Frequency-dependent force between ac-voltage-biased plates in electrolyte solutions

We analyze the frequency dependence of the force between ac-voltage-biased plates in electrolyte solutions. To this end we solve analytically the Poisson-Nernst-Planck transport model in the dilute concentration and low voltage regime for a 1:1 symmetric electrolyte with blocking electrodes under a dc+ac applied voltage. The total force, which is the resultant of the electric and osmotic forces, shows a complex dependence on plate separation, frequency, ion concentration, and compact layer properties, different from that predicted from electrostatic current models or equivalent circuit models, due to the relevance of the osmotic force contribution in almost the whole range of frequencies. For the total dc force, we show that it decays at fixed ion concentration, linearly with plate separation for separations larger than a few times the Debye screening length. This linear dependence is due to the assumption about the conservation of the number of ions in the system. Moreover, the 1ω and 2ω ac harmonics of the total force show a broad peak at intermediate frequencies; it is centered at about the inverse of the charging time of the double layer capacitance, and covers the frequency range between the inverse of the diffusion time and the inverse of the electrolyte dielectric relaxation time. Finally, the 1ω ac harmonic component attains its high frequency asymptotic value at frequencies much higher than the inverse of the electrolyte dielectric relaxation time due to the very slow relaxation of the osmotic 1ω harmonic component at high frequencies. The derived analytical expressions for the total force remain valid up to voltages of the order of the thermal voltage, as has been assessed by means of numerical calculations. The numerical calculations are also used to explore the onset of higher force harmonics for larger applied voltages. Understanding the frequency dependence of the force acting on voltage-biased plates in electrolyte solutions can be of relevance for electrical actuation strategies in microelectromechanical systems and for the interpretation of some emerging electric scanning probe force microscopy techniques operating in electrolyte solutions.