Latest publications

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

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

László Gránásy1,2, György Tegze1, Gyula Tóth3,1, 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

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 Tóth1,2, 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

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 Tóth3,1, 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
3Institute of Physics and Technology, University of Bergen, Allégaten 55, N-5007 Bergen, Norway
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 Tóth1,2, 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

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.

Phase field approach to heterogeneous nucleation in alloys

James A. Warren1, Tamás Pusztai2, László Környei3, László Gránásy2,4

1Metallurgy Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA
2Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, P.O. Box 49, Budapest H-1525, Hungary
3Department of Mathematics and Computational Sciences, Széchenyi István University, Győr 9026, Hungary
4BCAST, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom

We extend the phase field model of heterogeneous crystal nucleation developed recently [L. Gránásy et al., Phys. Rev. Lett. 98, 035703 (2007)] to binary alloys. Three approaches are considered to incorporate foreign walls of tunable wetting properties into phase field simulations: a continuum realization of the classical spherical cap model (called model A herein), a nonclassical approach (model B) that leads to ordering of the liquid at the wall and to the appearance of a surface spinodal, and a nonclassical model (model C) that allows for the appearance of local states at the wall that are accessible in the bulk phases only via thermal fluctuations. We illustrate the potential of the presented phase field methods for describing complex polycrystalline solidification morphologies including the shish-kebab structure, columnar to equiaxed transition, and front-particle interaction in binary alloys.

Topics: Heterogeneous nucleation

Advanced operator-splitting-based semi-implicit spectral method to solve the binary phase-field crystal equation with variable coefficients

György Tegze1, Gurvinder Bansel2, Gyula Tóth3,1, Tamás Pusztai1, Zhongyun Fan2, László Gránásy1,2

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

We present an efficient method to solve numerically the equations of dissipative dynamics of the binary phase-field crystal model proposed by Elder et al. [K.R. Elder, M. Katakowski, M. Haataja, M. Grant, Phys. Rev. B 75, 064107 (2007)] characterized by variable coefficients. Using the operator splitting method, the problem has been decomposed into sub-problems that can be solved more efficiently. A combination of non-trivial splitting with spectral semi-implicit solution leads to sets of algebraic equations of diagonal matrix form. Extensive testing of the method has been carried out to find the optimum balance among errors associated with time integration, spatial discretization, and splitting. We show that our method speeds up the computations by orders of magnitude relative to the conventional explicit finite difference scheme, while the costs of the pointwise implicit solution per timestep remains low. Also we show that due to its numerical dissipation, finite differencing can not compete with spectral differencing in terms of accuracy. In addition, we demonstrate that our method can efficiently be parallelized for distributed memory systems, where an excellent scalability with the number of CPUs is observed.

Topics: Phase field crystal

Phase-field approach to polycrystalline solidification including heterogeneous and homogeneous nucleation

Tamás Pusztai1, György Tegze1, Gyula Tóth2,1, László Környei3, Gurvinder Bansel4, Zhongyun Fan4, László Gránásy1,4

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
3Department of Mathematics and Computational Sciences, Széchenyi István University, Győr 9026, Hungary
4BCAST, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom

Advanced phase-field techniques have been applied to address various aspects of polycrystalline solidification including different modes of crystal nucleation. The height of the nucleation barrier has been determined by solving the appropriate Euler-Lagrange equations. The examples shown include the comparison of various models of homogeneous crystal nucleation with atomistic simulations for the single-component hard sphere fluid. Extending previous work for pure systems [Gránásy et al., Phys. Rev. Lett. 98, 035703 (2007)], heterogeneous nucleation in unary and binary systems is described via introducing boundary conditions that realize the desired contact angle. A quaternion representation of crystallographic orientation of the individual particles [outlined in Pusztai et al., Europhys. Lett. 71, 131 (2005)] has been applied for modeling a broad variety of polycrystalline structures including crystal sheaves, spherulites and those built of crystals with dendritic, cubic, rhombo-dodecahedral and truncated octahedral growth morphologies. Finally, we present illustrative results for dendritic polycrystalline solidification obtained using an atomistic phase-feld model.

Topics: Polycrystalline solidification

Phase field theory of interfaces and crystal nucleation in a eutectic system of fcc structure: I. Transitions in the one-phase liquid region

Gyula Tóth1,2, 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

The phase field theory PFT has been applied to predict equilibrium interfacial properties and nucleation barrier in the binary eutectic system Ag–Cu using double well and interpolation functions deduced from a Ginzburg-Landau expansion that considers fcc face centered cubic crystal symmetries. The temperature and composition dependent free energies of the liquid and solid phases are taken from Calculation of Phase Diagrams-type calculations. The model parameters of PFT are fixed so as to recover an interface thickness of 1 nm from molecular dynamics simulations and the interfacial free energies from the experimental dihedral angles available for the pure components. A nontrivial temperature and composition dependence for the equilibrium interfacial free energy is observed. Mapping the possible nucleation pathways, we find that the Ag and Cu rich critical fluctuations compete against each other in the neighborhood of the eutectic composition. The Tolman length is positive and shows a maximum as a function of undercooling. The PFT predictions for the critical undercooling are found to be consistent with experimental results. These results support the view that heterogeneous nucleation took place in the undercooling experiments available at present. We also present calculations using the classical droplet model classical nucleation theory CNT and a phenomenological diffuse interface theory DIT. While the predictions of the CNT with a purely entropic interfacial free energy underestimate the critical undercooling, the DIT results appear to be in a reasonable agreement with the PFT predictions.

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