论文标题
在具有空间效应的稳态泊松的渐近极限上
On the asymptotic limit of steady state Poisson--Nernst--Planck equations with steric effects
论文作者
论文摘要
当离子拥挤时，离子之间的空间排斥的影响变得很重要，并且应修改传统的泊松 - 波尔兹曼（PB）方程（无空间效应）。为此，我们研究具有空间效应（PNP-SERIC方程）的稳态泊松的渐近极限。根据空间效应的假设，我们将稳态PNP-SERTER方程转换为具有空间效应（PB-SERTER方程）的PB方程，该方程具有参数$λ$和正常数$λ_i$的$λ_i$（取决于离子和溶剂分子的半径）。 PB-STERIC方程的非线性项主要由代表溶剂分子浓度的兰伯特类型函数确定。当$λ= 0 $时，PB-STERIC方程成为常规的PB方程，但作为$λ> 0 $，一个大$λ$使空间排斥（离子和溶剂分子之间）更强。这促使我们发现PB-STERIC方程的渐近极限是$λ$变为无穷大。在Robin（或Neumann）边界条件下，我们从理论和数字上证明PB-STERIC方程具有独特的解决方案$ ϕ_λ $，该方程将修改后的PB（MPB）方程的解决方案$ ϕ^*$收敛，为$λ$倾向于Infinity。我们的结果表明，PB-STERIC方程的极限方程式（$λ$输入Infinity）是MPB方程，其形式与\ cite {1942Bikerman，1997borukhov，1997borukhov，2007 kilic，2007Kilic，2009Li，2009Li2,2013li}中的MPB方程相同（达到标量倍数）。因此，可以将PB-SERIC方程视为MPB方程的广义模型。
When ions are crowded, the effect of steric repulsion between ions becomes significant and the conventional Poisson--Boltzmann (PB) equation (without steric effect) should be modified. For this purpose, we study the asymptotic limit of steady state Poisson--Nernst--Planck equations with steric effects (PNP-steric equations). By the assumptions of steric effects, we transform steady state PNP-steric equations into a PB equation with steric effects (PB-steric equation) which has a parameter $Λ$ and positive constants $λ_i$'s (depend on the radii of ions and solvent molecules). The nonlinear term of PB-steric equation is mainly determined by a Lambert type function which represents the concentration of solvent molecules. As $Λ=0$, the PB-steric equation becomes the conventional PB equation but as $Λ>0$, a large $Λ$ makes the steric repulsion (between ions and solvent molecules) stronger. This motivates us to find the asymptotic limit of PB-steric equation as $Λ$ goes to infinity. Under the Robin (or Neumann) boundary condition, we prove theoretically and numerically that the PB-steric equation has a unique solution $ϕ_Λ$ which converges to solution $ϕ^*$ of a modified PB (mPB) equation as $Λ$ tends to infinity. Our results show that the limiting equation of PB-steric equation (as $Λ$ goes to infinity) is a mPB equation which has the same form (up to scalar multiples) as those mPB equations in \cite{1942bikerman,1997borukhov,2007kilic,2009li,2009li2,2013li,2011lu}. Therefore, the PB-steric equation can be regarded as a generalized model of mPB equations.
