Atomistic-to-Continuum Multiscale Modeling with Long-Range Electrostatic Interactions in Ionic Solids

This is the preprint of an article that will appear in the Journal of the Mechanics and Physics of Solids. (doi:10.1016/j.jmps.2013.09.025).

Atomistic-to-Continuum Multiscale Modeling with Long-Range Electrostatic Interactions in Ionic Solids

by Jason Marshall and Kaushik Dayal, Carnegie Mellon University

Abstract 

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 impor-

tantly 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. As a side

note, we examine an important issue that is critical to our method: namely, the dependence of the

computed polarization field on the choice of unit cell. Potentially, this is fatal to our coarse-graining

scheme; however, we show that consistently accounting for boundary charges leaves the continuum

electrostatic fields invariant to choice of unit cell.