Dynamics of charge monolayers on insulating liquid interfaces

Abstract

At most air-liquid interfaces stressed by a static perpendicular electric field, a monolayer of charge is induced that shields the field from the liquid. For relatively inviscid and highly insulating liquids, the electrohydrodynamics can dominate in determining the monolayer and field distributions associated with additional dynamic field components. Electric shear stresses lead to a convection of surface charge that distorts the field, much as though the liquid were conducting. A configuration for studying the monolayer dynamics is developed in which a uniform field is used to induce a uniform monolayer on the interface of a liquid layer having thickness c. Superimposed is a travelling wave of potential \[ {\rm Re}\,\hat{V}_0\exp i(\omega t - kz), \] imposed in a plane parallel to, and a t a distance a above, the interface. Mechanisms for charge redistribution are reflected in the frequency response of the field transmitted through the interface to a second plane bounding the liquid layer from below. A model is developed which accounts for the self-consistent electromechanics in terms of the lumped surface parameters of surface charge σ0 and surface mobility bs, and a bulk conductivity σ. According to this model, interfacial convection dominates a t low frequencies in attenuating the field induced below the interface. For layers which are thick compared with the viscous skin depth, there is a resonance in the response a t the frequency \[ \begin{array}{@{}l@{\qquad\qquad}c@{}} &\omega = \omega_i(2^{\frac{1}{3}}),\\ {\rm where} & \omega_i = \{k\sigma^2_0/\epsilon_0(\rho\eta)^{\frac{1}{2}}[\cot {\rm h}\, ka +(\epsilon/\epsilon_0)\cot {\rm h}\,kc]\}^{\frac{2}{3}} \end{array} \] and ρ, η, ε0 and ε are the liquid mass density, liquid viscosity, and permittivities of air and the liquid, respectively. Surface and bulk conduction, characterized by the relaxation time \[ \tau_b = \epsilon_0(\cot h\,ka + (\epsilon/\epsilon_0)\cot {\rm h}\,kc)/[\sigma_0b_sk + \sigma\cot {\rm h}\,kc], \] result in a broadening of the resonance which is appreciable even for ωτb ∼ 5. At frequencies high compared with both wi and i/Tb) the response is uninfluenced by the charge monolayer. Experiments substantiate the electroviscous resonance. A time-average surface force density, and associated steady convection of the liquid, is also theoretically shown to be a consequence of the phase shifts caused by the electroviscous resonance. This is qualitatively demonstrated by measurement of the steady shear stress induced on the plane bounding the liquid from below.