Many structural materials (metal alloys, polymers, minerals, etc.) are formed by quenching liquids into crystalline solids. This highly nonequilibrium 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. It is established that phase field methods naturally account for diffusional instabilities that are responsible for dendritic single-crystal growth. However, a generalization of this model is required to describe spherulitic growth patterns, and in the present paper we propose a minimal model of this fundamental crystal growth process. Our calculations indicate that the diversity of spherulitic growth morphologies 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 (GFN)]. This randomization in the orientation accounts for the isotropy of spherulitic growth at large length scales and long times. In practice, many mechanisms can give rise to GFN, and the present work describes and explores three physically prevalent sources of disorder that lead to this kind of growth. While previous phase field modeling elucidated two of these mechanisms—disorder created by particulate impurities or other static disorder or by the dynamic heterogeneities that spontaneously form in supercooled liquids (even pure ones)—the present paper considers an additional mechanism, crystalline branching induced by a misorientation-dependent grain boundary energy, which can significantly affect spherulite morphology. We find the entire range of observed spherulite morphologies can be reproduced by this generalized phase field model of polycrystalline growth.

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 fundamentals 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.

We present a phase field theory for binary crystal nucleation. In the one-component limit, quantitative agreement is achieved with computer simulations (Lennard-Jones system) and experiments (ice-water system) using model parameters evaluated from the free energy and thickness of the interface. The critical undercoolings predicted for Cu-Ni alloys accord with the measurements, and indicate homogeneous nucleation. The Kolmogorov exponents deduced for dendritic solidification and for “soft impingement” of particles via diffusion fields are consistent with experiment.

The phase field approach is used to model heterogeneous crystal nucleation in an undercooled pure liquid in contact with a foreign wall. We discuss various choices for the boundary condition at the wall and determine the properties of critical nuclei, including their free energy of formation and the contact angle as a function of undercooling. For particular choices of boundary conditions, we may realize either an analog of the classical spherical cap model or decidedly nonclassical behavior, where the contact angle decreases from its value taken at the melting point towards complete wetting at a critical undercooling, an analogue of the surface spinodal of liquid-wall interfaces.