THE ELECTROHYDRODYNAMICS OF THE CHARGED LIQUID JET ISSUING FROM AN ELECTRIFIED TAYLOR CONE. UNIVERSAL SCALING LAWS

A hybrid experimental–numerical approach to study the dynamics of capillary electrified jets, which uses a quasi-one-dimensional model and the experimentally measured shape of an actual liquid thread (Gañán-Calvo (1997) J. Fluid Mech. 335, 165–188) has been employed in this work to analyze the electrohydrodynamics of the liquid micro-jets issuing from Taylor cones. Different liquids have been used in this study, with electrical permittivities from 6.5 to 38 times the vacuum permittivity, and electrical conductivities ranging from 8.5 to 4.5e-4 S m^-1. Up to 25 different jet shapes corresponding to steady and absolutely stable conditions have been digitized, and the corresponding surface charge distribution, normal external and internal electric fields at the surface, the axial electric field (the slender approach allows to consider the axial electric field constant in the transversal direction), the liquid velocity distribution, the electric current convected by the surface and the one driven through the bulk by Ohmic conduction at each axial point have been calculated. In particular, one of the liquid jets analyzed corresponded to the onset of stability of the steady cone-jet mode, where we supply just the (minimum) liquid flow rate that the electrostatic suction effect at the cone-jet neck is able to withdraw at the minimum needle–electrode potential difference for a given stable cone elongation. This has revealed a surprising result: even in this critical situation, the inner normal electric displacement is at most a mere 15% of the outer one, and this happens only at one point of the whole cone-jet, located close to the point at which the convected electric current equals the current driven by bulk conduction (i.e. a little downstream of the cone-jet neck), being the inner displacement at other points of the jet and the cone hundreds of times smaller than the outer displacement. As one increases the liquid flow rate, the ratio of the maximum inner displacement to the outer displacement becomes proportionally smaller. This result clarifies for the first time the controversy about charge relaxation phenomena in cone-jet electrosprays, since it can be used to show from a physicochemical argument that the charge layer at the whole cone-jet surface is almost relaxed even at the onset of stability, at least for liquid permittivities of the order of the ones used in this study. This result also guarantees a homogeneous bulk conductivity along the hole cone-jet. Secondly, and similarly interesting, the kinetic energy per unit volume acquired by the liquid in the jet results independent of the flow rate for a given liquid and a cone elongation, explained by the fact that the normal electric field (or surface charge distribution) which provokes the main acceleration force (the electrostatic suction effect, at the cone-jet neck and the beginning of the jet) results independent of the flow rate as well. A universal scaling of the electro-hydrodynamic variables, jet size and total emitted electric current is proposed, and the experimental results are collapsed into a universal collection of distributions of non-dimensional variables along the axis. The resulting droplet size, also measured in the same experiments, scales as the jet radius, and the droplet charge results proportional to its surface, a result shown by many investigators but never explained. Other previously used electrohydro-dynamic hypotheses and scaling laws are discussed under these new results.