  We present a multiscale atomistic-to-continuum method for ionic crystals with
defects. Defects often play a central role in ionic and electronic solids, not
only to limit reliability, but more importantly to enable the functionalities
that make these materials of critical importance. Examples include solid
electrolytes that conduct current through the motion of charged point defects,
and complex oxide ferroelectrics that display multifunctionality through the
motion of domain wall defects. Therefore, it is important to understand the
structure of defects and their response to electrical and mechanical fields. A
central hurdle, however, is that interactions in ionic solids include both
short-range atomic interactions as well as long-range electrostatic
interactions. Existing atomistic-to-continuum multi-scale methods, such as the
Quasicontinuum method, are applicable only when the atomic interactions are
short-range. In addition, empirical reductions of quantum mechanics to density
functional models are unable to capture key phenomena of interest in these
materials. To address this open problem, we develop a multiscale atomistic
method to coarse-grain the long-range electrical interactions in ionic crystals
with defects. In these settings, the charge density is rapidly varying, but in
an almost-periodic manner. The key idea is to use the polarization density
field as a multiscale mediator that enables efficient coarse-graining by
exploiting the almost-periodic nature of the variation. In regions far from the
defect, where the crystal is close-to-perfect, the polarization field serves as
a proxy that enables us to avoid accounting for the details of the charge
variation. We combine this approach for long-range electrostatics with the
standard Quasicontinuum method for short-range interactions to achieve an
efficient multiscale atomistic-to-continuum method.
