Jet Breakup and Conic Plumes of the DC Taylor Cone Due to Induced Space Charge

Taylor first showed that the singular electric Maxwell pressure, due to induced polarization, at a cone can offset the singular azimuthal capillary pressure if this Taylor cone sustains a half angle of 49 degrees.  However, the mechanisms of how a microjet emits from the Taylor cone and breaks up into aerosols remain poorly understood despite their importance in electrospray sample delivery in proteomic mass spectrometry.  An earlier theory by Ganon-Calvo [1] estimates the Maxwell pressure on the jet using a current balance and predicts a jet radius R(z) that scales as (gK)^-1/6z^-1/8 in the axial z direction for a liquid jet of conductivity K and surface tension g.   By assuming a constant and unknown breakup length, his theory then predicts a breakup radius and drop size that scale as (gK)^-1/6. He was also able to produce two current-flow rate correlations for the jet that collapsed literature data. However, our high-speed imaging of the jet breakup shows a breakup length that is inversely proportional to surface tension g and a weak but important ln(K) scaling. We propose an alternative theory that captures the space charge induced by the difference between the gas potential (due to the Taylor cone) and an equipotential axis along the conducting jet using a nonlinear Gouy-Chapman theory. This new theory predicts an R(z) that scales as K^-1/4exp(-(gz)^1/2), with a drastically different z scaling from the Ganon-Calvo theory. The Maxwell pressure due to the induced space charge can overcome azimuthal capillary pressure to produce a breakup length that scales as (ln(K))²/g, which is consistent with our measurements. This first theory for the breakup length also predicts a universal potential drop, φ ~ ln(g⁴/K³)  across the cone-jet of a few volts , which is a function of the surface tension g and conductivity K of the fluid.  This potential corresponds to a Rayleigh-like radius for the jet at breakup and suggests a universal drop size, prior to Rayleigh fission, that scales as (gK)^-1/7, close to Ganon-Calvo's prediction . An important confirmation and prediction of this induced space charge theory is that the plume angle at the breakup point can be captured by the ratio of the axial field from Taylor's dominant harmonic and our radial field due to induced charge. We are able to explain the existence of multiple plume cones by relating the large-angle cones to higher order harmonics of the Taylor cone. Our theory also offers explicit flow-rate/current correlations consistent with Ganon-Calvo theory because of the importance of convective current due to the induced space charge. Induced space charge polarization dominates at the jet and determines the breakup location and the subsequent plume angles.

[1] A. M. Ganan-Calvo, Phys. Rev. Letts. 79, 217-220 (1997).