電雙層是電解質溶液中在帶電表面所形成的一種結構。在此研究中，我們考慮笛利克萊邊界條件，來研究有界物理域中MPB方程的二階近似，並分析它的解在德拜長度趨近於零的時候的漸進行為。我們發現在當解很靠近邊界(對應於帶電表面)的時候會形成邊界層(對應於電雙層)。其中總體積百分數$\nu \in (0,1)$在此現象中扮演很重要的角色。因為當$\nu$很小很靠近零時和$\nu$很靠近1時有不一樣的漸進行為。

In the electrolyte solutions, the electrical double layer (EDL) is formed near the charged surface. In this thesis, we revisit the modified Poisson--Boltzmann (MPB) equation over a bounded physical domain to analyze the asympotic behavior of electrostatic potential $
hi$ in the limit of zero Debye length. More precisely, under the Dirichlet boundary condition, we study the quadratic approximation of the equation of $
hi$ around $
hi=0$. As the small parameter $\epsilon$ (related to the Debye screening length) tends to zero, the solution (electrostatic potential) develops boundary layers (corresponding to EDL) near the boundary (corresponding to charge surface). In particular, the bulk volume fraction $\nu \in (0,1)$ plays a key role in the phenomenon. Because asympotic behaviors for $0<\nu<\frac{1}{3}$ and $\frac{1}{3} \leq \nu < 1$ are totally different. Under the dilute case $0<\nu<\frac{1}{3}$, we show that $
hi$ will approch to $0$ in the interior interval. On the other hand, under the high concentration case $\frac{1}{3} \leq \nu < 1$, we obtain a general solution which has different behavior from that in the case $0<\nu<\frac{1}{3}$.

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