帶可變介電系數之電荷守恆帕松波茲曼方程的邊界層解軸對稱之情形

Boundary layer solutions of charge conserving Poisson-Boltzmann equations with variable dielectric coefficients for radially symmetric case

呂治鴻(Jhih-Hong Lyu)

指導教授 ： 林太家

國立臺灣大學/理學院/數學研究所/碩士(2017年)

https://doi.org/10.6342/NTU201701477

摘要

在這篇文章中，我們研究關於帶可變介電系數的電荷守恆帕松波茲曼方程的漸近行為。當參數"趨近於零時，若方程所考慮的靜電位在邊界形成邊界層，具有這樣行為的解可以對應到電化學中的電雙層模型。我們在文章中指出邊界層形成的數學條件，並且給出內部電位與邊界電位滿足的方程組（參見定理1）。除此之外，我們還進一步在邊界層中考察方程的漸近逐點估計，藉由這樣的方法我們能指出在這個模型意義底下的邊界層的厚度以及提供電容的計算公式(1.7)。

另外，這篇文章也對非電中性的特殊情形做了一些初步的探討。這指出非電中性的電位行為在內部會趨平並且第一階展開項為log(1/eps^2)。因此這表明電解液在物理域的內部幾乎為電中性。

關鍵字

邊界層 ； 電荷守恆帕松波茲曼方程 ； 漸近逐點估計 ； 漸進行為

並列摘要

In this thesis, we study the asymptotic behavior of charge conserving Poisson-Boltzmann equation with variable dielectric coefficient. As the parameter " goes to zero, if electric double layer(EDL) appears at boundary, then this kind of solutions can be related to the electric double layer model in electrochemistry. In this paper, we give the condition of appearance of EDL and interior potential value and boundary potential value satisfy the system (cf. Theorem 1). In addition, we also study the asymptotic pointwise estimate of equation in boundary layer. By this method, we can compute the thickness of boundary layer in this model and provide the formula of capacitance (1:7).
Besides, the non-electroneutrality case is studied preliminarily. This study points out the potential behavior approaches at with leading order term log(1/eps^2) . Hence this shows that the electrolyte tends to electroneutrality in the physical bulk.

並列關鍵字

Boundary layer ； Charge conserving Poisson-Boltzmann equation ； Asymptotic pointwise estimate ； Asymptotic behavior

參考文獻

[1] Hainan Wang and Laurent Pilon, Accurate Simulations of Electric Double Layer Capacitance of Ultramicroelectrodes, J. Phys. Chem. C, 2011, 115 (33), pp 16711-16719.

[2] Barry Honig and Anthony Nicholls, Classical Electrostatics in Biology and Chemistry, Science, New Series, Vol. 268, No. 5214 (May 26, 1995), pp. 1144-1149.

[3] V. Barcilon, D. P. Chen, R. S. Eisenberg, and J. W. Jerome, Qualitative Properties of steady-state Poisson-Nernst-Planck systems: perturbation and simulation study, SIAM J. Appl. Math., 57(3),631-648, 1997.

[4] Chiun-Chang Lee, The charge conserving Poisson-Boltzmann equations: Existence, uniqueness, and maximum principle, Journal of Mathematical Physics 55, 051503 (2014).