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Taylor cone
Taylor cone
Taylor cone
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Photograph of a meniscus of polyvinyl alcohol in aqueous solution showing a fibre drawn from a Taylor cone by the process of electrospinning.

A Taylor cone refers to the cone observed in electrospinning, electrospraying and hydrodynamic spray processes from which a jet of charged particles emanates above a threshold voltage. Aside from electrospray ionization in mass spectrometry, the Taylor cone is important in field-emission electric propulsion (FEEP) and colloid thrusters used in fine control and high efficiency (low power) thrust of spacecraft.

History

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This cone was described by Sir Geoffrey Ingram Taylor in 1964 before electrospray was "discovered".[1] This work followed on the work of Zeleny[2] who photographed a cone-jet of glycerine in a strong electric field and the work of several others: Wilson and Taylor (1925),[3] Nolan (1926)[4] and Macky (1931).[5] Taylor was primarily interested in the behavior of water droplets in strong electric fields, such as in thunderstorms.

Formation

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Electrospray diagram depicting the Taylor cone, jet and plume

When a small volume of electrically conductive liquid is exposed to an electric field, the shape of liquid starts to deform from the shape caused by surface tension alone. The liquid becomes polarized [6] and as the voltage is increased the effect of the electric field becomes more prominent. This causes an intense electric field surrounding the liquid droplet[6] As this effect of the electric field begins to exert a similar magnitude of force on the droplet as the surface tension does, a cone shape begins to form with convex sides and a rounded tip. This approaches the shape of a cone with a whole angle (width) of 98.6°.[1] When a certain threshold voltage has been reached the slightly rounded tip inverts and emits a jet of liquid. This is called a cone-jet and is the beginning of the electrospraying process in which ions may be transferred to the gas phase. It is generally found that in order to achieve a stable cone-jet a slightly higher than threshold voltage must be used. As the voltage is increased even more, other modes of droplet disintegration are found. The term Taylor cone can specifically refer to the theoretical limit of a perfect cone of exactly the predicted angle or generally refer to the approximately conical portion of a cone-jet after the electrospraying process has begun.

Taylor cones can be stationary as cone-jets described previously, or transient which can form when droplets undergo Coulombic explosion.[7]

Theory

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Sir Geoffrey Ingram Taylor in 1964 described this phenomenon, theoretically derived based on general assumptions that the requirements to form a perfect cone under such conditions required a semi-vertical angle of 49.3° (a whole angle of 98.6°) and demonstrated that the shape of such a cone approached the theoretical shape just before jet formation. This angle is known as the Taylor angle. This angle is more precisely where is the first zero of (the Legendre function of order 1/2).

Taylor's derivation is based on two assumptions: (1) that the surface of the cone is an equipotential surface and (2) that the cone exists in a steady state equilibrium. To meet both of these criteria the electric field must have azimuthal symmetry and have dependence to counter the surface tension to produce the cone. The solution to this problem is:

where (equipotential surface) exists at a value of (regardless of R) producing an equipotential cone. The angle necessary for for all R is a zero of between 0 and which there is only one at 130.7099°. The complement of this angle is the Taylor angle.

References

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Taylor cone

View on Grokipedia
from Grokipedia
A Taylor cone is a conical deformation of the surface of a conducting induced by a sufficiently strong , forming an equilibrium shape at the interface between the fluid and air (or another ) where the electric stress balances the surface tension, with a characteristic semi-vertical angle of 49.3 degrees. This phenomenon arises when the applied electric field exceeds a critical value, causing the fluid meniscus to elongate and stabilize into a cone from whose apex a fine jet can be emitted under further perturbation. The Taylor cone was theoretically predicted and experimentally verified by British physicist Geoffrey Ingram Taylor in 1964, through analysis of water drops and oil-water interfaces subjected to high-voltage electric fields, where instability leads to conical protrusions and jet ejection. Taylor's model assumes a perfectly conducting fluid and derives the equilibrium angle from the condition that the tangential electric stress must vanish on the cone surface, ensuring hydrostatic equilibrium. Subsequent refinements, such as the leaky dielectric model for non-perfect conductors, have extended the theory to a broader range of fluids, confirming the cone's stability under specific voltage and geometry conditions. In practical applications, the Taylor cone is central to (ESI), a technique developed by John B. Fenn in the 1980s for ionizing large biomolecules in by generating charged droplets from the cone-jet, enabling soft ionization without fragmentation; this work earned Fenn the 2002 . ESI relies on the cone's formation at the tip of a charged , where from the ejected microdroplets leads to analyte ion release, revolutionizing proteomic and pharmaceutical analysis. Similarly, in , the Taylor cone initiates the ejection of a jet that thins into nanofibers upon and whipping under the , producing materials for , , and with diameters down to tens of nanometers. These processes operate in the stable cone-jet mode for uniform output, with flow rate and voltage tuned to maintain the cone's integrity.

Historical Development

Early Observations

In the , (William Thomson) laid foundational theoretical groundwork for understanding electric forces acting on liquids through his studies on and charged droplets, predicting how electrical stresses could influence the equilibrium and motion of liquid surfaces in . These predictions, derived from analyses of and water-based electrostatic generators, influenced subsequent experimental investigations into the deformation of electrified liquids. Early 20th-century experiments by John Zeleny advanced these ideas by systematically examining the behavior of charged menisci under s. In 1914, Zeleny constructed an apparatus featuring a horizontal glass capillary tube connected to a of conductive , such as or mercury, positioned between parallel metal plates to apply a . He observed that as the voltage increased, the meniscus deformed from a rounded shape into an elongated, pointed form, eventually becoming unstable and ejecting fine streams of charged droplets when the strength exceeded a critical threshold. By 1917, Zeleny extended this work to study the instability of electrified surfaces more broadly, noting precursors to steady "cone-jet" emission where the pointed meniscus led to repetitive droplet formation without measuring specific cone angles. Building on Zeleny's findings, researchers in reported additional observations of pointed deformations in charged liquids during high-voltage experiments. W. A. Macky investigated the deformation and bursting of suspended drops in strong , documenting how increasing caused drops to elongate into conical shapes before fissioning into smaller charged fragments. Similarly, in 1932, J. J. Nolan and J. G. O'Keeffe examined from drops, observing that electrified drops formed transient pointed tips from which charged sprays emanated, akin to early modes. In the 1950s, further reports highlighted cone-like shapes in setups with various liquids under high voltages. In 1955, V. G. Drozin's studies on the electrical dispersion of liquids as aerosols described menisci deforming into approximate conical profiles at the tip, leading to jet-like ejection of fine droplets, though without quantitative analysis of the . These pre-1964 observations collectively identified the conical deformation as a recurring feature in electrified liquids but lacked a unified theoretical framework, which later refined in 1964.

Taylor's Contribution

In 1964, G.I. Taylor published a seminal paper analyzing the equilibrium shape of a charged pendant drop subjected to electric stress, demonstrating that a conical interface between two fluids could exist stably under such conditions. This work built briefly on earlier observations by J. Zeleny regarding the instability of charged liquid droplets. Taylor's experiments involved a setup where a slowly evaporating conducting liquid, such as a soap solution, was formed as a pendant drop from a fine capillary tube by dipping and withdrawing under high voltages of approximately 5-10 kV, deforming into a cone-like shape in the electric field. He also examined oil-water interfaces to visualize the conical form more clearly, using photographic evidence to confirm the structure. The apparatus generated stable electric fields sufficient to balance surface tension against electrostatic forces, allowing the cone to persist as an equilibrium configuration at the threshold of instability, just prior to the onset of jet ejection and droplet disintegration. Through this combination of and , Taylor established that the stable cone maintains a precise semi-vertical of approximately 49.3 degrees, a geometric feature arising from the balance of electric and stresses. The conical shape he described, now known as the Taylor cone in recognition of his foundational contribution, represents a critical equilibrium state in electrohydrodynamic processes.

Formation Mechanism

Experimental Setup

The typical laboratory setup for generating a Taylor cone consists of a metallic or needle, often , with an inner diameter ranging from 0.1 to 1 mm, connected to a capable of delivering 2 to 20 kV. A or polar , such as or , is introduced into the capillary via a to control flow rates precisely at low levels, typically from nanoliters per minute (nL/min) to microliters per minute (μL/min), ensuring stable meniscus formation at the tip. A counter-electrode, such as a grounded metallic plate or collector, is positioned 5 to 50 cm from the tip to generate a uniform across the gap. The voltage is gradually increased until a threshold of 1 to 10 kV is reached, at which point the Taylor cone forms; this threshold depends on the liquid's and other properties. Ambient conditions, including and , influence cone stability by affecting liquid conductivity and rates. Due to the high voltages involved, safety measures are essential, including proper insulation, grounding, and shielding to mitigate risks of electrical discharge or arcing. Variations in the setup, such as capillaries, allow for the delivery of compound jets by injecting an inner liquid surrounded by an outer sheath fluid through nested needles. The resulting cone shape approximates the theoretical prediction from Taylor's analysis when operational parameters are optimized.

Physical Processes Involved

The formation of a Taylor cone begins with the application of an external to a meniscus, typically emerging from a , which induces charge accumulation at the -air interface. In or leaky , free charges migrate rapidly to the surface under the influence of the field, creating a tangential electric stress that deforms the initially spherical or hemispherical meniscus by pulling it toward the . This stress competes with the restoring force of , leading to an initial elongation of the meniscus into a more pointed shape. As the applied voltage increases, the meniscus progresses toward a conical , with the electric stress intensifying at the apex due to field concentration. The critical point occurs when the tangential electric stress balances the surface tension, approaching the Rayleigh limit for charge-induced , beyond which the interface would otherwise undergo fission similar to that in charged droplets. At this equilibrium, the cone achieves a stable shape characterized by a half-angle of approximately 49.3°, as measured in early experiments with conducting liquids. plays a key role here by providing hydrodynamic resistance that stabilizes the deforming interface against perturbations, preventing premature breakup and allowing the cone to maintain its form. Beyond this equilibrium, further increase in the leads to onset at the cone tip, where the highly concentrated field causes charge separation and emission of a fine jet from the apex. The charge relaxation time in conducting liquids, typically on the order of milliseconds (e.g., 1.6 ms for certain aqueous solutions), governs how quickly charges redistribute to sustain this process, ensuring the surface remains nearly . Liquid properties such as the dielectric constant influence the normal electric stress across the interface, while conductivity determines the rate of charge transport, both critically affecting cone stability; higher conductivity facilitates faster relaxation and more stable cones, whereas varying dielectric constants modulate the overall field penetration and deformation dynamics.

Theoretical Framework

Mathematical Model

The mathematical model for the Taylor cone originates from the equilibrium analysis of a conducting liquid meniscus under an applied , where the shape achieves static balance between electrostatic and capillary forces. At the interface, the normal electric stress σe=12ϵ0E2\sigma_e = \frac{1}{2} \epsilon_0 E^2—with ϵ0\epsilon_0 denoting the and EE the magnitude—must equal the capillary pressure γκ\gamma \kappa, where γ\gamma is the liquid's and κ=cot⁡αr\kappa = \frac{\cot \alpha}{r} is the of the cone surface, with rr the distance from the apex along the generator and α\alpha the semi-vertical . This balance ensures no net deforms the surface further. To derive the conical equilibrium shape, Taylor employed the infinite cone approximation, treating the meniscus as an axisymmetric cone extending indefinitely, which simplifies the scale-invariant nature of the stresses. The electric potential ϕ\phi external to the cone satisfies Laplace's equation ∇2ϕ=0\nabla^2 \phi = 0 in spherical coordinates (r,θ)(r, \theta), with the boundary condition of constant potential on the conducting cone surface (set to zero for simplicity). The self-similar solution takes the form ϕ=Ar1/2P−1/2(cos⁡θ)\phi = A r^{1/2} P_{-1/2}(\cos \theta), where AA is a constant and P−1/2P_{-1/2} is the Legendre function of non-integer order −1/2-1/2, yielding an electric field E∝r−1/2E \propto r^{-1/2} that scales appropriately with the cone's geometry. The cone's semi-vertical angle α\alpha (measured from the axis to the surface) is determined by requiring the angular variation of 12ϵ0E2\frac{1}{2} \epsilon_0 E^2 to match γcot⁡αr\gamma \frac{\cot \alpha}{r} precisely along the surface. The angle α\alpha is determined such that the computed electric stress from the potential matches the required capillary pressure γcot⁡αr\gamma \frac{\cot \alpha}{r} along the entire surface, yielding the characteristic value α≈49.3∘\alpha \approx 49.3^\circ. In Taylor's 1964 formulation, the surface field equation emerges as E=Vr1/2⋅f(α)E = \frac{V}{r^{1/2} \cdot f(\alpha)}, where VV is the applied potential and f(α)f(\alpha) derives from the Legendre function evaluation at the critical angle, confirming the r−1/2r^{-1/2} dependence essential for balance. While this model captures the ideal static cone, it relies on the infinite approximation, which limits applicability to finite-sized menisci; near the apex of real cones, finite dimensions introduce deviations in the potential distribution and stress, altering the local curvature and field enhancement.

Stability Analysis

The stability of the Taylor cone is analyzed by considering small perturbations around its equilibrium shape, extending classical Rayleigh-Plateau instabilities to include the effects of . In linear stability theory for electrified liquid jets emerging from the cone, axisymmetric perturbations (varicose modes) grow due to , but the axial stabilizes shorter wavelengths while destabilizing longer ones, leading to a modified where the growth rate σ satisfies σ ∝ k (1 - k^2 a^2) modified by an electric term proportional to the charge relaxation time. Non-axisymmetric perturbations (whipping modes) arise from tangential electric stresses, with growth rates increasing for azimuthal wavenumbers m ≥ 1, particularly in low-conductivity liquids where charge accumulation amplifies bending instabilities. These analyses reveal that the cone-jet transition occurs when the suppresses the Plateau-Rayleigh near the apex, allowing a steady jet to form before downstream instabilities dominate. The critical voltage for jet emission from the Taylor cone is determined by the electric Bond number, defined as Boe=ε0E2L2γBo_e = \frac{\varepsilon_0 E^2 L^2}{\gamma}, where ε0\varepsilon_0 is the , EE is the strength, LL is the characteristic length (L=γ/(ρg)L = \sqrt{\gamma / (\rho g)}
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