A nonlinear equation for ionic diffusion in a strong binary electrolyte.
A nonlinear equation for ionic diffusion in a strong binary electrolyte.
复制标题
强二元电解质中离子扩散的非线性方程。
DOI：
10.1098/rspa.2010.0028
复制
发表时间：
2010
期刊：
影响因子：
--
通讯作者：
Chen,Zhen
中科院分区：
文献类型：
--
作者：
Ghosal,Sandip;Chen,Zhen
考虑离子在强二元电解质中的一维电扩散问题。这种数学描述被称为泊松-能斯特-普朗克 (PNP) 系统，由每个物质的扩散方程组成，该方程通过泊松方程确定的自洽静电场而通过传输增强。该描述还与物理学中的其他重要问题相关，例如半导体结上的电子和空穴扩散以及等离子体中离子的扩散。如果浓度在德拜长度量级的距离内没有明显变化，则泊松方程可以用普朗克首先引入的局部电荷中性条件代替。然后可以证明，两种物质以相同的速率扩散，其共同的扩散率介于慢速物质和快速物质之间（双极扩散）。在这里，我们通过利用德拜长度与特征长度尺度的比率作为一个小的渐近参数，得出了一个更一般的理论。结果表明，任一物质的浓度都可以通过非线性偏微分方程来描述，该方程提供了比双极扩散的经典线性方程更好的近似值，但在适当的限度内减少到它。
The problem of the one-dimensional electro-diffusion of ions in a strong binary electrolyte is considered. The mathematical description, known as the Poisson–Nernst–Planck (PNP) system, consists of a diffusion equation for each species augmented by transport owing to a self-consistent electrostatic field determined by the Poisson equation. This description is also relevant to other important problems in physics, such as electron and hole diffusion across semiconductor junctions and the diffusion of ions in plasmas. If concentrations do not vary appreciably over distances of the order of the Debye length, the Poisson equation can be replaced by the condition of local charge neutrality first introduced by Planck. It can then be shown that both species diffuse at the same rate with a common diffusivity that is intermediate between that of the slow and fast species (ambipolar diffusion). Here, we derive a more general theory by exploiting the ratio of the Debye length to a characteristic length scale as a small asymptotic parameter. It is shown that the concentration of either species may be described by a nonlinear partial differential equation that provides a better approximation than the classical linear equation for ambipolar diffusion, but reduces to it in the appropriate limit.