  Understanding the underlying factors responsible for higher-than-anticipated
local field enhancements that trigger vacuum breakdown on pristine metal
surfaces is crucial for the development of devices capable of withstanding
intense operational fields. In this study, we investigate the behavior of
nominally flat copper electrode surfaces exposed to electric fields of hundreds
of MV/m. Our novel approach considers curvature-driven diffusion processes to
elucidate the formation of sharp breakdown precursors. To do so, we develop a
mesoscale finite element model that accounts for driving forces arising from
both electrostatic and surface-tension-induced contributions to the free
energy. Our findings reveal a dual influence: surface tension tends to mitigate
local curvature, while the electric field drives mass transport toward regions
of high local field density. This phenomenon triggers the growth of sharper
protrusions, ultimately leading to a rapid enhancement of local fields and,
consequently, system instability. Furthermore, we delineate supercritical and
subcritical regimes across a range of initial surface roughness. Our numerical
results align closely with experimentally reported data, predicting critical
precursor formation fields in the range of 200 MV/m to 500 MV/m.
