Nonlinear oscillations in an electrolyte solution under ac voltage

The response of an electrolyte solution bounded between two blocking electrodes subjected to an ac voltage is considered. We focus on the pertinent thin-double-layer limit, where this response is governed by a reduced dynamic model [L. Hojgaard Olesen, M. Z. Bazant, and H. Bruus, Phys. Rev. E 82, 011501 (2010)]. During a transient stage, the system is nonlinearly entrained towards periodic oscillations of the same frequency as that of the applied voltage. Employing a strained-coordinate perturbation scheme, valid for moderately large values of the applied voltage amplitude V, we obtain a closed-form asymptotic approximation for the periodic orbit which is in remarkable agreement with numerical computations. The analysis elucidates the nonlinear characteristics of the system, including a slow (logarithmic) growth of the zeta-potential amplitude with V and a phase straining scaling as V-1 ln V. In addition, an asymptotic current-voltage relation is provided, capturing the numerically observed rapid temporal variations in the electric current.