The effect of charge emission from electrified liquid cones

The formation of stable cones in electrified liquid interfaces was explained by Taylor as a balance between electrical and capillary tensions, where the electrostatic potential varies as ϕ ∼ r½ with the distance r from the cone tip. Although Taylor's predictions for the dependence of the onset voltage for cone formation on the liquid surface tension γ and the cone dimensions agree with observed trends, his conclusion that the cone semiangle α can only take the value α = αT = 49.3° does not. A more general theory free from this paradox is constructed for highly conducting fluids by accounting for the space charge of the droplets emanating from the cone apex, whose potential has the remarkable property of also obeying Taylor's r½ law. In this formulation, where the apex of a conical meniscus of semiangle α emits an angularly uniform opposed coaxial conical spray of semiangle π—β, both β and the spray current I turn out to be fixed as functions of α; namely, β = β(α), and I = 2πγKqG(α), where Kq and q are the droplet's electrical mobility and total charge, respectively. In experiments with 5% H2SO4 in 1-octanol, the observed sprays are approximately conical with an apex nearly touching the meniscus tip. The measured and predicted β(α) relations are in reasonable agreement in the range 46° > α > 32°, where the liquid cone is stable and the spray is visible, though the data fall clearly below the theoretical curve. The predicted spray current I is also in rough agreement with preliminary experiments. The analysis applies neither to sprays of large droplets with significant inertia, nor to liquid cones in vacuo.