Leaky Dielectric Liquid Research Articles

Electrohydrodynamic flows inside a neutrally buoyant leaky dielectric drop

We present an experimental investigation of electrohydrodynamic (EHD) flows within a neutrally buoyant drop with a radius of 2.25 mm. Utilizing particle image velocimetry and high-speed shadowgraphy, we measure the internal circulation and reported velocity profiles in the bulk and at the interface of the drop. Two leaky dielectric liquids, silicone and castor oils, are employed as the drop and as the external phase, allowing the analysis of two shape configurations: oblate and prolate. The strength of the applied uniform electric field (from 0.125 to 1.75 kV/cm) enables the analysis covering both the small-deformation limit (CaE≪1) and drops with larger deformations. Our measurements show good agreement with the leaky dielectric model (LDM) for the small-deformation cases. The flows begin at the interface as a result of jump in the electric stresses, leading then to four counter-rotating vortices inside the drop. At a permanent regime, the analytical solutions adequately predict the radial and tangential velocity components within the drop. However, a nuanced behavior is noticed for larger deformations, where the LDM theory underpredicts the internal circulation. Moreover, due to the increased deformation, a non-uniform azimuthal profile is observed for the velocity at the interface, vθ. Transient measurements of this velocity component enlighten the dynamic response of the EHD flows of the drop. Following the available analytical solutions, the dynamic response is governed by the timescale of the deformation of the drop, τdef=μa/γ. We propose a critical value of CaE≈0.1 below which the LDM adequately describes the velocity field in both quasi steady-state and transitory regimes.

Deformation transients of confined droplets within interacting electric and magnetic field environment

A theoretical exploration and an analytical model for the electro-magneto-hydrodynamics (EMHD) of leaky dielectric liquid droplets suspended in an immiscible confined fluid domain have been presented. The analytical solution for the system, under small deformation approximation in the creeping flow regime, has been put forward. The study of droplet deformation suggests that its temporal evolution is exponential and dependent on the electric and magnetic field interaction. Furthermore, the direction of the applied magnetic field relative to the electric field determines whether the magnetic force opposes or facilitates the interfacial net electrical force caused by the electric field. Validation of the proposed model at the asymptotic limits of vanishing magnetic field shows that the model accurately reduces to the case of the transient electrohydrodynamic model. We also propose a magnetic discriminating function (ϕM) to quantify the steady-state droplet deformation in the presence of interacting electric and magnetic fields. The change of droplets from a spherical shape to prolate and oblate spheroids corresponds to ϕM> 0 and < 0 regimes, respectively. It is shown that a substantial augmentation in the deformation parameter and the associated EMHD circulation within and around the droplet is achieved when aided by a low-magnitude magnetic field. The analysis also reveals the deformation lag and specific critical parameters that aid or suppress this lag behavior, discussed in terms of relevant non-dimensional parameters.

A spectral boundary integral method for simulating electrohydrodynamic flows in viscous drops

A weakly conducting liquid droplet immersed in another leaky dielectric liquid can exhibit rich dynamical behaviors under the effect of an applied electric field. Depending on material properties and field strength, the nonlinear coupling of interfacial charge transport and fluid flow can trigger electrohydrodynamic instabilities that lead to shape deformations and complex dynamics. We present a spectral boundary integral method to simulate droplet electrohydrodynamics in a uniform electric field. All physical variables, such as drop shape and interfacial charge density, are represented using spherical harmonic expansions. In addition to its exponential accuracy, the spectral representation affords a nondissipative dealiasing method required for numerical stability. A comprehensive charge transport model, valid under a wide range of electric field strengths, accounts for charge relaxation, Ohmic conduction, and surface charge convection by the flow. A shape reparametrization technique enables the exploration of significant droplet deformation regimes. For low-viscosity drops, the convection by the flow drives steep interfacial charge gradients near the drop equator. This introduces numerical ringing artifacts that we treat via a weighted spherical harmonic expansion, resulting in solution convergence. The method and simulations are validated against experimental data and analytical predictions in the axisymmetric Taylor and Quincke electrorotation regimes.

Effect of an axial electric field on the breakup of a leaky-dielectric liquid filament

We study experimentally and numerically the thinning of Newtonian leaky-dielectric filaments subjected to an axial electric field. We consider moderately viscous liquids with high electrical permittivity. We analyze the influence of the electric field on the formation of satellite droplets from the breakup of the filaments in the experiments. The electric force delays the free surface pinching. Two electrified filaments with the same minimum radius are thin at the same speed regardless of when the voltage is applied. The numerical simulations show that the polarization stress is responsible for the pinching delay observed in the experiments. Asymptotically close to the pinching point, the filament pinching is dominated by the diverging hydrodynamic forces. The polarization stress becomes subdominant even if this stress also diverges at this finite-time singularity.

Enhancement of electric field on bubble dispersion characteristics in leaky-dielectric liquid medium

Enhancement of the electric field on mixture has attached wide interests owing to its advantages in the terrestrial and space applications. In this paper, the dispersion characteristics of bubbles in leaky-dielectric liquid were investigated with a novel designed capillary electrode in the presence of a non-uniform electric field, employing high-speed photography and with a focus on the electrohydrodynamic (EHD) effects. The results show that the bubble dispersion is associated with three typical patterns in the manipulated parameter regions, including the dripping, mixed, and spraying patterns. With the increase of electrical Bond number (BoE), bubble size, characterized by the dimensionless bubble diameter ζ, is decreased significantly, while the evolution process of bubbles becomes faster and more complex. In particular, the size of the generated micro/nanobubbles in the spraying pattern is basically below 20 μm. The rise of the dimensionless flow rate (Q/Q0) increases ζ. It is found that the transition of bubble dispersion patterns depends strongly on the BoE, and the rise in Q/Q0 tends to slow down the transition process. In addition, an empirical correlation of ζ as a function of BoE and Q/Q0 is established for the dripping pattern of bubbles based on the present experiment data, which is believed to be of significance for the modeling of bubble dispersion under electric field. The bubble shapes, characterized by aspect ratios, show a diversity of distribution in the cross-section close to the capillary orifice, while terminally tending to follow the conventional formulation in the literature as the bubbles develop in the vertical direction.

Electrohydrodynamic-induced interactions between droplets

Dispersion of droplets in an emulsion is commonly seen in several chemical, pharmaceutical and petroleum industries. Electric field has been shown to affect the stability of these dispersions. We study the dynamics of a pair of leaky dielectric droplets in a leaky dielectric liquid in the presence of an externally applied electric field. A pair of droplets may coalesce or repel each other in the presence of an electric field. Interactions between a pair of drops have been shown to be governed by the ratio $\varepsilon _r/\sigma _r$ , where $\varepsilon _r$ and $\sigma _r$ are the ratios of drop to ambient fluid electric permittivities and conductivities, respectively. When inertia is neglected, the droplets approach each other if $\varepsilon _r/\sigma _r > 1$ , whereas droplets repel when $\varepsilon _r/\sigma _r < 1$ . However, inclusion of inertia permits interesting transient behaviour, where the droplets may attract due to the electrostatic dipole–dipole attraction even for $\varepsilon _r/\sigma _r < 1$ . The approach velocity then is governed by the electrostatic forces and varies as $1/h^4$ , where $h$ is the separation distance between the droplets, in contrast to being hydrodynamically driven as predicted in the Stokes flow limit by Baygents et al. (J. Fluid Mech., vol. 368, 1998, pp. 359–375). For compound droplets, interactions between droplets are essentially governed by the electrical properties of the outer droplet and the ambient fluid. However, transient dynamics may also result in the breakup of a compound droplet and lead to formation of single droplets.

Nonlinear dynamics of thin liquid films subjected to mixed-frequency electrical field

The nonlinear dynamics of the interface between both perfect and leaky dielectric liquid films, interposed between two parallel electrodes, are investigated under the effect of mixed-frequency electric fields. A coupled system of evolution equations is derived in dimensionless form, by employing the long-wave approximation. The linear stability analysis is implemented in accordance with the characteristics of each specific case, namely, the constant (DC) and the altering (AC) fields. In particular, the response of the system to the multi-mode AC electrical field is analyzed. Assisted by the conclusions of the theoretical investigation, the initial-boundary-value problem associated with the coupled system of evolution equations is solved numerically for several parameter sets. The system behavior is studied by monitoring the evolution process and by examining the steady/quasi-steady pillar formations in the nonlinear regime. The possibility to generate interface profiles of diverse topological forms, to manipulate their features, and to control the time dependent progress and the film rupture by imposing different combinations of frequencies and/or amplitudes of the corresponding mode is confirmed.

Effect of finite conductivity on the nonlinear behaviour of an electrically charged viscoelastic liquid jet

In this paper a one-dimensional numerical study on the nonlinear behaviour of an electrically charged jet of Oldroyd-B viscoelastic, Taylor–Melcher leaky dielectric liquid is carried out. The effect of surface charge level, axial wavenumber and finite conductivity on the nonlinear evolution of the jet is investigated. Different structures including beads-on-a-string with/without satellite droplets, quasi-spikes and spikes are detected, and their domains in the plane of the non-dimensional axial wavenumber and the electrical Bond number are illustrated. The underlying mechanisms in the formation of the structures are examined. It is found that tangential electrostatic force plays a key role in the formation of a quasi-spike structure. Decreasing liquid conductivity may lead to a decrease in the size of satellite droplets or even the complete removal of them from a beads-on-a-string structure, induce the transition from a beads-on-a-string to a quasi-spike structure or postpone the appearance of a spike. On the other hand, finite conductivity has little influence on filament thinning in a beads-on-a-string structure, owing to the fact that the electrostatic forces are of secondary importance compared with the capillary force. The difference between the finite conductivity, large conductivity and other cases is elucidated. An experiment is carried out to observe spike structures.

Faradaic reactions’ mechanisms and parameters in charging of oils

Leaky dielectric liquids, e.g. oils, constitute a class of conductors capable of being electrified to possess a net charge. Faradaic reactions have been recently demonstrated to be responsible for the electrification of such liquids as canola oil in electrostatic atomizers. Here we explain the fundamental chemical kinetics of the mechanisms responsible for charging of these oils at metallic electrodes and measure their parameters. Three main mechanisms of the electrode faradaic reactions of oil are recognized. (i) Cathodic reduction of protons resulting from fatty acid dissociation accompanied by anodic reactions with formation of metallo-organic deposits (metal salts of fatty acids). (ii) Redox reactions with participation of dissociated water molecules (the impurities) absorbed from the surrounding humidity. (iii) Redox reactions on a sharp pin electrode (either being a cathode or an anode) at higher voltages (with the magnitude above about 4 kV) leading to the Coulombic repulsion of charged oil from this electrode and the emergence of a toroidal vortex-like circulation in the oil bath. The electrochemical findings are corroborated by the results obtained by means of the Infrared spectroscopy and Raman spectroscopy of the electrode deposits, and the introduction of a novel method of measurement of the electric conductivity of oils used to establish the kinetic constants of the Frumkin-Butler-Volmer kinetics in the form of the Tafel plot.

Structure of the menisci of leaky dielectric liquids during electrically-assisted evaporation of ions

An understanding of the processes enabling field-assisted evaporation of ions from leaky dielectric liquids, i.e., liquids that are substantially less conductive than liquid metals, has historically been elusive in comparison to those of conventional electrohydrodynamic emission modes such as that of the cone-jet. While select ionic liquids have been shown to yield nearly monodisperse beams of molecular ions under certain conditions, the dearth of direct observation (visualization) and theoretical insight has precluded a fundamental appreciation for the inherent mechanics. In this paper, we present a family of equilibrium meniscus structures that shed measurable charge when the meniscus is large in relation to a characteristic emission scale. Such structures reside in a region of parameter space where empirical evidence suggests that steady emission may occur and also where stationary interfaces have not been reported before. In this regime, we show (i) that the macroscopic shape of the meniscus may vary only with the applied electric field; (ii) that the feeding flow is very germane to the emission characteristics, unlike liquid metal ion sources; and (iii) that while the balance of stresses governing the interface shape may in some cases be very similar to that of the classical Taylor cone, the widespread notion of a ubiquitous 49° half-angle is unfounded. Further study of this family may be helpful in elucidating a number of outstanding questions surrounding the pure ion mode.

Spreading of Thin Droplets of Perfect and Leaky Dielectric Liquids on Inclined Surfaces.

The spreading of droplets may be influenced by electric fields, a situation that is relevant to applications such as coating, printing, and microfluidics. In this work we study the effects of an electric field on the gravity-driven spreading of two-dimensional droplets down an inclined plane. We consider both perfect and leaky dielectric liquids, as well as perfectly and partially wetting systems. In addition to the effects of electric fields, we examine the use of thermocapillary forces to suppress the growth of the capillary ridge near the droplet front. Lubrication theory is applied to generate a set of coupled partial differential equations for interfacial height and charge, which are then solved numerically with a finite-difference method. Electric fields increase the height of the capillary ridge in both perfect and leaky dielectric droplets due to electrostatic pressure gradients that drive liquid into the ridge. In leaky dielectrics, large interfacial charge gradients in the contact-line region create shear stresses that also enhance ridge growth and the formation of trailing minor ridges. The coalescence of these ridges can significantly affect the long-time thinning rate of leaky dielectric droplets. In partially wetting liquids, electric fields promote the splitting of smaller droplets from the primary droplet near the receding contact line due to the interplay between electrostatic forces and disjoining pressure. Cooling from below and heating from above generates thermocapillary forces that counteract the effects of electric fields and suppress the growth of the capillary ridge. The results of this work have important implications for manipulating the spreading of droplets down inclined surfaces.

Using electric fields to induce patterning in leaky dielectric fluids in a rod-annular geometry

The stability and axisymmetric deformation of two immiscible, viscous, perfect or leaky dielectric fluids confined in the annulus between two concentric cylinders are studied in the presence of radial electric fields. The fields are set up by imposing a constant voltage potential difference between the inner and outer cylinders. We derive a set of equations for the interface in the long-wavelength approximation which retains the essential physics of the system and allows for interfacial deformations to be as large as the annular gap hence accounting for possible touchdown at the inner or outer electrode. The effects of the electric parameters are evaluated initially by performing a linear stability analysis which shows excellent agreement with the linear theory of the full axisymmetric problem in the appropriate long wavelength regime. The nonlinear interfacial dynamics are investigated by carrying out direct numerical simulations of the derived long wave models, both in the absence and presence of electric fields. For non-electrified thin layer flows (i.e. one of the layers thin relative to the other) the long-time dynamics agree with the lubrication approximation results found in literature. When the liquid layers have comparable thickness our results demonstrate the existence of both finite time and infinite time singularities (asymptotic touching solutions) in the system. It is shown that a two-side touching solution is possible for both the non-electrified and perfect dielectric cases, while only one-side touching is found in the case of leaky dielectric liquids, where the flattened interface shape resembles the pattern solutions found in literature. Meanwhile the finite-time singular solution agrees qualitatively with the experiments of~\cite{Reynolds1965}.

Electric field manipulation of particles in leaky dielectric liquids

Patchy colloidal capsules, i.e., hollow structures with permeable heterogeneous surfaces, possess advantageous properties when compared with their homogenous counterparts. For example, owing to specific interactions between patches they can self-assemble into complex structures to form functional materials. Fabrication of patchy colloidal capsules has long been theorized by scientists able to design different models, but the actual production of them remains a challenge. Here, we demonstrate how to manipulate colloidal particles on a surface of a droplet with electric fields. We present three examples of patchy structures formed via electrohydrodynamic microflows and electro-coalescence. In addition, we show electro-orientation of patchy capsules composed of both nonconductive polymeric particles and conductive silver coated particles. Finally, we discuss the importance of an appropriate selection of oils and particles.

The Taylor–Melcher leaky dielectric model as a macroscale electrokinetic description

While the Taylor–Melcher electrohydrodynamic model entails ionic charge carriers, it addresses neither ionic transport within the liquids nor the formation of diffuse space-charge layers about their common interface. Moreover, as this model is hinged upon the presence of non-zero interfacial-charge density, it appears to be in contradiction with the aggregate electro-neutrality implied by ionic screening. Following a brief synopsis published by Baygents & Saville (Third International Colloquium on Drops and Bubbles, AIP Conference Proceedings, vol. 7, 1989, American Institute of Physics, pp. 7–17) we systematically derive here the macroscale description appropriate for leaky dielectric liquids, starting from the primitive electrokinetic equations and addressing the double limit of thin space-charge layers and strong fields. This derivation is accomplished through the use of matched asymptotic expansions between the narrow space-charge layers adjacent to the interface and the electro-neutral bulk domains, which are homogenized by the strong ionic advection. Electrokinetic transport within the electrical ‘triple layer’ comprising the genuine interface and the adjacent space-charge layers is embodied in effective boundary conditions; these conditions, together with the simplified transport within the bulk domains, constitute the requisite macroscale description. This description essentially coincides with the familiar equations of Melcher & Taylor (Annu. Rev. Fluid Mech., vol. 1, 1969, pp. 111–146). A key quantity in our macroscale description is the ‘apparent’ surface-charge density, provided by the transversely integrated triple-layer microscale charge. At leading order, this density vanishes due to the expected Debye-layer screening; its asymptotic correction provides the ‘interfacial’ surface-charge density appearing in the Taylor–Melcher model. Our unified electrohydrodynamic treatment provides a reinterpretation of both the Taylor–Melcher conductivity-ratio parameter and the electrical Reynolds number. The latter, expressed in terms of fundamental electrokinetic properties, becomes $O(1)$ only for intense applied fields, comparable with the transverse field within the space-charge layers; at this limit the asymptotic scheme collapses. Surface-charge advection is accordingly absent in the macroscale description. Owing to the inevitable presence of (screened) net charge on the genuine interface, the drop also undergoes electrophoretic motion. The associated flow, however, is asymptotically smaller than that corresponding to the Taylor–Melcher circulation. Our successful matching procedure contrasts the analysis of Baygents & Saville, who considered more general electrolytes and were unable to directly match the inner and outer regions. We discuss this difference in detail.

Pin–Plate Electrode System for Emulsification of a Higher Conductivity Leaky Dielectric Liquid into a Low Conductivity Medium

In this experimental study we propose the use of highly nonuniform electric field to emulsify a leaky dielectric oil into another oil. Specifically, a pin–plate electrode system is used to emulsify castor oil into silicone oil. The method is suitable for the emulsification of a higher conductivity leaky dielectric oil dispersed in a lower conductivity medium and is suitable for an already existing emulsion, unlike electrospray methods. The process is stabilized by charging of the dispersed phase and the associated electrohydrodynamic flow. A balance of electrocoalescence and breakup leads to a stationary drop size distribution. A short process time indicates its suitability for continuous operation.

Stability of two immiscible leaky-dielectric liquids subjected to a radial electric field in an annulus duct

In this paper, we investigated the stability of a two coaxial leaky dielectric fluid system flowing in an annulus duct. A constant pressure gradient was applied to drive the flow in the duct. A radial electric field was imposed between the outer and inner surfaces of the duct. Linear stability analysis was employed to discuss the influences of electric field on the capillary and interface wave instabilities. The former instability is caused by surface tension and the latter is caused by viscosity stratification at the interface. It was found that, depending on the electrical permittivities and conductivities of the two liquids, the electric field either stabilized or destabilized the flow system. Apart from that, it was found that an external electric field could impede the capillary and interface wave instabilities. Influences of the inner radius of the duct, viscosity ratio, thickness ratio, and Reynolds number on the stability of the system were discussed as well.

Active structuring of colloidal armour on liquid drops.

Adsorption and assembly of colloidal particles at the surface of liquid droplets are at the base of particle-stabilized emulsions and templating. Here we report that electrohydrodynamic and electro-rheological effects in leaky-dielectric liquid drops can be used to structure and dynamically control colloidal particle assemblies at drop surfaces, including electric-field-assisted convective assembly of jammed colloidal ‘ribbons’, electro-rheological colloidal chains confined to a two-dimensional surface and spinning colloidal domains on that surface. In addition, we demonstrate the size control of ‘pupil’-like openings in colloidal shells. We anticipate that electric field manipulation of colloids in leaky dielectrics can lead to new routes of colloidosome assembly and design for ‘smart armoured’ droplets.

On the use of hypodermic needles in electrospray

We present an experimental analysis of electrospray where the standard feeding capillary was re- placed with a hypodermic needle. Both the current intensity and the jet diameter were measured for a variety of leaky-dielectric liquids. The results obtained with a feeding capillary and a hypodermic needle of similar dimen- sions were compared. An increase of the liquid conductivity was measured by assuming that the electric current obeys the same scaling law in both configurations. That increase was likely due to the electrochemical reaction taking place in the needle tip.

Electrohydrodynamic Drop Deformation by Strong Electric Fields: Slender-Body Analysis

Slender-body approximations are utilized to analyze drop elongation by a uniformly applied electric field. The Taylor--Melcher model of leaky-dielectric liquids is employed, with electrohydrodynamic flow animation by electrical shear stresses at the free surface. Using the drop slenderness as the small asymptotic parameter, separate asymptotic expansions of the pertinent fields are presented in “inner” and “outer” regions, respectively, corresponding to the drop cross-sectional and longitudinal scales, as well as an additional expansion in the drop phase. For a given shape, both the electric potential and flow field are calculated. Asymptotic matching is possible only for low drop viscosity. The normal-stress condition on the free surface provides a scaling relation between the slenderness parameter and the dimensionless electric field, expressed as a capillary number. The predicted slenderness scaling, inversely with the $6/7$-power of the electric field, is the same as that appropriate for dielectric li...

Taylor cones in a leaky dielectric liquid under an ac electric field

Conical points of a leaky dielectric drop surrounded by a dielectric gas in an external ac electric field are investigated. A novel class of steady conical tips depending on the permittivity ratio and applied signal frequency is presented. It is found that conical solutions with very small angles are possible (angles much smaller than the classical Taylor cone angle 49.3° for a conducting drop in a dc field); this result can be relevant to the observations of small cone angles in Chetwani, Maheshwari, and Chang experiments [N. Chetwani, S. Maheshwari, and H.-C. Chang, Phys. Rev. Lett. 101, 204501 (2008)].