Frigyes Podmaniczky¹, László Gránásy^1,2

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

We present recent advances in phase-field crystal (PFC) modeling of crystal nucleation in simple undercooled liquids, where dynamics is based on hydrodynamic density relaxation, as opposed to the diffusive dynamics in colloid systems. Herein, we address two-step nucleation, the nucleation of new grains at the growth front, and crystal nucleation on the surface of foreign particles in flow. Owing to the different numerical complexity of these problems, we consider three different realizations of the hydrodynamic approach that are optimized to deal with the individual tasks. We show that during two-step nucleation taking place at high supersaturations, amorphous and layered two-dimensional quasicrystalline domains form simultaneously in the first stage, which is then followed by bcc nucleation/transformation of the existing solid into the stable body-centered structure. Next, the formation of satellite crystals at the solid-liquid interface, observed in molecular dynamics simulations are studied. We find that the interference of density waves ahead of the rough growth front may assist the formation of such satellite crystals. Finally, we show that, under appropriate conditions, fluid flow may tear off heterogeneously nucleated crystal particles from the surface of a curved substrate, as envisaged for colloidal systems.

Frigyes Podmaniczky¹, László Gránásy^1,2

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

Two-step nucleation and subsequent growth processes were investigated in the framework of the single mode phase-field crystal model combined with diffusive dynamics (corresponding to colloid suspensions) and hydrodynamical density relaxation (simple liquids). It is found that independently of dynamics, nucleation starts with the formation of solid precursor clusters that consist of domains with noncrystalline ordering (ringlike projections are seen from certain angles), and regions that have amorphous structure. Using the average bond order parameter q¯6, we distinguished amorphous, medium range crystallike order (MRCO), and crystalline local orders. We show that crystallization to the stable body-centered cubic phase is preceded by the formation of a mixture of amorphous and MRCO structures. We have determined the time dependence of the phase composition of the forming solid state. We also investigated the time/size dependence of the growth rate for solidification. The bond order analysis indicates similar structural transitions during solidification in the case of diffusive and hydrodynamic density relaxation.

László Gránásy^1,2, Frigyes Podmaniczky¹, Gyula Tóth³

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.

Structural aspects of crystal nucleation in undercooled liquids are explored using a nonlinear hydrodynamic theory of freezing proposed recently, which is based on combining fluctuating hydrodynamics with the phase-field crystal (PFC) theory. It will be shown that unlike the usual PFC models of diffusive dynamics, within the hydrodynamic approach not only the homogeneous and heterogeneous nucleation processes are accessible, but also growth front nucleation, which leads to the formation of differently oriented grains at the front in highly undercooled systems. Formation of dislocations at the solid-liquid interface and the interference of density waves ahead of the crystallization front are responsible for the appearance of new orientations.

Frigyes Podmaniczky¹, Gyula Tóth², György Tegze¹, Tamás Pusztai¹, 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

We review recent advances made in modeling heteroepitaxy, two-step nucleation, and nucleation at the growth front within the framework of a simple dynamical density functional theory, the Phase-Field Crystal (PFC) model. The crystalline substrate is represented by spatially confined periodic potentials. We investigate the misfit dependence of the critical thickness in the StranskiKrastanov growth mode in isothermal studies. Apparently, the simulation results for stress release via the misfit dislocations fit better to the PeopleBean model than to the one by Matthews and Blakeslee. Next, we investigate structural aspects of two-step crystal nucleation at high undercoolings, where an amorphous precursor forms in the first stage. Finally, we present results for the formation of new grains at the solid-liquid interface at high supersaturations/supercoolings, a phenomenon termed Growth Front Nucleation (GFN). Results obtained with diffusive dynamics (applicable to colloids) and with a hydrodynamic extension of the PFC theory (HPFC, developed for simple liquids) will be compared. The HPFC simulations indicate two possible mechanisms for GFN.

László Gránásy^1,2, Frigyes Podmaniczky¹, Gyula Tóth³, György Tegze¹, 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.

Crystallization of supersaturated liquids usually starts by heterogeneous nucleation. Mounting evidence shows that even homogeneous nucleation in simple liquids takes place in two steps; first a dense amorphous precursor forms, and the crystalline phase appears via heterogeneous nucleation in/on the precursor cluster. Herein, we review recent results by a simple dynamical density functional theory, the phase-field crystal model, for (precursor-mediated) homogeneous and heterogeneous nucleation of nanocrystals. It will be shown that the mismatch between the lattice constants of the nucleating crystal and the substrate plays a decisive role in determining the contact angle and nucleation barrier, which were found to be non-monotonic functions of the lattice mismatch. Time dependent studies are essential as investigations based on equilibrium properties often cannot identify the preferred nucleation pathways. Modeling of these phenomena is essential for designing materials on the basis of controlled nucleation and/or nano-patterning.

László Gránásy^1,2, Gyula Tóth³

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.

Gyula Tóth¹, György Tegze², Tamás Pusztai², László Gránásy^2,3

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
³BCAST, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom

A simple dynamical density functional theory is used to investigate freezing of an undercooled liquid in the presence of a crystalline substrate. We find that the adsorption of the crystalline phase on the substrate, the contact angle, and the height of the nucleation barrier are nonmonotonic functions of the lattice constant of the substrate. We show that the free-growth-limited model of particle-induced freezing by Greer et al. [Acta Mater. 48, 2823 (2000)] is valid for larger nanoparticles and a small anisotropy of the interface free energy. Faceting due to the small size of the foreign particle or a high anisotropy decouples free growth from the critical size of homogeneous nuclei.

Gyula Tóth¹, Tamás Pusztai², György Tegze², 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

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.

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.

Heike Emmerich¹, Hartmut Löwen², Raphael Wittkowski², Thomas Gruhn¹, Gyula Tóth³, György Tegze⁴, László Gránásy^4,5

¹Lehrstuhl für Material- und Prozesssimulation, Universität Bayreuth, D-95440 Bayreuth, Germany
²Institut für Theoretische Physik II, Weiche Materie, Heinrich-Heine-Universität Düsseldorf, D-40225 Düsseldorf, Germany
³Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, U.K.
⁴Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, P.O. Box 49, Budapest H-1525, Hungary
⁵BCAST, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom

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 fol- lowing 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.