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.

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.

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.PACS numbers: 81.10. Aj, 82.60.Nh, 64.60.Qb Understanding alloy solidification is of vast practical and theoretical importance. While the directional geometry in which the solidification front propagates from a cool surface towards the interior of a hot melt is understood fairly well, less is known of equiaxial solidification that takes place in the interior of the melt. The latter plays a central role in processes such as alloy casting, hibernation of biological tissues, hail formation, and crystallization of proteins and glasses. The least understood stage of these processes is nucleation, during which seeds of the crystalline phase appear via thermal fluctuations. Since the physical interface thickness is comparable to the typical size of critical fluctuations that are able to grow to macroscopic sizes, these fluctuations are nearly all interface. Accordingly, the diffuse interface models lead to a considerably more accurate description of nucleation than those based on a sharp interface [1,2].The phase field theory, a recent diffuse interface approach, emerged as a powerful tool for describing complex solidification patterns such as dendritic, eutectic, and peritectic growth morphologies [3]. It is of interest to extend this model to nucleation and post-nucleation growth including diffusion controlled "soft-impingement" of growing crystalline particles, expected to be responsible for the unusual transformation kinetics recently seen during the formation of nanocrystalline materials [4].In this Letter, we develop a phase field theory for crystal nucleation and growth, and apply it to current problems of unary and binary equiaxial solidification.Our starting point is the free energy functionaldeveloped along the lines described in [5,6]. Here φ and c are the phase and concentration fields, f (φ, c) = W T g(φ) + [1 − P (φ)]f S + P (φ)f L is the local free energy density, W = (1 − c)W A + cW B the free energy scale, the quartic function g(φ) = φ 2 (1 − φ) 2 /4 that emerges from density functional theory [7] ensures the doublewell form of f , while the function P (φ) = φ 3 (10 − 15φ + 6φ 2 ) switches on and off the solid and liquid contributions f S,L , taken from the ideal solution model. (A and B refer to the constituents.) For binary alloys the model contains three parameters ǫ, W A and W B that reduce to two (ǫ and W ) in the one-component limit. They can be fixed if the respective interface free energy γ, melting point T f , and interface thickn...

We apply a simple dynamical density functional theory, the phase-field crystal (PFC) model of overdamped conservative dynamics, to address polymorphism, crystal nucleation, and crystal growth in the diffusion-controlled limit. We refine the phase diagram for 3D, and determine the line free energy in 2D and the height of the nucleation barrier in 2D and 3D for homogeneous and heterogeneous nucleation by solving the respective Euler-Lagrange (EL) equations. We demonstrate that, in the PFC model, the body-centered cubic (bcc), the face-centered cubic (fcc), and the hexagonal close-packed structures (hcp) compete, while the simple cubic structure is unstable, and that phase preference can be tuned by changing the model parameters: close to the critical point the bcc structure is stable, while far from the critical point the fcc prevails, with an hcp stability domain in between. We note that with increasing distance from the critical point the equilibrium shapes vary from the sphere to specific faceted shapes: rhombic dodecahedron (bcc), truncated octahedron (fcc), and hexagonal prism (hcp). Solving the equation of motion of the PFC model supplied with conserved noise, solidification starts with the nucleation of an amorphous precursor phase, into which the stable crystalline phase nucleates. The growth rate is found to be time dependent and anisotropic; this anisotropy depends on the driving force. We show that due to the diffusion-controlled growth mechanism, which is especially relevant for crystal aggregation in colloidal systems, dendritic growth structures evolve in large-scale isothermal single-component PFC simulations. An oscillatory effective pair potential resembling those for model glass formers has been evaluated from structural data of the amorphous phase obtained by instantaneous quenching. Finally, we present results for eutectic solidification in a binary PFC model.

We review recent advances made in the phase field modelling of polycrystalline solidification. Areas covered include the development of theory from early approaches that allow for only a few crystal orientations, to the latest models relying on a continuous orientation field and a free energy functional that is invariant to the rotation of the laboratory frame. We discuss a variety of phenomena, including homogeneous nucleation and competitive growth of crystalline particles having different crystal orientations, the kinetics of crystallization, grain boundary dynamics, and the formation of complex polycrystalline growth morphologies including disordered ('dizzy') dendrites, spherulites, fractal-like polycrystalline aggregates, etc. Finally, we extend the approach by incorporating walls, and explore phenomena such as heterogeneous nucleation, particle-front interaction, and solidification in confined geometries (in channels or porous media).

Microstructure plays an essential role in determining the properties of crystalline materials. A widely used method to influence microstructure is the addition of nucleating agents. Observations on films formed from clay-polymer blends indicate that particulate additives, in addition to serving as nucleating agents, may also perturb crystal growth, leading to the formation of irregular dendritic morphologies. Here we describe the formation of these 'dizzy dendrites' using a phase-field theory, in which randomly distributed foreign particle inclusions perturb the crystallization by deflecting the tips of the growing dendrite arms. This mechanism of crystallization, which is verified experimentally, leads to a polycrystalline structure dependent on particle configuration and orientation. Using computer simulations we demonstrate that additives of controlled crystal orientation should allow for a substantial manipulation of the crystallization morphology.

We use a simple density functional approach on a diffusional time scale, to address freezing to the body-centered cubic (bcc), hexagonal close-packed (hcp), and face-centered cubic (fcc) structures. We observe faceted equilibrium shapes and diffusion-controlled layerwise crystal growth consistent with twodimensional nucleation. The predicted growth anisotropies are discussed in relation with results from experiment and atomistic simulations. We also demonstrate that varying the lattice constant of a simple cubic substrate, one can tune the epitaxially growing body-centered tetragonal structure between bcc and fcc, and observe a Mullins-Sekerka-Asaro-Tiller-Grinfeld-type instability.

Dynamical density-functional simulations reveal structural aspects of crystal nucleation in undercooled liquids: The first appearing solid is amorphous, which promotes the nucleation of bcc crystals but suppresses the appearance of the fcc and hcp phases. These findings are associated with features of the effective interaction potential deduced from the amorphous structure.