On the Outcome of the Coulombic Fission of a Charged Isolated Drop

Abstract

Observations on the Coulombic fission of isolated drops of diameterDcharged near the Rayleigh limit show that they often form a transient Taylor cone through which many droplets much smaller thanDare emitted. Sometimes, however, the products of the explosion are only a few, and their size is comparable toD. We argue that the “fine fission” mode takes place under the same conditions generally leading to the formation of a steady Taylor cone; namely,Dhas to be much larger than the charge relaxation lengthdm= (γτ2/ρ)1/3(τ = ϵϵ0/Kis the electrical relaxation time; ϵ0is the electrical permittivity of vacuum; ϵ,K, γ, and ρ are the dielectric constant, electrical conductivity, surface tension, and density of the liquid). Otherwise, no Taylor cone may form and the explosion must proceed through a “rough fission” mode. Consequently, although drops of low conductivity liquids may break up into a few large and probably unequal fragments, more conducting drops are bound to explode with little mass loss, producing many very small and relatively monodisperse daughter droplets. For the case of polar liquids for whichD >> dm, we reason that the emissions from the exploding drop must be quasi-steady, with characteristics similar to those of steady electrified cone-jets supported on a capillary tube. In addition, the liquid flow rateQthrough the cone-jet forming on the exploding drop must be near the threshold value, which is on the order ofQm= γτ/ρ. This fixes approximately the size and charge of the fission products to be on the order ofdmand of the Rayleigh limit, respectively. No quantitative data are available for the size of the daughters from exploding polar liquids. Furthermore, the electrical conductivityKhas not been reported for the polar liquids whose Coulombic explosions have been studied so far. But the present predictions agree qualitatively with available measurements on the relative charge over mass loss in Coulomb fissions.