臺灣博碩士論文加值系統

類似將較小的微細粒子擴散至較大粒子表面進行反應的現象，在工業界或大自然中是相當常見的；而假定大粒子表面的反應速率為無限大，求其總反應速率常數的模擬，在過去已經有許多相關的論文發表，但是討論當表面反應速率為有限值情況的研究卻仍相當少。所以這個研究的目的是，藉由邊界取點法求解邊界值問題與蒙地卡羅法模擬的方式，來計算當空間中的懸浮粒子隨機分佈，且表面反應速率為有限值時，對其總反應速率常數的影響。在假設中懸浮粒子的表面反應速率並非無限大，可藉由一無因次常數P來控制整個反應的速率決定步驟；當P＝0時為擴散控制，當P→∞時則為表面反應控制。
由結果中，我們得到兩種截然不同的趨勢；當懸浮粒子隨機散佈於空間中時，所看到的結果是正規化總反應速率常數隨著懸浮粒子的體積分率的增加而上升，且當反應的速率決定步驟，漸漸的由擴散控制反應，轉變表面反應控制時，隨著改變的比重加大，將導致正規化總反應速率常數下降。
如果將懸浮粒子隨機散佈時所得的數據與規則排列情形的相比較，可以發現當總懸浮粒子的體積分率小於0.6時，對於規則排列的情形而言，懸浮粒子間仍有一定的距離存在，故其粒子間的遮蔽效應並不嚴重，因此可以發現在這樣的體積分率下，懸浮粒子呈隨機散佈排列，因遮蔽效應較明顯，因此其正規化總反應速率常數便會較低。但是一旦懸浮粒子的總體積分率超過0.6後，由數據中所反映出的情形，卻告訴我們懸浮粒子以規則排列時，其粒子間的遮蔽效應反而有略微超越粒子呈隨機散佈時的趨勢，不過基本上相差的結果還是相當有限，並不十分明顯。
由於懸浮粒子在隨機分佈時，很可能出現多顆粒子相互附著的情形，在此我們也模擬了，不同數目的粒子附著在一起的影響，發現當較多顆粒子附著在一起時，由於
相互的遮蔽效應較顯著，使得正規化總反應速率常數也隨著下降。
而在另一方面，當懸浮粒子數處於極稀薄狀態下，亦即是懸浮粒子群之相互競爭的效應可以忽略時，可以發現正規化總反應速率常數是隨著P值的上升而增加的。且當懸浮粒子以1維、2維及3維規則排列時，所得的正規化總反應常數，將與其懸浮粒子數(N)與排列維度有下列的關係。
當 D>1 當 D=1
且若將其圖形所得的斜率對P值作圖，也可以發現不論是何種方式排列所得到的斜率，其基本的趨勢幾乎都相同，而圖形大致可以分成三個區域：P=0~0.1，P=0.1-10，P=10-無限大；當P=0.1-10這個區域中，其斜率變化很大，而在另外兩個區域，卻相當平緩。而且在P=0~0.1這個區域，不同維度的排列下，所得到的斜率相差甚大，而在P=10-無限大這個區域斜率都趨近於0。

Processes involving diffusion of small entities onto the surface of much larger inclusions and incorporation of these small entities into the much larger inclusions, are commonly found in nature and industry. There were many researches by computer simulation for diffusion-limited processes, but the researches for the non-diffusion-limited were a few. In this research, we suppose that there are many randomly distributed particles in space. By using boundary collocation method and Monte-Carlo simulation I compute the normalized overall rate constants of reaction when the surface reaction rate is finite. We can control the surface reaction rate to be finite or not by a dimensionless parameter P. As P=0, the process is diffusion limited. And as P→∞, the process is reaction-limited.
When particles are randomly distributed in space, we see that normalized overall rate constants increase as increasing the volume fraction of particles, and normalized overall rate constants decrease as increasing the P value. If we compare normalized overall rate constants obtained from particles in random or in regular arrays, we see when the volume fraction of particles is smaller than 0.6, normalized overall rate constants of particles of random arrays are smaller.
But when particles in are a very dilute situation, normalized overall rate constants increase with increasing the P value. And when particles are arranged in a rod (1D), square (2D) or cube (3D), normalized overall rate constant and the number of particle (N) will have the following relation ( D: fractal number ) :
as D>1 as D=1
If we use the slopes from the equations and P value to make a plot, the plot can be separated into three regions：P=0~0.1, P=0.1-10, P=10-infinite. And the slopes decrease dramatically when P value change 0.1 from 10.

摘要1
Abstract3
目錄4
圖目錄6
表目錄7
第一章 緒論8
1-1 前言8
1-2 表面反應為速率瞬間反應之研究10
1-3 表面反應速率為有限值之研究12
第二章 多顆球形懸浮粒子─邊界取點法17
2-1 濃度分佈之推導、計算17
2-1-1 模型之說明17
2-1-2 邊界取點法說明20
2-2 正規化總反應速率常數21
2-2-1 正規化的標準21
2-2-2 平均濃度22
2-2-3 正規化總反應速率常數23
第三章 多顆球形懸浮粒子--蒙地卡羅模擬24
3-1 模型之說明24
3-2 微細粒子的運動26
3-2-1 首次通過時間理論26
3-2-2 擴散控制反應29
3-2-3 非擴散控制反應30
第四章 球形懸浮粒子於無窮稀薄環境之探討32
4-1 邊界取點法部分32
4-1-1 推導與計算32
4-1-2 正規化總反應速率常數33
4-2 蒙地卡羅模擬35
4-2-1 模擬方式35
4-2-2 擴散與非擴散控制反應37
第五章 結果與討論38
5-1 多顆球形懸浮粒─邊界取點法38
5-1-1 測試結果38
5-1-2 多顆球形懸浮粒子之總反應速率常數40
5-2 多顆球形懸浮粒子--蒙地卡羅模擬44
5-2-1 測試結果44
5-2-2 正規化總反應速率常數52
5-2-3 連續式加入球心之探討56
5-3 球形懸浮粒子於無窮稀薄環境58
5-3-1 邊界取點法58
5-3-2 蒙地卡羅模擬66
第六章 結論69
符號說明71
參考文獻73
附錄75
(1)多顆球形懸浮粒子於單位立方中--邊界取點法之數學式75
(2)多顆球形懸浮粒子於無限稀薄環境--邊界取點法之數學式77
(3)蒙地卡羅模擬--Psurvival之推導。78
(4)蒙地卡羅法─微細粒子若存活則亂射方式反射之說明80

Calef, D.F., & Deutch, J.M. (1983). Diffusion-controlled reactions. Annual Review of Physical Chemistry, 34, 493-524
Fixman., M. (1984) Absorption by Static Traps - Initial-Value and Steady-State Problems. Journal of ChemicalPhysics,81,3666-3677
Lee, S.B., Kim, I.C., Miller, C.A., & Torquato, S. (1989). Random-Walk Simulation of Diffusion-Controlled Processes Among Static Traps. Physical Review B, 39, 11833-11839
Lu, S.-Y. (1998a). Diffusion and Reaction in Regular Arrays of Spheres. Journal of Chemica Physics,109, 4985-4989
Lu, S.-Y. (1998b). Overall rate constants for dilute spheroid dispersions. Chemical Engineering Scencei. 54, 3917-3920
Lu, S.-Y., & Chen, T.C. (1999). Effects of Particle Polydispersity on Diffusion Controlled Rate Constants of Spherical Dispersions. Journal Chinese Institute Chemical Engineers, 30, 71-79
Lu, S.-Y. (1999). Rate constants of spherical dispersions: From diffusion-limited data to nondiffusion limited results. Journal of Chemical Physics, 110, 12263-12264
Lu, S.-Y. (2000). Diffusion and Reaction in Rectangular Arrays of Spheroids. Journal of Chemical Physics, 113, 1-10
Miller., C.A., Kim, I.C., & Torquato, S. (1991). Trapping and flow among random arrays of oriented spheroidal inclusions. Journal of Chemical Physics, 94, 5592-5598
Muthukumar., M. (1982). Concentration-Dependence of Diffusion Controlled Processes Among Static Traps. Journal of Chemical Physics, 76, 2667-2671
Richards., P.M. (1986). Diffusion to Finite-Size Traps. Physical Review Letters, 56, 1838-1841
Torquato, S., & Miller, C.A. (1989). Diffusion controlled reactions among spherical traps: Effect of polydispersity in trap size. Physical Review B, 40, 7101-7108
Torquato, S., & Yeong, C.L.Y. (1997). Universal Scaling for Diffusion Controlled Reactions among Traps, Journal of Chemical Physics,106, 8814-8820
Tsao, H.-K., Lu, S.-Y., & Tseng, C.-Y. (2001). Rate of Diffusion-Limited Reaction in a Cluster of Spherical Sinks. Journal of Chemical Physics, revision under review.
Tseng, C.-Y., & Tsao, H.-K. (2000). Translational Friction Coefficient of a Self-Similar Assembly. The 2000 CIChE Annual Meeting and Conferences, 21-25
Weiss, G.H. (1986). Overview of theoretical models for reaction rates. Journal of Statistical Physics, 94, 5592-5598
Zheng, L., & Chiew, Y.C. (1989). Computer Simulation of Diffusion Controlled in Dispersions of Spherical Sink. Journal of Chemical Physics, 90, 322-327
Zheng, L., & Chiew, Y.C. (1990). Computer Simulation of Steady-State Diffusion Controlled Reaction Rates on Dispersions of Static Sink: Effect of Sink Size. Journal of Chemical Physics, 93, 2658-2663