臺灣博碩士論文加值系統

在這篇文章中，研究主要分為三部分，在第一部分當中，我們提出了一個多域勒讓德擬譜補償法，主要針對定義在一般領域的浦松問題。這個方法特別之處在於所構建的離散拉普拉斯算子具有對稱以及正定的矩陣性質，因此確保了此方法的離散拉普拉斯逆算子的存在性。此外，我們也從中發現本數值的編制與格林函數方法之間存在對應關係。最後我們從多項數值實驗當中，如預期地觀察到近似解指數收斂的結果。

在第二部分當中，我們專注於修改經典的Bikerman模型，使得其能夠考量特定離子尺寸的效應。
此原因在於傳統的浦松 - 波爾茲曼和浦松 - 恩斯特 - 普朗克模型只能在離子濃度非常稀薄時才能準確地做出預測，然而實驗上所考慮的離子濃度通常都是不稀薄的，這使得其預測出的結果無法和實驗數據相比。而此問題正出在於，傳統的浦松 - 波爾茲曼和浦松 - 恩斯特 - 普朗克模型將每個離子視為單點，忽略了每個離子所具有的體積，這樣的假設使得當離子濃度不稀薄時，離子和離子之間可以相當地靠近，造成其所預測的結果不符合實驗上所觀測到的現象。因此，研究人員一直在嘗試修改此傳統的模型，使得其能夠考慮離子體積的影響。Bikerman模型正是眾多具有空間效應的修改模型當中，頗受歡迎的一種。然而，離子體積大小在原始的Bikerman模型中是被假設為一致的，。許多研究人員致力於擴展Bikerman模型使其能夠考慮不同離子體積大小。其中一種最簡單且直覺的方式，就是將離子尺寸修改成具有特定的離子尺寸。在這裡我們提出了證明，說明這樣看似簡單且直覺的作法，事實上違反了平均場晶格氣體模型當中計算相同的離子佔據位置熵的假設。我們在提出了正確的擴展版本，並基於此獲得了修改後的的浦松 - 波爾茲曼和浦松 - 恩斯特 - 普朗克模型。然而，這樣的修改雖然看似沒有問題，但當離子濃度稀薄時，其無法回到傳統的浦松 - 波爾茲曼和浦松 - 恩斯特 - 普朗克模型。

對於第三部分，我們對PNP空間模型採用漸近展開，並推導出在特定參數選定下，陽離子和陰離子在漸進展開中的領導項分別能夠回到上述我們提出修正後的Bikerman模型的陰離子以及陽離子濃度。

For the first study we present a multidomain Legendre pseudospectral penalty scheme for Poisson problems defined on general domains. We pay special attention upon constructing the discrete Laplace operator to possess certain matrix properties, so that the existence of the inverse of the pseudospectral penalty Laplace operator can be established. Furthermore, it is found that there are correspondences between the present numerical formulation and the analytic Green function approach. Numerical experiments are conducted and we observer exponential convergence of the approximation solutions as expected.

For the second study we focus on modification of Bikerman model with specific ion sizes.
Classical Poisson-Boltzman and Poisson-Nernst-Planck models can only work when ion concentrations are very dilute, which often mismatches experiments. Researchers have been working on the modification to include finite-size effect of ions, which is non-negligible when ion concentrations are not dilute. One of modified models with steric effect is Bikerman model, which is rather popular nowadays. It is based on the consideration of ion size and additional entropy term for solvent. However, ion size is universal in original Bikerman model, which did not consider specific ion sizes. Many researchers have worked on the extension of Bikerman model to have specific ion sizes. A straight forward way of doing so simply changes the universal ion size to specific ones. Here we prove this straight forward extension violates of the upholding of identical ion occupation site for entropy calculation based on mean-field lattice gas model. We derived a correct extension version in the current study , and obtained modified Poisson-Boltzmann and Poisson-Nernst-Planck models based on this. However, this version of modification, though seems perfect, can not reduce to classical Poisson-Boltzmann and Poisson-Nernst-Planck models when ion concentrations are dilute.

For the third part, we employ the formal asymptotic on PNP-steric model and deduce several parameter constraints so that the leading term of cp and cn which are the concentration of cation and anion respectively reduce to the Modified Bikerman model case.

致謝 i
中文摘要 ii
Abstract iv
1. Introduction 1
2. Formulation 4
2.1. Poisson equation and Green function 4
2.2. Basic concepts of the pseudospectral method 5
2.3. Single domain scheme for the Poisson problem 7
2.4. Green-function-like solution operator and Dirac- delta-like polynomial 12
2.5. Multidomain scheme for one-dimensional Poisson''s problem 13
2.6. Multidomain scheme for 2D Poisson''s equation 21
3. Three dimension scheme for Poisson problem 29
4. Numerical results 35
4.1. Eigenvalues and condition number of the operator 36
4.2. hp convergence 38
4.3. Problem involving material discontinuity 40
4.4. Curvilinear domain problems 42
4.5. Three dimensional results 46
4.6. Computational issues 49
5. Concluding remarks and further works 50
6. Appendix 50
6.1. Proof of the symmetric positive de nite property for 2D poisson problem
subject to exterior boundary condition 50
6.2. Proof of the symmetric poisitive de nite property for 2D multi-domain
poisson problem subject to interface boundary condition 62
6.3. Positive definite 71
7. Introduction 75
8. Review of Bikerman model 76
9. Extension of Bikerman model to employ speci c ion size 79
10. Correct extension of Bikermann model to include speci c ion sizes 80
11. Conclusion 84
12. Formal Asymptotic on PNP-steric equations 84
13. Recover to Modi ed Bikerman model 88
References 91

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