Phase field crystal

Phase-field crystal modeling of nucleation including homogeneous and heterogeneous processes, and growth front nucleation

László Gránásy1,2, Frigyes Podmaniczky1, Gyula Tóth3,1

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
3Institute of Physics and Technology, University of Bergen, Allégaten 55, N-5007 Bergen, Norway

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.

Topics: Phase field crystal

Phase-field crystal modeling of heteroepitaxy and exotic modes of crystal nucleation

Frigyes Podmaniczky1, Gyula Tóth2,1, György Tegze1, Tamás Pusztai1, László Gránásy1,3

1Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, P.O. Box 49, Budapest H-1525, Hungary
2Institute of Physics and Technology, University of Bergen, Allégaten 55, N-5007 Bergen, Norway
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.

Topics: Phase field crystal

Heterogeneous nucleation of/on nanoparticles: a density functional study using the phase-field crystal model

László Gránásy1,2, Frigyes Podmaniczky1, Gyula Tóth3,1, 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
3Institute of Physics and Technology, University of Bergen, Allégaten 55, N-5007 Bergen, Norway

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.

Topics: Heterogeneous nucleation, Phase field crystal

Crystallization: Colloidal suspense

László Gránásy1,2, Gyula Tóth3,1

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
3Institute of Physics and Technology, University of Bergen, Allégaten 55, N-5007 Bergen, Norway

Topics: Phase field crystal

Heterogeneous Crystal Nucleation: The Effect of Lattice Mismatch

Gyula Tóth1,2, György Tegze2, Tamás Pusztai2, László Gránásy2,3

1Institute of Physics and Technology, University of Bergen, Allégaten 55, N-5007 Bergen, Norway
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.

Topics: Heterogeneous nucleation, Phase field crystal

Amorphous Nucleation Precursor in Highly Nonequilibrium Fluids

Gyula Tóth1,2, Tamás Pusztai2, György Tegze2, Gergely Tóth3, László Gránásy2,4

1Institute of Physics and Technology, University of Bergen, Allégaten 55, N-5007 Bergen, Norway
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.

Topics: Phase field crystal

Faceting and Branching in 2D Crystal Growth

György Tegze1, Gyula Tóth2,1, László Gránásy1,3

1Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, P.O. Box 49, Budapest H-1525, Hungary
2Institute of Physics and Technology, University of Bergen, Allégaten 55, N-5007 Bergen, Norway
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.

Topics: Phase field crystal

Phase-field-crystal models for condensed matter dynamics on atomic length and diffusive time scales: an overview

Heike Emmerich1, Hartmut Löwen2, Raphael Wittkowski2, Thomas Gruhn1, Gyula Tóth3,4, György Tegze4, László Gránásy4,5

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
3Institute of Physics and Technology, University of Bergen, Allégaten 55, N-5007 Bergen, Norway
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.

Topics: Phase field crystal

Tuning the structure of non-equilibrium soft materials by varying the thermodynamic driving force for crystal ordering

György Tegze1, László Gránásy1,2, Gyula Tóth3,1, 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
3Institute of Physics and Technology, University of Bergen, Allégaten 55, N-5007 Bergen, Norway
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.

Topics: Phase field crystal

Polymorphism, crystal nucleation and growth in the phase-field crystal model in 2D and 3D

Gyula Tóth1,2, György Tegze2, Tamás Pusztai2, Gergely Tóth3, László Gránásy2,4

1Institute of Physics and Technology, University of Bergen, Allégaten 55, N-5007 Bergen, Norway
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.

Topics: Phase field crystal

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