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
One question - freely interpreted by a mathematician (Kusner, of course :-) -
which arose at the workshop was this:
Are the volume bounds, which allow a single bridge to break, applicable to
multiple bridges? Specifically, if we choose a volume per pad that is less
than the critical volume at which a single bridge is unstable, will a multiple
bridge confugration with the same volume per pad also be unstable?
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
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: