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Journal of Electrostatics 72 (2014) 330e335

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Journal of Electrostatics

journal homepage: www.elsevier .com/locate/elstat

Electrohydrodynamic cone-jet bridges: Stability diagram andoperating modes

Fangjie Liu, Chuan-Hua Chen*

Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC 27708, USA

a r t i c l e i n f o

Article history:Received 4 January 2014Received in revised form5 May 2014Accepted 29 May 2014Available online 12 June 2014

Keywords:ElectrohydrodynamicsTaylor cone-jetLiquid bridgeStability diagramOperating modesCapillary breakup

* Corresponding author. Tel.: þ1 919 660 5343; faxE-mail address: chuanhua.chen@duke.edu (C.-H. C

http://dx.doi.org/10.1016/j.elstat.2014.05.0040304-3886/© 2014 Elsevier B.V. All rights reserved.

a b s t r a c t

An electrohydrodynamic cone-jet bridge is formed when two opposing Taylor cones are bridged by aliquid jet. We used high-speed video imaging to systematically investigate the operating regimes of thecone-jet bridge established between a nozzle and a liquid pool that were closely separated. There was astability island for the cone-jet bridge in the voltage-flow rate operating diagram, and the stable bridgecould only be formed above a minimum flow rate and at an intermediate range of voltages. In the vicinityof the stability island, the cone-jet bridge broke up via a thinning or beading mode.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

An electrohydrodynamic cone-jet bridge is formed when twoclosely separated Taylor cones [1] are connected by a liquid column[2]. The bridging column evolves from the thin jet emitted from asingle cone, i.e. the conventional Taylor cone-jet [3e7]. However,the close presence of the counter cone fundamentally alters theproperties of the electrohydrodynamic jet, e.g. the bridging jet istypically dominated by Ohmic current and much larger in diametercompared to the conventional jet [2]. The cone-jet bridge is relatedto other forms of electrohydrodynamic liquid bridges [8e16], but itsmorphology is distinct with two opposing cones bridged by aslender jet. Although a non-electrical liquid bridge is unstablewhenthe length of the static bridge exceeds its circumference [17e20],electrohydrodynamic bridges can be stabilized by radial polariza-tion stresses to achieve a much larger aspect ratio [2,8,21e27].

For the conventional cone-jet, voltage (V) and flow rate (Q) arethemain external control variables, and the VeQ operating diagramis an indispensable guide for accessing the various modes of stableand unstable cone-jets [5,7,28e35]. For a givenworking fluid with afixed electrode separation, a minimum flow rate is needed to sus-tain a steady jet. Above the minimum flow rate, a steady cone-jetcan be maintained for an intermediate range of voltages. The

: þ1 919 660 8963.hen).

upper and lower voltage boundaries merge at the minimum flowrate, giving rise to a stability island on the VeQ map [5,7].

The main objective of this paper is to establish the VeQ oper-ating diagram and identify the operating modes of the cone-jetbridges. In addition to reporting the bridge morphologies withinthe stability island as a function of voltage and flow rate, we alsobriefly discuss the thinning/beading breakup modes [36e38] nearthe lower/upper stability boundaries.

2. Experimental setup

We adopted a nozzle-to-pool configuration to study the cone-jetbridge (Fig.1). A tapered glass nozzle (NewObjective TT360-150-50-N-5) with a length of 20 mm and an inner/outer diameter of 140/200mmat thenozzle exitwas groundedwith respect to a liquid pool,which was negatively electrified with a high-voltage source (Trek610E). The glass nozzle was above the center of the liquid pool andperpendicular to it. The non-conducting nozzle was affixed to atubing sleeve and attached to a metallic union for hydraulic andelectrical connections; see Fig. 1a of Ref. [39] for a detailed sche-matic. The liquid pool was prepared by filling the working fluid in arectangular container with a bottom wall made of aluminum(15mm� 15mm) and four glass walls on the sidewith a height of 6mm.A circular hole of 1mmdiameterwas drilled at the center of thebottomplate for active liquid drainage. Ethylene glycol (CASNo.107-21-1) was used as the working fluid with the conductivity doped to

Fig. 1. (a) Schematic (not to scale) of cone-jet bridge formation between a nozzle and aliquid pool. The glass nozzle was electrified via a metallic union at a voltage V withrespect to the liquid pool, and the working fluid was supplied and extracted at thesame flow rate Q to maintain a constant liquid volume. The dot-dashed line indicatedthe position of the pool surface prior to electrification. (b) A sample cone-jet bridge of10 mS/cm ethylene glycol at 2 kV and 300 mL/h, where the bridge was pinned on theouter rim of the nozzle. With respect to the undeformed pool surface, the nozzle-to-pool separation was maintained at 400 mm unless otherwise noted. Note that thethe tapered glass nozzle had an inner/outer diameter of 140/200 mm at the exit, andthe conical base on the nozzle could span the inner or outer rim depending on theoperating conditions.

Fig. 2. Formation process of the cone-jet bridge at 3.5 kV and 300 mL/h. Top row:emergence of the counter cone from the liquid pool; middle row: merging of thecounter cone-jet with the upstream Taylor cone-jet issued from the nozzle; bottomrow: receding of the Taylor cones toward the steady-state configuration of the liquidbridge.

F. Liu, C.-H. Chen / Journal of Electrostatics 72 (2014) 330e335 331

10 mS/cm by sodium borate, and ambient air was the surroundingdielectricmedium. Theworking fluidwas infused andwithdrawn atthe sameflowratewith twodual-mode syringe pumps (Legato 180).The separation between the nozzle exit and the surface of the un-deformed liquid pool was fixed at 400 mm unless otherwise noted,with care taken to account for the slight liquid overflow (to facilitateimaging) due topinning at the glass sidewalls. The resulting cone-jetbridgewasmonitored bya high-speed camera (PhantomV710) via along-distance microscope (Infinity K2 with a 4� objective), whichwas tilted 5� unless otherwise noted.

Our setup deviated from the initial report where the cone-jetbridge was horizontally oriented, and the counter cone wasattached to a textured substrate verticallyorientated forgravitationaldrainage [2]. In Fig.1, thebridgewasorientatedvertically to eliminategravitationally induced asymmetry particularly on each Taylor cone.The bridge volumewas held constant by extracting from the pool atthe same flow rate as that supplied to the nozzle. The active drainageof theworking fluid was incorporated to avoid complications arisingfrom a potential mismatch in the feeding and draining rates.

3. Results and discussions

We now discuss the conditions for stable cone-jet bridges byestablishing a VeQ stability diagram and documenting the oper-ating modes with respect to this diagram. We used 10 mS/cmethylene glycol as a model working fluid, although many otherworking fluids used in conventional electrospraying are alsoconducive to cone-jet bridges [2].

3.1. Formation process

As discussed in the original report [2], close electrode separationis crucial to the formation of the cone-jet bridge which evolves

from a single cone-jet. We established the cone-jet bridge with theliquid pool, which was closely positioned from the nozzle (Fig. 1).Although the liquid pool replaced the textured substrate used inRef. [2] as the counter electrode, the cone-jet bridge formationprocess was very similar.

The representative formation process in Fig. 2 could be dividedinto three steps: (i) when electrified, an upstream cone-jet [3,5]was issued from the nozzle while an inverse Taylor cone [1]developed from the liquid pool. (ii) When the inverse coneapproached the upstream cone at a close enough separation, asecond jet was issued from the inverse cone, and the two opposingcones bridged together after the two jets merged. Note that thedevelopment of the counter jet wasmuch clearer in Fig. 2 of Ref. [2],where the two opposing cones were slightly off-centered as a resultof the asymmetry due to gravitational drainage. (iii) The cone-jetbridge relaxed to a steady configuration with two receded Taylorcones connected by a slender liquid column. The actively drainedliquid pool offered not only a deformable interface leading to theinverse cone, but also a mechanism to maintain a constant volumefor the steady cone-jet bridge.

3.2. Operating diagram

To obtain the operating diagram for steady cone-jet bridges(“stable bridge” in Fig. 3), the flow rate was consistently reducedfrom 400 mL/h at 50 mL/h decrements. At each flow rate, the voltagewas consistently increased from low to high to identify the lowerand upper voltage boundaries, between which a steady cone-jetbridge could be formed. The consistent procedure, particularly inthe direction of voltage variation, was designed to minimize thehysteresis in the VeQ operating diagram [40,41]. Although thehysteresis in the operating voltage range is known to be significantfor single cone-jet [5,7], the hysteresis for the cone-jet bridge wasobserved to be small, typically within ±0.2 kV for both the lowerand upper boundaries.

For comparison, we also constructed the operating diagram forconventional single cone-jet in a similar fashion (“stable jet” inFig. 3). For this purpose, the setup in Fig. 1 was modified so that theliquid pool was eliminated and the nozzle exit was 400 mm apartfrom the top surface of the holed aluminum plate.With such a close

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0 100 200 300 400 500

Volta

ge(k

V)

Flow rate (μL/h)

stable jet

stable bridge

Fig. 3. Stability diagram of the cone-jet bridge (solid line) versus the single cone-jet(dashed line). Beyond the minimum flow rate required to form a stable cone-jetbridge, the stability island of the cone-jet bridge encompassed a much broaderrange of voltages for a fixed nozzle-to-pool separation of 400 mm.

Fig. 4. Morphologies of stable cone-jet bridges at the stability boundaries. The twoimages in each columnwere captured at a given flow rate (shown at the bottom) at theupper and lower voltage boundaries (indicated in the images) extracted from Fig. 3.

F. Liu, C.-H. Chen / Journal of Electrostatics 72 (2014) 330e335332

electrode separation, liquid accumulation on the counter electrodecould interfere with the upstream cone-jet; Hence, any excessliquid on the plate was frequently blown away in addition to beingextracted from the bottom hole.

Since the cone-jet bridge evolves from conventional cone-jet,their operating diagrams should be related. Qualitatively, the twooperating diagrams in Fig. 3 were similar. Like the VeQ diagram forthe single cone-jet, there was a stability island for the cone-jetbridge with a minimum flow rate of around 150 mL/h (stablecone-jet bridges could not be obtained at 100 mL/h or below).Above the minimum flow rate, there was an intermediate range ofvoltages within which stable bridges could be obtained. Quantita-tively, the lower boundaries of two stability islands more or lessoverlapped, but the operable voltage range was much broader forthe cone-jet bridge. Indeed, multi-jets [28] at voltages well abovethe upper boundary for a “stable jet” could be transformed into a“stable bridge” if an inverse conewas allowed to develop. The liquidbridges at such high fields were stabilized by radial polarizationstresses which counter balance the capillary forces [2].

Two pitfalls should be noted when comparing the operatingdiagrams in Fig. 3. First, the electrode separation of 400 mm wasnominal. The deformation of the liquid pool reduced the actualelectrode separation to be significantly smaller than this nominalvalue (Fig. 1). Therefore, the electrode separations between thesecases were not exactly equal. Second, the non-conducting nozzlewas electrified at a metallic union some 20 mm upstream and theelectrical resistance of this extra nozzle length had a much strongerinfluence in the case of cone-jet bridge, since the bridging jet wasmuch thicker and carried a much larger current; see Ref. [2] fordetailed discussions.

Fig. 5. Receding of the upstream Taylor cone at the nozzle exit from the outer rim tothe inner rim: (a) as the voltage was increased from the lower to upper stabilityboundary at a fixed flow rate of 400 mL/h, the conical base reached the inner cross-sectional area at around 2.3 kV. (b) As the flow rate was reduced at a fixed voltageof 2 kV, the conical base receded to the inner rim at 200 mL/h. The optical axis wastilted 20� to clearly show the conical base.

3.3. Stable morphologies

Within the stability island for the cone-jet bridge (Fig. 3), thebridge morphology exhibited a large variation. In Fig. 4, the imageswere captured at the lower and upper stability boundaries for eachgiven flow rate. At the upper boundary, the bridge diameterappeared to increase with increasing flow rate. However, this trendwas not apparent at the lower boundary. Compared to the mor-phologies at the lower boundary (bottom row in Fig. 4), the bridgewas thicker and the upstream cone was much more receded at theupper voltage boundary (top row). The notable exception was at

150 mL/h, approximately the minimum flow rate at which the lowerand upper boundaries were expected to merge together.

The receded meniscus from a wetting nozzle has rarely beenreported in conventional electrohydrodynamic systems, an excep-tion being the liquid bridges reported in Fig. 3c of Ref. [12]. To bettervisualize the meniscus close to the inner nozzle, the optical axiswas tilted 20� (Fig. 5). Apparent from the last images in Fig. 5a andb, the conical base receded to the inner rim of the nozzle at highvoltages and low flow rates, a trend consistent with Fig. 4. Note thatthe voltage and flow rate variations in Fig. 5 were all within thestability island.

3.4. Breakup processes

Even for the conventional cone-jet, the functioning modesoutside the stability island are very complex and not well under-stood [28,31,40,42]. For the cone-jet bridge, we only briefly discussthe breakup processes in the immediate vicinity of the stabilityisland.

Immediately below the lower stability boundary, the cone-jetbridge broke up in a “thinning” mode (Fig. 6). When the voltagewas slightly below the threshold for a stable bridge, a transientcone-jet bridge was formed from time to time. The transient bridge

Fig. 6. A representative breakup process in the thinning mode near the lower stabilityboundary. Without much axial variation, the diameter of the bridging filament uni-formly shrank down. The image sequence was captured at 1.74 kV, slightly below thelower boundary of 1.79 kV at 300 mL/h.

Fig. 8. Morphologies of stable cone-jet bridges as a function of nozzle-to-pool sepa-ration: (a) 300 mm; (b) 400 mm; (c) 500 mm. The left and right columns represent thelower and upper stability boundaries, respectively. The bottom of each image indicatesthe initial location of the unelectrified pool surface.

F. Liu, C.-H. Chen / Journal of Electrostatics 72 (2014) 330e335 333

experienced capillary oscillations in a manner similar to Fig. 3 ofRef. [2]. In the last cycle of the oscillations, the bridging jet uni-formity thinned down (at a decelerating pace) till the point ofdisappearance.

Immediately above the upper stability boundary, the cone-jetbridge broke up in a “beading” mode (Fig. 7). In this case, thetransient bridge also experienced capillary oscillations but a bea-ded structure gradually appeared after a few cycles of oscillations.The breakup process was markedly different from the thinningmode. In the beading mode, the jet diameter was highly nonuni-form in the axial direction and a localized point of breakup could beclearly identified.

Below the minimum flow rate (slightly to the left of the stabilityisland), the breakup mode was more or less a combination of thethinning and beading modes. However, a more systematic inves-tigation should be undertaken to map out the possible instabilitymodes.

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300 μm400 μm500 μm

3.5. Nozzle-to-pool separation

As evident in the bridging process in Fig. 2, close electrodeseparation was critical to the formation of the cone-jet bridge. Sofar, the nozzle-to-pool separation was fixed at 400 mm.

The bridge morphologies at different separations are shown inFig. 8. Steady bridges could be established when the separationwasbelow 500 mm or so. Above 500 mm, bridging was prohibited andonly single cone-jet could be obtained. Below 200 mm, cone-jetbridges were quickly destroyed by hydrodynamic bridging be-tween the nozzle and the pool. Note that the liquid bridges couldbecome off-centered at high fields. An example is shown in Fig. 8cat a voltage of 7.1 kV, which was the upper boundary at the sepa-ration of 500 mm.

The stability islands for three representative separations areshown in Fig. 9. These VeQ operating diagrams were qualitativelysimilar with larger separations requiring higher voltages. It isimportant to note again the nominal nature of the reported sepa-ration, which was between the nozzle exit and the undeformed

Fig. 7. A representative breakup process in the beading mode near the upper stabilityboundary. Although the diameter of the bridging filament was initially uniform in theaxial direction, a beaded structure developed and the axial variation eventually led tocapillary breakup. The image sequence was captured at 5.34 kV, slightly above theupper boundary of 5.30 kV at 300 mL/h.

pool surface. The actual electric field was strongly affected by theprotrusion of the Taylor cone and deformation of the pool as shownin Fig. 8, both effectively reducing the electrode separation.

To change the electric field which is the ultimate controllingparameter, either the voltage or separation can be varied. For avoltage and flow rate combination that initially yielded a stablecone-jet bridge, morphological transition akin to Fig. 5a withincreasing voltage was observed when the separation wasdecreased at a constant voltage. It is also interesting to note that thebridge broke up in the beading (thinning) mode similar to Fig. 7(Fig. 6) as the separation was decreased (increased) beyond thestability limit.

3.6. Discussions

Although the focus of this study of cone-jet bridges was theexperimental mapping of the stability diagram and operating

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Flow Rate (μL/h)

Fig. 9. Stability diagrams of the cone-jet bridge at different nozzle-to-pool separations.The nominal separation was between the nozzle exit and the unelectrified pool sur-face. The minimum flow rate required for stable cone-jet bridge was approximately150 mL/h at a separation of both 300 mm and 400 mm, and close to 250 mL/h at 500 mm.

F. Liu, C.-H. Chen / Journal of Electrostatics 72 (2014) 330e335334

modes, we shall make a few comments to better connect the pre-sent study with prior work.

The cone-jet bridge builds upon the conventional Taylor cone-jet [3e5]. The coupling between electric and flow fields withdeformable interfaces makes it very difficult to quantitativelyanalyze the cone-jet systems. Even the conversion from voltages toelectric fields is not straightforward as noted earlier. For this reason,our operating diagrams were presented on the voltage-flow rate(VeQ) map, similar to the those reported for conventional cone-jets[5]. Although the lower voltage boundary of the cone-jet bridgewas close to that of the Taylor cone-jet (Fig. 3), the minimum flowrates in the two systems did not seem to be directly related. Forexample, the minimum flow rate in conventional Taylor cone-jetstypically varies inversely with the electrical conductivity[7,40,43], except for highly viscous working fluids [41]; this well-known inverse dependence was not observed in cone-jet bridgeswhen the conductivity of ethylene glycol was varied from 0.3 mS/cmto 30 mS/cm.

The cone-jet bridge is related to a pinned liquid bridge [17,19]under an electric field [8,9] with two notable differences: thecone-jet bridge has an imposed flow rate, and the bridging jet isslender compared to the pinned bridge. A natural question iswhether a slender cone-jet bridge can be directly built from aconventional liquid bridge without an imposed flow rate. Weexplored different possibilities including quickly stretching a con-ventional bridge while imposing an electric field, turning off theflow rate after establishing a stable cone-jet bridge, and imposingopposite flow rates from two symmetrically positioned nozzles.Despite extensive trials, we were not able to continuously trans-form a conventional pinned liquid bridge into a cone-jet bridgewithout first experiencing a breakup of the bridge.

The breakup modes for the cone-jet bridge have been observedin other contexts of capillary breakupwith or without electric fields[36e38]. The thinning mode seems to be promoted by radial po-larization stresses [8,22,23], which can conceivably produce effectssimilar to those caused by axial extensions [38,44]. Although thebeadingmode has a non-electrical analog [17e19], its occurrence athigh fields is likely related to the accumulation of electrostaticcharges and the associated Coulombic repulsion [22,45,46].

4. Conclusions

In this paper, we experimentally documented the operatingconditions of cone-jet bridges using the nozzle-to-pool setup. Thecentral theme was the stability island for cone-jet bridges on thevoltage-flow rate (VeQ) map. Within the stability island, the cone-jet bridge took on complex morphologies depending on the oper-ating conditions. The conical base on the wettable nozzle couldeither span the outer or inner rim depending on the combination offield strength and flow rate. Outside the stability island, the bridgebroke up in the thinning (beading) mode at the lower (upper)stability boundary. Although the stability island qualitativelyresembled that of a conventional Taylor cone-jet, the stabilityregime for the cone-jet bridgewas considerably broader in terms ofthe operative voltage range, provided that the liquid bridge wasformed between two closely positioned Taylor cones.

Given the widespread use of the VeQ stability diagram in othercone-jet systems, our work is expected to provide useful guidelinesfor future applications of cone-jet bridges. Although we usedethylene glycol as the working fluid and air as the surroundingmedium, we stress that many other combinations are possible toform transient or steady cone-jet bridges. In addition to theemployment of other electrospraying working fluids in a gaseousenvironment [2], the surrounding medium could in principle bereplaced with dielectric liquids [47]. Our work therefore has

potential implications for electrocoalescence processes [48,49] inaddition to microfluidic systems [2,50].

Acknowledgements

This work was funded by an NSF CAREER Award (Grant No.CBET-08-46705). We thank D.B. Bober, W.J. Scheideler and Y. Zhaofor helpful discussions.

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