Chapter VI: Dynamic Cone Formation in Conductive Liquids: Inviscid Theory
6.1 Interfacial Cone Formation in Electrically Conductive Liquids
As already discussed in Chapter 5, if charged beyond a critical level, the interface of an electrically conductive liquid surrounded by an insulator (e.g., air, a vacuum or an- other dielectric medium) becomes susceptible to the influence of intense electric field.
Formation of small conical interfacial features, followed by ejection of liquid filaments or ions in the end, often accompanies the strong distortion of an electrified liquid after its initial loss of stability. This unique electrohydrodynamical phenomenon is often as- sociated with the terminology, Taylor cones, named after G. I. Taylor (1964)’s seminal calculation on the cone opening angle. Although the earliest recorded observation of these conical shapes can be traced back to the 16th century when Gilbert (1600) in his work De Magneto described the deformation of a water drop into a cone in the presence of a charged object and has fascinated scientists in many different disciplines for centuries since then.
It is now commonly accepted that the distinguishing factor which separates many cone- related phenomena into two major categories is liquid electrical conductivity. An infinite conductivity corresponds to the electrostatic limit of a perfect conductor or dielectrics where surface charges are instantaneously distributed on the interface and all the Maxwell forces act normally on the interface. Under such an assumption G. I. Taylor (1964) reported that a perfectly conductive liquid subject only to capillary and electrical forces in hydrostatic equilibrium must adopt the shape of a cone with a vertex angle of 98.6◦, known as the Taylor cone angle. On the contrary, for a medium of finite or high conductivity, charge relaxation, i.e. convection-driven departures of the surface charge density away from the value it would have on a perfect conductor, takes a finite amount of time (Saville,1997). The situation then becomes much complex as tangential Maxwell stresses may be present, leading to shear forces, hence tangential accelerations, on the charged liquid interface that are eventually responsible for jet emission observed in experiments.
Most researches on liquids of finite or high conductivity are oriented around the Taylor cone-jet configuration, a regime where a steady non-breaking jet issues continuously from the cone apex, eventually either breaking into a spray of charged drops, known as electrospray (Fernández de la Mora, 2007), or being drawn further as a continuous fiber (possibly whipping and bending) instead of a charged cloud, known as electrospin (Hohman et al., 2001a; Hohman et al., 2001b). The model of charge transport in
the literature concerning the theory of steady but not static cone-jet modes is largely based on the ohmic conduction (uniform electrical conductivity) and leaky dielectric model (ohmic liquid governed by Stokes equation) introduced and popularized by the reviews (Melcher and G. I. Taylor, 1969; Saville, 1997). The crucial role of charge transport in cone-jet phenomenon has been confirmed in a number of computational studies such as simulated electrohydrodynamic tip streaming of a leaky-dielectric liquid film (Collins, Jones, et al.,2007) and of a conducting drop subject to a uniform external electric field (Collins, Sambath, et al., 2013), progeny formation from critically charged conductive inviscid drops (Burton and Taborek, 2011), micro-jet emission from the tip of low-conductivity pendant drops subject to a step change in the electric field magnitude (Ferrera et al.,2013), first-droplet ejection from a parent capillary meniscus under volumetric relaxation of charge density in the hydrodynamic bulk (Gañán-Calvo, López-Herrera, Rebollo-Muñoz, et al., 2016; Gañán-Calvo, López-Herrera, Herrada, et al., 2018) and the transient electrohydrodynamic response of a liquid drop containing ions, to both small and large values of electric field (Pillai et al., 2016). It’s worth noting that the conic base of experimentally observed cone-jet configurations sometimes exhibits substantially smaller opening angles than the Taylor cone due to the presence of space charges carried by the jet which technically violates Taylor’s assumption of free- space electrostatics in the region exterior to the cone and inevitable causes repulsion between jet and conic base.
Despite large amount of literature and applications on cone-jet related physics (e.g., electrospray and electrospin), little has been known of the transient process that a round meniscus takes to morph into a conic cusp before any criterion for spontaneous ejection of ion or fluid is met. In particular it is the subject of this chapter to decipher the electrohydrodynamic mechanism underlying the rapid development of conic tips in perfectly conductive liquids. There are many practical reasons that it is important to understand the dynamic process of cone formation in addition to steady cone-jet. For instance, the electrohydrodynamic direct-writing technique (C. Chang, Limkrailassiri, and L. Lin, 2008; Huang et al., 2013), one of the emerging solutions to the increas- ing demand for micro/nano-scale manufacture, can be used to print nanofibers onto a large-area substrate in a direct, continuous, controllable and free-form fashion. The development of fine and steady jets at cone apex is known from experiments to correlate strongly with electrical conductivity: thread diameter approximately shrinks toward zero in a rate inversely proportional to a power of the electrical conductivity (Fernández de la Mora, 2007). Therefore knowledge of bulk hydrodynamics at the singular limit of infinite conductivity could shed light upon the onset of nano-/micro-sized jet emitted from liquid tips of high but finite electrical conductivity. There is also liquid metal ion source (LMIS) which was originally developed as the charge material of electrostatic droplet sprayers as a heavy charged particle source for electric space micropropulsion
(Bartoli et al., 1984; Rüdenauer, 2007; Tajmar et al., 2009). It was then discovered that LMIS device was a high brightness source of metal ions, capable of being focused to spots of nanometer dimensions. LMIS now becomes a fundamental component of fo- cused ion beam microscopy and micromachining (Orloff,2017), one of the most precise micro-/nano-fabrication tools at the present time. Once the local intensity of electric field reaches above certain threshold, it causes ions to begin to form through field evap- oration and field ionization of the metal atoms in the vapor, a process similar to the mechanism that inspired Fenn (1993) for his Nobel prize winning technology mass spec- trometry (Guerrero et al., 2007). The field threshold for common LMIS (e.g., gallium and indium) is on the order of109V/m(Orloff, Utlaut, and Swanson,2003). It is crucial to understand the hydrodynamic origin of highly curved cone apex that spontaneously forms in liquid metals which naturally exceeds the requirement for field ion evaporation.
In this chapter, we exclusively focus on the process of dynamic cone formation in per- fectly conductive liquids during a small time window long after initial destabilization and but shortly before the onset of other physics on smaller scales (e.g., secondary fluid ejection or field ion evaporation). In practice, the rapid development of conical cusps at liquid interface is notoriously difficult for experimental observations owning to its transcendental nature at small spatial dimensions and explosive fluid/ion ejection that ensues. As technology advances, accurate imaging and measurement of cone for- mation have only become available in the last few decades since G. I. Taylor (1964)’s original photography of oil/water interfaces. Although it still poses tremendous chal- lenges in high-speed and high-resolution imaging, appearance of the Taylor cone has been recognized in experiments involving a variety of highly conductive liquids including stroboscopic shadow photography of indium-gallium alloy wetting a liquid-metal cathode under high vacuum (Baskin et al.,1995), in situ observation ofAuGeliquid alloy and tin liquid metal ion source in a high-voltage transmission electron microscope (Driesel and Dietzsch,1996; Driesel, Dietzsch, and Mühle,1996), determination of onset voltage for ionic liquid EMI-BF at tip of a filled capillary tip of a microchined electrospray emitter (Krpoun and Shea, 2008), controlled meniscus evolution of deionized water in micro- gravity environment over a wide range of fluid properties (Elele et al., 2015) and free surfaces of cryogenic superfluid helium charged from below (Moroshkin et al.,2017).
Direct numerical simulations have also been developed over years to resolve final stages of cone formation that are not accessible to current experiment and imaging tech- niques. Suvorov and Zubarev (2004) employed standard finite difference discretization of the axisymmetric Navier-Stokes equation for free surface of liquid gallium in vac- uum under a flat electrode, where the governing fluid and electrostatic equations were transformed to a set of curvilinear coordinates and numerical grid adaptive to the evolv- ing liquid/vacuum interface. They reported that the formation of a conical singularity seemed to be universal, i.e. irrespective of initial surface shape (in their case a Gaussian
bump). Collins, Jones, et al. (2007) solved the Navier-Stokes equation coupled with surface charge transport modeled by ohmic conduction for a conducting film under a different electrode geometry (a metallic cylindrical rod protruding from upper plate).
Their finite element method adaptively remeshed the liquid domain and its interface with elliptic mesh generation. It was confirmed in the paper that when charge transport was turned off, no electrohydrodynamic tip streaming was observed at all and conic cusping singularity occurred instead. Giglio et al. (2008) simulated the spontaneous de- formation of an initially critically charged droplet by a spectral method utilizing spherical harmonic functions in a prolate spheroidal coordinate system. The liquid was assumed to be inviscid and incompressible with an irrotational flow. Their simulation (and ex- periment) suggested the droplet shape right before charge emission was remarkably well fitted by a “lemon” profile with two pointed ends, however, substantially narrower than the Taylor cone. Later, the identical simulation was repeated and refined by Burton and Taborek (2011) via a different yet much more accurate numerical method based on boundary integral formulation of harmonic potential. The ultrahigh resolution of their scheme revealed long-sought convergence to the Taylor cone angle at two conical ends of the droplet as well as decades of power-law behavior of tip electric field and curvature during the last stage of cone formation.
A common theme shared by these researches is that, the apical region of a round meniscus, after initial instability is triggered, continuously undergoes a self-sharpening and accelerating process towards a conic shape while simultaneously accumulating a significant amount of surface charges concentrating near the cone apex. Here is a summary of we have learned from these experiments and numerical simulations. First of all, cone formation in highly conductive liquid is unequivocally a dynamic process.
The continuously sharpening and simultaneous enhancement of the electric field near the apex are always accompanied by increasingly large bulk acceleration of the liquid.
Secondly, dynamic cone formation appears to be a runaway process that seldom halts on its own. It’s usually the non-hydrodynamic causes such as controlled quench-down of electrode potential, ejection of fluid filament/jet due to small-scale charge transport or field ion evaporation into ambient gas, that eventually mitigate the runaway process.
Thirdly, since none of the experiments or simulations were set up identical to each other or to pursue an idealization of G. I. Taylor (1964)’s electrostatic solution, formation of conic cusp at liquid/gas interface is very likely to be universal, i.e. not sensitive to geometric details of ambient electrode or initial liquid shape as long as the interface can be destabilized in the first place.
Theoretical understanding of this phenomenon is often credited to G. I. Taylor (1964)’s hydrostatic equilibrium between capillary and electrical forces. Despite simplicity of his static argument, the cone angle Taylor predicted has simulated a large amount of theoretical effort on understanding the steady cone-jet configuration, for instance, the
cone-jet mode of infinitely conductive liquids with a negligibly short jet that opens into the spray infinitesimally close to the cone apex (Fernández de la Mora,1992), the analytical cone-jet solution of infinitely long and thin, charged liquid jet issuing from the tip of Taylor cone (Gañán-Calvo, 1997), flow structure of the Taylor meniscus and emitted jet examined through singular perturbation methods in the limit of low flow rates (Cherney, 1999), various scaling laws between jet radius, current carried by the jet, liquid dielectric and electrical conductivity (Fernández de la Mora, 2007) and a recent extension of previous theories to account for variations of the cone angles (non- Taylor) by replacing the equipotential assumption of conic base with a finite electric current along the cone surface (Subbotin and Semenov,2015).
Technically these aforementioned theories are “dynamic” in the sense that surface charges are being steadily transported downstream along the jet. However, the over- all interface shape of the cone-jet is still stationary. Besides, none of these existing theories have addressed the spontaneous transition from a round meniscus to a conic cusp. Significant progress has been achieved in the study of non-stationary electrohy- drodynamic behavior since Zubarev (2001) pioneered a self-similar theory to capture universality and dynamic nature of conic cusp formation in perfectly conductive inviscid liquid. The novelty of his work lies in the self-similar scaling of an unsteady flow field which results in nontrivial inertia forces contributing to the stress balance at conic sur- face. The self-similar dynamics that Zubarev developed is also a fully nonlinear theory which is normally obscured if only infinitesimally weak disturbances, as in conventional linear stability analysis, of a Taylor cone under hydrostatic equilibrium, are considered (Sujatha et al.,1983; Chung, Cutler, and Miskovsky,1989). As correctly pointed out by Zubarev (2001), dynamic cone formation is the result of local balance between capillary, Maxwell (electrostatic) and inertia forces, which in his self-similar framework all blow up at the same rate as the conical singularity is approached. At the time, Zubarev was not able to obtain the exact solutions to his self-similar theory which requires sophisticated numerical techniques. Instead he employed asymptotic expansions of field variables to approximate the true solutions far away from the cone apex. Our work is an extension and an completion of Zubarev (2001)’s initial effort by introducing a more important yet still compatible leading order term in the asymptotic series, based on which a system of boundary integral equations are formulated to numerically compute the exact solutions conforming to the asymptotic behavior analytically derived.
In this chapter we hope to clarify the following: first of all, the classical Taylor cone is merely a geometric condition under which local capillary and Maxwell stresses scale with each other all the way up to the apex. The underlying liquid in the bulk is not necessarily static. Secondly, in inviscid regime formation of conic cusp is a dynamically self-similar process for perfectly conductive liquids, during which capillary, Maxwell and inertia forces all blow up in the vicinity of cone apex. The precise boundary conditions
(a) (b)
Sink flow No flow
Xc Xc
Figure 6.1: Common depictions of a dynamic cone: (a) A pulsating unstable liquid meniscus “glued” onto a far-field Taylor cone (dashed blue) in hydrostatic equilibrium subject to a specific electrode geometry (solid black) which coincides with one of the electric potential contours (dashed black). According to Chung, Cutler, and Miskovsky (1989), no liquid motion can be allowed far away from the conical pointXc. The unsta- ble meniscus oscillates up and down due to small vertical shifts of the electrode position.
(b) A self-reinforced meniscus converging to the Taylor cone angle due to a spherically symmetric sink flow (arrow) based on Zubarev (2001)’s self-similar mechanism.
(flow or electrostatics) of ambient environment are irrelevant compared to the dominant local scales. Thirdly, we wish to establish the multiplicity of dynamic cone formation.
The widespread spherically symmetric sink flow moving tangentially to the conic surface predicted by Zubarev (2001) is one of the many possibilities in the self-similar framework of inviscid cone formation. Our complete asymptomatic analysis independently shows that in addition to sink flow, pressure mismatch between capillary and Maxwell stresses induces a novel uplifting velocity field near cone apex, which we coin the“lifting” flow, with streamlines nearly vertical upward intercepting the conic surface at a finite angle.
The exact solution family, uncovered by our use of a patched boundary integral for- mulation, depends on two parameters reflecting the relative strength between capillary, Maxwell and inertia forces. Novel hydrodynamic patterns in the vicinity of cone apex such as counter flows, stagnation point and oscillatory pressure field during dynamic cone formation are also revealed for the first time.