  Oxygen vacancies have been identified to play an important role in
accelerating grain growth in polycrystalline perovskite-oxide ceramics. In
order to advance the fundamental understanding of growth mechanisms at the
atomic scale, classical atomistic simulations were carried out to investigate
the atomistic structures and oxygen vacancy formation energies at grain
boundaries in the prototypical perovskite-oxide material SrTiO$_3$. In this
work, we focus on two symmetric tilt grain boundaries, namely
$\Sigma$5(310)[001] and $\Sigma$5(210)[001]. A one-dimensional continuum model
is adapted to determine the electrostatic potential induced by charged lattice
planes in atomistic structure models containing grain boundaries and point
defects. By means of this model, electrostatic artifacts, which are inherent to
supercell models with periodic or open boundary conditions, can be taken into
account and corrected properly. We report calculated formation energies of
oxygen vacancies on all the oxygen sites across boundaries between two
misoriented grains, and we analyze and discuss the formation-energy values with
respect to local charge densities at the vacant sites.
