Eur. Phys. J. E (2019) 42: 37
https://doi.org/10.1140/epje/i2019-11800-5

Regular Article

Tonks-Frenkel instability in electrolyte under high-frequency AC electric fields

¹ Laboratory of Electro-Hydrodynamics of Micro- and Nanoscales, Department of Mathematics and Computer Science, Financial University, 350051, Krasnodar, Russia
² Institut de Mécanique et d’Ingénierie - TREFLE, UMR CNRS 5295, University of Bordeaux, 16 Avenue Pey-Berland, Pessac Cedex, France
³ Univ. Grenoble Alpes, Grenoble-INP, CNRS, LRP, 38000, Grenoble, France
⁴ Laboratory of General Aeromechanics, Institute of Mechanics, Moscow State University, 117192, Moscow, Russia

^* e-mail: ganchenko.ru@gmail.com

Received: 26 October 2018
Accepted: 25 February 2019
Published online: 26 March 2019

Abstract

The instability of an electrolyte surface to a high-frequency, 10 to 200kHz, electric field, normal to the interface is investigated theoretically. From a practical viewpoint, such a high frequency leads to the absence of undesired electrochemical reactions and provides an additional control parameter. The theory of unsteady electric double layer by Barrero and Ramos is exploited. At such a high frequency, which is much larger than the eigenfrequency of the mechanical system, the nonlinear mechanical term does not “feel” the fast part of the Coulomb force, but it feels its slower component. In fact, the system behaves as if the electric field were a DC field. The observed instability is qualitatively close to the Tonks-Frenkel instability. The problem of the linear stability of the 1D quiescent stationary solution is solved analytically. For the important limiting cases, simple analytical formulas are derived. The linear stability analysis is complemented by the DNS of the full nonlinear system of equations with broadband low-amplitude white-noise initial conditions. After a transition period, the linear instability mechanism filters out the broad spectrum except for a narrow band near the maximum growth rate in rather good agreement with the linear stability analysis. If the external field is large enough, the nonlinear evolution results in coherent structures with sharp tips resembling to a Taylor cone. An evaluation of the cone angle for different conditions gives its value of about 30^° to 60^° , which is smaller than the angle of 98.6^° for DC field and qualitatively corresponds to the experiments (L.Y. Yeo et al., Phys. Rev. Lett. 92, 133902 (2004)) for the high-frequency AC field and to the theoretical evaluation of the AC Taylor cones in E.A. Demekhin et al., Phys. Rev. E 84, 035301(R) (2011).