CALCULATION OF THE SHAPE OF THE LIQUID CONE IN A LIQUID-METAL ION-SOURCE

The shape of the liquid cone is determined by the electrohydrodynamic equations governing the liquid flow. We model the cone as a stationary steady-state axially-symmetric structure, neglect viscosity effects, and apply the Bernoulli equation to a free-surface stream line. The pressure difference across the liquid surface is given as the difference between the surface tension and the Maxwell stress. This relation, together with the Bernoulli and mass-conservation equations yields the equation for the shape of the cone. The equation contains the electric field intensity at the liquid surface, which is calculated by solving an appropriate Poisson equation. The space-charge distribution is calculated by modelling the charged-particle trajectories in the self-consistent electric field. The Poisson equation solution needs the surface to be given, so we use an iterative procedure in which we improve the assumed shape at each step, to satisfy the shape equation in the weighted residual sense. We use trial functions with no predetermined parameters; in particular the limiting cone angle is not pre-set. Results are as follows. The cone angle is close to the Taylor cone angle only for large needle radii. For smaller needle radii the cone angle is current dependent, the angle decreasing with increasing current. The protrusion length is also current dependent, but values are generally much smaller than those calculated by Kingham and Swanson. Current/voltage characteristics are steeper than those calculated by Kingham and Swanson and agree better with experiment.