Boundary Conditions for Atomistic Simulation of Material Defects

Atomistic simulations are an indispensable tool of modern materials science, solid state physics and chemistry, as they allow scientists to study individual atoms and molecules in a way that is impossible in laboratory experiments. Understanding atomistic processes opens up avenues for the manipulation of matter at the atomic scale in order to achieve superior material properties for applications in science and engineering.

One of the most common tasks of atomistic materials modelling is to determine properties of crystalline defects, including their atomic structure, formation, activation and ionisation energies, from which electronic and atomistic mechanisms of chemical reactivity, charge mobility, etc., can be directly discovered, and mesoscopic material properties or coarse-grained models (e.g., employed in kinetic Monte-Carlo, discrete dislocation dynamics, continuum fracture laws, transport simulations) can be derived.

Defects distort the surrounding host lattice, generating long-ranging elastic (and possibly also electrostatic) fields. Since practical schemes necessarily work in small computational domains they cannot explicitly resolve these far-fields but must employ artificial boundary conditions (e.g., periodic boundary conditions) to emulate the elastic bulk. This approximation gives rise to a simulation error that must be controlled and ideally balanced against other model and/or discretisation errors. For example, for a wide class of defects encompassing all (neutral) point defects and straight dislocations it is shown by Ehrlacher, Ortner and Shapeev (2016) that the geometry error decays with a universal rate O(N^{-1/2}) where N denotes the number of atoms in the computational cell. For a cubic scaling computational chemistry model, this slow rate is particularly severe. For cracks, it turns out that the standard models even yield schemes that are divergent in N.

This extremely slow rate of convergence or even divergence represents both a theoretical and computational challenge, which we propose to address in this project. Specifically, we will develop a hierarchy of high-accuracy boundary conditions for four common classes of defects: charge neutral point defects, dislocations, cracks, and charged defects. At its core, this research involves the development of a range of new analytical tools to describe elastic and polarisation fields in crystalline solids and how they are coupled to defect cores. The analytical results will feed directly back into materials simulation methodology through new algorithms and open source software. The effect of these new algorithms will be to enhance both the reliability and efficiency of atomistic simulation of materials, and enable simulation of particularly complex defect structures that have so far been inaccessible with conventional tools.

Boundary conditions are one of the most critical aspects of material defect simulations tasks. The proposed project develops a mathematical framework, within which one can develop boundary conditions with - in principle - arbitrary accuracy. Moreover, the project will develop numerical algorithms and simulation tools arising from this analysis.

The immediate beneficiaries of this work are both academic and industrial researchers who use atomistic material modelling software to study the properties of material defects. Indeed, the results of the project lead to more accurate and more reliable boundary conditions, or alternatively allow usage of much smaller computational domains without loss of accuracy. In particular, this development makes possible the use of more accurate and hence more expensive material models, or simply a faster turnaround time. The increased reliability also facilitates unsupervised simulations. In some cases, the new models we propose will enable the simulation of material defects that were previously inaccessible. Theoreticians will also benefit from the new methods and perspectives we propose, enabling extensions to other use-cases.

The broader impact of the research arises from its applications in academic and industrial materials science. In the end, our research allows researchers to obtain more reliable numerical results at lower computational cost. Materials science progress that will be achieved through improved simulation tools leads, for example, to improved manufacturing processes, and the development of new materials with superior properties (e.g., strength, weight, electrical and thermal conduction, optical response or even entirely new and unexpected properties). For example, such improvements in engineering materials and manufacturing processes have the potential to significantly reduce energy consumption. On the other hand, our work on charged defects directly feeds into research on energy storage and energy conversion technology as well as catalysis and green chemistry.