We derive a perturbation solution to the one-dimensional Poisson–Nernst–Planck (PNP) equations between parallel electrodes under oscillatory polarization for arbitrary ionic mobilities and valences. Treating the applied potential as the perturbation parameter, we show that the second-order solution yields a nonzero time-average electric field at large distances from the electrodes, corroborating the recent discovery of Asymmetric Rectified Electric Fields (AREFs) via numerical solution to the full nonlinear PNP equations [Hashemi Amrei et al., Phys. Rev. Lett., 2018, 121, 185504]. Importantly, the first-order solution is analytic, while the second-order AREF is semi-analytic and obtained by numerically solving a single linear ordinary differential equation, obviating the need for full numerical solutions to the PNP equations. We demonstrate that at sufficiently high frequencies and electrode spacings the semi-analytical AREF accurately captures both the complicated shape and the magnitude of the AREF, even at large applied potentials.

中文翻译：

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完整 Poisson-Nernst-Planck 方程的扰动解产生不对称的整流电场。

我们推导了任意离子迁移率和价数的振荡极化下平行电极之间的一维 Poisson-Nernst-Planck (PNP) 方程的扰动解。将施加的电位视为扰动参数，我们表明二阶解在距电极很远的距离处产生非零时间平均电场，这证实了最近通过数值解发现的不对称整流电场 (AREF)非线性 PNP 方程 [Hashemi Amrei等人。,物理。牧师莱特。, 2018 年, 121, 185504]。重要的是，一阶解是解析的，而二阶 AREF 是半解析的，通过数值求解单个线性常微分方程获得，无需 PNP 方程的完整数值解。我们证明，在足够高的频率和电极间距下，半分析 AREF 准确地捕获了 AREF 的复杂形状和幅度，即使在大施加电位下也是如此。