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.

Equilibrium height of a liquid in a gap between corrugated walls under spontaneous capillary penetration.

A simple and general method is presented to calculate the equilibrium surface of a liquid that penetrates spontaneously, due to capillarity, in the gap between two vertical corrugated plates. Several properties of the equilibrium solution are discussed and the results are backed by a qualitative experiment.

Breakup of a supported drop of a viscous conducting liquid in a uniform electric field.

Numerical computations and order-of-magnitude estimates are used to describe the time evolution of a drop of a very viscous liquid of finite electrical conductivity attached to a metallic plate which is suddenly subject to a uniform electric field. Under the action of the electric stresses induced at its surface, the drop elongates in the direction of the field, and charged droplets are emitted when the strength of the field is higher than a certain critical value. A stationary emission mode exists in which the attached drop develops a conical tip and a thin jet, with small droplets emitted from the end of the jet in a process that involves the formation of a long ligament. The flow rate and the electric current carried by the stream of droplets emitted in this mode are determined by the flow and the transfer of charge in the attached drop, in particular in a small region around its tip and in a leading stretch of the jet, where the solution is nearly stationary despite the transient character of the jet further downstream. A simplified analysis of the stationary regions is carried out to elucidate the effects of the physical properties of the liquid (electrical conductivity, permittivity, viscosity, and surface tension), the volume of the drop, and the strength of the applied field. For high electrical conductivities and applied fields well above its critical value, the electrical and viscous stresses are large compared to surface tension stresses, and their balance gives a flow rate proportional to the square of the applied field. The electric current is then that of a stationary electrified jet fed with this flow rate.

Model of the meniscus of an ionic-liquid ion source.

A simple model of the transfer of charge and ion evaporation in the meniscus of an ionic-liquid ion source working in the purely ionic regime is proposed on the basis of order-of-magnitude estimates which show that, in this regime, (i) the flow in the meniscus is dominated by the viscosity of the liquid and is affected very little by the mass flux accompanying ion evaporation, and (ii) the effect of the space charge around the evaporating surface is negligible and the evaporation current is controlled by the finite electrical conductivity of the liquid. The model predicts that a stationary meniscus of a very polar liquid undergoing ion evaporation is nearly hydrostatic and can exist only below a certain value of the applied electric field, at which the meniscus attains its maximum elongation but stays smooth. The electric current vs applied electric field characteristic displays a frozen regime of negligible ion evaporation at low fields and a conduction-controlled regime at higher fields, with a sharp transition between the two regimes owing to the high sensitivity of the ion evaporation rate to the electric field. A simplified treatment of the flow in the capillary or liquid layer through which liquid is delivered to the meniscus shows that the size of the meniscus decreases and the maximum attainable current increases when the feeding pressure is decreased, and that appropriate combinations of feeding pressure and pressure drop may lead to high maximum currents.

Natural convection in tilted cylindrical cavities embedded in rocks.

This paper presents a theoretical investigation of the low Rayleigh number conjugate natural convection in a slender tilted cylindrical cavity which is embedded in a solid that is subject to a uniform vertical temperature gradient. Two cases have been analyzed; a fluid-filled cavity and a cavity filled with a fluid-saturated porous medium. The temperature of the solid and the velocity, temperature, and pressure in the cavity have been determined by analytically solving the coupled problems within and around the cavity. The effect of the ratio of the thermal conductivity of the material in the cavity to the thermal conductivity of the solid on the structure of the convection flow is discussed. The theoretical results for convection in the fluid-filled cavity are shown to be in good agreement with experimental PIV measurements.

Liquid flow induced by ion evaporation in an electrified meniscus.

A simple model is proposed for the flow around the apex of a meniscus of a liquid undergoing ion evaporation in a vacuum under the action of a high electric field. The model includes a simplified description of the effect of the space charge surrounding the evaporating surface, and the idealizations that ion evaporation occurs at a constant surface field and that the electric field and viscous forces are negligible in the liquid. In agreement with known experimental and theoretical results for liquid metal ion sources, numerical solutions of the model problem show that the meniscus develops a protrusion and the current-voltage characteristic is linear in a range of voltages above an extinction voltage at which evaporation switches off. An oscillatory regime and transient evolutions ending in surface pinch-off and the emission of a drop are described, and the stabilizing effect of the pressure variations due to the evaporation flux is discussed. Asymptotic estimates for large evaporation flow rates are worked out.

Ion evaporation from the surface of a Taylor cone.

An analysis is carried out of the electric field-induced evaporation of ions from the surface of a polar liquid that is being electrosprayed in a vacuum. The high-field cone-to-jet transition region of the electrospray, where ion evaporation occurs, is studied taking advantage of its small size and neglecting the inertia of the liquid and the space charge around the liquid. Evaporated ions and charged drops coexist in a range of flow rates, which is investigated numerically. The structure of the cone-to-jet transition comprises: a hydrodynamic region where the nearly equipotential surface of the liquid departs from a Taylor cone and becomes a jet; a slender region where the radius of the jet decreases and the electric field increases while the pressure and the viscous stress balance the electric stress at the surface; the ion evaporation region of high, nearly constant field; and a charged, continuously strained jet that will eventually break into drops. Estimates of the ion and drop contributions to the total, conduction-limited current show that the first of these contributions dominates for small flow rates, while most of the mass is still carried by the drops.

Free-boundary problems describing two-dimensional pulse recycling and motion in semiconductors.

An asymptotic analysis of the Gunn effect in two-dimensional samples of bulk n GaAs with circular contacts is presented. A moving pulse far from contacts is approximated by a moving free boundary separating regions where the electric potential solves a Laplace equation with subsidiary boundary conditions. The dynamical condition for the motion of the free boundary is a Hamilton-Jacobi equation. We obtain the exact solution of the free-boundary problem (FBP) in simple one-dimensional and axisymmetric geometries. The solution of the FBP is obtained numerically in the general case and compared with the numerical solution of the full system of equations. The agreement is excellent so that the FBP can be adopted as the basis for an asymptotic study of the multidimensional Gunn effect.

Axisymmetric pulse recycling and motion in bulk semiconductors.

The Kroemer model for the Gunn effect in a circular geometry (Corbino disks) has been numerically solved. The results have been interpreted by means of asymptotic calculations. Above a certain onset dc voltage bias, axisymmetric pulses of the electric field are periodically shed by an inner circular cathode. These pulses decay as they move towards the outer anode, which they may not reach. As a pulse advances, the external current increases continuously until a new pulse is generated. Then the current abruptly decreases, in agreement with existing experimental results. Depending on the bias, more complex patterns with multiple pulse shedding are possible.