Renormalization group approach to multiscale modelling in materials science

TL;DR

This paper develops a multiscale modeling approach in materials science by applying renormalization group techniques to phase field crystal models, enabling efficient simulations across atomic to macroscopic scales.

Contribution

It introduces a systematic way to derive coarse-grained equations from PFC models using renormalization group methods, improving multiscale simulation capabilities.

Findings

Derivation of coupled phase and amplitude equations from PFC models.

Implementation of adaptive mesh refinement for multiscale simulations.

Enhanced modeling of microstructure formation across scales.

Abstract

Dendritic growth, and the formation of material microstructure in general, necessarily involves a wide range of length scales from the atomic up to sample dimensions. The phase field approach of Langer, enhanced by optimal asymptotic methods and adaptive mesh refinement, copes with this range of scales, and provides an effective way to move phase boundaries. However, it fails to preserve memory of the underlying crystallographic anisotropy, and thus is ill-suited for problems involving defects or elasticity. The phase field crystal (PFC) equation-- a conserving analogue of the Hohenberg-Swift equation --is a phase field equation with periodic solutions that represent the atomic density. It can natively model elasticity, the formation of solid phases, and accurately reproduces the nonequilibrium dynamics of phase transitions in real materials. However, the PFC models matter at the atomic scale, rendering it unsuitable for coping with the range of length scales in problems of serious interest. Here, we show that a computationally-efficient multiscale approach to the PFC can be developed systematically by using the renormalization group or equivalent techniques to derive appropriate coarse-grained coupled phase and amplitude equations, which are suitable for solution by adaptive mesh refinement algorithms.