碎形聚集體中擴散主導反應速率的研究

Diffusion-limited reaction in fractal aggregates

曾致堯(Chi-Yao Tseng)

指導教授 ： 曹恒光

國立中央大學/工學院/化學工程與材料工程學系/博士(2008年)

摘要

擴散主導反應在相當多的化學, 物理, 生物的動態系統中扮演了極為重要的角色。
先前有相當多的學者針對雙粒子系統, 得到各式各樣精確分析式, 半經驗式, 數值歸納式.. 等等的研究結果。但是在實際化學反應及生物環境中, 多數的系統卻是多體擴散反應, 存在著相當複雜及困難的交互現象. 如果利用兩粒子關係式來進行多體系統的數學解析運算, 這將是一個艱鉅且耗時的任務。
本論文目標將著重在單一粒徑分布之碎形結構內, 擴散主導的反應在聚集體內的研究, 希冀找出一個簡潔的有限規模尺度式來描述擴散反應的有效反應速率因子及其他細節。
本研究利用 multi-pole expansion 的數值方法, 來描述結構內粒子間因相對位置而產生的彼此影響擴散交互作用(diffusive interactions), 並佐以蒙地卡羅法來驗證 di-pole level 比 mono-pole level 有相當可靠的準確度及解算省時性。利用解矩陣特徵向量的加速技巧, 本研究方法能相當精確快速有效的得到收斂解算結果, 在可預估時間下的計算出一個數量內之反應性顆粒聚集體的平均單顆粒反應速率. 本研究方法可推及至內含10^5個組成顆粒的大規模之聚集體。
針對簡單對稱的結構, 如一維(1D)線形, 二維(2D)平面, 三維(3D)立方體, 及各種碎形維度的定規(deterministic)或是隨機(random)碎形, 吾人進行從小到大尺度的電腦數值實驗。依照K-R平均近似式(pre-averaging Kirkwood-Riseman approximation)的分析程序, 吾人推導得到, 聚集體的無因次化有效平均反應速率因子(effectiveness factor)會遵守與結構顆粒數目跟碎形維度的指數尺度式(power scaling laws). 在越大尺度的聚集體中也發現到, 當維度大於1, 外圍結構的屏蔽效應(screening effect)越明顯, 對有效反應速率因子的影響重大。當維度小於1時, 屏蔽效應的影響就有其限制。在維度為1時, 產生了從強烈屏蔽效應到有限屏蔽效應的轉折現象。
不僅在擴散反應上發現這種規則, 吾人也將此研究方法應用在多體聚集物在終端速度沈降的問題上, 也得到相當簡潔類似的尺度式來陳述平均沈降係數(mean drag coefficient)。因此本研究的結果可推廣到各式的輸送現象中, 如熱量傳導甚至到生物擴散系統中等等。

關鍵字

有限尺度率 ； 碎形維度 ； 擴散反應

並列摘要

Diffusion-limited reactions play a major role in many chemical, physical processes and biological systems.
The fundamental solution (as known as Smoluchowski theory) of diffusion includes reactions within homogeneous and isolated spherical sinks. Various systems with two-particle interaction have been widely studied, and successfully developed approximations with different analytical, numerical and experimental methods. On the contrary, in realistic systems, diffusion in many-body systems is the most complex problem found in engineering, science and nature. Therefore, this study intended to focus on the reaction rate for an aggregated cluster with immovable, reactive spherical sinks in a medium including diffusive reactants.
The effectiveness factor, h, defined as the ratio of the total reaction rate of the cluster to that of without diffusional interactions in a diffusion- limited reaction system, is evaluated for small clusters and fractals aggregated by mono-dispersed reactive spherical sinks in this thesis.
The method of multi-pole expansion involving dipole level is effective for a finite system constructed by reactive sinks. The numerical method is proven to be an accurate result within 1% deviation by comparing with the exact data from Monte Carlo simulations and computationally time-saving approximation. Matrix elimination and eigen-value solving technique are used to accelerate the computing speed to obtain an excellent semi-analytical agreement for potential matrix in dipole expansion.
Those conformations included structures in 1D (regular polygons and linear chains), 2D (squares), 3D (cubic arrays) and specified aggregates in fractal dimensions, each of which was considered and evaluated for effectiveness factor h from D<1 to D=3 . The number of fractal assembly can be as high as O(105). The scaling behavior of h is derived based on the generalized Kirkwood-Riseman pre-averaging approach. The asymptotic scaling behavior of the effectiveness factor with particle numbers N is h~N(1/D-1) for D>1, h~(lnN)-1 for D=1, and h~N 0 for D<1. The crossover behavior indicates 2 regimes. In the regime of D>1, the screening effect of diffusive interactions grows with size. In the regime of D<1, it is limited in a finite range and decays with decreasing D.
The asymptotic behavior for deterministic fractals was confirmed as it followed the similar scaling laws for the translation drag coefficient in the low Reynolds number flow regime. This approximation is applicable to other transport phenomena like heat conduction, and even biological diffusion systems.

並列關鍵字

finite size scaling law ； fractal dimension ； diffusion-limited reaction

參考文獻

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