5 research outputs found
Renormalization group approach to multiscale modelling in materials science
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