Electric Bond Number Research Articles

Periodic emission of droplets from an oscillating electrified meniscus of a low-viscosity, highly conductive liquid.

The generation of identical droplets of controllable size in the micrometer range is a problem of much interest owing to the numerous technological applications of such droplets. This work reports an investigation of the regime of periodic emission of droplets from an electrified oscillating meniscus of a liquid of low viscosity and high electrical conductivity attached to the end of a capillary tube, which may be used to produce droplets more than ten times smaller than the diameter of the tube. To attain this periodic microdripping regime, termed axial spray mode II by Juraschek and Röllgen [R. Juraschek and F. W. Röllgen, Int. J. Mass Spectrom. 177, 1 (1998)], liquid is continuously supplied through the tube at a given constant flow rate, while a dc voltage is applied between the tube and a nearby counter electrode. The resulting electric field induces a stress at the surface of the liquid that stretches the meniscus until, in certain ranges of voltage and flow rate, it develops a ligament that eventually detaches, forming a single droplet, in a process that repeats itself periodically. While it is being stretched, the ligament develops a conical tip that emits ultrafine droplets, but the total mass emitted is practically contained in the main droplet. In the parametrical domain studied, we find that the process depends on two main dimensionless parameters, the flow rate nondimensionalized with the diameter of the tube and the capillary time, q, and the electric Bond number B(E), which is a nondimensional measure of the square of the applied voltage. The meniscus oscillation frequency made nondimensional with the capillary time, f, is of order unity for very small flow rates and tends to decrease as the inverse of the square root of q for larger values of this parameter. The product of the meniscus mean volume times the oscillation frequency is nearly constant. The characteristic length and width of the liquid ligament immediately before its detachment approximately scale as powers of the flow rate and depend only weakly on the applied voltage. The diameter of the main droplets nondimensionalized with the diameter of the tube satisfies d(d)≈(6/π)(1/3)(q/f)(1/3), from mass conservation, while the electric charge of these droplets is about 1/4 of the Rayleigh charge. At the minimum flow rate compatible with the periodic regimen, the dimensionless diameter of the droplets is smaller than one-tenth, which presents a way to use electrohydrodynamic atomization to generate droplets of highly conducting liquids in the micron-size range, in marked contrast with the cone-jet electrospray whose typical droplet size is in the nanometric regime for these liquids. In contrast with other microdripping regimes where the mass is emitted upon the periodic formation of a narrow capillary jet, the present regime gives one single droplet per oscillation, except for the almost massless fine aerosol emitted in the form of an electrospray.

Development and Modeling of Melt Electrohydrodynamic-Jet Printing of Phase-Change Inks for High-Resolution Additive Manufacturing

This paper presents the development and modeling a high-resolution electrohydrodynamic-jet (EHD-jet) printing process using phase-change ink (i.e., wax), which is capable of producing sub-10 μm footprints (sub-10 fL in volume) for super-resolution additive manufacturing. In this study, we successfully apply EHD-jet printing for phase-change ink (wax), which is widely used as modeling and supporting material for additive manufacturing, to achieve micron-scale features. The resolution for single droplet on substrate is around 5 μm with the thickness in the range of 1–2 μm, which provides great potential in both high-resolution 3D printing and 2D drop-on-demand microfabrication. The droplet formation in EHD printing is modeled by finite element analysis (FEA). Two important forces in EHD printing, electrostatic force and surface tension force, are modeled separately by FEA. The droplet size is obtained by balancing the electrostatic force and surface tension of the pending droplets around meniscus apex. Furthermore, to predict the droplet dimension at different process conditions, a dimensionless scaling law is identified to describe the relationship between dimensionless droplet diameter and modified nondimensional electrical bond number. Finally, the droplets in-flight velocity and impact characteristics (e.g., Reynolds number and Weber number) are modeled using the results from FEA analysis.

Morphology of water electrosprays in the simple-jet mode

Experiments were conducted in order to study and characterize electrohydrodynamic atomization in the simple-jet mode for inviscid liquids. The operational window of this mode regarding the electric potential and liquid flow rate is presented. From the data it could be concluded that this mode can be divided by the characteristics of its breakup mechanism and that these characteristics are a function of the liquid Weber number and the electric Bond number for a given setup. Additionally we were also able to calculate the average charge per droplet and define the average size of primary and satellite droplets. The dispersion of the spray was also studied regarding its relation to the liquid Weber number and to the electric Bond number. We conclude that simple-jet mode electrosprays are a good option for applications which require monodisperse micrometer droplets with high throughput.

Dielectric Nanodroplets: Structure, Stability, Thermodynamics, Shape Transitions and Electrocrystallization in Applied Electric Fields

The response and properties of a dielectric nanodroplet with a 10 nm diameter made of formamide placed in a uniform electric field are explored with molecular dynamics simulations and an analytic free energy formulation. Increasing fields cause the initially spherical liquid droplet to undergo gradual prolate spheroidal elongation along the field direction that culminates in a shape instability and a first-order shape transition to a slender liquid needle at a field of ∼0.50 V/nm. This transition is accompanied by enhanced reorientation of the molecular dipoles along the field direction, with the elongated droplets exhibiting a field-induced ferroelectric state, unlike the state of the droplet under field-free conditions. For larger fields, we find further gradual enhancement of the molecular dipole reorientation and the onset of a first-order electro-crystallization transition at a field of 1.4375 V/nm. This transition is portrayed by a sharp increase in the positional order parameter, a significant decrease in the molecular diffusion, and further preferential reorientation of the dipoles. Both transitions exhibit hysteresis as the change in the electric field is reversed, that is, for decreasing fields. Data from the molecular dynamics simulation were used for determination of the field-dependent dielectric permittivity of the formamide droplet. An analytic dielectric continuum model is developed, and a free energy expression which includes polarization, surface, and saturation terms is formulated, yielding results for the variation of the droplet shape as a function of the applied field, or electric Bond number, in agreement with the results obtained from the molecular dynamics simulations. Proper inclusion of dielectric saturation effects in the continuum model, expressed in terms of an electric field-dependent dielectric permittivity, is found to play an important role.

Absolute-convective instability transition of low permittivity, low conductivity charged viscous liquid jets under axial electric fields

The linear spatiotemporal instability of a charged viscous jet of low permittivity, low conductivity liquid under axial electric field is studied here. The local convective-absolute (C-A) instability transition is explored in the four-dimensional parameter space of the Reynolds number, Weber number, electrical Bond number, and externally applied tangential electric field. The locations where the C-A instability transition occurs in the four-dimensional space are identified (i.e., determining the critical Weber number) for both the axisymmetric and the first non-axisymmetric modes, covering a wide range of Reynolds numbers, electrical Bond numbers, and tangential electric field values. In contrast with the case of an uncharged jet, the C-A transition can be suppressed by the electric field at large Reynolds numbers for the axisymmetric mode, whereas the C-A transition for the first non-axisymmetric mode is always promoted by the electrical Bond number and the tangential electric field. We delimit the regions where the C-A transition is (i) always non-axisymmetric (which leads to local whipping) independent of the value of the applied tangential electric field or (ii) always axisymmetric (which leads to local dripping) below a certain charge level (electrical Bond number). The regions where one can have axisymmetric or non-axisymmetric C-A transition depending on the values of the jet charge and applied electric field are also described. The comparison of theoretical predictions with published experiments strongly supports the validity of the models proposed. These results provide the fundamentals to predict whether jetting-dripping or jetting-whipping transition can be expected in applications involving liquids of poor electrical conductivity and permittivity like many polymers, oils, and several solvents.

Buoyancy-driven motion of a two-dimensional bubble or drop through a viscous liquid in the presence of a vertical electric field

The effect of an electric field on the buoyancy-driven motion of a two-dimensional gas bubble rising through a quiescent liquid is studied computationally. The dynamics of the bubble is simulated numerically by tracking the gas–liquid interface when an electrostatic field is generated in the vertical gap of the rectangular enclosure. The two phases of the system are assumed to be perfect dielectrics with constant but different permittivities, and in the absence of impressed charges, there is no free charge in the fluid bulk regions or at the interface. Electric stresses are supported at the bubble interface but absent in the bulk and one of the objectives of our computations is to quantify the effect of these Maxwell stresses on the overall bubble dynamics. The numerical algorithm to solve the free-boundary problem relies on the level-set technique coupled with a finite-volume discretization of the Navier–Stokes equations. The sharp interface is numerically approximated by a finite-thickness transition zone over which the material properties vary smoothly, and surface tension and electric field effects are accounted for by employing a continuous surface force approach. A multi-grid solver is applied to the Poisson equation describing the pressure field and the Laplace equation governing the electric field potential. Computational results are presented that address the combined effects of viscosity, surface tension, and electric fields on the dynamics of the bubble motion as a function of the Reynolds number, gravitational Bond number, electric Bond number, density ratio, and viscosity ratio. It is established through extensive computations that the presence of the electric field can have an important effect on the dynamics. We present results that show a substantial increase in the bubble’s rise velocity in the electrified system as compared with the corresponding non-electrified one. In addition, for the electrified system, the bubble shape deformations and oscillations are smaller, and there is a reduction in the propensity of the bubble to break up through increasingly larger oscillations.

Two-Dimensional Wave Motion on the Charged Surface of a Viscous Liquid

The wave motion on the charged surface of a viscous Newtonian liquid is solved as an initial-value problem. Both the leaky dielectric and perfect dielectric cases are considered. The amplitude of wave is assumed to be small. The electric field induced by surface charge is shown to have a generally destabilizing effect on surface wave. The neutral stability curve is drawn in the (G,Ne) plane (G: the gravitational bond number; Ne: the electrical Bond number). The Ohnesorge number, Taylor–Melcher number and permittivity ratio have little influence on the neutral stability curve. It is testified that the classical normal mode method cannot predict wave behaviour at small times.

Injection of bubbles in a quiescent inviscid liquid under a uniform electric field

Numerical computations and order of magnitude estimates are presented for the periodic generation and coalescence of bubbles due to the injection of a constant flow rate of a gas through a circular orifice at the bottom wall of an inviscid dielectric or very polar liquid that is at rest and subject to a uniform vertical electric field far from the orifice. The problem depends on five dimensionless parameters: a Bond number based on the radius of the orifice; Weber and electric Bond numbers whose square roots are dimensionless measures of the flow rate of gas and the applied electric field; the dielectric constant of the liquid; and the contact angle of the liquid with the bottom wall. The bubbles that grow quasi-statically at the orifice for small values of the Weber number are always elongated vertically by the electric stress that acts on their surface when an electric field is applied. The volume of these bubbles at detachment may reach a maximum at a certain value of the electric Bond number, if the Bond number is sufficiently small, or decrease monotonically with the electric Bond number if the Bond number is larger. In both cases the bubbling ceases to be periodic beyond a certain value of the electric Bond number, apparently giving way to more complex bubbling regimes, which are not investigated here. Bubble interaction and eventually coalescence occur when the Weber number is increased keeping the electric Bond number in the range of periodic bubbling. Different periodic regimes are described. It is shown that a moderate electric field may increase the value of the Weber number above which coalescence occurs without changing the shape of the bubbles much. A large electric field may suppress coalescence but it also favours the development of upward and downward jets that cross the bubbles and may cause their breakdown.

The impact of external electrostatic fields on gas–liquid bubbling dynamics

The effect of an applied electric potential on the dynamics of gas bubble formation from a single nozzle in glycerol was studied experimentally. Dry nitrogen was bubbled into glycerol through a nozzle having an electrified tip while pressure measurements were made upstream of the nozzle. As the applied electric potential was increased from zero, bubble size reduced, bubble shape became more spherical, and bubbling frequency increased. At constant gas flow, bubble-formation exhibited a classic period-doubling route to chaos with increasing potential. We defined an electric Bond number assuming that both the liquid and gas phases are conducting. This is in contrast to previous studies where one phase was considered a perfect conductor and the other one a perfect nonconductor or insulator. Although electric potential and gas flow appear to have similar effects on the period-doubling bifurcation process for this system, the relative impact of electrostatic forces, as measured in terms of electric Bond number for conducting liquid and gas phases, is smaller. However, the relative impact of electrostatic forces for the case of insulating liquid and conducting gas phases is comparable to flow forces. Further data collection is required for different nozzle geometries and liquid column heights in order to verify the relative impacts of electrostatic and flow forces, and would allow us to ascertain if electrostatic potential is a feasible manipulated variable for controlling this system.

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.

The effect of residual axial gravity on the stability of liquid columns subjected to electric fields

The stability criterion for almost cylindrical dielectric liquid bridges subjected to axial electric fields in the presence of residual axial gravity is obtained. In its absence, a perfectly cylindrical equilibrium solution is allowed for all values of the relevant parameters, which are the slenderness of the liquid bridge, the electrical Bond number and the relative permittivity between the outer and inner media. This basic solution is unstable beyond a critical slenderness which varies with the electrical parameters (Gonzblez et al. 1989). The destabilization takes place axisymmetrically. The inclusion of the gravitational Bond number as a new, small parameter may be treated by means of the Lyapunov-Schmidt Method, a wellknown projection technique that gives the local bifurcation diagram relating the admissible equilibrium amplitudes for the liquid bridge and the aforementioned parameters. As in the absence of applied electric field, the gravitational Bond number breaks the pitchfork diagram into two isolated branches of axisymmetric equilibrium solutions. The stable one has a turning point whose location determines the new stability criterion. Quantitative results are presented after solving the resulting set of linear recursive problems by means of the method of lines.

Dimensional Analysis of Continuous High-capacity Electrodispersion of Aqueous Based Liquids in an Organic Continuous Phase

Abstract High-intensity electric fields can be used to disperse aqueous-based solutions in a relatively nonconducting immiscible organic phase. Dimensional analysis of electrical dispersion performance from a single grounded nozzle between two charged electrodes can be characterised in terms of five dimensionless groups: an electrode height; nozzle-electrode distance; a nozzle Reynolds number; an electrical Bond number, which relates electrical to surface forces; and a Taylor number, which relates electrical to viscous forces. Experimental results on the electrodispersion of water in 2-ethyl-1-hexanol indicate that pulsed DC fields can accomplish electrodispersion utilizing a lower rms-voltage than steady DC fields. In addition, the pulsed-field-behavior varies with pulse frequency with 200 Hz fields being more effective for higher continuous-phase viscosities while 2000 Hz fields are more effective when the viscosity is lower. A steady DC field displays invariant behavior with changing viscosity. In the ...