聯的特徵參數通 式可知。 The theme of diffusion, accompanying a first-order reaction due to unevenly distributed imbedded catalysts is considered in this thesis. In order to avoid the difficulty arising from the units of various quantities, the reaction-diffusion equation is formulated in terms of reduced variables in such a way that the equation is dimensionless. We found, if the catalyst is linearly distributed, the analytic solution to the reaction- diff usion equation and, thus, the permeability, forward and reverse survivals, lag time, forward and reverse mean first- passage times can all be expressed in terms of Airy's functions. In case that catalysts are quadratically distributed, the Weber's functions are found to be relevant. How the above-mentioned diffusion parameters effected by the distribution is also discussed. We also give the formulation of the permeability and lag time for diffusion, accompanying first- order reactions, across a composite serial-parallel membrane. The crucial point in this formulation consists in the fact that the resultant admittance matrix of two membrane (A and B) in parallel is the sum of the counterparts of the component ones, and the transmission matrix of two membranes A, and B in series is the product of those two corresponding transmission matrix. With the help of these, permeability and lag time are easily formulated in conjunction with Siegel's theory. The directional symmetry of lag time with respect the flux direction is also proven, based on the fact of the determinant of the transmission matrix is unity. Finally we also propose a quick, easily to be coded algorithm to calculated the related diffusion parameters of a reaction- diffus ion across a quite general, heterogeneous membrane.