The conserved Swift-Hohenberg equation and crystallization

The phase-field model, also known as the conserved Swift-Hohenberg equation, provides a useful model of crystallization that is derivable from the more accurate dynamical density functional theory. I will survey the properties of this model focusing on spatially localized structures and their organization in parameter space. I will highlight the role played by conserved mass and discuss the role played by these structures in the thermodynamic limit in both one and two spatial dimensions. I will then discuss dynamic crystallization via a propagating crystallization front. Two types of fronts can be distinguished: pulled and pushed fronts, with different properties. I will demonstrate, via direct numerical simulation, that the crystalline structures deposited by a rapidly moving front are not in thermodynamic equilibrium and so become disordered as they age. I will conclude with a discussion of a two-wavelength generalization of the model that exhibits quasicrystalline order in both two and three dimensions and of the associated spatially localized structures with different quasicrystalline motifs. The possible role of metastable spatially localized structures in nucleating crystallization will be highlighted.