The hydrodynamic evolution of the surface of a liquid metal in the presence of an electric field is investigated using both analytical and numerical techniques. It is established that a free liquid surface with axial symmetry evolves with time into a cone-like shape, with the cone angle equal to Taylor's static value of 98.6°. The mechanism behind such interesting flow behaviour is that the system of electrohydrodynamic (EHD) equations has a self-similar asymptotic solution that generalizes Taylor's static result. The asymptotic solutions are found and the time-dependent behaviours of basic physical quantities (electric field strength, fluid velocity and surface curvature) near the singularity are established. The results and the analytical and numerical techniques used are thought to be useful in the development of time-dependent models of operating liquid metal ion sources and EHD sprayers.

Results of time-dependent modelling of electrohydrodynamic effects on the surface of a liquid metallic conductor are reported for a regime where no electron, ion or particle emission occurs. The Navier-Stokes equations, with free liquid boundaries subject to Maxwell field stress, surface-tension stress and viscous action, have been solved by a method that uses transformation of the interfaces into a rectangle; this overcomes a problem of surface oscillations that appeared using the marker-and-cell technique. The situation geometry is a deep unbounded surface with axial symmetry. With time, an almost flat surface evolves into a cone-like shape, with the angle of the cone depending on the initial shape of the surface. We describe this structure as a dynamic Taylor cone. The time-dependent profiles of the surface shape are in good agreement with experimental observations of this process. The calculations have also shown that, when the protrusion is formed, the time dependences of the surface radius of curvature, the electric field value at the protrusion apex and the axial velocity of the liquid metal, exhibit a run-away behaviour: the physical values become very large for a short time. As a cusp evolves on a surface, the Maxwell stress acting outwards becomes very large and overtakes the growth of both the surface tension and viscous stress acting inwards. Analysis of the time dependences of physical values can strongly assist the development of analytical treatments of such phenomena, and give insight into the problem of the dynamic description of operating liquid metal ion source atomisers. The development of numerical methods using transformation of the interfaces appears very useful for the treatment of problems in which the cathode or the anode significantly change shape. This situation occurs, for example, when a liquid surface is covered by a metal plasma and the evolution of the surface occurs in the context of a Langmuir shield.

Two-dimensional axisymmetric flow of a viscous, incompressible liquid metal with a free boundary subject to the action of an electric field is investigated in the context of liquid metal ion source operation. It is established that an almost flat surface evolves with time into a cone-like shape, with the cone angle equal to Taylor's static value of 98.6• . The growth time of the Taylor cone and the time-dependent behaviour of basic physical quantities (electric field strength, fluid velocity and surface curvature) near the cone apex are studied. A possible physical explanation of the effect of globule formation on the supporting needle of the liquid metal ion source has been put forward. The simulation is based on the method of adaptive numerical grid generation.

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It is well known experimentally that a liquid-metal ion source (LMIS) has a minimum stable emission current, typically around 1 to 2 µA. Attempts to run at lower current result in instability, or current extinction due to collapse of the liquid-metal cone (the "Taylor-Gilbert cone") that is essential to LMIS operation. The existence of a minimum current seems to be one of the factors that limit our ability to improve the resolution of a Focused Ion Beam machine by reducing the emission current and the spot size at the specimen. So there is interest in having good understanding of the origin of minimum current, and of relevant theory able to predict its size and dependence on relevant parameters.We have recently looked again at the existing theory of LMIS minimum current, developed by Beckman and others [1], and have discovered two theoretical difficulties with it. First, these authors assume that the emitter shape at low emission currents is a rounded Taylor cone. However, HVEM observations established long ago that at higher emission currents the emitter has the shape of 'a cusp on a cone'. Because there are electrostatic difficulties with the rounded Taylor cone assumption, it is theoretically probable that it also has this shape at lower currents.Second, it is well established that the ion emission process in the LMIS is "field evaporation (FEV)". This is also the emission process in the Atom-Probe Field Ion Microscope. It is know from basic work long ago, in the context of field ion microscopy, that the variant of FEV theory used in Beckman's minimum-current theory contains a disabling mathematical error.We have therefore reformulated minimum-current theory, for a LMIS with the shape of a cusp on a cone, using a FEV theory [2] that is known to be an approximation but which does often explain field evaporation results at low temperatures. Unfortunately, this theory does not yield plausible results for LMIS minimum current; the exact reasons for this are not currently understood. We have therefore reformulated FEV theory more generally, in terms of a second-order Taylor expansion of the FEV activation energy Q about the field F e at which Q becomes zero. Best existing FEV theory has the first-order Taylor coefficient zero, but it seems clear that to get a plausible theory of LMIS minimum current it is necessary to have a first-order Taylor coefficient that is significantly large.In unpublished work with D N Zurlev, one of us (RGF) has explored whether ion tunneling effects might give rise to a significant first-order Taylor coefficient in FEV theory, but we can find no convincing evidence of this. Further, no obvious cause of a significant first-order Taylor coefficient has yet emerged in the course of a re-exploration of the (simplified) analytical theory of the chargeexchange mechanism of FEV developed by one of us (RGF) some years ago (see Ref.[2]).A further possibility that remains to be explored in detail is that, at low emission currents, the shape of the LMIS emitter corresponds neither to a 'cusp on a c...