Many structural materials (metal alloys, polymers, minerals, etc.) are formed by quenching liquids into crystalline solids. This highly non-equilibrium process often leads to polycrystalline growth patterns that are broadly termed 'spherulites' because of their large-scale average spherical shape. Despite the prevalence and practical importance of spherulite formation, only rather qualitative concepts of this phenomenon exist. The present work explains the growth and form of these fundamental condensed matter structures on the basis of a unified field theoretic approach. Our phase field model is the first to incorporate the essential ingredients for this type crystal growth: anisotropies in both the surface energy and interface mobilities that are responsible for needle-like growth, trapping of local orientational order due to either static heterogeneities (impurities) or dynamic heterogeneities in highly supercooled liquids, and a preferred relative grain orientation induced by a misorientation-dependent grain boundary energy. Our calculations indicate that the diversity of spherulite growth forms arises from a competition between the ordering effect of discrete local crystallographic symmetries and the randomization of the local crystallographic orientation that accompanies crystal grain nucleation at the growth front (growth front nucleation or GFN). The large-scale isotropy of spherulitic growth arises from the predominance of GFN.

Here, we review the basic concepts and applications of the phase-field-crystal (PFC) method, which is one of the latest simulation methodologies in materials science for problems, where atomic-and microscales are tightly coupled. The PFC method operates on atomic length and diffusive time scales, and thus constitutes a computationally efficient alternative to molecular simulation methods. Its intense development in materials science started fairly recently following the work by Elder et al. [Phys. Rev. Lett. 88 (2002), p. 245701]. Since these initial studies, dynamical density functional theory and thermodynamic concepts have been linked to the PFC approach to serve as further theoretical fundaments for the latter. In this review, we summarize these methodological development steps as well as the most important applications of the PFC method with a special focus on the interaction of development steps taken in hard and soft matter physics, respectively. Doing so, we hope to present today's state of the art in PFC modelling as well as the potential, which might still arise from this method in physics and materials science in the nearby future.Keywords: phase-field-crystal (PFC) models, static and dynamical density functional theory (DFT and DDFT), condensed matter dynamics of liquid crystals and polymers, nucleation and pattern formation, simulations in materials science, colloidal crystal growth and growth anisotropy * Corresponding authors. Emails: heike.emmerich@uni-bayreuth. de, hlowen@thphy.uni-duesseldorf.de, and grana@szfki.hu

Most research into microstructure formation during solidification has focused on single-crystal growth ranging from faceted crystals to symmetric dendrites. However, these growth forms can be perturbed by heterogeneities, yielding a rich variety of polycrystalline growth patterns. Phase-field simulations show that the presence of particulates (for example, dirt) or a small rotational-translational mobility ratio (characteristic of high supercooling) in crystallizing fluids give rise to similar growth patterns, implying a duality in the growth process in these structurally heterogeneous fluids. Similar crystallization patterns are also found in thin polymer films with particulate additives and pure films with high supercooling. This duality between the static and dynamic heterogeneity explains the ubiquity of polycrystalline growth patterns in polymeric and other complex fluids.