Interaction of Potential Sources in Infinite 2D Arrays: Diffusion through Composite Membranes, Micro-Electrochemistry, Entrance Resistance, and Other Examples,Advanced Theory and Simulations

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Interaction of Potential Sources in Infinite 2D Arrays: Diffusion through Composite Membranes, Micro-Electrochemistry, Entrance Resistance, and Other Examples
Advanced Theory and Simulations ( IF 2.9 ) Pub Date : 2021-09-09 , DOI: 10.1002/adts.202100128
Andriy Yaroshchuk _{1,

2} , M. P. Bondarenko ₃

Affiliation

Regular 2D arrays of potential sources on impervious “screens” are a mathematical idealization for the description of a number of natural and/or technological processes. At steady state, they are described by Laplace equation with suitable boundary conditions. This study explains the evolution of boundary conditions from a given potential to a given potential gradient at infinity with increasing size of arrays and provides a criterion for micro- and macro-arrays in terms of distance to potential-defining surfaces. For regular macro-arrays, the problem is formulated and solved numerically by using cell models. The use of rigorous cell models requires 3D numerical simulations, but a simplified (cylindrical, 2D) cell model is shown to have an excellent accuracy. Numerical as well as theoretical analyses reveal a simple far-field behavior described by an asymptotic expression for an additional potential drop (applicable for sources whose size does not exceed about 40% of the intersource distance) dependent on just one numerical constant. This expression is used for the derivation of useful approximate formulae for several applications (diffusion resistance of composite membranes, limiting current in arrays of microelectrodes, entrance diffusion resistance in arrays of scarce and short nanopores) and compared with relevant interpolation formulae available in the literature.

中文翻译：

无限二维阵列中潜在源的相互作用：通过复合膜的扩散、微电化学、入口电阻和其他示例

不透水“屏幕”上潜在源的规则二维阵列是描述许多自然和/或技术过程的数学理想化。在稳态下，它们由具有合适边界条件的拉普拉斯方程描述。这项研究解释了随着阵列尺寸的增加，边界条件从给定的电位到给定的无穷远电位梯度的演变，并根据到电位定义表面的距离提供了微阵列和宏阵列的标准。对于常规的宏阵列，问题是通过使用单元模型来制定和数值解决的。使用严格的细胞模型需要 3D 数值模拟，但简化的（圆柱形，2D）细胞模型显示出具有出色的准确性。数值分析和理论分析揭示了一种简单的远场行为，该行为由附加电位下降的渐近表达式描述（适用于大小不超过源间距离约 40% 的源），仅依赖于一个数值常数。该表达式用于推导几种应用的有用近似公式（复合膜的扩散阻力、微电极阵列中的限制电流、稀少和短纳米孔阵列中的入口扩散阻力），并与文献中可用的相关插值公式进行比较。

更新日期：2021-11-09

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