Latest publications

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

Physical Review Letters 103, 035702, (2009)
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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.

Journal of Physical Chemistry B - Condensed Matter Materials Surfaces Interfaces and Biophysical 113, 5141-5148, (2009)
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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

Physical Review B Condensed Matter and Materials Physics 79, 014204, (2009)
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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, 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
3Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, U.K.

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

Journal of Computational Physics 228, 1612-1623, (2009)
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Phase-field approach to polycrystalline solidification including heterogeneous and homogeneous nucleation

Tamás Pusztai1, György Tegze1, Gyula Tóth2, 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
2Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, U.K.
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

Journal of Physics-Condensed Matter 20, 404205, (2008)
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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, 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 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.

Journal of Chemical Physics 127, 074709, (2007)
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Phase field theory of interfaces and crystal nucleation in a eutectic system of fcc structure: II. Nucleation in the metastable liquid immiscibility region

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

In the second part of our paper, we address crystal nucleation in the metastable liquid miscibility region of eutectic systems that is always present, though experimentally often inaccessible. While this situation resembles the one seen in single component crystal nucleation in the presence of a metastable vapor-liquid critical point addressed in previous works, it is more complex because of the fact that here two crystal phases of significantly different compositions may nucleate. Accordingly, at a fixed temperature below the critical point, six different types of nuclei may form: two liquid-liquid nuclei: two solid-liquid nuclei; and two types of composite nuclei, in which the crystalline core has a liquid "skirt", whose composition falls in between the compositions of the solid and the initial liquid phases, in addition to nuclei with concentric alternating composition shells of prohibitively high free energy. We discuss crystalline phase selection via exploring/identifying the possible pathways for crystal nucleation.

Journal of Chemical Physics 127, 074710, (2007)
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Phase field theory of heterogeneous crystal nucleation

László Gránásy1,2, Tamás Pusztai1, D Saylor, James A. Warren3

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
3Metallurgy Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA

The phase field approach is used to model heterogeneous crystal nucleation in an undercooled pure liquid in contact with a foreign wall. We discuss various choices for the boundary condition at the wall and determine the properties of critical nuclei, including their free energy of formation and the contact angle as a function of undercooling. For particular choices of boundary conditions, we may realize either an analog of the classical spherical cap model or decidedly nonclassical behavior, where the contact angle decreases from its value taken at the melting point towards complete wetting at a critical undercooling, an analogue of the surface spinodal of liquid-wall interfaces.

Physical Review Letters 98, 035703, (2007)
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Komplex kristálymorfológiák modellezése három dimenzióban

Tamás Pusztai1, G Bortel, 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

Fizikai Szemle 56, 412-415, (2006)

Phase field theory of nucleation and polycrystalline pattern formation

László Gránásy1,2, Tamás Pusztai1, T Börzsönyi

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 review our recent modeling of crystal nucleation and polycrystalline growth using a phase field theory. First, we consider the applicability of phase field theory for describing crystal nucleation in a model hard sphere fluid. It is shown that the phase field theory accurately predicts the nucleation barrier height for this liquid when the model parameters are fixed by independent molecular dynamics calculations. We then address various aspects of polycrystalline solidification and associated crystal pattern formation at relatively long timescales. This late stage growth regime, which is not accessible by molecular dynamics, involves nucleation at the growth front to create new crystal grains in addition to the effects of primary nucleation. Finally, we consider the limit of extreme polycrystalline growth, where the disordering effect due to prolific grain formation leads to isotropic growth patterns at long times, i.e., spherulite formation. Our model of spherulite growth exhibits branching at fixed grain misorientations, induced by the inclusion of a metastable minimum in the orientational free energy. It is demonstrated that a broad variety of spherulitic patterns can be recovered by changing only a few model parameters.

Handbook of Theoretical and Computational Nanotechnology 525-572, (2006)
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