The current emitted by highly conducting Taylor cones

Abstract

When a liquid meniscus held at the exit of a metallic capillary tube is charged to a high voltage V, the free surface often takes the form of a cone whose apex emits a steady microjet, and thus injects a certain charge I and liquid volume Q per unit time into the surrounding gas. This work deals with liquids with relatively large conductivities K, for which the jet diameter d_j is much smaller than the diameter d_n of the capillary tube. In the limit d_j/d_n → 0, the structure of the jet (d_j and I, in particular) becomes independent of electrostatic parameters such as V or the electrode configuration, being governed mostly by the liquid properties and flow rate Q. Furthermore, the measured current is given approximately by I = f(ε) (γQK/ε)^½ for a wide variety of liquids and conditions (ε, and γ are, respectively, the dielectric constant of the liquid and the coefficient of interfacial tension; f(ε) is shown in figure 11). The following explanation is proposed for this behaviour. Convection associated with the liquid flow Q transports the net surface charge towards the cone tip. This upsets the electrostatic surface charge distribution slightly at distances r from the apex large compared to a certain charge relaxation length λ, but substantially when r ∼ λ. When the fluid motion is modelled as a sink flow, λ is of the order of r^* = (Qεε₀/K)^$\frac13$ (ε₀ is the electrical permittivity of vacuum). If, in addition, the surface charge density is described through Taylor's theory, the corresponding surface current convected towards the apex scales as I_s ∼ (γQK/ε)^½, as observed for the spray current. The sink flow hypothesis is shown to be realistic for sufficiently small jet Reynolds numbers. In a few photographs of ethylene glycol cone jets, we find the rough scaling d_j ∼ 0.4r^* for the jet diameter, which shows that the jet forms as soon as charge relaxation effects set in. In the limit ε [Gt ] 1, an upper bound is found for the convected current at the virtual cone apex, which accounts for only one-quarter of the total measured spray current. The rest of the charge must accordingly reach the head of the jet by conduction through the bulk.