Atomistic-to-continuum multiscale modeling with long-range electrostatic interactions in ionic solids

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 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 multiscale 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 longrange 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.

Effects of Polydispersity on Structuring and Rheology in Flowing Suspensions

“…The FLD method further increased the efficiency by using fast approximate methods for the far-field interaction [8,9]. The broader idea of developing multiscale methods for long-range interactions by decomposing into far-and near-field and then using multipole expansions for the far-field has been studied theoretically and numerically in the context of bubbly fluids [42] and electromagnetic interactions [43][44][45].…”

Cutoff-Based Modeling of Coulomb Interactions for Atomistic-to-Continuum Multiscale Methods

“…Existing atomistic-to-continuum multiscale methods, such as the quasicontinuum (QC) method [43], are applicable almost exclusively in cases where the atomic interactions are short-ranged. So far there has been only one approach, to the best of our knowledge, that has demonstrated, however only qualitatively, the application of atomistic-to-continuum methods for ionic crystals [31]. The method involves coarse-graining of the long-range Coulomb interactions in ionic crystals.…”

“…The Wolf summation method uses direct cutoff-based truncated sum along with a correction term. This approach is conceptually different from that of [31] and incurs a significantly lower computational complexity. We choose the fully nonlocal QC method from the existing collection of multiscale methods for demonstration.…”

An analytical model for porous single crystals with ellipsoidal voids

“…Lode angle Z second-order shape tensor J fourth-order hydrostatic tensor P,P fourth-order Eshelby tensors Q,Q fourth-order microstructural tensors S,Ŝ fourth-order tensors used to defined the modified variational effective stress potential 2000; Marshall and Dayal, 2014). Such anisotropic matrix systems have known slip directions and contain usually a small volume fraction of impurities.…”