Latest publications

Selected issues of phase-field crystal simulations

Heike Emmerich1, László Gránásy2,3, 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.

Amorphous Nucleation Precursor in Highly Nonequilibrium Fluids

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

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.

Topics: Phase field crystal

Faceting and Branching in 2D Crystal Growth

György Tegze1, Gyula Tóth2, László Gránásy1,3

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.

Topics: Phase field crystal

Ginzburg-Landau-type multiphase field model for competing fcc and bcc nucleation

Gyula Tóth1, J.R. Morris2, László Gránásy3,4

1Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, U.K.
2Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830, USA
3Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, P.O. Box 49, Budapest H-1525, Hungary
4BCAST, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom

We address crystal nucleation and fcc-bcc phase selection in alloys using a multiphase field model that relies on Ginzburg-Landau free energies of the liquid-fcc, liquid-bcc, and fcc-bcc subsystems, and determine the properties of the nuclei as a function of composition, temperature, and structure. With a realistic choice for the free energy of the fcc-bcc interface, the model predicts well the fcc-bcc phase-selection boundary in the Fe-Ni system.

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

György Tegze1, László Gránásy1,2, 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.

Topics: Phase field crystal

Phase-field crystal modelling of crystal nucleation, heteroepitaxy and patterning

László Gránásy1,2, György Tegze1, Gyula Tóth3, 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.

A simple dynamical density functional theory, the phase-field crystal (PFC) model, was used to describe homogeneous and heterogeneous crystal nucleation in two-dimensional (2D) monodisperse colloidal systems and crystal nucleation in highly compressed Fe liquid. External periodic potentials were used to approximate inert crystalline substrates in addressing heterogeneous nucleation. In agreement with experiments in 2D colloids, the PFC model predicts that in 2D supersaturated liquids, crystalline freezing starts with homogeneous crystal nucleation without the occurrence of the hexatic phase. At extreme supersaturations, crystal nucleation happens after the appearance of an amorphous precursor both in two and three dimensions. Contrary to expectations based on the classical nucleation theory, it is shown that corners are not necessarily favourable places for crystal nucleation. Finally, it is shown that by adding external potential terms to the free energy, the PFC theory can be used to model colloid patterning experiments.

Topics: Phase field crystal

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

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

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.

Topics: Phase field crystal

Classical density functional theory methods in soft and hard matter PREFACE

Mikko Haataja1, László Gránásy2,3, Hartmut Löwen4

1Department of Mechanical and Aerospace Engineering, Institute for the Science and Technology of Materials (PRISM) and Program in Applied and Computational Mathematics (PACM), Princeton University, Princeton NJ 08544, USA
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

Herein we provide a brief summary of the background, events and results/outcome of the CECAM workshop ‘Classical density functional theory methods in soft and hard matter’ held in Lausanne between October 21 and October 23 2009, which brought together two largely separately working communities, both of whom employ classical density functional techniques: the soft-matter community and the theoretical materials science community with interests in phase transformations and evolving microstructures in engineering materials. After outlining the motivation for the workshop, we first provide a brief overview of the articles submitted by the invited speakers for this special issue of Journal of Physics: Condensed Matter, followed by a collection of outstanding problems identified and discussed during the workshop.

Diffusion-controlled anisotropic growth of stable and metastable crystal polymorphs in the phase-field crystal model

György Tegze1, László Gránásy1,2, Gyula Tóth3, Frigyes Podmaniczky1, A Jaatinen4, T Ala-Nissila4, 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.
4Department of Applied Physics, Helsinki University of Technology, Post Office Box 1100, FI-02015 TKK, Finland

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 two- dimensional 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.

Topics: Phase field crystal

Crystal nucleation in the hard-sphere system revisited: Critical test of theoretical approaches

Gyula Tóth1, László Gránásy2,3

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

The hard-sphere system is the best known fluid that crystallizes: the solid-liquid interfacial free energy, the equations of state, and the height of the nucleation barrier are known accurately, offering a unique possibility for a quantitative validation of nucleation theories. A recent significant downward revision of the interfacial free energy from 0.61kT/s^2 to 0.56 kT/s^2 [Davidchack, R.; Morris, J. R.; Laird, B. B. J. Chem. Phys. 125, 094710 (2006)] necessitates a re-evaluation of theoretical approaches to crystal nucleation. This has been carried out for the droplet model of the classical nucleation theory (CNT), the self-consistent classical theory (SCCT), a phenomenological diffuse interface theory (DIT), and single- and two-field variants of the phase field theory that rely on either the usual double-well and interpolation functions (PFT/S1 and PFT/S2, respectively) or on a Ginzburg-Landau expanded free energy that reflects the crystal symmetries (PFT/GL1 and PFT/GL2). We find that the PFT/GL1, PFT/GL2, and DIT models predict fairly accurately the height of the nucleation barrier known from Monte Carlo simulations in the volume fraction range of 0.52 < f < 0.54, whereas the CNT, SCCT, PFT/S1, and PFT/S2 models underestimate it significantly.

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