EVOLVER files for modelling simple bridges, and multiple bridges among singly- or doubly-periodic arrays of sessile or pendant solder droplets


Listed below are some sample files for Ken Brakke's Surface Evolver and scripts to model the instability of bridges between adjacent sessile or pendant droplets attached to nearby pads: as various parameters (the size and spacing of the pads, the volumes of the drops, the surface tension and density of the solder) are adjusted, the bridges either develop a Rayleigh instability and break, into several disconnected solder beads ("good"); or else they evolve to a stable equilibrium bridge, and thus a short-circuit ("bad"). Sessile drops might arise in reflow soldering of ball-grid-arrays, whereas pendant drops could come from wave soldering of pin-arrays. These files were created in June 1994 when Rob Kusner (a mathematician from the University of Massachusetts at Amherst) was visiting the Geometry Center at the time of the first workshop for solder joint design.


Real experiments with molten-solder-like materials (mercury) in similar configurations were demonstrated by Tim Singler (a physicist from SUNY Binghamton) at the same workshop, though these concerned only the sessile case. He found that sessile bridges, once formed, are very difficult to break via the Rayleigh instability, even as the droplet volume is greatly reduced. Kusner's experiments with the Evolver corroborate this. In the case of pendant droplets, there do not seem to be any physical experiments carried out yet, though Evolver experiments suggest it is much easier to break the bridges.


One question - freely interpreted by a mathematician (Kusner, of course :-) - which arose at the workshop was this: In other words, is the worst case of bridge formation the simplest? Since this qualitative question can be asked independent of physical constants, this seemed like a reasonable mathematical question, which might have a clearcut answer. And the practical consequences would be that detailed physical experiments (like Singler's) could confidently focus on the single bridge case. Of course, it is not entirely clear whether this geometry applies exactly to any specific devices, though again, the qualitative features should be instructive.


It may be difficult to give an exact analytic solution, but Kusner's preliminary conclusions from experimenting with his Evolver files (below) are that symmetric multiple bridges do require even larger volumes to form (or remain stable) than single bridges.


The following Evolver files include scripts that are executed by typing "doit" (no quotes) at the "enter command:" prompt. Information is obtained with the "inf" command. Volume per pad can be assessed and altered using the "v" and "set body target" commands within Evolver: