  We consider the dynamics of overdamped MEMS devices undergoing the pull-in
instability. Numerous previous experiments and numerical simulations have shown
a significant increase in the pull-in time under DC voltages close to the
pull-in voltage. Here the transient dynamics slow down as the device passes
through a meta-stable or bottleneck phase, but this slowing down is not well
understood quantitatively. Using a lumped parallel-plate model, we perform a
detailed analysis of the pull-in dynamics in this regime. We show that the
bottleneck phenomenon is a type of critical slowing down arising from the
pull-in transition. This allows us to show that the pull-in time obeys an
inverse square-root scaling law as the transition is approached; moreover we
determine an analytical expression for this pull-in time. We then compare our
prediction to a wide range of pull-in time data reported in the literature,
showing that the observed slowing down is well captured by our scaling law,
which appears to be generic for overdamped pull-in under DC loads. This
realization provides a useful design rule with which to tune dynamic response
in applications, including state-of-the-art accelerometers and pressure sensors
that use pull-in time as a sensing mechanism. We also propose a method to
estimate the pull-in voltage based only on data of the pull-in times.
