Latest publications

Analysis of Ginzburg-Landau-type models of surfactant-assisted liquid phase separation

Gyula Tóth1, Bjørn Kvamme2

1Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, U.K.
2Institute of Physics and Technology, University of Bergen, Allégaten 55, N-5007 Bergen, Norway

In this paper diffuse interface models of surfactant-assisted liquid-liquid phase separation are addressed. We start from the generalized version of the Ginzburg-Landau free-energy-functional-based model of van der Sman and van der Graaf. First, we analyze the model in the constant surfactant approximation and show the presence of a critical point at which the interfacial tension vanishes. Then we determine the adsorption isotherms and investigate the validity range of previous results. As a key point of the work, we propose a new model of the van der Sman/van der Graaf type designed for avoiding both unwanted unphysical effects and numerical difficulties present in previous models. In order to make the model suitable for describing real systems, we determine the interfacial tension analytically more precisely and analyze it over the entire accessible surfactant load range. Emerging formulas are then validated by calculating the interfacial tension from the numerical solution of the Euler-Lagrange equations. Time-dependent simulations are also performed to illustrate the slowdown of the phase separation near the critical point and to prove that the dynamics of the phase separation is driven by the interfacial tension.

Topics: Emulsion

Phase-Field Modeling of Solidification in Light-Metal Matrix Nanocomposites

Tamás Pusztai1, László Rátkai1, Attila Szállás1, 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

The quantitative phase-field approach has been adapted to model solidification in the presence of Metal Matrix Nanocomposites (MMNCs) in a single-component liquid. Nanoparticles of fixedsize and shape are represented by additional fields. The corresponding equations of motion are assumed to ensure relaxation dynamics, and can be supplemented by random forces (realizing Brownian motion) or external fields. The nanoparticles are characterized by two model parameters: their mobility and the contact angle they realize with the solid-liquid interface. We investigate the question how grain size distribution can be influenced by heterogeneous nucleation on the nanoparticles and by the front-particle interaction. We explore, furthermore, how materials and process parameters, such as temperature, density and size/shape distribution of the nanoparticles, influence microstructure evolution.

Topics: Heterogeneous nucleation

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

László Gránásy1,2, 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.

Topics: Heterogeneous nucleation, Phase field crystal

Crystallization: Colloidal suspense

László Gránásy1,2, 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.

Topics: Phase field crystal

Nonlinear hydrodynamic theory of crystallization

Gyula Tóth1, László Gránásy2,3, György Tegze2

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

We present an isothermal fluctuating nonlinear hydrodynamic theory of crystallization in molecular liquids. A dynamic coarse-graining technique is used to derive the velocity field, a phenomenology which allows a direct coupling between the free energy functional of the classical density functional theory and the Navier–Stokes equation. In contrast to the Ginzburg–Landau type amplitude theories, the dynamic response to elastic deformations is described by parameter-free kinetic equations. Employing our approach to the free energy functional of the phase-field crystal model, we recover the classical spectrum for the phonons and the steady-state growth fronts. The capillary wave spectrum of the equilibrium crystal–liquid interface is in good qualitative agreement with the molecular dynamics simulations.

Phase-Field Modeling of Polycrystalline Solidification: From Needle Crystals to Spherulites-A Review

László Gránásy1,2, László Rátkai1, Attila Szállás1, Bálint Korbuly1, Gyula Tóth3, László Környei4, 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 Mathematics and Computational Sciences, Széchenyi István University, Győr 9026, Hungary

Advances in the orientation-field-based phase-field (PF) models made in the past are reviewed. The models applied incorporate homogeneous and heterogeneous nucleation of growth centers and several mechanisms to form new grains at the perimeter of growing crystals, a phenomenon termed growth front nucleation. Examples for PF modeling of such complex polycrystalline structures are shown as impinging symmetric dendrites, polycrystalline growth forms (ranging from disordered dendrites to spherulitic patterns), and various eutectic structures, including spiraling two-phase dendrites. Simulations exploring possible control of solidification patterns in thin films via external fields, confined geometry, particle additives, scratching/piercing the films, etc. are also displayed. Advantages, problems, and possible solutions associated with quantitative PF simulations are discussed briefly.

Topics: Polycrystalline solidification, Spiral eutectic dendrites

Free energy of the bcc-liquid interface and the Wulff shape as predicted by the phase-field crystal model

Frigyes Podmaniczky1, Gyula Tóth2, 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
2Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, U.K.
3BCAST, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom

The Euler-Lagrange equation of the phase-field crystal (PFC) model has been solved under appropriate boundary conditions to obtain the equilibrium free energy of the body centered cubic crystal-liquid interface for 18 orientations at various reduced temperatures in the range ε∈[0,0.5]. While the maximum free energy corresponds to the {100} orientation for all ε values, the minimum is realized by the {111} direction for smaller ε(<0.13), and by the {211} orientation for higher ε. The predicted dependence on the reduced temperature is consistent with the respective mean field critical exponent. The results are fitted with an eight-term Kubic harmonic series, and are used to create stereographic plots displaying the anisotropy of the interface free energy. We have also derived the corresponding Wulff shapes that vary with increasing ε from sphere to a polyhedral form that differs from the rhombo-dodecahedron obtained previously by growing a bcc seed until reaching equilibrium with the remaining liquid.

Spiraling eutectic dendrites

Tamás Pusztai1, László Rátkai1, Attila Szállás1, 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

Eutectic dendrites forming in a model ternary system have been studied using the phase-field theory. The eutectic and one-phase dendrites have similar forms, and the tip radius scales with the interface free energy as for one-phase dendrites. The steady-state eutectic patterns, appearing on these two-phase dendrites, include concentric rings and single- to multiarm spirals from which the fluctuations choose; a stochastic phenomenon characterized by a peaked probability distribution. The number of spiral arms correlates with the tip radius and the kinetic anisotropy.

Topics: Spiral eutectic dendrites

Heterogeneous Crystal Nucleation: The Effect of Lattice Mismatch

Gyula Tóth1, György Tegze2, Tamás Pusztai2, 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

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

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, 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
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.

Topics: Phase field crystal

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