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. Phase field models, enhanced by optimal asymptotic methods and adaptive mesh refinement, cope with this range of scales, and
provide a very efficient way to move phase boundaries. However, they fail to preserve memory of the underlying crystallographic anisotropy. Elder and Grant have convincingly shown how one can use the phase field crystal (PFC) equation -- a conserving analogue of the Swift-Hohenberg equation -- to create field equations with periodic solutions that model elasticity,
the formation of solid phases, and accurately reproduce the nonequilibrium dynamics of phase transitions in real materials. In this talk, I show that a computationally-efficient multiscale approach to the PFC can be developed
systematically by using the renormalization group or equivalent techniques to derive the appropriate coupled phase and amplitude equations, which can
then be solved by adaptive mesh refinement algorithms.
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