Important phenomena in materials processing, such as dendritic growth during solidification, involve a wide range of length scales from the atomic level up to product dimensions. The phase-field approach, enhanced by optimal asymptotic methods and adaptive mesh refinement, copes with a part of this range of scales, from few tens of microns to millimeters, and provides an effective continuum modeling technique for moving boundary problems. A serious limitation of the usual representation of the phase-field model however, is that it fails to keep track of the underlying crystallographic anisotropy, and thus is unable to capture lattice defects and model polycrystalline microstructure without non-trivial modifications. The phase-field crystal (PFC) model on the other hand, is a phase field equation with periodic solutions that represent the atomic density. It natively incorporates elasticity, and can model formation of polycrystalline films, dislocation motion and plasticity, and nonequilibrium dynamics of phase transitions in real materials. Because it describes matter at the atomic length scale however, it is unsuitable for coping with the range of length scales in problems of serious interest. This thesis takes a first step towards developing a unified multiscale approach spanning all relevant lengths, from the nanoscale up, by combining elements from the phase-field and phase-field crystal modeling approaches, perturbative renormalization group theory, and adaptive mesh refinement.; A chapter of this thesis also examines the effect of confinement on dendritic growth, during equiaxed solidification in a pure material and the directional solidification of a dilute binary alloy, using phase-field models.
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