Radial deformation and disintegration of an electrified liquid jet

Destabilization and subsequent breakup of electrified liquid jets is a basic electrohydrodynamic (EHD) phenomenon appearing in applications, such as electrospraying, electrospinning, printing, painting, and microfluidic and nanofluidic devices.^1–4 The possibly involved factors, including inertia, viscosity, capillarity, rheological properties, electrification manner, electrical properties, heat and mass transfer, external excitation, and ambient medium, make the instability nature of jets extraordinarily diverse.^5,6 For instance, depending on the electrification level, a jet undergoes axisymmetric instability (varicose mode) or non-axisymmetric instability (kink mode), thereby influencing the outcome in real life.^7–9 Since Rayleigh’s work,¹⁰ jet instability has been studied for over a century.^11–17 Nevertheless, there are still many open problems that attract researchers to explore.⁵ 

In electrospraying, needle–plane electrode geometry is commonly utilized, with the needle being perpendicular to the plane.^18,19 Under the action of the electric field, a liquid meniscus is stretched into a pointed cone, and at the tip of the cone, a thin jet is emitted (Taylor cone–jet mode).^20–23 The jet is destabilized and breaks up into droplets at some distance downstream. The Taylor cone–jet mode is used to generate monodisperse droplets down to micro/nanometer size, which has tremendous applications in various fields.^24,25 In addition to the needle–plane geometry, the needle–cylinder electrode configuration, which creates a radial electric field, has also been considered in the observation of disintegration of drops/jets.²⁶ In this Letter, we will present a vivid tip-streaming pattern, i.e., radial multi-cone–jet mode, formed by a liquid jet subjected to a radial electric field. The linear instability analyses showed that there exists a critical wavenumber k_c: a radial electric field enhances the axisymmetric instability of a jet when the wavenumber k > k_c, while it suppresses the jet instability when k < k_c.^13,15,16 The nonlinear numerical simulations demonstrated that a radial electric field may induce large radial deformations of primary droplets in a jet (in this case, electric stresses work against surface tension).^14,27–29 On the other hand, interestingly, an imposed radial electric field may delay the pinch-off of a jet (stabilize the jet) under some circumstances.^28,30

In our experiment, a cylindrical liquid jet was injected into the air from a stainless steel needle of inner diameter 0.21 mm, as sketched in Fig. 1(a). The cylindrical coordinate system (r, θ, z) is utilized to describe the problem, where r, θ, and z are the radial, azimuthal, and axial coordinates, respectively. The needle was connected to a high-voltage dc electric source to charge the jet. An exact radial electric field can be realized by positioning a concentric cylindrical grounded electrode around the jet.²⁶ Here, to facilitate the observation, two plane grounded electrodes were used instead, which were placed parallelly sandwiching the jet at a distance of R₀ = 4.7 mm. The distribution of the electric field in the rθ plane is shown in Fig. 1(b). It is justified that at the jet surface the electric field is nearly radial and uniform.²⁹ To minimize the edge effects, the plane electrodes were made large enough (8 cm wide and 12 cm high) to surround the entire jet (whose length is typically 1–3 cm). A high-speed camera (Photron SA5) with a macrolens in front of it was used to record the deformation of the jet. The frame rate was 2 × 10⁴ to 10⁵ fps, and the exposure time was 1 µs. The jet was backlighted by using a collimated LED light source. The liquid used was pure ethanol (density ρ ∼ 785 kg/m³, dynamic viscosity μ ∼ 1.1 mPa s, surface tension γ ∼ 0.022 N/m, relative electrical permittivity $ε$_r ∼ 24.3, and electrical conductivity σ ∼ 1.35 × 10⁻⁷ S/m).

The jet velocity U is of the order of 1 m/s. The jet radius R is typically 100 μm. The time scales characterizing jet dynamics are estimated: the hydrodynamic convection time t_h = R/U ∼ 10⁻⁴ s, the viscous time t_v = μR/γ ∼ 5 × 10⁻⁶ s, the capillary time $tc=ρR3/γ∼1.9×10−4s$⁠, and the electrical relaxation time t_e = $ε$_r$ε$₀/σ ∼ 1.6 × 10⁻³ s, where $ε$₀ = 8.85 × 10⁻¹² F/m is the electrical permittivity of vacuum. Apparently, t_v is two or three orders of magnitude smaller than the others, implying that viscosity plays a secondary role in jet dynamics. The Ohnesorge number defined as $Oh=tv/tc=μ/ρRγ$⁠, with a value of 0.026, also reflects this fact. t_h is comparable to t_c, suggesting the equal importance of inertia and capillarity. Moreover, the Weber number $We=ρU2R/γ=tc2/th2$⁠, which represents the relative magnitude of inertial to capillary force, is of the order of one. t_e is much larger than the other time scales. That is, charges cannot relax instantaneously at the jet surface as the jet deforms. To describe such a poorly conducting liquid, it is appropriate to introduce the Taylor–Melcher leaky dielectric theory.^31,32 The nondimensional number representing the relative magnitude of capillary to electrostatic force, i.e., the electric Bond number defined as $χ=ε0V02/γR(ln(R/R0))2$ (V₀: the electric voltage imposed on the jet)³⁰ ranges from 0 to ∼9.8 in our experiment.

Figures 1(c)–1(e) demonstrate shadowgraph images of the jet. When the flow rate Q is small, the jet length is short, and a side view of the entire jet can be obtained. In the absence of electric field (the electric voltage V₀ is equal to 0), end-pinching breakup occurs with an almost spherical droplet formed at the end of the jet³³ [see Fig. 1(c)]. With the radial electric field imposed, the droplet stretches in the direction of the electric field and grows into a pointed spike shape [see Fig. 1(d)]. At a sufficiently large electric field, tip-streaming occurs at the edge of the pointed spike, as shown in Fig. 1(e).

By varying the flow rate Q and the electric voltage V₀, distinct jet configurations were observed, which can be categorized roughly into three groups: regular jets (RJ), irregular jets without tip-streaming (IRJ), and irregular jets with tip-streaming (IRJS). For a regular jet at a sufficiently large flow rate, waves on the jet surface resemble harmonic ones, gentle and elegant, as shown in Figs. 2(a) and 2(b). This is the scenario at small voltages, in which the Rayleigh capillary instability dominates. Comparing Fig. 2(b) with Fig. 2(a), one can find that the applied radial electric field shortens the jet length, accelerates the perturbation growth, and diminishes the axial wavelength. The dominant wavenumber k_max corresponding to the maximum growth rate can be predicted by linear theory.^15,16,28 Naturally, k_max is greater than the critical wavenumber k_c. Hence, the radial electric field has a destabilizing effect on the jet, in agreement with our experimental observation. At larger voltages, droplets deform in the radial direction visibly, and moreover, the droplet at the end of the jet possesses a much larger width-to-height ratio than in the zero voltage case (IRJ) [see Fig. 2(c)]. (It should be noted that the criterion chosen to distinguish between RJ and IRJ is empirical but consistent in order that the border between them in general reflects some facts, including the ratio between the growth rate of the waves and the jet velocity.) At even larger voltages [see Fig. 2(d)], the droplet prior to pinch-off is further stretched and ultra-fine liquid jets are ejected from the sharp edge of the droplet (IRJS). During this nonlinear process, droplets driven by the radial electric field first evolve into a pointed spike structure, which can be well predicted by the slender body theory.^29,34,35 However, at later stages, radial bulk flow becomes profound and the deformation of droplets possesses an evident two-dimensional characteristic.²⁸ Due to the radial flow, droplets are further elongated in the radial direction and are more flattened, resembling a thin disk.

The domains of different configurations are illustrated in the Q–V₀ diagram in Fig. 2(e). For a fixed flow rate, as the voltage increases, RJ, IRJ, and IRJS take place successively. There exists a minimum voltage for the onset of IRJ or IRJS. The borders between adjacent groups can be fitted as a straight line approximately [see the dashed and dash-dotted lines in Fig. 2(e)]. With a narrow domain, IRJ can be regarded as a transition mode from RJ to IRJS. It is also found that larger flow rates require larger voltages for the occurrence of IRJS. This is understandable, considering that as the flow rate increases, the breakup length of the jet increases, and the droplets shed faster from the jet, with their charging time reduced. In order to deform the droplets in the shorter time, a higher growth rate of the perturbations is required. This can be achieved by imposing a higher voltage.

Time evolution of the jet profile is depicted in Fig. 3(a). (The corresponding video can be found in the supplementary material.) At the initial time, the voltage is imposed. Driven by the radial electrostatic force, the droplet starts to deform. At t = 0.1 ms, it develops into a shower head-like shape with thick center and thin edge. Charges accumulate at the edge, causing large electric stresses. At t = 0.11 ms, the electric stresses overcome the other factors and induce the occurrence of Coulombic disintegration, with tens of ultra-fine secondary jets being ejected from the edge of the droplet. These tiny jets carry away mass and charge and lower the electric field strength at the droplet surface. With the attenuation of the electric field, secondary jets get fewer and fewer. Meanwhile, the droplet shrinks to a multi-lobe shape; see the frames for t = 0.14–0.3 ms. At t = 0.38 ms, it departs from the jet. In the zoomed-in image of t = 0.41 ms, interestingly, the liquid ligament between primary droplets undergoes secondary instability and breaks up into a queue of small progeny droplets, as indicated by the horizontal arrow. The radially traveling secondary jets also undergo capillary instability and break up into progeny droplets that are orders of magnitude smaller than the parent one, as indicated by the slanted arrow.

The feature of disintegration of the jet is discernible in the closeup images in Figs. 3(b)–3(f). The picture is like this: first, Taylor cones are formed at the edge of a shower head-like droplet, then a secondary jet is ejected from the tip of each cone, and finally, the secondary jets break up into smaller droplets. We term this tip-streaming pattern the radial multi-cone–jet mode. A top-view schematic of this mode is presented in Fig. 3(g). Sometimes, disintegration of the second droplet already occurs before the shedding of the first one [see Fig. 3(e)]. Disintegration also occasionally happens to satellite droplets between primary ones [see Fig. 3(f)]. Compared with those steady or periodic dripping, jetting, and assorted nonlinear dynamic phenomena arising when a liquid is forced through a nozzle at a constant flow rate,^{21,22,36–38} the radial multi-cone–jet mode is truly unsteady, whose lifetime is extremely short. Furthermore, to make it happen repeatedly to a droplet seems an insoluble problem.

It is of interest to note that the radial multi-cone–jet mode is a complexity of three instabilities: (1) the jet ejected from the needle undergoes axisymmetric instability, with deformed droplets detached from it; (2) the droplets in the jet undergo non-axisymmetric instability, resulting in an even distribution of Taylor cones at their edges in the equatorial plane; (3) the secondary jets ejected from the Taylor cones experience capillary instability, breaking up into progeny droplets. Similar EHD lateral/equatorial streaming phenomena were observed by other researchers.^39,40 By imposing a uniform electric field aligned with its poles, a liquid drop suspended in another liquid may undergo interfacial instability and shed concentric rings from its equator, and these rings eventually break up into droplets via capillary instability.³⁹ In contrast, in our case, capillarity acts as a resistance and electric stresses have to overcome capillary force to trigger the formation of the radial multi-cone–jet mode, although capillarity indeed prompts the instability of the jet at the initial times.^29,35 According to hydrostatics, if the local surface charge density q exceeds the critical value $qc=2ε0γ∇⋅n$⁠, where ∇ · n is twice the local mean curvature (at q_c, the electrostatic force balances the capillary force), a charged droplet of infinite conductivity will be destabilized, being radially stretched.³⁵ Another interesting phenomenon is that, upon imposing a uniform electric field, a pendant droplet in a moderately viscous outer liquid deforms laterally forming a jellyfish-like structure and disintegration into fine jets/droplets occurs at the rim of the structure (splashing mode).⁴⁰ 

The number of secondary jets, denoted by N, is not a constant. From Fig. 3(b) to 3(d), N increases. Figure 3(d) demonstrates that a droplet is unnecessarily stretched to a great extent to favor the occurrence of the multi-cone–jet mode. Since the droplets got blocked by the jet, we failed to count N accurately, but tried to keep the error within ±3. The result is shown in Fig. 3(h), where N ranges from 10 to 50. Generally, a smaller flow rate or a larger voltage results in a larger N. So what is the underlying factor determining the magnitude of N? Considering that the formation of the cones is associated with the non-axisymmetric instability of a droplet,^41,42 the azimuthal wavenumber n of the dominant non-axisymmetric mode is supposed to determine the number of cones and thereby the number of secondary jets N. To verify the relationship between N and n, linear and nonlinear instability analyses of electrified droplets will help. One feasible way to get the value of n is measuring the distance between two adjacent cones l_m and the radius of the deformed droplet in the equatorial plane R_m and then estimating the azimuthal wavenumber through n = 2πR_m/l_m. As Q decreases or V₀ increases, droplets are more charged, and as a result, higher-order non-axisymmetric modes become dominant, giving rise to an increase in n.

Images of the droplets detached from the jet are shown in Fig. 4(a). (The corresponding video can be found in the supplementary material.) Below the long slanted arrow are uncharged droplets (the voltage has not been imposed yet), which oscillate around a spherical shape during falling, with a relatively small amplitude. The small amplitude oscillations of a liquid droplet can be predicted via linear analysis.^43,44 Above the long arrow are charged droplets, which are stretched in the direction of the electric field and evolve into oblate spheroids with remarkably large deformations. Moreover, these highly deformed droplets may emit fine jets (see t = 0.75 ms and 0.9 ms). After losing mass and charge, the droplets begin to shrink (see t = 1.1–1.38 ms). Linear theory predicts that when the amount of charge carrying by a perfectly conducting liquid drop in vacuum or air exceeds $8πε0γR3$ (the Rayleigh limit), the drop is coulombically unstable.⁴¹ On the other hand, the nonlinear oscillations of drops of low or arbitrary viscosity with or without electric fields/charges have been studied by many researchers.^45–48 It was shown that Coulombic disintegration also occurs above the Rayleigh limit.⁴⁹ Coulombic disintegration is usually observed at two poles of a liquid drop.^17,50 However, in our experiment, it appears at the equator of a drop.

The aspect ratio is defined as ξ = D/H, where H is the height and D is the diameter of a droplet, as depicted in Fig. 4(a). The dependence of ξ on V₀ and Q is illustrated in Fig. 4(b). For a fixed Q, ξ generally increases with increasing V₀; for a fixed V₀, ξ decreases as Q increases. Both trends can be attributed to the fact that with the smaller Q or the larger V₀, the charge level of a droplet is higher. Most intriguingly, there is a threshold V₀ (∼3 kV), above which a sudden increase in ξ is observed. It is deduced that beyond this threshold value of the voltage, the electrostatic force overcomes the capillary force and prompts large deformations of droplets. An alternative parameter measuring the degree of deformation is δ = (D − H)/(D + H).^20,50 The maximum δ here is around 0.78, much larger than those in the prolate drop case.^17,50 Large deformations result in large surface-to-volume ratios, which can assist mixing and mass/heat transfer.

The radial multi-cone–jet mode found in this work may contribute to the reduction of droplet size in applications. Moreover, its rich dynamics has not yet been understood well. We hope this Letter will inspire more relevant research from both theoretical and technical aspects.

See the supplementary material for the videos of the deformation processes of the jet and the droplets.

This work was supported by the National Natural Science Foundation of China (Project Nos. 11772328 and 11621202). We thank all the referees for their constructive comments. The high-speed camera was provided by Experimental Center of Engineering and Material Sciences of USTC. We thank Professor Erqiang Li and Professor Ting Si for lending us the light source and the macrolens.