Heike Emmerich¹, László Gránásy^2,3, Hartmut Löwen⁴

¹Lehrstuhl für Material- und Prozesssimulation, Universität Bayreuth, D-95440 Bayreuth, Germany
²Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, P.O. Box 49, Budapest H-1525, Hungary
³BCAST, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom
⁴Institut für Theoretische Physik II, Weiche Materie, Heinrich-Heine-Universität Düsseldorf, D-40225 Düsseldorf, Germany

In this contribution our focus is on the phase-field crystal method, which can be viewed as the youngest methodology in the field of interface computation based on recent work by Elder et al. (Phys. Rev. Lett. 88, 245701 (2002)). It bridges the gap between the molecular simulation approaches and the phase-field approach by operating on diffusive time scales yet atomic length scales. Here we review the fundaments of the phase-field crystal method as well as different models established so far with the aim to capture the main features of the wide range of phase diagrams found in materials science more and more comprehensively.

György Tegze¹, Gyula Tóth², László Gránásy^1,3

¹Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, P.O. Box 49, Budapest H-1525, Hungary
²Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, U.K.
³BCAST, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom

Using atomic scale time-dependent density functional calculations we confirm that both diffusion-controlled and diffusionless crystallization modes exist in simple 2D systems. We provide theoretical evidence that a faceted to nonfaceted transition is coupled to these crystallization modes, and faceting is governed by the local supersaturation at the fluid-crystalline interface. We also show that competing modes of crystallization have a major influence on mesopattern formation. Irregularly branched and porous structures are emerging at the crossover of the crystallization modes. The proposed branching mechanism differs essentially from dendritic fingering driven by diffusive instability.

György Tegze¹, László Gránásy^1,2, Gyula Tóth³, Jack F. Douglas⁴, Tamás Pusztai¹

¹Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, P.O. Box 49, Budapest H-1525, Hungary
²BCAST, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom
³Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, U.K.
⁴Polymers Division, National Institute of Standards and Technology,Gaithersburg, MD, 20899, USA

The present work explores the ubiquitous morphological changes in crystallizing systems with increasing thermodynamic driving force based on a novel dynamic density functional theory. A colloidal 'soft' material is chosen as a model system for our investigation since there are careful colloidal crystallization observations at a particle scale resolution for comparison, which allows for a direct verification of our simulation predictions. We particularly focus on a theoretically unanticipated, and generic, morphological transition leading to progressively irregular-shaped single crystals in both colloidal and polymeric materials with an increasing thermodynamic driving force. Our simulation method significantly extends previous 'phase field' simulations by incorporating a minimal description of the 'atomic' structure of the material, while allowing simultaneously for a description of large scale crystal growth. We discover a 'fast' mode of crystal growth at high driving force, suggested before in experimental colloidal crystallization studies, and find that the coupling of this crystal mode to the well-understood 'diffusive' or 'slow' crystal growth mode (giving rise to symmetric crystal growth mode and dendritic crystallization as in snowflakes by the Mullins-Sekerka instability) can greatly affect the crystal morphology at high thermodynamic driving force. In particular, an understanding of this interplay between these fast and slow crystal growth modes allows us to describe basic crystallization morphologies seen in both colloidal suspensions with increasing particle concentration and crystallizing polymer films with decreasing temperature: compact symmetric crystals, dendritic crystals, fractal-like structures, and then a return to compact symmetric single crystal growth again.

Gyula Tóth¹, György Tegze², Tamás Pusztai², Gergely Tóth³, László Gránásy^2,4

¹Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, U.K.
²Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, P.O. Box 49, Budapest H-1525, Hungary
³Institute of Chemistry, Eötvös University, P.O. Box 32, H-1518 Budapest, Hungary
⁴BCAST, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom

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.