A perturbation solution to the full Poisson-Nernst-Planck equations yields an asymmetric rectified electric field

• Published in: Soft matter Vol. 16; no. 3; pp. 752 - 762
• Main Authors: Hashemi, Aref, Miller, Gregory H, Bishop, Kyle J. M, Ristenpart, William D
• Format: Journal Article
• Language: English
• Published: Cambridge Royal Society of Chemistry 14.08.2020
• Subjects: Asymmetry; Differential equations; Electric fields; Electrode polarization; Electrodes; Nonlinear equations; Ordinary differential equations; Perturbation; Spatial distribution
• ISSN: 1744-683X; 1744-6848
• DOI: 10.1039/d0sm00417k

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 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. 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.