Leaky Dielectric Model Research Articles

Ion evaporation from Taylor cones of propylene carbonate mixed with ionic liquids

A combined experimental and numerical approach is used to extract information on the kinetics of ion evaporation from the region of high electric field around the tip of a Taylor cone of the neutral solvent propylene carbonate (PC) mixed with two ionic liquids. On the numerical side, the electric field on the surface of the liquid is computed in the absence of evaporation by solving the electrohydrodynamic problem in this region within the framework of the leaky dielectric model. These computations justify the approximate (2% max error) scaling Emax = β Ek for the maximum electric field on the surface, with Ek = γ1/2 ϵ0−2/3 (K/Q)1/6 for 0.111 < K < 0.888 S m−1 and a numerical value of β ≈ 0.76. Here γ is the surface tension of PC, ϵ0 is the electrical permittivity of vacuum, and K and Q are the liquid electrical conductivity and flow rate. On the experimental side, 16 different propylene carbonate solutions with either of the ionic liquids 1-ethyl-3-methylimidazolium tetrafluoroborate (EMI-BF4) or EMI-bis(trifluoro-methylsulfonyl)imide (EMI-Im) are electrosprayed in a vacuum from a single Taylor cone, and their emissions of charged drops and ions are analysed by time-of-flight mass spectrometry at varying liquid flow rates Q. The sprays contain exclusively drops at large Q, both for small and for large electrical conductivities K, but enter a mixed ion–drop regime at sufficiently large K and small Q. Interestingly, the mixtures containing 10% and 15% (vol) EMI-Im exhibit no measurable ion currents at high Q, but approach a purely ionic regime (almost no drops) at small Q. The charge/mass ratio for the drops produced in these two mixtures increases continuously with decreasing Q, and gets very close to ionic values. Measured ion currents are represented versus computed maximum electric fields Emax on the liquid surface to infer ion evaporation kinetics. Comparison of measured ion currents with predictions from ion evaporation theory yields an anomalously low activation energy (~1.1 eV). This paradox appears to be due to alteration of the pure conj–eet electric field in the scaling laws used for the pure cone–jet regime, due to the substantial ion current density arising even when the ion current is relatively small. Elimination of this interference would require future ion current measurements in the 10–100 pA level. The electrical propulsion characteristics of the emissions from these liquids are determined and found to be excellent, particularly for 10% and 15% (vol) EMI-Im.

Bulk electroconvective instability at high Péclet numbers

Bulk electroconvection pertains to flow induced by the action of a mean electric field upon the residual space charge in the macroscopic regions of a locally quasielectroneutral strong electrolyte. For a long time, controversy has existed in the literature as to whether quiescent electric conduction from such an electrolyte into a uniform charge-selective solid, such as a metal electrode or ion exchange membrane, is stable with respect to bulk electroconvection. While it was recently claimed that bulk electroconvective instability could not occur, this claim pertained to an aqueous, low-molecular-weight electrolyte characterized by an order-unity electroconvection Péclet number. In this paper, we show that the bulk electroconvection model transforms into the leaky dielectric model in the limit of infinitely large Péclet number. For the leaky dielectric model, conduction of the above-mentioned type is unstable, and so it is in the bulk electroconvection model for sufficiently large Péclet numbers. Such instability is sensitive to the ratio of the diffusivity of the cations to the anions. For infinite Péclet number, the case with equal ionic diffusivities is a bifurcation point separating stable and unstable regimes at the low-current limit. Further, for a cation-selective solid, when the Péclet number is finite and the anions are much more diffusive than the cations, an unreported bulk electroconvective instability is possible at low current. At higher currents and large Péclet numbers, we found that the system is unstable for all cation-to-anion diffusivity ratios, but passes from a monotonic instability to an oscillatory one as this ratio passes through unity.

Stationary coaxial electrified jet of a dielectric liquid surrounded by a conductive liquid

The electrohydrodynamic problem describing the evolution of a coaxial jet of two immiscible Newtonian liquids injected through concentric orifices into a region of uniform electric field is formulated in the framework of the leaky dielectric model, and the dimensionless parameters governing the flow are identified. A simplified model is proposed combining a quasi-uni-directional approximation for the flow and the transport of electric charge with a fully numerical evaluation of the electric field. Results of this model are compared with boundary element solutions of the full governing equations for viscosity-dominated flows. The current/flow rate characteristic of a coaxial viscosity-dominated jet is computed and the ranges of flow rates in which a stationary axisymmetric jet is realized are discussed in the case of an outer liquid of finite electrical conductivity surrounding a dielectric liquid, in which the electric shear that stretches the jet is concentrated at its outer surface. The dependence of the flow on the applied electric field and the flow rates of the liquids, as well as the effects of the viscosity and electrical conductivity of the inner liquid, and of the surface tension of the inner surface, are discussed.

Electrohydrodynamic Flows in Nonhomogeneous Liquids

Small-scale electrohydrodynamic flows were investigated in a configuration where the motive force derives from charge generated in an electrolyte composition gradient. Using a thin Hele-Shaw cell filled with aqueous electrolyte, we studied flow through a disc-shaped bolus whose conductivity differed from that of the surrounding solution. When an ac electric field was applied, a thin layer of free charge arose from differences in conductivity between the bolus and its surroundings. The action of the field on the charge produced bulk flow, and since material interfaces were absent, the bolus deformed continuously. Deformations were prolate or oblate, depending on the conductivity of the bolus relative to the surrounding electrolyte. Modeling the flow with the Hele-Shaw formalism showed how electrical forces produce motion. To indicate the sense of the deformation-prolate or oblate with respect to the applied field-a discriminating function was derived. The discriminating function reduces to the Taylor-Melcher leaky dielectric model under the appropriate limiting conditions. Experiment and theory are in good agreement. The flows described here occur in the absence of rigid interfaces and equilibrium charge layers. Accordingly, they offer a means of controlling microfluidic flow and mixing.

Modeling and characterization of a carbon fiber emitter for electrospray ionization

A novel microscale emitter employing a pointed carbon fiber (CF) can be used for electrospraying in mass spectrometric (MS) analysis. The carbon fiber is located coaxial with a fused silica capillary tube of 360 µm OD and 75 µm ID with its sharp tip extending 30 µm beyond the tube terminus. The electrospray ionization (ESI) process is simulated using a computational fluid dynamics (CFD) code based on the Taylor–Melcher leaky-dielectric fluid model for solving the electrohydrodynamics and the volume of fluid (VOF) approach for tracking the liquid–gas interface. The CFD code is first validated for a conventional geometry and then used to simulate the CF emitter based ESI model. The simulated current–flow and current–voltage results are in good agreement with experimental results for the CF emitter. The effects of emitter geometry, potential difference, flow rate and the physical properties of the liquid on the electrospray behavior of the CF emitter are thoroughly investigated. The spray current and jet diameter are correlated with the flow rate, potential difference and physical properties of the liquid and the correlation results are quantitatively compared with the results reported in the literature.

Electrohydrodynamic instability of the interface between two fluids confined in a channel

The stability of the interface between two dielectric fluids confined between parallel plates subjected to a normal electric field in the zero Reynolds number limit is studied analytically using linear and weakly nonlinear analyses, and numerically using a thin-layer approximation for long waves and the boundary element technique for waves with wavelength comparable to the fluid thickness. Both the perfect dielectric and leaky dielectric models are studied. The perfect dielectric model is applicable for nonconducting fluids, whereas the leaky dielectric fluid model is applicable to fluids where the time scale for charge relaxation, (ϵϵ0∕σ), is small compared to the fluid time scale (μR∕Γ), where ϵ0 is the dielectric permittivity of the free space, ϵ and σ are the dielectric constant and the conductivity of the fluid, μ and Γ are the fluid viscosity and surface tension, and R is the characteristic length scale. The linear stability analysis shows that the interface becomes unstable when the applied potential exceeds a critical value, and the critical potential depends on the ratio of dielectric constants, electrical conductivities, thicknesses of the two fluids, and surface tension. The critical potential is found to be lower for leaky dielectrics than for perfect dielectrics. The weakly nonlinear analysis shows that the bifurcation is supercritical in a small range of ratio of dielectric constants when the wavelength is comparable to the film thickness, and subcritical for all other values of dielectric constant ratio in the long-wave limit. The thin-film and boundary integral calculations are in agreement with the weakly nonlinear analysis, and the boundary integral calculation indicates the presence of a secondary subcritical bifurcation at a potential slightly larger than the critical potential when the instability is supercritical. When a mean shear flow is applied to the fluids, the critical potential for the instability increases, and the flow tends to alter the nature of the bifurcation from subcritical to supercritical.

Droplet Deformation in Dc Electric Fields: The Extended Leaky Dielectric Model

The leaky dielectric model (LDM) was extended to large droplet distortions in dc electric fields. The resulting extended LDM (ELDM) reduces to the LDM for small droplet aspect ratios and to the pure dielectric model when the ratio of droplet and matrix conductivities equals the inverse ratio of their permittivities. The ELDM distinguishes between two types of phenomena possible at high electric fields: continuous deformation and hysteresis. For droplets deforming parallel to the electric field, the relationship that distinguishes between the two phenomena is a function of the droplet and matrix conductivities and viscosities but not of their permittivities. For droplets deforming perpendicular to the electric field, the relationship is a function of the droplet permittivities and conductivities but depends only slightly on the ratio of their viscosities. Some of the predictions of the LDM and the ELDM were compared with our own data and with data from the literature. For the systems that deformed parallel to the field direction, the ELDM not only predicted the data qualitatively but also predicted the data quantitatively when the experimental errors in its input parameters were taken into account, whereas the older LDM did not even predict the qualitative trend of the data. For the systems that deformed perpendicular to the field direction, however, the ELDM predicted the observed the aspect ratios in only one out of the four systems examined. In the other three systems, the LDM appeared to give reasonable predictions when either the ratio of the matrix/droplet viscosities was relatively small or the value of total charge relaxation time was relatively large. Thus, the applicability of the ELDM, as presently formulated, appears to be limited in the case of deformations perpendicular to the electric field.

Electrohydrodynamics of a drop under nonaxisymmetric electric fields

We consider the electrohydrodynamics of a spherical drop in a nonaxisymmetric electric field, which can be approximated by the sum of a uniform field and a linear straining field. We obtain the analytic solution of the three-dimensional flow fields inside and outside a drop for the Stokes flow regime by using Lamb's general solution and the leaky dielectric model. With the analytic solution, the dielectrophoretic migration velocity of a drop is obtained as a function of the type and the frequency of the imposed electric field. The direction of drop motion is found to be parallel to the dielectrophoretic force. The analytic solution is also used to investigate the characteristics of the interfacial flow under various nonaxisymmetric electric fields. While investigating the interfacial flow, we find a surface vortex structure under certain nonaxisymmetric electric fields, which is found to be related to the chaotic mixing inside the drop. Finally, we consider the chaotic features of three-dimensional flows inside the drop under static nonaxisymmetric electric fields.

Linear stability analysis of thin leaky dielectric films subjected to electric fields

An intriguing process, known as lithographically induced self-assembly (LISA), is initiated by positioning a template parallel to a flat silicon wafer-coated with a thin polymeric film and then raising the temperature above the glass transition/melting temperature of the film. Electric fields exert a force on charges induced at the polymer–air interface, placing the film in tension. The static equilibrium that results is unstable to disturbances with wavelengths for which the electrostatic force overcomes the surface tension. Flow ensues, generating a pattern in the film with periodicity reflecting the characteristic length of the instability. Though the initiation of the process as outlined above is generally accepted, the forces guiding the evolution of the film into the remarkably periodic microstructures observed are not. Our goal here is to create a sound understanding of the mechanism through quantitative modeling to facilitate the conversion of these microstructures into nano-structures. Given the apparent importance of conductivity in the film we adopt the “leaky dielectric model”, which also allows for re-distribution of charges on the interfaces, and undertake a linear stability analysis to explore the effects of various process parameters, particularly the conductivity and the film thickness. The linear stability analysis with the leaky dielectric model for the polymer film yields growth exponents and characteristic wavenumbers much larger than that for the perfect dielectric model. The differences are striking in that the slightest conductivity increases the growth exponent by a factor of 2–20 and decreases the fastest growing wavelength by a factor of 2–4.

Linear oscillations and stability of a liquid bridge in an axial electric field

Small amplitude oscillations of viscous, capillary bridges are studied in the presence of an electric dc field. The electric field is proposed as a means to maintain bridges longer than their perimeter and of uniform cylindrical shape. This is desired in the fabrication of semiconductor crystals. The material of the bridge and the surrounding medium is modeled either as a perfect or as a leaky dielectric. The frequency and the damping rate of the oscillations are calculated numerically by solving a generalized eigenvalue problem. It is shown that they depend on the ratios of the dielectric constants, ε=εin/εout, and conductivities, S=σin/σout, of the two materials, the aspect ratio of the bridge, Λ=πR̃/L̃, the ratio of viscous to the capillary force, Oh=Re−1, which can also be viewed as the inverse Reynolds number of the flow, and, finally, the electrical Bond number, Cel, which is the ratio of the electric stresses to the capillary force. The stability limit of an initially cylindrical bridge is determined with respect to varicose disturbances. In agreement with previous studies it is shown that, if both materials are perfect dielectrics, application of an electric field has a stabilizing effect on the bridge, in the sense that the minimum value, Λmin, of the aspect ratio for the bridge to remain stable drops below 0.5, irrespective of the specific value of the ratio ε. If both materials are leaky dielectrics, bridge stability is determined by the sign of (S−ε) and (S−1)(ε−1), with the positive sign indicating bridge stabilization. The factor (S−ε) arises due to the appearance of a tangential electric stress in the perturbed state for leaky dielectrics. For both cases of leaky and perfect dielectrics, the most unstable mode is the one leading to amphora shaped bridges. It was also found that, when application of an electric field stabilizes the bridge, leaky dielectrics require a lower field than perfect dielectrics and that a large enough field tends to stabilize the bridge for almost the entire range of values of the aspect ratio Λ. These findings concur with earlier analytical results for the stability of jets in longitudinal electric fields and, in conjunction with certain experimental observations, point to the usefulness of the leaky dielectric model pertaining to the stability of bridges.

Electrohydrodynamic effects on the deformation and orientation of a liquid capsule in a linear flow

The role of a uniform electric field on the deformation and orientation of a liquid capsule with a viscoelastic membrane is considered analytically in the small deformation limit. The capsule is freely suspended either in a quiescent fluid or in a shear flow. The viscoelasticity of the membrane is taken into account by the Kelvin–Voigt model and the electrohydrodynamic flow is analyzed on the basis of the leaky dielectric model. In this article, we consider three different prototype models of capsules; viz., a neo-Hookean (incompressible isotropic) membrane, a red blood cell-type (area-preserving) membrane, and an interfacial-tension droplet. The deformed capsule shape from its initial sphericity and its orientaion are determined from the linearized governing equations and boundary conditions in the limit of small deformations. The asymptotic theory shows that the degree of capsule deformation induced by a uniform electric field alone is independent of the surface viscosity of the capsule as well as the viscosity ratio between the two fluids inside and outside the capsule. Meanwhile, in the presence of an imposed shear flow, the degree of deformation depends on the surface viscosity with preserving still the independence of the viscosity ratio. For an illustrative purpose, experimental results for the role of a uniform electric field on the orientation of an interfacial-tension droplet in a shear flow are discussed briefly.

The electrohydrodynamic stability of a liquid bridge: microgravity experiments on a bridge suspended in a dielectric gas

The electrohydrodynamic stability of a liquid bridge was studied in steady and oscillatory axial electric fields with a novel apparatus aboard a space shuttle. To avoid interphase transport, which complicates matters in terrestrial, matched-density systems, the experiments focused on a liquid column surrounded by a dielectric gas. The micro-gravity acceleration level aboard the spacecraft kept the Bond number small; so interface deformation by buoyancy was negligible. To provide microgravity results for comparison with terrestrial data, the behaviour of a castor oil bridge in a silicone oil matrix liquid was studied first. The results from these experiments are in excellent agreement with earlier work with isopycnic systems as regards transitions from a perfect cylinder to the amphora shape and the separation of an amphora into drops. In addition, the location of the amphora bulge was found to be correlated with the field direction, contrary to the leaky dielectric model but consistent with earlier results from terrestrial experiments. Next, the behaviour of a bridge surrounded by a dielectric gas, sulphur hexa fluoride (SF6), was investigated. In liquid–gas experiments, electrohydrodynamic ejection of liquids from ‘Taylor cones’ was used to deploy fluid and form bridges by remote control. Experiments with castor oil bridges in SF6 identified the conditions for two transitions: cylinder–amphora, and pinch-off. In addition, new behaviour was uncovered with liquid–gas interfaces. Contrary to expectations based on perfect dielectric behaviour, castor oil bridges in SF6 could not be stabilized in AC fields. On the other hand, a low-conductivity silicone oil bridge, which could not be stabilized by a DC field, was stable in an AC field.

Electrohydrodynamic behaviour of a drop subjected to a steady uniform electric field at finite electric Reynolds number

The electrohydrodynamic flow associated with a fluid drop in an electric field is a consequence of the tangential electric stress at the fluid interface. The tangential viscous stress due to the electrohydrodynamic flow arises to just balance the tangential electric stress at the fluid interface so that the traction boundary condition is satisfied. Influenced by both the local electric stress and viscous stress, the drop interface may exhibit various shapes. The presence of fluid flow also leads to charge convection phenomena. The relative significance of the charge convection effect is usually measured in terms of the electric Reynolds number, Re E , defined as the ratio of the timescales of charge convection by flow and that for charge relaxation by ohmic conduction. This work presents a quantitative analysis of the charge convection effects in a framework of the leaky dielectric model at finite Re E , which has not been considered in previous investigations. Axisymmetric steady flows driven by an applied uniform electric field about a deformable fluid drop suspended in an immiscible fluid are studied by computational means of the Galerkin finite–element method with supplementary asymptotic analysis. The results of finite–element computations are in general agreement with the prediction by the asymptotic analysis for spherical drops at vanishingly small Re E . A common effect of charge convection is found to reduce the intensity of electrohydrodynamic flow. As a consequence, oblate drops are predicted to be less deformed in an electric field when charge convection is taken into account. The prolate drops are often associated with an equator–to–pole flow, which convects charges toward the poles to form a charge distribution resembling that in a highly conducting drop immersed in an insulating medium. Therefore, charge convection tends to enhance the prolate drop deformation. In many cases, charge convection effects are found to be significant even at apparently small Re E , corresponding to the charge relaxation time–scale about 10 −3 s, suggesting that many experimental results reported in the previous publications could have been influenced by charge convection effects.

Electrohydrodynamic deformation and interaction of drop pairs

The motion of two drops in a uniform electric field is considered using the leaky dielectric model. The drops are assumed to have no native charge and a dielectrophoretic effect favours translation of the drops toward one another. However, circulatory flows that stem from electrohydrodynamic stresses may either act with or against this dielectrophoretic effect. Consequently, both prolate and oblate drop deformations may be generated and significant deformation occurs near drop contact owing to enhancement of the local electric field. For sufficiently widely spaced drops, electrohydrodynamic flows dominate direct electrical interactions so drops may be pushed apart, though closely spaced drops almost always move together as a result of the electrical interaction or deformation.

Characteristics of Electrorheological Responses in an Emulsion System

The characteristics of electrorheological (ER) responses in an emulsion system have been examined for the first time. The emulsion system consists of a chlorinated paraffin oil and a silicone oil; these two liquids have similar dielectric constants, but the former liquid is 1000 times more conductive than the latter liquid. The mismatch in electric properties between the dispersed phase and the dispersing medium can be reversed in an emulsion by inverting the phase distribution. Transient ER responses in the emulsion system have been studied over a range of shear rates following step changes (on/off) of an external dc electric field. The relative response strength characterizing the ER effect increases with the field strength and with the volume fraction of silicone oil, but decreases with increasing shear rate imposed. The paraffin oil-in-silicone oil emulsion exhibits a response much stronger than that observed in the phase-reversed silicone oil-in-paraffin oil emulsion. No negative ER response is detected in the silicone oil-in-paraffin oil emulsion. With an optical microscope, very different microscopic response characteristics are observed for the phase-reversed emulsions under static conditions, and this can be explained according to the leaky dielectric liquid model of Taylor.

ELECTROHYDRODYNAMICS: The Taylor-Melcher Leaky Dielectric Model

▪ Abstract Electrohydrodynamics deals with fluid motion induced by electric fields. In the mid 1960s GI Taylor introduced the leaky dielectric model to explain the behavior of droplets deformed by a steady field, and JR Melcher used it extensively to develop electrohydrodynamics. This review deals with the foundations of the leaky dielectric model and experimental tests designed to probe its usefulness. Although the early experimental studies supported the qualitative features of the model, quantitative agreement was poor. Recent studies are in better agreement with the theory. Even though the model was originally intended to deal with sharp interfaces, contemporary studies with suspensions also agree with the theory. Clearly the leaky dielectric model is more general than originally envisioned.

Dielectrophoresis of a deformable fluid particle in a nonuniform electric field.

In nonuniform electric fields, the dielectrophoretic behavior and the interface shape of uncharged fluid particles freely suspended in an immiscible fluid may be significantly influenced by the induced electrohydrodynamic flow. This work presents a theoretical investigation of the primary effects of electrohydrodynamic flow associated with the dielectrophoresis of a deformable fluid particle with a leaky dielectric model. The electrohydrodynamic flow is shown to either enhance or hinder the dielectrophoretic motion of fluid particles. A variety of shapes of deformable fluid particles are predicted. \textcopyright{} 1996 The American Physical Society.

A computational analysis of electrohydrodynamics of a leaky dielectric drop in an electric field

Axisymmetric steady flows driven by an electric field about a deformable fluid drop suspended in an immiscible fluid are studied within the framework of the leaky dielectric model. Deformations of the drop and the flow fields are determined by solving the nonlinear free-boundary problem composed of the Navier-Stokes system governing the flow field and Laplace's system governing the electric field. The solutions are obtained by using the Galerkin finite-element method with an elliptic mesh generation scheme. Under conditions of creeping flow and vanishingly small drop deformations, the results of finite-element computations recover the asymptotic results. When drop deformations become noticeable, the asymptotic results are often found to underestimate both the flow intensity and drop deformation. By tracking solution branches in parameter space with an arc-length continuation method, curves in parameter space of the drop deformation parameter D versus the square of the dimensionless field strength E usually exhibit a turning point when E reaches a critical value Ec. Along such a family of drop shapes, steady solutions do not exist for E > Ec. The nonlinear relationship revealed computationally between D and E2 appears to be capable of providing insight into discrepancies reported in the literature between experiments and predictions based on the asymptotic theory. In some special cases with fluid conductivities closely matched, however, drop deformations are found to grow with E2 indefinitely and no critical value Ec is encountered by the corresponding solution branches. For most cases with realistic values of physical properties, the overall electrohydrodynamic behaviour is relatively insensitive to effects of finite-Reynolds-number flow. However, under extreme conditions when fluids of very low viscosities are involved, computational results illustrate a remarkable shape turnaround phenomenon: a drop with oblate deformation at low field strength can evolve into a prolate-like drop shape as the field strength is increased.

The electrohydrodynamic deformation of drops suspended in liquids in steady and oscillatory electric fields

When an electric field is applied to a drop suspended in another liquid the drop deforms. The relation between the applied field and the mode and magnitude of the deformation have been studied extensively. Nevertheless, Torza, Cox & Mason (1971) found that quantitative agreement between the leaky dielectric theory (Taylor 1966) and experiment is quite poor. Here we describe results from a new series of experiments. Drops suspended in weakly conducting liquids were deformed into spheroids with both steady and oscillatory fields. Drop deformation, interfacial tension, and the electrical properties of the fluids were measured for each system to provide a definitive test of the theory. The agreement between the leaky dielectric model and our results for drop deformations in steady fields is much improved over previous results, although discrepancies remain for some systems. Drop deformations in oscillatory fields consist of steady and oscillatory parts because of the quadratic dependence on the field strength. Measurements of the steady part at 60 Hz, where the oscillatory deformation is negligible, are in excellent agreement with the theory. The effects of frequency on the steady deformation were studied by measuring oblate deformations at a series of frequencies and field strengths; the agreement with theory is good. Finally, the time-dependent total deformation was measured under conditions where both parts of the deformation are commensurate. Good agreement was found between the measured and predicted maximum and minimum deformations. Nevertheless, only a small range of fluid properties could be studied owing to the need to avoid droplet sedimentation.