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When applying an oscillatory electric potential to an electrolyte solution, it is commonly assumed that the choice of which electrode is grounded or powered does not matter because the time-average of the electric potential is zero. Recent theoretical, numerical, and experimental work, however, has established that certain types of multimodal oscillatory potentials that are "non-antiperodic" can induce a net steady field toward either the grounded or powered electrode [Hashemi et al., Phys. Rev. E 105, 065001 (2022)]. Here, we elaborate on the nature of these steady fields through numerical and theoretical analyses of the asymmetric rectified electric field (AREF) that occurs in electrolytes where the cations and anions have different mobilities. We demonstrate that AREFs induced by a non-antiperiodic electric potential, e.g., by a two-mode waveform with modes at 2 and 3 Hz, invariably yields a steady field that is spatially dissymmetric between two parallel electrodes, such that swapping which electrode is powered changes the direction of the field. Additionally, using a perturbation expansion, we demonstrate that the dissymmetric AREF occurs due to odd nonlinear orders of the applied potential. We further generalize the theory by demonstrating that the dissymmetric field occurs for all classes of zero-time-average (no dc bias) periodic potentials, including triangular and rectangular pulses, and we discuss how these steady fields can tremendously change the interpretation, design, and applications of electrochemical and electrokinetic systems.