Elena Castro-Hernández

Pablo García-Sánchez

Say Hwa Tan

Alfonso M Gañán-Calvo

Jean-Christophe Baret

Antonio Ramos
email: ramos@us.es

Breakup length of AC electrified jets in a microfluidic flow-focusing junction

Keywords: AC electric field, flow focusing, microfluidics, jet

Electroactuation of liquid-liquid interfaces offers promising methods to actively modulate droplet formation in droplet-based microfluidic systems. Here, flow-focusing junctions are coupled to electrodes to control droplet production in the well-known jetting regime. In this regime, a convective instability develops leading to droplet formation at the end of a thin and uniform, long liquid finger. We show that in AC electric fields the jet length is a function of both the magnitude of the applied voltage and the electrical parameters such as the frequency of the AC field and the conductivity of the dispersed phase. We explain that dependency using a simple transmission line model along

Introduction

Droplet-based microfluidics provides new means to miniaturize and automatize biochemical reactions for ultrahigh throughput screening applications [START_REF] Miller | Directed evolution by in vitro compartmentalization[END_REF][START_REF] Huebner | Microdroplets: a sea of applications?[END_REF][START_REF] Seemann | Droplet-based microfluidics[END_REF][START_REF] Guo | Droplet microfluidics for high-throughput biological assays[END_REF]. The controlled production and manipulation of monodisperse droplets of well-defined volume is of key interest to many applications ranging from diagnostics in emulsion PCR [START_REF] Pekin | Quantitative and sensitive detection of rare mutations using droplet-based microfluidics[END_REF] and cell screening [START_REF] Agresti | Ultrahigh-throughput screening in drop-based microfluidics for directed evolution[END_REF][START_REF] Debs | Functional single-cell hybridoma screening using droplet-based microfluidics[END_REF] to drug screening [START_REF] Miller | Highresolution dose-response screening using dropletbased microfluidics[END_REF]. At the core of the technology lies droplet production. Several systems are compatible with ultra-high throughput droplet production. T-junction [START_REF] Thorsen | Dynamic pattern formation in a vesicle-generating microfluidic device[END_REF], flow-focusing [START_REF] Anna | Microscale tipstreaming in a microfluidic flow focusing device[END_REF], co-flow [START_REF] Cramer | Drop formation in a co-flowing ambient fluid[END_REF] and step emulsification [START_REF] Dangla | Droplet microfluidics driven by gradients of confinement[END_REF] geometries provide a reliable and controllable environment for monodispersed droplet generation. However, the unceasing needs for enhanced functionalities and control has driven the development of alternative methods to actively control droplet generation. These methods are based on external perturbation, through thermal [START_REF] Tan | Thermally controlled droplet formation in flow focusing geometry: formation regimes and effect of nanoparticle suspension[END_REF], magnetic [START_REF] Nguyen | Manipulation of ferrofluid droplets using planar coils[END_REF][START_REF] Tan | Generation and manipulation of monodispersed ferrofluid emulsions: The effect of a uniform magnetic field in flow-focusing and t-junction configurations[END_REF], acoustic [START_REF] Ramo | Fields and waves in communication electronics[END_REF], pneumatic [START_REF] Abate | Valve-based flow focusing for drop formation[END_REF] or electrical [START_REF] Kim | Controlled production of emulsion drops using an electric field in a flow-focusing microfluidic device[END_REF][START_REF] Malloggi | Electrowetting -a versatile tool for controlling microdrop generation[END_REF] modulation of the liquid properties. To date, electrical control has proven to be the most robust, reliable and rapid mode for microdroplets actuation in microfluidic devices. Sorting droplets with electric field is achieved at rates up to 30 kHz in microfluidics [START_REF] Sciambi | Accurate microfluidic sorting of droplets at 30 khz[END_REF]) and is the central ingredient in commercial Fluorescence-Activated cell sorters which function at similar throughputs [START_REF] Ashcroft | Commercial high speed machines open new opportunities in high throughput flow cytometry (htfc)[END_REF]. Electric fields offer promissing routes to control droplet production. Using the combined effects of focused forces of hydrodynamic and electrical origin in a direct current (DC) was early proposed by [START_REF] Gañán-Calvo | Device for the production of capillary jets and micro-and nanometric particles. Pats no EP1479446 (A1)[END_REF][START_REF] Gañán-Calvo | Capillary jets and micro/nanometric particles production device uses electrohydrodynamic force, fluid-dynamic force and specific geometry to produce micro or nano capsules which are disintegrated to form drops of micro or nano size. Pats no ES2199048 (A1) & (B1)[END_REF][START_REF] Gañán-Calvo | Device for the production of capillary jets and micro-and nanometric particles. Pats no EP1479446 (A1)[END_REF][START_REF] Gañán-Calvo | Device for the production of capillary jets and micro-and nanometric particles. Pats no EP1479446 (A1)[END_REF] followed by [START_REF] Kim | Controlled production of emulsion drops using an electric field in a flow-focusing microfluidic device[END_REF]. The use of an alternating current (AC) by [START_REF] Malloggi | Electrowetting -a versatile tool for controlling microdrop generation[END_REF] and Tan et al. (2014b) has provided further avenues for exploration: well-defined microdroplets of desired sizes can be manipulated and formed repeatedly in the order of milli-seconds or faster, in operating regimes where the absence of electric fields or the use of DC ones does not allow that control. Indeed, we have demonstrated a new method of droplet generation in microfluidic devices using an AC voltage mediated electric field (Tan et al., 2014b), where microdroplet production is electrically controlled in the dripping or jetting regime [START_REF] Cubaud | Capillary threads and viscous droplets in square microchannels[END_REF] depending on the applied voltage, electrode configuration and physical properties of the fluids. This method of control has also been extended to demonstrate reliable droplet generation and manipulation via a musical interpretation of droplet based microfluidics (Tan et al., 2014a).

In both DC [START_REF] Gañán-Calvo | Device for the production of capillary jets and micro-and nanometric particles. Pats no EP1479446 (A1)[END_REF][START_REF] Kim | Controlled production of emulsion drops using an electric field in a flow-focusing microfluidic device[END_REF] or AC (Tan et al., 2014b) electroflow-focusing, three different regimes are identified : Dripping, axisymmetric jetting and unstable regime. The dripping regime is characterized by the generation of droplets close to the orifice, within a distance equal to one orifice diameter. In contrast, when jetting occurs, the droplets are produced at the end of a jet that extends at least three orifice diameters [START_REF] Gañán-Calvo | Generation of steady liquid microthreads and micron-sized monodisperse sprays in gas streams[END_REF][START_REF] Anna | Formation of dispersions using Flow Focusing in microchannels[END_REF]. In an AC field the unstable regime is observed at low values of the frequency field. Here the jet breaks in a random manner producing polydisperse droplets and approaching alternatively both PDMS walls without any characteristic periodicity. The transitions between these regimes are not fully understood although a qualitative behavior has been identified for the relevant parameters of this problem. If the high-voltage is applied to the downstream pair of electrodes while the others are grounded, the only transition that is observed is from the jetting to the unstable regimes (Tan et al., 2014b). This is the configuration that will be study in this paper.

In this work, we investigate the role of electric field in the jetting regime. The effect of viscosities of the continuous phase fluids, applied frequencies and voltages are studied with a focus on the breakup length of the jet. A minimal model is also proposed to describe the length of the jet in the corresponding electrical parameters. In addition, the transition between the axisymmetric jet and the unstable droplet production has been observed at low frequency and experimentally characterized.

Experimental setup

Soft lithography techniques are used to manufacture a microfluidic flow-focusing junction by replica molding in polydimethylsiloxane (PDMS, Dow Corning, relative permittivity ε r,PDMS = 2.5). Microfluidic channels in the device are 100 µm wide and 35 µm tall, w and h respectively in figure 1. Two pairs of electrodes are patterned around the junction as microfluidic channels and produced using the microsolidics technique [START_REF] Siegel | Cofabrication of electromagnets and microfluidic systems in poly(dimethylsiloxane)[END_REF] as described in Tan et al. (2014b). The spacing between the electrodes and the fluidic channel is d e = 35 µm over a length l e = 2.2 mm (figure 1). The PDMS device is plasma bonded to the nonconductive side of an Indium Tin Oxide glass (ITO, thickness 1 mm, ε r,glass = 7.5). The conductive side of the ITO glass is used as a counter electrode.

Water-in-oil (W/O) droplets are formed by focusing an aqueous stream (dispersed phase) with two side oil streams (continuous phase). The inner and outer volumetric flow rates, Q i and Q o , are controlled using a syringe pump (Model 33, Harvard Apparatus). The dispersed phase is an aqueous solution of NaCl in Milli-Q water with a viscosity η i = 1 cP and an electrical conductivity varying between κ = 3 × 10 -4 S/m and κ = 3 × 10 -3 S/m. The continuous phase is mineral oil (RTM14, Sigma Aldrich) with a viscosity between η o = 1 cP and η o = 100 cP. The relative permittivity of mineral oil is ε r,o = 2.1 and its electrical conductivity is negligible (κ o < 10 -10 S/m), i.e. it is considered here as a perfect insulator. A 5 % (w/w) of a non-ionic surfactant (Span 80, Sigma Aldrich) is added to the continuous phase lowering the equilibrium surface tension of the liquid-liquid interface from σ = 40 mN/m to σ = 5 mN/m, this value being independent of the NaCl concentration.

In principle, with the configuration of the three electrodes (upstream pairs, downstream pairs and ITO) and the two possible connections generates 2 3 possible connnections. However, removing the trivial cases where all electrodes have the same voltage and making use of symmetries, only three different electrical configurations remain (Tan et al., 2014b). Here, we focus on the single case where the high-voltage is applied to the downstream pair of electrodes, while the others were grounded. This configuration guarantees that the incoming liquid has zero potential (Tan et al., 2014b). As a consequence, there is an applied AC potential difference between the inner liquid emerging from the upstream electrodes and the downstream electrodes. A sinusoidal voltage with frequencies ranging from f = 1 kHz to f = 50 kHz (TGA1244, TTi) is amplified from V = 0 V to V = 1000 V (PZD700A, Trek). In the following discussion, all voltages are peak to peak. The setup is placed on an inverted microscope (Eclipse Ti-U, Nikon) connected to a high-speed camera (Phantom v7.3) with a resolution of 800 × 256 px 2 when operated at an acquisition rate of 10 4 fps. The jet diameter, d j , and jet length, l j , of at least 100 images are measured via image processing (Matlab, Mathworks and ImageJ).

Experimental results

We perform series of experiments varying the outer viscosities (η o = 1, 10 and 100 cP), the inner conductivities (κ = 3 × 10 -4 , 1 × 10 -3 and 3 × 10 -3 S/m), the frequency of the AC field (f = 0 -50 kHz), the inner to outer flow rate ratio (Q i /Q o = 0.125, 0.25 and 0.5) and the voltage (V = 0, 250, 750 and 1000 V). The frequency is fixed at 40 kHz and, according to previous results (Tan et al., 2014b), a dripping to jetting transition is reached when the voltage amplitude increases. A further increase in voltage results in longer jets and, for η o = 10 cP, a jet of 200 µm in length is obtained when the maximum voltage supplied by the amplifier is applied (1000 V). The same qualitative behavior is found for the two other viscosities with the remarkable observation that, for η o = 100 cP, we observe slender jets and, in some cases, longer than the channel length (3 mm). An analogous phenomenon has been reported by some authors without the presence of an electric field [START_REF] Gañán-Calvo | Focusing capillary jets close to the continuum limit[END_REF][START_REF] Gañán-Calvo | Unconditional jetting[END_REF][START_REF] Utada | Absolute instability of a liquid jet in a coflowing stream[END_REF][START_REF] Cubaud | Capillary threads and viscous droplets in square microchannels[END_REF][START_REF] Cordero | Quantitative analysis of the dripping and jetting regimes in co-flowing capillary jets[END_REF][START_REF] Castro-Hernández | Slender-body theory for the generation of micrometre-sized emulsions through tip streaming[END_REF], and explained via more or less elaborated spatiotemporal stability analyses [START_REF] Guillot | Stability of a jet in confined pressure-driven biphasic flows at low Reynolds numbers[END_REF][START_REF] Guillot | Stability of a jet in confined pressure-driven biphasic flows at low Reynolds number in various geometries[END_REF][START_REF] Gañán-Calvo | Unconditional jetting[END_REF]. A remarkable result found was that an unconditionally stable jetting can be obtained independently of the jet diameter (theoretically, down to the continuum limit) for asymptotically low Reynolds numbers when a properly defined Capillary number Ca * = (η i η o ) 1/2 U i /σ is above a certain critical threshold, being U i the velocity of the continuous phase surrounding the jet (Gañán-Calvo, 2008) (strictly speaking, U i should be the velocity at the jet surface). In that case, even with liquids begetting very small inner capillary number Ca i = η i U i /σ (e.g. liquid metals like mercury), one may obtain jets and droplets down to nanometric size as long as Ca * is above the limit threshold, which so far has been experimentally confirmed at different instances [START_REF] Gañán-Calvo | Focusing capillary jets close to the continuum limit[END_REF][START_REF] Gopalan | Turbulent shearing of crude oil mixed with dispersants generates long microthreads and microdroplets[END_REF]. Here, we show that the simple use of an AC voltage dramatically elongates the jets and might provide a technological advantage for manufacturing processes where structures with large aspect ratios are required.

Ancient experience shows that very long jets are easily obtained when a high viscosity liquid is stretched by sustained forces of many different origins [START_REF] Eggers | Physics of liquid jets[END_REF]. However, as the capillary number of the developing jet decreases, so does its length in a more or less drastic fashion, depending on the nature of the surrounding environment. The available methods to get thin, very long jets of low viscosity liquids like water virtually reduce to zero when the diameter ranges sought for decrease down to the micrometric range or below, unless a sufficiently viscous environment is used. In this case, experimentally confirmed results from spatiotemporal stability analyses [START_REF] Gañán-Calvo | Focusing capillary jets close to the continuum limit[END_REF][START_REF] Gañán-Calvo | Unconditional jetting[END_REF] predict convective instability (jetting) down to the molecular length scale if the geometrically averaged capillary number Ca * is above 0.2 (approx.), for η o /η i > 0.1. Thus, for the sake of efficiency, we fix the velocity of the surrounding environment, keeping the outer flow rate as Q o = 400 µl/h in this study. Assuming that the radial diffusion of mo- 3a, 4a, 5a). AC signals of amplitude V = 1000 V were applied and, for all conductivities, longer jets are found as the frequency of the signal was reduced (sequence from b to e in the figures). For a fixed frequency longer jets are obtained for liquids with higher conductivities. Figures 3f,4f and 5f show that the unstable regime is observed for frequencies low enough (f ≤ f mess ≃ 3 kHz).

Apparently, this transition frequency to the unstable regime does not depend much on water conductivity for η o = 100 cP. This is different from the observations reported by Tan et al. (2014b), where a transition from jetting to unstable drop production was found for a frequency which linearly increases with water conductivity for η o = 30 cP. In the present experiments, the unstable regime might be a different phenomenon since a long jet is formed and remains when the instability takes place.

From this moment the jet wets the wall and produces liquid filaments that eventually break into droplets of different sizes. A possible reason for the instability is that the jet at the center of the channel is a situation potentially unstable. The jet is attracted to both electrodes, due to the voltage difference with respect to them, and any nonsymmetrical perturbation can be amplified if the electric field is high enough. The appearance of this instability at a certain frequency and below lacks of a theoretical explanation.

Figure 6 shows the jet length, l j , as a function of signal frequency for the three water conductivities. The jet length is measured from the begining of the downstream electrodes. If we plot the dimensional quantities l j versus f /κ, all data points collapse onto a single master curve as depicted in the inset of figure 6. The figure also shows the best-fit curve with power -1/2. This key clue provided by the experimental data will be confirmed by theoretical analysis (see section 4).

We have also tested smaller values of the outer flow rate: Q o = 100 µl/h and Q o = 50 µl/h. When the AC electric field is applied smaller values of l j are obtained, as expected, and the same trends are found. Interestingly, f mess increases from

f mess ≃ 3 kHz to f mess ≃ 9 kHz when Q o is decreased from Q o = 400 µl/h to Q o = 50 µl/h.
Besides, we have performed experiments for outer viscosities η o = 1 cP and η o = 10 cP. In these cases in absence of an electric field the dripping regime takes place, as predicted by the spatiotemporal stability analysis [START_REF] Herrada | Spatiotemporal instability of a confined capillary jet[END_REF]. With these values of the viscosities, shorter jets are obtained unless very high values of the outer velocities are used. As a consequence, these experiments were not adequate to our study of long jets. Nevertheless, a relevant behavior was noted: if the rest of the parameters are fixed (Q i , Q o , κ, V and f ) when η o is decreased f mess increases. Particularly, for the case η o = 1 cP the unstable regime was always present. However, the spatiotemporal stability analysis of our system in the presence of electric fields (DC or AC) and a detailed account of the parametrical dependence of those features is out of the scope of present work and the object of a subsequent one.

Discussion

Let us summarize the main dimensionless numbers involved in this problem. Once the viscosity ratio and flow rate ratio are fixed, it can be seen from the table 1 that the inner and outer Reynolds numbers are small, which tell us that inertial effects are negligible as it is usual in these types of geometries. Here U i = 4Q i /πd 2 j is the average velocity of the dispersed phase. The electrical Bond number B e is defined as the time-averaged electrical pressure on the interface ε o E 2 rms /2 divided by the capillary pressure σ/ℓ, where ℓ is a typical radius of curvature. A characteristic value of the electrical Bond number for the cone-jet transition is B e,cone ∼ 0.5, taking as typical cone radius of curvature ℓ = 20 µm and a characteristic electric field on the cone surface E rms = 3.5×10 6 V/m. Since the radius of the jet is measured experimentally and almost constant d j /2 = 6.5 µm and the electric field on the surface of the jet is E rms = 2.3 × 10 7 V/m (computed numerically using finite elements), the electrical Bond number for the jet becomes B e,jet ∼ 6. This indicates that in the cone-to-jet transition B e changes by an order of magnitude. While the inner and outer capillary numbers are below unity, the electrical Bond number of the jet is much greater, reassuring the fact that the electric forces are responsible for the formation of such long jets. The experiments show that there is a long jet when

η i /η o = 0.01 Q i /Q o = 0.125 Re o = ρ o U o h/η o ≃ 0.01 Re i = ρ i U i d j /η i ≃ 0.7 Ca o = η o U o /σ ≃ 0.5 Ca i = η i U i /σ ≃ 0.02 B e,cone = ε o E 2 rms ℓ/2σ ≃ 0.5 B e,jet = ε o E 2 rms d j /4σ ≃ 6
there is both a high applied voltage and sufficient low frequency. When either the applied voltage is zero or the frequency is high, there is a very short jet that breaks into drops rapidly. Based upon these experimental facts, we can think that the jet breaks into drops when the electric field amplitude around the jet tip is below a certain value. Let us study the voltage difference between the jet and the channel electrodes as a function of distance along the jet. This voltage difference is maximum at the channel entrance and decreases for increasing distance inside the channel. Eventually, this voltage difference would be zero when the jet voltage adapts to the surrounding voltage and we can think that near this point is when the jet breaks into drops. We are going to use concepts of transmission line, or distributed element, theory [START_REF] Ramo | Fields and waves in communication electronics[END_REF] applied to a very long jet, ideally infinitely long. This transmission line model is valid when there is translational symmetry and the characteristic length along the line is much greater than the characteristic length in transverse direction. Let V 0 be the potential difference between the jet and the electrodes at the channel entrance (z = 0). The potential drop in the axial direction along a resistive jet is

V (z) -V (z + dz) = dz κπa 2 I(z) ⇒ - ∂V ∂z = I κπa 2 (1)
where I(z) is the current intensity carried by the jet at z, κ is the liquid conductivity and a = d j /2 is the jet radius. From charge conservation, the current intensity at z is equal to the current at z + dz plus the displacement current leaving the jet interface

I(z) = I(z + dz) + iωCV (z) ⇒ - ∂I ∂z = iωCV (2)
where C is the capacitance per unit of length. The equation that describes the potential is finally

∂ 2 V ∂z 2 = iωC κπa 2 V (3)
with boundary conditions of V (z = 0) = V 0 and V (z → ∞) = 0. Therefore, according to the transmission line model, the potential along the jet is

V (z) = V 0 exp[-(1 + i)z/δ], where δ = a √ 2κπ ωC (4)
Strictly speaking, when the jet is finite the boundary condition should be zero current at the tip of the jet [START_REF] Baret | Electroactuation of fluid using topographical wetting transitions[END_REF][START_REF] Baret | Transport dynamics in open microfluidic grooves[END_REF]. However, the penetration length δ is not affected by this choice since it is the characteristic length scale of the transmission line equation (3).

The capacitance per unit of length C is here obtained numerically using the finite element solver COM-SOL and taking into account the dimensions of our system in a cross-section (see Figure 1b). The numerically obtained value for C is 5.1 × 10 -11 F/m. The approximation is valid when δ ≫ a, i.e. when the characteristic axial length is much greater than the transverse length. For instance, for κ = 10 -3 S/m, a = 6.5 µm, f = 5 kHz, we get δ = 410 µm and the condition is fulfilled. This value of δ is of the order of the jet lengths observed in experiments. Interestingly, δ is inversely proportional to the square root of frequency, in agreement with the observed trend for jet length with frequency. Figure 7 shows jet length in units of radius a as a function of nondimensional frequency Ω = ωC/κ for the three conductivities, and the functions δ/a and 2δ/a. As can be seen, the jet breaks into drops at a distance from the entrance between δ and 2δ. Since E 2 rms is proportional to |V | 2 = V 2 0 exp(-2z/δ), the electrical Bond number decreases from a value B e,jet ∼ 6 at the entrance (z = 0) to a value B e,jet ∼ 6 exp(-4) ≈ 0.1 when the jet has already broken into drops (at z = 2δ). Therefore, the jet breaks into drops when the electrical pressure is much smaller than the capillary pressure. It should be noted that from an electrical point of view, the existence of a conducting jet inside the downstream channel is favored because it increases the electrical energy stored in the system, which is at constant potential.

In the model we have performed several approximations that we now justify. We have neglected the convection of charge by the moving jet interface in front of the ohmic current through the jet bulk. The ratio between convection and conduction currents is known as the electric Reynolds number [START_REF] Melcher | Electrohydrodynamics: a review of the role of interfacial shear stresses[END_REF]. In our case, the convective current at a certain axial location is I conv = 2πaq s U i . Here q s is the induced surface charge on the jet given approximately by ε o E n , with E n the outer normal electric field. The ohmic current along the jet is I ohm = πa 2 κE z where E z is the axial electric field. Therefore, the electric Reynolds number is I conv /I ohm ∼ ε o δU i 2/κa 2 , where we have taken into account that E n /E z ∼ δ/a in our model. For the frequency of the instability (3kHz) and for the smallest conductivity (0.3 mS/m), the ratio between convected and ohmic currents is of the order of 0.05 and we can safely ignore the motion of the jet when computing the fields. We have also neglected the displacement current through the jet bulk in front of the ohmic current. This is valid if ε i ω/κ ≪ 1, which is the case for the signal frequencies we are dealing with, ε i ω/κ ∼ 4 × 10 -3 for f = 1 kHz, and κ = 10 -3 S/m. In fact, the model can easily incorporate the displacement current by using a complex conductivity, κ = κ + iε i ω.

Conclusions

In this work we have experimentally studied AC electrified jets in a microfluidic flow-focusing junction. We have shown that in AC electric fields the jet length is a function of both the magnitude of the applied voltage and the electrical parameters such as the frequency of the AC field and the conductivity of the dispersed phase. We have explained this dependency using a simple transmission line model along the liquid jet. The penetration length of the AC electric field along a conducting cylinder determines the length of the liquid jet: the scaling law of the jet length vs the electrical parameters obtained experimentally is fully compatible with our minimal theoretical modeling. In addition, we have observed that the jets become unstable rather abruptly below a certain frequency. In this study, we have shown that AC voltages can elongate dramatically liquid jets, which can be a technological advantage for manufacturing processes where structures with large aspect ratios are required.

  Fig. 1 a) Sketch of a microfluidic flow-focusing junction under an AC electric field. The electrodes are in black and the aqueous phase in grey. b) Cross section of the device at the level of the downstream electrodes.

Fig. 2

 2 Fig. 2 Series of images showing the effect of increasing the voltage on the jet length for Q i = 50 µl/h, Q o = 400 µl/h, η o = 10 cP, κ = 1 × 10 -3 S/m and f = 40 kHz : a)V = 250 V; b)V = 750 V; c)V = 1000 V.

Figure 2

 2 Figure2shows a series of images for a given flow rate ratio, water conductivity and an outer fluid viscosity η o = 10 cP. The frequency is fixed at 40 kHz and, according to previous results(Tan et al., 2014b), a dripping to jetting transition is reached when the voltage amplitude increases. A further increase in voltage results in longer jets and, for η o = 10 cP, a jet of 200 µm in length is obtained when the maximum voltage supplied by the amplifier is applied (1000 V). The same qualitative behavior is found for the two other viscosities with the remarkable observation that, for η o = 100 cP, we observe slender jets and, in some cases, longer than the channel length (3 mm). An analogous phenomenon has been reported by some authors without the presence of an electric field[START_REF] Gañán-Calvo | Focusing capillary jets close to the continuum limit[END_REF][START_REF] Gañán-Calvo | Unconditional jetting[END_REF][START_REF] Utada | Absolute instability of a liquid jet in a coflowing stream[END_REF][START_REF] Cubaud | Capillary threads and viscous droplets in square microchannels[END_REF][START_REF] Cordero | Quantitative analysis of the dripping and jetting regimes in co-flowing capillary jets[END_REF][START_REF] Castro-Hernández | Slender-body theory for the generation of micrometre-sized emulsions through tip streaming[END_REF], and explained via more or less elaborated spatiotemporal stability analyses[START_REF] Guillot | Stability of a jet in confined pressure-driven biphasic flows at low Reynolds numbers[END_REF][START_REF] Guillot | Stability of a jet in confined pressure-driven biphasic flows at low Reynolds number in various geometries[END_REF][START_REF] Gañán-Calvo | Unconditional jetting[END_REF]. A remarkable result found was that an unconditionally stable jetting can be obtained independently of the jet diameter (theoretically, down to the continuum limit) for asymptotically low Reynolds numbers when a properly defined Capillary number Ca * = (η i η o ) 1/2 U i /σ is above a certain critical threshold, being U i the velocity of the continuous phase surrounding the jet (Gañán-Calvo, 2008) (strictly speaking, U i should be the velocity at the jet surface). In that case, even with liquids begetting very small inner capillary number Ca i = η i U i /σ (e.g. liquid metals like mercury), one may obtain jets and droplets down to nanometric size as long as Ca * is above the limit threshold, which so far has been experimentally confirmed at different instances[START_REF] Gañán-Calvo | Focusing capillary jets close to the continuum limit[END_REF][START_REF] Gopalan | Turbulent shearing of crude oil mixed with dispersants generates long microthreads and microdroplets[END_REF]. Here, we show that the simple use of an AC voltage dramatically elongates the jets

Fig. 3

 3 Fig. 3 Series of images showing the effect of decreasing the frequency field on the jet length for Q i = 50 µl/h, Q o = 400 µl/h, η o = 100 cP, κ = 3 × 10 -4 S/m and V = 1000 V: a)V = 0 V; b)f = 50 kHz; c)f = 30 kHz; d)f = 5 kHz; e)f = 3 kHz; f)f = 1 kHz.

Fig. 4 Fig. 5

 45 Fig. 4 Series of images showing the effect of decreasing the frequency field on the jet length for Q i = 50 µl/h, Q o = 400 µl/h, η o = 100 cP, κ = 1 × 10 -3 S/m and V = 1000 V: a)V = 0 V; b)f = 50 kHz; c)f = 30 kHz; d)f = 10 kHz; e)f = 5 kHz; f)f = 1 kHz.

Fig. 6

 6 Fig. 6 Jet length versus frequency field for different values of the conductivity for Q i = 50 µl/h, Q o = 400 µl/h, η o = 100 cP and V = 1000 V. The inset shows l j versus f /κ.

Fig. 7

 7 Fig. 7 Dimensionless jet length as a function of nondimensional frequency for different values of the conductivity for Q i = 50 µl/h, Q o = 400 µl/h, η o = 100 cP and V = 1000 V.

Table 1

 1 Values of the dimensionless relevant parameters involved in this problem.

Acknowledgements

The authors are grateful to the DAAD (German Academic Exchange Service) for the financial support. They would also like to acknowledge financial support

by the ERC (FP7/2007-2013 /ERC Grant agreement 306385-SofI) and from Spanish Government Ministry MEC under Contract No. FIS2011-25161.