Nonlinear electrohydrodynamics of a surfactant-laden leaky dielectric drop

TL;DR

The study develops a three-dimensional, small-deformation theory for a surfactant-laden leaky dielectric drop in a uniform DC electric field, explicitly retaining surface charge convection to capture the Quincke rotation within the Taylor–Melcher framework. It derives a coupled nonlinear ODE system for the dipole moment, drop shape, and surfactant distribution, solved with RK4 and pseudo-arclength continuation, and shows how diffusion (via the elasticity number $El$ and diffusion parameter $ ext{zeta}$) modulates the Quincke threshold and deformation in both Taylor and Quincke regimes. The results reveal that weakly diffusing surfactants can lower the Quincke-rotation threshold and even suppress hysteresis, while strong diffusion tends to homogenize the surface tension and damp the rotational dynamics; elastic effects can either enhance or suppress deformation and rotation depending on the diffusion regime. The work provides a predictive framework for electrohydrodynamic control of surfactant-laden drops and lays groundwork for incorporating interfacial rheology and nondilute surfactant effects in complex emulsions.

Abstract

A nonlinear three-dimensional small-deformation theory is presented for a leaky dielectric drop coated with a dilute monolayer of insoluble apolar surfactant and subjected to a uniform DC electric field. The theory is developed within the framework of the Taylor--Melcher leaky dielectric model, and builds on previous work by retaining surface charge convection in the charge conservation equation. Solving the problem in three dimensions and retaining charge convection allows us to capture the transition to Quincke rotation, a symmetry breaking instability wherein a drop begins rotating at a steady angular velocity when the applied electric field strength exceeds a critical value. We derive a system of coupled nonlinear ordinary differential equations for the drop shape, dipole moment, and surfactant distribution, which we solve numerically. We discuss the combined effects of charge convection and surfactant in the Taylor regime -- in which the field strength is too weak to induce Quincke rotation and the drop adopts an axisymmetric spheroidal shape. In the Quincke regime, we find that the presence of a weakly-diffusing surfactant results in a lower critical electric field than that for a drop with uniform surfactant coverage. Varying the elasticity number, which quantifies the variation of the surface tension as a function of the surfactant concentration, can either increase or decrease the critical field strength depending on the diffusivity of the surfactant. Additionally, we find that the experimentally observed hysteresis in the angular velocity of the drop can disappear when surfactant diffusion is sufficiently weak.

Figures (11)

• Figure 1: An uncharged, neutrally buoyant, leaky dielectric drop coated with a dilute monolayer of insoluble, apolar surfactant immersed in an immiscible leaky dielectric fluid. A uniform DC electric field is applied in the $z$ direction, $\bm{E}_{0}=E_{0}\bm{\hat{e}}_{z}.$
• Figure 2: Evolution of (a) the tangential electric (yellow lines), hydrodynamic (blue lines) and Marangoni (dark red lines) stresses and (b) the deformation of a prolate A drop, for $\zeta=1$ (dotted lines), $\zeta=10$ (dashed lines), $\zeta=100$ (dash-dotted lines) and $\zeta=1000$ (solid lines). In these figures, $S=1/3$, $Q=1,$$\lambda=10,$${\rm El}=0.5$, ${\rm Re_{E}=1},$$\rm{Ca_{E}}=0.2.$
• Figure 3: Dependence of the steady deformation parameter $D_{Q}$ of (a) a prolate A drop with $S=1/3$, $Q=1$, (b) a prolate B drop with $S=1/3$, $Q=7/2,$ and (c) an oblate drop with $S=1$, $Q=2$ on $\zeta$ for $\rm{El}=$ 0 (black lines), 0.5 (blue lines), 1 (red lines) and 2 (purple lines). The solid lines are the results obtained when retaining charge convection, while the dashed lines are the results obtained when neglecting charge convection. (d--f) show the strength of the surface charge distribution, and (g--i) show the strength of the surface tension variations. In all cases, $\rm{Ca_{E}}=0.2,$$\rm{Re_{E}}=1,$$\lambda=10.$
• Figure 4: Bifurcation diagrams showing (a) the angular velocity $\Omega$ and (b) the deformation $D_{Q}$ of the drop as functions of the scaled electric field strength $E_{0}/E_{\mathrm{c,s}}$ for eleven values of $\zeta=0.01,$$0.1,$$0.5,$$1,$$5,$$10,$$20,$$30,$$50,$$100,$$1000,$ where solid blue lines denote stable solutions and dashed red lines denote unstable solutions. The arrows indicate the directions of increasing $\zeta.$ The insets in (a) show streamlines around the drop, and the insets in (b) show the corresponding surface tension as a function of $\theta.$ In all cases, $\rm{El}=0.5$ and the other parameter values are those used by salipante2010electrohydrodynamics, given in Table \ref{['parametervalues']}.
• Figure 5: Parameter $\eta$ given by \ref{['etadef']} as a function of the scaled electric field strength $E_{0}/E_{\mathrm{c,s}}$ for the same parameter values as those used in Figure \ref{['bifurcations']}. The arrow indicates the direction of increasing $\zeta.$